2.3. General Molecular Transport Equation For Momentum, Heat, and
Mass Transfer
2.3A General Molecular Transport Equation and General Property Balance
1. Introduction: ( Reading)
2. General Molecular Transport Equation. All three of transport processes (Momentum,
heat or mass) are characterized by the same general type of transport equation.
Rate of a transfer process = driving force/ resistance
That means: A driving force is needed to overcome a resistance in order to transport a
property. For molecular transport or diffusion of a property:
 z  
d
dz
Where Ψz is the flux of the property as amount of property being transferred per
unit time per cross-sectional area perpendicular to the z direction of flow (amount
of property/s.m2). δ is a proportionality constant called diffusivity (m2/s), Γ is
concentration of the property (amount of property/m 3), and z in the direction of flow
(m).
z2
2
z1
1
 z  dz    d 
z  
 ( 2  1 )
z2  z1
Concentration of Property, 
For steady state, Ψz is constant. Rearranging:


z Flux
z1
z2
Example 2.3-1: ( See the textbook or write it down from the board)
3. General property balance for unsteady state
To account for a property transported in the entire system, a general balance (conservation
equation) for this property at unsteady state is needed. for one-dimensional assumption,
say z, and unit volume, z(1) m3 (see the fig.):
In z|z
Out =z|z+z
z
z
Z+z
Unit
area
(rate of property in) + (rate of generation of property)
=(rate of property out) + (rate of accumulation of property). For 1.0 m2 cross-sectional
area:
Input rate= (z|z)*1 amount of property/s and output = (z|z+z)*1. the rate of generation =
R(z*1) where R is rate of generation of property/s.m3. the accumulation term is:
Rate of accumulation of property =

( z *1) . Thus,
t
(z|z)*1 + R(z*1) = (z|z+z)*1 +

( z *1)
t
Dividing by z, and letting z go to zero,
  z

R
t
z
Differentiating the equation above,  z  
d
, gives:
dz
 z
 2
  2 .
z
t
Thus, with assumption of  is constant,:

 2
 2  R
t
z
If there is no generation,

 2
 2
t
z
NOTE that the final equation relates the concentration of the property to position z and
time t.
2.3B Introduction to Molecular Transport
Molecular transport or molecular diffusion of a property such as momentum, mass, or heat
occurs in a fluid due to rapid random movements of individual molecules.
A net flux of a property from high to low concentration will occur.
1. Momentum transport and Newton’s law
vx
z
x
Fluid is flowing in x direction parallel to a solid surface:
the fluid has x-directed momentum, (kg.m/s). Momentum concentration is vx
(Momentum/m3)= ((kg.m/s)/ m3).
random diffusion of molecules => equal no. of molecules is moving in each direction
between the faster-and slower- moving layer of molecules. Thus, momentum is transferred
in the z direction from the faster- to the slower- moving layer. The momentum transport
equation is written as follows:
 zx  
d (vx  )
dz
zx: flux of x-directed momentum in the z direction, (kg.m/s)/s.m2, v: the momentum
diffusivity, m2/s, = and is the viscosity, kg/m.s
2. Heat transport and Fourier’s law: it can be written as:
d (  c pT )
qz
 
A
dz
qz/A: the heat flux, J/s.m2, : the thermal diffusivity, m2/s, and cpT: concentration of heat
(thermal energy), J/m3. when there is a temperature gradient, energy transfer by molecules
moving equally between the hot and colder region.
3. Mass transport and Fick’s law: Fick’s law can be written as:
J  zx   DAB
dC A
dz
J*AZ: the flux of A, kgmol A/s.m2, DAB: molecular diffusivity of the molecule A in B, m2/s,
and CA: concentration of A in kgmol A/m3. when there is a concentration gradient, mass
transfer by molecules moving equally between the high and low concentration region.
NOTE: Compare all the above three equations with the general equation of molecular
transport.