DERIVATION OF BASIC EQUATION

1 Definitions

Basic dimensions Concentration [M] Mass Mass per unit [L] Length volume [T] Time [M·L-3]

Mass Flow Rate Flux Mass per unit Mass flow rate time through unit area [M·T-1] [M·L-2 ·T-1] 2 The Transport Equation

Mass balance for a control volume where the transport occurs only in one direction (say x-direction)

Mass Mass entering leaving the the control control volume volume

Δx

Positive x direction

3 The Transport Equation

The mass balance for this case can be written in the following form

⎡ ofChange ⎤ ⎢ ⎥ ⎢ theinmass ⎥ ⎡Mass entering ⎤ ⎡Mass leaving ⎤ ⎢ ⎢control volume ⎥ = the control ⎥ − ⎢ the control ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ in timea ⎥ ⎣⎢ volume in Δt ⎦⎥ ⎣⎢volume in Δt ⎦⎥ ⎢ ⎥ ⎣ interval Δt ⎦

∂C V JA ⋅−⋅= JA Equation 1 ∂t 1 2 4 The Transport Equation

A closer look to Equation 1

∂C V ⋅ JA ⋅−⋅= JA ∂t 1 2 Flux Flux Concentration Area [M·L-2 ·T-1] Area [M·L-2 ·T-1] over time Volume [L3] [L2] [L2] [M·L-3 ·T-1]

[L3]·[M·L-3·T-1] = [M·T-1] [L2]·[M·L-2·T-1] = [M·T-1] Mass over time Mass over time 5 The Transport Equation

Change of mass in unit volume (divide all sides of Equation 1 by the volume)

∂C A A J ⋅−⋅= J Equation 2 ∂t V 1 V 2 Rearrangements ∂C A ()−⋅= JJ Equation 3 ∂t V 21

6 The Transport Equation

A.J1 A.J2

Δx Positive x direction ∂J The flux is changing in x direction with gradient of ∂x Therefore ∂J JJ += ⋅ Δx Equation 4 12 ∂x 7 The Transport Equation

∂C A ()−⋅= JJ Equation 3 ∂t V 21

∂J JJ += ⋅ Δx Equation 4 12 ∂x

∂C A ⎛ ⎛ ∂J ⎞⎞ ⎜ ⎜JJ 11 +−⋅= Δ⋅ x ⎟⎟ Equation 5 ∂t V ⎝ ⎝ ∂x ⎠⎠ 8 The Transport Equation

Rearrangements

∂C A ⎛ ⎛ ∂J ⎞⎞ ⎜ ⎜JJ 11 +−⋅= Δ⋅ x ⎟⎟ Equation 5 ∂t V ⎝ ⎝ ∂x ⎠⎠ V A 1 x =⇒Δ= Equation 6 A V Δx

∂C 1 ⎛ ∂J ⎞ = ⎜ JJ 11 −−⋅ Δ⋅ x⎟ Equation 7 ∂t Δx ⎝ ∂x ⎠ 9 The Transport Equation

Rearrangements ∂C 1 ⎛ ∂J ⎞ = ⎜ JJ 11 −−⋅ Δ⋅ x⎟ Equation 7 ∂t Δx ⎝ ∂x ⎠

Finally, the most general transport equation in x direction is: ∂C ∂J −= Equation 8 ∂t ∂x 10 The Transport Equation

We are living in a 3 dimensional space, where the same rules for the general mass balance and transport are valid in all dimensions. Therefore

x1 = x 3 ∂C ∂ x2 = y Equation 9 −= ∑ Ji ∂t ∂x =1i i x3 = z

∂C ⎛ ∂ ∂ ∂ ⎞ Equation 10 −= ⎜ Jx + Jy + Jz ⎟ ∂ ⎝ ∂xt ∂y ∂z ⎠ 11 The Transport Equation

• The transport equation is derived for a conservative tracer (material) • The control volume is constant as the time progresses • The flux (J) can be anything (flows, dispersion, etc.)

12 The Advective Flux

13 The advective flux can be analyzed with the simple conceptual model, which includes two control volumes. Advection occurs only towards one direction in a time interval.

Δx

I II

Particle x 14 Δx is defined as the distance, which a particle can pass in a time interval of Δt. The assumption is that the particles move on the direction of positive x only.

Δx

I II

Particle x 15 Δx

I II

Particle x The number of particles (analogous to mass) moving from control volume I to control volume II in the time interval Δt can be calculated using the Equation below, where = Δ⋅ ⋅ AxCQ Equation 11 where Q is the number of particles (analogous to mass) passing from volume I to control volume II in the time interval Δt [M], C is the concentration of any material dissolved in in control volume I [M·L-3], Δx is the distance [L] and A 2 is the cross section area between the control volumes [L ]. 16

Δx

Number of I II = ⋅ Δ ⋅ AxCQ particles passing from I to II in Δt

Particle x Division by time: Q ⋅ Δ ⋅ AxC Number of = particles Δt Δt passing from I to II in unit time Division by cross-section area:

Q Δx Number of particles passing from I to II JADV == ⋅ C ⋅ ΔtA Δt in unit time per unit area = FLUX

⎛ Δx ⎞ ∂x Advective ADV = limJ ⎜ ⋅ C⎟ = ⋅ C flux →Δ 0t ⎝ Δt ⎠ ∂t Equation 1217 The Advective Flux

∂x J = ⋅ C Equation 12 ADV ∂t

18 The Dispersive Flux

19 The dispersive flux can be analyzed with the simple conceptual model too. This conceptual model also includes two control volumes. Dispersion occurs towards both directions in a time interval.

Δx Δx

I II

Particle x 20 Δx is defined as the distance, which a particle can pass in a time interval of Δt. The assumption is that the particles move on positive and negative x directions. In this case there are two directions, which particles can move in the time interval of Δt. Δx Δx

I II

Particle x 21 Another assumption is that a particle does not change its direction during the time interval of Δt and that the probability to move to positive and negative x directions are equal (50%) for all particles. Therefore, there are two components of the dispersive mass transfer, one from the control volume I to control volume II and the second from the control volume II to control volume I

q1 I II q2

x 22

q1 I II q2

x 50 % probability Equation 13 1 = ⋅ 1 ⋅ Δ ⋅ AxC5.0q Equation 14 2 = ⋅ 2 ⋅ Δ ⋅ AxC5.0q

Equation 15 = q-qQ 21

Equation 16 = ⋅ Δx5.0Q ⋅ ( − CCA 21 ) 23 Δx Δx Number of particles passing from I to II in Δt I II = ⋅ ⋅Δ ⋅ ( − CCAx5.0Q ) 21 Divide Equation 16 by time Particle x Number of particles passing from I to Q ⋅ ⋅Δ ⋅ ( − CCAx5.0 21 ) = II in unit time Δt Δt ⎛ ⎛ ∂C ⎞⎞ ⎜ ⎜CCAx5.0 11 +−⋅⋅Δ⋅ Δ⋅ x⎟⎟ Equation 17 Q ⎝ ∂x ⎠ = ⎝ ⎠ ∂C Δt Δt CC 12 += ⋅ Δx Equation 19 Equation 18 ∂x ⎛ ∂C ⎞ ⎜ CCAx5.0 11 −−⋅⋅Δ⋅ Δ⋅ x⎟ Q ⎝ ∂x ⎠ ∂C = x5.0 ⋅Δ⋅− Δ⋅ x Δt Δt Q ∂x JDISP == Divide Equation 20 Δ⋅ tA Δt by ∂C Equation 22 Area x5.0 ⋅Δ⋅− A Δ⋅ x Q = ∂x Number of particles passing from I to Δt Δt II in unit time per unit area = FLUX Equation 21 24 ∂C x5.0 ⋅Δ⋅− Δ⋅ x Q J == ∂x ⋅ ΔtA DISP Δt Equation 22

(Δ⋅ x5.0 )2 ∂C JDISP −= ⋅ Equation 23 Δt ∂x ∂C DISP DJ ⋅−= (Δ⋅ x5.0 )2 D = ∂x Equation 25 Equation 24 Δt

5.0 → [ ] ⎫ 2 2 2 2 ⎪ ()Δ⋅ x5.0 ⎡[][⋅ L ⎤ ] 1-2 LxLx []()→Δ⇒→Δ []LxLx ⎬ D =⇒ → ⎢ ⎥ []⋅= TL ⎪ Δt ⎣ []T ⎦ T t →Δ []T ⎭ 25 Dispersion

Molecular Turbulent diffusion Longitudinal dispersion

GENERALLY

Molecular diffusion << Turbulent diffusion << Longitudinal dispersion26 Ranges of the Dispersion Coefficient (D)

27 The Dispersive Flux

∂C DJ ⋅−= Equation 25 DISP ∂x

28 THE ADVECTION-DISPERSION EQUATION FOR A CONSERVATIVE MATERIAL

29 The Advection-Dispersion Equation

∂C ∂ General −= J Equation 8 transport ∂ ∂xt equation ∂x Equation 12 Advective J = ⋅ C flux advection ∂t ∂C Equation 25 Jdispersion D ⋅−= Dispersive ∂x flux

= advection + JJJ dispersion Equation 26 ∂C ∂ Equation 27 −= ()advection + JJ dispersion ∂ ∂xt 30 The Advection-Dispersion Equation ∂C ∂ −= ()+ JJ Equation 27 ∂ ∂xt advection dispersion ∂C ∂ ∂ −= J − J Equation 28 ∂ ∂xt advection ∂x dispersion ∂x ∂C J = ⋅ C J D ⋅−= advection ∂t dispersion ∂x ∂C ∂ ⎛ ∂x ⎞ ∂ ⎛ ∂C ⎞ −= ⎜ ⋅ C⎟ − ⎜ D ⋅− ⎟ Equation 29 ∂ ∂xt ⎝ ∂t ⎠ ∂x ⎝ ∂x ⎠ 31 The Advection-Dispersion Equation

∂C ∂ ⎛ ∂x ⎞ ∂ ⎛ ∂C ⎞ −= ⎜ ⋅ C⎟ − ⎜ D ⋅− ⎟ Equation 29 ∂ ∂xt ⎝ ∂t ⎠ ∂x ⎝ ∂x ⎠

Velocity u in x direction ∂C ∂2C u ⋅ D ⋅− ∂x ∂x2

∂C ∂C ∂2C Equation 30 u ⋅−= D ⋅+ 2 ∂t ∂x ∂x 32 Dimensional Analysis of the Advection-Dispersion Equation

∂C ∂C ∂2C u ⋅−= D ⋅+ Equation 30 ∂t ∂x ∂x2

Concentration Velocity times [L2·T-1]·[M·L-3·L-2] over time concentration over space = [M·L-3 ·T-1] [M·L-3 ·T-1] [L·T-1]·[M·L-3·L-1] = [M·L-3 ·T-1] 33 The Advection-Dispersion Equation

We are living in a 3 dimensional space, where the same rules for the general mass balance and transport are valid in all dimensions. Therefore x = x, u = u, D = D 3 ⎛ 2 ⎞ 1 1 1 x ∂C ⎜ ∂C ∂ C ⎟ ui ⋅−= Di ⋅+ x = y, u = v, D = D ∂t ∑⎜ ∂x 2 ⎟ 2 2 2 y =1i ⎝ i ∂xi ⎠ x = z, u = w, D = D Equation 31 3 3 3 z

∂C ∂C ∂2C ∂C ∂2C ∂C ∂2C u ⋅−= D ⋅+ v ⋅− D ⋅+ w ⋅− D ⋅+ ∂t ∂x x ∂x2 ∂y y ∂y2 ∂z z ∂z2 Equation 32 34 THE ADVECTION-DISPERSION EQUATION FOR A NON CONSERVATIVE MATERIAL

35 The Advection-Dispersion Equation for non conservative materials

∂C ∂C ∂2C ∂C ∂2C ∂C ∂2C u ⋅−= D ⋅+ v ⋅− D ⋅+ w ⋅− D ⋅+ ∂t ∂x x ∂x2 ∂y y ∂y2 ∂z z ∂z2 Equation 32

∂C ∂C ∂2C ∂C ∂2C u ⋅−= D ⋅+ v ⋅− D ⋅+ ∂t ∂x x ∂x2 ∂y y ∂y2 ∂C ∂2C w ⋅− D ⋅+ ⋅+ Ck ∂z z ∂z2 ∑ Equation 33 36 The Transport Equation for non conservative materials with sedimentation ∂C ∂C ∂2C ∂C ∂2C u ⋅−= D ⋅+ v ⋅− D ⋅+ ∂t ∂x x ∂x2 ∂y y ∂y2 ∂C ∂2C Equation 33 w ⋅− D ⋅+ ⋅+ Ck ∂z z ∂z2 ∑

∂C ∂C ∂2C ∂C ∂2C u ⋅−= Dx ⋅+ 2 v ⋅− Dy ⋅+ 2 ∂t ∂x ∂x ∂y ∂y Equation 34 ∂C ∂2C ∂C w ⋅− D ⋅+ Ck −⋅+ υ ⋅ ∂z z ∂z2 ∑ sedimentation ∂z 37 Transport Equation with all Components

∂C ∂C ∂2C ∂C ∂2C u ⋅−= Dx ⋅+ v ⋅− Dy ⋅+ Equation 35 ∂t ∂x ∂x2 ∂y ∂y2 ∂C ∂2C ∂C w ⋅− D ⋅+ Ck −⋅+ υ ⋅ ∂z z ∂z2 ∑ sedimentation ∂z ± external sources sinksand Sedimentation in z direction

• External loads • Interaction with bottom

• Other sources and sinks 38 Dimensional Analysis of Components

⎛ ∂C ⎞ ⎜ ⎟ ∑ ⋅= Ck ⎝ ∂t ⎠reaction kinetics [M·L-3]·[T-1]=[M·L-3 ·T-1]

⎛ ∂C ⎞ ∂C ⎜ ⎟ −= υsedimentation ⋅ ⎝ ∂t ⎠sedimentation ∂z [L·T-1]·[M·L-3·L-1]=[M·L-3 ·T-1]

⎛ ∂C ⎞ ⎜ ⎟ ±= external sources sinksand ⎝ ∂t ⎠external Must be given in [M·L-3 ·T-1] 39 Advection-Dispersion Equation with all components ∂C ∂C ∂2C ∂C ∂2C u ⋅−= D ⋅+ v ⋅− D ⋅+ ∂t ∂x x ∂x2 ∂y y ∂y2 ∂C ∂2C w ⋅− D ⋅+ ∂z z ∂z2 ∂C Ck −⋅+ υ ⋅ ∑ sedimentation ∂z ± external sources sinksand Equation 35 40