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Differential Equations of Mass Transfer

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Differential Equations of Mass Transfer Differential equations of mass transfer 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 – 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 – out + generation – 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, Δx Δy Δz (Fig. 1) Fig. 1 1 Apply the law of conservation of mass on this control volume [in – out = accumulation] direction in out in - out x 휌v푥 ∆푦∆푧⃒푥 휌v푥 ∆푦∆푧⃒푥+Δ푥 (휌v푥⃒푥 − 휌v푥⃒푥+Δ푥) ∆푦∆푧 y 휌v푦 ∆푥∆푧⃒푦 휌v푦 ∆푥∆푧⃒푦+Δ푦 (휌v푦⃒푦 − 휌v푦⃒푦+Δ푦) ∆푥∆푧 z 휌v푧 ∆푥∆푦⃒푧 휌v푧 ∆푥∆푦⃒푧+Δ푧 (휌v푧⃒푧 − 휌v푧⃒푧+Δ푧) ∆푥∆푦 휕푚 휕휌∆푥∆푦∆푧 휕흆 Accumulation = = = ∆푥∆푦∆푧 휕푡 휕푡 휕푡 Write the above terms in the overall equation [in – out = accumulation (rate of change)] 휕흆 (휌v ⃒ − 휌v ⃒ ) ∆푦∆푧 + (휌v ⃒ − 휌v ⃒ ) ∆푥∆푧 + (휌v ⃒ − 휌v ⃒ ) ∆푥∆푦 = ∆푥∆푦∆푧 푥 푥 푥 푥+Δ푥 푦 푦 푦 푦+Δ푦 푧 푧 푧 푧+Δ푧 휕푡 Dividing each term in the above equation by ∆푥∆푦∆푧: (휌v푥⃒푥− 휌v푥⃒푥+Δ푥) (휌v푦⃒푦− 휌v푦⃒푦+Δ푦) (휌v푧⃒푧− 휌v푧⃒푧+Δ푧) 휕흆 + + = ∆푥 ∆푦 ∆푧 휕푡 Take the limit as Δx, Δy, and Δz approach zero: 휕 휕 휕 휕휌 − 휌v - 휌v - 휌v = 휕푥 푥 휕푦 푦 휕푧 푧 휕푡 휕 휕 휕 휕휌 ∴ 휌v + 휌v + 휌v + = 0 휕푥 푥 휕푦 푦 휕푧 푧 휕푡 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): 휕휌 ∇. 휌⃗⃗⃗⃗v + = 0 휕푡 2 Special forms of the general continuity equation: a. Steady state conditions ∇. 휌⃗⃗⃗⃗v = 0 b. Constant density (either steady state or not) ∇. v⃗ = 0 c. Steady state, one dimensional flow (assume in x direction) and constant density 푑v x = 0 푑푥 (2) Component continuity equation The component continuity equation takes two forms depending on the units of concentration; (i) mass continuity equation and (ii) molar continuity equation. Component continuity equation Mass units Molar units What is the importance of the component differential equation of mass transfer? It is used to get (describe) the concentration profiles, the flux or other parameters of engineering interest within a diffusing system. (i) Component mass continuity equation: Consider the control volume, Δx Δy Δz (Fig. 1) Apply the law of conservation of mass to this control volume: [in – out + generation – consumption = accumulation] Note: in the total continuity equation there is no generation or consumption terms 3 direction in out in - out x 푛퐴,푥 ∆푦∆푧⃒푥 푛퐴,푥 ∆푦∆푧⃒푥+Δ푥 (푛퐴,푥⃒푥 − 푛퐴,푥⃒푥+Δ푥) ∆푦∆푧 y 푛퐴,푦 ∆푥∆푧⃒푦 푛퐴,푦 ∆푥∆푧⃒푦+Δ푦 (푛퐴,푦⃒푦 − 푛퐴,푦⃒푦+Δ푦) ∆푥∆푧 z 푛퐴,푧 ∆푥∆푦⃒푧 푛퐴,푧 ∆푥∆푦⃒푧+Δ푧 (푛퐴,푧⃒푧 − 푛퐴,푧⃒푧+Δ푧) ∆푥∆푦 휕푚 휕흆 ∆푥∆푦∆푧 휕흆 Accumulation = = 푨 = ∆푥∆푦∆푧 푨 휕푡 휕푡 휕푡 If A is produced within the control volume by a chemical reaction at a rate 푟퐴 (mass/(volume)(time) Rate of production of A (generation) = 푟퐴 ∆푥∆푦∆푧 Put all terms in the equation: in – out + generation – consumption = accumulation 휕휌퐴 (푛 ⃒ − 푛 ⃒ ) ∆푦∆푧 + (푛 ⃒ − 푛 ⃒ ) ∆푥∆푧 + (푛 ⃒ − 푛 ⃒ ) ∆푥∆푦 + 푟 ∆푥∆푦∆푧 = ∆푥∆푦∆푧 퐴,푥 푥 퐴,푥 푥+Δ푥 퐴,푦 푦 퐴,푦 푦+Δ푦 퐴,푧 푧 퐴,푧 푧+Δ푧 퐴 휕푡 Dividing each term in the above equation by ∆푥∆푦∆푧: (푛퐴,푥⃒푥− 푛퐴,푥⃒푥+Δ푥) (푛퐴,푦⃒푦− 푛퐴,푦⃒푦+Δ푦) (푛퐴,푧⃒푧− 푛퐴,푧⃒푧+Δ푧) 휕휌퐴 + + + 푟 = ∆푥 ∆푦 ∆푧 퐴 휕푡 Take the limit as Δx, Δy, and Δz approach zero: 휕 휕 휕 휕휌 − 푛 - 푛 - 푛 + 푟 = 퐴 휕푥 퐴,푥 휕푦 퐴,푦 휕푧 퐴,푧 퐴 휕푡 휕 휕 휕 휕휌 푛 + 푛 + 푛 + 퐴 − 푟 = 0 (1) 휕푥 퐴,푥 휕푦 퐴,푦 휕푧 퐴,푧 휕푡 퐴 Equation 1 is the component mass continuity equation and it can be written in the form: 휕휌 ∇. 푛⃗ + 퐴 − 푟 = 0 (2) 퐴 휕푡 퐴 (The above equation can be written in different coordinate systems since it is written in a vector form) 4 But from Fick’s law 푛퐴 = −휌퐷퐴퐵∇휔퐴 + 휌퐴v Substitute in equation 2 by this value we can get the equation: 휕휌 −∇. 휌퐷 ∇휔 + ∇. 휌 v + 퐴 − 푟 = 0 (3) 퐴퐵 퐴 퐴 휕푡 퐴 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: 푛퐴 푏푦 푁퐴;휌 푏푦 푐; 휔퐴 푏푦 푦퐴 ; 휌퐴 푏푦 푐퐴 푎푛푑 푟퐴 푏푦 푅퐴 The different forms of the component molar continuity equation: 휕 휕 휕 휕푐 푁 + 푁 + 푁 + 퐴 − 푅 = 0 (4) 휕푥 퐴,푥 휕푦 퐴,푦 휕푧 퐴,푧 휕푡 퐴 Equation 4 is the component molar continuity equation and it can be written in the form: 휕푐 ∇. 푁⃗⃗ + 퐴 − 푅 = 0 (5) 퐴 휕푡 퐴 But from Fick’s law 푁퐴 = −푐퐷퐴퐵∇푦퐴 + 푐퐴V Substitute in equation 5 by this value we can get the equation: 휕푐 −∇. 푐퐷 ∇푦 + ∇. 푐 V + 퐴 − 푅 = 0 (6) 퐴퐵 퐴 퐴 휕푡 퐴 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. 5 Special forms of the component continuity equation 1. If the density and diffusion coefficient are constant (assumed to be constant) For mass concentration equation 3 becomes 휕휌 −퐷 ∇2휌 + 휌 ∇. v + v ∇. 휌 + 퐴 − 푟 = 0 퐴퐵 퐴 퐴 퐴 휕푡 퐴 For constant density ∇. v = 0 [from the total continuity equation] (page 3 no. b) 휕휌 ∴ −퐷 ∇2휌 + v ∇. 휌 + 퐴 − 푟 = 0 퐴퐵 퐴 퐴 휕푡 퐴 and for molar concentration equation 6 becomes 휕푐 ∴ −퐷 ∇2푐 + V ∇. 푐 + 퐴 − 푅 = 0 퐴퐵 퐴 퐴 휕푡 퐴 2. If there is no consumption or generation term and the density and diffusion coefficient are assumed constant For mass concentration: 휕휌 ∴ −퐷 ∇2휌 + v ∇. 휌 + 퐴 = 0 퐴퐵 퐴 퐴 휕푡 For molar concentration: 휕푐 −퐷 ∇2푐 + V ∇. 푐 + 퐴 = 0 퐴퐵 퐴 퐴 휕푡 3. If there is no fluid motion, no consumption or generation term, and constant density and diffusivity For mass concentration: 휕휌 ∴ 퐴 = 퐷 ∇2휌 휕푡 퐴퐵 퐴 6 For molar concentration: 휕푐 퐴 = 퐷 ∇2푐 (7) 휕푡 퐴퐵 퐴 Equation 7 referred to as Fick’s second law of diffusion Fick’s second ‘‘law’’ 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 푁퐴 is equal in magnitude but acting in the opposite direction to 푁퐵 푑푦 푁 = −푐퐷 퐴 + 푦 (푁 + 푁 ) 퐴 퐴퐵 푑푧 퐴 퐴 퐵 If 푁퐴 = −푁퐵 the bulk motion term will be cancelled from the above equation 4. If there is no fluid motion, no consumption or generation term, constant density and diffusivity and steady state conditions For mass concentration: 2 ∇ 휌퐴 = 0 7 For mass concentration: 2 ∇ 푐퐴 = 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. 푎푡 푡 = 0 푐퐴 = 푐퐴표 (푚표푙푎푟 푢푛푖푡푠) 푎푡 푡 = 0 휌퐴 = 휌퐴표 (푚푎푠푠 푢푛푖푡푠) where 푐퐴표 and 휌퐴표 are constant (defined values) 8 Note: Initial conditions are associated only with unsteady-state or pseudo-steady-state processes. 2. Boundary conditions: It means the concentration is specified (known) at a certain value of coordinate (x, y or z). Concentration is expressed in terms of different units, for example, molar concentration 푐퐴, mass concentration 휌퐴, gas mole fraction 푦퐴, liquid mole fraction 푥퐴, etc. Types of boundary conditions: (1) The concentration of the transferring species A at a boundary surface is specified. (The concentration of the species is known at the interface) Examples: I. For a liquid mixture in contact with a pure solid A, (liquid-solid interface) the concentration of species A in the liquid at the surface is the solubility limit of A in the liquid, 푐퐴푠 II.
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