Compressible Flow Along Pipelines A

October, 21st, 2014 Des. Ind. Plant, 2

Compressible Flow along pipelines a. Consider a pipe (diameter D, lenght L) used to transport gas (molar mass M, ratio of specific heat evaluated at constant pressure and at constant volume given by γ = cp/cv) from point 1 to point 2. 1. Derive the relation between specific mass flow rate G and pressure at the upstream/downstream section of the pipeline. Assume isothermal transformation for the gas (p/ρ = C) along the pipeline. Since the pipeline works in steady state conditions,m ˙ = cost and if the pipe diameter is constant along the length of pipe, also G = const. The flow along the line is described by Benrnoulli equation:

2 v dp dlv d + + gdh = dws (1) 2 ρ − ρ

2 where gdh 0, dws = 0 (no compressor along the line) and dlv/ρ = 2fdxv /D as for incompressible ﬂow. Bernoulli should≃ be integrated between point 1 and 2, but it can not be integrated in the form of Equation 1 because we do not know the law of variation of velocity along the pipe, v(x), which is required to quantify the viscous losses. To integrate that term, we need to rewrite Bernoulli as:

2 2 2 2 v dp 2f(v ρ )dx ρ d + = (2) 2 ρ − D which, considering G = ρv, becomes

ρ2G2 1 2fG2dx d + ρdp = (3) 2 ρ2 − D

Integration of the ﬁrst term gives

2 2 2 2 2 2 ρ G 1 ρ G 1 2 dρ 2 ρ2 2 p2 2 p1 d 2 = ( 2) 3 dρ = G = G ln = G ln = G ln (4) 1 2 ρ 1 2 − ρ − ρ − ρ1 − p1 p2 Z Z where we considered the relationship between p and ρ in isothermal ﬂow. Integration of the second term gives

2 2 pM M 2 2 ρdp = dp = (p2 p1) (5) 1 1 RT 2RT − Z Z Integration of the right hand side gives

2 2fG2dx L =2f G2 (6) 1 D D Z The integral version of Bernoulli equation for isothermal ﬂow along the pipeline is:

2 p1 M 2 2 L 2 G ln + (p2 p1)+2f G =0= F (G, p1,p2) (7) p2 2RT − D

from which we calculate

M 2 2 2RT (p1 p2) G = − = f(p1,p2) (8) v ln p1 +2f L u p2 D u t the relationship we were looking for. If we assume that the upstream pressure p1 is fixed, the specific flow rate is a function of the downstream pressure p2 only. p2 can varies in the range [0 : p1]. G is positive definite, and is null for both p2 = p1 and p2 = 0. For the Weierstass theorem it has maximum in the range [0 : p1]. To find the maximu we should calculate dG(p2)/dp2 = 0. Since the function G(p2) is rather complex, we can use Equation 7 to derive the derivative of G in a simpler way. Considering the differential of F (G, p1,p2) we get

1 ⌣ 2014 [email protected] ·· October, 21st, 2014 Des. Ind. Plant, 2

∂F ∂F dG ∂F/∂p2 dF (G, p2)= dG + dp2 =0 = (9) ∂G ∂p2 → dp2 − ∂F/∂G

G from which d = 0 only if ∂F/∂p2 = 0. Therefore: dp2

2 ∂F G M M 2 = + p2 =0 G = p2 = √ρ2p2 (10) ∂p2 − p2 RT → rRT

and the velocity of gas at the outlet section is v2 = G/ρ2 = p2/ρ2 = vsound. When G is maximum, the velocity of gas at the outlet section is the sound speed (evaluated in isothermal conditions). Substituting the value of p Gmax in Equation 7 we obtain

2 2 M 2 p1 Mp2 p1 L M 2 p2 ln + 1 +2f p2 = 0 (11) RT p2 2RT − p2 D RT " #

which can be simpliﬁed as

2 p1 1 p1 L ln + 1 +2f = 0 (12) p2 2 − p2 D " #

This is a funtion of (p1/p2) only and gives the value of the critical pressure ratio corresponding to G = Gmax. As already discussed for the efflux from a tank, when the gas velocity at the outlet section equals the speed of sound, if the outer pressure is lowered there is no way for this information to propagate upstream along the pipe toward the inlet section. Therefore, once the critical flow is established, no further variation of G is expected even if p2 is reduced and G = Gmax. 2. Derive the formula for the specific mass flow rate at critical conditions when the gas undergoes adiabatic (irreversible, i.e. non isoentropic) transformation moving along the pipe.

In this case, the integral form of Bernoulli will be given by:

2 p1 L G2 ln + ρdp +2f G2 = 0 (13) p2 1 D Z where we need a relationship between ρ and p along the pipeline which is valid for irreversible adiabatic transformation. We cannot use p/ργ = cost which is only valid for revesible adiabatic. Nevertheless, we can derive an alternative relationship from the total energy conservation equation: p 1 d(e + + v2 + gh) = dq + dw (14) ρ 2 s

e is the internal energy, q is the heat ﬂux added to the control volume of gas and ws is the mechanical work made on the control volume. If we deﬁne the enthalpy as H = e + p/ρ, the total energy equation becomes 1 d(H + v2 + gh) = dq + dw (15) 2 s

where we can neglect gh, dws = 0 (there is no compressor on the pipeline) and dq = 0 (for adiabatic ﬂow). We obtain 2 1 2 dv d(H + v )=0 cpdT + = 0 (16) 2 → 2

where, according to Mayer equation, cp is given by γ R c = (17) p γ 1 M −

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and according to ideal gas law dT = M/Rd(p/ρ) and

γ p 1 d + dv2 =0 (18) γ 1 ρ 2 −

2γ p G2 + = C = cost (19) γ 1 ρ ρ2 − Equation 19 gives a relationship between p and ρ which is valid for each point along the pipe and can be used to integrate the term in Equation 13. From 19 we caan write

G2 γ 1 p = C − ρ (20) − ρ2 2γ and G2 G2 γ 1 2G2 G2 γ 1 G2 γ 1 dp = ( 2) ρ + C − dρ = + C − dρ = + C − dρ (21) − − ρ3 − ρ2 2γ ρ2 − ρ2 2γ ρ2 2γ The integral in Bernoulli becomes

2 2 2 G γ 1 γ 1 2 ρ2 C 2 2 ρdp = ρ 2 + C − dρ = − G ln + (ρ2 ρ1) (22) 1 1 ρ 2γ 2γ ρ1 2 − Z Z Further simpliﬁcations led to

2 2 2 γ 1 2 ρ2 γ G ρ2 ρdp = − G ln + p1ρ1 + 1 (23) 1 2γ ρ1 γ 1 2 ρ1 − Z " − !#

and the ﬁnal form for integral Bernoulli equation for adiabatic irreversible ﬂow:

2 2 ρ1 γ +1 γ 1 2 ρ2 L 2 G ln + p1ρ1 + − G 1 +4f G = 0 (24) ρ2 γ γ ρ1 − D " #

which is again an implicit function F (G, ρ1,ρ2) = 0. If we consider the upstream condition ﬁxed, F (G, ρ2)=0 describes the variation of G as the downstream condition change. Following the same procedure as before, we can ﬁnd the condition for which G = Gmax from

dG dG ∂F/dρ2 ∂F =0 = = 0 (25) dρ2 → dρ2 − ∂F/∂G → dρ2

which gives

2 ∂F G γ +1 γ 1 2 2 = + ρ1p1 + − G 2 ρ2 = 0 (26) dρ2 − ρ2 γ 2γ ρ 1 and using Equation 19 to rewrite the term in parentesis, we get

2 2 G γ +1 γ 1 G 2γ p2 = − 2 + 2ρ2 (27) ρ2 γ 2γ ρ2 γ 1 ρ2 − and G2 γ +1 γ 1 G2 − =2p2 2 =2ρ2p2 Gmax = √γp2ρ2 (28) ρ2 γ − γ → γ →

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The last equation indicates that when the specific flow rate is maximum, G is a function of pressure and density at the outlet section and the formula is the same as that calculated for the flow exiting from a tank under adiabatic conditions. The velocity of gas at the outlet section turns out to be v2 = G/ρ2 = γp2/ρ2 which is the speed of sound. Substituting the value of Gmax into Equation 24 we obtain the equation which gives the link between densities at the upstream and downstream section of the pipe at critical flow conditions:p

2 ρ1 1 ρ1 L γ ln + 1 +4f = 0 (29) ρ2 2 − ρ2 D γ +1 " #

which together with

p1 ρ1 γ +1 γ 1 ρ2 = − (30) p2 ρ2 2 − 2 ρ1

allows to link upstream and downstream conditions of the pipe. Equation 29 is the analogous of the critical pressure ratio calculated for isothermal transport. Equation 30 is derived from Equation 19 using the result of Equation 29.