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The differential equation for mass transfer is obtained by applying the law of ... Steady state, one dimensional flow (assume in x direction) and constant ...
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Definition:
The differential equations of mass transfer are general equations describing mass transfer in all directions and at all conditions.
How is the differential equation obtained?
The differential equation for mass transfer is obtained by applying the law of conservation of mass (mass balance) to a differential control volume representing the system.
The resulting equation is called the continuity equation and takes two forms:
(1) Total continuity equation [in โ out = accumulation] (this equation is obtained if we applied the law of conservation of mass on the total mass of the system) (2) Component continuity equation[in โ out + generation โ consumption = accumulation] (this equation is obtained if we applied the law of conservation of mass to an individual component) (1) Total continuity equation
Consider the control volume, ฮx ฮy ฮz (Fig. 1)
Fig. 1
Apply the law of conservation of mass on this control volume [in โ out = accumulation] direction in out in - out x (^) ๐v๐ฅ โ๐ฆโ๐งโ (^) ๐ฅ ๐v๐ฅ โ๐ฆโ๐งโ (^) ๐ฅ+ฮ๐ฅ (๐v๐ฅ โ ๐ฅ โ ๐v๐ฅโ ๐ฅ+ฮ๐ฅ) โ๐ฆโ๐ง
y (^) ๐v๐ฆ โ๐ฅโ๐งโ (^) ๐ฆ ๐v๐ฆ โ๐ฅโ๐งโ (^) ๐ฆ+ฮ๐ฆ (๐v๐ฆ โ ๐ฆ โ ๐v๐ฆ โ ๐ฆ+ฮ๐ฆ) โ๐ฅโ๐ง z (^) ๐v๐ง โ๐ฅโ๐ฆโ (^) ๐ง ๐v๐ง โ๐ฅโ๐ฆโ (^) ๐ง+ฮ๐ง (๐v๐ง โ ๐ง โ ๐v๐ง โ ๐ง+ฮ๐ง) โ๐ฅโ๐ฆ
Accumulation = ๐๐๐๐ก = ๐๐โ๐ฅโ๐ฆโ๐ง๐๐ก = โ๐ฅโ๐ฆโ๐ง ๐๐๐๐ก
Write the above terms in the overall equation [in โ out = accumulation (rate of change)]
(๐v๐ฅโ ๐ฅ โ ๐v๐ฅ โ ๐ฅ+ฮ๐ฅ) โ๐ฆโ๐ง + (๐v๐ฆ โ ๐ฆ โ ๐v๐ฆ โ ๐ฆ+ฮ๐ฆ) โ๐ฅโ๐ง + (๐v๐ง โ ๐ง โ ๐v๐งโ ๐ง+ฮ๐ง) โ๐ฅโ๐ฆ = โ๐ฅโ๐ฆโ๐ง ๐๐๐๐ก
Dividing each term in the above equation by โ๐ฅโ๐ฆโ๐ง:
(๐v๐ฅโ ๐ฅโ ๐v๐ฅ โ๐ฅ+ฮ๐ฅ) โ๐ฅ +^
(๐v๐ฆ โ๐ฆโ ๐v๐ฆ โ๐ฆ+ฮ๐ฆ) โ๐ฆ +^
(๐v๐ง โ๐งโ ๐v๐ง โ๐ง+ฮ๐ง) โ๐ง =^
๐๐ ๐๐ก
Take the limit as ฮx, ฮy, and ฮz approach zero:
โ (^) ๐๐ฅ๐ ๐v๐ฅ - (^) ๐๐ฆ๐ ๐v๐ฆ - (^) ๐๐ง๐ ๐v๐ง = ๐๐๐๐ก
โด (^) ๐๐ฅ๐ ๐v๐ฅ+ (^) ๐๐ฆ๐ ๐v๐ฆ + (^) ๐๐ง๐ ๐v๐ง + ๐๐ ๐๐ก = 0
The above equation is the general total continuity equation (the velocity distribution can be obtained from this equation)
It can be written in the following form (this form can be used in all coordination system):
โ. ๐vโโโโ + ๐๐๐๐ก = 0
direction in out in - out x (^) ๐๐ด,๐ฅ โ๐ฆโ๐งโ (^) ๐ฅ ๐๐ด,๐ฅ โ๐ฆโ๐งโ (^) ๐ฅ+ฮ๐ฅ (๐๐ด,๐ฅ โ ๐ฅ โ ๐๐ด,๐ฅโ ๐ฅ+ฮ๐ฅ) โ๐ฆโ๐ง y (^) ๐๐ด,๐ฆ โ๐ฅโ๐งโ (^) ๐ฆ ๐๐ด,๐ฆ โ๐ฅโ๐งโ (^) ๐ฆ+ฮ๐ฆ (๐๐ด,๐ฆ โ ๐ฆ โ ๐๐ด,๐ฆ โ ๐ฆ+ฮ๐ฆ) โ๐ฅโ๐ง z (^) ๐๐ด,๐ง โ๐ฅโ๐ฆโ (^) ๐ง ๐๐ด,๐ง โ๐ฅโ๐ฆโ (^) ๐ง+ฮ๐ง (๐๐ด,๐ง โ ๐ง โ ๐๐ด,๐ง โ ๐ง+ฮ๐ง) โ๐ฅโ๐ฆ
Accumulation = ๐๐๐๐ก = ๐๐๐จโ๐ฅโ๐ฆโ๐ง๐๐ก = โ๐ฅโ๐ฆโ๐ง ๐๐ ๐๐ก๐จ
If A is produced within the control volume by a chemical reaction at a rate ๐๐ด (mass/(volume)(time)
Rate of production of A (generation) = ๐๐ด โ๐ฅโ๐ฆโ๐ง
Put all terms in the equation: in โ out + generation โ consumption = accumulation
(๐๐ด,๐ฅ โ๐ฅ โ ๐๐ด,๐ฅโ ๐ฅ+ฮ๐ฅ) โ๐ฆโ๐ง + (๐๐ด,๐ฆโ ๐ฆ โ ๐๐ด,๐ฆ โ๐ฆ+ฮ๐ฆ) โ๐ฅโ๐ง + (๐๐ด,๐ง โ๐ง โ ๐๐ด,๐งโ ๐ง+ฮ๐ง) โ๐ฅโ๐ฆ + ๐๐ด โ๐ฅโ๐ฆโ๐ง = โ๐ฅโ๐ฆโ๐ง ๐๐ ๐๐ก๐ด
Dividing each term in the above equation by โ๐ฅโ๐ฆโ๐ง:
(๐๐ด,๐ฅโ ๐ฅโ ๐๐ด,๐ฅโ ๐ฅ+ฮ๐ฅ) โ๐ฅ +^
(๐๐ด,๐ฆ โ๐ฆโ ๐๐ด,๐ฆ โ๐ฆ+ฮ๐ฆ) โ๐ฆ +^
(๐๐ด,๐งโ ๐งโ ๐๐ด,๐งโ ๐ง+ฮ๐ง) โ๐ง +^ ๐๐ด=^
๐๐๐ด ๐๐ก
Take the limit as ฮx, ฮy, and ฮz approach zero:
โ (^) ๐๐ฅ๐ ๐๐ด,๐ฅ - (^) ๐๐ฆ๐ ๐๐ด,๐ฆ - (^) ๐๐ง๐ ๐๐ด,๐ง + ๐๐ด= ๐๐ ๐๐ก๐ด
๐ ๐๐ฅ ๐๐ด,๐ฅ^ +^
Equation 1 is the component mass continuity equation and it can be written in the form:
โ. ๐โ๐ด + ๐ ๐๐ก๐๐ด โ ๐๐ด = 0 (2)
(The above equation can be written in different coordinate systems since it is written in a vector form)
But from Fickโs law
๐๐ด = โ๐๐ท๐ด๐ตโ๐๐ด + ๐๐ดv
Substitute in equation 2 by this value we can get the equation:
โโ. ๐๐ท๐ด๐ตโ๐๐ด + โ. ๐๐ดv + ๐ ๐๐ก๐๐ด โ ๐๐ด = 0 (3)
Equation 3 is a general equation used to describe concentration profiles (in mass basis) within a diffusing system.
(ii) Component molar continuity equation
Equations 1, 2 and 3 can be written in the form of molar units to get the component continuity equation in molar basis by replacing:
๐๐ด ๐๐ฆ ๐๐ด;๐ ๐๐ฆ ๐; ๐๐ด ๐๐ฆ ๐ฆ๐ด ; ๐๐ด ๐๐ฆ ๐๐ด ๐๐๐ ๐๐ด ๐๐ฆ ๐ ๐ด
The different forms of the component molar continuity equation:
๐ ๐๐ฅ ๐๐ด,๐ฅ^ +^
Equation 4 is the component molar continuity equation and it can be written in the form:
โ. ๐โโ๐ด + ๐ ๐๐ก๐๐ด โ ๐ ๐ด = 0 (5)
But from Fickโs law
๐๐ด = โ๐๐ท๐ด๐ตโ๐ฆ๐ด + ๐๐ดV
Substitute in equation 5 by this value we can get the equation:
โโ. ๐๐ท๐ด๐ตโ๐ฆ๐ด + โ. ๐๐ดV + ๐ ๐๐ก๐๐ด โ ๐ ๐ด = 0 (6)
Equation 6 is a general equation used to describe concentration profiles (in molar basis) within a diffusing system.
Note: you may be given the general form and asked to apply specific conditions to get a special form of the differential equation.
For molar concentration:
๐๐๐ด ๐๐ก =^ ๐ท๐ด๐ตโ
Equation 7 referred to as Fickโs second law of diffusion
Fickโs second โโlawโโ of diffusion written in rectangular coordinates is
Note: what are the conditions at which there is no fluid motion? (bulk motion = 0.0)
The assumption of no fluid motion (bulk motion) restricts its applicability to:
a) Diffusion in solid b) Stationary (stagnant) liquid c) Equimolar counterdiffusion (for binary system of gases or liquids where ๐๐ด is equal in magnitude but acting in the opposite direction to ๐๐ต ๐๐ด = โ๐๐ท๐ด๐ต^ ๐๐ฆ ๐๐ง ๐ด+ ๐ฆ๐ด(๐๐ด + ๐๐ต)
If ๐๐ด = โ๐๐ต the bulk motion term will be cancelled from the above equation
For mass concentration:
โ^2 ๐๐ด = 0
For mass concentration:
โ^2 ๐๐ด = 0
Note: see page 438 in the reference book for the differential equation of mass transfer in different coordinate systems.
The general differential equation for mass transfer of component A, or the equation of continuity of A, written in rectangular coordinates is
๏ Initial and Boundary conditions
To describe a mass transfer process by the differential equations of mass transfer the initial and boundary conditions must be specified.
Initial and boundary conditions are used to determine integration constants associated with the mathematical solution of the differential equations for mass transfer
1. Initial conditions:
It means the concentration of the diffusing species at the start (t = 0) expressed in mass or molar units.
๐๐ก ๐ก = 0 ๐๐ด = ๐๐ด๐ (๐๐๐๐๐ ๐ข๐๐๐ก๐ )
๐๐ก ๐ก = 0 ๐๐ด = ๐๐ด๐ (๐๐๐ ๐ ๐ข๐๐๐ก๐ )
where ๐๐ด๐ and ๐๐ด๐ are constant (defined values)
(ii) for solutions where species A is only weakly soluble in the liquid, Henryโs law may be used to relate the mole fraction of A in the liquid to the partial pressure of A in the gas ๐๐ด = ๐ป โ ๐ฅ๐ด H: Henryโs constant (iii) at gas solid interface
๐๐ด๐ ๐๐๐๐ = ๐ โ ๐๐ด ๐๐ด๐ ๐๐๐๐ : is the molar concentration of A within the solid at the interface in units of kg mol/m^3 and ๐๐ด is the partial pressure of gas phase species A over the solid in units of Pa. ๐: solubility constant (partition coefficient) (2) A reacting surface boundary is specified
๐๐ด โ ๐ง=0 = ๐๐๐๐ด^ ๐๐
where ๐๐ is a surface reaction rate constant with units of m/s. n is the reaction order
Note: the reaction may be so rapid that ๐๐ด๐ = 0 if species A is the limiting reagent in the chemical reaction.
(3) The flux of the transferring species is zero at an impermeable boundary
๐๐ด โ ๐ง=0 = โ๐ท๐ด๐ต^ ๐๐ ๐๐งโ๐ด ๐ง=0 = 0
where the impermeable boundary or the centerline of symmetry is located at z = 0
(4) The convective mass flux at the boundary surface is specified
๐๐ดโ ๐ง=0 = ๐ (๐๐ด๐ โ ๐๐ดโ ) Where k is the convection mass transfer coefficient
Solved problems:
Problem 1:
The following sketch illustrates the gas diffusion in the neighborhood of a catalytic surface. Hot gases of heavy hydrocarbons diffuse to the catalytic surface where they are cracked into lighter compounds by the reaction: H โ 2L, the light products diffuse back into the gas stream.
a. Reduce the general differential equation for mass transfer to write the specific differential equation that will describe this steady-state transfer process if the catalyst is considered a flat surface. List all of the assumptions you have made in simplifying the general differential equation. b. Determine the Fickโs law relationship in terms of only compound H and insert it into the differential equation you obtained in part (a). c. Repeat the solution for spherical catalyst surface.
Solution
a. The specific differential equation
Assumptions: steady state, unidirectional mass transfer.
๐ ๐๐ฅ ๐๐ด,๐ฅ^ +^
Apply these assumptions on the general equation we get the specific differential equation:
๐ ๐๐ง ๐๐ป,๐ง^ + ๐ ๐ป^ = 0
b. Fickโs law relationship in terms of only compound H
๐๐ป = โ๐๐ท๐ป๐ฟ^ ๐๐ฆ ๐๐ง ๐ป+ ๐ฆ๐ป(๐๐ป + ๐๐ฟ)
Apply these assumptions on the general differential equation:
โ. ๐โโ๐ด + ๐๐ ๐๐ก ๐ดโ ๐ ๐ด = 0
โด โ. ๐โโ๐ด = 0
For spherical coordinates and mass transfer in r-direction
1 ๐^2
b. The simplified differential form of Fickโs equation for water vapor (species A)? Assume water is A and air is B
๐๐ด = โ๐๐ท๐ด๐ต^ ๐๐ฆ ๐๐ ๐ด+ ๐ฆ๐ด(๐๐ด + ๐๐ต)
๐๐ต = 0
๐๐ด = โ (^) (1 โ ๐ฆ๐๐ท๐ด๐ต ๐ด)
Problem 3:
A large deep lake, which initially had a uniform oxygen concentration of 1kg/m^3 , has its surface concentration suddenly raised and maintained at 9 kg/m^3 concentration level. Reduce the general differential equation for mass transfer to write the specific differential equation for
a. the transfer of oxygen into the lake without the presence of a chemical reaction; b. the transfer of oxygen into the lake that occurs with the simultaneous disappearance of oxygen by a first-order biological reaction.
Solution:
Assume oxygen = A and water = B
a. the transfer of oxygen into the lake without the presence of a chemical reaction
Basic assumptions:
But
๐๐ด = โ๐๐ท๐ด๐ต^ ๐๐ฆ ๐๐ง ๐ด+ ๐๐ด๐
Since the liquid is stationary
โด ๐๐ด = โ๐๐ท๐ด๐ต^ ๐๐ฆ ๐๐ง๐ด
๐๐๐ด ๐๐ง = โ๐ท๐ด๐ต
b. The transfer of oxygen into the lake that occurs with the simultaneous disappearance of oxygen by a first-order biological reaction.
Basic assumptions:
๐ท๐ด๐ต^ ๐
where ๐๐is the reaction rate constant
But ๐๐ฆ = 0 (based on the assumptions)
โด ๐๐ด,๐ฆ = โ๐ท๐ด๐ต^ ๐๐ ๐๐ฆ๐ด
Substitute by the values of ๐๐ด,๐ฅ and ๐๐ด,๐ฆ in equation (i)
โ๐ท๐ด๐ต^ ๐
๐๐ฅ^2 + ay
๐๐ฆ^2 ] = ay
Boundary conditions:
Note:
The specific differential equation of mass transfer for a given system can be obtained by two methods:
Supplementary data:
The general differential equation for mass transfer of component A, in rectangular coordinates is