Transport Phenomena

The key-concept behind the study of transport phenomena is the idea that heat transfer, mass transfer and fluid flow are analogous. In some cases, they can be described in a way where they resemble each other in their mathematics/algebra. This is what I want to cover today. In other cases a 'heat-and-mass transfer analogy' such as the 'J-factor' or 'Chilton-Colburn analogy' is used to calculate, say, mass transfer based on empirical heat tranfer correllations, and I may cover this type of approach in future blogs.

Undergraduate chemical engineering courses often introduce the concept of mass transfer by using a heat transfer analogy. A comparison might be drawn between Fick's law for mass transfer and Fourier's law for heat transfer. The student learns that "the driving force for heat transfer is temperature difference, the driving force for mass transfer is concentration difference*". As the student learns more about mass transfer and heat transfer the similarities between the two can often become more-and-more opaque. 

But with CFD, we're often interested in problems where mass transfer, heat transfer and fluid flow are influencing each other. Many chemical engineering problems fall into this category. Instead a generalised representation of transport equations can be made. Any variable, whether it's concentration, velocity or enthalpy/temperature, can be represented by phi (Φ).

Equation 1: Generalised transport equation. Where the variables rho (ρ), t, uj, xj have their usual meanings (density, time, xj = distance, uj = velocity, j defines direction (1,2,3) ). Capital gamma (Γ) = diffusivity coefficient, SΦ = The source term(s) for variable Φ.

1. The term (ρΦ) denotes the amount of extensive property available in a unit volume. This rate-of-change is only important for transient-phenomenon. ∂(ρΦ)/∂t is called the transient term, and accounts for the accumulation of Φ inside the control volume (chemical engineers will recognise the connotations between accumulation and unsteady-state mass balance!).
2. ∂(ρujΦ)/∂xj is called the convection term.  uj is the velocity, it can be thought of as a flow per unit area. If ∂xj is the length of the control volume, then the convection term is like the amount of Φ being brought into & out-of the control volume by the flow.
3. The term Γ represents diffusion. For mass transfer in laminar flow, this is just molecular diffusion. For heat, this is heat conduction in the fluid. For fluid flow, this is viscosity (“viscosity is the diffusion of momentum”). This is where turbulence can make a big difference. In laminar flows Γ can be measured by experiments, say, in a lab environment. In turbulent flows these become highly dependent on the flow and this is no longer possible – this is where turbulence modelling comes in.
4. Source terms account for any creation or destruction of the variable Φ. For mass transfer, this may be used to describe reactions consuming reactants & generating products. But it is not limited to chemical transformations – source terms would also describe phase change or adsorption. For heat transfer similarly, this may represent heat transfer on a wall or from a phase change for example. In momentum transfer, pressure gradient ∂P/∂xj is a source term. Any term that can’t be cast as a convection or diffusion term is considered a source term.

	Φ	Γ	SΦ net source
Conservation of mass	1	0	0
Mass transfer	ω	ρDeff	R
Momentum	ui	μeff	-∂P/∂xj + ρBi +Sui
Energy (enthalpy)	h	keff/Cp	Q

Table 1: where omega ω = mass fraction, Deff effective diffusivity, R = rate of generation of species (eg: by reaction), ui velocity in direction i(1,2,3), µeff = effective viscosity, -∂P/∂xj = pressure gradient, ρBi = buoyancy forces, centrifugal forces, particle drag etc. (Sui may represent eg: Stoke’s Stress Laws), h = enthalpy, keff = thermal conductivity of fluid, Cp = heat capacity of fluid, Q is a heat source, such as heat of a reaction.

The Transport equations are a fundamental part of CFD. Whilst the concept that heat, mass and momentum are similar is much older, the representation in Equation 1, the generalised transport equation, is attributed to the work of Prof. DB Spalding and Prof. SV Patankar, who are also responsible for the SIMPLE algorithm. The equations are often written out in full, but at the moment I’m finding the table-version a little easier-on-the-eyes.

* I prefer to think of this phrase instead as "the driving force for mass transfer is the concentration's 'distance-to-equilibrium' ". I find this helps when thinking about problems such as two-film theory, where the concentration difference at equilibrium for say, a separation oil-and-water, is defined by the water-octanol partition coefficient. 

References

Date, AW. 2005. Introduction to Computational Fluid Dynamics. Cambridge: Cambridge University Press.

Kim, D. Kim, D. Lee, KS. 2015. Frosting Model for Predicting Macroscopic and Local Frost Behaviours on a Cold Plate. International Journal of Heat and Mass Transfer. 82. p.135-142.