6. Mass Transfer
6.1 Introduction
It is important to distinct between mass transfer
and the bulk fluid motion (or fluid flow). Fluid flow
occurs on a macroscopic level as a fluid is transported
from one location to another. Mass transfer requires the
presence of two regions at different chemical
composition, and mass transfer refers to the movement
of a chemical species from a high concentration region
Figure (1) Nature tends to equalize things by
toward a lower concentration one relative to other
forcing a flow from the high to the low
chemical species present in the medium. The primary concentration
driving force for fluid flow is the pressure difference,
whereas for mass transfer it is the concentration difference. Therefore, there is no mass transfer
in homogenous medium. As shown in figure (1), the species flow is always in the direction of
decreasing concentration; that is from the region of high concentration to the region of low
concentration. The rate of flow of a species is proportional to the concentration gradient dC/dx,
and the area A normal to flow direction and is expressed as:
Flow rate ∝ (Normal area)×(Concentration gradient)
or
𝑑𝐶𝐴
𝑚̇𝑑𝑖𝑓𝑓 = −𝐷𝐴𝐵 𝐴
𝑑𝑥
Where DAB is the proportional constant and is called the
diffusion constant. The foregoing equation is similar to
Fourier’s law of conduction. This equation, tell us how
much the rate of mass of component A, at which species
A diffused in the mixture of A and B, of course in the
direction of lower value concentration of A in this
mixture.
Mass Transfer can occur in liquids and solids as
well as in gas. Typically diffusion rates in gases much
higher than they are in liquids and much higher in
liquids than in solid. Accordingly, diffusion coefficients
in gas mixtures are a few orders of magnitude larger
Figure (2) Some examples of mass transferthat
involve a liquid and /or a solid
than these of liquid or solid solution, see figure (2).
6.2 Analogy between Heat and Mass Transfer
The mechanisms of heat and mass transfer are
analogous to each other and thus the heat transfer
knowledge may be helpful to solve mass transfer
problems. In the following sections, the similarity
between heat and mass transfer is explained.
Temperature
The driving force for heat transfer is the temperature
difference. In contrast, the driving force for mass
1
Figure (3) Analogy between heat and mass transfer
transfer is the concentration difference. According to this analogy, the temperature is a measure
of heat concentration, see figure (3).
Conduction
As it is known, heat is transferred by conduction,
convection and radiation. Mass, however, is transferred by
conduction (called diffusion) and convection only and
there is no radiation mass transfer. The rate of heat
conduction in a direction x is proportional to the
temperature dradient dT/dx in that direction and is
expressed by Fourier’s law of conduction as:
𝑄̇𝑐𝑜𝑛𝑑 = −𝑘 𝐴
𝑑𝑇
𝑑𝑥
Where k is the thermal conductivity of the medium and A
is the area normal to the direction of heat transfer.
(4) Analogy between heat
Likewise, the rate of mass diffusion 𝑚̇𝑑𝑖𝑓𝑓 of a chemical Figure
conduction and mass diffusion
species A in a stationary medium in direction x is
proportional to the concentration gradient dc/dx in that direction and is expressed by Fick’s law
of diffusion as: (see figure (4))
𝑚̇𝑑𝑖𝑓𝑓 = −𝐷𝐴𝐵 𝐴
𝑑𝐶𝐴
𝑑𝑥
Where 𝐷𝐴𝐵 is the diffusion coefficient (or mass diffusivity) of species in mixture and 𝐶𝐴 is the
concentration of the species in the mixture at that location.
Heat Generation
Similar to heat generation throughout the medium, some mass transfer problems involve
chemical reaction that occur within the medium and result in the generation of a species
throughout. The rate of generation may vary from point to point in the medium. Such reactions
that occur within the medium are called homogeneous reaction. In contrast, some chemical
reactions produce species at the surface, such reaction is called heterogeneous reaction and are
analogous to specified surface heat flux.
Convection
Like heat convection, mass convection is the mass transfer mechanism between a surface and a
moving fluid that involves both mass diffusion and bulk fluid motion. In mass convection, we
define a concentration boundary layer in an analogous manner to the thermal boundary layer.
The rate of heat convection for external flow was expressed conveniently by Newton’s law of
cooling as:
𝑄̇𝑐𝑜𝑛𝑣 = ℎ𝑐𝑜𝑛𝑣 𝐴𝑠 (𝑇𝑠 − 𝑇∞ )
Likewise, the rate of mass convection can be expressed as
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𝑚̇𝑐𝑜𝑛𝑣 = ℎ𝑚𝑎𝑠𝑠 𝐴𝑠 (𝐶𝑠 − 𝐶∞ )
Where ℎ𝑚𝑎𝑠𝑠 is the mass transfer coefficient. 𝐴𝑠 is the surface area and (𝐶𝑠 − 𝐶∞ ) denotes the
concentration difference across the concentration boundary layer.
6.3 Mass Diffusion
6.3.1 Mass Concentration
As it is mentioned previously and according to Fick’s law of mass diffusion, the species
concentration plays the main role in mass transfer. The two following ways are commonly used
in expressing species concentration:
1. Mass basis
On mass basis, concentration is expressed in terms of density (or mass concentration),
which is mass per unit volume. Considering a small volume V at a location within the
mixture, the densities of a species (subscript i) and of the mixture (no subscript) at that
location are given by:
Partial density of species i :
𝜌𝑖 = 𝑚𝑖 /𝑉
Total density of mixture:
𝜌=
𝑚
𝑉
=∑
𝑚𝑖
𝑉
= ∑ 𝜌𝑖
Mass concentration can also expressed in dimensionless form in terms of mass fraction w
as
𝑚
𝑚𝑖 /𝑉
𝜌
Mass fraction of species i:
𝑤𝑖 = 𝑚𝑖 = 𝑚/𝑉
= 𝜌𝑖
As it is clear, wi varies from 0 to 1 and the conservation of mass requires that the sum of
the mass fraction of constituents of mixture be equal to 1 (∑ 𝑤𝑖 = 1).
2. Mole Basis
On mole basis, concentration is expressed in terms of molar concentration (molar
density) as:
Partial molar concentration of species i : 𝐶𝑖 = 𝑁𝑖 /𝑉
𝐶=
Total molar concentration of mixture :
𝑁
𝑉
=∑
𝑁𝑖
𝑉
= ∑ 𝐶𝑖
And dimensionless form of concentration can be expressed in terms of mole fraction y as
Mole fraction species:
𝑦𝑖 =
𝑁𝑖
𝑁
=
𝑁𝑖 /𝑉
𝑁/𝑉
=
𝐶𝑖
𝐶
As it is seen, the summation of mole fraction of species is equal to 1 (∑ 𝑦𝑖 = 1) and
single mole fraction yi takes a value from 0 to 1. The mass m and mole number N of a
substance are related to each other by:
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𝑚 = 𝑁 𝑀 (or for a unit volume, 𝜌 = 𝐶 𝑀), where M is the molar mass (molecular
𝜌
𝜌
weight) of substance. Accordingly; 𝑐𝑖 = 𝑀𝑖 (for species) and 𝑐 = 𝑀 (for mixture). Where
𝑖
M is the molar mass of the mixture which can be determined by:
𝑀=
𝑚 ∑ 𝑚𝑖 ∑ 𝑀𝑖 𝑁𝑖
𝑁𝑖
=
=
= ∑ 𝑀𝑖 = ∑ 𝑦𝑖 𝑀𝑖
𝑁
𝑁
𝑁
𝑁
The mass and mole fractions of species i of a mixture are related to each other by:
𝑤𝑖 =
𝜌𝑖 𝐶𝑖 𝑀𝑖
𝑀𝑖
= ×
= 𝑦𝑖
𝜌
𝐶 𝑀
𝑀
For perfect gas: The pressure fraction, is given through the following relations:
𝑃𝑖 𝑁𝑖 𝑅𝑢 𝑇/𝑉 𝑁𝑖
=
=
= 𝑦𝑖
𝑃 𝑁 𝑅𝑢 𝑇/𝑉
𝑁
Therefore, the pressure fraction of species i of an ideal gas mixture is equivalent to the
mole fraction of that species.
6.3.2 Fick’s Law of Diffusion
Stationary Medium consisting of Two Species
The linear relationship between the rate of diffusion of mass and the
concentration gradient proposed by Fick is known as Fick’s Law of
diffusion and can be expressed as:
Mass flux = Constant of proportionality × Concentration gradient
has different forms, according to the expression of concentration (ρ, C,
w, y), thus one can write mathematically (as shown in figure (5) :
Mass basis:
𝐽𝑑𝑖𝑓𝑓,𝐴 = −𝜌 𝐷𝐴𝐵
Mole basis: 𝐽𝑑𝑖𝑓𝑓,𝐴 = −𝐶 𝐷𝐴𝐵
𝑑𝜌𝐴 /𝜌
𝑑𝑥
𝑑𝐶𝐴 /𝐶
𝑑𝑥
= −𝜌 𝐷𝐴𝐵
= −𝐶 𝐷𝐴𝐵
𝑑𝑤𝐴
(kg/m2.s)
𝑑𝑥
𝑑𝑦𝐴
𝑑𝑥
(kmole/m2.s)
For the special case, when ρ and C can be assumed constant, the
foregoing equations of diffusive mass flux and diffusive molar flux
take simpler form as:
Mass basis:
Mole basis:
𝐽𝑑𝑖𝑓𝑓,𝐴 = − 𝐷𝐴𝐵
𝐽𝑑𝑖𝑓𝑓,𝐴 = − 𝐷𝐴𝐵
𝑑𝜌𝐴
𝑑𝑥
𝑑𝐶𝐴
𝑑𝑥
(kg/m2.s)
(kmole/m2.s)
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Figure (5) Various expression
of Fick’s law of diffusion for a
binary mixture.
This assumption of ρ = constant and C = constant, is appropriate for solid and dilute solution but
for gas mixture and concentrated liquids solutions, it is not true.
The mass diffusivity (binary diffusion coefficient) DAB is a transport property as thermal
diffusivity and momentum diffusivity (kinematic viscosity) v.
The diffusion coefficients are usually determined experimentally.
For dilute gases at ordinary pressure, DAB has the following relations:
𝐷𝐴𝐵 ∝
𝑇 3/2
𝐷𝐴𝐵,1
or
𝑃
𝐷𝐴𝐵,2
𝑃
3/2
𝑇
= 𝑃2 × (𝑇1 )
1
2
Table (1) gives the diffusion coefficient of some gases in air at 1atm pressure. The binary
diffusion coefficient for several binary gas mixtures and solid and liquid solutions are given in
Tables (2) and Table (3). As it is clear, from these tables, the diffusion coefficients, in general,
are highest in gases and lowest in solids. Also, it is noted, that the diffusion coefficients increase
with temperature.
For water vapor diffuses in air, there is an empirical formula states, also table (4), that:
𝐷𝐻2 𝑜−𝐴𝑖𝑟 = 1.87 × 10−10
𝑇 2.072
(m2/s)
𝑃
280 K < T < 450 K
Where P is the total pressure in atm and T is the temperature in K.
6.4 Boundary Conditions
The two common types of boundary conditions are :
1. Specified species concentration, which is corresponding to specified temperature.
2. Specified species flux, which is corresponding to specified heat flux.
Despite their apparent similarity, an important difference
exists between temperature and concentration;
temperature is necessarily a continuous function, but
concentration, in general is not. For water-air interface,
the concentration of air on the two sides is obviously very
different as shown in figure (6).In fact the concentration
of air in water is close to zero. Likewise, the
concentration of water on the two sides of water air
interface is also different even when air is saturated.
Accordingly, in mass transfer boundary condition, not
enough the location but the side of this boundary of great
importance (e. g. liquid side or gas side).
Figure (6) The concentration of species on the
two sides of liquid-gas (or solid-gas or solidliquid)ninterface are usually not the same.
Using Fick’s Law, the constant species flux boundary condition for a diffusing species A at
the boundary x = 0 is:
−𝐶 𝐷𝐴𝐵
𝑑𝑦𝐴
𝑑𝑥
= 𝐽𝐴,0
or
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−𝜌 𝐷𝐴𝐵
𝑑𝑤𝐴
𝑑𝑥
= 𝐽𝐴,0
For impermeable surface, no mass can be diffused, and
accordingly (see figure (7) :
𝑑𝑦𝐴 (0) 𝑑𝑤𝐴 (0)
=
=0
𝑑𝑥
𝑑𝑥
To apply the specified concentration boundary
condition, the concentration of species at the boundary
must be known. This information is usually obtained
from the requirement that thermodynamics equilibrium
must exist at the interface of the two phases of species.
In case of air-water interface, the concentration values
of water vapor in the air are easily determined from
saturation data.
Figure (7) An impermeable surface in mass transfer
is analogous to an insulated surface in heat transfer.
The situation is similar at solid-liquid interface. Again, at given temperature, only a certain
amount of solid can be dissolved in liquid, and the solubility of the solid in liquid is determined
from the requirement that thermodynamic equilibrium exists between the solid and the solution
at the interface. The solubility represents the maximum amount of solid that can be dissolved in a
liquid at specified temperature. Table (5) gives solubility data for sodium chloride (Na Cl) and
Calcium bicarbonate [Ca (h CO3)2] at various temperature. For a solid diffuses in a liquid, the
mass fraction of salt at the interface and at liquid side is given by:
𝑤𝑠𝑎𝑙𝑡,𝑙𝑖𝑞𝑢𝑖𝑑 𝑠𝑖𝑑𝑒 =
𝑚𝑠𝑎𝑙𝑡
𝑠𝑜𝑙𝑢𝑏𝑖𝑙𝑖𝑡𝑦
=
𝑚
𝑚𝑠𝑎𝑙𝑡 + 𝑚𝑙𝑖𝑞𝑢𝑖𝑑
Whereas the mass fraction of salt in the pure solid salt is wsalt,solid side =1.0.
Most gases are weakly soluble in liquids (such as air in water) and for dilute solutions the
mole fractions of a species i in the gas and liquid phases at the interface are observed to be
proportional to each other. That is, yi,gas side ∝ yi,liquid side or Pi,gas side ∝ P yi, liquid side since yi, gas side =
Pi,gas side / P for ideal gas mixtures. This is known as Henry’s law and is expressed as:
𝑦𝑖,𝑙𝑖𝑞𝑢𝑖𝑑 𝑠𝑖𝑑𝑒 =
𝑃𝑖,𝑔𝑎𝑠 𝑠𝑖𝑑𝑒
𝐻
(at interface)
Where H is Henry’s constant, its values for a number of aqueous solutions are given in Table (6).
as it is clear from the Table (6), the dissolved gases decrease with increasing temperature. Also,
from equation the concentration of a gas dissolved in a liquid is proportional to the partial
pressure of the gas.
For highly dissolved gases in liquid, the mole fraction in liquid side can be given, according to
Raoult’s law, as:
𝑃𝑖,𝑔𝑎𝑠 𝑠𝑖𝑑𝑒 = 𝑦𝑖,𝑔𝑎𝑠 𝑠𝑖𝑑𝑒 𝑃 = 𝑦𝑖,𝑙𝑖𝑞𝑢𝑖𝑑 𝑠𝑖𝑑𝑒 𝑃𝑖,𝑠𝑎𝑡 (𝑇)
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Where Pi, sat(T) is the saturation pressure of the species i at the interface temperature and P is the
total pressure on the gas phase side.
Gases may also dissolve in solids. The concentration of gas species i in the solid at the interface
Ci,solid side is proportional to the partial pressure of species i in the gas Pi, gas side on the gas side of
the interface and is expressed as:
𝐶𝑖,𝑠𝑜𝑙𝑖𝑑 𝑠𝑖𝑑𝑒 = T × 𝑃𝑖,𝑔𝑎𝑠 𝑠𝑖𝑑𝑒
(kmole/m3)
Where T is the solubility. Expressing the pressure in bars and noting that the unit of molar
concentration is kmole of species i per m3, the unit of solubility is kmole/m3.bar. Solubility data
for selected gas-solid combinations are given in Table (7).
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