1. CHEM-E7130 Process Modeling Lecture 5

2. Lecture 5 outline When accurate mass transfer models should be used?
Origins of diffusion from molecular collisions (Maxwell-Stefan formulation) including non-conventional driving forces
Matrices, basic operations and matrix functions
Solution of diffusion model with boundary conditions (film model)

3. Various degrees of modeling rigour can be used depending on the relative importance
When mass transfer is of less importance, a simple model is sufficient
How does concentration vary as a function of time?
k is a “relaxation parameter”, determining how rapidly equilibrium is approached. It has also a physical meaning. What is it?

4. One of the most important things to understand is the order of the process
Mass transfer is (approximately) a first order process → exponential “decay” towards equilibrium
spot an error in the equation!

5. This level is reasonable in preliminary analysis and estimation
Why to use more complicated models?
Specify control volume for material balances correctly
Mechanistic model for mass transfer coefficient and driving forces (extrapolation!)
Understanding the physics of diffusion

6. For mass transfer rate, a model is needed for mass transfer flux and mass transfer area.
mol/s
Flux
mol/m2s
Area m2
Only calculation of mass transfer flux, N, is considered in this lecture. A can be obtained from
geometry
correlations
population balances

7. Mass transfer flux Mass transfer flux needed in the material balances
Diffusion flux, movement with respect to average molar velocity
Convection (advection), flow with respect to the control volume boundary
Nt is the total flux (sum of all fluxes)

8. Fick’s law (diffusion model)
Here it is simply assumed that there is a linear response (flux) when composition gradients are present. The proportionality coefficient is the diffusion coefficient.
What is the difference in the two forms above?
When they are equivalent?

9. Mass transfer with diffusion and convection, example
Liquid A is evaporating in a tube. Air flushes component A away from the top of the tube. Air does not dissolve in liquid A
Why?
What?
At the liquid surface, partial pressure of A is its vapor pressure

10. Evaporation in tube, example
Mass transfer flux
diffusion
convection
What is the expression for flux (solve NA)? a) b) d) c)
Tips: assume NA, c and D as constants. Out of these NA is unknown (to be solved). Flux NA schetced in the figure would be negative. This is a separable 1st order differential equation

11. Evaporation in tube, example

12. Evaporation in tube, example
ct and D assumed constants
So answer b was correct. Actually c is the same result.
You can also pick the correct solution by deducing

13. Problems in the Fick’s law (diffusion model)
Is diffusion coefficient a component property? Which things affect its numerical value?
Diffusion coefficient is not a property of a single component. In the Fick’s law, it depends on the concentrations of other components, and for systems containing more than 2 components, also driving forces. In addition, T, p…

14. Maxwell-Stefan diffusion model more fundamental than Fick’s law
The sum of the forces acting on a system
The rate of change of linear momentum of the system 
Average amount of momentum exchanged in a single collision
Number of collisions of molecules of type i per unit volume
The rate of change of linear momentum of the molecules of type i per unit volume = x
What are the variables that affect on these?

15. Maxwell-Stefan diffusion model
Number of collisions of molecules of type i and j per unit volume 
xixj
Also velocity, density, and cross-sectional (collision) area affect
Average amount of momentum exchanged in a single collision =

16. After some manipulation…
Dimensionless driving force, e.g. mole fraction gradient in ideal gases
We need fluxes, and typically mole fractions are known. Either they are guessed and fluxes are used to check the material balance or they are known from the previous time step and fluxes are calculated for the source term for time rate of change. Driving force can be calculated from the state variables and other known system properties (electric field, pressure differences, centrifugal forces etc)
This is the Maxwell-Stefan diffusion equation in a classical form
What do we need for the material balance and what is typically known (from the state variables)?

17. Matrices
Matrices are ”table of numbers” with certain mathematical operations. Matrices are linear operators.
row number
column number
Matrix multiplication

18. Example: Parameters for a quadratic polynomial
y’s are known at three points of x:
(x1, y1), (x2, y2), (x3, y3)
Is this a system of linear equations, i.e. can matrices be used?
How would the matrix formulation look like?

19. Parameters for a quadratic polynomial
Write in a matrix form
And solve (efficient algorithms exist for matrix inversion)
Least squares solution, if more measurements than parameters

20. Maxwell-Stefan diffusion model in matrix form

21. Composing M-S diffusion matrix
Elements of the B matrix
(size (n-1)(n-1))

22. Calculating the fluxes
driving force
diffusion
”friction”
diffusion
coefficient
driving force
mass transfer
diffusion
convection

23. Matrices again Or This looks familiar
component index whose flux is considered
component index for driving force contribution Or

24. Matrices again Component 1 flux, diffusion coefficient and gradient
Cross-coefficients; effect of other gradients

25. Special phenomena in multicomponent systemsJ
-x
Ideal two-component system

26. Special phenomena in multicomponent systemsJi
Osmotic diffusion
3 or more components
Normal diffusion
Diffusion barrier
-x
Normal diffusion
Reverse diffusion
What happens in the green triangle?

27. Other driving forces
Starting from irreversible thermodynamics, the following generalized driving force can be derived
driving force
chemical potential
Total pressure gradient
external force (electric field etc.)
For example, ultrasentrifuge. What is it?
It causes a strong pressure gradient, that separates dense molecules from less dense (e.g. separation of uranium isotopes)

28. Let’s go back to ”easy” systems, 2 ideal components
Continuity equation (differential material balance)
mass transfer
diffusion
convection
What to do with this in practical process modeling problems where we want to calculate mass transfer, e.g. between two phases?

29. Film theory
Assume one-dimensional mass transfer without reaction. Interface is in equilibrium
Fluid phase
Equilibrium at the interface
Other fluid or solid phase
Discuss with your friend what happens near the interface. How to deal with two films (on two sides), and how would the mole fraction profiles look like?

30. Film theory Steady state, one dimension, only two componentsxI
Boundary conditions l xb

31. Stationary diffusion without convection
Steady state, one dimension, two components
Solution with film model boundary conditions
Diffusion

32. Stationary diffusion without convection
Mass transfer coefficient

33. Stationary diffusion with convection
Steady state, one dimension, two components
Can be easily solved for mole fraction profiles with film model boundary conditions. dx/dz inserted into Fick’s law
”High flux correction”
Diffusion flux

34. Multicomponent formulation (linearized theory)
Easy:
Just add brackets!

35. Matrix functions
When multicomponent diffusion models are solved in various practical cases, matrix functions are needed
For example high flux corrections or mass transfer coefficient correlations:

36. Matrix functions
In general, matrix functions can be calculated with similarity transformation
Eigenvectors
full matrix
Eigenvalues
diagonal matrix
Matlab: [P,Lambda] = eig(X)

37. Matrix functions Matlab: Y = expm(X)
Also series expansions can be used. For example
Matlab: Y = expm(X)

38. Solution of a two-film model with rigorous Maxwell-Stefan model
Given:
bulk compositions
thermodynamic properties (equilibrium model, diffusion coefficients, etc.)
Unknown: mole fractions at the interface, total flux Nt

39. Solution of a two-film model
mass transfer direction
1. Guess interfacial compositions xI and total flux Nt xb
yI=KxI xI yb
2. calculate gas phase interfacial compositions
3. calculate mass transfer rate factor for both phases

40. Solution of a two-film model
mass transfer direction xb
yI=KxI
4. calculate diffusion fluxes xI yb
5. and mass transfer fluxes

41. Solution of a two-film model
mass transfer direction xb
yI=KxI
n-1 independent
equations xI yb
1 equation
n+1 unknowns and n+1 equations!
1 equation

42. Some common assumptions
Linearized mass transfer theory was already used (constant D and ct)
K values and [D] are calculated with bulk compositions
Fluid nonidealities neglected in driving force
Mass transfer coefficient matrix from binary coefficients, or by matrix approximations

43. Summary Mass transfer is a process of approaching chemical equilibrium
A simple, ”engineering”, interpretation to mass transfer coefficient is the speed of approach to equilibrium
Maxwell-Stefan equations are based on conservation of momentum

44. Summary
In multicomponent systems, components can diffuse against their gradients due to interactions with other components
Generally, solution of multicomponent mass transfer models requires matrix function calculations
Film theory provides boundary conditions for practical mass transfer flux calculations