Cambridge University Engineering Department
Engineering Tripos Part IIA
Module 3A6: Heat and Mass Transfer
Some Notes on Mass Transfer
Dr. Christos Markides
cnm24@cam.ac.uk
NOTE: Unless otherwise stated, the Einstein Summation Convention is used throughout, with
.
CNM
22 of February, 2008
nd
1. Background:
The definition of the Material or Substantive Derivative operator is:
Conservation of mass, or continuity,
or, in vector calculus notation, and thus in any (all) co-ordinate system(s):
Then, for any scalar φ, and using Eqn. 2:
In one, planar dimension, with v = w = 0, Eqns. 2 and 3 become:
In one, cylindrical dimension, with uθ = uz = 0, Eqn. 3 becomes:
In one, spherical dimension, with uθ = uφ = 0, Eqn. 3 becomes:
For one-dimensional, incompressible flows, Eqns. 5, 6 and 7 become, respectively:
In addition, conservation of species (mass) gives,
where Fick’s law has been used for the molecular diffusive mass flux
, in
exactly the same way that Fourier’s law of heat conduction was used for the molecular diffusion
(conduction) of heat. The reader is asked to refer also to Section 2: “Diffusive (Fick’s law) and
advective mass/molar flux”.
Note that, if we sum Eqn. 9 throughout, over all species “i”:
Hence, we see that Eqn. 9 is an equation for the conservation of mass of species “i”, such that if we
sum over all “i”, we get back to the equation for the conservation of the total mass of all species.
In vector calculus notation, and thus in any (all) co-ordinate system(s):
Note that, similarly to above, if we sum over all “i”:
In one, planar, cylindrical or spherical dimension, Eqn. 10 becomes, respectively:
In this course you will concentrate on one-dimensional, incompressible flows and fluids with
constant D. In these situations, Eqns. 11 simplify to,
From left to right in Eqns. 12, the terms represent: unsteadiness, advection, volumetric species mass
generation/chemical reaction (also the chemical source term), and molecular diffusion.
You will deal with both non-reactive (
) and reactive (
) mass transfer. In reactive mass
transfer you will be presented with homogeneous and heterogeneous reactions, and further you will
only encounter these reactions in steady, stagnant conditions, i.e. in absence of unsteadiness and
advection. Heterogeneous reactions do not occur throughout the volume of the fluid, so that in this
case,
, as with non-reactive mass transfer.
In non-reactive mass transfer you will be expected to solve steady diffusion with non-zero advection,
or non-zero bulk flow, that is,
steady diffusion with zero advection, or stagnant mass transfer with no bulk flow,
and unsteady diffusion, but never with advection:
In the latter case, you will only ever be presented with the planar geometry of Eqn. 15a, and further
will not be expected to actually solve Eqn. 15a and obtain the error function solution itself. Instead,
you might face a question in which you are asked to obtain the result by being guided through it,
and/or to use this result.
In reactive mass transfer (homogeneous reactions) you will only be expected to treat so-called
“reactive-diffusive” problems, which are steady and with no advection, resulting in:
2. Diffusive (Fick’s law) and advective mass/molar flux:
So far we have considered governing equations of Yi, the mass fraction of species “i”. The solutions,
dealt with in detail in the following Sections, will yield the spatial distributions of Yi, and in the case of
unsteady mass transfer the temporal evolution of these distributions.
However, one might in fact be interested in the mass flux of species “i”, denoted by Gi, and which is
defined as the mass flow rate of “i” per unit cross-sectional flow area, A:
It can be seen from above, that given Yi, the definition of Gi is of limited usefulness, since we still
require ui. Instead, it is best to think of the total mass flux of species “i” as the sum of an advective
part and a diffusive part,
and where the diffusive contribution is given by Fick’s law, here presented in vector calculus
notation,
such that, in one-dimension:
Note that, by summing over Eqn. 18, we can obtain a further property of the diffusive mass flux:
This existence of two contributions to the mass flux can also be seen directly by considering Eqn. 9:
In addition, and similarly to above, one can define the molar flux of species “i”, denoted by Ji:
where c is the total molar concentration and ci is the molar concentration of species “i”, and,
in an analogous way to,
with n being number of moles and m mass.
3. Steady, 1-D, incompressible, non-reacting flows and fluids with constant D:
For steady, planar one-dimensional, incompressible flows and fluids with constant D, and by using
Eqn. 8a, Eqn. 12a simplifies to:
For steady, cylindrical one-dimensional, incompressible flows and fluids with constant D, and by
using Eqn. 8b, Eqn. 12b simplifies to:
For steady, spherical one-dimensional, incompressible flows and fluids with constant D, and by using
Eqn. 8c, Eqn. 12c simplifies to:
Furthermore, for steady, one-dimensional, incompressible, non-reacting flows and fluids with
constant D, Eqn. 21 that is the same as Eqn. 13a, becomes,
Eqn. 22 that is the same as Eqn. 13b, becomes,
and Eqn. 23 that is the same as Eqn. 13c, becomes:
4. Steady, 1-D, incompressible, non-reacting stagnant mass transfer with constant D:
For steady, one-dimensional, incompressible, non-reacting stagnant mass transfer with constant D,
all velocities are zero, u = ur = 0, and Eqns. 24, 25 and 26 that are the same as Eqns. 14a, 14b and 14c
collapse to, respectively:
It is straightforward to integrate twice, and apply (two) suitable boundary conditions, in order to
obtain the distributions of Yi in each case:
5. Unsteady, 1-D planar, incompressible, non-reacting stagnant mass transfer with constant D:
The solution of Eqn. 15a,
is identical to the error function solution for unsteady, one-dimensional planar heat transfer in a
solid with constant properties, with the thermal diffusivity α being replaced here by the molecular
diffusivity D:
6. Steady, 1-D, incompressible, reacting stagnant mass transfer with constant D:
In reactive mass transfer one finds homogeneous reactions, which are volumetric (that is, they occur
throughout the volume of the fluid), and heterogeneous reactions that are not volumetric, but
surface phenomena (in that they occur on the surface of catalysts). Note that not all catalysis results
in heterogeneous reactions. The former appear as relations for
in Eqns. 16, but the latter as
boundary conditions, with
. Hence, for heterogeneous reactions, the governing equations look
exactly like Eqns. 27, in Section 4: “Steady, 1-D, incompressible, non-reacting stagnant mass transfer
with constant D”, and the solutions look exactly like Eqns. 28 in the same section.
For homogeneous reactions, the governing equations are:
A chemical reaction between two reactants R1 and R2, producing a product P, is an expression of
conservation of mass:
You will be expected to be familiar with zeroth and first order chemical reactions. In these reactions
we have,
where
is the molar production rate of P (in kmols/s), k is the reaction rate constant, and ci is the
molar concentration of species “i” (in kmols/m3). You must be careful that the units of kj depend on
the order of the reaction, such that in Eqn. 31a these are kmols/s, whereas in Eqn. 31b they are m3/s.
It is customary to deal with molar quantities in chemical kinetics, but we need to obtain the mass
reaction rate
of the various species (reactants and products) if we are to proceed here. This is
straightforward,
where MWi is the molecular weight of species “i”.
Furthermore,
and
can be obtained from Eqns. 31 depending on the type of the reaction. One final point is that
in first order reactions, you will need to turn ci into a mass based quantity, since the molar reaction
rate in Eqn. 31 involves ci,
and Eqns. 31 become:
Hence, for a zeroth order reaction, Eqns. 16 become,
where the sign of the chemical source term is positive for the product, “i” being “P”, and negative for
the reactants, “i” being “R1”, or “R2”.
The solution of Eqn. 36a, for geometries with planar symmetry, is,
For A and B we can use suitable boundary conditions for Yi, such as at x = 0, Yi,o, or
, Yi,∞.
The solution of Eqn. 36b, for cylindrically symmetric geometries, is,
The solution of Eqn. 36c, for spherically symmetric geometries, is,
Similarly, for a first order reaction, we can substitute the chemical source term from Eqn. 35b into
Eqns. 16. For example, in order to find YR1 in a planar geometry:
For a spherical geometry, we can make the following substitution:
Note that, in all solutions of second order linear, ordinary differential equations, the exponentials can
written as sinhs and coshs:
, 2A = D + C, 2B = D – C.