Mathematical Theorems

Appendix A Mathematical Theorems

The mathematical theorems needed in order to derive the governing model equations are deﬁned in this appendix.

A.1 Transport Theorem for a Single Phase Region

The transport theorem is employed deriving the conservation equations in continuum mechanics. The mathematical statement is sometimes attributed to, or named in honor of, the German Mathematician Gottfried Wilhelm Leibnitz (1646Ð1716) and the British ﬂuid dynamics engineer Osborne Reynolds (1842Ð1912) due to their work and contributions related to the theorem. Hence it follows that the transport theorem, or alternate forms of the theorem, may be named the Leibnitz theorem in mathematics and Reynolds transport theorem in mechanics. In a customary interpretation the Reynolds transport theorem provides the link between the system and control volume representations, while the Leibnitz’s theorem is a three dimensional version of the integral rule for differentiation of an integral. There are several notations used for the transport theorem and there are numerous forms and corollaries.

A.1.1 Leibnitz’s Rule

The Leibnitz’s integral rule gives a formula for differentiation of an integral whose limits are functions of the differential variable [7, 8, 22, 23, 45, 55, 79, 94, 99]. The formula is also known as differentiation under the integral sign.

H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1361 © Springer International Publishing Switzerland 2014 1362 Appendix A: Mathematical Theorems

b(t) b(t) d ∂f (t, x) db da f (t, x) dx = dx + f (t, b) − f (t, a) (A.1) dt ∂t dt dt a(t) a(t)

The ﬁrst term on the RHS gives the change in the integral because the function itself is changing with time, the second term accounts for the gain in area as the upper limit is moved in the positive axis direction, and the third term accounts for the loss in area as the lower limit is moved. The formal derivation of the Leibnitz’s rule can be found elsewhere [8, 45].

A.1.2 Leibnitz Theorem

A three dimensional extension of the Leibnitz rule for differentiating an integral is relevant for the derivation of the governing transport equations.1 In the material (Lagrangian) representation of continuum mechanics a representative particle of the continuum occupies a point in the initial configuration of the continuum at time t = 0 and has the position vector ξ = (ξ1, ξ2, ξ3).Inthisξ-space the coordinates are called the material coordinates. In the Eulerian representation the particle position vector in r-space is defined by r = (r1, r2, r3). The coordinates r1, r2, r3 which gives the current position of the particle are called the spatial coordinates. Let ψ(r, t) be any scalar, vector or tensor function of time and position and V(t) a material volume. We may then define a variable Ψ(t) as the volume integral [2]: Ψ(t) = (ρψ) dv (A.2) V(t)

It is desired to ﬁnd an expression for differentiating the integral Ψ(t) with time: DΨ(t) D = (ρψ)(r, t) dv (A.3) Dt Dt V(t)

The integral is over the material volume V(t) that is a function of t, hence we cannot take the differentiation through the integration sign. However, if the integration were with respect to a volume in the material ξ- coordinates it would be possible to interchange the differentiation and integration, since D/Dt is deﬁned as differentiation with respect to time keeping ξ constant. A transformation of the volume from r-space to ξ-space allows us to do the desired

1 The theory has been outlined by Truesdell and Toupin [98] (p. 347), Aris [2] (p. 84), Malvern [62] (p. 210), Slattery [87] (p. 17), Slattery [89] (p. 21), Bird et al. [7] (p. 732), Fan and Zhu [40] (p. 167), Kundu [55] (p. 75), Delhaye and Achard [22] (p. 9), Delhaye [23] (p. 42), Bouré and Delhaye [12] (pp. 1Ð37), Whitaker [99], Donea and Huerta [27] (Sects.1.3Ð1.4), Donea et al. [28], and Collado [20]. Appendix A: Mathematical Theorems 1363 operation. The position vector in r-space, r = (r1, r2, r3), is transformed as r = r(ξ, t) in ξ-space. If the coordinate system is changed from r-space to ξ-space, the volume element changes according to:

∂(r1, r2, r3) dv = dξ1dξ2ξ3 = Jdv0 (A.4) ∂(ξ1, ξ2, ξ3) in which dξ1dξ2ξ3 denotes the material volume dv0 about a given point ξ at the initial instant. The Jacobian determinant of transformation between the material and ﬁxed coordinate systems, is deﬁned by:

∂(r , r , r ) dv J = 1 2 3 = (A.5) ∂(ξ1, ξ2, ξ3) dv0

The quantity J may be thought of as the ratio of an elementary material volume to its initial volume. The differentiation of the integral Ψ(t) with respect to time can then be proven: DΨ D D = (ρψ)(r, t) dv = (ρψ)(r(ξ, t), t) Jdv0 Dt Dt Dt V0 V(t) D(ρψ) DJ = J + (ρψ) dv Dt Dt 0 V0 D(ρψ) = + (ρψ)(∇·v) Jdv Dt 0 (A.6) V0 D(ρψ) = + (ρψ)(∇·v) dv Dt V(t) ∂(ρψ) = +∇·((ρψ)v) dv ∂t V(t) in which we have adopted, without proof, the following Lemma2:

DJ = J(∇·v) (A.7) Dt Applying the divergence theorem (A.19) to the second integral on the RHS of (A.6) we get a particular three dimensional form of the Leibnitz theorem: DΨ D ∂(ρψ) = (ρψ)(r, t) dv = dv + (ρψ)v · n da (A.8) Dt Dt ∂t V(t) V(t) A(t)

2 The proofs of the transport theorem are given by Slattery [90]. 1364 Appendix A: Mathematical Theorems where A(t) is the surface of V(t), and v represents the velocity of the control surface with respect to the coordinate reference frame. This kinematical transport theorem is due to Reynolds [78]. For this reason it is sometimes referred to as the Reynolds theorem. The given theorem can be extended to a general case considering an arbitrary geometric volume with a closed surface moving with an arbitrary velocity vS. Truesdell and Toupin [98] (p. 347) presented the corollary that the above relation remains valid if we replace the derivatives with respect to time while following material particles, D/Dt, by derivatives with respect to time while following fictitious system particles d/dt, and the velocity vector for a material particle v by the velocity vector for a fictitious system particle vS. Let us thus consider a geometric volume VS(t), not necessarily a material volume, which is moving in space and bounded by a closed surface AS(t). At a given point belonging to this surface, n is the unit normal vector outwardly directed. The speed of the displacement of the surface at that point is denoted by vS · n. The generalized Leibnitz theorem enables the time derivative of the volume integral to be transformed into the sum of a volume integral and a surface integral [12, 23, 55]: d ∂(ρψ) (ρψ)(r, t) dv = dv + (ρψ)v · n da (A.9) dt ∂t S VS(t) VS(t) AS(t) in which vS is the velocity of the points on the control volume surface with respect to the coordinate reference frame. Slattery [90] named this mathematical statement the generalized transport theorem. This kinematical transport theorem was asserted, not proven, by Reynolds [78]. Comparing (A.8) and (A.9) we note that to make these relations coincide the total time derivative must be specified equal to the substantial time derivative. In this way the substantial derivative may be considered a special kind of the total time derivative [7, 79], and thus the Reynolds transport theorem is a special kind of the Leibnitz theorem. In the case that the integral boundaries are fixed, the surface integral vanishes because the surface velocity is zero [22, 23, 38, 55]: d ∂f fdv = dv (A.10) dt ∂t V V

A.1.3 Reynolds Theorem

In fluid mechanics the laws governing the fluid motion are expressed using both system concepts in which we consider a given mass of the fluid, and control volume concepts in which we consider a given volume [45, 69, 96, 100]. Basically the Appendix A: Mathematical Theorems 1365 physical laws are defined for a system, thus we need a mathematical link between control volume and system concepts to convert the governing equations to apply to a specific region rather than to individual masses. The Reynolds transport theorem is precisely the analytical tool required to transform the laws from one representation to the other. Let Ψsys(t) be an extensive property of the system at time t, ψ(r, t) is the corresponding intensive property. If Vsys(t) denotes a system material volume at time t, CV a control volume, and CS the control volume surface, the extensive system property can be defined by:

Ψsys(t) = ρψ dv (A.11)

Vsys(t)

The corresponding extensive control volume property is deﬁned by:

ΨCV (t) = ρψ dv (A.12) CV

The system is defined by the fluid mass within the control volume at the initial time t. The values of the analogous extensive properties of the system and the fluid within control volume are thus equal at this time, Ψsys(t0) = ΨCV (t0). A short time later a portion of the system fluid may have exited from the control volume and some of the surrounding fluid may have entered the control volume. Then we seek to determine how the rate of change of Ψsys within the system is related to the rate of change of ΨCV within the control volume at any instant. Based on a physical understanding of the concepts of the system and control volume motion, the kinematic Reynolds transport theorem relating system concepts to control volume concepts can be derived by geometrical analysis. The most general form of the Reynolds transport theorem is defined for an arbitrary moving and deforming control volume [20]: DΨsys(t) D D = ρψ dv = ρψ dv + ρψw · n da (A.13) Dt Dt Dt Vsys(t) CV(t) CS(t) in which w is the velocity of the fluid at the control surface with respect to the control surface, and ρ is the fluid density. For a fixed control volume, the integral limits are fixed thus the order of differentiation and integration may be interchanged, so the substantial derivative of the CV integral in (A.13) can be written in the equivalent form [100]: D ∂(ρψ) ρψ dv = dv (A.14) Dt ∂t CV CV 1366 Appendix A: Mathematical Theorems

Hence, for a fixed control volume, the Reynolds theorem (A.13) reduces to: DΨsys(t) D ∂(ρψ) = ρψ dv = dv + ρψv · n da (A.15) Dt Dt ∂t Vsys(t) CV CS in which the velocity v represents the velocity of the fluid at the control surface with respect to the control surface that coincides with the coordinate reference frame. This form of the Reynolds transport theorem is a special version of the more general mathematical statement (A.13). However, the latter version is particularly important because fixed control volumes are commonly employed in fluid mechanics and reactor modeling. In Chap. 1 the derivation of the governing Eulerian equations for single phase flow is performed employing (A.15). Let us now consider the Leibnitz theorem (A.8) and the Reynolds theorem (A.13). The volume integral on the RHS of the Reynolds theorem is defined over a control volume CV(t) which coincides with the geometric volume on the LHS of Leibnitz theorem at the considered instant t in time. At that instant the integrals cover precisely the same space, so we can substitute the Leibnitz theorem (A.8) expression for differentiating the integral into the Reynolds theorem (A.13). The Reynolds theorem can then be written as [20]: DΨsys(t) ∂ = (ρψ) dv + (ρψ)v · n da (A.16) Dt ∂t CV(t) CS(t)

The vector quantity v = w + v represents the velocity of the ﬂuid at the control surface with respect to the coordinate reference frame. This is the conventional Reynolds transport theorem.

A.2 Gauss Theorem

In general, consider a geometric volume, V(t), bounded by a closed surface, A(t), which may either be material or not, moving or not. At a given point belonging to this surface A(t), the unit normal vector n is outwardly directed. For any scalar ( f ), vector or tensor ﬁelds (f), Gauss theorem3 enable a surface integral to be transformed into a volume integral according to the following relation:

3 The Gauss’ theorem is also known as the divergence theorem, Green’s theorem, and Ostrogradsky’s theorem [99]. In particular, the vector form of Gauss’s theorem is normally referred to as the divergence theorem [55]. Appendix A: Mathematical Theorems 1367

Gauss’ theorem for a scalar For single phase ﬂows: f n da = ∇fdv (A.17) A(t) V(t)

For two phase ﬂows we need to consider the interface, and a modiﬁed form of the theorem is applied (e.g., [22, 23]). Now we may consider a geometric volume, Vk(t), bounded by a closed surface, Ak(t) + AI (t), which may either be material or not, moving or not. At a given point belonging to this surface Ak(t) + AI (t), the unit normal vector n is outwardly directed. f n da + f n da = ∇fdv (A.18)

Ak(t) AI (t) Vk (t)

Gauss’ theorem for a vector or tensor For single phase ﬂows: f · n da = ∇·f dv (A.19) A(t) V(t)

For two phase ﬂows we need to consider the interface, and a modiﬁed form of the theorem is applied. Now we may consider a geometric volume, Vk(t), bounded by a closed surface, Ak(t) + AI (t), which may either be material or not, moving or not. At a given point belonging to this surface Ak(t) + AI (t), the unit normal vector n is outwardly directed. f · n da + f · n da = ∇·f dv (A.20)

Ak (t) AI (t) Vk (t)

A.3 Surface Theorems

These theorems can be found in the books of [2, 67][89] (p. 73) [33], pp. 48Ð52 and in the papers of [13, 24], pp. 428Ð432, and pp. 436Ð438.

A.3.1 Leibnitz Transport Theorem for a Surface

The surface transport theorem ([89], p. 73) can be used to reformulate the term: 1368 Appendix A: Mathematical Theorems d I ρ ψ da (A.21) dt I I AI (t) where ψI is any scalar-, vector-, or tensor function of time and position on the dividing surface. The indicated integration is to be performed over the dividing surface in its current conﬁguration AI . We allow AI or the limits of the this integration to be a function of time. Slattery ([89], p. 74) presents the surface transport theorem: d d (ρ ψ ) I (ρ ψ ) da = I I I + ρ ψ ∇ · v da (A.22) dt I I dt I I I I AI (t) AI (t)

Note that this theorem can be reformulated into more generalized forms, but the formulation given here is used in most papers on reactor modeling.

A.3.2 Gauss Theorem for a Surface

In a 3D space, the Gauss theorems enable the transformation of a surface integral into a volume integral. Similarly, in a 2D space, the Gauss theorems enable the transformation of a line integral into a surface integral. The surface divergence theorem can be expressed in a generic form as follows [33]:

RI NI dl = ∇I · (eI RI )da = (∇I RI − (∇I · nI )RI nI ) da (A.23) AI (t) lI (t) AI where RI is a generic field of arbitrary tensorial order. The generic identity may be specialized by inserting a dot- or cross-product operational sign into identical tensorial positions on both sides of the equality sign. The scalar fields RI is defined by RI NI . The frequently used forms of the theorem are given as illustrative examples:

For interfacial vector ﬁelds

fI · NI dl = ∇I · fI da (A.24) AI (t) lI (t) where NI is the unit normal vector at a given point belonging to the curve l(t),the boundary of AI . The vector NI is directed outward the area AI and located in the tangent plane. ∇I is the surface del operator. fI is a vector tangent to the surface AI . Appendix A: Mathematical Theorems 1369

For interfacial tensor ﬁelds

fI · NI dl = ∇I · fI da (A.25) AI (t) lI (t) where fI is any interfacial tensor ﬁeld.

For interfacial scalar ﬁelds

fI dl = ∇I · (eI fI )da = (∇I fI − (∇I · nI )fI nI ) da (A.26)

lI AI AI where fI is any interfacial scalar ﬁeld, eI is the dyadic idemfactor ([33], p. 46), and ∇I · nI is the surface divergence (equal to twice the mean curvature).

Static Force Balance at a Fluid Interface Brenner ([13], pp. 428Ð432, and pp. 436Ð438), Middleman [66] (pp. 39Ð42) and Edwards et al. ([33], pp. 48Ð52) address the basic nature of macro-scale interfacial force balances at an arbitrary curved fluid in the state of hydrostatic equilibrium (a state that serves as a standard from which non-equilibrium interfacial transport processes depart). Analogous to 3D fluid continua, applying Newton second law the macro-scale fluid interfaces are acted upon by two fundamental types of forces, surface body forces per unit area, fI , and surface contact stress component, TI · NI , being a force per unit length. The apparent surface body force densities denote forces originating outside of the 2D interface itself, whereas the apparent surface stresses denote forces acting lineally by virtue of intimate contact between adjacent 2D interfacial fluid elements. Considering an interface existing in a state of hydrostatic equilibrium, an integral force balance over an element of area, AI , lying on a fluid interface may be expressed as :

FI da + TI · dl = 0 (A.27)

AI lI where dl = NI dl is an outwardly directed, differential line element to the closed contour lI of the area domain AI . To derive the differential counterpart of the integral balance, starting from (A.27) we need to express the second term (line integral) as an area integral. Thereafter, analogous to the 3D approach used in Chap.1 deriving the governing conservation equations, the differential form emerges requiring that the resulting relation is valid for an arbitrary chosen surface domain so that the integrand itself must vanish. To transform the line integral in (A.27) to a surface integral, the version of (A.23) that is defined inserting the dot product sign is relevant. Accordingly, introducing the 1370 Appendix A: Mathematical Theorems dot product and a tensor field into (A.23), a specific version of the surface divergence theorem can be derived:

RI · dl = ∇I · (eI · RI )da = ∇I · RI da lI AI AI

= (RI ·∇I · eI + eI ·∇I · RI ) da = (RI · 2HI nI +∇I · RI ) da

AI AI (A.28)

T = ·∇ =∇ ∇ · = where the following relationships have been used: eI eI , eI I I , I eI 2HI nI , and nI · TI = 0. To proceed we need to determine the nature of the interface defining the surface pressure- or stress tensor. For the general non-equilibrium circumstances TI is a non- isotropic and non-symmetric tensor, containing six independent components. Four of these components are analogous to the normal- and shear stresses in 3D fluids, while the so-called bending forces have no analog in ordinary 3D fluids. Fortunately, in practice most fluid interfaces are assumed to be inviscid and isotropic. For an isotropic interface existing in a state of hydrostatic equilibrium the surface-excess pressure tensor is given by TI = σI eI , where the scalar σI is the interfacial tension. This scalar quantity is an apparent macro-scale property of the physicochemical system, generally interpreted as the 2D analog of the thermodynamic pressure, p,for 3D continua. From thermodynamic analysis it is concluded that σI dependents only upon macro-scale pressure, temperature and interfacial composition at the point rI . eI denotes the dyadic idem-factor ([33], p. 46), defined as eI = e − nI nI , with the dyadic e being the 3D spatial idem-factor. For the special case of RI = TI ,(A.28) reduces to:

TI · dl = ∇I · (eI · TI )da = ∇I · TI da = ∇I · (σI eI )da lI AI AI AI (A.29) = (σI ∇I · eI + eI ·∇I σI ) da = (2HI σI nI +∇I σI ) da

AI AI

Thereby, for an isotropic interfacial tensor existing in a state of hydrostatic equilibrium (A.27) yields,

(FI +∇I · TI ) da = (FI + 2HI σI nI +∇I σI ) da = 0(A.30)

AI AI

As the choice of local surface domain, AI , is arbitrary, and since the ﬁeld variables FI (rI ) and TI (rI ) are independent of this choice, this requires at each point (rI ) that: Appendix A: Mathematical Theorems 1371

FI +∇I · TI = FI +∇I σI + 2HI σI nI = 0 (A.31) which constitutes the local surface force balance at a static fluid interface.4 Never- theless, the net surface tension force on a closed surface equals zero [32, 70, 71, 77]. Finally, we note that the surface body force term constitutes the sum of the surface-excess body force and the bulk-phase body force vector densities. The surface-excess body force is the 2D analog of continuum body forces in 3D fluids (e.g., gravitational force, electromagnetic force, etc). This force is often neglected. The bulk-phase body force has no counterpart for 3D fluids, as it denotes the stresses applied intimately at the interface by the surrounding 3D bulk phases. The normal component of this force equals the pressure difference between the two bulk phases, a relationship often referred to as the YoungÐLaplace equation.

4 This is the analog of the corresponding hydrostatic equation:

ρg −∇·T = 0 (A.32) for a 3D ﬂuid continuum, where the isotropic stresses are given by T = pe, with p = p(r) the thermodynamic pressure. Note also that in typical textbooks on mechanical engineering this relation will be given as T =−pe, as the hydrostatic equation yields:

ρg +∇·T = 0 (A.33) Appendix B Equation of Change for Temperature for a Multicomponent System

The equation of change for temperature can be derived from the enthalpy equation. For completeness, in this appendix we outline the procedure described by Bird et al. ([8], Problem 19D.2., pp. 608Ð609) and Instructor’s resource CD-ROM to accom- pany TRANSPORT PHENOMENA, Second Edition, pp. 15Ð19 and pp. 16Ð19.

B.1 The Problem Deﬁnition

In this appendix we will derive the equation of change for temperature: DT N T ∂ρ Dp ρCP =−∇·q − σ :∇v + Jc · gc − Dt = ρ ∂T Dt c 1 p,ω (B.1) N ¯ Jc Rc + hc ∇·( ) − ( ) c=1 Mωc Mωc starting out from the enthalpy equation:

Dh N Dp ρ =−∇·q − σ :∇v + J · g + (B.2) Dt c c Dt c=1

In the next section the sequence of steps for the derivation of (B.1)from(B.2) will be described in detail.

B.2 Deriving the Equation of Change for Temperature

For a closed system the enthalpy is given by Hˆ = Hˆ (P, T), while for an open system that can exchange mass with its surroundings the total enthalpy depends also on the possible changes in masses Mc of each component c. This implies that for an open

H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1373 © Springer International Publishing Switzerland 2014 1374 Appendix B: Equation of Change for Temperature

ˆ ˆ system the enthalpy is written as H = H(T, p, M1, M2, M3, ..., MN ), where N is the number of chemical components. An extensive quantity can be divided by the mass of the system constituting a new variable defining a specific quantity ([54], p. 103). The specific enthalpy (per unit mass) is then expressed as:

N ˇ H = H(T, p, ω1, ω2, ω3,...,ωN−1) = ωcHc (B.3) c=1

ˇ ∂Hˆ where the partial speciﬁc enthalpy of species c is given by H = ( ) , , = c ∂Mc M p T ∂(MH) ( ) , , . Note also that these intensive quantities may be computed both per ∂Mc M p T unit mass, per unit volume, or per mole. In the given notation the LHS of (B.2) can be reformulated in terms of a complete differential if we consider the enthalpy per unit mass to be a thermodynamic function of T, p and the ﬁrst (N − 1) mass fractions:

− ∂H ∂H N1 ∂H dH = dp + dT + dωc (B.4) ∂p , ∂T , ∂ωc , , T ω p ω c=1 p T ω and then apply the principle of local instantaneous equilibrium:

− Dh ∂h Dp ∂h DT N1 ∂h Dω = + + c (B.5) Dt ∂p , Dt ∂T , Dt ∂ωc , , Dt T ω p ω c=1 p T ω

The capital letters for the quantities (e.g., H = H(T, p, ω1, ω2, ω3, ..., ωN−1)) indi- cate that we are considering the variable as a thermodynamic function, whereas the corresponding variables used in continuum mechanics are deﬁned by lower case letters (e.g., h = h(t, r, T, p, ω1, ω2, ω3, ..., ωN−1)). Dωc Next, the term Dt can be eliminated from (B.5) using the transport equation for the chemical species: Dω ρ c =−∇J + R (B.6) Dt c c

Dωc The coefficient in front of Dt in (B.5) is usually reformulated in terms of molar quantities known from thermodynamic theory. This is a rather complex task, thus a detailed description of this part of the model derivation is given shortly. The theoretical basis for the model derivation used at this point originates from thermodynamics. The corresponding variables used in continuum mechanics are then defined in analogy to the thermodynamic quantities applying the principle of local instantaneous equilibrium, in line with the approach adopted above obtaining (B.5)from(B.4). Following the same approach, the thermodynamic quantities and Dωc relations needed in order to reformulate the coefficient in front of Dt in (B.5) will be Appendix B: Equation of Change for Temperature 1375 described before we introduce the corresponding extensions approved in continuum mechanic theory. If we first consider the enthalpy variable as a thermodynamic quantity, the total ˆ ˆ enthalpy, H = H(T, p, M1, M2, M3, ..., MN ), and the specific enthalpy (per unit mass), H = H(T, p, ω1, ω2, ω3, ..., ωN−1), can be defined characterizing an open thermodynamic system. Since enthalpy is an extensive thermodynamic property, we may write: ˆ H(M1, M2, M3, ..., MN ) = MH(ω1, ω2, ω3, ..., ωN−1) (B.7) in which the Mα are the masses of the various species, M is the sum of the Mα’s, and ˆ ωα = Mα/M are the corresponding mass fractions. Both H and H are understood to be functions of T, p and as well as of composition (i.e., T and p have been left out for simplicity in the mathematical manipulation below). By use of the chain rule of partial differentiation we find for α = N: ˆ N−1 ( ) ∂ω ( ) ∂H = ∂ MH β + ∂ MH ∂M ∂Mα β=1 ∂ωβ ω ,M ∂Mα M ∂M ω ∂Mα M Mγ γ γ γ γ − N1 ∂H ∂ Mβ/M ∂M = M + H · 1 β=1 ∂ωβ ω ∂Mα ∂M ω γ Mγ γ N−1 ∂H 1 ∂M M ∂M = M β − β + H · · 2 1 1 = ∂ωβ M ∂Mα M ∂Mα β 1 ωγ N−1 ∂H δ M = M αβ − β + H 2 = ∂ωβ M M β 1 ωγ N−1 ∂H Mβ = δαβ − + H = ∂ω M β 1 β ωγ (B.8) The corresponding expression for α = N can be found in a similar way:

ˆ N−1 ( ) ∂ω ( ) ∂H = ∂ MH β + ∂ MH ∂M ∂MN β=1 ∂ωβ ω ,M ∂MN M ∂M ω ∂M N M Mγ γ γ γ γ N−1 ∂H 1 ∂M M ∂M = M β − β + H · 2 1 = ∂ωβ M ∂MN M ∂MN β 1 ωγ N−1 ∂H M = M − β + H 2 = ∂ωβ M β 1 ωγ N−1 ∂H M = − β + H = ∂ω M β 1 β ωγ (B.9) The subscript ωγ means that all other mass fractions should be held constant. 1376 Appendix B: Equation of Change for Temperature

Subtraction then gives for α = N:

∂Hˆ − ∂Hˆ ∂Mα ∂MN Mγ Mγ N− N− 1 M 1 M = ∂H δ − β + H −[ ∂H − β + H] ∂ωβ αβ M ∂ωβ M β=1 ωγ β=1 ωγ N− 1 M M (B.10) = ∂H δ − β + β ∂ωβ αβ M M β=1 ωγ N−1 = ∂H δ = ∂H ∂ωβ αβ ∂ωα β=1 ωγ ωγ

In the last paragraph the coefficient of dωc in (B.4) has been expressed in terms of species masses rather than species mass fractions by use of thermodynamic theory and a complete differential regarding the specific enthalpy to be a function of T, p and the first (N − 1) mass fractions. These species mass based functions may then more easily be converted to the appropriate molar quantities which we can obtain from thermodynamic models. On the equivalent molar basis5 the total enthalpy content (or the enthalpy) Hˆ may ˆ ˆ be defined by H = MH = nH. This means that H(T, p, M1, M2, M3, ..., MN ) may ˆ equivalently be written as H(T, p, n1, n2, n3, ..., nN ). The specific molar enthalpy (per mole) is then expressed as:

N H = H(T, p, x1, x2, x3, ..., xN−1) = xcHc (B.11) c=1

∂Hˆ where the partial (speciﬁc) molar enthalpy of species c is given by H = ( ) , , = c ∂nc n p T ∂(nH) ˇ ∂Hˆ 1 ∂Hˆ ( )n,p,T . Thus, we can easily see that Hc = ( )M,p,T = ( )n,p,T = ∂nc ∂Mc Mwc ∂nc 1 ∗ Hc. The speciﬁc molar enthalpy for an ideal gas is denoted by H , and for ideal Mwc N = ∗ = ∗ = ∗ gases we have the following relations: H H xcHc and Hc Hc . c=1 The LHS of (B.2) can now be expressed as:

⎛ ⎞ ∂ 1 Dh = ⎝ 1 − ρ ⎠ Dp + DT ρ Dt ρ ρ T ∂T Dt CP Dt p,ω (B.12) − N1 ˆ ˆ + [ ∂h − ∂h ] (−∇ · J + R ) ∂Mc ∂MN c c c=1 M M

5 For further studies of these thermodynamic quantities the reader is referred to standard thermodynamic—and continuum mechanic textbooks [7, 8, 54, 56, 79, 81]. Appendix B: Equation of Change for Temperature 1377 where we have applied the principle of local instantaneous equilibrium to obtain an Dωc expression for the coefﬁcient in front of Dt in (B.5) based on the result we obtained from thermodynamic theory as given in (B.10). = ˆ = = Because of the relations Mc ncMwc and h Mh nh, the differential quotients in the last term in (B.12) can be rewritten in terms of partial molar quantities: ⎛ ⎛ ⎞ ⎞ 1 Dh 1 ∂ ρ Dp DT ρ = ρ ⎝ − T ⎝ ⎠ ⎠ + ρC Dt ρ ∂T Dt P Dt p,ω N− 1 1 ∂hˆ 1 ∂hˆ + [ − ] (−∇ · Jc + Rc) M ∂nc M ∂nN c=1 wc wN ⎛ ⎛ n⎞ ⎞ n 1 1 ∂ ρ Dp DT = ρ ⎝ − T ⎝ ⎠ ⎠ + ρC ρ ∂T Dt P Dt p,ω − N1 h¯ h¯ + [ c − N ] (−∇ · J + R ) M M c c c=1 wc wN ⎛ ⎛ ⎞ ⎞ 1 1 ∂ ρ Dp DT = ρ ⎝ − T ⎝ ⎠ ⎠ + ρC ρ ∂T Dt P Dt p,ω − − N1 h¯ N1 h¯ + c (−∇ · J + R ) − (−∇ · J + R ) N M c c c c M c=1 wc c=1 wN ⎛ ⎛ ⎞ ⎞ 1 1 ∂ ρ Dp DT = ρ ⎝ − T ⎝ ⎠ ⎠ + ρC ρ ∂T Dt P Dt p,ω − N1 J R h¯ + h¯ −∇ · c + c + (−∇ · J + R ) N c M M N N M c=1 wc wc wN ⎛ ⎛ ⎞ ⎞ 1 1 ∂ ρ Dp DT = ρ ⎝ − T ⎝ ⎠ ⎠ + ρC ρ ∂T Dt P Dt p,ω − N1 J R J R + h¯ −∇ · c + c + h¯ −∇ · N + N c M M N M M c=1 wc wc wN wN ⎛ ⎛ ⎞ ⎞ ∂ 1 N ⎝ 1 ⎝ ρ ⎠ ⎠ Dp DT ¯ Jc Rc = ρ − T + ρCP + hc −∇ · + ρ ∂T Dt Dt = Mwc Mwc p,ω c 1 (B.13)

N N where we have used the relations that Jc = 0 and Rc = 0. c=1 c=1 Inserted into (B.2) this result coincides with the temperature equation on the form (B.1). Appendix C Governing Equations for Single Phase Flow

C.1 The Mass Based Equation of Continuity

Vector notation

∂ρ +∇·(ρv) = 0(C.1) ∂t

Cartesian Coordinates (x, y, z)

∂ρ ∂ ∂ ∂ + (ρv ) + (ρv ) + (ρv ) = 0(C.2) ∂t ∂x x ∂y y ∂z z

Cylindrical Coordinates (r, θ,z)

∂ρ 1 ∂ 1 ∂ ∂ + (rρv ) + (ρv ) + (ρv ) = 0(C.3) ∂t r ∂r r r ∂θ θ ∂z z

H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1379 © Springer International Publishing Switzerland 2014 1380 Appendix C: Governing Equations for Single Phase Flow

Spherical Coordinates (r, θ, φ)

∂ρ 1 ∂ 2 1 ∂ 1 ∂ + (r ρvr) + (ρv sin θ) + (ρv ) = 0(C.4) ∂t r2 ∂r2 r sin θ ∂θ θ r sin θ ∂φ φ

C.2 The Equation of Motion

Vector notation

∂ (ρv) +∇·(ρvv) =−∇p −∇·σ + ρg (C.5) ∂t

Cartesian Coordinates (x, y, z)

x-component: ∂ ∂ ∂ ∂ (ρv ) + (ρv v ) + (ρv v ) + (ρv v ) ∂t x ∂x x x ∂y y x ∂z z x ∂p ∂σ ∂σyx ∂σ =− − xx − − zx + ρg (C.6) ∂x ∂x ∂y ∂z x y-component:

∂ ∂ ∂ ∂ (ρv ) + (ρv v ) + (ρv v ) + (ρv v ) ∂t y ∂x x y ∂y y y ∂z z y ∂p ∂σxy ∂σyy ∂σzy =− − − − + ρg (C.7) ∂y ∂x ∂y ∂z y Appendix C: Governing Equations for Single Phase Flow 1381

z-component:

∂ ∂ ∂ ∂ (ρv ) + (ρv v ) + (ρv v ) + (ρv v ) ∂t z ∂x x z ∂y y z ∂z z z ∂p ∂σ ∂σyz ∂σ =− − xz − − zz + ρg (C.8) ∂z ∂x ∂y ∂z z

Cylindrical Coordinates (r, θ,z)

r-component:

v2 ∂ 1 ∂ 1 ∂ ρ θ ∂ (ρvr) + (rρvrvr) + (ρvθvr) − + (ρvzvr) ∂t r ∂r r ∂θ r ∂z (C.9) ∂p 1 ∂ 1 ∂ σ ∂ =− − (rσ ) − (σ ) + θθ − (σ ) + ρg ∂r r ∂r rr r ∂θ θr r ∂z zr r

θ-component:

∂ 1 ∂ 1 ∂ ρv v ∂ (ρv ) + (rρv v ) + (ρv v ) + r θ + (ρv v ) = ∂t θ r ∂r r θ r ∂θ θ θ r ∂z z θ 1 ∂p 1 ∂ 1 ∂ ∂ (C.10) − − (r2σ ) − (σ ) − (σ ) + ρg r ∂θ r2 ∂r rθ r ∂θ θθ ∂z zθ θ

z-component:

∂ 1 ∂ 1 ∂ ∂ (ρv ) + (rρv v ) + (ρv v ) + (ρv v ) = ∂t z r ∂r r z r ∂θ θ z ∂z z z ∂p 1 ∂ 1 ∂ ∂ (C.11) − − (rσ ) − (σ ) − (σ ) + ρg ∂z r ∂r rz r ∂θ θz ∂z zz z 1382 Appendix C: Governing Equations for Single Phase Flow

Spherical Coordinates (r, θ, φ)

r-component:

∂ 1 ∂ 2 1 ∂ 1 ∂ (ρvr) + (r ρvrvr) + (ρv vr sin θ) + (ρv vr) ∂t r2 ∂r r sin θ ∂θ θ r sin θ ∂φ φ ρ(v2 + v2) θ φ ∂p 1 ∂ 2 1 ∂ − =− − (r σrr) − (σ sin θ) r ∂r r2 ∂r r sin θ ∂θ θr 1 ∂ σθθ + σφφ − (σ ) + + ρgr (C.12) r sin θ ∂φ φr r

θ-component:

v v ∂ 1 ∂ 2 1 ∂ 1 ∂ ρ r θ (ρv ) + (r ρvrv ) + (ρv v sin θ) + (ρv v ) + ∂t θ r2 ∂r θ r sin θ ∂θ θ θ r sin θ ∂φ φ θ r 2 ρv cot θ 1 ∂p 1 ∂ 1 ∂ 1 ∂ − φ =− − (r3σ ) − (σ sin θ) − (σ ) r r ∂θ r3 ∂r rθ r sin θ ∂θ θθ r sin θ ∂φ φθ σ cot θ − rθ + σ + ρg (C.13) r r φφ θ

φ-component:

v v ∂ 1 ∂ 2 1 ∂ 1 ∂ ρ r φ (ρv ) + (r ρvrv ) + (ρv v sin θ) + (ρv v ) + ∂t φ r2 ∂r φ r sin θ ∂θ θ φ r sin θ ∂φ φ φ r ρv v cot θ 1 ∂p 1 ∂ 1 ∂ − θ φ =− − (r3σ ) − (σ sin θ) r r sin θ ∂φ r3 ∂r rφ r sin θ ∂θ θφ 1 ∂ σ + 2σ cot θ − (σ ) − rφ θφ + ρg (C.14) r sin θ ∂φ φφ r φ

C.2.1 The Viscous Stress Tensor for Newtonian Fluids

Vector notation

T 2 σ =−μ ∇v + (∇v) + μ∇·ve, where e = eiejδij (C.15) 3 i j Appendix C: Governing Equations for Single Phase Flow 1383

C.2.2 Cartesian Coordinates (x, y, z):

∂vx 2 σxx =−μ 2 − (∇·v) (C.16) ∂x 3

∂vy 2 σyy =−μ 2 − (∇·v) (C.17) ∂y 3

∂vz 2 σzz =−μ 2 − (∇·v) (C.18) ∂z 3

∂v ∂vy σ = σ =−μ x + (C.19) xy yx ∂y ∂x

∂vy ∂v σ = σ =−μ + z (C.20) yz zy ∂z ∂y

∂v ∂v σ = σ =−μ z + x (C.21) zx xz ∂x ∂z

∂v ∂vy ∂v ∇·v = x + + z (C.22) ∂x ∂y ∂z 1384 Appendix C: Governing Equations for Single Phase Flow

C.2.3 Cylindrical Coordinates (r, θ,z):

∂vr 2 σrr =−μ 2 − (∇·v) (C.23) ∂r 3

1 ∂v v 2 σ =−μ 2 θ + r − (∇·v) (C.24) θθ r ∂θ r 3

∂vz 2 σzz =−μ 2 − (∇·v) (C.25) ∂z 3

∂ v 1 ∂v σ = σ =−μ r θ + r (C.26) rθ θr ∂r r r ∂θ

∂v 1 ∂v σ = σ =−μ θ + z (C.27) θz zθ ∂z r ∂θ

∂v ∂v σ = σ =−μ z + r (C.28) zr rz ∂r ∂z

1 ∂ 1 ∂ ∂v ∇·v = (rv ) + (v ) + z = 0 (C.29) r ∂r r r ∂θ θ ∂z Appendix C: Governing Equations for Single Phase Flow 1385

C.2.4 Spherical Coordinates (r, θ, φ):

∂vr 2 σrr =−μ 2 − (∇·v) (C.30) ∂r 3

1 ∂v v 2 σ =−μ 2 θ + r − (∇·v) (C.31) θθ r ∂θ r 3

1 ∂v v v cot θ 2 σ =−μ 2 φ + r + θ − (∇·v) (C.32) φφ r sin θ ∂φ r r 3

∂ v 1 ∂v σ = σ =−μ r θ + r (C.33) rθ θr ∂r r r ∂θ

sin θ ∂ v 1 ∂v σ = σ =−μ φ + θ (C.34) θφ φθ r ∂θ sin θ r sin θ ∂φ

1 ∂v ∂ v σ = σ =−μ r + r φ (C.35) φr rφ sin θ ∂φ ∂r r

v 1 ∂ 2 1 ∂ 1 ∂ φ ∇·v = (r vr) + (v sin θ) + (C.36) r2 ∂r r sin θ ∂θ θ r sin θ ∂φ 1386 Appendix C: Governing Equations for Single Phase Flow

C.3 The Equation of Heat in Terms of T and Flux q

Vector notation

∂T T ∂ρ Dp ρCp + v ·∇T =−∇·q − − σ :∇v ∂t ρ ∂T p,ω Dt N ¯ Q (C.37) hs Rref ,r + ∇·j + −H , Mω s Mw Rref r s=1 s r=1 ref

Cartesian Coordinates (x, y, z)

∂T ∂T ∂T ∂T ∂qx ∂qy ∂qz ρCp + vx + vy + vz =− + + ∂t ∂x ∂y ∂z ∂x ∂y ∂z N ¯ − T ∂ρ Dp − σ :∇v + hs ∂jsx + ∂jsy + ∂jsz ρ ∂T , Dt Mωs ∂x ∂y ∂z (C.38) p ω s=1 Q Rref ,r + −H , Mw Rref r r=1 ref

Cylindrical Coordinates (r, θ,z)

∂T ∂T v ∂T ∂T 1 ∂ 1 ∂q ∂q ρC + v + θ + v =− (rq ) + θ + z p ∂t r ∂r r ∂θ z ∂z r ∂r r r ∂θ ∂z N ¯ T ∂ρ Dp hs 1 ∂ 1 ∂jθ,s ∂jz,s − − σ :∇v + (rjr,s) + + ρ ∂T , Dt M r ∂r r ∂θ ∂z p ω s=1 ωs Q Rref ,r + −H (C.39) M Rref ,r r=1 wref Appendix C: Governing Equations for Single Phase Flow 1387

Spherical Coordinates (r, θ, φ)

∂T ∂T vθ ∂T vφ ∂T ρCp + vr + + ∂t ∂r r ∂θ r sin φ ∂φ 1 ∂ 1 ∂ 1 ∂q T ∂ρ Dp =− r2q − (q ) − φ − − σ :∇v 2 r θ sin θ r ∂r r sin θ ∂θ r sin θ ∂φ ρ ∂T p,ω Dt N ¯ hs 1 ∂ 2 1 ∂ 1 ∂jφ,s + (r jr,s) + (j , sin θ) + M r2 ∂r r sin θ ∂θ θ s r sin θ ∂φ s=1 ωs Q R , + ref r −H (C.40) M Rref ,r r=1 wref

C.3.1 The Fourier Law of Heat Conduction

Vector notation

q =−k∇T (C.41)

Cartesian Coordinates (x, y, z)

x − component: y − component: z − component: ∂T ∂T ∂T (C.42) q =−k q =−k q =−k x ∂x y ∂y z ∂z 1388 Appendix C: Governing Equations for Single Phase Flow

Cylindrical Coordinates (r, θ,z)

r − component: θ − component: z − component: ∂T 1 ∂T ∂T (C.43) q =−k q =−k q =−k r ∂r θ r ∂θ z ∂z

Spherical Coordinates (r, θ, φ)

r − component: θ − component: φ − component: ∂T 1 ∂T 1 ∂T (C.44) qr =−k q =−k q =−k ∂r θ r ∂θ φ r sin θ ∂φ

C.3.2 The Viscous Dissipation Term

Vector notation

σ : ∇v

Cartesian Coordinates (x, y, z)

∂vx ∂vx ∂vx ∂vy ∂vy σ :∇v = σxx + σxy + σxz + σyx + σyy ∂x ∂y ∂z ∂x ∂y ∂vy ∂v ∂v ∂v + σ + σ z + σ z + σ z (C.45) yz ∂z zx ∂x zy ∂y zz ∂z Appendix C: Governing Equations for Single Phase Flow 1389

Cylindrical Coordinates (r, θ,z)

∂vr 1 ∂vr vθ ∂vr ∂vθ σ :∇v = σrr + σrθ − + σrz + σθr ∂r r ∂θ r ∂z ∂r 1 ∂vθ vr ∂vθ ∂vz 1 ∂vz + σθθ + + σθz + σzr + σzθ r ∂θ r ∂z ∂r r ∂θ ∂v + σ z (C.46) zz ∂z

Spherical Coordinates (r, θ, φ)

∂vr 1 ∂vr vθ 1 ∂vr vφ σ :∇v = σrr + σrθ − + σrφ − ∂r r∂θ r rsin θ ∂φ r ∂vθ 1 ∂vθ vr 1 ∂vθ vφ cot θ + σθr + σθθ + + σθφ − ∂r r ∂θ r r sin θ ∂φ r ∂v ∂v ∂v v v + σ φ + σ 1 φ + σ 1 φ + r + θ cot θ φr ∂r φθ r ∂θ φφ r sin θ ∂φ r r (C.47)

C.4 The Species A Mass Balance in Terms of Flux jA

Vector notation

∂(ρω ) A +∇·(ρvω ) =−∇·j + R (C.48) ∂t A A A 1390 Appendix C: Governing Equations for Single Phase Flow

Cartesian Coordinates (x, y, z)

∂ ∂ ∂ ∂ (ρωA) + (ρωAvx) + (ρωAvy) + (ρωAvz) ∂t ∂x ∂y ∂z ∂j ∂jAy ∂j =− Ax + + Az + R (C.49) ∂x ∂y ∂z A

Cylindrical Coordinates (r, θ,z)

∂ 1 ∂ 1 ∂ ∂ (ρω ) + (rρω v ) + (ρω v ) + (ρω v ) ∂t A r ∂r A r r ∂θ A θ ∂z A z 1 ∂ 1 ∂ ∂j (C.50) =− (rj ) + (j ) + Az + R r ∂r Ar r ∂θ Aθ ∂z A

Spherical Coordinates (r, θ, φ)

∂ 1 ∂ 1 ∂ 1 ∂ ( ) + (r2 v ) + ( v ) + ( v ) ρωA 2 ρωA r ρωA θ sin θ ρωA φ ∂t r ∂r r sin θ ∂θ r sin θ∂φ 1 ∂ 2 1 ∂ 1 ∂jAφ =− (r jAr) + (j sin θ) + + RA (C.51) r2 ∂r r sin θ ∂θ Aθ r sin θ ∂φ

C.5 The Mass Based Fick’s Law of Binary Diffusion

Vector notation

jA =−ρDAB∇ωA (C.52) Appendix C: Governing Equations for Single Phase Flow 1391

Cartesian Coordinates (x, y, z)

x-component: y-component: z-component:

∂ω ∂ω ∂ω j =−ρD A j =−ρD A j =−ρD A (C.53) Ax AB ∂x Ay AB ∂y Az AB ∂z

Cylindrical Coordinates (r, θ,z)

r-component: θ-component: z-component:

∂ω 1 ∂ω ∂ω j =−ρD A j =−ρD A j =−ρD A (C.54) Ar AB ∂r Aθ AB r ∂θ Az AB ∂z

Spherical Coordinates (r, θ, φ)

r-component: θ-component: φ-component:

∂ωA 1 ∂ωA 1 ∂ωA jAr =−ρDAB j =−ρDAB j =−ρDAB (C.55) ∂r Aθ r ∂θ Aφ r sin θ ∂φ

C.6 The Mole Based Equation of Continuity

Vector notation

N ∂c ∗ +∇·(cv ) = r (C.56) ∂t s s=1 1392 Appendix C: Governing Equations for Single Phase Flow

Cartesian Coordinates (x, y, z)

N ∂c ∂ ∗ ∂ ∗ ∂ ∗ + (cv ) + (cv ) + (cv ) = r (C.57) ∂t ∂x x ∂y y ∂z z s s=1

Cylindrical Coordinates (r, θ,z)

N ∂c 1 ∂ ∗ 1 ∂ ∗ ∂ ∗ + (rcv ) + (cv ) + (cv ) = r (C.58) ∂t r ∂r r r ∂θ θ ∂z z s s=1

Spherical Coordinates (r, θ, φ)

N ∂c 1 ∂ 2 ∗ 1 ∂ ∗ 1 ∂ ∗ + (r cv ) + (cv sin θ) + (cv ) = rs (C.59) ∂t r2 ∂r2 r r sin θ ∂θ θ r sin θ ∂φ φ s=1

∗ C.7 The Species A Mole Balance in Terms of Flux JA

Vector notation

∂(cx ) ∗ ∗ A +∇· cv x =−∇·J + r (C.60) ∂t A A A Appendix C: Governing Equations for Single Phase Flow 1393

Cartesian Coordinates (x, y, z)

∂ ( ) + ∂ ( v∗) + ∂ ( v∗) + ∂ ( v∗) cxA cxA x cxA y cxA z ∂t ∂x ∂y ∂z ∗ ∗ ∗ ∂J ∂JAy ∂J =− Ax + + Az + r (C.61) ∂x ∂y ∂z A

Cylindrical Coordinates (r, θ,z)

∂ ( ) + 1 ∂ ( v∗) + 1 ∂ ( v∗) + ∂ ( v∗) cxA rcxA r cxA θ cxA z ∂t r ∂r r ∂θ ∗ ∂z 1 ∂ 1 ∂ ∂J (C.62) =− (rJ∗ ) + (J∗ ) + Az + r r ∂r Ar r ∂θ Aθ ∂z A

Spherical Coordinates (r, θ, φ)

∂ 1 ∂ 1 ∂ 1 ∂ (cx ) + (r2cx v∗) + (cx v∗ ) + (cx v∗) A 2 A r A θ sin θ A φ ∂t r ∂r r sin θ ∂θ r sin θ∂φ ∂J∗ 1 ∂ 2 ∗ 1 ∂ ∗ 1 Aφ =− (r J ) + (J sin θ) + + rA (C.63) r2 ∂r Ar r sin θ ∂θ Aθ r sin θ ∂φ

C.8 The Mole Based Fick’s Law of Binary Diffusion

Vector notation

∗ =− ∇ JA cDAB xA (C.64) 1394 Appendix C: Governing Equations for Single Phase Flow

Cartesian Coordinates (x, y, z)

x-component: y-component: z-component:

∗ ∂x ∗ ∂x ∗ ∂x J =−cD A J =−cD A J =−cD A (C.65) Ax AB ∂x Ay AB ∂y Az AB ∂z

Cylindrical Coordinates (r, θ,z)

r-component: θ-component: z-component:

∗ ∂x ∗ 1 ∂x ∗ ∂x J =−cD A J =−cD A J =−cD A (C.66) Ar AB ∂r Aθ AB r ∂θ Az AB ∂z

Spherical Coordinates (r, θ, φ)

r-component: θ-component: φ-component:

∗ ∂xA ∗ 1 ∂xA ∗ 1 ∂xA J =−cDAB J =−cDAB J =−cDAB Ar ∂r Aθ r ∂θ Aφ r sin θ ∂φ (C.67)

Multicomponent Mass Diffusion Flux Models

C.9 The Mass Based Maxwell-Stefan Flux Model

Vector notation

n Jk Mwws e − ρgws∇ln(Mw) − ρg∇ws Mw D k=1 k sk = = i s Js n (C.68) wk Mw e Mw D k=1 k sk k=s Appendix C: Governing Equations for Single Phase Flow 1395

Cartesian Coordinates (x, y, z)

n Jk,x ∂ ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D ∂x ∂x k=1 k sk = = k s Js,x n (C.69) wk Mw e Mw D k=1 k sk k=s

n Jk,y ∂ ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D ∂y ∂y k=1 k sk = = k s Js,y n (C.70) wk Mw e Mw D k=1 k sk k=s

n Jk,z ∂ ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D ∂z ∂z k=1 k sk = = k s Js,z n (C.71) wk Mw e Mw D k=1 k sk k=s

Cylindrical Coordinates (r, θ,z)

n Jk,r ∂ ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D ∂r ∂r k=1 k sk = = k s Js,r n (C.72) wk Mw e Mw D k=1 k sk k=s 1396 Appendix C: Governing Equations for Single Phase Flow

n Jk,θ 1 ∂ 1 ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D r ∂θ r ∂θ k=1 k sk = = k s Js,θ n (C.73) wk Mw e Mw D k=1 k sk k=s

n Jk,z ∂ ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D ∂z ∂z k=1 k sk = = k s Js,z n (C.74) wk Mw e Mw D k=1 k sk k=s

Spherical Coordinates (r, θ, φ)

n Jk,r ∂ ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D ∂r ∂r k=1 k sk = = k s Js,r n (C.75) wk Mw e Mw D k=1 k sk k=s

n Jk,θ 1 ∂ 1 ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D r ∂θ r ∂θ k=1 k sk = = k s Js,θ n (C.76) wk Mw e Mw D k=1 k sk k=s

n Jk,φ 1 ∂ 1 ∂ws Mwws e − ρgws ln(Mw) − ρg Mw D rSinφ ∂φ rSinφ ∂φ k=1 k sk = = k s Js,φ n (C.77) wk Mw e Mw D k=1 k sk k=s Appendix C: Governing Equations for Single Phase Flow 1397

C.10 The Mass Based Dusty Gas Flux Model

Vector notation

n 2 Jk vmρgMw M ws e − e − ρg (ws∇Mw + Mw∇ws) w Mw D D k=1 k sk sk = = k s Js n (C.78) 2 wk Mw M e + e w Mw D D k=1 k sk sk k=s

Cartesian Coordinates (x, y, z)

n 2 Jk,x vmρgMw ∂Mw ∂ws M ws e − e − ρg ws + Mw w Mw D D ∂x ∂x k=1 k sk sk = = k s Js,x n (C.79) 2 wk Mw M e + e w Mw D D k=1 k sk sk k=s

n 2 Jk,y vmρgMw ∂Mw ∂ws M ws e − e − ρg ws + Mw w Mw D D ∂y ∂y k=1 k sk sk = = k s Js,y n (C.80) 2 wk Mw M e + e w Mw D D k=1 k sk sk k=s

n 2 Jk,z vmρgMw ∂Mw ∂ws M ws e − e − ρg ws + Mw w Mw D D ∂z ∂z k=1 k sk sk = = k s Js,z n (C.81) 2 wk Mw M e + e w Mw D D k=1 k sk sk k=s 1398 Appendix C: Governing Equations for Single Phase Flow

Cylindrical Coordinates (r, θ,z)

n v 2 Jk,r m,r ρgMw ∂Mw ∂ws M ws e − e − ρg ws + Mw w Mw D D ∂r ∂r k=1 k sk sk = = k s Js,r n (C.82) 2 wk Mw M e + e w Mw D D k=1 k sk sk k=s

n v 2 Jk,θ m,θρgMw 1 ∂Mw 1 ∂ws M ws e − e − ρg ws + Mw w Mw D D r ∂θ r ∂θ k=1 k sk sk = = k s Js,θ n (C.83) 2 wk Mw M e + e w Mw D D k=1 k sk sk k=s

n v 2 Jk,z m,zρgMw ∂Mw ∂ws M ws e − e − ρg ws + Mw w Mw D D ∂z ∂z k=1 k sk sk = = k s Js,z n (C.84) 2 wk Mw M e + e w Mw D D k=1 k sk sk k=s

Spherical Coordinates (r, θ, φ)

n v 2 Jk,r m,r ρgMw ∂Mw ∂ws M ws e − e − ρg ws + Mw w Mw D D ∂r ∂r k=1 k sk sk = = k s Js,r n (C.85) 2 wi Mw M e + e w Mw D D k=1 k sk sk k=s Appendix C: Governing Equations for Single Phase Flow 1399 n v 2 Jk,θ m,θρgMw 1 ∂Mw 1 ∂ws M ws e − e − ρg ws + Mw w Mw D D r ∂θ r ∂θ k=1 k sk sk = = k s Js,θ n (C.86) 2 wi Mw M e + e w Mw D D k=1 k sk sk k=s

n v 2 Jk,φ m,φρgMw 1 ∂Mw 1 ∂ws M ws e − e − ρg ws + Mw w Mw D D rSinθ ∂φ rSinθ ∂φ k=1 k sk sk = = k s Js,φ n 2 wi Mw M e + e w Mw D D k=1 k sk sk k=s (C.87) Appendix D Alternative Two-Fluid Model Granular Material Kinetic Theory Closures

Two families of two-fluid model closures have emerged from the kinetic theory of granular flow (KTGF). The first family of closures is based on the modeling work of Gidaspow [42] and the second family of closures is based on the modeling work of Simonin et al. [3Ð6, 25, 47, 48, 68, 82, 83, 86]. The kinetic theory of granular flow (KTGF) derived by Gidaspow [42] for dense beds has been reviewed in Chap. 4. The KTGF derived by Simonin et al. is outlined in this appendix. This theory might be viewed as an extention of the Chapman-Enskog theory presented by Gidaspow [42]. With these to families of closure models at hand, numerous models for fluidized bed reactor flows have been established based on solving the average continuity, momentum and granular temperature or turbulent kinetic energy equations. How- ever, the majority of the papers published on this topic still focus on cold flow gas-particle flows, intending to develop closures that are able to predict the important flow phenomena observed analyzing experimental data. Very few attempts have been made to predict the performance of chemical reactive processes using this type of models.

Alternative Two-Fluid Model Closures

According to Enwald and Almstedt [36], the existing ensemble averaged two-ﬂuid model closures for bubbling beds, developed by Simonin et al. (e.g., [3Ð6, 25, 47, 48, 68, 82, 83, 86]), Drew [31], Drew and Lahey [30], and the group at Chalmers University of Technology (e.g., [35Ð37, 74Ð76]), are frequently divided into four different model classes. With increasing model complexity, these model versions are: • Constant Particle Viscosity (CPV) models. • Particle Turbulence (PT) models. • Particle and Gas Turbulence (PGT) models. • Particle and Gas Turbulence with Drift Velocity (PGTDV) models.

H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1401 © Springer International Publishing Switzerland 2014 1402 Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures

The continuity and momentum equations that are common for these model versions are listed below. The model equations adopted for non-reactive mixtures can be deduced from the more general formulations (3.297) and (3.300), respectively. The continuity equation used is expressed as: ∂ α ρ Xk +∇· α ρ Xk v Xk ρk = 0(D.1) ∂t k k k k k

The momentum equation employed is given by: ∂ α ρ Xk v Xk ρk +∇· α ρ Xk v Xk ρk v Xk ρk t k k k k k k k ∂ (D.2) Re,X =−∇· Xk + k + Xk + αk Tk αkTk αk ρk g MkI

In order to separate the average of products into products of average, weighted averaged values are commonly introduced. The phasic—and mass averages have been deﬁned by (3.281) and (3.282), respectively. Hence, the instantaneous velocity is decomposed into a weighted mean component and a ﬂuctuation component in accordance with (3.283). The Reynolds stress tensor of phase k is given by:

, Re Xk = Xk Xkρk Tk ρk vk vk (D.3)

The total stress tensor is conventionally decomposed into a pressure term and a viscous stress term. The average total stress term in the momentum equations may thus be re-written as:

X X X −∇ · (αkTk k ) =−∇·[αk(pk k e +σk k )] X X (D.4) =−∇(αkpk k ) −∇·(αkσk k )

The viscous stress tensor of both phases can be modeled using the rigorous Newtonian strain-stress relation:

X X ρ X ρ 1 X ρ σ k =−μ , ∇·v k k e − 2μ (S k k − ∇·v k k e) (D.5) k B k k k k 3 k where μB,k represents the bulk viscosity of phase k (kg/ms). The average strain rate tensor is deﬁned by:

1 S Xk ρk = (∇v Xk ρk + (∇v Xkρk )T ) (D.6) k 2 k k

X It was explained in Chap. 4 that the particulate phase pressure, pp p , consists of three effects, one kinetic contribution corresponding to momentum transport caused X by particle velocity ﬂuctuation correlations, pp,kin p , one collisional contribution Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures 1403

X caused by particle interaction, pp,coll p , and one being a contribution from the gas X phase pressure, pg p . The pressure in the particulate phase is thus given by:

Xp Xp Xp Xp αppp = αppp,kin + αppp,coll + αppg (D.7)

The particulate phase total stress tensor can then be written as:

X X X −∇ · (αpTp p ) =−∇(αppp,kin p ) −∇(αppp,coll p ) (D.8) X X X −pg p ∇αp − αp∇pg p −∇·(αpœp p )

The interfacial momentum transfer to phase k is deﬁned by [30, 35]:

MkI =Tk ·∇Xk (D.9)

This relation can be reformulated adopting one out of several possible modeling approaches. The conventional continuum mechanical approach for re-writing the interfacial momentum transfer terms for dispersed ﬂows was outlined in Sect.3.4.3. Hence, an alternative approach for calculating the interfacial momentum transfer terms based on kinetic or probabilistic theories, as proposed by Simonin et al. is examined in this section. The gradient of the phase indicator function, which appears in (D.9), was deﬁned by (3.292). The expression for MkI then becomes [30, 35]:

MkI =Tk ·∇Xk=−Tk · nkδk (D.10) as we recall that ∇Xk = (∂Xk/∂n)nk, where (∂Xk/∂n) =−δk. He and Simonin [48] argued that to find a relation for the drag force acting on a single sphere in a suspension, the velocity field of the undisturbed flow is needed. They derived a momentum equation for an undisturbed flow based on probabilistic arguments. Based on the momentum equations for the disturbed and undisturbed flow, they derived an expression for the interfacial momentum transfer. The interfacial momentum transfer term was thus decomposed as follows: Tp · npδp=−Tg · ngδg=−Tg · ngδg−δTg · ngδg (D.11) =Tg ·∇Xg−δTg · ngδg≈pge ·∇Xg−δTg · ngδg =pg∇Xg−δTg · ngδg = ∇ + ∇ − · ≈ ∇ + pg αg pg Xg δTg ngδg pg αg Fg where Tg denotes the total stress tensor of the undisturbed flow. The undisturbed pressure is decomposed in accordance with the Reynolds proce- = + dure pg pg pg. The interfacial momentum transfer terms for the disturbed flow are approximated by the steady drag force: 1404 Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures

X ρ X ρ Fg ≈−δTg · ngδg≈Xpρpvr/τgp≈Xpρpvr p p /τgp p p X X ρ X ρ (D.12) ≈ αpρp p vr p p /τgp p p where τgp is the particle relaxation time. We reiterate that for a dispersed ﬂow Fp the macroscopic generalized drag force normally contains numerous contributions, as outlined in Chap. 5. However, for gas- solid ﬂows the lift force fL, the virtual mass force fV , and the Besset history force fB components are usually neglected [35]. The conventional generalized drag force given by (5.7) thus reduces to:

Fp ≈ Np(fD + fL + fV + fB) ≈ NpfD (D.13) where the forces in the brackets on the right hand side are the forces acting on a single particle in a suspension and Np is the number of particles per unit volume. The generalized drag force is then expressed as:

Xpρp Fg =−Fp =− (vg − vp) (D.14) τgp in which the particle relaxation time τgp is deﬁned by:

1 3 ρg = CD|vg − vp| (D.15) τgp 4dp ρp

The averaged drag force was approximated by [3]:

Xpρp Fg=−Fp=− (vg − vp) τgp (D.16) ≈− 3 α C Xpρp ρ Xp |v |Xpρp v Xpρp 4dp p D g r r

The average drag coefﬁcient used is [41]: 17.3 − . Xpρp = + . 1 8 CD X ρ 0 336 αg (D.17) Rep p p

An alternative parameterization for the drag coefﬁcient is given by [26, 42]. The particle Reynolds number is given by:

ρ Xp |v |Xpρp d Xpρp g r p Rep = (D.18) μg

The average particle relaxation time is given by:

X 4 ρp p dp Xpρp = τgp X X ρ X ρ (D.19) 3 ρg g CD p p |vr| p p Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures 1405

The resulting decomposition of the interfacial momentum transfer term is equivalent to the conventional closure outlined in Sect. 3.4.3, and adopted by several investi- gations on gas solids flow [35, 47, 48, 74]. Nevertheless, as for the conventional formulation, several simplifying assumptions are invoked in this model closure as well. Most important, the viscous terms in the undisturbed flow were neglected, the average undesturbed pressure is approximated by the mean pressure of the bulk phase, and the terms involving correlations of the pressure fluctuations are negligible in gas-solid flows. Moreover, additional closures are needed for the stress tensors, the fluctuating terms and the mean relative velocity. According to Bel F’dhila and Simonin [6], the average of the relative velocity between each particle and the surrounding fluid vr can be expressed as a function of the mean relative velocity and a drift velocity due to the correlation between the instantaneous distribution of the particles and the large scale turbulent fluid motion with respect to the particle diameter:

Xpρpvr=Xpρpvp−Xpρpvg (D.20)

Introducing weighted velocity variables in the ﬁrst and second term in this equation, while decomposing the instantaneous velocity in the third term into its weighted and ﬂuctuating components, we obtain:

Xpρp = Xpρp − Xpρp − Xpρp vr Xpρp vp Xpρp vg Xpρpvg (D.21)

Dividing all terms by the factor Xpρp, we get:

Xpρp = Xpρp − Xpρp − / vr vp vg Xpρpvg Xpρp (D.22)

The last term on the right hand side of (D.22) is deﬁned as the drift velocity:

= Xpρp = / vdrift vg Xpρpvg Xpρp (D.23)

where vg is the gas ﬂuctuating velocity. X ρ The term |vr| p p is the average relative velocity length that is approximated by [47, 48]: | |Xpρp ≈ Xpρp · Xpρp + · Xpρp vr vr vr vr vr (D.24)

where vr is the ﬂuctuating relative velocity. · Xpρp The term vr vr is determined by:

· Xpρp = ( + − ) vr vr 2 kp kg kgp (D.25)

2 2 where kg represents the turbulent kinetic energy of the gas phase (m /s ), kgp the 2 2 gas-particle ﬂuctuation covariance (m /s ), kp = 3θp/2 the turbulent kinetic energy 1406 Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures

2 2 analogue of the particulate phase (m /s ), and θp the granular temperature of the particle phase (m2/s2).

The Constant Particle Viscosity (CPV) model

The first attempts at describing the gas-particle flows in fluidized beds were per- Re,Xk formed using rather simple models neglecting both the Reynolds stresses, Tk in X (D.3), and the kinetic pressure-gradient term, αppp,kin p ,in(D.8). No turbulence models are thus used for any of the phases. The momentum equation for the gas phase is thus given by: ∂ Xg Xgρg Xg Xgρg Xgρg αgρg vg +∇· αgρg vg vg ∂t

Xg Xg Xg =−αg∇pg −∇· αgσg + αgρg g +Fg (D.26)

The viscous stress tensor of the gas phase is modeled using a reduced form of the Newtonian strain-stress relation (D.5): 1 σ Xg =−2μ S Xgρg − ∇·v Xgρg e (D.27) g g g 3 g

The bulk viscosity is set to zero for the continuous gas phase, in line with what is common practice for single phase ﬂows. The momentum equation for the particulate phase is written as:

∂ Xp Xpρp Xp Xpρk Xpρp αpρp vp +∇· αpρp vp vp ∂t X X =−∇· αgTp p + αpρp p g +MpI

X X X =−∇ αppp p −∇· αpœp p + αpρp p g +MpI

X X X X =−∇ αppp,coll p − αp∇pg p −∇· αpœp p + αpρp p g +Fp (D.28) where Fp is given by (D.16)to(D.25). Since no turbulence closure is employed in the CPV model, (D.24) is approximated as: Xpρp X ρ X ρ |vr| ≈ vr p p ·vr p p , (D.29) and the average relative velocity vector (D.22) is approximated by:

Xpρp Xpρp Xpρp vr =vp −vg (D.30) Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures 1407

The particle collisional pressure-gradient term in (D.8) is approximated by [42]:

Xp ∇(αppp,coll ) ≈−G(αg)∇αg (D.31)

This term is often referred to as a particleÐparticle interaction force and has the effect of keeping the particles apart above a maximum possible particle packing. The particleÐparticle interaction coefﬁcient G(αg) is named the modulus of elasticity. A survey of different particleÐparticle interaction force models are given by Massoudiet et al. [65]. Enwald and Almstedt [36] adopted a relation for the particleÐparticle interaction force proposed by Bouillard et al. [11]:

Xp ∗ ∇(αppp,coll ) ≈−G0 exp(−c(αg − α ))∇αg (D.32)

∗ ∗ where G0, c and α are empirical constants. Enwald and Almstedt [36]setα = 0.46 to limit the voidage from decreasing below this value. The other two constants were 2 chosen as G0 = 1.0(kg/ms ) and c = 500. In the CPV model the viscous stress tensor for the particulate phase is modeled using a simpliﬁed version of the Newtonian strain-stress relation (D.5), similar to that employed for the gas phase: 1 Xp Xpρp Xpρp σp =−2μp Sp − ∇·vp e (D.33) 3

In this particular model version, the bulk viscosity is set to zero for the particulate phase. The particle viscosity variable, μp, is set to a constant value. Enwald and Almstedt [36] used a particle viscosity value of μp ≈ 1.0(kg/m s), being representative for the experimental data presented by Clift and Grace [19] (p. 77).

The Particle Turbulence (PT) models

Over the years the CPV model has been shown not to be appropriate to represent important details of certain gas-particle flows. For this reason more rigorous closures have been developed for the total stress tensor of the particulate phase, intending to obtain better representations of the physical phenomena involved. The PT model represents an extension of the basic CPV model and contains extended closures for the particle collisional pressure and the particleÐparticle velocity correlation terms, as well as simple attempts to account for some of the gas-particle interaction phenomena. For the gas phase, on the other hand, the same set of transport equations as for the CPV model are employed. The particulate phase continuity equation is also the same, but the momentum equation for the particulate phase is modified. To model the particle velocity fluctuation covariances caused by particleÐparticle collisions and particle interactions with the interstitial gas phase, the concept of 1408 Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures kinetic theory of granular flows is adapted (see Chap. 4). This theory is based on an analogy between the particles and the molecules of dense gases. The particulate phase is thus represented as a population of identical, smooth and inelastic spheres. In order to predict the form of the transport equations for a granular material the classical framework from the kinetic theory of dense gases is used [17]. However, as explained by Peirano and Leckner [75], to derive the closure laws for the fluxes that occur in these equations the method of Grad [44] was preferred to that of Chapman- Enskog [17]. It is thus emphasized that the work of Simonin et al. is based on the results of Jenkins and Richman [52] that derived the necessary closure laws using the Grad’s 13 moment system for a dense gas of inelastic spheres. It follows that the transport equations discussed in this section are derived from the classical results of the kinetic theory for dense gases [17], in combination with Grad’s theory [44]. Bear in mind that in Chap. 4 the Chapman-Enskog method was used, so the closure laws obtained by Simonin et al. are similar but not completely identical to those given earlier. Moreover, He and Simonin [47, 48] considered the early models developed by Jenkins and Richman [52] appropriate for granular flows in vacuum, but inaccurate in the dilute zones of the bed where the interstitial gas phase fluctuations may affect the particles. He and Simonin [47, 48] thus extended the kinetic theory of granular materials in vacuum to take into account the influence of the interstitial gas. In the PT model the extended momentum balance for the particle phase yields: ∂ Xp Xpρp +∇· Xp Xpρp Xpρp ∂t αp ρp vp αp ρp vp vp X X X =−αp∇ pg p −∇ αp[pp,kin p +pp,coll p ] (D.34) , X Re Xp X −∇ · αpσp p + αpTp + αpρp p g +Fp where Fp is given by (D.16)to(D.25). However, in accordance with the turbulence closure employed in the PT model, the relative velocity covariance term (D.25) therein is approximated by: · Xpρp = vr vr 2kp (D.35)

Moreover, the average relative velocity vector (D.22) is approximated by (D.30), as for the CPV model, because the drift velocity is neglected. The effective stress tensor of the particulate phase can be expressed by an analogy to Newton’s law of viscosity for viscous ﬂuids (D.5), adopting the well known gradient and Boussinesq hypotheses modeling the Reynolds stresses (D.3):

, X Re Xp X ρ −αpσp p − αpTp = αpμB,p∇·vp p p e (D.36) + Xpρp − 1 ∇· Xpρp 2αpμp,eff Sp 3 vp e Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures 1409

In the PT model the particle pressure terms in (D.8) are modeled by:

Xp Xp Xp −∇ αp[pp,kin +pp,coll ] =−∇ αpρp (1 + 2αpg0(1 + e))θp (D.37)

The transport equation for the granular temperature θp, written in terms of the turbulent kinetic energy analogue of the particulate phase kp, is given by [47, 48, 75]: ∂ Xp Xp Xpρp αpρp kp +∇· αpρp vp kp ∂t =∇· Xp ( coll + t )∇ − σ Xp + ReXp :∇ Xpρp αp ρp Kp Kp kp αp p αp Tp vp

α ρ Xp e2 − 1 p p Xp − (2kp − kgp) + αpρp kp (D.38) Xpρp c τgp 3τp

= 3 where kp 2 θp represents the turbulent kinetic energy analogue of the particulate 2/ 2 coll 2/ t phase (m s ), Kp the collisional diffusion coefficient (m s), Kp the turbulent 2/ c diffusion coefficient (m s), e the restitution coefficient, and τp the particleÐparticle collision time (s). Furthermore, it is emphasized that the gas-particle covariance kgp is set to zero in the PT-model. The bulk viscosity μB,p and the effective dynamic viscosity of the particulate phase μp,eff are given by: 4 θ Xp p μB,p = dpαpρp g (1 + e) (D.39) 3 0 π

= Xp ( coll + t ) μp,eff ρp νp νp (D.40)

coll t in which νp and νp are the collisional and turbulent viscosities of the particulate phase. The collisional and turbulent viscosity values were calculated from [47, 48, 75]: coll 4 t θp ν = g (1 + e)(ν + dp ) (D.41) p 5 0 p π

t t 2 τgp 2 B ν = kgp + θp(1 + αpg0A) / + (D.42) p Xpρp c 3 τgp τgp τp where A = 2(1 + e)(3e − 1)/5 and B = (1 + e)(3 − e)/5. The average particle X ρ relaxation time τgp p p is obtained from (D.19). Moreover, it is emphasized that 1410 Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures the gas-particle covariance kgp and the interaction time between the particle motion t and the gas phase velocity fluctuations τgp are set to zero in the PT-model. In the formulation of the transport equations, several characteristic time scales are defined. In this framework these time scales are considered fundamental in the classification and the understanding of the dominant mechanisms in the suspension flow. The particle relaxation time τgp was already defined in (D.15). The particleÐ c particle collision time τp , is defined by: d c = p π τp (D.43) 24αpg0 θp

The radial distribution function g0 accounts for the probability of particle contact. A possible parameterization is given by [60]:

−2.5αp,max g0 = (1 − αp/αp,max) (D.44) where αp,max is the maximum packing of the particulate phase (≈ 0.64). Alternative parameterizations for g0 can be found in [14, 26, 42, 61, 72, 75]. The collisional and turbulent diffusion coefﬁcients are modeled by [75, 83]:

θ (1 + α g C) t = p p 0 Kp (D.45) 9 + D Xpρp c 5τgp τp coll 6 t 4 θp K = αpg (1 + e)( K + dp ) (D.46) p 0 5 p 3 π where C = 3(1 + e)2(2e − 1)/5 and D = (1 + e)(49 − 33e)/100.

The Particle and Gas Turbulence (PGT) model

The PGT model represents an extension of the PT models in that the gas turbulence is taken into account by including the Reynolds stress tensor in the momentum equation for the gas phase. The turbulence model used for the gas phase is similar to the standard single phase k- turbulence model presented in Sect. 1.3.5, although additional generation and dissipation terms may be added to consider the presence of particles. In the PGT model the drift velocity is neglected. In the PGT momentum equations the average drag force Fp is given by (D.16) to (D.25). Moreover, the average relative velocity vector (D.22) is approximated by (D.30), as for the CPV and PT models, because the drift velocity is neglected in the PGT model too. In the momentum equation for the gas phase the Reynolds stress tensor is approximated by the gradient and Boussinesq hypotheses and given by: Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures 1411

Re,X g = Xg Xgρg Tg ρg vg vg (D.47) = 2 Xg − t − 1 ∇· Xgρg 3 ρg kge 2μg Sg 3 vg e

t where μg is the dynamic turbulent viscosity of the gas phase. The viscous stress tensor used is given by (D.27). Simonin and Viollet [86] calculated the dynamic viscosity for the gas phase from a modiﬁed k − ε model. The time scale of the large eddies of the gas phase ﬂow was given by: t = /( ) τg 3Cμkg 2g (D.48)

t = Xg The dynamic turbulent viscosity of the gas phase ﬂow was given by μg 2 ρg t / kgτg 3, in accordance with the standard single phase turbulence theory presented in Sect. 1.3.5. The transport equation that was used for the turbulent kinetic energy of the gas phase is written as [3, 83, 86]: ∂ μt Xg +∇· Xg Xgρg =∇· g ∇ αg ρg kg αg ρg vg kg αg σ kg ∂t k (D.49) , Re Xg X ρ X −αgTg :∇vg g g − αgρg g g + Πkg

3 where Πkg represents the gas-particle interaction phenomena (kg/ms ). This interaction term is modeled by: α ρ Xp p p Xpρp Πkg = − 2kg + kgp + vdrift ·vr (D.50) x Xpρp τgp

However, in the PGT model the drift velocity vdrift is neglected and set to zero. The X ρ average particle relaxation time τgp p p is obtained from (D.19). The transport equation for the dissipation rate of the gas-phase turbulent kinetic energy is given by [3, 83, 86]: ∂ μt Xg Xg Xg g αgρg g +∇· αgρg vg g =∇· αg ∇g ∂t σ (D.51) , g Re Xg Xgρg Xg −αg C1αgTg :∇vg + C2ρg g + Πg kg

4 where Πg denotes the interaction term in the g equation (kg/ms ). This interaction term is modeled by: g Πg = Cε3 Πkg (D.52) kg 1412 Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures

The parameter values chosen in the gas phase turbulence model are the same as those used for the standard single phase k- model (see Sect. 1.3.5). The additional interaction term parameter is set at a fixed value, C3 = 1.3, as suggested by Elghobashi and Abou-Arab [34]. For the particulate phase, the PT-model equations that were described in Appen- dix D are used with minor extensions. That is, in the PGT-model the transport equation for kp (D.38) contains the gas-particle fluctuation covariance, kgp, to take into account the effect of the gas phase turbulence. The effective particle phase viscosity is still obtained from (D.40). In addition, the turbulent viscosity of the particulate phase is calculated from (D.42) in which kgp is obtained from a separate balance equation. The interaction time between the t particle motion and the gas velocity fluctuations τgp, is modeled as suggested by Csanady [21]: τ t t = g τgp (D.53) X ρ X ρ 1 + 1.45 3vr p p ·vr p p /2kg

The transport equation for the gas-particle ﬂuctuation covariance is given by [83]: ∂ Xp Xp Xpρp αpρp kgp +∇· αpρp vp kgp ∂t t X νgp X Xpρ (D.54) =∇· αpρp p ∇kgp − αpρp p v v :∇vp p σk g p − Xp :∇ Xgρg − Xp + Π αp ρp vgvp vg αp ρp gp gp

t 2/ where νgp denotes the gas-particle turbulent viscosity (m s), εgp the dissipation rate 2 3 of the gas-particle fluctuation covariance (m /s ), and Πgp the interaction term in 3 the kgp model (kg/ms ). The dissipation rate of the gas-particle fluctuation covariance gp and the gas- t particle turbulent viscosity νgp are defined by:

= / t gp kgp τgp (D.55)

t = t / νgp kgpτgp 3 (D.56) The gas-particle ﬂuctuation correlation tensor vg vp is expressed by: 1 t 1 v v = k pe − ν S p− tr(S p)e (D.57) g p 3 g gp g 3 g

The average gas-particle strain rate tensor is given by: Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures 1413 1 Xgρg Xpρp T S p= ∇v + (∇vp ) (D.58) g 2 g

The interaction term in (D.54) is modeled by: X X X αpρp p αpρp p αpρp p Πgp =− (1 + )kgp − 2kg − 2 kp . (D.59) x Xpρp Xg Xg τgp αg ρg αg ρg

The Particle and Gas Turbulence Model with Drift Velocity (PGTDV) model

The PGTDV model consists of the same equations as the PGT model described in Appendix D, the only difference being that the drift velocity is considered in the PGTDV model. The drift velocity vdrift is included in (D.22) and (D.50). The drift velocity takes into account the dispersion effect due to the particle transport by the ﬂuid turbulence. From the limiting case of particles with diameter tending towards zero, for which the drift velocity reduces to single turbulence correlation between the volumetric fraction of the dispersed phase and the turbulent velocity ﬂuctuations of the continuous phase. The drift velocity: vdrift is modeled as [25]: = t 1 ∇ − 1 ∇ vdrift Dgp αg αp (D.60) αg αp

t Based on semi-empirical analysis, the fluid-particle turbulent dispersion tensor, Dgp, is expressed in terms of the covariance between the turbulent velocity fluctuations of the two phases and a fluid particle turbulent characteristic time:

t = t / Dgp τgpkgp 3 (D.61)

The model assumes that the particles are suspended in a homogeneous field of gas turbulence. It is mentioned, although not used in the model evaluation by Enwald and Almstedt t [36], that a much simpler closure for the binary turbulent diffusion coefficient Dgp has been derived by Simonin [82] by an extension of Tchen’s theory. This simple closure has been used by Simonin and Viollet [84], Simonin and Flour [85] and Mudde and Simonin [68] simulating several dispersed two-phase flows.

Initial and Boundary Conditions

To simulate a rectangular fluidized bed reactor the bed vessel dimensions have to be specified first. The vessel used for validation has a rectangular cross section [36]. The bed vessel was 0.3 (m) wide, 2.22 (m) high and 0.2 (m) deep. 1414 Appendix D: Alternative Two-Fluid Model Granular Material Kinetic Theory Closures

Proper boundary conditions are generally required for the primary variables like the gas and particle velocities, pressures and volume fractions at all the vessel boundaries as these model equations are elliptic. Moreover, boundary conditions for the granular temperature of the particulate phase is required for the PT, PGT and PGTDV models. For the models including gas phase turbulence, i.e., PGT and PGTDV, additional boundary conditions for the turbulent kinetic energy of the gas phase, as well as the dissipation rate of the gas phase and the gas-particle fluctuation covariance are required. The boundary conditions for the primary variables are normally specified adopting the standard single phase flow approaches. For some of the variables like the turbulence properties and the volume fractions one has to use empirical or semi-empirical information obtained from experiments to approximate the boundary values. The specification of the velocities at the inlet may require special attention to consider the different geometries of the gas distributors. The initial conditions are generally specified in correspondence with the state of a fluidized bed operating at minimum fluidization conditions. The bed height at minimum fluidization conditions is then set to Lmf , and the gas volume fraction is set to αmf at the bed levels below Lmf and unity in the freeboard. The pressure profile in the bed is initialized using the Ergun [39] equation, whereas the pressure in the freeboard is set to the operational pressure at the outlet. The horizontal velocity components of both phases and the vertical particle velocity component are set to s / zero. The vertical interstitial gas velocity in the bed is normally initiated as Umf αmf , s and Umf in the freeboard. The gas density is initiated by use of the ideal gas law requiring that the gas pressure, species composition and temperature are known. When turbulence is considered, kg, kp and g are frequently set to small but non-zero values. kgp is set to zero. To obtain an asymmetrical flow, as observed for real cases, particular flow pertur- bations are generally introduced for a short time period as the flow develops in time from the start. Small jets at the bottom are often used for this purpose.

Model Evaluations Enwald and Almstedt [36] assessed the four different two-fluid model closures given above to investigate the effect of the gas phase turbulence, drift velocity and three dimensionality on the fluid dynamics of a bubbling fluidized bed. A few characteristics features of the different models were observed. The CPV model results generally deviated from those obtained by the more rigorous model versions. Nevertheless, the CPV model results were often in better agreement with the experimental data than the other model predictions. Comparing the PGT and PT model results it was observed that at atmospheric conditions the gas phase turbulence did not have any significant effect on the bed behavior. However, at higher pressures significant changes in the results were observed. Moreover, the drift velocity included in the most advanced model version PGTDV did not have any noticeable effect on the results at any pressure. Furthermore, strictly grid independent solutions were not obtained, and the three-dimensional effects were considered considerable. Appendix E Integral and Constitutive Equations

In this appendix, the integral theorems needed in order to solve particular integrals in the derivation of the three-ﬂuid model with a KTGF stress closure are deﬁned. All of these integrals can be evaluated using procedures given by Chapman and Cowling [16]. Most of the necessary integral theorems are provided in their book [16], but the third and fourth integral theorems are not given by Chapman and Cowling [16]. For this reason, a brief summary of the derivation of these two theorems is given after the list of theorems.

E.1 Integral Equations

The ﬁrst integral corresponds to Chapman and Cowling [16] Eq. (1.42, 4): 1 F(C)CCdC = e F(C)C2dC (E.1) 3

The second integral corresponds to Chapman and Cowling [16] Eq. (16.8, 3): 2 2π 2 kk(gij · k) dk = (2gijgij + g e) (E.2) 15 ij

The third integral is derived based on the procedure given by Chapman and Cowling [16] (see derivation at the end of this subsection): n 2π n−1 (gij · k) kdk = g gij (E.3) n + 2 ij

The fourth integral is derived based on the procedure given by Chapman and Cowling [16] (see derivation at the end of this subsection):

H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1415 © Springer International Publishing Switzerland 2014 1416 Appendix E: Integral and Constitutive Equations n 2π n (gij · k) dk = g (E.4) n + 1 ij

The ﬁfth integral corresponds to Chapman and Cowling [16] Eq. (16.8,4): ν · g (ν · )( · )2 = π ij ( + 2 ) + ( ν + ν ) kk k gij k dk gijgij gije gij gij gij (E.5) 12 gij where ν is any vector indipendent of ψ and ψ. The sixth integral of type (a) corresponds to Chapman and Cowling [16]Eq.(1.4, 2): ∞ √ − 2 π 1 3 5 r − 1 −( + )/ e αC CrdC = · · · ··· α r 1 2 (E.6a) 0 2 2 2 2 2 where r is a positive even integer. The sixth integral of type (b) corresponds to Chapman and Cowling [16]Eq.(1.4,3):

∞ − 2 1 −( + )/ r − 1 e αC CrdC = α r 1 2 ! (E.6b) 0 2 2 where r is an odd integer greater than −1.

On the Derivation of Integral Theorems for the Collision Integrals

Several integrals with respect to k, i.e., the unit vector pointing from the centre of particle one to the centre of particle two at collision, must be solved evaluating the collision integrals. The integrals can be evaluated using the approach of Chapman and Cowling [16] at p. 319. Consider the three mutually perpendicular unit vectors h, i and j, and let h be the unit vector in the direction of g12. Deﬁne the polar angles ψ and φ of k with respect to h as axis and the plane of i and j as the initial plane, illustrated in Fig. E.1.The unit vector k can then be deﬁned by:

k = h cos ψ + i sin ψ cos φ + j sin ψ sin φ (E.7)

This unit vector deﬁnition implies that:

g12 · k = g12cosψ (E.8) where the relative velocity is deﬁned by (2.111) and the element of solid angle is expressed on the form (2.650). Appendix E: Integral and Constitutive Equations 1417

Fig. E.1 The coordinate system

Scalar integrals

The fourth type of scalar integrals transform as: 2π π/2 n n (k · g12) dk = (g12 cos ψ) sin ψdψdφ 0 0 π/2 n = 2πg21 (cos ψ) sin ψdψ 0 2π = (E.9) n + 1

The remainding integral can simply be solved by the integration by parts theorem. The integration limits for ψ and φ are respectively [0, π/2] and [0, 2π] since the integration should be done only where (g21 · k) is positive. Thus, integrals involving odd powers in sin φ or cos φ might be neglected.

Vector integrals

The third type of scalar integrals transform as: 2π π/2 n n (k · g12) kdk = h cos ψ(g12 cos ψ) sin ψdψdφ 0 0 π/2 = n ( )n+1 = 2π n 2πg21 h cos ψ sin ψdψ g12h 0 n + 2 1418 Appendix E: Integral and Constitutive Equations

2π − = gn 1g (E.10) n + 2 12 12

E.2 Constitutive Equation Calculation and Approximations

coll E.2.1 Calculation of Collisional Pressure Tensor, pi

The collisional pressure tensor naturally divides into two separate integrals when expanding the pair distribution function by truncated Taylor series. From Eq. (4.349), coll = coll,1+ coll,2 the integral could thus be separated into two parts pi pi pi . The pressure coll,1 coll,2 tensors pi and pi are deﬁned by: N m m coll,1 = 1 3 i j ( + ) ( · )2 pi dijg0,ij 1 eij fifjkk gij k dV (E.11) 2 mc,ij j=1 gij·k>0

N (0) coll,2 1 4 mimj (0) (0) fi 2 p = d g , (1 + e ) f f k ·∇ kk(g · k) dv (E.12) i ij 0 ij ij i j (0) ij 4 mc,ij j=1 fj gij·k>0

Using Eqs. (4.345) and (E.2), Eq. (E.11) becomes: coll,1 N 1 mimjninj 1 3 = 3 , ( ) 2 ( + ) pi j=1 2 dijg0 ij m , Θ Θ 1 eij 120π c ij i j × [− 1 2 − 1 2]( + 2 ) (E.13) exp Ci Cj 2gijgij gije dcidcjdVij 2Θi 2Θj gij·k>0 where dVij = δ(mi − mi)δ(Ti −Ti)δ(ω1,i −ω1,i)δ(ω2,i −ω2,i) ··· ( − ) ( − ) ( − ) ( − ) ( − ) ··· δ ωNξ,i ωNξ,i δ mj mj δ Tj Tj δ ω1,j ω1,j δ ω2,j ω2,j ( , − , ) , , , , ··· , , δ ωNξ j ωNξ j dmidmjdTidTjdω1 idω1 jdω2 idω2 j dωNξ idωNξ j, and dvij = 1[15, 57, 59]. It is noted that gij = Cij + vij, and vij lead to the binary particle momentum coupling terms. To approximate these terms, Chao [15] assumed that the binary particle momentum coupling is mainly determined by the particleÐparticle drag, thus vij was neglected when calculating the collisional pressure tensor, i.e. gij ≈ Cij. Appendix E: Integral and Constitutive Equations 1419

Equation (E.13) was thus written as:

N coll,1 1 3 mimjninj 1 3 p = d g , ( ) 2 (1 + e ) i 2 ij 0 ij Θ Θ ij = 120π mc,ij i j j 1 × [− 1 2 − 1 2]( + 2 ) exp Ci Cj 2CijCij Cije dCidCjdvij (E.14) 2Θi 2Θj gij·k>0

Manipulations the terms in the last parenthesis using dyadic products: = + − − 2 = · = CijCij CiCi CjCj CiCj CjCi and scalar products: Cij Cij Cij 2 + 2 − · Ci Cj 2Ci Cj,Eq.(E.14) becomes: N + coll,1 1 eij 3 mimjninj 1 3 1 2 1 2 = d , ( ) 2 exp(− C − C ) pi 2 ijg0 ij i j 120π mc,ij ΘiΘj 2Θi 2Θj j=1 gij·k>0 × ( + + 2 + 2 ) 2CiCi 2CjCj Ci e Cj e dCidCjdVij (E.15)

Applying Eq. (E.1), Eq. (E.15) becomes:

+ coll,1 N 1 eij 3 mimjninj 1 3 p = = d g , ( ) 2 i j 1 120π2 ij 0 ij mc,ij ΘiΘj × (− 1 2 − 1 2)(5 2 + 5 2 ) (E.16) exp Ci Cj Ci e Cj e dCidCjdvij 2Θi 2Θj 3 3 gij·k>0

By assuming spherical symmetry, the integral can be carried out in parts using (4.139):

+ coll,1 N 1 eij 3 mimjninj 1 3 80π2 p = = d g , ( ) 2 × e i j 1 120π2 ij 0 ij mc,ij ΘiΘj 3 × (− 1 2) 4 (− 1 2) 2 exp Ci Ci dCi exp Cj Cj dCjdvij 2Θi 2Θj gij·k>0 (E.17) + (− 1 2) 2 (− 1 2) 4 exp Ci Ci dCi exp Cj Cj dCjdvij 2Θi 2Θj gij·k>0

The two integrals in this relation can be written on the form that can be solved by (E.6a) with r = 2 and r = 4, respectively. The integrals in (E.17) can thus be computed as exempliﬁed below: ( )1/2 (− 1 2) 4 =3 2π Θ5/2, = exp Ci Ci dCi i with r 4 (E.18) 2Θi 4 gij·k>0 1420 Appendix E: Integral and Constitutive Equations ( )1/2 (− 1 2) 2 = 2π Θ3/2, = exp Cj Cj dCj j with r 2 (E.19) 2Θj 2 · > gij k 0

dvij =1(E.20)

It is seen that by using the same modeling approach as employed in Sect. 4.1.5 for the mono particle counterparts (4.127), the integral approximations yield [15]:

N m m n n coll,1 = π 3 i j i j ( + )(Θ + Θ ) pi dijg0,ij 1 eij i j e (E.21) 3 mc,ij j=1

coll,2 Applying (E.5), the second part of the collisional pressure tensor, pi ,(E.12) can be expressed as: N m m f coll,2 = π 4 i j ( + ) ·∇ j ( + 2 )/ pi dijg0,ij 1 eij fifj gij ln gijgij gije gij = 48 mc,ij fi j 1 g ·k>0 ij fj fj + gij(gij∇ ln +∇ln gij) dcidcjdvij (E.22) fi fi

f in which ∇ ln j is deﬁned by Eq. (A.6) from [64]: fi

fj nj 1 1 2 3 1 1 2 3 ∇ ln ≈∇ln + ( C − )∇Θj − ( C − )∇Θi f n 2 Θ2 j Θ 2 Θ2 i Θ i i j j i i (E.23) 1 1 + ∇vj · Cj − ∇vi · Ci Θj Θi

In the study of Chao [15], only the last terms in (E.23) related to ∇vi and ∇vj were taken into account, thus Eq. (E.22) becomes:

N m m coll,2 = π 4 i j + pi dijg0,ij 1 eij (E.24) 48 mc,ij j=1 × · 1 ∇ · − 1 ∇ · + 2 / fifj gij vj Cj vi Ci gijgij gije gij Θj Θi g ·k>0 ij 1 1 1 1 + gij gij ∇vj · Cj − ∇vi · Ci + ∇vj · Cj − ∇vi · Ci gij Θj Θi Θj Θi

× dcidcjdvij ·∇ · = :∇ ·∇ · = :∇ Using the relations gij vj Cj gijCj vj and gij vi Ci gijCi vi from Chapman and Cowling [16], Eq. (E.24) becomes: Appendix E: Integral and Constitutive Equations 1421

N m m coll,2 = π 4 i j + pi dijg0,ij 1 eij (E.25) 48 mc,ij j=1 × 1 :∇ − 1 :∇ + 2 / fifj gijCj vj gijCi vi gijgij gije gij Θj Θi g ·k>0 ij 1 1 1 1 + gij gij ∇vj · Cj − ∇vi · Ci + ∇vj · Cj − ∇vi · Ci gij Θj Θi Θj Θi

× dcidcjdvij

Invoking the simplifying approximation gij ≈ Cij,Eq.(E.25) becomes:

N m m coll,2 = π 4 i j + pi dijg0,ij 1 eij (E.26) 48 mc,ij j=1 × 1 :∇ − 1 :∇ + 2 / fifj CijCj vj CijCi vi CijCij Cije Cij Θj Θi C ·k>0 ij 1 1 1 1 + Cij Cij ∇vj · Cj − ∇vi · Ci + ∇vj · Cj − ∇vi · Ci Cij Θj Θi Θj Θi

× dCidCjdvij

Using W · a = a · WT, a relation corresponding to Eq. (1.32, 2) in the book of Chapman and Cowling [16]) (in which W is a tensor, a is a vector, and WT denotes the tensor conjugate to W), Eq. (E.26) becomes:

N m m coll,2 = π 4 i j + pi dijg0,ij 1 eij (E.27) 48 mc,ij j=1 × 1 :∇ − 1 :∇ + 2 / fifj CijCj vj CijCi vi CijCij Cije Cij Θj Θi C ·k>0 ij

1 T 1 T 1 1 + Cij Cij Cj · ∇vj − Ci · (∇vi) + ∇vj · Cj − ∇vi · Ci Cij Θj Θi Θj Θi

× dCidCjdvij

Applying Cij = Ci − Cj and (E.1), relation (E.27) can be written as:

N m m coll,2 =− π 4 i j + pi dijg0,ij 1 eij (E.28) 48 mc,ij j=1 × 4 1 2∇· + 1 2∇· fifj Cj vj Ci vi Cije 9 Θj Θi Cij·k>0 C2 2 1 j T 1 Ci T + Cij ∇vj + ∇vj + Cij ∇vi + (∇vi) dCidCjdvij 3 Θj 3 Θi 1422 Appendix E: Integral and Constitutive Equations

To proceed we need to approximate the unknown function Cij in some appropriate manner. A truncated series expansion might be sufﬁcient. Following the approach proposed by Chao [15], a zero order truncated series expansion is deduced based on the binomial theorem [80] as follows. The unknown function Cij was ﬁrst expressed in an alternative form:

=[( − ) · ( − )]1/2 =[ 2 + 2 − · ]1/2 Cij Ci Cj Ci Cj Ci Cj 2Ci Cj (E.29)

Noticing that with statistical collisional symmetry (i.e., Ci · Cj = 0), the last term in the bracket will approach zero. Hence, a zero order truncated series expansion was proposed on the form:

≈[ 2 + 2]1/2 ≈ + + 1/2 1/2 Cij Ci Cj Ci Cj α1Ci Cj (E.30)

The value of the parameter α1 was approximated by the value obtained with Ci ≈ Cj, giving α1 ≈−0.586. The advantage of using this approximate modeling approach is that it provides a relatively simple relation with a limited number of terms. A higher order series expansion would be more computational expensive. ≈ + − . 0.5 0.5 Employing the resulting approximate relation Cij Ci Cj 0 586Ci Cj as given by Chao [15], and the integral theorem (E.6a), the second collisional pressure tensor part (E.28) might be expressed as:

N √ coll,2 2π mimjninj p =− d4g , (1 + e ) i 72 ij 0 ij mc,ij ij j=1 4 0.5 0.5 0.25 0.25 × (∇vj + ∇vj + ∇·vje)(3Θ + 4Θ − 1.955Θ Θ ) (E.31) 3 i j i j × (∇ + ∇ + 4 ∇· )( Θ0.5 + Θ0.5 − . Θ0.25Θ0.25) vi vi 3 vie 3 j 4 i 1 955 i j

The sum of the ﬁrst part of the collisional presure tensor (E.21) and the second part (E.31) can be casted into the standard form of the total collisional pressure tensor closure relation (4.350). It is noticed that when N = 1, the proposed constitutive equations for the collisional pressure tensor reduce to those for a mono-particle bed. In particular, relation (E.21) reduces to:

coll,1 = ( + ) 2 Θ pi 2 1 ei giαi ρi ie (E.32)

coll,1 The resulting relation for the ﬁrst tensor part, pi , is identical to Eq. (9.229) in the book of Gidaspow [42]. It is further noted that Manger [64] did use a different series expansion approach (of higher order) approximating the collisional pressure tensor components for binary mixtures based on the work of Gidaspow [42]. Their modeling approach was outlined in the thesis by Manger [64], Sects. 7.4.2 and 7.2. Appendix E: Integral and Constitutive Equations 1423

E.2.2 Calculating the Collisional Particle–Particle Drag, Fij

Neglecting the second part, the momentum source term Eq. (4.356) becomes: N N m m Ω ( ) =− i j 2( + ) ( · )2 gij ci dij 1 eij g0,ij k gij kfifkdv (E.33) mc,ij j=1 j=1 gij·k>0

Using the integral Eq. (E.3), Eq. (E.33) becomes: N N πm m Ω ( ) =− i j 2( + ) gij ci dij 1 eij g0,ij gijgijdcidcjdvij (E.34) 2mc,ij j=1 j=1 gij·k>0

= + Following the approach of Manger [64], and by use of gij Cij vij, the part related to Cij cancel out. The remaining part of the term (E.34) becomes: N N πm m Ω ( ) =− i j 2( + ) gij ci dij 1 eij g0,ijvij gijdcidcjdVij (E.35) 2mc,ij j=1 j=1 gij·k>0

To approximate the unknown function gij = Cij + vij in some appropriate manner a truncated series expansion might be applicable. Following the approach proposed by Chao [15], a zero order truncated series expansion is deduced based on the binomial theorem [80]. The unknown function gij was ﬁrst expressed in an alternative form:

= ( · )1/2 =[( + ) · ( + )]1/2 =[ 2 + v2 + · ]1/2 gij gij gij Cij vij Cij vij Cij ij 2vij Cij (E.36)

Noticing that with statistical collisional symmetry (i.e., vij · Cij = 0), the last term in the bracket will approach zero. Hence, a zero order truncated series expansion was proposed on the form:

≈[ 2 + v2]1/2 ≈ + v + 1/2v1/2 gij Cij ij Cij ij α2Cij ij (E.37)

The value of the parameter α2 was approximated by the value obtained with Ci ≈ Cj, giving α1 ≈−0.586. 1/2 v1/2 Moreover, an approximation of Cij is required (whereas ij is known from the solution of the governing equations). Following the same approximation method as outlined above, Chao [15] deduced the necessary relation considering that:

1/2 =( · )1/4 Cij Cij Cij (E.38) =[( − ) · ( − )]1/4 =[ 2 + 2 − · ]1/4 Ci Cj Ci Cj Ci Cj 2Ci Cj (E.39) 1424 Appendix E: Integral and Constitutive Equations

Noticing that with statistical collisional symmetry (i.e., Ci · Cj = 0), the last term in the bracket will approach zero. Hence, a zero order truncated series expansion was proposed on the form:

1/2 ≈[ 2 + 2]1/4 ≈ 1/2 + 1/2 + 1/4 1/4 Cij Ci Cj Ci Cj α3Ci Cj (E.40)

The value of the parameter α3 was approximated by the value obtained with Ci ≈ Cj, giving α3 ≈−0.811. By employing the resulting approximate relation given by Chao [15]:

≈ v + + − . 0.5 0.5 − . v0.5( 0.5 + 0.5 − . 0.25 0.25), gij ij Ci Cj 0 586Ci Cj 0 586 12 Ci Cj 0 811Ci Cj (E.41) and the integral theorem (E.6a) and (E.6b), the source term integration (E.35) result is:

N N m m n n Ω ( ) =− i j i j 2( + ) gij ci dij 1 eij g0,ijvij mc,ij j=1 j=1 √ × ( Θ + Θ − Θ0.25Θ0.25 2π i 2π j 2 i j π 0.5 0.25 0.25 0.125 0.125 + vij − 1.135v (Θ + Θ − 0.8Θ Θ ). (E.42) 2 ij i j i j

E.2.3 Calculation of the Collisional Source Term of Granular Temperature Equation

Equation(4.363) can be divided into two separate integrals:

N N m m d2 Ω ( 1 2) = i j ij ( 2 − ) ( · )3 gij ci g0,ij eij 1 k gij fifjdkdcidcjdVij 2 4mc,ij j=1 j=1 gij·k>0 N 3 mimjd f + ij ( 2 − ) ( · )3 ·∇ j g0,ij eij 1 k gij fifjk ln dkdcidcjdvij (E.43) 8mc,ij fi j=1 gij·k>0

For the ﬁrst integral part, the inner integral over k is determined by use of (E.4), 3 π 3 (k · gij) dk = g (E.44) 2 ij

For the second integral part, the inner integral over k is determined by use of (E.3), Appendix E: Integral and Constitutive Equations 1425 3 2π 2 (k · gij) kdk = g gij (E.45) 5 ij

Hence, by use of these two integral results, Eq. (4.363) becomes:

N N πm m d2 Ω ( 1 2) = i j ij ( 2 − ) 3 gij ci g0,ij eij 1 gijfifjdcidcjdVij 2 8mc,ij j=1 j=1 gij·k>0 N 3 πmimjd f + ij ( 2 − ) ∇ j · 2 g0,ij eij 1 fifj ln gijgijdcidcjdvij (E.46) 20mc,ij fi j=1 gij·k>0

By use of Eq. (E.23), but only taking into account the terms related to ∇vi and ∇vj, Eq. (E.46) is reduced to:

N N πm m d2 Ω ( 1 2) = i j ij ( 2 − ) 3 gij ci g0,ij eij 1 gijfifjdcidcjdvij 2 8mc,ij j=1 j=1 gij·k>0 N πm m + i j 3 ( 2 − ) 2( 1 ·∇ − 1 ·∇ ) · dijg0,ij eij 1 fifjgij Cj vj Ci vi gij 20mc,ij Θj Θi j=1 gij·k>0

× dcidcjdvij (E.47)

·∇ · = :∇ Moreover, by application of Cj vj gij Cjgij vj from Chapman and Cowling ≈ [16], the simplifying approximation gij Cij, the relative peculiar velocity deﬁnition Cij = Ci − Cj and (E.1), relation (E.47) can be expressed as:

N 1 2 Ωg C ij 2 i j=1 N πm m d2 = i j ij ( 2 − ) 3 g0,ij eij 1 CijfifjdCidCjdvij 8mc,ij j=1 Cij·k>0 N πm m − i j 3 ( 2 − ) 2 dijg0,ij eij 1 fifjCij = 60mc,ij j 1 C ·k>0 ij C2 2 j Ci × ∇·vj − ∇·vi dCidCjdvij (E.48) Θj Θi

3 An appropriate approximation of the unknown function Cij is required. Following the approach proposed by Chao [15], a zero order truncated series expansion was 3 deduced based on the binomial theorem [80] as follows. The unknown function Cij was ﬁrst expressed in an alternative form: 1426 Appendix E: Integral and Constitutive Equations

3 = ( · )3/2 =[ 2 + 2 − · ]3/2 Cij Cij Cij Ci Cj 2Ci Cj (E.49)

Noticing that with statistical collisional symmetry, the last term in the bracket vanishes. A truncated series expansion based on the binomial theorem was thus employed: 3 ≈[ 2 + 2]3/2 ≈ 3 + 3 + ( 2 + 2) Cij Ci Cj Ci Cj α4 Ci Cj CiCj (E.50)

The value of the parameter α4 was approximated by the value obtained with Ci ≈ Cj, giving α4 ≈ 0.414. 3 ≈ 3 + 3 + . ( 2 + 2 ) By applying the approximate relation Cij Ci Cj 0 414 CiCj Ci Cj from Chao [15], and by employing the integral theorem (E.6a) and (E.6b), the source term (E.48) can be expressed as:

N 1 2 Ωg c ij 2 i j=1 N ( 2 − ) 2 π eij 1 dijg0,ijmimjninj . . . . = (6.4(Θ1 5 + Θ1 5) + 2.(Θ Θ0 5 + Θ0 5Θ )) (E.51) 8(m + m ) i j i j i j j=1 i j N π 3 mimjninj 2 + d g , (1 − e )(∇·v (15Θ + 9Θ ) +∇·v (15Θ + 9Θ )) 60 ij 0 ij m + m ij i i j j j i j=1 i j

It is noted that when N = 1, the source term closure (E.51) reduces to the following relation: . Θ 0.5 Ω (1 2) = ( 2 − ) 2 Θ 3 72 i − . ∇· . gij ci 3 ei 1 αi ρig0,i i 0 8 vi (E.52) 2 di π

This result is similar to Eq. (9.213) in the book of Gidaspow [42]. The latter relation was approximated as (4.179): Θ 0.5 Ω 1 2 = ( 2 − ) 2 Θ 4 i −∇· . ci 3 ei 1 αi ρig0,i i vi (4.179) 2 di π

E.2.4 Calculation of the Granular Heat Conduction in the Granular Temperature Equation

Following the approach of Manger [64], the collisional dynamics relation (4.366)is inserted into the collisional heat ﬂux relation (4.365), and by separating the integrals four ﬂux contributions are obtained (i.e., corresponding to Eqs. (A.55)Ð(A.58) in the thesis of Manger [64]). Moreover, the terms (A.55)Ð(A.57) are neglected, thus only Appendix E: Integral and Constitutive Equations 1427

Eq. (A.58) is taken into account. The term (A.58) is written as:

N m m d4 col = i j ij ( + ) qi g0,ij eij 1 4mc,ij j=1 (E.53) f × ·∇ j ( · )2( · ) fifjk ln k gij k k Ci dkdcidcjdvij fi gij·k>0

Applying the integral theorem (E.5), the closure (E.53) can be expressed as:

N 4 πmimjd col = ij ( + ) qi g0,ij eij 1 fifjgij (E.54) = 48mc,ij j 1 g ·k>0 ij fj fj fj × 2∇ln · Cigij +∇ln · gijCi + Ci · gij∇ln dcidcjdvij fi fi fi ≈ Invoking the simplifying approximation gij Cij and the deﬁnition of the relative peculiar velocity Cij = Ci − Cj, and applying the theorem (E.1), relation (E.54) can be expressed as:

N 4 πmimjd f col = ij ( + ) 2∇ j qi g0,ij eij 1 fifjgijCi ln dcicjdvij (E.55) 24mc,ij fi j=1 gij·k>0

Employing (E.23), considering the terms related to ∇Θi and ∇Θj only, (E.55) can be expressed as: N πm m d4 C2 i j ij 2 j 3 col = , ( + ) − ∇Θ qi g0 ij eij 1 fifjgijCi 2 j 48mc,ij Θ Θj j=1 · > j gij k 0 (E.56) C2 3 − i − ∇Θ dc c dv Θ2 Θ i i j ij i i

Application of the following approximations for the relative particle velocity gij ≈ Cij ≈ + − . 0.5 0.5 and the relative pecular velocity Cij Ci Cj 0 586Ci Cj obtained from Chao [15], and by use of the integral theorem (E.6a), the ﬁnal form of (E.56) can be written as: N 2πm m n n col =− i j i j ( + ) 4 Θ ∇Θ qi 1 eij g0,ijdij i i (E.57) 5(mi + mj) j=1 1428 Appendix E: Integral and Constitutive Equations

It is further noted that when N = 1, the closure relation (E.57) reduces to: Θ qcol =−2.13α2ρ (1 + e )g d i ∇Θ (E.58) i i i i i i π i which is very similar to the mono-particle relation (9.217) in the book of Gidaspow [42]: Θ qcol =−2α2ρ (1 + e )g d i ∇Θ . (4.174) i i i i i i π i Appendix F Trondheim Bubble Column Model

The 2D steady-state two-fluid model presented in this section is based on the early work by Torvik and Svendsen [97], Svendsen et al. [95], and Jakobsen [51]. The two-fluid model was derived based on the time-after volume averaging procedure, described in Sect.3.4.4. The model was later implemented in a commercial code PHOENICS. The input files specifying the calculations have been deposited in the PHOENICS library of two-phase flow examples [9]. This early bubble column model is included in this book because it is considered particularly useful for educational purposes. The model derivation is outlined in Cartesian coordinates. The governing equations are then more conveniently written in vector notation (vector symbolism). For practical applications and simulations these vector equations are converted into cylindrical coordinates and finally reduced to the 2D axi-symmetric bubble column problem. The axi-symmetric model is discretized by use of the IPSA-SIMPLEC solution algorithm in Sect. F.4.1.

F.1 Model Formulation

The elementary Two-ﬂuid model derivation is outlined in this section.

Conservation of mass

The instantaneous volume averaged conservation of mass of the continuous liquid and dispersed gas phases were expressed as:

∂ (α ρ ) +∇·(α ρ v ) = Γ (F.1) ∂t l l l l l l

∂ (α ρ ) +∇·(α ρ v ) = Γ (F.2) ∂t g g g g g g

H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1429 © Springer International Publishing Switzerland 2014 1430 Appendix F: Trondheim Bubble Column Model

The void fractions must fulﬁll the compatibility condition:

αl + αg = 1(F.3)

The mass exchange terms must satisfy the constraint:

Γl + Γg = 0. (F.4)

Conservation of momentum

The instantaneous volume averaged Navier-Stokes equations for the two phases are: ∂ ∂ ∂pl ∂ ∂vl,i ∂vl,j (α ρ v , ) + (α ρ v , v , ) =−α + α μ ( + ) ∂t l l l i ∂x l l l j l i l ∂x ∂x l l ∂x ∂x j i j j i ∂ 2 ∂vl,k − δijαl( μl − μbl) ∂xj 3 ∂xk + + C ρlαlgi Fl,i (F.5)

∂ ∂ ∂pg ∂ ∂vg,i ∂vg,j (α ρ v , ) + (α ρ v , v , ) =−α + α μ ( + ) ∂t g g g i ∂x g g g j g i g ∂x ∂x g g ∂x ∂x j i j j i ∂ 2 ∂vg,k − δijαg( μg − μbg) ∂xj 3 ∂xk + + C ρgαggi Fg,i (F.6) where μbl is the bulk viscosity of the liquid, and μbg is the bulk viscosity of the gas. The surface tension force was neglected so the interfacial momentum transfer terms satisfy: C + C = Fl Fg 0(F.7)

The pressure inside individual bubbles may vary, but this was assumed to have no relation to the ﬂow of the continuous phase. The volume averaged pressure of the two phases were assumed to be equal, pl = pg. The dispersed phase approximation neglects internal ﬂow inside the dispersed phases. The viscous terms of the gas equation were thus neglected. The bulk viscosity terms were also neglected since they are generally small, as discussed in Chap.2. These assumptions and approximations simplify the momentum equations. The instantaneous volume averaged liquid phase equation was written: Appendix F: Trondheim Bubble Column Model 1431

v ∂ ∂ ∂p ∂ ∂vl,i ∂ l,j (αlρlvl,i) + (αlρlvl,jvl,i) =−αl + αlμl + ∂t ∂xj ∂ xi ∂xj ∂xj ∂xi (F.8) ∂ 2 ∂vl,k C − δijα μ + ρ α gi + F ∂xj 3 l l ∂xk l l l,i

The corresponding gas phase equation was written:

∂ ( v ) + ∂ ( v v ) =− ∂p + + C . αgρg g,i αgρg g,j g,i αg ρgαggi Fg,i (F.9) ∂t ∂xj ∂xi

Turbulence Modeling

Reynolds decomposition and time averaging were then applied to the instantaneous variables in the volume average model equations. However, it was assumed that none of the densities fluctuate. The terms of fluctuating quantities with order higher than two were considered small compared to those of first and second order and thus neglected. Developed versions of the gradient and Boussinesq hypotheses were employed to model the second-order covariance terms. The liquid phase volume fraction-velocity covariance and the Reynolds stresses, for example, were approximated by:

v =−νl,t ∂αl αl l,j (F.10) σαl,t ∂xj and

v v = σt + 2 l,j l,i l,ij kδij 3 νl,t ∂vl,i ∂vl,j 2 ∂vl,k 2 =− + − δij + kδij (F.11) σ , ∂x ∂x 3 ∂x 3 l t j i k νl,t ∂vl,i ∂vl,j 2 νl,t ∂vl,k =− + + δij + k σl,t ∂xj ∂xi 3 σl,t ∂xk

t The turbulent Schmidt numbers (σψ) were included for all variables that are modeled via the gradient and Boussinesq hypotheses. These Schmidt numbers were set to 1.0. The only exception was σ = 1.3.

Turbulence Modeling of the Liquid Phase Continuity Equation

Time averaging the volume averaged liquid mass balance (F.1) gives for the transient term: ∂ ∂ ∂ (ρ α˜ ) = (ρ (α + α)) = (ρ α ) (F.12) ∂t l l ∂t l l l ∂t l l 1432 Appendix F: Trondheim Bubble Column Model

For the convective terms yield:

∂ ( ˜ v˜ ) = ∂ ( ( + )(v + v )) = ∂ ( ( v + v )) ρlαl l,i ρl αl αl l,i l,i ρl αl l,i αl l,i ∂xi ∂xi ∂xi ∂ νl,t ∂αl = ρlαlvl,i − ρl . (F.13) ∂xi σαl,t ∂xi

The Modeled Liquid Phase Continuity Equation

For none reactive ﬂow calculations there are no mass exchange between the phases, the continuity equation was reduced to:

∂ ∂ ∂ μl,t ∂αl (αlρl) + αlρlvl,i = (F.14) ∂t ∂xi ∂xi σαl,t ∂xi

In vector notation: ∂ μl,t (αlρl) +∇·(αlρlvl) =∇· ∇αl . (F.15) ∂t σαl,t

Turbulence Modeling of the Liquid Phase Momentum Equation

Time averaging the volume averaged momentum balance (F.8) gives for the transient term:

∂ ( ˜ v˜ ) = ∂ ( ( + )(v + v )) = ∂ ( ( v + v )) αlρl l,i ρl αl αl l,i l,i ρl αl l,i αl l,i ∂t ∂t ∂t (F.16) ∂ ≈ (ρ α v , ) ∂t l l l i The turbulent covariance terms were neglected. The transient term did serve as a means of under-relaxation. Modeling the convection terms:

∂ ( ˜ v˜ v˜ ) = ∂ ( ( + )(v + v )(v + v )) αlρl l,j l,i ρl αl αl l,j l,j l,i l,i ∂xj ∂xj ∂ = (ρ (α v , v , + α v v , + α v v , + α v v )) ∂x l l l j l i l l,j l i l l,i l j l l,j l,i j ∂ νl,t ∂αl = αlρlvl,jvl,i − ρl vl,i ∂xj σαl,t ∂xj Appendix F: Trondheim Bubble Column Model 1433

νl,t ∂αl νl,t ∂vl,i ∂vl,j − ρl vl,j − αlρl ( + ) σ , ∂x σ , ∂x ∂x αl t i lt j i 2 νl,t ∂vl,k + δijαlρl( + k) (F.17) 3 σl,t ∂xk

The time after volume averaged pressure-volume fraction term was modeled as:

∂p˜ ∂ ∂p ∂p α˜ = α + α (p + p) = α + α l ∂x l l ∂x l ∂x l ∂x i i i i (F.18) ∂p νl,t ∂αl ∂vl,i ∂ νl,t ∂αl = αl + ρl + ρlvl,j ∂xi σαl,t ∂xj ∂xj ∂xj σαl,t ∂xi

The pressure-volume fraction covariance terms were deduced from the instantaneous steady state equations for conservation of mass and momentum of the continuous phase [51]. The instantaneous steady state continuity was written as: ∂ ( v + v + v + v ) = ρl αl l,i αl l,i αl l,i αl l,i 0 (F.19) ∂xi

The time after volume averaged continuity equation became: ∂ ( v + v ) = ρl αl l,i αl l,i 0(F.20) ∂xi

The steady state equation for the instantaneous mass ﬂuctuations was found by subtracting (F.20) from (F.19) giving: ∂ ( v + v + v − v ) = ρl αl l,i αl l,i αl l,i αl l,i 0 (F.21) ∂xi

The instantaneous void fractions always satisfy (F.3), hence:

+ + + = αl αl αg αg 1(F.22)

Time averaging the instantaneous volume averaged compatibility relation gives:

αl + αg = 1 (F.23)

By subtracting the time averaged compatibility relation (F.23)from(F.22), the ﬂuc- tuations were shown to satisfy: + = αl αg 0(F.24)

Since the two phases share the same pressure, the phasic pressureÐvolume fraction covariance terms were related through: 1434 Appendix F: Trondheim Bubble Column Model

∂p =− ∂p αl αg (F.25) ∂xi ∂xi

The pressure-volume fraction covariance terms appear with opposite signs in the equations for the gas and liquid phases. The pressure-volume fraction covariance terms thus describe momentum transfer fluxes between the phases. A steady state equation for the instantaneous momentum fluctuations was found by subtracting the time averaged steady state equation from the instantaneous steady state equation. The resulting steady state relations were then simplified by the use / of (F.22) and (F.24). The resulting equation was then multiplied through with αl αl and time averaged. Covariances of fluctuating quantities with order larger than two were assumed small compared to the other terms and thus neglected. The fluctuating interphase forces and viscous interaction terms were also neglected. The factor:

2 αl 2 (F.26) αl is now a common factor for three of the remaining terms in the equation for the pressureÐvolume fraction covariance terms. By assuming that this pre-factor term was close to zero, all the terms with this prefactor were neglected. The pressureÐ volume fraction covariance terms were then approximated by:

v ∂v ∂p =− v ∂ l,i − v l,i αl ρlαl l,j ρl l,jαl (F.27) ∂xi ∂xj ∂xj

The ﬁrst term on the RHS can be modeled using the gradient and Boussinesq hypotheses. The covariance within the second term was written as:

v ∂α ∂ l,i = ∂ ( v ) − v l αl αl l,i l,i (F.28) ∂xj ∂xj ∂xj

A weakly justiﬁed approximation was then introduced [51]:

∂v l,i ≈ ∂ ( v ) αl αl l,i (F.29) ∂xj ∂xj

The ﬁnal equation for the turbulent pressureÐvolume fraction covariance terms was then obtained: v ∂p =− v ∂ l,i − v ∂ ( v ) αl ρlαl l,j ρl l,j αl l,i (F.30) ∂xi ∂xj ∂xj

The closure is completed by use of the gradient and Boussinesq hypotheses on the form (F.10). Appendix F: Trondheim Bubble Column Model 1435

Time averaging the volume averaged gravity force yield:

˜ = ( + ) = αlρlgi αl αl ρlgi αlρlgi (F.31)

The time after volume averaged viscous shear terms became:

∂ ∂v˜l,i ∂v˜l,j (α˜ lμl( + )) ∂xj ∂xj ∂xi ∂(v , + v ) ∂(v , + v = ∂ ( + )( l i l,i + l j l,j ) μl αl αl ∂xj ∂xj ∂xi (F.32) v v , ∂v , ∂v , ∂ , = ∂ (∂ l i + l j ) + ( l i + l j ) μlαl μlαl ∂xj ∂xj ∂xi ∂xj ∂xi

∂ ∂vl,i ∂vl,j ≈ (μlαl( + )) ∂xj ∂xj ∂xi and ⎛ ⎞ v + v ∂ 2 ∂v˜ , 2 ∂ ⎜ ∂ l,k l,k ⎟ − ˜ l k =− ⎝ + ⎠ δijαlμl μl αl αl ∂xj 3 ∂xk 3 ∂xi ∂xk v ∂v (F.33) =−2 ∂ ∂ l,k + l,k μlαl μlαl 3 ∂xi ∂xk ∂xk 2 ∂ ∂vl,k ≈− μlαl 3 ∂xi ∂xk

To make the notation used for the latter term consistent with the other equations, index j is substituted for k : ∂ 2 ∂v˜l,j 2 ∂ ∂vl,j − δijα˜lμl ≈− μlαl (F.34) ∂xj 3 ∂xj 3 ∂xi ∂xj

In abbreviated form the viscous shear can thus be written as:

˜ V = V + V Fl,i Fl,i Fl,i (F.35)

V in which the Fl,i terms are ignored. 1436 Appendix F: Trondheim Bubble Column Model

The modeled viscous term was thus approximated as: v , ∂v , ∂v , ˜ V ≈ V ≈ ∂ ∂ l i + l j − 2 ∂ l j Fl,i Fl,i μlαl μlαl (F.36) ∂xj ∂xj ∂xi 3 ∂xi ∂xj

The time averaged interphase forces were modeled as:

˜ C = C + C Fl,i Fl,i Fl,i (F.37)

C It is noted that Fl,i is made up of terms which are of order higher than one in the ﬂuctuating quantities, so by time averaging they are not zero. The interphase forces and the turbulence modeling of the drag force are described shortly in this section.

The Modeled Liquid Phase Momentum Equation

After all the modeled terms were substituted into the averaged conservation equation for momentum, the balance equation became:

∂ ∂ (α ρ v , ) + (α ρ v , v , ) ∂t l l l i ∂x l l l j l i " #$ % " j #$ % Transient term Convection term ∂ νl,t ∂αl νl,t ∂αl νl,t ∂vl,i ∂vl,j = (ρl vl,i + ρl vl,j + αlρl ( + )) ∂x σ , ∂x σ , ∂x σ , ∂x ∂x " j αl t j αl t #$i l t j i % From convection term ∂ 2 νl,t ∂vl,k ∂p νl,t ∂αl ∂vl,i − ( δijαlρl(k + )) −αl − ρl ∂x 3 σ , ∂x ∂x σ , ∂x ∂x " j #$ l t k % " i #$αl t j j% From convection term From pressvre term − v ∂ ( νl,t ∂αl ) + ρl l,j "ρl#$αlg%i ∂xj σαl,t ∂xi " #$ % Gravity term From pressvre term ∂ ∂v , ∂vl,j ∂ 2 ∂v , + (α μ ( l i + )) − ( α μ l k ) + FC + FC (F.38) ∂x l l ∂x ∂x ∂x l l ∂x " l,i #$ l,%i " j j #$i i 3 k % Interphasial forces term Viscovs terms

The viscous terms were joined with the turbulent diffusion terms that come from the μ , ν , modeling of the convection terms. The notations μ , = μ + l t and ν , = ν + l t l eff l σl,t l eff l σl,t are used, giving: Appendix F: Trondheim Bubble Column Model 1437

∂ ∂ (αlρlvl,i) + (αlρlvl,jvl,i) ∂t ∂xj

∂ νl,t ∂αl νl,t ∂αl ∂vl,i ∂vl,j = (ρl vl,i + ρl vl,j + αlρlνl,eff( + )) ∂xj σα ,t ∂xj σα ,t ∂xi ∂xj ∂xi l l (F.39) ∂ 2 ∂vl,k ∂p νl,t ∂αl ∂vl,i − ( αlρl(k + νl,eff )) − αl − ρl ∂xi 3 ∂xk ∂xi σαl,t ∂xj ∂xj − v ∂ ( νl,t ∂αl ) + + C + C ρl l,j ρlαlgi Fl,i Fl,i ∂xj σαl,t ∂xi

These equations can be written in vector notation giving:

∂ (α ρ v ) +∇·(α ρ v v ) ∂t l l l l l l l νl,t νl,t T =∇·(ρl ∇αlvl + ρl vl∇αl + αlρlνl,eff(∇vl + (∇vl) )) σαl,t σαl,t (F.40) 2 νl,t −∇( αlρl(k + νl,eff∇·vl)) − αl∇p − ρl ∇αl ·∇vl 3 σαl,t − ·∇( νl,t ∇ ) + + C + C ρlvl αl ρlαlg Fl Fl σαl,t

This equation can be transformed into cylindrical coordinates as shown shortly in this appendix. The resulting equations in 2D cylindrical are listed in Sect.F.4.

The Modeled Gas Phase Continuity Equation

The continuity equation was modeled in an identical manner to that of the liquid phase. The result is:

∂ ∂ ∂ μg,t ∂αg (αgρg) + (αgρgvg,i) = ( ) (F.41) ∂t ∂xi ∂xi σαg,t ∂xi

In vector notation this is:

∂ μg,t (αgρg) +∇·(αgρgvg) =∇·( ∇αg). (F.42) ∂t σαg,t

The Modeled Gas Phase Momentum Equation

The turbulence modeling of the transient and convective terms of the momentum balance (F.9) gives analogous results to that of the liquid phase. The pressure term is determined from [51]: 1438 Appendix F: Trondheim Bubble Column Model

∂p =− ∂p αg αl (F.43) ∂xi ∂xi

The gravitational and interphase forces are also analogous to those of the liquid phase. The momentum balance was thus be written as: ∂ ∂ (α ρ v , ) + (α ρ v , v , ) ∂t g g g i ∂x g g g j g i " #$ % " j #$ % Transient term Convection term ∂ νg,t ∂αg νg,t ∂αg νg,t ∂vg,i ∂vg,j = (ρg vg,i + ρg vg,j + αgρg ( + )) ∂x σ , ∂x σ , ∂x σ , ∂x ∂x " j αg t j αg t #$i g t j i % From convection term ∂ 2 νg,t ∂vg,k ∂p νl,t ∂αl ∂vl,i − ( δijαgρg(k + )) −αg + ρl ∂x 3 σ , ∂x ∂x σ , ∂x ∂x " j #$ g t k % " i #$αl t j j% From convection term From pressvre term + v ∂ ( νl,t ∂αl ) + + C + C ρl l,j ρgαggi Fg,i Fg,i ∂x σ , ∂x " #$ % " #$ % " j #$ αl t i % Gravity term Interphasial forces term From pressvre term (F.44) In vector notation this equation becomes:

∂ αgρgvg +∇· αgρgvgvg ∂t

νg,t νg,t νg,t T =∇· ρg ∇αgvg + ρg vg∇αg + αgρg ∇vg + ∇vg σαg,t σαg,t σαg,t 2 νg,t νl,t −∇ αgρg k + ∇·vg − αg∇p + ρl ∇αl ·∇vl 3 σαg,t σαl,t ν , + ·∇ l t ∇ + + C + C ρlvl αl ρgαgg Fg Fg σαl,t (F.45) The mass and momentum conservation equations are written in cylindrical coordinates in a later section in this appendix.

The Liquid Phase Turbulence Model

The k − model is chosen as the turbulence model [58]. It is assumed that the turbulence inside the dispersed phase (gas bubbles) does not affect the liquid phase turbulence. Both k and are determined from transport equations, as described in Sect. 8.4.4. Appendix F: Trondheim Bubble Column Model 1439

In vector notation the equations are:

∂ μl,t (αlρlk) +∇·(αlρlvlk) =∇·(αl ∇k) + αl(Pk + Pb − ρlε) (F.46) ∂t σk and

∂ μl,t (αlρlε) +∇·(αlρlvlε) =∇·(αl ∇) + αl (Cε1(Pk + Pb) − Cε2ρlε) (F.47) ∂t σε k

The turbulent production terms due to ﬂuid shear and bubble movement are expressed as: T Pk = μl,t(∇vl + (∇vl) ) :∇vl (F.48) and Pb = CbFD,l · (vl − vg) (F.49)

The Cb parameter takes values between 0 and 1, and generally depends on bubble size and shape, and on the turbulent length scale. The empirical coefﬁcients in the turbulence model were kept equal to the standard values for the original single phase model.

Interphase Forces

The interphase forces considered were steady drag, added (virtual) mass and lift. The steady drag force on a collection of dispersed bubbles with a given average diameter was described by (5.28) and (5.14). The transversal lift force was determined by the conventional model (5.45), whereas the added mass force was approximated by (5.92). Jakobsen [51] used a drag coefﬁcient formulation for the bubbles formulated by Johansen and Boysan [53] based on the terminal velocities for ellipsoidal bubbles given by Clift et al. [18]: 0.622 C = (F.50) D 1.0 + . Eo 0 235

By applying turbulence modeling to the drag force, negative transversal forces arise. The resulting transversal force was written as [51]: (v − v ) v v i 3 μl Cτ dS 2 l g k ∂ l,i ∂ l,k F , = αlαg CD ( ) (−νl,t( + ))(1 − δik) T g 4 d2 (1 + τL ) ν ∂x ∂x S tP l ReP k i (F.51) where ReP is the time averaged turbulent particle Reynolds number: 1440 Appendix F: Trondheim Bubble Column Model (v + v − v − v )2 dS l,i l,i g,i g,i ReP = . (F.52) νl

Variable Local Bubble size model

Jakobsen [51] developed a simple model for the bubble size postulating that the bubble diameter was proportional to the turbulence length scale determined by the k-ε turbulence model. The bubble diameter was thus approximated by:

3 k 2 d = C (F.53) s SMD ε in which CSMD = 0.04 was considered a constant system parameter tuned to the air-water system.

F.2 Tensor Transformation Laws

The two-fluid model has been derived in Cartesian coordinates. However, for working problems it is often more natural to use curvilinear coordinates like cylindrical and spherical coordinates. In reactor modeling cylindrical coordinates are of particular interest because many reactors have the shape of a tube. In this section we are thus primarily interested in knowing how to convert the various differential operations written in Cartesian coordinates into vector notation and from thence into curvilinear coordinates. The first operation is relatively easy to perform since the elementary operators can be found in many introductory textbooks on fluid mechanics. The second operation can also be achieved in a rigorous manner provided that we know, for the coordinate being used, two mathematical characteristics6: The expressions for ∇ and the spatial derivatives of the unit vectors in curvilinear coordinates. In the following subsections we define the formulas that are necessary to convert the equations from the general vector notation to cylinder coordinates. Finally, the governing equations for the two-fluid model are given in cylinder coordinates.

F.2.1 Curvilinear Coordinate Systems

The position of a point P in any coordinate system may be speciﬁed by three coordinates . To determine these we must ﬁrst establish a frame of reference by

6 The textbooks by Bird et al. [7], Aris [2], Malvern [63], Slattery [88], Irgens [49, 50] and Borisenko and Tarapov [10] may be consulted for thorough-going studies of the extensive theory of vector and tensor analysis. Appendix F: Trondheim Bubble Column Model 1441 taking any point O as the origin and drawing through it three lines, the coordinates. A reference frame with origin O consists of three base vectors pointing in three different directions which do not all lie in the same plane. The set of three base vectors is called a basis. A coordinate system is said to be curvilinear if its coordinate curves are not straight lines. A characteristic property of curvilinear coordinate systems is that the orientation of the axes vary from point to point. Consid- ering the general non-orthogonal curvilinear coordinate systems the base vectors are not orthogonal and need not be the same at different points in space. In these non-orthogonal curvilinear coordinate systems, two distinct frames of basis vectors exist at any point. One frame follows the coordinate lines, i.e., the covariant basis vectors are tangents to the coordinate curves. In the other frame, the contravari- ant basis vectors are normal to the coordinate surface. The orthogonal curvilinear coordinates can be considered a special case of the non-orthogonal curvilinear coordinates. These orthogonal coordinate systems are characterized by tangential basis vectors to the coordinate lines which are mutually perpendicular at every point. In this book only orthogonal coordinate systems are considered. The Cartesian coordinate system is defined by three mutually orthogonal unit vectors with equal units of measurement. The unit vectors may then be thought of as lines of unit length lying along the three axes. The orthogonal curvilinear coordinate systems, like the cylindrical and spherical coordinates, are defined by three mutually orthogonal unit vectors with unequal units of measurement. Considering a generalized orthogonal coordinate system, the orthogonal curvilinear coordinates are defined as qα.InthisO-system the base vectors eα are defined as unit vectors along the coordinates. The position of the point P is given by the coordinates, or by the position vector r = r(qα, t).

F.2.2 The Tensor Concept

A scalar is a quantity associated with a point in space, whose specification requires just one number. For example, the fluid density, mass fraction, temperature, pressure and work are all scalar quantities. Scalars can be compared only if they have the same physical dimensions. Scalars measured in the same system of units are said to be equal if they have the same magnitude and sign. A vector is an entity that possesses both magnitude and direction and obeys certain laws. For example, velocity, acceleration, force are all vectors. Two vectors are equal if they have the same direction and the same magnitude. Moreover, a direction has to be specified in relation to a given frame of reference and this frame of reference is just as arbitrary as the system of units in which the magnitude is expressed. We distinguish therefore between the vector as an entity and its components which allow us to reconstruct it in a particular system of reference. Second-order tensors (also called second-rank tensors7) are next in order of complexity after scalars and vectors. Scalars and vectors are both special cases of the more general mathematical entity called a tensor of order n, whose specification

7 The rank of a particular tensor is the number of array indices required to describe such a quantity. 1442 Appendix F: Trondheim Bubble Column Model in any given coordinate system requires 3n numbers for 3D tensors, these are called the components of the tensor. In this way, we may consider that scalars are tensors of order 0, with 30 = 1 components, and vectors are tensors of order 1, with 31 = 3 components. By a second-order tensor is thus meant a quantity uniquely specified by 32 = 9 numbers denoting the components of the tensor. Higher-order tensors can naturally be defined too, but the second-order tensors are the ones of primary interest in this book. For brevity, we often use the word tensor to mean second-order tensor. The stress tensor is customarily considered the primary tensor in fluid mechanics. A fluid stress has the units of a force per unit area. This tensor is thus an entity associated with two directions (those of the force and the normal to the area). The Cauchy’s stress principle establishes that the stress in fluids can be represented by a tensor (e.g., Aris [2], Chap. 5; Borisenko and Tarapov [10], Sect. 2.4.2; Slattery [88], Sect. 2.2.2). Basically, Cauchy’s stress principle asserts that f/δA, tends towards a finite limit as δA → 0. This limit is called the stress vector. To elucidate the nature of the stress system at a point P one considers a fluid element with shape like a small tetrahedron with three of its faces perpendicular to the coordinate axes, while the fourth has an area δA with normal n. The three faces are mutually orthogonal and coincide with a set of Cartesian coordinate planes intersecting at z.Bythelawof action and reaction, the stress forces acting on the inside faces of the tetrahedron are equal and opposite to those acting on the outside faces. Applying the principle of local equilibrium then shows that if the tetrahedral fluid element shrink in volume toward a point P, the net surface force will approach zero. It follows that the stress vector can be re-written as f =−(f1n1 + f2n2 + f3n3). With the convention that −Tmk is the kth component of the stress vector acting upon the positive side of the plane zm = constant, we project f onto the axes of the system:

f =−niTijej, (F.54) where Tij is a matrix of nine stress components constituting a tensor. This allow us to write (F.54)as: f =−n · T (F.55)

Thus, in order to describe completely the state of stress at a point in a continuum, we must specify the stress tensor T. A key property of a tensor is the transformation law of its components. This law expresses the way in which the tensor components in one coordinate system are related to its components in another coordinate system. The precise form of this transformation law is a consequence of the physical or geometric meaning of the tensor.

F.2.3 Coordinate Transformation Prerequisites

In this section we explain how to determine ∇ and the spatial derivatives of the unit vectors in cylindrical coordinates. Appendix F: Trondheim Bubble Column Model 1443

In Cartesian coordinates the position vector is given by:

r = xex + yey + zez (F.56)

The nabla operator yields:

∂ ∂ ∂ ∇= e + e + e (F.57) ∂x x ∂y y ∂z z

The gradient of a scalar ﬁeld ψ is deﬁned by:

∂ψ ∂ψ ∂ψ ∇ψ = e + e + e (F.58) ∂x x ∂y y ∂z z

The divergence of a vector ﬁeld v is deﬁned by:

∂v ∂vy ∂v ∇·v = x + + z (F.59) ∂x ∂y ∂z

The divergence of a tensor ﬁeld σ is deﬁned by:

∂ ∂ ∇·σ = ei · (σjkejek) = ek σik (F.60) ∂xi ∂xi i k

The Laplacian of a scalar ﬁeld ψ is deﬁned by:

∂2ψ ∂2ψ ∂2ψ ∇2ψ = + + (F.61) ∂x2 ∂y2 ∂z2

The Curl of a vector ﬁeld v is deﬁned by:

ex ey ez ∇× = ∂ ∂ ∂ v ∂x ∂y ∂z (F.62) vx vy vz

The governing equations can be transformed directly from Cartesian coordinates into cylindrical coordinates without considering the vector notation. In this appendix the relationships between the Cartesian coordinates and the cylindrical coordinates are deﬁned solely, but the method of coordinate transformation is generic and can thus be applied to any orthogonal coordinate system. In cylindrical coordinates, instead of locating a point in space by x,y,z as in Carte- sian coordinates, we designate the coordinates of the point by r,θ,z. The Cartesian coordinates are related to the cylindrical coordinates by [7]:

x = r cos θ, y = r sin θ, z = z (F.63) 1444 Appendix F: Trondheim Bubble Column Model

To convert the derivatives of scalars with respect to x,y,z into derivatives with respect to r,θ,z, the chain rule of partial differentiation is used. The derivative operators are thus related as: ∂ ∂ sin θ ∂ ∂ ∂ cos θ ∂ ∂ ∂ = cos θ − , = sin θ + , = (F.64) ∂x ∂r r ∂θ ∂y ∂r r ∂θ ∂z ∂z

Trigonometrical arguments lead to the following relations between the unit vectors:

er = cos θex + sin θey, eθ =−sin θex + cos θey, ez = ez (F.65)

From these equations one can derive the formulas required for the spatial derivatives of the unit vectors er,eθ,ez.

The given trigonometrical relationships may also be solved for ex,ey,ez, giving:

ex = cos θer − sin θeθ, ey = sin θer + cos θeθ, ez = ez (F.66)

To obtain the formula for ∇ in cylindrical coordinates we employ the deﬁnition of the ∇-operator in Cartesian coordinates (F.57), eliminate the Cartesian unit vectors by (F.66) and eliminate the Cartesian derivative operators by (F.64). The resulting formula for the ∇ operator in cylindrical coordinates can then be used to calculate all the necessary differential operators in cylindrical coordinates provided that the spatial derivatives of the unit vectors er,eθ,ez are used to differentiate the unit vectors on which ∇ operates.

The spatial derivatives of the unit vectors er,eθ,ez can be determined from (F.65):

∂ ∂ ∂ e = 0 e = 0 e = 0 (F.67) ∂r r ∂r θ ∂r z ∂ ∂ ∂ e = e e =−e e = 0 (F.68) ∂θ r θ ∂θ θ r ∂θ z ∂ ∂ ∂ e = 0 e = 0 e = 0 (F.69) ∂z r ∂z θ ∂z z

By use of (F.56), (F.63) and (F.68) the position vector can be transformed into cylindrical coordinates: r = rer(θ) + zez. (F.70)

F.2.4 Orthogonal Curvilinear Coordinate Systems and Differential Operators

In this section the relevant differential operators are deﬁned for generalized orthogonal curvilinear coordinate systems. Let (q1,q2,q3) be curvilinear orthogonal coordinates connected with the Cartesian coordinates (x,y,z) by the vector relation Appendix F: Trondheim Bubble Column Model 1445 r = r(q1, q2, q3), where r is the radius vector of the point P considered. The Carte- sian coordinates are then related to the generalized curvilinear coordinates by:

x = x(q1, q2, q3), y = y(q1, q2, q3), z = z(q1, q2, q3) (F.71)

If the Jacobian is nonzero, ∂(x, y, z) = 0, (F.72) ∂(q1, q2, q3) then q1 = q1(x, y, z), q2 = q2(x, y, z), q3 = q3(x, y, z) (F.73) form a basis for the orthogonal curvilinear coordinate system.

The basis vectors (not necessarily unit vectors) in the generalized curvilinear coordinate system are deﬁned as (e.g., [1], p. 193; [101], p. 6; [7], p. 737):

∂r ∂r ∂xk ∂xk ∂qα gα = = = ek or ek = gα (F.74) ∂qα ∂xk ∂qα ∂qα ∂xk where xk is the Cartesian coordinates, and gα are the tangent basis vectors which are tangents to the coordinate lines. From these basis vectors we can deﬁne the Lamé coefﬁcients denoting the length of the basis vectors (also named scale factors) and expressed by: √ =| |=|∂r |= = · = ∂r · ∂r = ∂xk ∂xk hqα gα hα gα gα (F.75) ∂qα ∂qα ∂qα ∂qα ∂qα

The unit tangent vectors to the coordinate lines qα can thus be determined by:

gα gα gα eα = = √ = (F.76) |gα| gαgα hα

For orthogonal coordinate systems, we can write: √ ∂xk ∂xk gα · gβ = = 0, for α = β. (F.77) ∂qα ∂qβ

We may then present a generalization of (F.75), the relation for hα: ∂xk ∂xk = hαδαβ (F.78) ∂qα ∂qβ 1446 Appendix F: Trondheim Bubble Column Model

Moreover, since: ∂qα ∂qα ∂xk = = δαβ (F.79) ∂qβ ∂xk ∂qβ

The equality relation between the last two terms in the above expression can be rewritten as: − ∂q ∂x 1 α = k (F.80) ∂xk ∂qα

Combining the latter three relationships, we obtain:

− ∂q 1 ∂qα β = ∂xk ∂xk = 1 2 δαβ (F.81) ∂xk ∂xk ∂qα ∂qβ hα

In Cartesian coordinates the position vector (F.56) is expressed in terms of the unit base vectors ex,ey,ez, hence a position vector increment dr between two infinitely close points yields: dr = dxex + dyey + dzez. The base vectors gα in the curvilinear system, called the natural basis of the curvilinear system (also called covariant base vectors), is defined such that the same position vector increment dr is given in terms of the curvilinear increments dgα by: dr = dgαeα. The distance element in curvilinear coordinate systems is then computed as the square of the element of arc length between the two infinitely close points:

2 (ds) =|dr|=dr · dr = gαγdqαdqγ (F.82) where gαγ = eα · eγ is called the metric tensor. The basic quantities√ describing√ an orthogonal√ coordinate system are the metric coefﬁcients h1 = g11, h2 = g22, h3 = g33, which satisfy the formula:

2 2 2 2 (ds) = dr · dr = (h1dq1) + (h2dq2) + (h3dq3) (F.83)

For the cylindrical coordinates the metric coefﬁcients can be determined from (F.75): ∂r ∂r ∂r ∂r ∂r ∂r h = · = 1, h = · = r, h = · = 1(F.84) r ∂r ∂r θ ∂θ ∂θ z ∂z ∂z in which we have used the following relations deduced from (F.70):

∂r ∂r ∂e ∂r = e , = r r = re , = e (F.85) ∂r r ∂θ ∂θ θ ∂z z

The del operator can then be written in a generalized form using (F.56), (F.74) and (F.81): Appendix F: Trondheim Bubble Column Model 1447 ∂ () ∂q ∂ () ∂q 1 ∂ () ∇=e = α g β = δ g k ∂x ∂x α ∂q ∂x h2 αβ α ∂q k k β k α α β (F.86) 1 ∂ () 1 ∂ () = g = e h2 α ∂q h α ∂q α α α α α α where e denotes the unit vectors in the Cartesian coordinate system and e = gα is k α hα the unit tangent vectors to the coordinate lines qα. The resulting expression for the nabla operator (F.86) are then employed to deduce the transformation formulas for the gradient, divergence, and curl operators in any orthogonal curvilinear coordinate system [49]:

1 ∂ψ ∇ψ = e (F.87) h α ∂q α α α 1 ∂(v e ) ∇·v = e · β β (F.88) h α ∂q α α α 1 ∂(v e ) curl(v) = rot(v) =∇×v = e × β β (F.89) h α ∂q α α α 1 ∂(v e ) ∇v = e β β (F.90) h α ∂q α α α 1 ∂(σ e eγ) ∇·σ = e βγ β (F.91) h α ∂q α α α

The vector v is presented in terms of its physical components vα in such a way that:

v = vα eα (F.92) where eα is the unit tangent vector to the coordinate lines in the orthogonal qα -system. The orthogonal curvilinear unit vectors just introduced obey certain laws, which are used in the subsequent paragraphs. The scalar or dot product of two unit vectors yields: eα · eβ = δαβ (F.93)

The vector or cross product of two unit vectors is deﬁned by:

3 eα × eβ = αβγeγ (F.94) γ=1

Many formulas in tensor analysis are expressed compactly in terms of the Kronecker delta, δαβ, and the alternating unit tensor, αβγ. These entities are deﬁned as: 1448 Appendix F: Trondheim Bubble Column Model & +1ifα = β δαβ = (F.95) 0ifα = β ⎧ ⎨⎪+1ifαβγ = 123, 231, or 312 αβγ = −1ifαβγ = 321, 132, or 213 . (F.96) ⎩⎪ 0 if any two indices are alike

F.2.5 Differential Operators in Cylindrical Coordinates

In this section the cylindrical coordinate transforms are deduced from the formulas presented in the preceding subsections. From (F.92) we recognize that a vector in cylindrical coordinates yields:

v = vr er + vθ eθ + vz ez (F.97) where er, eθ, and ez are the unit vectors in the radial, azimuthal, and axial directions, respectively. The nabla-operator is given by (F.86):

1 ∂ ∂ 1 ∂ ∂ ∇= e = e + e + e . (F.98) h α ∂q r ∂r θ r ∂θ z ∂z α α α

Gradient of a scalar

The gradient of a scalar ψ is:

1 ∂ψ ∂ψ 1 ∂ψ ∂ψ grad (ψ) =∇ψ = e = e + e + e (F.99) h α ∂q r ∂r θ r ∂θ z ∂z α α α which is a vector.

Divergence of a vector

If v is a vector, the divergence of v is:

1 ∂ div (v) =∇·v = e · v e (F.100) h α ∂q β β α α α ∂ 1 ∂ ∂ = e · v e + e · v e + e · (v e ) r ∂r β β θ r ∂θ β β z ∂z k k Appendix F: Trondheim Bubble Column Model 1449

∂vr 1 ∂ ∂vz = + eθ · (vrer + vθeθ) + ∂r r ∂θ ∂z ∂vr 1 ∂er ∂vr ∂ ∂vz = + eθ · vr + er + (vθeθ) + ∂r r ∂θ ∂θ ∂θ ∂z ∂vr 1 ∂vr ∂ ∂vz = + eθ · vreθ + er + (vθeθ) + ∂r r ∂θ ∂θ ∂z v v v = ∂ r + 1 v + ∂ θ + ∂ z ∂r r r ∂θ ∂z 1 ∂ 1 ∂v ∂v = (rv ) + θ + z r ∂r r r ∂θ ∂z which is scalar.

Gradient of a vector

Let v represent a vector, the gradient of v is:

1 ∂ grad (v) =∇v = e v e h α ∂q β β α α α ∂ = e (v e + v e + v e ) r ∂r r r θ θ z z 1 ∂ + e (v e + v e + v e ) r θ ∂θ r r θ θ z z ∂ + e (v e + v e + v e ) z ∂z r r θ θ z z ∂v ∂v ∂v = e e r + e e θ + e e z r r ∂r r θ ∂r r z ∂r 1 ∂v 1 1 ∂v 1 1 ∂v + e e r + e e v ++ e e θ − e e v + e e z r θ r ∂θ r θ θ r r θ θ ∂θ r θ r θ r θ z ∂θ ∂v ∂v ∂v + e e r + e e θ + e e z z r ∂z z θ ∂z z z ∂z ∂vr ∂vθ ∂vz = erer + ereθ + erez ∂r ∂r ∂r ∂ 1 ∂v 1 1 ∂v 1 1 ∂v + e e r − v + e e θ + v + e e z θ r ∂θ r ∂θ r θ θ θ r ∂θ r r θ z r ∂θ ∂v ∂v ∂v + e e r + e e θ + e e z (F.101) z r ∂z z θ ∂z z z ∂z

The product is a second-order tensor, or a dyadic product. 1450 Appendix F: Trondheim Bubble Column Model

Divergence of a second-order tensor

The unit dyads may be multiplied with each other and with the unit vectors:

eα : eβ = δακδβγ (F.102)

eαeβ · eγ = eαδβγ (F.103)

eα · eβeγ = δαβeγ (F.104)

eαeβ · eγeκ = δβγeαeκ (F.105)

Then, if A is a second order tensor, or a dyad, the divergence of A is:

1 ∂ div (A) =∇·A = Aγκeγeκ · eα α hα ∂qα ∂ 1 ∂ ∂ = Aγκeγeκ · er + Aγκeγeκ · e + Aγκeγeκ · ez ∂r r ∂θ θ ∂z ∂ ∂ = Aγκ eγeκ · er + Aγκ eγeκ · er ∂r ∂r 1 ∂ 1 ∂ + Aγκ eγeκ · e + Aγκ eγeκ · e r ∂θ θ r ∂θ θ ∂ ∂ + Aγκ eγeκ · ez + Aγκ eγeκ · ez ∂z ∂z ∂ ∂ ∂ = (Arr) er + (A ) e + (Azr) ez ∂r ∂r θr θ ∂r 1 ∂ 1 ∂ 1 ∂ + (Arθ) er + (Aθθ) eθ + Azθ ez r ∂θ r ∂θ r ∂θ 1 1 ∂e + A e + A k e · e r kr k r kθ ∂θ θ θ ∂ ∂ ∂ + (Arz) er + A e + (Azz) ez ∂z ∂z θz θ ∂z ∂ ∂ ∂ = (Arr) er + (A ) e + (Azr) ez ∂r ∂r θr θ ∂r 1 ∂ 1 ∂ 1 ∂ + (A ) er + (A ) e + A ez r ∂θ rθ r ∂θ θθ θ r ∂θ zθ 1 1 1 1 1 + Arrer + A e + Azrez + A e − A er r r θr θ r r θθ θ r θθ ∂ ∂ ∂ + (Arz) er + Aθz eθ + (Azz) ez ∂z ∂z ∂z ∂ 1 ∂ 1 1 ∂ = er (Arr) + (A ) + Arr − A + (Arz) ∂r r ∂θ rθ r r θθ ∂z ∂ 1 ∂ 1 1 ∂ + e (A ) + (A ) + A + A + A θ ∂r θr r ∂θ θθ r θr r rθ ∂z θz Appendix F: Trondheim Bubble Column Model 1451 ∂ 1 ∂ 1 ∂ + ez (Azr) + A + Azr + (Azz) ∂r r ∂θ zθ r ∂z 1 ∂ 1 ∂ ∂ 1 = er (rArr) + (A ) + (Arz) − A r ∂r r ∂θ rθ ∂z r θθ 1 ∂ 1 ∂ ∂ 1 + e (rA ) + (A ) + A + A θ r ∂r θr r ∂θ θθ ∂z θz r rθ 1 ∂ 1 ∂ ∂ + ez (rAzr) + A + (Azz) (F.106) r ∂r r ∂θ zθ ∂z which is a vector.

The Laplacian of a Scalar Field

If we take the divergence of the gradient of the scalar function ψ, as is done for the pressure ﬁeld formulating an equation for the pressure, we obtain:

∇2ψ =∇·∇ψ =∇·grad (ψ) ∂ 1 ∂ ∂ ∂ψ 1 ∂ψ ∂ψ = e + e + e · e + e + e r ∂r θ r ∂θ z ∂z r ∂r θ r ∂θ z ∂z ∂2ψ ∂ 1 ∂ψ 1 ∂ ∂ψ 1 ∂2ψ ∂2ψ = + e · e + e · e + + 2 r θ θ r 2 2 2 ∂r ∂r r ∂θ r ∂θ ∂r r ∂θ ∂z ∂2ψ 1 ∂ψ ∂e ∂ 1 ∂ψ = + e · θ + e · e ∂r2 r r ∂θ ∂r r θ ∂r r ∂θ 1 ∂ ∂ψ 1 ∂2ψ ∂2ψ + e · e + + (F.107) θ r ∂θ r ∂r r2 ∂θ2 ∂z2 ∂2ψ 1 ∂α ∂e ∂ ∂ψ 1 ∂2α ∂2ψ = + e · r + e + + ∂r2 θ r ∂r ∂θ r ∂θ ∂r r2 ∂θ2 ∂z2 ∂2ψ 1 ∂α 1 ∂2ψ ∂2ψ = + e · e + + ∂r2 θ r ∂r θ r2 ∂θ2 ∂z2 2 2 2 = ∂ ψ + 1 ∂ψ + 1 ∂ ψ + ∂ ψ 2 2 2 2 ∂r r ∂r r ∂θ ∂z 1 ∂ ∂ψ 1 ∂2ψ ∂2ψ = r + + r ∂r ∂r r2 ∂θ2 ∂z2

The result is a scalar.

The Curl of a Vector Field

1 ∂ vβeβ curl (v) = rot (v) =∇×v = e × h α ∂q α α α 1452 Appendix F: Trondheim Bubble Column Model

∂ vβeβ 1 ∂ vβeβ ∂ vβeβ = e × + e × + e × r ∂r r θ ∂θ z ∂z ∂ 1 ∂ = e × (v e + v e + v e ) + e × (v e + v e + v e ) r ∂r r r θ θ z z r θ ∂θ r r θ θ z z ∂ + ez × (vrer + vθeθ + vzez) ∂z ∂er ∂vr ∂eθ ∂vθ ∂ez ∂vz = er × vr + er + vθ + eθ + vz + ez ∂r ∂r ∂r ∂r ∂r ∂r 1 ∂er ∂vr ∂eθ ∂vθ ∂ez ∂vz + eθ × vr + er + vθ + eθ + vz + ez r ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂er ∂vr ∂eθ ∂vθ ∂ez ∂vz + ez × vr + er + vθ + eθ + vz + ez ∂z ∂z ∂z ∂z ∂z ∂z ∂vr ∂vθ ∂vz = er × er + eθ + ez ∂r ∂r ∂r 1 ∂vr ∂vθ ∂vz + eθ × vreθ + er − vθer + eθ + ez r ∂θ ∂θ ∂θ ∂vr ∂vθ ∂vz + ez × er + eθ + ez ∂z ∂z ∂z ∂vθ ∂vz 1 ∂vr ∂vz = er × eθ + ez + eθ × er − vθer + ez ∂r ∂r r ∂θ ∂θ ∂v ∂v + e × r e + θ e z ∂z r ∂z θ

The vector or cross product of two unit vectors was deﬁned by (F.94), hence in cylindrical coordinates the following relations are valid:

er × eθ = rθγeγ = rθzez = ez (F.108) er × ez = rzθeθ =−eθ (F.109) eθ × er = θrzez =−ez (F.110) eθ × ez = θzrer = er (F.111) ez × er = zrθeθ = eθ (F.112) ez × eθ = zθrer =−er (F.113)

Introducing these relations into the above expression for the Curl operator gives:

∂vθ ∂vz 1 ∂vr vθ 1 ∂vz ∂vr ∂vθ curl (v) = ez − eθ − ez + ez + er + eθ − er ∂r ∂r r ∂θ r r ∂θ ∂z ∂z 1 ∂v ∂v ∂v ∂v ∂v 1 ∂v v = e z − θ + e r − z + e θ − r + θ r r ∂θ ∂z θ ∂z ∂r z ∂r r ∂θ r Appendix F: Trondheim Bubble Column Model 1453 1 ∂v ∂v ∂v ∂v 1 ∂ (rv ) 1 ∂v = e z − θ + e r − z + e θ − r . r r ∂θ ∂z θ ∂z ∂r z r ∂r r ∂θ

F.2.6 Differential Operators Required for the Two-ﬂuid Model

In this section the govening equations are transformed from vector notation to cylindrical coordinates.

Gradient of a scalar

The pressure gradient is one of the gradients of a scalar variable that are part of the equations in question. In cylindrical coordinates it is determined as:

∂p 1 ∂p ∂p ∇p = e + e + e . (F.114) ∂r r r ∂θ θ ∂z z

Divergence of a vector

In all the scalar equations a convective term on the form ∇·(αlρlvl) is included. In cylindrical coordinates this term is expressed as:

1 ∂ 1 ∂ ∂ ∇·(α ρ v ) = (rα ρ v , ) + (α ρ v , ) + (α ρ v , ). (F.115) l l l r ∂r l l l r r ∂θ l l l θ ∂z l l l z

Divergence of a gradient

In the scalar equations there are also terms like ∇·( μl,t ∇α ). This term is identical in σt l form to the previous one, since the gradient is just a vector. In cylindrical coordinates the term is written as: μl,t 1 ∂ μl,t ∂αl 1 ∂ μl,t 1 ∂αl ∂ μl,t ∂αl ∇· ∇αl = r + + . σt r ∂r σt ∂r r ∂θ σt r ∂θ ∂z σt ∂z (F.116)

Divergence of a dyad

Generally, the divergence of a dyad or 2. order tensor, say σ, is transformed as: 1454 Appendix F: Trondheim Bubble Column Model ∇·σ = 1 ∂ ( σ ) + 1 ∂ (σ ) − 1 σ + ∂ (σ ) r ∂r r rr r ∂θ θr r θθ ∂z zr er + 1 ∂ (σ ) + 1 ∂ ( σ ) + 1 σ + ∂ (σ ) r ∂θ θθ r ∂r r rθ r θr ∂z zθ eθ (F.117) + 1 ∂ ( σ ) + 1 ∂ (σ ) + ∂ (σ ) r ∂r r rz r ∂θ θz ∂z zz ez

The dyad v v is symmetric. In cylindrical coordinates ∇·(αρvv) is given as: 1 ∂ 1 ∂ 1 ∂ ∇·(αρvv) = (rαρvrvr) + (αρv vr) − αρv v + (αρvzvr) er r ∂r r ∂θ θ r θ θ ∂z 1 ∂ 1 ∂ 1 ∂ + (αρv v ) + (rαρvrv ) + αρv vr + (αρvzv ) e r ∂θ θ θ r ∂r θ r θ ∂z θ θ 1 ∂ 1 ∂ ∂ + (rαρvrvz) + (αρv vz) + (αρvzvz) ez (F.118) r ∂r r ∂θ θ ∂z

The terms similar to ∇·( μl,t ∇α v) are also divergences of a dyad. ∇α v is not σα,t symmetric.

μ , ∇·( l t ∇α v) σ , α t 1 ∂ μl,t ∂α 1 ∂ μl,t 1 ∂α μl,t 1 ∂α ∂ μl,t ∂α = (r vr) + ( vr) − vθ + ( vr) er r ∂r σ , ∂r r ∂θ σ , r ∂θ σ , r2 ∂θ ∂z σ , ∂z α t α t α t α t 1 ∂ μl,t 1 ∂α 1 ∂ μl,t ∂α μl,t 1 ∂α ∂ μl,t ∂α + ( vθ) + (r vθ) + vr + ( vθ) eθ r ∂θ σ , r ∂θ r ∂r σ , ∂r σ , r2 ∂θ ∂z σ , ∂z α t α t α t α t 1 ∂ μl,t ∂α 1 ∂ μl,t 1 ∂α ∂ μl,t ∂α + (r vz) + ( vz) + ( vz) ez (F.119) r ∂r σα,t ∂r r ∂θ σα,t r ∂θ ∂z σα,t ∂z

The term ∇·( μl,t v∇α) is very similar to the previous one: σα,t

μ , ∇·( l t v∇α) σα,t 1 ∂ μ , ∂α 1 ∂ μ , ∂α μ , 1 ∂α ∂ μ , ∂α = (r l t v ) + ( l t v ) − l t v + ( l t v ) e r θ 2 θ z r r ∂r σα,t ∂r r ∂θ σα,t ∂r σα,t r ∂θ ∂z σα,t ∂r + 1 ∂ ( μl,t v 1 ∂α) + ∂ ( μl,t v 1 ∂α) + 1 μl,t v ∂α + μl,t 1v ∂α θ r 2 r θ r ∂θ σα,t r ∂θ ∂r σα,t r ∂θ r σα,t ∂θ σα,t r ∂r ∂ μl,t 1 ∂α + ( vz ) eθ ∂z σα,t r ∂θ 1 ∂ μl,t ∂α 1 ∂ μl,t ∂α ∂ μl,t ∂α + (r vr ) + ( vθ ) + ( vz ) ez (F.120) r ∂r σα,t ∂z r ∂θ σα,t ∂z ∂z σα,t ∂z

In cylindrical coordinates the term ∇·(αμ∇v) is expressed as: Appendix F: Trondheim Bubble Column Model 1455 ∇· αμ∇v 1 ∂ ∂v 1 ∂ 1 ∂v v 1 1 ∂v v = rαμ r + αμ r − θ − αμ θ + r r ∂r ∂r r ∂θ r ∂θ r r r ∂θ r ∂ ∂v + αμ r e ∂z ∂z r 1 ∂ 1 ∂v v 1 1 ∂v v 1 ∂ ∂v + αμ θ + r + αμ r − θ + rαμ θ r ∂θ r ∂θ r r r ∂θ r r ∂r ∂r ∂ ∂v + αμ θ e ∂z ∂z θ 1 ∂ ∂v 1 ∂ ∂v ∂ ∂v + rαμ z + αμ z + αμ z e r ∂r ∂r r ∂θ ∂θ ∂z ∂z z (F.121) The divergence of the transpose of the gradient of a vector ∇(αμ(∇v)T ) in cylindrical coordinates is written: T ∇· αμ ∇v 1 ∂ ∂vr 1 ∂ ∂vθ 1 1 ∂vθ vr ∂ ∂vz = rαμ + αμ − αμ + + αμ er r ∂r ∂r r ∂θ ∂r r r ∂θ r ∂z ∂r 1 ∂ 1 ∂vr v 1 ∂ 1 ∂v vr 1 ∂v + rαμ − θ + αμ θ + + αμ θ r ∂r r ∂θ r r ∂θ r ∂θ r r ∂r ∂ 1 ∂vz + αμ e ∂z r ∂θ θ 1 ∂ ∂vr 1 ∂ ∂vθ ∂ ∂vz + rαμ + αμ + αμ ez. (F.122) r ∂r ∂z r ∂θ ∂z ∂z ∂z

Gradient of a vector

The gradient of v is part of the momentum equations. In cylindrical coordinates it is expressed as:

∂vr ∂vθ ∂vz ∇v = erer + ereθ + erez ∂r ∂r ∂r 1 ∂vr vθ 1 ∂vθ vr 1 ∂vz + − e e + + e e + e e (F.123) r ∂θ r θ r r ∂θ r θ θ r ∂θ θ z ∂v ∂v ∂v + r e e + θ e e + z e e . ∂z z r ∂z z θ ∂z z z 1456 Appendix F: Trondheim Bubble Column Model

The dot product ∇α ·∇v

In cylindrical coordinates this term is written as: ∂α ∂vr 1 ∂α 1 ∂vr vθ ∂α ∂vr ∇α ·∇v = + − + er ∂r ∂r r ∂θ r ∂θ r ∂z ∂z ∂α ∂vθ 1 ∂α 1 ∂vθ vr ∂α vθ + + + + eθ (F.124) ∂r ∂r r ∂θ r ∂θ r ∂z z ∂α ∂v 1 ∂α 1 ∂v ∂α ∂v + z + z + z e . ∂r ∂r r ∂θ r ∂θ ∂z ∂z z

μ The dot product v ·∇( l,t ∇α) σα,t

In cylindrical coordinates this term is written as:

μ , v ·∇( l t ∇α) σ , α t ∂ μl,t ∂α 1 ∂ μl,t ∂α 1 μl,t ∂α ∂ μl,t ∂α = vr ( ) + vθ ( ) − vθ + vz ( ) er ∂r σ , ∂r r ∂θ σ , ∂r r2 σ , ∂θ ∂z σ , ∂r α t α t α t α t ∂ μl,t 1 ∂α 1 μl,t ∂α 1 ∂ μl,t 1 ∂α ∂ μl,t 1 ∂α + vr ( ) + vθ + vθ ( ) + vz ( ) eθ ∂r σ , r ∂θ r σ , ∂r r ∂θ σ , r ∂θ ∂z σ , r ∂θ α t α t α t α t ∂ μl,t ∂α 1 ∂ μl,t ∂α ∂ μl,t ∂α + vr ( ) + vθ ( ) + vz ( ) ez. (F.125) ∂r σα,t ∂z r ∂θ σα,t ∂z ∂z σα,t ∂z

F.3 Two-Fluid Equations in cylindrical coordinates

By use of the transformations deﬁned above, the governing 3D equations are written in cylindrical coordinates.

Liquid Phase Continuity Equation in Cylindrical Coordinates

∂ 1 ∂ 1 ∂ ∂ (α ρ ) + (rα ρ v , ) + (α ρ v , ) + (α ρ v , ) ∂t l l r ∂r l l l r r ∂θ l l l θ ∂z l l l z (F.126) 1 ∂ μ , ∂α 1 ∂ μ , ∂α ∂ μ , ∂α = (r l t l ) + ( l t l ) + ( l t l ). r ∂r σαl,t ∂r r ∂θ σαl,t ∂θ ∂z σαl,t ∂z Appendix F: Trondheim Bubble Column Model 1457

Liquid Phase Radial Momentum Balance in Cylindrical Coordinates

∂ α ρ v , ∂t l l l r 1 ∂ 1 ∂ 1 ∂ + rα ρ v , v , + α ρ v , v , − α ρ v , v , + α ρ v , v , r ∂r l l l r l r r ∂θ l l l θ l r r l l l θ l θ ∂z l l l z l r 1 ∂ μl,t ∂αl 1 ∂ μl,t 1 ∂αl 1 μl,t ∂αl = r v , + v , − v , r ∂r σ ∂r l r r ∂θ σ r ∂θ l r r2 σ ∂θ l θ αl,t αl,t αl,t ∂ μl,t ∂αl 1 ∂ μl,t ∂αl 1 ∂ μl,t ∂αl + v , + r v , + v , ∂z σ l r ∂z r ∂r σ ∂r l r r ∂θ σ l θ ∂r αl,t αl,t αl,t v 1 μl,t ∂αl ∂ μl,t ∂αl 1 ∂ ∂ l,r − v , + v , + rα μ , r2 σ l θ ∂θ ∂z σ l z ∂r r ∂r l l eff ∂r αl,t αl,t 1 ∂ 1 ∂vl,r vl,θ 1 1 ∂vl,θ vl,r + α μ , − − α μ , + r ∂θ l l eff r ∂θ r r l l eff r ∂θ r ∂ ∂vl,r 1 ∂ ∂vl,r 1 ∂ ∂vl,θ + α μ , + rα μ , + α μ , ∂z l l eff ∂z r ∂r l l eff ∂r r ∂θ l l eff ∂r 1 1 ∂vl,θ vl,r ∂ ∂vl,z − α μ , + + α μ , r l l eff r ∂θ r ∂z l l eff ∂r ∂ 2 1 ∂ 1 ∂vl,θ ∂vl,z − α ρ k + ν , rv , + + ∂r 3 l l l eff r ∂r l r r ∂θ ∂z ∂p μ , ∂α ∂v , 1 ∂α 1 ∂v , v , ∂α ∂v , − α − l t l l r + l l r − l θ + l l r l ∂r σ ∂r ∂r r ∂θ r ∂θ r ∂z ∂z αl,t ∂ μl,t ∂αl 1 ∂ μl,t ∂αl 1 μl,t ∂αl − v , − v , + v , l r ∂r σ ∂r l θ r ∂θ σ ∂r l θ r2 σ ∂θ αl,t αl,t αl,t μ − v ∂ l,t ∂αl + + C + C . l,z αlρlgr Fl,r Fl,r (F.127) ∂z σαl,t ∂r

Liquid Phase Azimuthal Momentum Balance in Cylindrical Coordinates

∂ (α ρ v , ) ∂t l l l θ 1 ∂ 1 ∂ 1 ∂ + (rα ρ v , v , ) + (α ρ v , v , ) + α ρ v , v , + (α ρ v , v , ) r ∂r l l l r l θ r ∂θ l l l θ l θ r l l l θ l r ∂z l l l z l θ 1 ∂ μl,t ∂αl μl,t 1 ∂αl 1 ∂ μl,t 1 ∂αl = (r v , ) + v , + ( v , ) l θ 2 l r l θ r ∂r σαl,t ∂r σαl,t r ∂θ r ∂θ σαl,t r ∂θ ∂ μl,t ∂αl ∂ μl,t 1 ∂αl 1 μl,t ∂αl 1 μl,t ∂αl + ( v , ) + ( v , ) + v , + v , l θ l r 2 l r l θ ∂z σαl,t ∂z ∂r σαl,t r ∂θ r σαl,t ∂θ r σαl,t ∂r 1458 Appendix F: Trondheim Bubble Column Model

v 1 ∂ μl,t 1 ∂αl ∂ μl,t 1 ∂αl 1 ∂ ∂ l,θ + ( vl,θ ) + ( vl,z ) + (rαlμl,eff ) r ∂θ σαl,t r ∂θ ∂z σαl,t r ∂θ r ∂r ∂r 1 ∂ 1 ∂vl,θ vl,r 1 1 ∂vl,r vl,θ ∂ ∂vl,θ + (α μ , ( + )) + α μ , ( − ) + (α μ , ) r ∂θ l l eff r ∂θ r r l l eff r ∂θ r ∂z l l eff ∂z 1 ∂ 1 ∂vl,r vl,θ 1 ∂ 1 ∂vl,θ vl,r αlμl,eff ∂vl,θ + (rα μ , ( − )) + (α μ , ( + )) + r ∂r l l eff r ∂θ r r ∂θ l l eff r ∂θ r r ∂r ∂ 1 ∂vl,z ∂ 2 1 ∂ 1 ∂vl,θ ∂vl,z + (α μ , ) − ( α ρ (k + ν , ( (rv , ) + + ))) ∂z l l eff r ∂θ ∂θ 3 l l l eff r ∂r l r r ∂θ ∂z v v v v ∂p μl,t ∂αl ∂ l,θ 1 ∂αl 1 ∂ l,θ l,r ∂αl ∂ l,θ − αl − ( + ( + ) + ) ∂θ σαl,t ∂r ∂r r ∂θ r ∂θ r ∂z ∂z ∂ μl,t 1 ∂αl 1 μl,t ∂αl 1 ∂ μl,t 1 ∂αl − vl,r ( ) − vl,θ − vl,θ ( ) ∂r σαl,t r ∂θ r σαl,t ∂r r ∂θ σαl,t r ∂θ μ − v ∂ ( l,t 1 ∂αl ) + + C + C . l,z αlρlgr Fl,θ Fl,θ (F.128) ∂z σαl,t r ∂θ

Liquid Phase Axial Momentum Balance in Cylindrical Coordinates

∂ 1 ∂ 1 ∂ ∂ (α ρ v , ) + (rα ρ v , v , ) + (α ρ v , v , ) + (α ρ v , v , ) ∂t l l l z r ∂r l l l r l z r ∂θ l l l θ l z ∂z l l l z l z 1 ∂ μl,t ∂αl 1 ∂ μl,t 1 ∂αl ∂ μl,t ∂αl = (r vl,z) + ( vl,z) + ( vl,z) r ∂r σαl,t ∂r r ∂θ σαl,t r ∂θ ∂z σαl,t ∂z 1 ∂ μl,t ∂αl 1 ∂ μl,t ∂αl ∂ μl,t ∂αl + (r vl,r) + ( vl,θ) + ( vl,z) r ∂r σαl,t ∂z r ∂θ σαl,t ∂z ∂z σαl,t ∂z 1 ∂ ∂vl,z 1 ∂ ∂vl,z ∂ ∂vl,z + (rα μ , ) + (α μ , ) + (α μ , ) r ∂r l l eff ∂r r ∂θ l l eff ∂θ ∂z l l eff ∂z 1 ∂ ∂vl,r 1 ∂ ∂vl,θ ∂ ∂vl,z + (rα μ , ) + (α μ , ) + (α μ , ) r ∂r l l eff ∂z r ∂θ l l eff ∂z ∂z l l eff ∂z ∂ 2 1 ∂ 1 ∂vl,θ ∂vl,z − ( α ρ (k + ν , ( (rv , ) + + ))) ∂z 3 l l l eff r ∂r l r r ∂θ ∂z v v v − ∂p − μl,t (∂αl ∂ l,z + 1 ∂αl ∂ l,z + ∂αl ∂ l,z ) αl 2 ∂z σαl,t ∂r ∂r r ∂θ ∂θ ∂z ∂z ∂ μl,t ∂αl 1 ∂ μl,t ∂αl ∂ μl,t ∂αl − (vl,r ( ) + vl,θ ( ) + vl,z ( )) ∂r σαl,t ∂z r ∂θ σαl,t ∂z ∂z σαl,t ∂z + + C + C . αlρlgz Fl,z Fl,z (F.129) Appendix F: Trondheim Bubble Column Model 1459

Turbulence Model

The transport equation for turbulent kinetic energy in cylindrical coordinates is written as: ∂ 1 ∂ 1 ∂ ∂ (α ρ k) + (rα ρ v , k) + (α ρ v , k) + (α ρ v , k) ∂t l l r ∂r l l l r r ∂θ l l l θ ∂z l l l z 1 ∂ μl,eff ∂k 1 ∂ μl,eff 1 ∂k ∂ μl,eff ∂k = (rαl ) + (αl ) + (αl ) r ∂r σk ∂r r ∂θ σk r ∂θ ∂z σk ∂z + αl(Pk + Pb − ρlε) (F.130)

The transport equation for the turbulent energy dissipation rate in cylindrical coordinates is written as: ∂ 1 ∂ 1 ∂ ∂ (α ρ ε) + (rα ρ v , ε) + (α ρ v , ε) + (α ρ v , ε) ∂t l l r ∂r l l l r r ∂θ l l l θ ∂z l l l z 1 ∂ μl,eff ∂ε 1 ∂ μl,eff 1 ∂ε ∂ μl,eff ∂ε = (rαl ) + (αl ) + (αl ) r ∂r σ ∂r r ∂θ σε r ∂θ ∂z σε ∂z ε + α (C (P + P ) − C ρ ε) (F.131) l k 1 k b 2 l where 2 2 2 ∂vl,r 1 ∂vl,θ vl,r ∂vl,z P =2μ , + + + k l t ∂r r ∂θ r ∂z 2 2 2 ∂vl,z ∂vl,r 1 ∂vl,z ∂vl,θ ∂vl,θ 1 ∂vl,r vl,θ + μ , + + + + + − l t ∂r ∂z r ∂θ ∂z ∂r r ∂θ r (F.132) and

Pb = Cb FD,r(vg,r − vl,r) + FD,θ(vg,θ − vl,θ) + FD,z(vg,z − vl,z) . (F.133)

Gas phase equations in cylindrical coordinates

The gas phase continuity and momentum equations are almost identical to those for the liquid phase, and are not repeated to save space. 1460 Appendix F: Trondheim Bubble Column Model

F.4 The 2D axi-symmetric Bubble Column Model

Liquid phase equations in cylindrical coordinates:

In an axi-symmetric case the liquid phase continuity equation simpliﬁes as:

∂ 1 ∂ ∂ (α ρ )+ (rα ρ v , ) + (α ρ v , ) ∂t l l r ∂r l l l r ∂z l l l z 1 ∂ μ , ∂α ∂ μ , ∂α = (r l t l ) + ( l t l ) (F.134) r ∂r σαl,t ∂r ∂z σαl,t ∂z

The corresponding radial liquid phase momentum balance reduces as: ∂ 1 ∂ ∂ αlρlvl,r + rαlρlvl,rvl,r + αlρlvl,zvl,r ∂t r ∂r ∂z 1 ∂ μl,t ∂αl ∂ μl,t ∂αl = r v , + v , r ∂r σ ∂r l r ∂z σ ∂z l r αl,t αl,t 1 ∂ μl,t ∂αl ∂ μl,t ∂αl + r v , + v , r ∂r σ ∂r l r ∂z σ ∂r l z αl,t αl,t 1 ∂ ∂vl,r 1 vl,r ∂ ∂vl,r + rαlμl,eff − αlμl,eff + αlμl,eff r ∂r ∂r r r ∂z ∂z 1 ∂ ∂vl,r 1 vl,r ∂ ∂vl,z + rαlμl,eff − αlμl,eff + αlμl,eff r ∂r ∂r r r ∂z ∂r ∂ 2 1 ∂ ∂vl,z − αlρl k + νl,eff rvl,r + ∂r 3 r ∂r ∂z ∂p μ , ∂α ∂v , ∂α ∂v , − α − l t l l r + l l r l ∂r σ ∂r ∂r ∂z ∂z αl,t ∂ μl,t ∂αl ∂ μl,t ∂αl − vl,r + vl,z ∂r σαl,t ∂r ∂z σαl,t ∂r + + C + C αlρlgr Fl,r Fl,r (F.135)

Using the product rule of calculus a few terms cancel out and the equation yields:

∂ 1 ∂ ∂ (α ρ v , ) + (rα ρ v , v , ) + (α ρ v , v , ) ∂t l l l r r ∂r l l l r l r ∂z l l l z l r vl,r ∂ μl,t ∂αl ∂ μl,t ∂αl = (r ) + vl,r ( ) r ∂r σαl,t ∂r ∂z σαl,t ∂z 1 ∂ ∂vl,r 1 vl,r ∂ ∂vl,r + (rα μ , ) − α μ , + (α μ , ) r ∂r l l eff ∂r r l l eff r ∂z l l eff ∂z Appendix F: Trondheim Bubble Column Model 1461

1 ∂ ∂vl,r 1 vl,r ∂ ∂vl,z + (rα μ , ) − α μ , + (α μ , ) r ∂r l l eff ∂r r l l eff r ∂z l l eff ∂r ∂ 2 1 ∂ ∂vl,z − ( α ρ (k + ν , ( (rv , ) + ))) ∂r 3 l l l eff r ∂r l r ∂z μl,t ∂αl 1 ∂ ∂vl,z ∂p + [ (rvl,r) + ]−αl σαl,t ∂r r ∂r ∂z ∂r + + C + C αlρlgr Fl,r Fl,r (F.136)

Assuming axi-symmetry the azimuthal liquid phase momentum balance vanishes completely. The liquid phase axial momentum balance becomes:

∂ 1 ∂ ∂ (α ρ v , ) + (rα ρ v , v , ) + (α ρ v , v , ) = ∂t l l l z r ∂r l l l r l z ∂z l l l z l z 1 ∂ μl,t ∂αl ∂ μl,t ∂αl (r vl,z) + ( vl,z) r ∂r σαl,t ∂r ∂z σαl,t ∂z 1 ∂ μl,t ∂αl ∂ μl,t ∂αl + (r vl,r) + ( vl,z) r ∂r σαl,t ∂z ∂z σαl,t ∂z 1 ∂ ∂vl,z ∂ ∂vl,z + (rα μ , ) + (α μ , ) r ∂r l l eff ∂r ∂z l l eff ∂z 1 ∂ ∂vl,r ∂ ∂vl,z + (rα μ , ) + (α μ , ) r ∂r l l eff ∂z ∂z l l eff ∂z ∂ 2 1 ∂ ∂vl,z − ( α ρ (k + ν , ( (rv , ) + ))) ∂z 3 l l l eff r ∂r l r ∂z ∂p μl,t ∂αl ∂vl,z ∂αl ∂vl,z − αl − ( + ) ∂z σαl,t ∂r ∂r ∂z ∂z ∂ μl,t ∂αl ∂ μl,t ∂αl − (vl,r ( ) + vl,z ( )) ∂r σαl,t ∂z ∂z σαl,t ∂z + + C + C αlρlgz Fl,z Fl,z (F.137)

Using the product rule of calculus a few terms cancel out and the equation yields:

∂ 1 ∂ ∂ (α ρ v , ) + (rα ρ v , v , ) + (α ρ v , v , ) ∂t l l l z r ∂r l l l r l z ∂z l l l z l z 1 ∂ μl,t ∂αl ∂ μl,t ∂αl = vl,z (r ) + vl,z ( ) r ∂r σαl,t ∂r ∂z σαl,t ∂z μl,t ∂αl 1 ∂ ∂vl,z + ( (rvl,r) + ) σαl,t ∂z r ∂r ∂z 1 ∂ ∂vl,z ∂ ∂vl,z + (rα μ , ) + (α μ , ) r ∂r l l eff ∂r ∂z l l eff ∂z 1462 Appendix F: Trondheim Bubble Column Model

1 ∂ ∂vl,r ∂ ∂vl,z + (rα μ , ) + (α μ , ) r ∂r l l eff ∂z ∂z l l eff ∂z ∂ 2 1 ∂ ∂vl,z − ( α ρ (k + ν , ( (rv , ) + ))) ∂z 3 l l l eff r ∂r l r ∂z ∂p − α + α ρ g + FC + FC (F.138) l ∂z l l z l,z l,z

A generalized transport equation for a scalar quantity in the liquid phase can be formulated as: ∂ 1 ∂ ∂ (α ρ φ ) + (rα ρ v , φ ) + (α ρ v , φ ) ∂t l l l r ∂r l l l r l ∂z l l l z l

1 ∂ ∂φl ∂ ∂φl = (rα Γ , ) + (α Γ , ) + S , (F.139) r ∂r l φl eff ∂r ∂z l φl eff ∂z φ l

The particular scalar transport equation for the liquid phase turbulent kinetic energy is written: ∂ 1 ∂ ∂ (α ρ k)+ (rα ρ v , k) + (α ρ v , k) ∂t l l r ∂r l l l r ∂z l l l z 1 ∂ μl,eff ∂k ∂ μl,eff ∂k = (rαl ) + (αl ) + αl(Pk + Pb − ρlε) r ∂r σk ∂r ∂z σk ∂z (F.140) where v v v v v ∂ l,r 2 ∂ l,z 2 l,r 2 ∂ l,r ∂ l,z 2 P = μ , (2[( ) + ( ) + ( ) ]+( + ) ) (F.141) k l t ∂r ∂z r ∂z ∂r and

Pb = Cb(FD,z(vg,z − vl,z) + FD,r(vg,r − vl,r)) (F.142)

The particular transport equation for the liquid phase turbulent energy dissipation rate is written: ∂ 1 ∂ ∂ (α ρ ε) + (rα ρ v , ε) + (α ρ v , ε) ∂t l l r ∂r l l l r ∂z l l l z 1 ∂ μl,eff ∂ε ∂ μl,eff ∂ε ε = (rαl ) + (αl ) + αl (C1(Pk + Pb) − C2ρlε). r ∂r σε ∂r ∂z σ ∂z k (F.143) Appendix F: Trondheim Bubble Column Model 1463

Gas phase equations in cylindrical coordinates:

The gas phase mass balance equation for the axisymmetric case is:

∂ 1 ∂ ∂ (α ρ )+ (rα ρ v , ) + (α ρ v , ) ∂t g g r ∂r g g g r ∂z g g g z 1 ∂ μ , ∂α ∂ μ , ∂α (F.144) = (r g t g ) + ( g t g ) r ∂r σαg,t ∂r ∂z σαg,t ∂z

The radial gas phase momentum balance equation is:

∂ 1 ∂ ∂ (α ρ v , ) + (rα ρ v , v , ) + (α ρ v , v , ) ∂t g g g r r ∂r g g g r g r ∂z g g g z g r 1 ∂ μg,t ∂αg ∂ μg,t ∂αg = (r vg,r) + ( vg,r) r ∂r σαg,t ∂r ∂z σαg,t ∂z 1 ∂ μg,t ∂αg ∂ μg,t ∂αg + (r vg,r) + ( vg,z) r ∂r σαg,t ∂r ∂z σαg,t ∂r 1 ∂ μg,t ∂vg,r 1 μg,t vg,r ∂ μg,t ∂vg,r + (rαg ) − αg + (αg ) r ∂r σg,t ∂r r σg,t r ∂z σg,t ∂z 1 ∂ μg,t ∂vg,r 1 μg,t vg,r ∂ μg,t ∂vg,z + (rαg ) − αg + (αg ) r ∂r σg,t ∂r r σg,t r ∂z σg,t ∂r ∂ 2 νg,t 1 ∂ ∂vg,z − ( αgρg(k + ( (rvg,r) + ))) ∂r 3 σg,t r ∂r ∂z ∂p μl,t ∂αl ∂vl,r ∂αl ∂vl,r − αg + ( + ) ∂r σαl,t ∂r ∂r ∂z ∂z ∂ μl,t ∂αl ∂ μl,t ∂αl + vl,r ( ) + vl,z ( ) ∂r σαl,t ∂r ∂z σαl,t ∂r + + C + C αgρggr Fg,r Fg,r (F.145)

Assuming axi-symmetry the azimuthal gas phase momentum balance vanishes completely. The axial gas phase momentum balance equation is:

∂ 1 ∂ ∂ (α ρ v , ) + (rα ρ v , v , ) + (α ρ v , v , ) (F.146) ∂t g g g z r ∂r g g g r g z ∂z g g g z g z 1 ∂ μg,t ∂αg ∂ μg,t ∂αg = (r vg,z) + ( vg,z) r ∂r σαg,t ∂r ∂z σαg,t ∂z 1 ∂ μg,t ∂αg ∂ μg,t ∂αg + (r vg,r) + ( vg,z) r ∂r σαg,t ∂z ∂z σαg,t ∂z 1 ∂ μg,t ∂vg,z ∂ μg,t ∂vg,z + (rαg ) + (αg ) r ∂r σg,t ∂r ∂z σg,t ∂z 1464 Appendix F: Trondheim Bubble Column Model

1 ∂ μg,t ∂vg,r ∂ μg,t ∂vg,z + (rαg ) + (αg ) r ∂r σg,t ∂z ∂z σg,t ∂z ∂ 2 νg,t 1 ∂ ∂vg,z − ( αgρg(k + ( (rvg,r) + ))) ∂z 3 σg,t r ∂r ∂z ∂p μl,t ∂αl ∂vl,z ∂αl ∂vl,z − αg + ( + ) ∂z σαl,t ∂r ∂r ∂z ∂z ∂ μl,t ∂αl ∂ μl,t ∂αl + (vl,r ( ) + vl,z ( )) ∂r σαl,t ∂z ∂z σαl,t ∂z + + C + C αgρggz Fg,z Fg,z

A generalized transport equation for a scalar quantity in the gas phase can be formulated: ∂ 1 ∂ ∂ (α ρ φ ) + (rα ρ v , φ ) + (α ρ v , φ ) t g g g r r g g g r g z g g g z g ∂ ∂ ∂ (F.147) 1 ∂ ∂φg ∂ ∂φg = (rα Γ , ) + (α Γ , ) + S , . r ∂r g φg t ∂r ∂z g φg t ∂z φ g

F.4.1 Discretization of the Trondheim Bubble Column Model

In this section the Trondheim Bubble Column model is discretized using the finite volume technique on a staggered grid. The Trondheim Bubble Column model consists of a two-dimensional two-fluid model written in cylindrical coordinates. The governing equations are defined in appendixF. In the first subsection, the discretization of the continuity equations is outlined. The discretization of these equations follows the basic principles as outlined earlier for single phase flow. In the second subsection the generic part of the discretization procedure, being similar for all the transport equations, is outlined for a generalized variable ψ of phase k. The discretization concepts applied are the same as those used for the generic equation for single phase flow. In the third section the discretization scheme used for the volume fraction is presented. The volume fraction of the gas phase is calculated using a combination of the two continuity equations based on a scheme given by Spalding [93]. In the fourth section the discretization procedure used for the momentum equations is presented. The momentum equation was solved using the SIMPLEC method (the SIMPLE- Consistent approximation) by van Doormal and Raithby [29], and the PEA method (Partial Elimination Algorithm) by Spalding [91, 92]. The pressure-correction was calculated by use of the liquid phase continuity equation only, based on the scheme given by Grienberger [46] and the single phase algorithm of Patankar [73]. The two-fluid model discretization procedure outlined in this appendix is to a large extent based on the single phase flow algorithm implemented in the pioneering Appendix F: Trondheim Bubble Column Model 1465

Fig. F.1 Staggered Cartesian grid arrangement in the scalar cell notation. The scalar variables are located in the cell centers, while the velocity components are centered around the cell faces. The velocity components are located in the centers of their own grid cell volumes (not shown) which are staggered in one dimension compared to the scalar grid

TEACH-T code [43]. In order to simulate two phase bubble driven ﬂow in bubble columns, the original TEACH-T code was extended to enable solution of the two-ﬂuid model described previously in this appendix.

Uniform Staggered Grid Arrangement

The ﬁrst step in any discretization procedure is to deﬁne the grid to be used by dividing the computational domain into a number of grid cells and distribute the variables on the grid. In this work a uniform staggered grid arrangement was used as sketched in Fig. F.1 using the scalar cell notation. The indices of the nodes (i,j) vary between 1 and NI and 1 and NJ in the z- and r- directions, respectively. Boundary nodes are located at i = 1, NI and j = 1, NJ, thus the equations are solved in the range 2 < i < NI − 1 and 2 < j < NJ − 1.

Scalar Grid Cell Deﬁnition The uniform distance between two node points in the axial direction is given by:

Hight of reactor L Δz = = (F.148) (number of discretization points − 2) NI − 2 1466 Appendix F: Trondheim Bubble Column Model

The location of the different points in the z-direction can then calculated as follows, for i > 1: zi = zi−1 + Δz where z1 =−Δz/2 (F.149)

For i ≥ 1:

δzEP,i = zi+1 − zi, (F.150) δzPW,i = zi − zi−1 (F.151) 1 δzEW,i = (δzEP,i + δzPW,i) (F.152) 2 The uniform distance between two node points in the radial direction is given by:

Radius of reactor R Δr = = (F.153) (number of discretization points − 2) NJ − 2

The location of the node points in the r-direction can then calculated as follows, for j > 1: rj = rj−1 + Δr where r1 =−Δr/2 (F.154)

For j ≥ 1:

δrNP,j = rj+1 − rj, (F.155) δrPS,j = rj − rj−1 (F.156) 1 δrNS,j = (δrNP,j + δrPS,j) (F.157) 2 In cylindrical coordinates, a typical three dimensional volume integral of a function ψ(r, θ, z) would be of the form [8](A.8): z2 θ2 r2 ψ(r, θ, z)rdrdθ dz (F.158) z1 θ1 r1

Integration can also be performed on one of the surfaces of the coordinate system. In cylindrical coordinates there are three different kinds of surfaces, deﬁned by keeping one of the coordinates constant at the time: z2 θ2 ψ(r0, θ, z)r0 dθ dz, On the surface r = r0. (F.159) z1 θ1 z2 r2 ψ(r, θ0, z) dr dz, On the surface θ = θ0. (F.160) z1 r1 θ2 r2 ψ(r, θ, z0)rdrdθ, On the surface z = z0. (F.161) θ1 r1 Appendix F: Trondheim Bubble Column Model 1467

In two-dimensional problems, the volume and surface integrals reduce accordantly. The scalar grid cell surface areas and cell volume are deﬁned as follows:

1 A = Ae =rp × δrNS,j = rj × (δrNP,j + δrPS,j), (F.162) w 2 v 1 An =rn × δzEW,i = r + × (δzEP,i + δzPW,i), (F.163) j 1 2 v 1 As =rs × δzEW,i = r × (δzEP,i + δzPW,i) (F.164) j 2 and 1 1 ΔV = rp × δzEW,i × δrNS,j = rj × (δzEP,i +δzPW,i) × (δrNP,j +δrPS,j) (F.165) 2 2 Staggered Axial Velocity Component (w) Grid Cell Deﬁnition The location of the different points in the z-direction are calculated as follows, for i > 1: w 1 w Δz = (zi + zi− ) where z = 0 (F.166) 2 1 1 For i ≥ 1:

w = w − w, δzEP,i zi+1 zi (F.167) w = w − w , δzPW,i zi zi−1 (F.168) w 1 w w δz , = (δz , + δz , ) (F.169) EW i 2 EP i PW i The radial position of the grid for the axial velocity component (w) coincides with that of the scalar grid cell. The staggered axial velocity component grid cell surface areas and cell volume are deﬁned by:

1 A = Ae =rp × δrNS,j = rp × (δrNP,j + δrPS,j), (F.170) w 2 w v 1 w w An =rn × δz , = r + × (δz , + δz , ), (F.171) EW i j 1 2 EP i PW i w v 1 w w As =rs × δz , = r × (δz , + δz , ) (F.172) EW i j 2 EP i PW i and

w 1 w w 1 ΔV = rp × δz , × δrNS,j = rj × (δz , +δz , ) × (δrNP,j +δrPS,j) (F.173) EW i 2 EP i PW i 2 Staggered Radial Velocity Component (v) Grid Cell Deﬁnition 1468 Appendix F: Trondheim Bubble Column Model

The axial position of the grid cell for the radial velocity component (v) coincides with that of the scalar grid cell deﬁned above. The location of the different points in the r-direction can then calculated as follows, for j > 1:

v 1 r = (rj + rj− ) where r = 0 (F.174) j 2 1 1 For j ≥ 1:

v = v − v, δrNP,j rj+1 rj (F.175) v = v − v δrPS,j rj rj−1 (F.176) v 1 v v δr , = (δr , + δr , ) (F.177) NS j 2 NP j PS j For the radial velocity component (v) grid cell, the surface areas and cell volume are deﬁned by:

v v v 1 v v A = Ae =r × δr , = r × (δr , + δr , ), (F.178) w p NS j j 2 NP j PS j 1 An =rn × δzEW,i = rj+ × (δzEP,i + δzPW,i), (F.179) 1 2 1 As =rs × δzEW,i = rj × (δzEP,i + δzPW,i) (F.180) 2 and

v v v 1 1 v v ΔV = r ×δzEW,i ×δr , = r × (δzEP,i +δzPW,i)× (δr , +δr , ) (F.181) p NS j j 2 2 NP j PS j To discretize the governing equations using the finite volume method, the differential equations are written on the integral form by integrating over a grid cell volume. The volume integrals of the convective and diffusive flux terms are transformed into surface integrals by use of the Gauss theorem. It is further assumed that the source term is constant throughout the grid cell volume, and the fluxes are uniform over the cell faces. The resulting semi-discrete equation is then integrated over a time step. After the integrations all the terms are divided by the time step length Δt.The transient term is approximated by the midpoint rule, hence it is considered an average value representative for the whole grid cell volume. The order of the time and volume integration is interchangeable. The solution of the discretized forms of the governing equations can be based on general geometrical dimensions such as δzEP, δzPW , δrNP and δrPS which are valid for non-uniform grids as well as for uniform grids. However, the implementation of the discretized equations on uniform grids into a computer code can be simplified adopting a grid dependent notation which is faster to compute. Appendix F: Trondheim Bubble Column Model 1469

Fig. F.2 A sketch of a scalar Cartesian grid cell showing the distribution of the variables in the grid and the conﬁguration of the staggered velocity grids. In cylindrical coordinates equivalent grid cells can be deﬁned

F.4.2 The Continuity Equation

The phasic continuity equations are deﬁned by:

∂ 1 ∂ ∂ 1 ∂ ∂α ∂ ∂α (α ρ ) + (rα ρ v ) + (α ρ w ) = (r Γ k ) + (Γ k ) + S ∂t k k r ∂r k k k ∂z k k k r ∂r ∂r ∂z ∂z

The continuity equation is re-written on the integral form, integrated in time and over a grid cell volume in the non-staggered grid for the scalar variables sketched in Fig. F.2. The transient terms are discretized with the implicit Euler scheme.

o ((αkρk)P − (αkρk) )ΔV P + Δz((rα ρ v ) − (rα ρ v ) ) Δt k k k n k k k s ∂α ∂α +r Δr((α ρ w ) − (α ρ w ) ) = Δz((rΓ k ) − (rΓ k ) ) (F.182) p k k k e k k k w ∂r n ∂r s ∂α ∂α +r Δr((Γ k ) − (Γ k ) ) + SΔV p ∂z e ∂z w

The gradient terms were then approximated by the central difference scheme. For simplicity, the variables δrPN ,δrSP,δzPE, δzWP, A, C, D and F are commonly introduced. The resulting equation can thus be written as:

( ) Δ ( )o Δ αkρk P V αkρk P V + C − C + C − C = + D (α , − α , ) Δt n s e w Δt n k N k P − Ds(αk,P − αk,S) + De(αk,E − αk,P) − Dw(αk,P − αk,W ) + SΔV (F.183) 1470 Appendix F: Trondheim Bubble Column Model

The novel variables are deﬁned as follows:

Cn = An Fn = An(αkρkvk)n, C = A F = A (α ρ v ) , s s s s k k k s (F.184) Ce = Ae Fe = Ae(αkρkwk)e, Cw = Aw Fw = Aw(αkρkwk)w

The C-variables represent the convective ﬂuxes through the grid cell surfaces and are normally approximated by the central difference scheme:

Γ = n , Dn An δr ΓNP = s , Ds As δr ΓPS (F.185) = e , De Ae δz ΓEP D = A w w w δzPW

The D-variables represent the generalized diffusion conductance and are related to the diffusive fluxes through the grid cell surfaces. In order to approximate these terms the gradients of the transported properties and the diffusion coefficients Γ are required. The property gradients are normally approximated by the central difference scheme. In a uniform grid the diffusion coefficients are obtained by linear interpolation from the node values (i.e., using arithmetic mean values):

1 Γn = (ΓP + ΓN ), 2 1 Γs = (ΓP + ΓS), 2 (F.186) 1 Γe = (ΓP + ΓE), 2 1 Γ = (ΓP + ΓW ) w 2 We can re-organize (F.183)to:

Cn − Cs + Ce − Cw = transient + mC1 + mC2 (F.187) where (α ρ )o ΔV ( ) Δ = k k P − αk ρk P V transient Δt Δt mC1 = Dnαk,N + Dsαk,S + Deαk,E + Dwαk,W (F.188) mC2 =−(Dn + Ds + De + Dw) αk,P

The LHS of (F.187) can be recognized as part of the aP-factor of the discretized equations for all the other variables. To keep the aP-coefﬁcient always positive during the iterative process, the LHS terms can be substituted by the RHS terms. Appendix F: Trondheim Bubble Column Model 1471

F.4.3 The Generalized Equation

∂ 1 ∂ ∂ (α ρ ψ ) + (rα ρ v ψ ) + (α ρ w ψ ) ∂t k k k r ∂r k k k k ∂z k k k k 1 ∂ μk,t ∂ ∂ μk,t ∂ = (rαkρk ) + (αkρk ) r ∂r σαk,t ∂r ∂z σαk,t ∂z

The finite volume discretization of the generalized multi-fluid equation coincides with the corresponding equation for single phase flows as outlined in the preceding subsections.

The Transient term

The transient term is discretized using the implicit Euler scheme. ∂ (α ρ ψ ) dt dz r dr = (α ρ ψ ) − (α ρ ψ )o ΔV ∂t k k k k k k P k k k P ΔV Δt ΔV ΔV = (α ρ ψ ) Δt − (α ρ ψ )o Δt Δt k k k P Δt k k k P ΔV = (α ρ ψ ) Δt − ao ψo Δt (F.189) Δt k k k P P k,P where ΔV ao = (α ρ )o . (F.190) P Δt k k P

The Convection terms

Radial direction: 1 ∂ (rα ρ v ψ )dzrdrdt= Δz[(rα ρ v ψ ) − (rα ρ v ψ ) ]Δt r ∂r k k k k k k k k n k k k k s Δt ΔV

= An(αkρkvkψk)nΔt − As(rαkρkvkψk)sΔt = AnFnψnΔt − AsFsψsΔt = CnψnΔt − CsψsΔt (F.191) where F = (α ρ v ) n k k k n (F.192) Fs = (αkρkvk)s 1472 Appendix F: Trondheim Bubble Column Model

Axial direction: ∂ (α ρ w ψ )dz r dr dt = rΔr[(α ρ w ψ ) − (α ρ w ψ ) ]Δt ∂z k k k k k k k k e k k k k w Δt ΔV

= Ae(αkρkwkψk)eΔt − Aw(αkρkwkψk)wΔt = AeFeψeΔt − AwFwψwΔt = CeψeΔt − CwψwΔt (F.193) where F = (α ρ w ) e k k k e . (F.194) Fw = (αkρkwk)w

The Diffusion terms

The the diffusion terms are discretized using the central-difference scheme.

Radial direction: 1 μ , ∂ ( k eff ∂ψk ) ∂r rαk ∂r rdr dz dt r σψ Δt ΔV μ , μ , = Δ [( k eff ∂ψk ) − ( k eff ∂ψk ) ]Δ z rαk ∂r n rαk ∂r s t σψ σψ (F.195) ψk,N − ψk,P ψk,P − ψk,S = AnΓn Δt − AsΓs Δt δrPN δrSP = Dn(ψk,N − ψk,P)Δt − DS(ψk,P − ψk,S)Δt where μk,eff μk,eff Γ = (α ) and Γ = (α ) n k σ n s k σ s ψ ψ (F.196) AnΓn AsΓs Dn = and Ds = δrPN δrSP

Axial direction: ∂ μk,eff ∂ψk (αk )dz rdr dt ∂z σψ ∂z Δt ΔV μ , μ , = Δ k eff ∂ψk Δ − Δ k eff ∂ψk Δ rp r αk ∂z t rp r αk ∂z t (F.197) σψ e σψ w ψk,E − ψk,P ψk,P − ψk,W = AeΓe Δt − AwΓw Δt δzEP δzWP = De(ψk,E − ψk,P)Δt − Dw(ψk,P − ψk,W )Δt Appendix F: Trondheim Bubble Column Model 1473 where μk,eff μk,eff Γ = (α ) and Γ = (α ) w k σ w e k σ e ψ ψ (F.198) AeΓn AwΓs De = and Dw = . δzEP δzWP

The Source terms

The source terms are approximated by the midpoint rule, in which S is considered an average value representative for the whole grid cell volume.

Upwind Discretized Form of the Generalized Equation

After the given approximations of the terms have been substituted into the generic equation, and dividing all the terms by Δt, the balance equation yields: Δ V o o (α ρ ) ψ , − a ψ + C ψ , − C ψ , + C ψ , − C ψ , Δt k k P k P P k,P n k n s k s e k e w k w = Dn(ψk,N − ψk,P) − Ds(ψk,P − ψk,S) + De(ψk,E − ψk,P) − Dw(ψk,P − ψk,W ) + SΔV (F.199)

By use of the upwind scheme for the convective terms, the generalized transport equation becomes: ΔV (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV ψ , Δt k k P N S E W n s e w P 1 k P = + + + + Δ + o o aN ψk,N aSψk,S aEψk,E aW ψk,W SC,1 V aPψk,P (F.200)

The discretized equation can then be written on the standard algebraic form:

aPψP = aN ψN + aSψS + aEψE + aW ψW + b (F.201) in which the coefﬁcients are deﬁned as follows:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] aE = De + max[−Ce, 0] aW = Dw + max[Cw, 0] = + 0 0 b SC,1 aPψk,P ΔV a = (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV P Δt k k P N S E W n s e w P 1 (F.202) 1474 Appendix F: Trondheim Bubble Column Model

To avoid negative coefficients, the relation for the coefficient aP can be modified using the continuity Eq. (F.187). The result is:

( )0 Δ αkρk P V a = + a + a + a + a + mC + mC − S , ΔV (F.203) P Δt N S E W 1 2 P 1

The negative mC2-term must then be moved to the RHS of the discretized transport equation and included as part of the b-term (multiplied with ψν , the value at the ∗ ∗ k previous iteration). The alternative aP and b coefﬁcients are deﬁned by:

∗ =− ν + + 0 0 b mC2ψk,P SC,1 aPψk,P ∗ ΔV (F.204) a = (α ρ )0 + a + a + a + a + mC − S , ΔV. P Δt k k P N S E W 1 P 1

F.4.4 The Liquid Phase Radial Momentum Balance

The radial momentum balance is given in (F.135).

∂ 1 ∂ ∂ (αlρlvl) + (rαlρlvlvl) + (αlρlwlvl) ∂t r ∂r ∂z vl ∂ μl,t ∂αl ∂ μl,t ∂αl = r + vl r ∂r σα , ∂r ∂z σα , ∂z l t l t 1 ∂ ∂vl 1 vl ∂ ∂vl + rαlμl,eff − αlμl,eff + αlμl,eff r ∂r ∂r r r ∂z ∂z v v 1 ∂ ∂ l 1 l ∂ ∂wl (F.205) + rαlμl,eff − αlμl,eff + αlμl,eff r ∂r ∂r r r ∂z ∂r ∂ 2 1 ∂ ∂wl − αlρl k + νl,eff (rvl) + ∂r 3 r ∂r ∂z μ , 1 ∂p + l t ∂αl ∂ ( v ) + ∂wl − ∂r ∂r r l ∂z αl σαl,t r ∂r + + C + C αlρlgr Fl,r Fl,r

The transient term

t+Δt ∂ (α ρ v )dt dV = (α ρ v ) − (α ρ v )o ΔV (F.206) ∂t l l l l l l P l l l P ΔV t Appendix F: Trondheim Bubble Column Model 1475

Fig. F.3 A sketch of the staggered v-grid cell in Cartesian coordinates. The figure shows the distribution of the variables in this grid, and the configuration of the staggered w-velocity grid and the non-staggered scalar grid. In cylindrical coordinates equivalent grid cells can be defined

To approximate the scalar grid cell variables α and ρ at the staggered velocity grid cell nodes, arithmetic interpolation is needed: 1 (αlρlvl)P = (αl,Pρl,P) + (αl,Sρl,S) vl,P (F.207) 2 o 1 o o o (α ρ v ) = (α , ρ , ) + (α , ρ , ) v , (F.208) l l l P 2 l P l P l S l S l P

The convection terms

The locations of the node points in the staggered grid for the v-velocity are shown in Fig. F.3. Radial direction: 1 ∂ (rα ρ v v ) rdrdzdt= Δz (rα ρ v ) v , − (rα ρ v ) v , Δt r ∂r l l l l l l l n l n l l l s l s Δt ΔV

= Cnvl,nΔt − Csvl,sΔt (F.209)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used: 1476 Appendix F: Trondheim Bubble Column Model

1 Cn = An [FN + FP] 2 1 Cs = As [FP + FS] 2 1 FN = (α ρ v )N = [α , ρ , + α , ρ , ]v , (F.210) l l l 2 l N l N l P l P l N 1 FP = (α ρ v )P = [α , ρ , + α , ρ , ]v , l l l 2 l P l P l S l S l P 1 FS = (α ρ v )S = [α , ρ , + α , ρ , ]v , l l l 2 l S l S l SS l SS l S Axial direction: ∂ (α ρ w v )dzrdrdt=[(r Δrα ρ w ) v , − (r Δrα ρ w ) v , ]Δt ∂z l l l l l l l e l e l l l w l w Δt ΔV

= Cevl,eΔt − Cwvl,wΔt (F.211)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

1 Ce = Ae [FE + FSE] 2 1 C = A [FW + FSW ] w w 2 1 FE = (αlρlv)E = [αl,Eρl,E + αl,Pρl,P]vl,E 2 (F.212) 1 FSE = (α ρ v)SE = [α , ρ , + α , ρ , ]v , l l 2 l S l S l SE l SE l SE 1 FW = (α ρ v)W = [α , ρ , + α , ρ , ]v , l l 2 l P l P l W l W l P 1 FSW = (α ρ v)SW = [α , ρ , + α , ρ , ]v , . l l 2 l S l S l SW l SW l S

The Diffusion terms

The location of the node points in the staggered grid for the v-velocity are shown in Fig. F.3.

Radial direction: 1 ∂ ∂vl ∂vl ∂vl (rα μ , ) rdr dz dt = Δz[(rα μ , ) − (rα μ , ) ]Δt r ∂r l l eff ∂r l l eff ∂r n l l eff ∂r s Δt ΔV Appendix F: Trondheim Bubble Column Model 1477 vl,N − vl,P vl,P − vl,S = AnΓn Δt − AsΓs Δt δrNP δrPS = Dn(vl,N − vl,P)Δt − Ds(vl,P − vl,S)Δt (F.213)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

AnΓn Dn = δrNP AsΓs Ds = (F.214) δrPS Γn = (αlμl,eff )n = αl,Pμl,eff ,P Γs = (αlμl,eff )s = αl,Sμl,eff ,S

Axial direction: ∂ ∂v ∂v ∂v (α μ , )dz rdr dt =[r Δr(α μ , ) − r Δr(α μ , ) ]Δt ∂z l l eff ∂z p l l eff ∂z e p l l eff ∂z w Δt ΔV vl,E − vl,P vl,P − vl,W = AeΓe Δt − AwΓw Δt δzEP δzPW = De(vl,E − vl,P)Δt − Dw(vl,P − vl,W )Δt (F.215)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

AeΓe De = δzEP AwΓw Dw = δzPW Γe = (αlμl,eff )e 1 = [α , μ , , + α , μ , , + α , μ , , + α , μ , , ] 4 l P l eff P l E l eff E l S l eff S l SE l eff SE Γw = (αlμl,eff )w 1 = [α , μ , , + α , μ , , + α , μ , , + α , μ , , ]. (F.216) 4 l P l eff P l W l eff W l S l eff S l SW l eff SW

The source terms

The the source terms are approximated by the midpoint rule, in which S is considered an average value representative for the whole grid cell volume. The derivatives are 1478 Appendix F: Trondheim Bubble Column Model represented by an abbreviated Taylor series expansion, usually a central difference expansion of second order is employed. Term 1 on the RHS of the momentum equation: vl ∂ μl,t ∂αl vl,PΔVΔt ∂αl ∂αl r dVdt = rμ , − rμ , r r r v 1 ( + ) l t r l t r ∂ σαl,t ∂ rP δrNP δrPS ∂ n ∂ s Δt ΔV 2 (F.217) This term is implemented through the source term SC as: vl,PΔVΔt ∂αl ∂αl S , = r μ , − r μ , (F.218) C 1 v 1 ( + ) P l P ∂r S l S ∂r rP 2 δrNP δrPS n s

To approximate derivatives of scalar grid cell variables and scalar grid cell variables at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

1 1 ( ) − ( ) (αl,N + αl,P) − (αl,P + αl,S) ∂αl = αl N αl P = 2 2 ∂r δr δr n NP NP (F.219) 1 1 ( ) − ( ) (αl,P + αl,S) − (αl,S + αl,SS) ∂αl = αl P αl S = 2 2 ∂r s δrPS δrPS

Term 2 on the RHS of the momentum equation: ∂ μ , ∂α v , ΔVΔt μ , ∂α μ , ∂α v l t l dV dt = l P l t l − l t l l 1 ( + ) ∂z σαl,t ∂z δzEP δzPW σαl,t ∂z e σαl,t ∂z w Δt ΔV 2 (F.220) This term is implemented through the source term SC as: vl,PΔVΔt ∂αl ∂αl S , = (μ ) − (μ ) (F.221) C 2 1 ( + ) l e ∂z l w ∂z 2 δzEP δzPW e w

To approximate derivatives of scalar grid cell variables and scalar grid cell variables at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

1( + ) − 1( + ) ∂α (α ) − (α ) αl,E αl,SE αl,E αl,SE l = l E l P = 2 2 (F.222) ∂z e δzPE δzPE Appendix F: Trondheim Bubble Column Model 1479

1 1 ( ) − ( ) (αl,P + αl,S) − (αl,W + αl,SW ) ∂αl αl P αl W 2 2 ( )w = = ∂z δzWP δzWP 1 (μ )e = (μ , + μ , + μ , + μ , ) l 4 l P l E l S l SE 1 (μ ) = (μ , + μ , + μ , + μ , ) l w 4 l P l W l S l SW The 3rd and 6th terms on the RHS of the momentum equation are identical and are approximated as follows:

1 v ∂ ( ∂ l ) ∂r rαlμl,eff ∂r dV dt ΔtΔV r Δ Δ V t ∂v ∂v (F.223) = [(rα μ , l ) − (rα μ , l ) ] 1 l l eff ∂r n l l eff ∂r s rv (δrv + δrv ) P 2 NP PS

This term is implemented through the source term SC as: ΔVΔt vl,N − vl,P vl,P − vl,S S , = r α , μ , ( ) − r α , μ , ( ) C 3 v 1 ( v + v ) P l P l P δrv S l S l S δrv rP 2 δrNP δrPS NP PS (F.224) The scalar grid cell variables at the staggered velocity grid cell surface points coincide with center nodes, hence no interpolation is needed. The 4th and 7th terms on the RHS of the momentum equation are identical. They are approximated as follows: 1 vl vl,P − , =− ( , ) Δ Δ 2 αlμl eff dVdt 2 αlμl eff P 2v V t Δ Δ r r r t V P (F.225) 1 vl,P =−2 × (α , μ , , + α , , μ , ) ΔVΔt 2 l P l eff P l eff S l S 2,v rP

This term is implemented through the source term Sp as:

1 1 SP, =−2 × (α , μ , , + α , μ , , ) ΔVΔt (F.226) 1 2 l P l eff P l S l eff S 2,v rP

To approximate scalar grid cell variables at the staggered velocity grid cell surface points, arithmetic interpolation is used:

1 (μ )P = (μ , + μ , ) (F.227) l 2 l P l S 1480 Appendix F: Trondheim Bubble Column Model

The 5th term on the RHS of the momentum equation:

v ∂ ( ∂ l ) ∂z αlμl,eff ∂z dV dt ΔtΔV Δ Δ V t ∂v ∂v (F.228) = [(α μ , l ) − (α μ , l ) ] 1 l l eff ∂z e l l eff ∂z w (δzEP + δzPW ) 2

This term is implemented through the source term SC as:

ΔVΔt ∂vl ∂vl S , = [(α μ ) ( ) − (α μ ) ( ) ] (F.229) C 4 1 ( + ) l l e ∂z e l l w ∂z w 2 δzEP δzPW

To velocity derivatives are approximated by use of the central difference scheme:

v v − v ( ∂ l ) = lE lP ∂z e δzPE v v , − v , (F.230) ( ∂ l ) = l P l W ∂z w δzWP

The 8th term on the RHS of the momentum equation: ∂ ( ∂wl ) ∂z αlμl,eff ∂r dV dt ΔtΔV ΔVΔt = ( ∂wl ) − ( ∂wl ) (F.231) αlμl,eff ∂r e αlμl,eff ∂r w 1 v (δzPW + δr ) 2 EP

The 8th term is implemented through the source term SC as: ΔVΔt (wl,E − wl,SE) (wl,P − wl,S) S , = Γ − Γ (F.232) C 5 1 ( + v ) e 1 ( v + v ) w 1 ( v + v ) 2 δzPW δrEP 2 δrNP δrPS 2 δrNP δrPS

Term 9A on the RHS of the momentum equation: ∂ 2 2 ΔVΔt − ( α ρ k) dV dt =− [(α ρ k) −(α ρ k) ] (F.233) ∂r l l × 1 ( v + v ) l l n l l s 3 3 δrNP δrPS Δt ΔV 2

This term is implemented through the source term SC as:

2 ΔVΔt S , = [α , ρ , k − α , ρ , k ] (F.234) C 6 1 ( v + v ) l P l P P l S l S S 3 2 δrNP δrPS

The scalar grid cell variables at the staggered velocity grid cell surface points coincide with center nodes, hence no interpolation is needed. Appendix F: Trondheim Bubble Column Model 1481

Term 9B on the RHS of the momentum equation: ∂ 2 1 ∂(rvl) − ( α ρ ν , ) dV dt ∂r 3 l l l eff r ∂r Δt ΔV 2 ΔVΔt αlμl,eff ∂(rv ) αlμl,eff ∂rv =− l − l (F.235) × 1 ( v + v ) r ∂r r ∂r 3 2 δrNP δrPS n s

The 9B-th term is implemented through the source term SC as: 2 ΔVΔt αl,Pμl,eff ,P ∂(rvl) αl,W μl,eff ,W ∂(rvl) S , =− | − | C 7 × 1 ( v + v ) r ∂r n r ∂r s 3 2 δrNP δrPS P S (F.236) where v v ( v ) (rv ) − (rv ) r vl,N − r vl,P ∂ r l | = l N l P = N P ∂r n δrv δrv NP v NPv (F.237) ( v ) (rv ) − (rv ) r vl,P − r vl,S ∂ r l | = l P l S = P S ∂r s v v δrPS δrPS

Term 9C on the RHS of the momentum equation: ∂ 2 ∂wl − ( α ρ ν , ) dV dt ∂r 3 l l l eff ∂z Δt ΔV Δ Δ (F.238) 2 V t ∂w ∂w =− (α μ , l ) − (α μ , l ) 1 l l eff ∂z n l l eff ∂z s 3 × (δrv + δrv ) 2 NP PS

The term 9C is implemented through the source term SC as: 2 ΔVΔt (wl,E − wl,P) SC,8 =− αl,Pμl,eff ,P × 1( v + v ) 1( + ) 3 δrNP δrPS δzEP δzPW 2 2 (F.239) 2 ΔVΔt (wl,P − wl,W ) + [αl,W μl,eff ,W ] 1 v v 1 3 × (δr + δr ) (δzEP + δzPW ) 2 NP PS 2 Term 10 on the RHS of the momentum equation: μl,t ∂αl 1 ∂ ∂wl μl,t ∂αl 1 ∂ ∂wl ( (rvl) + ) dV dt = ( )P[ (rvl) + ]PΔVΔt σα,t ∂r r ∂r ∂z σα,t ∂r r ∂r ∂z Δt ΔV (F.240) This term is implemented through the source term SC as: 1482 Appendix F: Trondheim Bubble Column Model

μ , , μ , , 1 ( l t P + l t S ) ( v − v ) 2 σα,t σα,t ∂αl 1 rP n rS s ∂wl SC,9 = ( )P[ v + ( )P]ΔVΔt (F.241) σα,t ∂r rP δrPS ∂z

To approximate derivatives of velocity variables and scalar grid cell variables at the staggered velocity grid cell node points, arithmetic interpolation is frequently used:

(α ) − (α ) α , − α , ( ∂αl ) = l n l s = l P l S ∂r P 1 1 (δrv + δrv ) (δrv + δrv ) 2 NP PS 2 NP PS 1 (μ )P = (μ , + μ , ) l 2 l P l S (F.242) 1( + ) − 1( + ) (w ) − (w ) wE wSE wP wS ( ∂wl ) = l e l w = 2 2 . ∂z P 1 1 (δzPW + δzEP) (δzPW + δzEP) 2 2

Pressure force:

∂p 1 (PP − PS) − α dV dt =− (α , + α , ) ΔVΔt (F.243) l ∂r l P l S 1 ( v + v ) 2 δrNP δrPS Δt ΔV 2

This term is implemented through the source term SC as:

(PP − PS) SC,10 =−(αl)P ΔVΔt (F.244) δrPS

To approximate scalar grid cell variables at the staggered velocity grid cell node points, arithmetic interpolation is frequently used:

1 (α )P = (α , + α , ). (F.245) l 2 l P l S

Gravity force:

There is no gravity force in radial direction.

Added Mass force: ∂vl ∂vl ∂vg ∂vg − α α ρ fv[w + v − (w + v )] dV dt l g l l ∂z l ∂r g ∂z g ∂r Δt ΔV (F.246) v v v v =− ( ) ( ) ( ) [ ∂ l + v ∂ l − ( ∂ g + v ∂ g )] Δ Δ fv αl P αg P ρl P wl ∂z l ∂r wg ∂z g ∂r P V t Appendix F: Trondheim Bubble Column Model 1483

This term is implemented through the source term SC as:

1 1 1 SC, =−fv (α , + α , ) (α ,P + α ,S) (ρ , + ρ , ) 11 2 l P l S 2 g g 2 l P l S v v v v (F.247) ×[ ∂ l + v ∂ l − ( ∂ g + v ∂ g )] Δ Δ wl ∂z l ∂r wg ∂z g ∂r P V t

To approximate derivatives of velocity variables and scalar grid cell variables at the staggered velocity grid cell node points, arithmetic interpolation is frequently used:

1 (α )P = (α , + α , ) l 2 l P l S 1 (w )P = (v , + v , + v , + v , ) l 4 l P l S l E l SE 1 1 (v , + v , ) − (v , + v , ) v v , − v , l P l N l P l S ( ∂ l ) = l n l s = 2 2 (F.248) ∂r P 1 1 (δrv + δrv ) (δrv + δrv ) 2 NP PS 2 NP PS 1 1 (v , + v , ) − (v , + v , ) v v , − v , l P l E l P l W ( ∂ l ) = l e l w = 2 2 . ∂z P 1 1 (δzPW + δzEP) (δzPW + δzEP) 2 2

Transversal force: − ( − ) ∂wl αlαgρlCL wl wg ∂r dV dt ΔtΔt (F.249) =−( ) ( ) ( ) Δ ( ∂wl ) Δ Δ αl P αg P ρl PCL wP ∂r P V t

This term is implemented through the source term SC as:

1 1 1 ∂wl SC, =− (α , + α , ) (α ,P + α ,S) (ρ , + ρ , )CLΔwP( )PΔVΔt 12 2 l P l S 2 g g 2 l P l S ∂r (F.250) To approximate derivatives of velocity variables and scalar grid cell variables at the staggered velocity grid cell node points, arithmetic interpolation is used:

1 (α )P = (α , + α , ) l 2 l P l S ΔwP = (wl − wg)P 1 = (wl,P − wg,P + wl,E − wg,E + wl,S − wg,S + wl,SE − wg,SE) 4 1 1 − (w , + w , ) − (w , + w , ) ∂wl = wl,n wl,s = 2 l P l E 2 l S l SE ∂r 1 ( v + v ) 1 ( v + v ) P 2 δrNP δrPS 2 δrNP δrPS 1484 Appendix F: Trondheim Bubble Column Model

= − 3 νl,t CDCτ . CL fL τ (F.251) 4 νl (1 + L )Re tp p

Steady drag force:

3 CD αlαgρl |vl − vg|(vl − vg) dV dt ≈ (K)P(vl,P − vg,P)ΔVΔt (F.252) 4 dS Δt ΔV

To approximate scalar grid cell variables at the staggered velocity grid cell node points, arithmetic interpolation is used:

1 (K)P = (KP + KS) 2 3 CD 3 CD KP =[ αlαgρl |vl − vg|]P =[αlαgCW ]P = αl,Pαg,Pρl,P( )P|vl − vg|P 4 dS 4 dS (F.253)

The relative speed at the center node in the staggered v-grid cell volume is approximated by: 1 1 2 |v − v |P ≈ (w , + w , ) − (w ,P + w ,S) l g 2 l P l S 2 g g / (F.254) 1 1 2 1 2 + (v , + v , ) − (v ,P + v ,W ) 2 l P l W 2 g g

To deal with the strong coupling between the phasic momentum equations, the partial elimination algorithm (PEA)-method proposed by Spalding [91, 92] is frequently used. To outline the PEA-method, the discretized momentum equations for the gas phase at the e location in the staggered grid for the w variable (i.e., between the P and E grid points in the scalar grid) is re-written pulling the drag force out of the source term: ag,nvg,n = ag,nbvg,nb + K(vl,n − vg,n)ΔV + Sg (F.255) nb

One part of the drag term is put on the LHS of the equation, thus we may write: vg,n(ag,n + KΔV) = ag,nbvg,nb + KΔVvl,n + Sg (F.256) nb

The vg,n is then given by: Appendix F: Trondheim Bubble Column Model 1485 + Δ v + nb ag,nbwg,nb K V l,n Sg vg,n = (F.257) ag,n + KΔV

In the same way we can write for the liquid velocity (shown shortly): v + Δ v + nb al,nb l,nb K V g,n Sl vl,n = (F.258) al,n + KΔV

Combining these two equations to take out vl,n in the gas calculations and vg,n in the liquid calculations we get: v + Δ v + nb al,nb l,nb K V g,n Sl a , v , + KΔV + Sg nb g nb g nb al,n+KΔV vg,n = (F.259) ag,n + KΔV and ag,nbvg,nb+KΔVvl,n+Sg a , v , + KΔV nb + S nb l nb l nb ag,n+KΔV l vl,n = (F.260) al,n + KΔV

In the equation for phase k all the vk,n terms can be regrouped on the left, hence for k = g, l we get:

(KΔV)2 vg,n(ag,n + KΔV − ) = ag,nbvg,nb al,n + KΔV nb v + nb al,nb l,nb Sl + KΔV + Sg (F.261) al,n + KΔV

(KΔV)2 vl,n(al,n + KΔV − ) = al,nbvl,nb ag,n + KΔV nb v + nb ag,nb g,nb Sg + KΔV + Sl (F.262) ag,n + KΔV v + ( Δ )2 This means that the modified coupling terms KΔV ak,i k,i Sk and KΔV − K V ak,n+KΔV ak,n+KΔV must be calculated separately for the momentum balance components in each phase, then transferred and employed in the corresponding equation for the other phase. In particular, after the coupling terms have been received from the phase where they are calculated, the modified terms are added to the source Se and the coefficient ae terms in the other phase, respectively. In this case, Sk denotes the sum of all source terms except the drag force. 1486 Appendix F: Trondheim Bubble Column Model

Algebraic discretization equation

Finally, after dividing all the terms by Δt, the discretized equation can be written on the standard algebraic form:

v = v + v + v + v + aP l,P aN l,N aS l,S aE l,E aW l,W bvl (F.263)

The coefﬁcients are deﬁned by:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] aE = De + max[−Ce, 0] a = D + max[C , 0] W w w = + 0 v0 b SC,m aP l,P m ΔV a = (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV P Δt l l P N S E W n s e w P q q (F.264)

To avoid negative coefficients, the relation for the coefficient aP can be modified using the continuity Eq. (F.187). The negative mC2-term is then moved to the RHS of the ∗ discretized transport equation and included as part of the b-term. The alternative aP and b∗ coefficients are defined by: ∗ =− vν + + 0 v0 b mC2 , m SC,m,l a , l P P l P ∗ ΔV (F.265) a = (α ρ )0 + a + a + a + a + mC − S , , ΔV. P Δt l l P N S E W 1 q P q l

F.4.5 Liquid Phase Axial Momentum Balance

The axial component of the momentum balance is given in (F.137).

∂ 1 ∂ ∂ (α ρ w ) + (rα ρ v w ) + (α ρ w w ) ∂t l l l r ∂r l l l l ∂z l l l l 1 ∂ μl,t ∂αl ∂ μl,t ∂αl = wl (r ) + wl ( ) r ∂r σαl,t ∂r ∂z σαl,t ∂z μl,t ∂αl 1 ∂ ∂wl + ( (rvl) + ) σαl,t ∂z r ∂r ∂z 1 ∂ ∂wl ∂ ∂wl + (rα μ , ) + (α μ , ) r ∂r l l eff ∂r ∂z l l eff ∂z Appendix F: Trondheim Bubble Column Model 1487

Fig. F.4 A staggered Cartesian w-grid cell, the distribution of the variables in this grid and the conﬁguration of the staggered v-velocity grid cell and the non-staggered scalar grid. In cylindrical coordinates equivalent grid cells can be deﬁned

1 ∂ ∂vl ∂ ∂wl + (rα μ , ) + (α μ , ) r ∂r l l eff ∂z ∂z l l eff ∂z ∂ 2 1 ∂ ∂wl − ( α ρ (k + ν , ( (rv ) + ))) ∂z 3 l l l eff r ∂r l ∂z ∂p − α + α ρ g + FC + FC (F.266) l ∂z l l z l,z l,z

In the FVM, the integral form of the momentum equation is used. The differential equation is thus integrated in time and over a grid cell volume in the staggered grid for the w-velocity sketched in Fig. F.4.

The Transient term

t+Δt ∂ (α ρ w ) dt dV = (α ρ w ) − (α ρ w )o ΔV (F.267) ∂t l l l l l l P l l l P ΔV t

To approximate the scalar grid cell variables α and ρ at the staggered velocity grid cell nodes, arithmetic interpolation is needed:

1 (α ρ w )P = [(α , ρ , ) + (α , ρ , )]w , (F.268) l l l 2 l P l P l W l W l P 1488 Appendix F: Trondheim Bubble Column Model

o 1 o o o (α ρ w ) = [(α , ρ , ) + (α , ρ , ) ]w , . (F.269) l l l P 2 l P l P l W l W l P

The convection term

The locations of the node points in the staggered grid for the w-velocity component are shown in Fig. F.4. Radial direction: 1 ∂ (rα ρ v w ) rdrdzdt= Δz[(rα ρ v w ) − (rα ρ v w ) ]Δt r ∂r l l l l l l l l n l l l l s Δt ΔV

= Cnvl,nΔt − Csvl,sΔt (F.270)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

1 Cn = AnFn = An [FN + FNW ] 2 1 Cs = AnFn = As [FS + FSW ] 2 1 FN = (αlρlvl)N = [αl,N ρl,N + αl,Pρl,P]vl,N 2 (F.271) 1 FNW = (α ρ v )NW = [α , ρ , + α , ρ , ]v , l l l 2 l W l W l NW l NW l NW 1 FS = (α ρ v )S = [α , ρ , + α , ρ , ]v , l l l 2 l S l S l P l P l P 1 FSW = (α ρ v )SW = [α , ρ , + α , ρ , ]v , l l l 2 l W l W l SW l SW l W Axial direction: ∂ (α ρ w w )dzrdrdt=[r Δr(α ρ w w ) − r Δr(α ρ w w ) ]Δt ∂z l l l l p l l l l e p l l l l w Δt ΔV

= Cevl,eΔt − Cwvl,wΔt (F.272)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

1 Ce = AeFe = Ae [FE + FP] (F.273) 2 1 C = A F = A [FP + FW ] w w w w 2 Appendix F: Trondheim Bubble Column Model 1489

1 FE = (α ρ w )E = [α , ρ , + α , ρ , ]w , l l l 2 l P l P l E l E l E 1 FP = (α ρ w )P = [α , ρ , + α , ρ , ]w , l l l 2 l P l P l W l W l P 1 FW = (α ρ w )W = [α , ρ , + α , ρ , ]w , . l l l 2 l W l W l WW l WW l W

The diffusion terms

The locations of the node points in the staggered grid for the w-velocity are shown in Fig. F.4. Radial direction: 1 ∂ ∂wl (rα μ , ) rdr dz dt r ∂r l l eff ∂r Δt ΔV ∂wl ∂wl = (Δzrαlμl,eff )n − (Δzrαlμl,eff )s Δt ∂r ∂r wl,N − wl,P wl,P − wl,S = AnΓn Δt − AsΓs Δt δrNP δrPS = Dn(wl,N − wl,P)Δt − Ds(wl,P − wl,S)Δt (F.274)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

AnΓn Dn = δrNP AsΓs Ds = δrPS 1 Γn = (α μ , )n = (α , μ , + α , μ , + α , μ , + α , μ , ) l l eff 4 l P l P l N l N l W l W l NW l NW 1 Γs = (α μ , )s = (α , μ , + α , μ , + α , μ , + α , μ , ) (F.275) l l eff 4 l P l P l S l S l W l W l SW l SW Axial direction: ∂ ∂wl (α μ , )dz rdr dt ∂z l l eff ∂z Δt ΔV ∂wl ∂wl =[r Δr(α μ , ) − r Δr(α μ , ) ]Δt p l l eff ∂z e p l l eff ∂z w 1490 Appendix F: Trondheim Bubble Column Model w , − w , w , − w , = A Γ l E l P Δt − A Γ l P l W Δt e e 1 w w 1 (δzPW + δzEP) (δzPW + δzEP) 2 2 = De(wl,E − wl,P)Δt − Dw(wl,P − wl,W )Δt (F.276)

To approximate the scalar grid cell mass ﬂuxes at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used:

A Γ D = e e e 1 (δzPW + δzEP) 2 A Γ D = w w w 1 (F.277) (δzPW + δzEP) 2 Γe = (αlμl,eff )e = αl,Pμl,P Γw = (αlμl,eff )w = αl,W μl,W .

The source terms

The the source terms are approximated by the midpoint rule, in which S is considered an average value representative for the whole grid cell volume. The derivatives are represented by an abbreviated Taylor series expansion, usually a central difference expansion of second order is employed. Term 1 on the RHS of the momentum equation: 1 ∂ μl,t ∂αl wl r dV dt r ∂r σαl,t ∂r Δt ΔV w μ , ∂α μ , ∂α ΔVΔt = P r l t l − r l t l (F.278) 1 rP σαl,t ∂r n σαl,t ∂r s (δrNP + δrPS) 2

This term is implemented through the source term SC as: wP v μl,t ∂αl v μl,t ∂αl ΔVΔt S , = r −r (F.279) C 1 r N σ ∂r P σ ∂r 1 ( + ) P αl,t n n αl,t s s 2 δrNP δrPS

1( + ) − 1( + ) (α ) − (α ) αl,N αl,NW αl,P αl,W ( ∂αl ) = l N l P = 2 2 ∂r n δrNP δrNP (F.280) 1( + ) − 1( + ) (α ) − (α ) αl,P αl,W αl,S αl,SW ( ∂αl ) = l P l S = 2 2 ∂r s δrPS δrPS

Term 2 on the RHS of the momentum equation: ∂ μl,t ∂αl μl,t ∂αl μl,t ∂αl ΔVΔt wl dV dt = wl,P − ∂z σαl,t ∂z σαl,t ∂z e σαl,t ∂z w δzPW Δt ΔV (F.281) This term is implemented through the source term SC as: ΔVΔt μl,t,P ∂αl μl,t,W ∂αl SC,2 = wl,P − (F.282) δzPW σαl,t ∂z e σαl,t ∂z w where

1( + ) − 1( + ) (α ) − (α ) αl,E αl,P αl,P αl,W ( ∂αl ) = l E l P = 2 2 ∂z e 1 1 (δzPW + δzEP) (δzPW + δzEP) 2 2 (F.283) 1( + ) − 1( + ) (α ) − (α ) αl,P αl,W αl,W αl,WW ( ∂αl ) = l P l W = 2 2 ∂z w 1 1 (δzPW + δzEP) (δzPW + δzEP) 2 2 Term 3 on the RHS of the momentum equation: μ , 1 l t ∂αl ∂ ( v ) + ∂wl ∂z ∂r r l ∂z dV dt ΔtΔV σαl,t r μ , 1 (rv ) − (rv ) (F.284) = l t ∂αl l n l s + ( ∂wl ) ΔVΔt ∂z 1 ∂z P σαl,t P rP (δrNP + δrPS) 2

This term is implemented through the source term SC as: v 1 (v + v ) − v 1 (v + v ) μl,t ∂αl 1 rN 2 l,N l,NW rP 2 l,P l,W , = Δ Δ SC 3 1 V t σα , ∂z rP (δr + δr ) l t P 2 NP PS μl,t ∂αl ∂wl + ( )PΔVΔt (F.285) σαl,t ∂z P ∂z 1492 Appendix F: Trondheim Bubble Column Model

To approximate derivatives of scalar grid cell variables and scalar grid cell variables at the staggered velocity grid cell surface points, arithmetic interpolation is frequently used: 1 (μ , )P = (μ , + μ , ) l t 2 l P l W (α ) − (α ) α , − α , ( ∂αl ) = l e l w = l P l W ∂z P δzPW δzPW (F.286) 1 1 (v , + v , ) − (v , + v , ) v , − v , l P l E l P l W ( ∂wl ) = l e l w = 2 2 ∂z P δzPW δzPW

Term 4 on the RHS of the momentum equation: 1 ∂ ∂wl (rα μ , ) dV dt r ∂r l l eff ∂r Δt ΔV ΔVΔt ∂wl ∂wl = (αlrμl,eff )n − (αlrμl,eff )s (F.287) rP ∂r ∂r

This term is implemented through the source term SC as: Δ Δ − − = V t Γ v wl,N wl,P − Γ v wl,P wl,S ] SC,4 nrN srP (F.288) rP δrNP δrPS

Term 5th and 7th terms on the RHS of the momentum equation are identical and both of them are approximated as follows: ∂ ∂wl ΔVΔt ∂wl ∂wl (αlμl ) dV dt = (αlμl )e − (αlμl )w (F.289) ∂z ∂z δzPW ∂z ∂z Δt ΔV

This term is implemented through the source term SC as: ΔVΔt (wl,E − wl,P) (wl,P − wl,W ) S , = α , μ , − α , μ , (F.290) C 5 δz l P l P 1 ( + ) l W l W 1 ( + ) PW 2 δzEP δzPW 2 δzEP δzPW

The staggered velocity grid cell surface points coincide with center nodes in the scalar grid, so no interpolation is needed.

Term 6 on the RHS of the momentum equation: 1 ∂ ∂vl (rα μ , ) dV dt r ∂r l l eff ∂z Δt ΔV Appendix F: Trondheim Bubble Column Model 1493 ΔVΔt ∂vl ∂vl = (αlrμl,eff )n − (αlrμl,eff )s (F.291) rP ∂z ∂z

This term is implemented through the source term SC as: Δ Δ v − v v − v = V t Γ v l,N l,NW − Γ v l,P l,W SC,6 nrN srP (F.292) rP δzPW δzPW

Term 8A on the RHS of the momentum equation: ∂ 2 2 ΔVΔt − αlρlkl dV dt =− [(αlρlkl)e − (αlρlkl)w] ∂z 3 3 δzPW Δt ΔV 2 ΔVΔt =− [αl,Pρl,Pkl,P − αl,W ρl,W kl,W ] (F.293) 3 δzPW

This term is implemented through the source term SC as:

2 ΔVΔt SC,7 = [αl,W ρl,W kW − αl,Pρl,PkP] (F.294) 3 δzPW

The staggered velocity grid cell surface points coincide with center nodes in the scalar grid, so no interpolation is needed.

Term 8B on the RHS of the momentum equation: ∂ 2 1 ∂(rvl) − α ρ ν , dV dt ∂z 3 l l l eff r ∂r Δt ΔV 2ΔVΔt αlμl,eff ∂(rv ) αlμl,eff ∂(rv ) =− l − l (F.295) 3 δzPW r ∂r e r ∂r w

This term is implemented through the source term SC as: 2ΔVΔt αl,Pμl,eff ,P ∂(rvl) αl,W μl,eff ,W ∂(rvl) SC,8 =− |e − |w (F.296) 3 δzPW rP ∂r rP ∂r

The staggered velocity grid cell surface points coincide with center nodes in the scalar grid, so no interpolation is needed for the scalar grid variables. The velocity derivatives are approximated by central difference discretizations:

v v ∂(rv ) (rv ) − (rv ) r vl,N − r vl,P l | = l ne l se = N P (F.297) ∂r e 1 1 (δrNP + δrPS) (δrNP + δrPS) 2 2 1494 Appendix F: Trondheim Bubble Column Model

v v ( v ) ( v ) − ( v ) r v , − (r v , ∂ r l | = r l nw r l sw = NW l NW P l W ∂r w 1 1 (δrNP + δrPS) (δrNP + δrPS) 2 2 Term 8C on the RHS of the momentum equation: 2 − ∂ ∂wl ∂z αlρlνl,eff ∂z dV dt ΔtΔV 3 (F.298) 2ΔVΔt =− ( ∂wl ) − ( ∂wl ) αlμl,eff ∂z e αlμl,eff ∂z w 3 δzPW

This term is implemented through the source term SC as:

2ΔV (wl,E − wl,P) SC,9 =− αl,Pμl,eff ,P 3 δzPW 1 (δzEP + δzPW ) 2 (F.299) ( − ) 2ΔV wl,P wl,W + αl,W μl,eff ,W 3 δzPW 1 (δzEP + δzPW ) 2 The staggered velocity grid cell surface points coincide with center nodes in the scalar grid, so no interpolation is needed for the scalar grid variables. The velocity gradients are approximated by central difference expansions.

Pressure force

∂p 1 (PP − PW ) − αl dV dt =− (αl)P ΔVΔt (F.300) ∂z 2 δzPW Δt ΔV

This term is implemented through the source term SC as:

1 PP − PW SC,10 =− (αl,P + αl,W ) ΔV (F.301) 2 δzPW

To approximate scalar grid cell variables at the staggered velocity grid cell node points, arithmetic interpolation is frequently used:

1 (α )P = (α , + α , ). (F.302) l 2 l P l W Appendix F: Trondheim Bubble Column Model 1495

Gravity force

αlρlg dV dt = g(αlρl)PΔVΔt (F.303) Δt ΔV

This term is implemented through the source term SC as:

1 SC, = g (α , ρ , + α , ρ , )ΔVΔt (F.304) 11 2 l P l P l W l W To approximate scalar grid cell variables at the staggered velocity grid cell node points, arithmetic interpolation is frequently used:

1 (α ρ )P = (α , ρ , + α , ρ , ). (F.305) l l 2 l P l P l W l W

Added Mass force

∂wl ∂wl ∂vg ∂vg − α αgρ fv w + v − vg + vg dV dt l l l ∂z l ∂r ∂z ∂r ΔtΔV

=−fv (α ) α (ρ ) l P gP l P

∂wl ∂wl ∂wg ∂wg × wl + (vl) − wg + vg ΔVΔt ∂z ∂r ∂z ∂r P (F.306) This term is implemented through the source term SC as:

1 1 1 S , =−fv (α , + α , ) (α , + α , ) (ρ , + ρ , ) C 12 2 l P l W 2 g P g W 2 l P l W (F.307) ∂wl ∂wl ∂vg ∂vg × wl + vl − wg + (vg) ΔVΔt ∂z ∂r ∂z ∂r P

To approximate scalar grid cell variables at the staggered w-velocity grid cell center node point, arithmetic interpolation is frequently used. The radial velocity component is discretized in the staggered v-grid cell volume and need to be interpolated to the w-grid cell center node point. The derivatives of the w-velocity component is approximated by a central difference scheme. When needed, arithmetic interpolation is used for the velocity components as well.

1 (α )P = (α ,P + α ,W ) (F.308) g 2 g g 1 (ρ )P = (ρ , + ρ , ) l 2 l P l W 1496 Appendix F: Trondheim Bubble Column Model

1 (v )P = (v , + v , + v , + v , ) l 4 l P l N l W l NW 1 1 − (wl,P + wl,N ) − (wl,P + wl,S) ∂wl = wl,n wl,s = 2 2 ∂r P 1 1 (δrNP + δrPS) (δrNP + δrPS) 2 2 1 1 − (wl,P + wl,E) − (wl,P + wl,W ) ∂wl = wl,e wl,w = 2 2 ∂z P δzPW δzPW

The corresponding terms for the gas phase are discretized in the same way.

Steady Drag force

The steady drag term is treated in the same way as described for the radial velocity component.

Algebraic discretization equation

After dividing all the terms by Δt, the discretized equation can be written on the standard algebraic form:

= + + + + aPwl,P aN wl,N aSwl,S aEwl,E aW wl,W bwl (F.309)

The coefﬁcients are deﬁned by:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] aE = De + max[−Ce, 0] a = D + max[C , 0] W w w = + 0 v0 b SC,m aP l,P m ΔV a = (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV P Δt l l P N S E W n s e w P q q (F.310)

To avoid negative coefficients, the relation for the coefficient aP can be modified using the continuity Eq. (F.187). The negative mC2-term is then moved to the RHS of the ∗ discretized transport equation and included as part of the b-term. The alternative aP and b∗ coefficients are defined by: Appendix F: Trondheim Bubble Column Model 1497 ∗ =− ν + + 0 0 b mC2wl,P SC,m,l aPwl,P m ∗ = ΔV ( )0 + + + + + − Δ . (F.311) aP Δt αlρl P aN aS aE aW mC1 SP,q,l V q

F.4.6 The Gas Phase Radial Momentum Balance

The radial component of the momentum balance for the gas phase is given in (F.145):

∂ 1 ∂ ∂ αgρgvg + rαgρgvgvg + αgρgwgvg ∂t r ∂r ∂z 1 ∂ μg,t ∂αg ∂ μg,t ∂αg = r vg + vg r ∂r σα , ∂r ∂z σα , ∂z g t g t 1 ∂ μg,t ∂αg ∂ μg,t ∂αg + r vg + wg r ∂r σα , ∂r ∂z σα , ∂r g t g t 1 ∂ μg,t ∂vg 1 μg,t vg ∂ μg,t ∂vg + rαg − αg + αg r ∂r σ , ∂r r σ , r ∂z σ , ∂z g t g t g t 1 ∂ μg,t ∂vg 1 μg,t vg ∂ μg,t ∂wg + rαg − αg + αg r ∂r σ , ∂r r σ , r ∂z σ , ∂r g t g t g t

∂ 2 νg,t 1 ∂ ∂wg − αgρg k + rvg + ∂r 3 σ , r ∂r ∂z g t ∂p μ , ∂α ∂v ∂α ∂v − α + l t l l + l l g ∂r σ ∂r ∂r ∂z ∂z αl,t ∂ μl,t ∂αl ∂ μl,t ∂αl + vl + wl ∂r σαl,t ∂r ∂z σαl,t ∂r + + C + C αgρggr Fg,r Fg,r

In the FVM, the integral form of the momentum equation is used. The differential equation is thus integrated in time and over a grid cell volume in the staggered grid for the v-velocity. The transient, convective and diffusive terms are discretized just like the corresponding terms in the liquid phase (i.e., see Sect.F.4.4).

The source terms

The the source terms are approximated by the midpoint rule, in which S is considered an average value representative for the whole grid cell volume. The derivatives are 1498 Appendix F: Trondheim Bubble Column Model represented by an abbreviated Taylor series expansion, usually a central difference expansion of second order is employed. Term 1 on the RHS of the momentum equation is split into two parts, term 1a and term 1b:

1 ∂ μg,t ∂αg ∂ μg ∂αl 1 μg ∂αl ∂ r vg =−vg − rvg (F.312) r ∂r σαg,t ∂r ∂r σαg,t ∂r r σαg,t ∂r ∂r

By use of (8.11), the gas-volume fractions has been substituted with the liquid volume fraction. The gradient of the gas phase volume fraction is thus related to the liquid phase volume fraction in accordance with:

∇αg =∇(1 − αl) =−∇αl (F.313)

Term 1a on the RHS of the momentum equation: ∂ μg ∂αl − vg dV dt (F.314) ∂r σα , ∂r Δ Δ g t t V v , ΔVΔt μ μ =− g P g ∂αl − g ∂αl 1 σα , ∂r σα , ∂r (δr + δr ) g t n g t s 2 NP PS

The term 1a is implemented through the source term SC as: vg,PΔVΔt ∂αl ∂αl S , =− μ , − μ , (F.315) C 1 1 ( + ) g P ∂r g S ∂r 2 δrNP δrPS n s

The staggered velocity grid cell surface points coincide with center nodes in the scalar grid, so no interpolation is needed for the scalar grid variables. The scalar gradient terms in the staggered velocity cell volume are approximated by central difference expansions and arithmetic interpolation:

1( + ) − 1( + ) (α ) − (α ) αl,N αl,P αl,P αl,S ( ∂αl ) = l N l P = 2 2 ∂r n δrNP δrNP (F.316) 1( + ) − 1( + ) (α ) − (α ) αl,P αl,S αl,S αl,SS ( ∂αl ) = l P l S = 2 2 ∂r s δrPS δrPS

Term 1b on the RHS of the momentum equation: 1 μg ∂α ∂ 1 μg ∂α [ rvg − rvg ] − l rv dV dt =− l n s ΔVΔt g 1 r σαg,t ∂r ∂r r σαg,t ∂r (δrNP + δrPS) ΔtΔV P 2 (F.317) Appendix F: Trondheim Bubble Column Model 1499

This term is implemented through the source term SC as: 1 v + v − 1 v + v μg P ∂αl rP 2 g,P g,N rS 2 g,P g,S S , =− ΔVΔt (F.318) C 2 rv ∂r 1 ( + ) P P 2 δrNP δrPS

The scalar variables at the staggered velocity grid cell surface points is usually obtained by arithmetic interpolation. The scalar gradient terms in the staggered velocity cell volume are approximated by central difference expansions. The scalar properties at the staggered grid cell surface points coincide with the scalar grid central nodes: 1 μg = μg,P + μg,S (F.319) P 2 ( ) − ( ) − ∂αl = αl n αl s = αl,P αl,S ∂r 1 ( + ) 1 ( + ) P 2 δrNP δrPS 2 δrNP δrPS

Term 2 on the RHS of the momentum equation is split into two parts, term 2a and term 2b: ∂ μg,t ∂αg ∂ μg,t ∂αl μg,t ∂αl ∂vg vg =−vg − (F.320) ∂z σαg,t ∂z ∂z σαg,t ∂z σαg,t ∂z ∂z

Term 2a on the RHS of the momentum equation: ∂ μg,t ∂αl − vg dV dt (F.321) ∂z σαg,t ∂z Δt ΔV v , ΔV μ , μ , =− g P [ g t ∂αl − g t ∂αl ] 1 (δz + δz ) σα , ∂z σα , ∂z 2 PW EP g t e g t w

This term is implemented through the source term SC as: vg,PΔV ∂αl ∂αl S , =− μ , − μ , (F.322) C 3 1 ( + ) g t ∂z g t ∂z 2 δzPW δzEP e w

To approximate the scalar variables at the staggered velocity grid cell surface points, arithmetic interpolation is frequently applied. The derivatives are approximated by a central difference scheme:

1 1 ∂α (α ) − (α ) (α , + α , ) − (α , + α , ) l = l E l P = 2 l E l SE 2 l E l SE (F.323) ∂z e δzEP δzEP ( ) − ( ) 1 (α + α ) − 1 (α + α ) ∂αl = αl P αl W = 2 l,P l,S 2 l,W l,SW ∂z w δzPW δzPW 1500 Appendix F: Trondheim Bubble Column Model

1 (μg)e = (μ , + μ , + μ , + μ , ) 4 g P g E g S g SE 1 (μg)w = (μ , + μ , + μ , + μ , ) 4 g P g W g S g SW Term 2b on the RHS of the momentum equation: μ , ∂v μ , (v , − v , ) − g t ∂αl g =− g t ∂αl g e g w Δ Δ dV dt 1 V t (F.324) σαg,t ∂z ∂z σαg,t ∂z (δzEP + δzPW ) Δt ΔV P 2

This term is implemented through the source term SC as:

1 (v + v ) − 1 (v + v ) ∂αl 2 g,P g,E 2 g,P g,W S , =−(μ , ) ΔVΔt (F.325) C 4 g t ∂z P 1 ( + ) 2 δzEP δzPW where ( ) − ( ) (∂αl ) = αl e αl w ∂z P 1 ( + ) 2 δzEP δzPW 1 1 (α , + α , + α , + α , ) − (α , + α , + α , + α , ) = 4 l P l E l S l SE 4 l P l W l S l SW 1 ( + ) 2 δzEP δzPW (F.326)

Term 3 on the RHS of the momentum equation is split into two parts, term 3a and term 3b: 1 ∂ μg,t ∂αg vg ∂ μg,t ∂αl μg,t ∂αl ∂vg r vg =− r − (F.327) r ∂r σαg,t ∂r r ∂r σαg,t ∂r σαg,t ∂r ∂r

Term 3a on the RHS of the momentum equation: μg,t ∂αl μg,t ∂αl (r )n − (r )s v ∂ μ , ∂α v , σα , ∂r σα , ∂r − g r g t l dV dt =− g P g t g t ΔVΔt v 1 ( + ) r ∂r σαg,t ∂r rP δrNP δrPS Δt ΔV 2 (F.328) This term is implemented through the source term SC as:

μg,t,P ∂αl μg,t,S ∂αl v rP ( )n − rS ( )s g,P σαg,t ∂r σαg,t ∂r S , =− ΔVΔt (F.329) C 5 rv 1 ( + ) P 2 δrNP δrPS where Appendix F: Trondheim Bubble Column Model 1501 1 1 ( ) − ( ) (α , + α , ) − (α , + α , ) ∂αl = αl N αl P = 2 l N l P 2 l P l S ∂r n δrNP δrNP (F.330) 1 1 ( ) − ( ) (α , + α , ) − (α , + α , ) ∂αl = αl P αl S = 2 l P l S 2 l S l SS ∂r s δrPS δrPS

Term 3b on the RHS of the momentum equation: μg,t ∂α ∂vg μg,t ∂α (vg,n − vg,s) − l dV dt =− l ΔV 1 (F.331) σαg,t ∂r ∂r σαg,t P ∂r P (δrNP + δrPS) ΔtΔV 2

This term is implemented through the source term SC as: 1 (v + v ) − 1 (v + v ) μg,t ∂αl g,N g,P g,P g,S S , =− 2 2 ΔVΔt (F.332) C 6 σ ∂r 1 ( + ) αg,t P P 2 δrNP δrPS

The staggered grid variables are expressed in terms of the node values in the scalar grid:

1 (μg) = (μ , + μ , ) (F.333) P 2 g P g S ( ) − ( ) α − α ∂αl = αl n αl s = l,P l,S ∂r 1 ( + ) 1 ( + ) P 2 δrNP δrPS 2 δrNP δrPS

Term 4 on the RHS of the momentum equation is split into two parts, term 4a and term 4b: ∂ μg,t ∂αg ∂ μg,t ∂αl μg,t ∂αl ∂wg wg =−wg − (F.334) ∂z σαg,t ∂r ∂z σαg,t ∂r σαg,t ∂r ∂z

Term 4a on the RHS of the momentum equation:

μg,t ∂αl μg,t ∂αl ( )e − ( )w ∂ μg,t ∂αl σαg,t ∂r σαg,t ∂r −w dV dt =−w , ΔVΔt g g P 1 (F.335) ∂z σαg,t ∂r (δzEP + δzPW ) ΔtΔV 2

This term is implemented through the source term SC as:

μg,t ∂αl μg,t ∂αl ( )e − ( )w σαg,t ∂r σαg,t ∂r S , =−(w ) ΔVΔt (F.336) C 7 g P 1 ( + ) 2 δzEP δzPW

The staggered grid variables are expressed in terms of the node values in the scalar grid: 1 (w )P = (v ,P + v ,S + v ,E + v ,SE) (F.337) g 4 g g g g 1502 Appendix F: Trondheim Bubble Column Model

Term 4b on the RHS of the momentum equation: μg,t ∂α ∂wg μg,t ∂α (wg)e − (wg)w − l dV dt =− l ΔVΔt 1 (F.338) σαg,t ∂r ∂z σαg,t ∂r P (δzEP + δzPW ) ΔtΔV 2

This term is implemented through the source term SC as: μg,t ∂αl (wg)e − (wg)w S , =− ΔVΔt (F.339) C 8 1 ( + ) σαg,t ∂r P δzEP δzPW 2 1 1 μ , ∂α (vg,E + vg,SE) − (vg,P + vg,S) =− g t l 2 2 ΔVΔt σ ∂r 1 ( + ) αg,t P 2 δzEP δzPW

The 5th and 8th terms on the RHS of the radial component of the momentum equation for the gas phase are identical and discretized in the same way as the corresponding terms in the liquid phase equation, as discussed in Sect. F.4.4:

μg,t ∂vg μg,t ∂vg (rαg )n − (rαg )s 1 ∂ μ , ∂v ΔVΔt σα , ∂r σα , ∂r rα g t g dV dt = g t g t g v 1 ( + ) r ∂r σαg,t ∂r rP δrNP δrPS Δt ΔV 2 (F.340) This term is implemented through the source term SC as:

v v Δ Δ ( ∂ g ) − ( ∂ g ) V t rPαg,Pμg,P ∂r n rSαg,Sμg,S ∂r s S , = (F.341) C 9 rv 1 ( + ) P 2 δrNP δrPS

The scalar variables at the staggered grid surface are expressed in terms of the node values in the scalar grid. The derivatives of staggered grid variables are approximated by use of the central difference scheme: ∂v v ,N − v ,P g = g g (F.342) ∂r n δrNP ∂v v , − v , g = g P g S ∂r s δrPS

The 6th and 9th terms on the RHS of the radial component of the momentum equation for the gas phase are identical and discretized in the same way as the corresponding terms in the liquid phase equation, as discussed in Sect. F.4.4. μ , v μ , v , − 1 g t g =− g t g P Δ Δ 2 αg dV dt 2 αg 2,v V t r σαg,t r σαg,t P r Δt ΔV P Appendix F: Trondheim Bubble Column Model 1503

1 μg,t,P μg,t,S vg,P =−2 × α ,P + α ,S ΔVΔt (F.343) 2 g σ g σ 2,v αg,t αg,t rP

This term is implemented through the source term SP as: 1 μg,t,P μg,t,S ΔVΔt , =− × , + , Sp 1 2 αg P αg S 2v (F.344) 2 σαg,t σαg,t rP in which the scalar variables at the staggered grid surface are expressed in terms of the node values in the scalar grid. Term 7 on the RHS of the radial momentum equation is discretized in the same way as the corresponding term in the liquid phase equation, as discussed in Sect.F.4.4. The formulation is not repeated here. Term 10 on the RHS of the radial momentum equation is discretized in the same way as the corresponding term in the liquid phase equation, as discussed in Sect.F.4.4: ∂ (α μg,t ∂wg ) dV dt ∂z g σα , ∂r ΔtΔV g t = ΔVΔt ( μg,t ∂wg ) − ( μg,t ∂wg ) 1 αg e αg w (δzPW +δzEP) σαg,t ∂r σαg,t ∂r (F.345) 2 ( − ) ( − ) = ΔVΔt Γ wg,E wg,SE − Γ wg,P wg,S 1 ( + ) e 1 ( + ) w 1 ( + ) 2 δzPW δzEP 2 δrNP δrPS 2 δrNP δrPS

This term is implemented through the source term SC as: ΔVΔt (wg,E − wg,SE) (wg,P − wg,S) SC,10 = Γe − Γw (F.346) S , 1 ( + ) 1 ( + ) EW P 2 δrNP δrPS 2 δrNP δrPS in which the scalar variables at the staggered grid surface are expressed in terms of the node values in the scalar grid. Term 11A on the RHS of the radial momentum equation is discretized in the same way as the corresponding term in the liquid phase equation, as discussed in Sect. F.4.4: Δ Δ − ∂ 2 =−2 V t (( ) − ( ) ) αgρgk dV dt 1 αgρgk n αgρgk s ∂r 3 3 (δrNP + δrPS) Δt ΔV 2 (F.347) This term is implemented through the source term SC as:

2 ΔV SC, = [α ,Pρ ,PkP − α ,Sρ ,SkS] (F.348) 11 3 1 ( + ) g g g g 2 δrNP δrPS 1504 Appendix F: Trondheim Bubble Column Model in which the scalar variables at the staggered grid surface are expressed in terms of the node values in the scalar grid. Term 11B on the RHS of the radial momentum equation is discretized in the same way as the corresponding term in the liquid phase equation, as discussed in Sect. F.4.4: ∂ 2 μg,t 1 ∂(rvg) − αgρg dV dt (F.349) ∂r 3 σαg,t r ∂r ΔtΔV μ , μ , α g t α g t Δ g σα , ∂(rv ) g σα , ∂rv =−2 V g t g − g t g 3 1 ( + ) r ∂r r ∂r 2 δrNP δrPS n s

This term is implemented through the source term SC as:

μg,t,P μg,t,W αg,P αg,W 2 ΔV σαg,t ∂(rvg) σαg,t ∂(rvg) S , =− − ∗ C 12 3 1 ( + ) r ∂r r ∂r 2 δrNP δrPS P n S s (F.350) where v v ∂(rv ) (rv ) − (rv ) r v , − r v , g = g N g P = N g N P g P (F.351) ∂r n δrNP δrNP v v ∂(rv ) (rv ) − (rv ) r v , − r v , g = g P g S = P g P S g S ∂r s δrPS δrPS

Term 11C on the RHS of the radial momentum equation is discretized in the same way as the corresponding term in the liquid phase equation, as discussed in Sect.F.4.4: ∂ 2 μg,t ∂wg − αgρg dV dt ∂r 3 σαg,t ∂z ΔtΔV 2 ΔVΔt μg,t ∂wg μg,t ∂wg (F.352) =− αg − αg 3 1 + σαg,t ∂z n σαg,t ∂z s 2 δrNP δrPS

This term is implemented through the source term SC as: 2 ΔV μg,t,P (wg,E − wg,P) S , =− α , C 13 1 g P 1 (F.353) 3 (δr + δr ) σα , (δz + δz ) 2 NP PS g t 2 EP PW 2 ΔV μg,t,W (wg,P − wg,W ) + α , 3 1 ( + ) g W σ 1 ( + ) 2 δrNP δrPS αg,t 2 δzEP δzPW

Term 13A on the RHS of the momentum equation: μ , ∂α ∂v μ , ∂α ∂v l t l l dV dt = l t l l ΔVΔt (F.354) σαl,t ∂r ∂r σαl,t ∂r ∂r P ΔtΔV Appendix F: Trondheim Bubble Column Model 1505

This term is implemented through the source term SC as: 1 μl,t,P μl,t,S ∂αl ∂vl SC,14 = + ΔVΔt (F.355) 2 σαl,t σαl,t ∂r ∂r P

The scalar variables at the staggered grid center node are obtained by arithmetic interpolation of the node values in the scalar grid. Moreover, the velocity gradient is approximated by use of the central difference scheme and the surface values are obtained by arithmetic interpolation of the node values in the stagged velocity grid:

1 1 ∂v v , − v , (vl,P + vl,N ) − (vl,P + vl,S) ( l ) = l n l s = 2 2 (F.356) ∂r P 1 ( + ) 1 ( + ) 2 δrNP δrPS 2 δrNP δrPS

Term 13B on the RHS of the momentum equation: μ , ∂α ∂v μ , ∂α ∂v l t l l dV dt = ΔVΔt l t l l (F.357) σαl,t ∂z ∂z σαl,t ∂z ∂z P Δt ΔV

This term is implemented through the source term SC as: 1 (v + v ) − 1 (v + v ) μl,t ∂αl 2 l,E l,P 2 l,P l,W S , = ΔVΔt (F.358) C 15 σ ∂z 1 ( + ) αl,t P 2 δzEP δzPW where ( ) − ( ) ∂αl = αl e αl w ∂z 1 ( + ) P 2 δzEP δzPW 1 1 (α , + α , + α , + α , ) − (α , + α , + α , + α , ) = 4 l P l E l S l SE 4 l P l W l S l SW 1 ( + ) 2 δzEP δzPW (F.359)

Term 14 on the RHS of the momentum equation: ( ∂αl ) − ( ∂αl ) ∂ μl,t ∂αl μl,t ∂r n μl,t ∂r s v dV dt = v , ΔVΔt (F.360) l l P 1 ( + ) ∂r σαl,t ∂r δrNP δrPS Δt ΔV 2

This term is implemented through the source term SC as: ( ∂αl ) − ( ∂αl ) μl,P ∂r n μl,S ∂r s S , = v ΔVΔt (F.361) C 16 P 1 ( + ) 2 δrNP δrPS 1506 Appendix F: Trondheim Bubble Column Model in which the scalar variables at the staggered grid center node are obtained by arithmetic interpolation of the node values in the scalar grid. Term 15 on the RHS of the momentum equation:

μl,t ∂αl μl,t ∂αl ( )e − ( )w ∂ μ , ∂α σα , ∂r σα , ∂r w l t l dV dt = (w ) l t l t ΔVΔt l l P 1 ( + ) ∂z σαl,t ∂r δzEP δzPW Δt ΔV 2 (F.362) This term is implemented through the source term SC as: ( ∂αl ) − ( ∂αl ) μl,t ∂r e μl,t ∂r w S , = (w ) ΔVΔt. (F.363) C 17 l P 1 ( + ) 2 δzEP δzPW

Pressure force

The pressure force term is treated in the same way as described for the radial liquid velocity component.

Added mass force

The added mass force term is treated in the same way as described for the radial liquid velocity component.

Transversal force

The transversal force term is treated in the same way as described for the radial liquid velocity component.

Steady drag force

The steady drag force term is treated in the same way as described for the radial liquid velocity component.

Algebraic discretization equation

After dividing all the terms by Δt, the discretized equation can be written on the standard algebraic form:

v = v + v + v + v + aP g,P aN g,N aS g,S aE g,E aW g,W bvg (F.364) Appendix F: Trondheim Bubble Column Model 1507

The coefﬁcients are deﬁned by:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] aE = De + max[−Ce, 0] a = Dw + max[Cw, 0] W = + 0 v0 b SC,m aP g,P m ΔV a = (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV P Δ g g P N S E W n s e w P q t q (F.365)

To avoid negative coefficients, the relation for the coefficient aP can be modified using ∗ the continuity equation, as shown for the liquid phase equations. The alternative aP and b∗ coefficients are defined by: ∗ =− vν + + 0 v0 b mC2 g,P SC,m,g aP g,P m Δ (F.366) ∗ V 0 a = (α ρ ) + a + a + a + a + mC − S , , ΔV P Δ g g P N S E W 1 P q g t q

F.4.7 The Gas Phase Axial Momentum Balance

The axial momentum balance for gas is given in (F.146).

∂ 1 ∂ ∂ αgρgwg + rαgρgvgwg + αgρgwgwg ∂t r ∂r ∂z 1 ∂ μg,t ∂αg ∂ μg,t ∂αg = r wg + wg r ∂r σα , ∂r ∂z σα , ∂z g t g t 1 ∂ μg,t ∂αg ∂ μg,t ∂αg + r vg + wg r ∂r σα , ∂z ∂z σα , ∂z g t g t 1 ∂ μg,t ∂wg ∂ μg,t ∂wg + rαg + αg r ∂r σ , ∂r ∂z σ , ∂z g t g t 1 ∂ μg,t ∂vg ∂ μg,t ∂wg + rαg + αg r ∂r σ , ∂z ∂z σ , ∂z g t g t ∂ 2 νg,t 1 ∂ ∂wg − αgρg k + rvg + ∂z 3 σ , r ∂r ∂z g t ∂p μl,t ∂αl ∂wl ∂αl ∂wl − αg + + ∂z σαl,t ∂r ∂r ∂z ∂z 1508 Appendix F: Trondheim Bubble Column Model ∂ μl,t ∂αl ∂ μl,t ∂αl + vl + wl ∂r σαl,t ∂z ∂z σαl,t ∂z + + C + C αgρggz Fg,z Fg,z

The transient, convective and the diffusive terms are discretized just like the corresponding terms in the liquid phase as shown in Sect. F.4.5. Only the novel source terms found only in the axial gas phase momentum equation are considered in this section.

The source terms

The source terms are approximated by the midpoint rule, in which S is considered an average value representative for the whole grid cell volume. The derivatives are represented by an abbreviated Taylor series expansion, usually a central difference expansion of second order is employed. Term 1 on the RHS of the momentum equation is split into two parts, term 1a and term 1b: 1 ∂ μg,t ∂αg wg ∂ μg,t ∂αl μg,t ∂αl ∂wg r wg =− r − (F.367) r ∂r σαg,t ∂r r ∂r σαg,t ∂r σαg,t ∂r ∂r

Term 1a on the RHS of the momentum equation: − wg ∂ r μg,t ∂αl dV dt r ∂r σα , ∂r ΔtΔV g t Δ Δ (F.368) =− wg,P V t r μg,t ∂αl − r μg,t ∂αl 1 ( + ) σα , ∂r σα , ∂r rP 2 δrNP δrPS g t n g t s

This term is implemented through the source term SC as: wg,PΔVΔt v μg,t ∂αl v μg,t ∂αl , =− − SC 1 1 rN rP r (δr + δr ) σα , ∂r σα , ∂r P 2 NP PS g t n n g t s s (F.369) where 1 μ , = μ , + μ , + μ , + μ , (F.370) l t n 4 l P l N l W l NW 1 μ , = μ , + μ , + μ , + μ , l t s 4 l P l S l W l SW 1 1 ( ) − ( ) α , + α , − α , + α , ∂αl = αl N αl P = 2 l N l NW 2 l P l W ∂r n δrNP δrNP Appendix F: Trondheim Bubble Column Model 1509

1 1 ( ) − ( ) α , + α , − α , + α , ∂αl = αl P αl S = 2 l P l W 2 l S l SW ∂r s δrPS δrPS

Term 1b on the RHS of the momentum equation: μ ,t ∂α ∂w μ ,t ∂α (w ,n − w ,s) − g l g dV dt =− g l g g ΔVΔt (F.371) 1 ( + ) σαg,t ∂r ∂r σαg,t ∂r P δrNP δrPS Δt ΔV 2

This term is implemented through the source term SC as: 1 ( + ) − 1 ( + ) 1 μg,t,P μg,t,W ∂αl 2 wP wN 2 wP wS SC, =− + ΔVΔt 2 2 σ σ ∂r 1 ( + ) αg,t αg,t P 2 δrNP δrPS (F.372) where

∂αl (αl)n − (αl)s ( )P = ∂r SNS,P 1 1 (α , + α , + α , + α , ) − (α , + α , + α , + α , ) = 4 l P l N l W l NW 4 l P l S l W l SW 1 ( + ) 2 δrNP δrPS (F.373)

Term 2 on the RHS of the momentum equation is split into two parts, term 2a and term 2b: ∂ μg,t ∂αg ∂ μg,t ∂αl μg,t ∂αl ∂wg wg =−wg − (F.374) ∂z σαg,t ∂z ∂z σαg,t ∂z σαg,t ∂z ∂z

Term 2a on the RHS of the momentum equation:

μg,t ∂αl μg,t ∂αl ( )e − ( )w ∂ μg,t ∂αl σαg,t ∂z σαg,t ∂z − wg dV dt =−wg,P ΔVΔt ∂z σαg,t ∂z δzPW Δt ΔV (F.375) This term is implemented through the source term SC as:

μg,t,P ∂αl μg,t,W ∂αl ( )e − ( )w σαg,t ∂z σαg,t ∂z SC,3 =−wg,P ΔVΔtΔt (F.376) δzPW where 1510 Appendix F: Trondheim Bubble Column Model 1 1 ( ) − ( ) (α , + α , ) − (α , + α , ) ∂αl = αl E αl P = 2 l E l P 2 l P l W ∂z 1 ( + ) 1 ( + ) e 2 δzPW δzEP 2 δzPW δzEP (F.377) 1 1 ( ) − ( ) (α , + α , ) − (α , + α , ) ∂αl = αl P αl W = 2 l P l W 2 l W l WW ∂z 1 ( + ) 1 ( + ) w 2 δzPW δzEP 2 δzPW δzEP

Term 2b on the RHS of the momentum equation: μ , , ∂α ∂w μ , , ∂α (w , − w , ) − g t P l g dV dt =− g t P l g e g w ΔVΔt (F.378) σαg,t ∂z ∂z σαg,t ∂z P δzPW Δt ΔV

This term is implemented through the source term SC as: 1 ( + ) − 1 ( + ) 1 μg,t,P μg,t,W ∂αl 2 wg,E wg,P 2 wg,P wg,W SC,4 =− + ΔVΔt 2 σαg,t σαg,t ∂z P δzPW (F.379) where ∂α (α ) − (α ) (α ) − (α ) l = l e l w = l P l W (F.380) ∂z P δzPW δzPW

Term 3 on the RHS of the axial momentum equation is split into two parts, term 3a and term 3b: 1 ∂ μg,t ∂αg ∂ μg ∂αl 1 μg ∂αl ∂ r vg =−vg − (rvg) (F.381) r ∂r σαg,t ∂z ∂r σαg,t ∂z r σαg,t ∂z ∂r

Term 3a on the RHS of the momentum equation: ∂ μ ∂α ∂α ∂α ΔVΔt − v g l dV dt =−(v ) μ l − μ l g g P g g 1 ( + ) ∂r σαg,t ∂z ∂z n ∂z s δrNP δrPS Δt ΔV 2 (F.382) This term is implemented through the source term SC as: ∂αl ∂αl ΔVΔt S , =−v , μ − μ (F.383) C 5 g P g ∂z g ∂z 1 ( + ) n s 2 δrNP δrPS where 1 (v )P = (v ,P + v ,N + v ,W + v ,NW ) (F.384) g 4 g g g g 1 (μ , )n = (μ , + μ , + μ , + μ , ) l t 4 l P l N l W l NW Appendix F: Trondheim Bubble Column Model 1511

1 (μ , )s = (μ , + μ , + μ , + μ , ) l t 4 l P l S l W l SW 1 1 (α , + α , ) − (α , + α , ) ∂αl = 2 l P l N 2 l W l NW ∂z n δzPW 1 1 (α , + α , ) − (α , + α , ) ∂αl = 2 l P l S 2 l W l SW ∂z s δzPW

Term 3b on the RHS of the axial momentum equation:

μg,t 1 μ , ∂α ∂ σα , ∂α (rv ) − (rv ) − g t l (rv ) dV dt =− g t l l n l s ΔVΔt g 1 ( + ) r σαg,t ∂z ∂r r ∂r P δrNP δrPS Δt ΔV 2 (F.385) This term is implemented through the source term SC as:

μg,t,P μg,t,S 1 ( + ) 1 (v + v ) − 1 (v + v ) 2 σαg,t σαg,t ∂αl 2 N NW 2 P W S , =− ΔVΔt C 6 r ∂r 1 ( + ) P P 2 δrNP δrPS (F.386) where ( ) − ( ) ∂αl = αl n αl s + ) ∂r P δrNP δrPS 1 1 (α , + α , + α , + α , ) − (α , + α , + α , + α , ) = 4 l P l N l W l NW 4 l P l S l W l SW (δrNP + δrPS) (F.387)

Term 4 on the RHS of the axial momentum equation is split into two parts, term 4a and term 4b: ∂ μg,t ∂αg ∂ μg,t ∂αl μg,t ∂αl ∂wg wg =−wg − (F.388) ∂z σαg,t ∂z ∂z σαg,t ∂z σαg,t ∂z ∂z

Term 4a on the RHS of the momentum equation:

μg,t ∂αl μg,t ∂αl ( )e − ( )w ∂ μg,t ∂αl σαg,t ∂z σαg,t ∂z − wg dV dt =−wg,P ΔVΔt ∂z σαg,t ∂z δzPW Δt ΔV (F.389) This term is implemented through the source term SC as: 1512 Appendix F: Trondheim Bubble Column Model

μg,t,P ∂αl μg,t,W ∂αl ( )e − ( )w σαg,t ∂z σαg,t ∂z SC,7 =−wg,P ΔVΔt (F.390) δzPW

Term 4b on the RHS of the momentum equation: μ ,t ∂α ∂w μ ,t ∂α w ,e − w , − g l g dV dt =− g l g g w ΔVΔt (F.391) σαg,t ∂z ∂z σαg,t ∂z P δzPW Δt ΔV

This term is implemented through the source term SC as: 1 ( + ) − 1 ( + ) 1 ∂αl 2 wg,E wg,P 2 wg,P wg,W SC,8 =− (μg,P + μg,W ) ΔVΔt 2 ∂z P δzPW (F.392) The 5th to 9th terms on the RHS of the axial component of the gas momentum equation are discretized in the same way as the corresponding equations in the liquid phase equation, discussed in Sect. F.4.5. Term 11A on the RHS of the axial momentum equation: μl,t ∂αl ∂wl μl,t ∂αl ∂wl dV dt = ( )PΔVΔt (F.393) σαl,t ∂r ∂r σαl,t ∂r ∂r Δt ΔV

This term is implemented through the source term SC as: 1 μl,t,P μl,t,W ∂αl ∂wl SC,14 = + ΔVΔt (F.394) 2 σαl,t σαl,t ∂r P ∂r P where 1 1 ∂w w , − w , (wl,P + wl,N ) − (wl,P + wl,S) l = l n l s = 2 2 (F.395) ∂r 1 ( + ) 1 ( + ) P 2 δrNP δrPS 2 δrNP δrPS

Term 11B on the RHS of the axial momentum equation: μ , ∂α ∂w μ , ∂α ∂v l t l l dV dt = l t l l ΔVΔt (F.396) σαl,t ∂z ∂z σαl,t ∂z ∂z P Δt ΔV

This term is implemented through the source term SC as: 1 μl,t,P μl,t,W ∂αl ∂vl SC,15 = + ΔVΔt (F.397) 2 σαl,t σαl,t ∂z P ∂z P

Term 12 on the RHS of the axial component of the momentum equation: Appendix F: Trondheim Bubble Column Model 1513 ∂ μ , ∂α (v ) ΔVΔt μ , ∂α μ , ∂α v l t l dV dt = l P l t l − l t l l 1 ( + ) ∂r σαl,t ∂z δrNP δrPS σαl,t ∂z n σαl,t ∂z s Δt ΔV 2 (F.398) This term is implemented through the source term SC as: (vl)PΔVΔt μl,t,P ∂αl μl,t,S ∂αl S , = − (F.399) C 16 1 ( + ) σ ∂r σ ∂r 2 δrNP δrPS αl,t n αl,t s

Term 13 on the RHS of the axial component of the momentum equation: ∂ μl,t ∂αl wl,PΔVΔt μl,t ∂αl μl,t ∂αl wl dV dt = − ∂z σαl,t ∂z δzPW σαl,t ∂z e σαl,t ∂z w Δt ΔV (F.400) This term is implemented through the source term SC as: wl,PΔV μl,t,P ∂αl μl,t,W ∂αl SC,17 = − . (F.401) δzPW σαl,t ∂z e σαl,t ∂z w

Pressure force term

The pressure force term in the axial component of the gas momentum equation is discretized in the same way as described when considering the liquid momentum equation in Sect. F.4.5.

Gravity force term

The gravity force term in the gas momentum equation is discretized in the same way as described when considering the liquid momentum equation in Sect. F.4.5.

Virtual mass force term

The virtual mass force term in the gas momentum equation is discretized in the same way as described when considering the liquid phase momentum equation in Sect. F.4.5.

Steady drag force term

The steady drag force term in the axial component of the gas momentum equation is treated in the same way as described when considering the liquid momentum equation, as discussed in Sect. F.4.5. 1514 Appendix F: Trondheim Bubble Column Model

Algebraic discretization equation

After dividing all the terms by Δt, the discretized equation can be written on the standard algebraic form:

= + + + + aPwg,P aN wg,N aSwg,S aEwg,E aW wg,W bwg (F.402)

The coefﬁcients are deﬁned by:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] aE = De + max[−Ce, 0] a = D + max[C , 0] W w w = + 0 0 b SC,m aPwg,P m ΔV a = (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV P Δt g g P N S E W n s e w P q q (F.403)

To avoid negative coefficients, the relation for the coefficient aP can be modified using ∗ the continuity equation, as shown for the liquid phase equations. The alternative aP and b∗ coefficients are defined by: ∗ =− ν + + 0 0 b mC2wg,P m SC,m,g aPwg,P (F.404) ∗ = ΔV ( )0 + + + + + − Δ . aP Δt αgρg P aN aS aE aW mC1 q SP,q,g V

F.4.8 Turbulent Kinetic Energy

The equation for the turbulent kinetic energy is discretized in accordance with the generalized equation in Sect.F.4.3, with ψ = k and phase k = l.

The source terms

αl(Pk + Pb − ρlε) dV dt = αl(Pk + Pb − ρlε)ΔVΔt (F.405) Δt ΔV where

SC,1ΔVΔt =[αl,PPkΔV + αl,PPb]ΔVΔt (F.406) Sp,1ΔVΔt = αl,Pρl,PεΔVΔt (F.407)

The production terms are approximated as: v v v ∂ l 2 ∂wl 2 l 2 ∂ l ∂wl 2 P = μ , , (2[( ) + ( ) + ( ) ]+( + ) ) k l t P ∂r P ∂z P r P ∂z ∂r P where v (v ) − (v ) v − v ∂ l = l n l s = l,N l,S ∂r 1 (δr + δr ) 1 (δr + δr ) P 2 NP PS 2 NP PS ( ) − ( ) − ∂wl = wl e wl w = wl,E wl,P 1 1 ∂z P (δPW + δEP) (δPW + δEP) 2 2 v 1 (v + v ) l = 2 l,P l,N 1 v v r P (r + r ) 2 P N v 1 (v + v + v + v ) − 1 (v + v + v + v ) ∂ l = 4 l,P l,E l,N l,NE 4 l,P l,W l,N l,NW 1 ∂z P (δPW + δEP) 2 1 1 (w , + w , + w , + w , ) − (w , + w , + w , + w , ) ∂wl = 4 l P l E l N l NE 4 l P l W l N l NW ∂r 1 ( + ) P 2 δrNP δrPS (F.408) and P = C [F , ((w ) − (w ) ) + F , ((v ) − (v ) )] b d D z g P l P D r g P l P 1 1 = C F , (w , + w , ) − (w , + w , ) d D z 2 g P g E 2 l P l E (F.409) + ( 1 (v + v ) − 1 (v + v . FD,r 2 g,P g,N 2 l,P l,N

Algebraic discretization equation

After dividing all the terms by Δt, the discretized equation can be written on the standard algebraic form:

= + + + + aPkl,P aN kl,N aSkl,S aEkl,E aW kl,W bkl (F.410) 1516 Appendix F: Trondheim Bubble Column Model

The coefﬁcients are deﬁned by:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] aE = De + max[−Ce, 0] a = D + max[C , 0] W w w = + 0 0 b SC,m aPkl,P m ΔV a = (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV P Δt l l P N S E W n s e w P q q (F.411)

To avoid negative coefficients, the relation for the coefficient aP can be modified using the continuity equation, as shown for the liquid phase velocity equations. The ∗ ∗ alternative aP and b coefficients are defined by: ∗ =− ν + + 0 0 b mC2kl,P SC,m aPkl,P m ∗ = ΔV ( )0 + + + + + − Δ (F.412) aP Δt αlρl P aN aS aE aW mC1 SP,q,l V q

The convective and diffusive ﬂuxes are approximated in the following way:

= ( v ) = 1 ( + )v Cn An αlρl r n An 2 αN ρN αPρP r,N = ( v ) = 1 ( + )v Cs As αlρl r s As 2 αPρP αSρS r,P = ( v ) = 1 ( + )v Ce Ae αlρl z e Ae 2 αEρE αPρP z,E = ( v ) = 1 ( + )v Cw Aw αlρl z w Aw 2 αPρP αW ρW z,P Γ D = An n n δrNP Γ D = As s s δrPS Γ (F.413) D = Ae e e δzEP Γ D = Aw w w δzPW μl,eff 1 Γn = ( )n = (αN μN + αN μN ) σk 2σk μl,eff 1 Γs = ( )s = (αPμP + αSμS) σk 2σk μl,eff 1 Γw = ( )w = (αW μW + αPμP) σk 2σk μl,eff 1 Γe = ( )e = (αPμP + αW μW ). σk 2σk Appendix F: Trondheim Bubble Column Model 1517

F.4.9 Turbulent Kinetic Energy Dissipation Rate

The equation for the turbulent energy dissipation rate is discretized in accordance with the generalized equation in Sect. F.4.3, with ψ = and phase k = l.

The source term

The source terms in the equation for the turbulent kinetic energy dissipation rate are implemented through the source terms SP and SC in the following way: α (C (P + P ) − C ρ ε) dV dt =[α (C (P + P ) − C ρ ε)] ΔVΔt l k 1 k b 2 l l k 1 k b 2 l P Δt ΔV (F.414) where S , ΔVΔt =[α C P + α C P ] ΔVΔt (F.415) C 1 l k 1 k l k 1 b P ε S , ΔVΔt =[α ρ C ] ΔVΔt (F.416) p 1 l l 2 k P

The production terms Pk and Pb are deﬁned in Sect.F.4.8.

Algebraic discretization equation

After dividing all the terms by Δt, the discretized equation can be written on the standard algebraic form:

= + + + + aPεl,P aN εl,N aSεl,S aEεl,E aW εl,W bεl (F.417)

The coefﬁcients are deﬁned by:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] aE = De + max[−Ce, 0] a = D + max[C , 0] W w w = + 0 0 b SC,m aPkl,P m ΔV a = (α ρ ) + a + a + a + a + C − C + C − C − S , ΔV P Δt l l P N S E W n s e w P q q (F.418) 1518 Appendix F: Trondheim Bubble Column Model

To avoid negative coefficients, the relation for the coefficient aP can be modified using the continuity equation, as shown for the liquid phase velocity equations. The ∗ ∗ alternative aP and b coefficients are defined by: ∗ =− ν + + 0 0 b mC2εl,P SC,m aPεl,P m ∗ = ΔV ( )0 + + + + + − Δ (F.419) aP Δt αlρl P aN aS aE aW mC1 SP,q,l V q

The convective and diffusive ﬂuxes are approximated in the following way:

= ( v ) = 1 ( + )v Cn An αlρl r n An 2 αN ρN αPρP r,N = ( v ) = 1 ( + )v Cs As αlρl r s As 2 αPρP αSρS r,P = ( v ) = 1 ( + )v Ce Ae αlρl z e Ae 2 αEρE αPρP z,E = ( v ) = 1 ( + )v Cw Aw αlρl z w Aw 2 αPρP αW ρW z,P Γ D = An n n δrNP Γ D = As s s δrPS Γ (F.420) D = Ae e e δzEP Γ D = Aw w w δzPW μ , Γ = ( l eff ) = 1 (α μ + α μ ) n σ n 2σ N N N N μ , Γ = ( l eff ) = 1 (α μ + α μ ) s σ s 2σ P P S S μ , Γ = ( l eff ) = 1 (α μ + α μ ) w σ w 2σ W W P P μ , Γ = ( l eff ) = 1 (α μ + α μ ). e σ e 2σ P P W W

F.4.10 Volume Fraction

The gas volume fraction is calculated from the continuity equation for phase k which is discretized by the scheme proposed by Spalding [93]. The continuity equation for phase k (k = l, g) is derived in AppendixF:

∂ 1 ∂ ∂ (αkρk) + (rαkρkvk,r) + (αkρkvk,z) ∂t r ∂r ∂z 1 ∂ μ , ∂α ∂ μ , ∂α = r k t k + k t k + S r ∂r σαk ,t ∂r ∂z σαk,t ∂z

In the FEM, this equation is integrated in time and over a grid cell volume. The resulting terms are then approximated in accordance with the approach presented for the generalized equation. The derivatives of the volume fraction in the diffusive terms Appendix F: Trondheim Bubble Column Model 1519 are approximated by central differences and for the convection terms the upwind scheme is employed. The discretized liquid phase continuity equation (k = l) can then be expressed as: ρlΔV α , + ( [C , ]+D ) + ( [−C , ]+D ) l P Δ max n 0 n max s 0 s t

+ (max[Ce, 0]+De) + (max[−Cw, 0]+Dw) ( )o Δ αlρl P V = + α , (max[−C , 0]+D ) + α , (max[C , 0]+D ) Δt l N n n l S s s + αl,E(max[−Ce, 0]+De) + αl,W (max[Cw, 0]+Dw) + SΔV (F.421) where = = ( v ) = v 1 ( + ) Cn AnFn An ρl l n An r,N 2 ρl,P ρl,N 1 Cs = AsFs = As(ρlvl)s = Asvr,P (ρl,P + ρl,S) 2 (F.422) = = ( ) = 1 ( + ) Ce AeFe Ae ρlwl e Aewl,E 2 ρl,P ρl,E = = ( ) = 1 ( + ) Cw AwFw Aw ρlwl w Anwl,P 2 ρl,P ρl,W and Γ D = An n n δrNP Γ D = As s s δrPS Γ D = Ae e e δzEP Γ D = Aw w w δzPW Γ = ( μl,t ) = 1 μl,t,P + μl,t,N n σ n 2 σ σ (F.423) αl,t αl,t αl,t Γ = ( μl,t ) = 1 μl,t,P + μl,t,S s σ s 2 σ σ αl,t αl,t αl,t Γ = ( μl,t ) = 1 μl,t,P + μl,t,W w σ w 2 σ σ αl,t αl,t αl,t μl,t 1 μl,t,P μl,t,E Γe = ( )e = + σαl,t 2 σαl,t σαl,t

For convenience two new variables m and Sl,1 are introduced, hence the equation can be written in a more compact form: αl,P ml,i,ovt − αl,i,inml,i,in − Sl,1 = Rl = 0 (F.424) i i where 1520 Appendix F: Trondheim Bubble Column Model ρlΔV m , , = + (max[C , 0]+D ) + (max[−C , 0]+D ) l i ovt Δt n n s s i + (max[C , 0]+D ) + (max[−C , 0]+D ) e e w w αl,i,inml,i,in = αl,N (max[−Cn, 0]+Dn) + αl,S(max[Cs, 0]+Ds) i + αl,E(max[−Ce, 0]+De) + αl,W (max[Cw, 0]+Dw) ( )o Δ αlρl P V S , = (F.425) l 1 Δt A similar equation can be obtained for the gas phase as well. If both equations are solved for αP yields: + + i αl,i,inml,i,in Sl i αg,i,inmg,i,in Sg αl,P + αg,P = + = 1 (F.426) i ml,i,ovt i mg,i,ovt

With minor manipulation of the equation, we get: ml,i,ovt i ( α , , m , , + S ) m , , + ( α , , m , , + S ) m , , = i l i in l i in l i g iovt i g i in g i in g i l i ovt i mg,i,ovt (F.427) This relation is used to substitute for the i ml,i,ovt term in (F.424), hence we get: ( + ) + ( + ) i αl,i,inml,i,in Sl i mg,i,ovt i αg,i,inmg,i,in Sl i ml,i,ovt αl,P m , , i g i ovt = αl,i,inml,i,in + Sl. (F.428) i

Algebraic discretization equation

The algebraic equation that must be solved for the gas volume fraction variable can thus be written as:

aPαl,P = aN αl,N + aSαl,S + aEαl,E + aW αl,W + Sl (F.429)

The coefﬁcients are deﬁned as follows:

aN = Dn + max[−Cn, 0] aS = Ds + max[Cs, 0] Appendix F: Trondheim Bubble Column Model 1521

aE = De + max[−Ce, 0] aW = Dw + max[Cw, 0] (αlρl)PΔV SC = + SΔV Δt ( + ) + ( + ) i αl,i,inml,i,in Sl i mg,i,ovt i αg,i,inmg,i,in Sg i ml,i,ovt aP = i mg,i,ovt (F.430)

F.4.11 The Pressure-Velocity Correction Equations

The pressure correction equation is derived from the liquid continuity equation and the liquid velocity correction equation formulas. The SIMPLE Consistent (SIM- PLEC) -approximation proposed by van Doormal and Raithby [29] is used to derive the velocity correction formulas. The continuity equation for the liquid phase is given in appendixF. The discretization of this equation is discussed in Sect.F.4.10. The discretized form of the continuity equation thus yields:

[( ) − ( )o ]Δ αlρl P αlρl P V + A (α ρ v , ) Δt n l l l r n − As(αlρlvl,r)s + Ae(αlρlvl,z)e − Aw(αlρlvl,z)w = Dn(αl,N − αl,P) − Ds(αl,P − αl,S) + De(αl,E − αl,P) − Dw(αl,P − αl,W ) + SΔV (F.431)

The pressure correction is given by the difference between the correct pressure, p, and the guessed pressure, p∗. The velocity correction v is given by the difference between the correct velocity, v, and the guessed velocity v∗:

p = p∗ + p (F.432) v = v∗ + v

The relationship between the liquid velocity correction at the grid cell surface e and the pressure corrections, i.e., the velocity correction formula, is given as:

A α , (p − p ) = e l e P E = ( − ) wl,e de pP pE (F.433) al,e − anb nb where Aeαl,e de = (F.434) al,e − al,nb nb 1522 Appendix F: Trondheim Bubble Column Model

The correction formulas for the liquid velocity components in the other directions, as well as the corresponding correction formulas for the gas phase, can be deduced in a similar manner.

Algebraic discretization equation

The transport equation for the pressure corrections is obtained substituting all the velocity components in (F.431) by the sum of the guessed and corrected velocities, and then substituting the velocity corrections by the corresponding pressure corrections employing the liquid phase velocity correction formulas. The resulting algebraic equation to be solved is written:

= + + + + aP pP aN pN aS pS aE pE aW pW b (F.435)

The coefﬁcients are given as:

1 aN = An(α ρ )ndn = Andn (α , ρ , + α , ρ , ) l l 2 l P l P l N l N 1 aS = As(α ρ )sds = Asds (α , ρ , + α , ρ , ) l l 2 l P l P l S l S 1 aE = Ae(α ρ )ede = Aede (α , ρ , + α , ρ , ) l l 2 l P l P l E l E 1 aW = A (α ρ ) d = A d (α , ρ , + α , ρ , ) w l l w w w w 2 l P l P l W l W aP = aN + aS + aE + aW o ((αlρl)P − (αlρl) )ΔV ∗ ∗ ∗ b = P − A (α ρ v ) + A (α ρ v ) − A (α ρ v ) Δt n l l l,r n s l l l,r s e l l l,z e + ( v∗ ) + ( − ) − ( − ) + ( − ) Aw αlρl l,z w Dn αl,N αl,P Ds αl,P αl,S De αl,E αl,P − Dw(αl,P − αl,W ) + SCΔV. (F.436)

References

1. Anderson DA, Tannehill JC, Pletcher RH (1984) Computational fluid mechanics and heat transfer. Hemisphere , New York 2. Aris R (1962) Vectors, tensors, and the basic equations of fluid mechanics. Dover Inc, New York 3. Balzer G, Boëlle A, Simonin O (1995) Eulerian gas-solid flow modelling of dense fluidized bed. Fluidization VIII, international symposium of the engineering foundation, Tours, pp 1125Ð1134, 14Ð19 May 1995 4. Balzer G, Simonin O (1993) Extension of Eulerian gas-solid flow modeling to dense fluidized beds. Rapport HE-44/93.13, Laboratorie National d’Hydraulique, EDF, Chatou, France 5. Balzer G, Simonin O (1996) Turbulent eddy viscosity derivation in dilute gas-solid turbulent flows. 8th workshop on two-phase flow predictions, Merseburg, Germany, 26Ð29 Mar 1996 Appendix F: Trondheim Bubble Column Model 1523

6. Bel F’dhila R, Simonin O (1992) Eulerian prediction of a turbulent bubbly flow downstream of a sudden pipe expansion. Proceedinds of 6th workshop on two-phase flow predictions, Erlangen, FRG, pp 264Ð273, 30 MarchÐ2 April 1992 7. Bird RB, Stewart WE, Lightfoot EN (1960) Transport phenomena. Wiley, New York 8. Bird RB, Stewart WE, Lightfoot EN (2002) Transport phenomena, 2nd edn. Wiley, New York 9. Boisson N, Malin MR (1996) Numerical prediction of two-phase flow in bubble columns. Int J Numer Meth Fluids 23:1289Ð1310 10. Borisenko AI, Tarapov IE (1979) Vector and tensor analysis with applications (trans and edited: Silverman RA). Dover Publication Inc, New York 11. Bouillard JX, Lyczkowski RW, Folga S, Gidaspow D, Berry GF (1989) Hydrodynamics of erosion of heat exchanger tubes in fluidized bed combustor. Can J Chem Eng 67(2):218Ð229 12. Boure` JA, Delhaye JM (1982) General equations and two-phase flow modeling. In: Hetsroni G (ed) Handbook of multiphase systems, section 1.2, McGraw-Hill, New York, pp 1-36Ð1-95 13. Brenner H (1979) A Micromechanical Derivation of the Differential Equation of Interfacial Statics. Journal of Colloid and Interface Science 68(3):422Ð439 14. Carnahan NF, Starling KE (1969) Equation of state for nonattracting rigid spheres. J Chem Phys 51(2):635Ð636 15. Chao Z (2012) Modeling and simulation of reactive three-phase flows in fluidized bed reactors: application to the SE-SMR process. Doctoral thesis, the Norwegian Univerity of Science and Technology, Trondheim, Norway 16. Chapman S, Cowling TG (1970) The mathematical theory of non-uniform gases, 3rd edn. Cambridge University Press, Cambridge 17. Chapman S, Cowling TG (1970) The mathematical theory of non-uniform gases, 3rd edn. Cambridge Mathematical Library, Cambridge 18. Clift R, Grace JR, Weber ME (1978) Bubble drops, and particles. Academic Press, New York 19. Clift R, Grace JR (1985) Continuous bubbling and slugging. In: Davidson JF, Clift R, Harrison D (eds) Fluidization, Academic Press, London 20. Collado FJ (2007) Reynolds transport theorem for two-phase flow. Appl Phys Lett 90:024101 21. Csanady GT (1963) Turbulent diffusion of heavy particles in the atmosphere. J Atm Sci 20:201Ð208 22. Delhaye JM, Achard JL (1977) On the averaging operators introduced in two-phase flow. In: Banerjee S, Weaver JR (eds) Transient two-phase flow. Proceeding of CSNI specialists meeting, Toronto, 3Ð4 Aug 1997 23. Delhaye JM (1981) Basic equations for two-phase flow modeling. In: Bergles AE et al (eds) Two-phase flow and heat transfer in the power and process industries. Hempsherer Publishing, Washington 24. Delhaye JM (1974) Jump Conditions and Entropy Sources in Two-Phase Systems: Local Instant Formulation. Int J Multiphase Flow 1:395Ð409 25. Deutsch E, Simonin O (1991) Large Eddy Simulation applied to the motion of particles in steady homogeneous turbulence. Turbulence Modification in Multiphase Flow, ASME FED 1:34Ð42 26. Ding J, Gidaspow D (1990) A bubbling fluidization model using kinetic theory of granular flow. AIChE J 36(4):523Ð538 27. Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, Chichester 28. Donea J, Huerta A, Ponthot J-P, Rodríguez-Ferran A (2004) Arbitrary Lagrangian-Eulerian Methods. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of Computational Mechan- ics, Volume 1: Fundamentals, Chapter 14, pp 1Ð25, John Wiley & Sons Ltd ISBN: 0-470- 84699-2. 29. van Doormal JP, Raithby GD (1984) Enhancement of the SIMPLE method for predicting incompressible fluid flows. Numer Heat Transfer 7:147Ð163 30. Drew DA, Lahey RT Jr (1993) Analytical modeling of multiphase flow. Ed Roco MC particulate two-phase flow, chapter, Butterworth-Heinemann, Boston 16:509Ð566 31. Drew DA (1983) Mathematical modeling of two-phase flow. Ann Rev Fluid Mech 15:261Ð291 1524 Appendix F: Trondheim Bubble Column Model

32. Edwards CH Jr, Penny DE (1982) Calculus and analytic geometry. Prentice-Hall Inc, Engle- wood Cliffs, New Jersey 33. Edwards DA, Brenner H, Wasan DT (1991) Interfacial transport processes and rheology. Butterworth-Heinemann, Boston 34. Elghobashi SE, Abou-Arab TW (1983) A two-equation turbulence model for two-phase flows. Phys Fluids 26:931Ð938 35. Enwald H, Peirano E, Almstedt A-E (1996) Eulerian two-phase flow theory applied to fluidization. Int J Multiphase Flow 22(Supplement):21Ð66 36. Enwald H, Almstedt AE (1999) Fluid dynamics of a pressurized fluidized bed: comparison between numerical solutions from two-fluid models and experimental results. Chem Eng Sci 54:329Ð342 37. Enwald H, Peirano E, Almstedt A-E, Leckner B (1999) Simulation of a bubbling fluidized bed. Experimental validation of the two-fluid model and evaluation of a parallel multiblock solver. Chem Eng Sci 54:311Ð328 38. Enwald H, Peirano E, Almstedt AE (1996) Eulerian two-phase flow theory applied to fluidization. Int J Multiphase Flow 22:21Ð66, Suppl 39. Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48(2):89Ð94 40. Fan L-S, Zhu C (1998) Principles of gas-solid flows. Cambridge University Press, Cambridge 41. Gibilaro LG, Di Felice RI, Waldran SP (1985) Generalized friction factor and drag coefficient correlations for fluid-particle interactions. Chem Eng Sci 40:1817Ð1823 42. Gidaspow D (1994) Multiphase flow and fluidization-continuum and kinetic theory descrip- tions. Academic Press, Harcourt Brace & Company Publishers, Boston 43. Gosman AD, Ideriah FJK (1976) TEACH-T: a general computer program for two dimensional turbulent recirculating flows. Mechanical engineering department, Imperial College, London 44. Grad H (1949) On the kinetic theory of rarified gases. Comm Pure Appl Math 2(4):331Ð407 45. Greenberg MD (1978) Foundations of applied mathematics. Prentice-Hall Inc, Englewood Cliffs 46. Grienberger J (1992) Untersuchung und Modellierung von Blasensäulen. Doctering thesis, Der Technischen Fakultät der Universität Erlangen-Nürnberg, Germany 47. He J, Simonin O (1993) Non-equilibrium prediction of the particle-phase stress tensor in vertical pneumatic conveying. Gas-Solid Flows, ASME FED 166:253Ð263 48. He J, Simonin O (1994) Modélisation numérique des écoulements gaz-solides en conduite verticale. Rapport HE-44/94/021A, Laboratoire National d’Hydraulique, EDF,Chatou, France 49. Irgens F (2001) Kontinuumsmekanikk. Institutt for mekanikk, thermo- og fluiddynamikk, Norges teknisk- naturvitenskapelige universitet, Trondheim. 50. Irgens F (1982) Kontinuumsmekanikk Del III: Tensoranalyse. Tapir, Trondheim 51. Jakobsen HA (1993) On the modelling and simulation of bubble column reactors using a two- fluid model. Doctering thesis, the Norwegian Institute of Technology, Trondheim, Norway 52. Jenkins JT, Richman MW (1985) Grad’s 13 moment system for a dense gas of inelastic spheres. Arch Ratio Mech Anal 87:355Ð377 53. Johansen ST, Boysan F (1988) Fluid Dynamics in Bubble Stirred Ladles: Part 2. Mathematical Modelling. Met Trans B 19:755Ð764 54. Kuiken GDC (1995a) Thermodynamics of irreversible processes. Applications to diffusion and rheology. Wiley, Chichester 55. Kundu PK (1990) Fluid mechanics. Academic Press Inc, San Diego 56. Kuo KK (1986) Principles of combustion. Wiley, New York 57. Lathouwers D, Bellan J (2001) Modelling of dense gas-solid reactive mixtures applied to biomass pyrolysis in a fluidized bed. Int J Multiphase Flow 27:2155Ð2187 58. Launder BE, Spalding DB (1972) Mathematical models of turbulence. Academic Press, Lon- don 59. Lindborg H (2008) Modeling and simulation of reactive two phase flows in fluidized beds. Doctering thesis, the Norwegian University of Science and Technology, Trondheim, Norway 60. Lun CKK, Savage SB (1986) The effect of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials. Acta Mechanica 63:15Ð44 Appendix F: Trondheim Bubble Column Model 1525

61. Ma D, Ahmadi G (1986) An equation of state for dense rigid sphere gases. J Chem Phys 84(6):3449Ð3450 62. Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall Inc, Englewood Cliffs 63. Malvern LE (1969) Introduction to the mechanics of a continuum medium. Prentice-Hall Inc, Englewood Cliffs 64. Manger E (1996) Modelling and simulation of gas/solid flow in curvilinear coordinates. Doctering thesis, Norwegian University of Science and Technology, Porsgrunn 65. Massoudi M, Rajagopal KR, Ekmann JM, Mathur MP (1992) Remarks on the modeling of fluidized systems. AIChE J 38(3):471Ð472 66. Middleman S (1998) An introduction to fluid dynamics: principles of analysis and design. Wiley, New York 67. Miller CA, Neogi P (1985) Interfacial phenomena: equilibrium and dynamic effects. Marcel Dekker Inc, New York and Basel 68. Mudde RF, Simonin O (1999) Two- and three-dimensional simulations of a bubble plume using a two-fluid model. Chem Eng Sci 54:5061Ð5069 69. Munson BR, Young DF, Okiishi TH (2002) Fundamentals of fluid mechanics, 4th edn. Wiley,New York 70. Ni J, Beckermann C (1990) A two-phase model for mass, momentum, heat, and species transport during solidification. In: Charmchi M, Chyu MK, Joshi Y,Walsh SM (eds) Transport phenomena in material processing, New York. ASME HTD-VOL. 132:45Ð56 71. Nobari MR, Jan Y-J, Tryggvason G (1996) Head-on collision of droplets—a numerical inves- tigation. Phys Fluids 8(1):29Ð42 72. Ogawa S, Umemura A, Oshima N (1980) On the equations of fully fluidized granular materials. J Appl Math Phys 31:483Ð493 73. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing Corpora- tion, New York 74. Peirano E (1996) The Eulerian/Eulerian formulation applied to gas-particle flows. Report A96Ð218, ISSN 0281Ð0034, Department of Energy Conversion, Chalmers Univesity of Tech- nology, Sweden 75. Peirano E (1998) Modelling and simulation of turbulent gas-solid flow applied to fluidization. PhD thesis, Chalmers Univesity of Technology, Sweden 76. Peirano E, Leckner B (1998) Fundamentals of turbulent gas-solid flows applied to circulating fluidized bed combustion. Prog Energy Combust Sci 24:259Ð296 77. Prosperetti A, Jones AV (1984) Pressure forces in dispersed two-phase flow. Int J Multiph Flow 10(4):425Ð440 78. Reynolds O (1903) Papers on mechanical and physical subjects-the sub-mechanics of the Universe, collected work, vol. III. Cambridge University Press, Cambridge 79. Rosner DE (1986) Transport processes in chemically reacting flow systems. Butterworths, Boston 80. Rottmann K (1960) Mathematische Formelsammlung, 2nd edn. BI-Wissenschaftsverlag, Bib- liographisches Institut Mannheim, Deutschland 81. Sandler SI (1999) Chemical and engineering thermodynamics, 3rd edn. Wiley, New York 82. Simonin O (1990) Eulerian formulation for particle dispersion in turbulent two-phase flows. In: Sommerfeld M, Wennerberg P (eds) Fifth workshop on two-phase flow predictions. Erlangen, FRG, pp 156Ð166 83. Simonin O (1995) Two-fluid model approach for turbulent reactive two-phase flows. Sum- mer school on numerical modelling and prediction of dispersed two-phase flows, IMVU, Merseburg, Germany 84. Simonin O, Viollet PL (1990) Prediction of an oxygen droplet pulversization in a compressible subsonic coflowing hydrogen flow. Numerical methods for multiphase flows. ASME FED 91:73Ð82 85. Simonin O, Flour I (1992) An Eulerian approach for turbulent reactive two-phase flows loaded with discrete particles. In: Sommerfeld M Sixth workshop on two-phase flow predictions. Erlangen, pp 61Ð62 1526 Appendix F: Trondheim Bubble Column Model

86. Simonin O, Viollet PL (1989) Numerical study on phase dispersion mechanisms in turbulent bubbly flows. Proceeding of international conference on mechanics of two-phase flows, Taipei, Taiwan, 12Ð15 June 1989 87. Slattery JC (1972) Momentum, energy, and mass transfer in Continua, 2nd edn. McGraw-Hill Kogakusha LTD, Tokyo 88. Slattery JC (1972) Momentum, energy, and mass transfer in continua. McGraw-Hill Book Company, New York 89. Slattery JC (1990) Interfacial transport phenomena. Springer, New York 90. Slattery JC (1999) Advanced transport phenomena. Cambridge University Press, Cambridge 91. Spalding DB (1977) The calculation of free-convection phenomena in gas-liquid mixtures. ICHMT seminar, (1976) In: Turbulent buoyant convection. Hemisphere, Washington, pp 569Ð 586 92. Spalding DB (1980) Numerical computation of multiphase fluid flow and heat transfer. In: Morgan K, Taylor C (ed) Recent advances in numerical methods in fluids, Pineridge Press, Swansea, pp 139Ð167 93. Spalding DB (1981) IPSA 1981: new developments and computed results. Report HTS/81/2, Imperial College of Science and Technology, London 94. Stull RB (1988) An introduction to boundary layer meteorology. Kluwer Academic Publishers, Dordrecht 95. Svendsen HF, Jakobsen HA, Torvik R (1992) Local flow structures in internal loop and bubble column reactors. Chem Eng Sci 47(13Ð14):3297Ð3304 96. Thompson PA (1972) Compressible-fluid dynamics. McGraw-Hill Inc, New York 97. Torvik R, Svendsen HF (1990) Modeling of slurry reactors—a fundamental approach. Chem Eng Sci 45(8):2325Ð2332 98. Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S Handbuch der Physik, vol III/1, principles of classical mechanics and field theory, Springer, Berlin 99. Whitaker S (1968) Introduction to fluid mechanics. Prentice-Hall Inc, Englewood Cliffs 100. White FM (1999) Fluid mechanics, 4th edn. McGraw-Hill, Boston 101. Zapryanov Z, Tabakova S (1999) Dynamics of bubble, drops and rigid particles. Kluwer academic publishers, Dordrecht Index

A wall lift force, 919 Activity coefficient, 804 Bulk expansion coefficient, 69 Affinity, thermodynamic forces, 64 Agitation, 809 Algebraic-slip mixture model, 505 C Angular momentum balance, 66 Capillary number, 708 Averaging, 428 Cauchy equation of motion, 251 area averaging, 86, 90, 93, 512 Chemical reaction engineering (CRE), 370, ensemble averaging, 118, 464 789 statistics, 117 Chemical reaction equilibrium, 796 time averaging, 117, 454 Chilton-Colburn relation, 768 time- after volume averaging, 477 Classical thermodynamics, 37 volume averaging, 117, 431 Closure law constitutive, 687 constitutive equations, 687 coordinate invariance, 688 B correct low concentration limits, 688 Balance equations, 5 dimensional invariance, 688 Balance laws, 6 entropy inequality, 688 Balance principle, 11 equipresence, 687 Bernoulli equation, 82 interfacial momentum transfer, 688 Blending, 809 material frame indifference, 687 Bond number, 707 objectivity, 687 Bubble column phase separation, 688 applications, 892 topological, 687 bubble wall friction force, 919 transfer, 687 conventional modeling, 892 well-posedness, 687 Danckwerts boundary conditions, 894 Complete differential, 54 design, 883, 890 Compressible Experimental characterization, 886 flow, 3 flow regimes, 886 fluid, 4 fluid dynamic modeling, 895 Concentration diffusion, 19 forces on bubbles, 918 Conservative forces, 46 hydrodynamics, 883 Constitutive equations, 6 turbulence modeling, 920 Continuous stirred tank reactor model (CSTR), turbulent dispersion, 919 371

H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1527 © Springer International Publishing Switzerland 2014 1528 Index

Continuous surface force (CSF), 385 fiber-optic probe technique, 1318 Continuum mechanics, 5 high-speed camera imaging, 1343 Continuum surface stress (CSS), 385 hot-film anemometry, 1276 Control volume laser diffraction technique, 1335 arbitrary Lagrangian-Eulerian (ALE), 9 laser-Doppler anemometry, 1279 Eulerian, 9 model validation, 1275 Lagrangian, 9 particle image velocimetry, 1330 material, 9 phase-Doppler anemometry, 1290 Control volume approach, 8, 11 photography imaging, 1343 CRE Extent of reaction, 57, 801 batch reactor, 795 Danckwerts boundary conditions, 796 Ergun equation, 793 F Gibbs reactor, 807 Fanning friction factor, 86, 518 simplified models, 795 Fick’s law, 733 superficial velocity, 791 Fluid mechanics, 3 Curvature Fluid particle diameter mean curvature of surface, 382 drag diameter, 693 principal curvatures of a surface, 382 Sauter mean diameter (SMD), 693 principal radii of curvature, 382 surface diameter, 693 Curvilinear coordinate systems, 1440 volume equivalent diameter, 693 Fluidization, 1005 bubble rise velocity, 1032 D bubble size, 1037 Damko¨hler number, 838 bubbling fluidization, 1008 Danckwerts boundary conditions, 1040, 1042, 1050 CFB overall pressure balance, 1051 Darcy friction factor, 92, 518 channeling, 1009 Darcy’s law, 329 cyclone, 1010 Darcy-Weisbach equation, 828 Davidson-Harrison model, 1035 Degrees of freedom, 193 dense phase fluidization, 1008 Diffusion mixture model, 507 dilute transport fluidization, 1009 Dirac delta function, 383 dipleg, 1010, 1011 Dirichlet boundary conditions, 1098 downcomer, 1013, 1014, 1051, 1052 Dispersed flows, 373 fast fluidization, 1009 Dispersion reactor models, 371 flow regimes, 1007 axial, 98 fluidized bed advantages, 1029 heterogeneous, 522 fluidized bed combustor, 1020 pseudo-homogeneous, 523 fluidized bed disadvantages, 1029 Drift-flux model, 510 fluidized bed types, 1017 Dufour effect, 42, 271 freeboard region, 1008, 1009, 1011, 1024, 1051 gas distributor, 1010, 1037 E gas-solid separator, 1013 Eötvös number, 707 gasifier, 1021, 1023, 1025 Embedded interface method, EI, 377, 394 Geldart classification of particles, 1006 Ergodic hypothesis, 118 gulf streaming, 1034 Eulerian-Eulerian models, 373 heat exchanger, 1010 Eulerian-Lagrangian models, 373, 374 hopper, 1014, 1016, 1051 Excess property, 407 lean phase fluidization, 1009 Experimental minimum fluidization, 1007, 1018, 1031 CARPT technique, 1296 minimum fluidization velocity, 1031 computed tomography, 1300 particle drift velocity, 1405, 1413 conductivity probe technique, 1315 regenerator, 1014, 1015, 1022, 1026 Index 1529

riser, 1013Ð1016, 1018, 1020, 1022Ð1024, granular temperature dissipation, 559 1026, 1050, 1051 hindrance effect, 633 standpipe, 1011, 1014, 1022 impermeable wall, 582 three-zone model, 1044 inelastic binary particle collisions, 554 transport reactor, 1014 inelastic particle collisions, 539 turbulent fluidization, 1008 Maxwell’s transport equation, 556 two-zone model, 1031, 1039 Maxwell-Enskog equation, 637 two-zone theory, 1018 Maxwellian distribution of particles, 551 van Deemter two-zone model, 1042 modulus of elasticity, 598, 610, 1407 wake region, 1037 moment method, 542 Forced diffusion, 20 pair distribution function, 539, 545, 549, Fourier’s law, 733 646 Free surface flow, 382 particle and gas turbulence (PGT) model, Front tracking method, FT, 377 1401 Fugacity, 802 particle and gas turbulence with drift Fugacity coefficient, 804 velocity (PGTDV) model, 1401 particle drift velocity, 587 particle fluctuating velocity energy, 558 G particle mean speed, 553 Galilean transformation, 64 particle relaxation time, 609 Gauss’ theorem, 1366 particle turbulence (PT) model, 1401 perturbation function for particles, 568 divergence theorem, 1366 quasi-static flow regime, 538 Green’s theorem, 1366 radial distribution function, 546 Ostrgradsky’s theorem, 1366 rapid flow regime, 538 surface, 1368 reactive flows, 583, 594, 622, 625Ð627, Generalized Eulerian transport equation, 11 630, 669 Generalized transport theorem, 413 restitution coefficient, 584 Granular flow species effective diffusivity, 621 binary particle density segregation, 662 specific heat for dilute granular fluid, 579 binary particle segregation, 589, 661 Tchen theory, 587 binary particle size segregation, 664 thermal heat conductivity, 619 Boltzmann equation, 540, 541 total binary drag, 633 cold flow, 542, 602, 612, 631, 661, 663 total granular pressure tensor, 577 collision cylinder, 543 total translational energy equation, 558 collision operator formula, 564 undisturbed flow method, 585 collisional pressure, 541 velocity fluctuations, 538 collisional rate of change, 543 velocity moments, 555 constant particle viscosity (CPV) model, wall heat transfer coefficient, 620 584, 1401 wall velocity slip, 582 crossing trajectory effect, 586 dense phase approach, 541 dilute heat conductivity, 653 H dilute phase approach, 541 Hagen-Poiseuille law, 121 dilute viscosity, 569, 653 Heat of reaction, 59 dissipation by collisional in-elasticity, 539 Heaviside function, 391 Enskog’s transport equation, 565 High resolution models, 374 fluid particle interactions, 541 Hydraulic diameter, 92 frictional binary drag, 591 frictional pressure tensor, 575 granular flow, 537 I granular heat flux, 578 Ill-posed model system, 524 granular material, 537 Incompressible, 3 granular temperature, 538, 558 flow, 4, 67 1530 Index

fluids, 4 apse-line, 226 Integro-differential equation, 209 BBGKY-hierarchy, 204 Interface model bi-molecular collision rate, 236 macroscopic 2D dividing surface, 405 Boltzmann equation, 183, 187, 205, 209, microscopic 3D transition region, 405 210, 239, 240, 247 Interfacial coupling, 375 Boltzmann equation, collision term, 237 four way, 375 Boltzmann equation, gas dynamics, 250 one way, 375 Boltzmann equation, molecular collision, two way, 375 215, 217 Interfacial transfer Boltzmann equation, multicomponent, 266 Chilton-Colburn analogy, 759 Boltzmann equation, solution, 258 combined mass transfer flux, 727 Boltzmann equation, summation invariant, contact area, 694 253 engineering thermal radiation formula, 779 Boltzmann stosszahlansatz, 216 engineering transfer coefficients, 745 bulk density, 635 Fick’s second law, 736 Burnett equations, 187 film theory, 726, 747 canonical transformation, 202 Frössling equation, 769 center of force, 224, 228, 229, 234, 237, gas-side mass transfer coefficient, 745 240, 346 gradient hypothesis, 761 center of mass frame, 222, 223, 237, 347 heat transer coefficient concept, 740 Chapman-Enskog method, 187, 258 heat transfer models, 723 classical mechanics, 190 heat transport, 732, 764 Clausius, 187 interfacial area concentration, 694 coefficient of restitution, 221 jump conditions, 723 collision cylinder, 239, 316 laminar boundary layer theory, 753 collision impact parameter, 240 liquid side mass transfer coefficient, 744 configuration space, 200 mass transfer coefficient concept, 740 continuum hypothesis, 341 mass transfer models, 723 dense gas, 345, 543 mass transport, 732 differential collision cross section, 228, 232 mixture heat transfer, 730 diffusive flux vector, 267 Newton’s law of cooling, 728 dilute gas, 183, 189 penetration theory, 750 dilute gas hypothesis, 340 Reynolds analogy, 759 elastic collision, 206 species mass transfer, 728 Enskog equation, 188, 250, 350 steady diffusion, 734 Enskog equation for dense gas, 345 surface-renewal theory, 749 Enskog equation, multicomponent, 268 transport coefficients, 734 Enskog expansion method, 258 turbulent boundary layer theory, 759 Enskog’s theory, 183 unsteady diffusion, 736 entropy, 257 Irreversible thermodynamics, 37 equation of change, 246 Isentropic, 84 equation of change for multicomponent Isothermal compressibility, 68 mixture, 268 equilibrium flow, 183 equipartition theorem, 246 J Euler equations, 260 Jump condition formulation, 378 flux vector, 241, 242 H-theorem, 187, 216, 254, 255, 258 hydrostatic pressure, 244 K ideal gas law, 341, 342, 344 Kinetic theory, 186, 205 impact parameter, 229 χ-factor, 546 in-elastic collision, 206, 221 absolute Maxwellian, 256 inverse molecular collision, 225 Index 1531

Jacobian determinant, 212 self-diffusion, 337 joint-probability density, 204 shear stresses, 243 kinetic temperature, 344 solid angle, 226 laboratory frame, 217, 219 temperature, 246 Liouville equations, 202Ð204 thermodynamic pressure, 243 Liouville law, 240, 348 total scattering cross section, 232 Liouville theorem, 201, 202, 207, 210, 247 transport properties, 331, 338 Lioville equation, 200 velocity distribution function, 186, 207 local equilibrium, 215 Knudsen number, 340 local Maxwellian, 256 Maxwell molecules, 187 Maxwell’s equations of transfer, 187 L Maxwell’s moment equation, 187 Lagrange multiplier, 800 Maxwell’s transport equation, 188 Least squares method, 1099 Maxwell-Boltzmann equation, 183, 245 Leibnitz theorem, 1364 Maxwellian average, 208, 246, 250 Leibnitz’ theorem Maxwellian distribution function, 259 surface, 1367 Maxwellian molecules, 207 Leibnitz’s integral rule, 1361 Maxwellian velocity distribution, 254, 256 Level set method, LS, 377, 390 mean free path, 183, 187, 331, 334 mean pressure, 243 molecular billiard ball model, 206 M molecular chaos, 215 Mach number, 72 molecular collision, 212 Macro mixing, 837 molecular collision density, 235 Maker and cell method, MAC, 377, 379 molecular collision frequency, 235, 236 Mass diffusion molecular flux/intensity, 229 Bosanquet formula, 304 molecular hard sphere model, 206 Bulk diffusion, 322 molecular interaction potential, 205, 212, diffusion barrier, 291 218, 219 diffusion velocity, 266 molecular state vector, 208 dusty gas model, 269, 303, 305 mono-atomic gas, 253 Fick’s law, 269, 270 multicomponent diffusion, 267 forced diffusion, 299 multicomponent fluid dynamics, 269 generalized Fick’s law, 269, 270 multicomponent mass diffusion, 269 Knudsen diffusion, 304, 322 multicomponent mixtures, 264 Knudsen diffusivity, 304 mutual diffusion, 337 Maxwell’s diffusion force, 316 Navier-Stokes equations, 264 Maxwell’s friction coefficient, 315 non-equilibrium flow, 183 Maxwell-Stefan model, 269, 288, 290 normal stresses, 243 ordinary diffusion, 299 pair distribution function, 216, 234, 239 osmotic diffusion, 291 peculiar speed, 335, 344 pellet pore structure models, 325 peculiar velocity, 209 pressure diffusion, 299 perturbation function, 270 reverse diffusion, 291 phase space, 200Ð203, 207, 212 surface diffusion, 322 Poincare` theorem, 201, 212 thermal diffusion, 299 population balance, 358 Wilke model, 269, 300 pressure tensor, 243, 263 Wilke-Bosanquet model, 269, 304 range of molecular interaction, 218 Mean free path, 5 reduced mass, 222, 236 Mechanics, 184 relative molecular velocity, 223 added mass force, 689, 716 scattering angles, 225 angular momentum, 226 scattering cross section, 228 BBO equation, 689 1532 Index

Besset history force, 689 Newtonian mechanics, 190 body force, 689 non-linear dynamics, 184 centrifugal force, 192 normal stresses, 243 centripetal force, 192 particle drag coefficient relations, 699 chaos, 184 phase space, 207 classical mechanics, 184 phase trajectory, 200 conservative force, 191 Poisson bracket, 202 continuum mechanics, 184 pressure gradient term, 689 Coriolis force, 192 pressure tensor, 243 creeping flow, 708 quantum mechanics, 184 deviatoric stresses, 243 Saffman lift force, 700 drag coefficient definition, 697 shape regime map bubble/drops, 708 drag force, 689 shear stresses, 243 dynamics, 184 solid mechanics, 184 ensemble, 200 standard drag curve, 698 ergodic hypothesis, 186 standard steady drag, 696 external pressure gradient force force, 689 statics, 184 fluid mechanics, 184 statistical mechanics, 185, 190 form drag, 691, 694 steady drag force, 689 friction drag, 691, 694 Stokes flow, 694 Galileo law, 190 terminal velocity of air bubbles in water, Galileo’s principle, 191 710 generalized coordinates, 192, 193 thermodynamic pressure, 244 generalized drag force, 689, 691, 694 Tomiyama lift force, 713 generalized velocities, 194 virtual mass force, 689, 716 gravity force, 689 viscous stresses, 243, 244 Hamilton’s equations of motion, 198 wake lift of deforming bubble, 713 Hamilton’s integral principle, 193 wall interaction force, 689 Hamilton’s variational principle, 194 wall lift force, 714 Hamiltonian, 219 Method of manufactured solutions, 1091 Hamiltonian mechanics, 190, 197, 202, 207 Micro mixing, 837 history force, 721 Mixing, 809 holonomic constraints, 192 Mixture model, 501 homogeneity of time, 197 Molar heat of formation, 60 homogenity of space, 197 Momentum balance, 26 hydrostatic pressure force, 689 Morton number, 708 interfacial momentum transfer due to phase Moving bed, 1005 change, 722 Multifluid model, 376, 425 isotropy of space, 197 Multiphase control volume, 407 kinematics, 184 kinetics, 184 Lagrangian function, 194, 232 N Lagrangian mechanics, 190, 193 Neumann boundary conditions, 1098 lift force, 689, 692 Newton’s law, 733 Magnus lift force, 700 Newton’s second law, 374 Maxey-Riley equation, 705 Newton’s third law, 376 mean pressure, 243 Number density of particles, 693 momentym transfer, 762 Numerical methods Newton, 190 approximation function, 1099 Newton’s first law, 190, 191 arithmetic mean values, 1177 Newton’s second law, 190, 191, 689 basis function, 1099 Newton’s third law, 190, 191 boundary-value problems, 1095 Newton’s viscousity law, 245 boundedness, 1094 Index 1533

central difference scheme, 1133 Rayleigh-Ritz method, 1099 collocation method, 1099, 1102 Runge-Kutta methods, 1126 convergent, 1093 strong form, 1110 deferred correction method, 1135 tau method, 1099 density-based methods, 1116 test function, 1099 domain decomposition parallelization trail function, 1099 method, 1261 TVD schemes, 1138 FCT schemes, 1137 upwind differencing scheme, 1132 finite difference method, 1096 von Neumann method, 1093 finite volume method, 1098 weak form, 1110 fractional step methods, 1117 weight function, 1099 Galerkin method, 1099, 1107 Nusselt number, 746 Gauss-Seidel point iteration method, 1247 initial value problem, 1095 initial-boundary-value problems, 1095 O Jacobi point iteration method, 1247 Onsager Jacobi preconditioner, 1251 phenomenological coefficients, 274 Krylov subspace methods, 1250 reciprocal relations, 281 least-squares method (LSQ), 1106 method of lines, 1122 P method of moments, 1108 Packed bed reactor method of weighted residuals, 1089, 1099 dispersion models, 1061, 1062 multigrid solvers, 1256 heterogeneous reactor model, 1067 multistep methods, 1126 hot spot, 1058 numerical accuracy, 1094 methanol process, 1070 numerical stability, 1093 multi-bed, 1058 ODE solution methods, 1124 multi-tube, 1058 orthogonal collocation, 1100 pseudo-homogeneous models, 1062 PBE runaway, 1058 collocation method, 1233 SE-SMR, 1078 direct quadrature method of moments single-bed, 1058 (DQMOM), 1189, 1196 steam methane reforming, 1073 finite volume method, 1189, 1210 Partial molar enthalpy, 59 fixed pivot method, 1189, 1205 Partial specific enthalpy, 58 Galerkin method, 1233, 1234 Particle Reynolds number, 708 least squares method, 1189 Pellet equations, 321, 328 least-squares method (LSQ), 1222 Perimeter, 92 mean value theorem, 1207 Phase change, interfacial momentum transfer orthogonal collocation method, 1235 closure, 722 quadrature method of moments Phase field model (PF), 398 (QMOM), 1189, 1192 Plug flow reactor model, PFR, 371 sectional method, 1202 Polar coordinate frame, 224 sectional quadrature method of Population balance moments, 1189, 1198 breakage probability, 963 standard method of moments, 1189, coalescence density closures, 979 1190 coalescence time, 954 tau method, 1233, 1234 collision time, 955 weighted residual methods, 1211 diffusion terms, 971 predictor-corrector methods, 1126 effective swept volume rate, 948 pressure-based methods, 1116 macroscopic birth and death term closures, projection methods, 1117 946 quadrature formulas, 1119 microscopic birth and death term closures, QUICK scheme, 1134 971 1534 Index

modeling frameworks, 942 Thermodynamics, 37, 39, 254 moment transform of PBE, 995 activity, 283 multiple properties, 994 Carnot cycle, 188 size property, 991, 992 chemical reaction equilibrium, 796 time after volume average PBE, 970 Clausius, 188 Pressure diffusion, 20 entropy, 188, 277 equilibrium, 277 excess Gibbs energy, 285 R first law, 40 Radii of curvature, 412 fugasity, 283 Reactor flow characteristics, 372 Gibbs free energy, 276, 277 Realizable models, 1094 Gibbs-Duhem equation, 276, 277 Reversible adiabatic, 84 Helmholtz energy, 277 Reynolds number, 746 irreversible thermodynamics, 256, 277 Robin boundary conditions, 1098 latent heat of vaporization of mixtures, 730 non-ideal gas, 344 partial mass chemical potential, 276 S partial mass property, 278 Scalar quantity, 1441 partial molar property, 278 Separated flows, 373 second law, 61, 63, 188 Sheerwood number, 746 statistical thermodynamics, 186 Soret effect, 42, 271 transport property, 331 Specific chemical potentials, 62 van der Waal equation of state, 345 Specific enthalpy, 52 Torque, 701, 817, 818 Specific entropy, 39 Transport processes, 187 Specific molar enthalpy, 58 Turbulence Speed of sound, 72 k-ε model, 137 Stanton number, 746 auto-covariance, 105 Steady drag force, 691 Steady flow, 67 autocorrelation coefficient, 105 Superficial velocity, 523 Batchelor spectrum, 839 Surface tension, 417 Boussinesq turbulent viscosity hypothesis, static force balance, 1369 761, 764 Surface theorem, 413 buffer layer, 124 Symmetry of stress tensor, 66 coherent structures, 103 System approach, 8 cross-term stress, 168 definitions, 99 dispersion force, 919 T eddy concept, 105 Temperature equation, 1377 eddy turnover time, 112 Tensor quantity, 1441 eddy viscosity hypothesis, 920 Tensor transformation laws, 1440 energy cascade, 106 Thermal diffusion, 20 energy dissipation rate, 109, 112 Thermal radiation, 771 energy spectrum, 104, 114 absorptivity, 777 Eulerian longitudinal integral length scale, blackbody, 776 108 emissivity, 776 Eulerian transverse integral length scale, gray surface, 778 108 incident, 775 friction velocity, 123 irradiation, 775 gradient transport hypothesis, 160, 761 Kirchhoff’s law, 778 homogeneous turbulence, 107 Lambert’s cosine law, 776 inner wall layer, 123 radiosity, 775 intensity, 119 Stefan-Boltzmann law, 776 isotropic turbulence, 107 Index 1535

Kolmogorov five-third law, 115 transverse autocorrelation function, 108 Kolmogorov hypotheses, 113 turbulent dispersion, 919 Kolmogorov microscales, 113 two-point correlation function, 108 Kolmogorov similarity hypothesis, 114 universal velocity profile, 763 Kolmogorov structure function, 115 velocity-defect law, 127 Kolmogorov two-third-law, 115 viscous sub-layer, 124 Kolmogorov-Prandtl relationship, 141 wall functions, 149 Lagrangian integral time scale, 106 Turbulent impeller, 811 Lagrangian microscale, 106 Two-fluid large eddy simulation, LES, 160 continuity, 418 law of the wall, 126, 760, 763 energy balance, 421 Leonard stresses, 168 internal energy, 422 local isotropic turbulence, 113 momentum balance, 420 log-law sublayer, 124 species mass balance, 419 longitudinal autocorrelation function, 108 mixing length, 105 modulation, 921 V one-point two-time correlation, 105 Variable density flow, 75 outer wall layer, 123 Vector quantity, 1441 overlap wall region, 123 Viscous stress tensor, 29 passive scalar spectra, 838 power law velocity profile, 121 Volume of fluid method, VOF, 377, 380 Prandtl’s Mixing length model, 122 Piecewise Linear Interface Construction, residual stresses, 168 PLIC, 385 Reynolds analogy, 764 Simple Line Interface Calculation, SLIC, Reynolds averaging, 104, 128 385 Reynolds stress models, 132 SOLA-VOF method, 382 rms-velocity, 110 Smagorinsky constant, 170 Smagorinsky eddy-viscosity model, 169 W standard k-ε model parameters, 142 Weber number, 708 statistical theory, 104 Well-posed model system, 524 sub-grid-scale, SGS, 163 Well-posedness, 523 Taylor’s hypothesis, 111 Whole field formulation, 378, 383