Heat and Mass Transfer Analogy

By: David Frisbie

Summary

The topic of relating heat and mass transfer dates back to Wilhelm Nusselt’s original paper The Basic Laws of Heat Transfer in which he discussed the analogy between heat and mass transfer in evaporative processes.  Since then, the fundamental laws and constitutive relations have been well formulated and documented.  The analogy is based on the notion that just as a temperature gradient constitutes the driving potential for heat transfer, a species concentration gradient in a mixture provides the driving potential for mass transfer [3].  The laws of mass transfer show the relation between the flux of the diffusing substance and the concentration gradient forcing the mass transfer.  Because diffusion occurs only in mixtures, its evaluation must involve an analysis of the effect of each component in the system.  A summary of the conductive and convective heat transfer governing equations and boundary conditions as they relate to the respective mass transfer equations can be seen below.

Summary of Equation Relations:

Constitutive Equations

 Heat Transfer Mass Transfer Fourier’s Law of Heat Conduction       = Heat flux (W/m2) k = Thermal Conductivity of the Material (W/m K) T = Temperature of the Material (K) Fick’s Law of Diffusion     = Mass Flux (kg/m2 s) DAB = Binary Diffusion Coefficient (m2/s) CA = Concentration of Species A mfA = Mass Fraction of Species A

Conductive Heat Transfer

 Heat Transfer Mass Transfer Heat Diffusion Equation     = Heat source/sink term (W/m2 s2) k = Thermal Conductivity of the Material (W/m K) = Density (kg/m3) = Constant pressure specific heat (J/kg K) Mass Diffusion Equation     = Heat source/sink term (W/m2 s2) DAB = Binary Diffusion Coefficient (m2/s) = Density (kg/m3) = Density (kg/m3) mfA = Mass Fraction of Species A Boundary Conditions Boundary Conditions       = Constant Species Flux at the Surface

Convective Heat Transfer

 Heat Transfer Mass Transfer Local Heat Flux     h = Local Heat Transfer Coefficient (W/m2 K)   Heat Rate   = Average Heat Transfer Coefficient   As = Surface Area Associated with Local Mass Flux   hm = Local Mass Transfer Coefficient (m/s)   Mass Diffusion Rate   = Average Mass Transfer Coefficient As = Surface Area Associated with

Conduction Example [3]

Consider a one-dimensional, steady state diffusion of a catalytic surface reaction, in which the molar flux of species A is desired to be known.  Figure 1 represents a plate in a stationary fluid in which a chemical reaction at the surface produces a species A at a rate (molar rate of production).  In order to apply one-dimensional steady state analysis, a few assumptions must be made.  The first being the rate of species transfer from the surface  must be equal to the surface reaction rate .  Secondly, that species A leaves the surface as a result of one-dimensional transfer through a thin boundary at the reaction boundary.  The mole fraction of species A at y = L (xA,L) refers to the mainstream conditions.  The final assumption being that fluid B has to be stationary.

Figure 1 – Diffusion of A from Catalytic Surface

The governing equation for the system:

With boundary conditions:

xA(L) = xA,L

From the assumption , and application of Fick’s Law,

The surface reaction rate  depends on the concentration at the surface CA(0).  The reaction rate is of the form:

,

where is the reaction rate constant (m/s).

The surface boundary condition above then becomes:

Solving the governing equation using the boundary conditions:

,

Therefore, at the surface, the result is as follows:

,

and from this the molar flux can be determined to be:

,

Convection Example

Consider the two dimensional problem of the convective thermal boundary layer for flow over a flat plate, as in Figure 2.

Figure 2 – Thermal Boundary Layer over a Flat Plate

where Us, Ts are the free stream, constant values, Tw is the temperature at the wall, q’’ is the heat flux at the wall, is the thickness of the thermal boundary layer.

From the thermal energy equation:

This can be approximated by scaling to become:

If we look at the rough size of the terms, and keeping in mind that in most boundary layer flows, typically:

And in many flows, we expect the advection terms in the x-direction and thermal diffusion terms to be of most importance, then we can approximate

~  ~  ~

Knowing that the Reynold’s number and Prandtl number are respectively defined as:

The relation above then becomes:

~

Now consider the equivalent scaling analysis for the corresponding diffusion problem, as in Figure 3.

Figure 3 – Concentration Boundary Layer Over a Flat Plate

where Cs, Us are the free stream constant values, Cw is the concentration at the wall, m’’ is the mass flux at the wall.

From the diffusion equation:

This can be approximated by scaling to become:

If we look at the rough size of the terms, and keeping in mind that in most boundary layer flows, typically :

And in many flows, we expect the advection terms in the x-direction and thermal diffusion terms to be of most importance, then we can approximate

~  ~  ~

Knowing that the Reynold’s number and Schmidt number are respectively defined as:

The above equation then becomes:

~

From these two separate analyses, it can be seen that thermal and mass transfer solutions of the temperature and concentration profiles can be related through their respective non-dimensional ratios of momentum and thermal/mass diffusivities.  The Lewis number is therefore defined as the measure of the relative thermal and boundary layer thicknesses [3].

Therefore, if there is a case in which the thermal diffusivity is on the order of the diffusion coefficient of species A in B, then the differential equations involving heat and mass transfer can be approximated as analogous.  Similarly, if the Schmidt and Prandtl number are of the same order, then the solutions can be approximated as analogous.  For example, for air at atmospheric pressure and at T = 273 K, the Pr = .717.  An appropriate analogous mass transfer system would be that of pure oxygen diffusing in air, in which the corresponding Sc = .74  [4].

Conclusion

Further analysis can be performed to expand such equations to more specific situations in both diffusion and convective mass transfer.  For example, the basic mass transfer equations and one dimensional steady state diffusion problems can be broadened into multidimensional problems, problems of different coordinate systems, as well as transient problems.  Similarly, the mass transfer equations and convection mass transfer basics can easily be applied to external or internal multidimensional problems as seen in most heat transfer analysis.

Final Project

Introduction

Heat transfer is the notion of thermal energy in transit due to a temperature difference.  That is, whenever there is a temperature difference in a medium or between mediums, heat transfer must occur [1].  There are three main modes of heat transfer: conduction, convection, and radiation.  Conduction refers to the heat transfer that occurs when a temperature gradient exists within a stationary medium.  Convection refers to heat transfer through a fluid due to macroscopic motion of particle.  Radiation refers to the energy emitted by all surfaces of finite temperature and the net heat transfer between two surfaces at different temperatures [2].

Similarly, mass transfer is mass in transit due to a concentration difference.  Therefore, if there is a difference in the concentration of some species in a mixture, mass transfer must occur [3].  Just as the driving potential for heat transfer is a temperature gradient, a species concentration gradient is the driving potential for mass transport of that species.  The analogy of heat transfer to mass transfer is correlated to the individual modes by which they are influenced.  Mass transfer by due to fluid flow relates to convective heat transfer, and mass transfer by diffusion is analogous to conduction heat transfer.  This paper will discuss the derivation and relation between heat and mass transfer.

Balance Equations

The balance equations (or conservation equations) are based around the flux of any state variable flowing into and out of a differential volume and hold true for any material.

Conservation of Mass:

Conservation of Linear Momentum:

Conservation of Energy:

Where: t is time, ρ is density,  is fluid velocity,  is the stress Tensor,  is the body force, e is the internal energy,  is the heat flux vector, r is the volumetric energy supply.

In order to complete the description of any material, a constitutive equation must be introduced to couple with the balance equations.  As in the case of heat transfer, Fourier’s Law of Heat Conduction is used as the constitutive relation. Similarly, in regards to mass transfer, the constitutive equation used is Fick’s Law of Diffusion [3].

Fourier’s Law of Heat Conduction:

Where k is the thermal conductivity of the material W/(m * K), T is temperature.

Fick’s Law of Diffusion:

,

where D is the diffusion coefficient for species A, CA is the concentration of species A (kg/m3), ρ is the density of the mixture (kg/m3), DAB is the diffusion coefficient for species A in B, mfA is the mass fraction of A in mixture A+B.

Fourier’s law is the simplest form of a general energy flux law and is restricted to cases in which the system is completely described by only the temperature gradient.  Fick’s Law is similarly restricted to situations in which only a concentration gradient only exists [4].  However, other potential gradients in the system (i.e. pressure gradients and body forces), coupled with the concentration gradient can induce mass diffusion.  When such potential gradients are considered, the diffusion equation is affected, as in the case if we were to consider concentration and temperature gradients.  The resulting diffusion equation as developed by Chapman and Cowling [4] is known as the thermo-diffusion, or Soret, effect:

,

where M1 and M2 are the molecular masses of the respective substances A and B, M2 is the square of the mean of the molecular mass, KT is the thermal diffusive coefficient.

Conductive Heat and Mass Transfer

The main objective in heat conduction problems is to determine the temperature distribution throughout a medium.  By defining an infinitesimally small control volume (in Cartesian coordinates) as shown in Figure 4 and applying the first law of thermodynamics, the result is a differential equation whose solution provides the temperature distribution for the medium [5].

Figure 4 – Conduction Analysis Differential Control Volume [3].

From the first law of thermodynamics:

,

where the energy source and energy storage terms are given respectively by:

Applying the first law to the system in Figure 1:

When applying a Taylor series expansion on the conduction heat rates, the equation becomes:

Finally, applying the evaluated conduction heat rates from Fourier’s Law, the equation becomes the general form of the heat conduction equation:

From this governing equation, the temperature distribution can easily be determined for a simplified one-dimensional system if the physical conditions at the boundaries of the medium are be known.  Because the heat equation is of second order in the spatial directions, two boundary conditions need to be expressed to describe the system.  Only initial condition is also needed, because the system is first order in time.  There are three forms of boundary conditions, and when applied to a simplified one dimensional system, are usually specified at the surface [6].

The boundary condition of the first kind (or Dirichlet boundary condition) corresponds to the situation when a surface temperature is constant.

The boundary condition of the second kind (or Neumann boundary condition) corresponds to the constant heat flux qs” at the surface.

or for an adiabatic surface (or symmetry condition),

The boundary condition of the third kind corresponds to the existence of convection at the surface,

To continue the analogy of heat transfer to mass transfer, we can now consider the application of the law of conservation of species to determine the mass distribution throughout a medium.  By defining an infinitesimally small control volume (in Cartesian coordinates) as shown in Figure 5 and applying the law of conservation of species, the result is a differential equation whose solution provides the mass distribution for the medium.

Figure 5 – Diffusion Analysis Differential Control Volume [3].

From conservation principle:

,

where the generation and storage terms are defined respectively by,

,

where  is the rate of increase of the mass of species A per unit volume of the mixture and  is the density of species A.

Applying conservation of species to Figure 2 with the similar Taylor series approximations and reductions:

Finally, applying Fick’s Law to the individual spatial directions, the equation becomes the general form of the mass diffusion equation:

,

where  is the density of species A, DAB is the diffusion coefficient of species A in B, and mfA is the mass fraction of species A in B.

Or, in term of the molar concentration, the derivation yields:

,

where CA is the concentration of species A, and xA is the molar fraction of species A.

To determine the species concentration distribution in a medium, the physical conditions existing at the boundaries must be known.  Two kinds of boundary conditions are normally seen in species diffusion problems and are analogous to the first two heat transfer conditions [6].

The first boundary condition corresponds to when the species concentration at the surface is a known constant,

The second boundary condition corresponds to the existence of a constant species flux at the surface using Fick’s Law, in general,

,

where  is the constant species flux at the surface [7].

Convective Heat and Mass Transfer

Convective heat transfer is bounded by its description as the energy transfer between a surface and a fluid moving over the surface.  While the mechanism of diffusion does contribute to this transfer, the dominant contribution is made by the bulk motion of particles [3].  Considering the flow depicted by Figure 6; a fluid of velocity u and T flowing over a flat plate of length L and area As maintained at temperature Ts.

Figure 6 – Convection Heat Transfer Analysis Flow Conditions

If such conditions are that Ts > T, then convection heat transfer will occur from the surface to the fluid.  The local heat flux may be expressed by the same boundary condition noted in the heat conduction section,

,

where h is the heat transfer coefficient (W/m2 * K).

However, it is known that the flow conditions vary along the length of the plate, therefore, the heat rate can be obtained by integrating the local flux over the entire surface and we then define an average heat transfer coefficient for the entire surface [2].

Equivalent results can be determined for convection mass transfer where a fluid of species concentration CA,∞ flows over a surface which the species concentration is maintained at some value CA,s.  If CA,s > CA,∞ then there will exist a concentration gradient and transfer of the species by convection will occur.  Species A is typically a vapor that is transferred into a gas stream due to evaporation or sublimation at a liquid or solid surface [3].  In this case the molar flux of species A is given by,

,

where hm is the convection mass transfer coefficient (m/s) and the molar concentrations CA,s and CA,∞ have units of (kg*mol/m3).  As before, the flow conditions vary along the length of the plate, therefore, the mass diffusion rate can be obtained by integrating the local flux over the entire surface and we then define an average convection mass transfer coefficient  for the entire surface,

The species transfer can also be expressed as a mass flux and mass transfer rate [3],

Conclusion

The derivations described have only briefly related the previously known heat transfer equations to their mass transfer equivalents, and these can be seen in the table below.   However, further analysis can be performed to expand such equations to more specific situations in both diffusion and convective mass transfer.  For example, the basic mass transfer equations and one dimensional steady state diffusion problems can be broadened into multidimensional problems or into different coordinate systems.  Similarly, the mass transfer equations and convection mass transfer basics can easily be applied to external or internal multidimensional problems as seen in most heat transfer analysis.

Works Cited

[1]  DeWitt, David. Introduction to Heat Transfer.  New York: John Wiley & Sons, 2002.

[2]  Welty, James.  Fundamentals of Momentum, Heat and Mass Transfer.  New York: John

Wiley & Sons, 1969.

[3]  Incropera, Frank.  Fundamentals of Heat and Mass Transfer.  New York: John Wiley &

Sons,1985.

[4]  Kays, William.  Convective Heat and Mass Transfer.  New York: McGraw Hill, 2005.

[5]  Cussler, E.L. Diffusion Mass Transfer in Fluid Systems.  New York: Cambridge University

Press, 1997.

[6]  Skelland, A.H.P. Diffusional Mass Transfer.  New York: John Wiley & Sons, 1975.

[7]  Hines, Anthony.  Mass Transfer – Fundamentals and Applications.  New Jersey: Prentice-

Hall, Inc., 1985.