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Dept of Chemical and Biomolecular Engineering CN2125 Heat and Mass Transfer

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Dept of Chemical and Biomolecular Engineering CN2125 Heat and Mass Transfer

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    1. Dept of Chemical and Biomolecular Engineering CN2125 Heat and Mass Transfer Dr. Praveen Linga, E5-02-23, 6601-1487 (Radiation, Mass Transfer)

    2. Tutorial/HW Week #9 WWWR Chapters 26-27 ID Chapter 14 Tutorial #9 WWWR #26.17, 26.27, 27.6, 27.22 & ID #14.33 To be discussed during the week of Mar 19 - 23, 2012. By either volunteer or class list. Homework #9 (self practice) WWWR # 26.22, 27.16 & ID #14.39

    3. Molecular Diffusion General differential equation One-dimensional mass transfer without reaction

    4. Unimolecular Diffusion Diffusivity of gas can be measured in an Arnold diffusion cell

    5. Assuming Steady state, no reaction, and diffusion in z-direction only We get And since B is a stagnant gas,

    6. Thus, for constant molar flux of A, when NB,z = 0, with boundary conditions: at z = z1, yA = yA1 at z = z2, yA = yA2 Integrating and solving for NA,z

    7. since the log-mean average of B is we get This is a steady-state diffusion of one gas through a second stagnant gas;

    8. For film theory, we assume laminar film of constant thickness d, then, z2 – z1 = d and But we know So, the film coefficient is then

    9. To determine concentration profile, if isothermal and isobaric, integrated twice, we get

    10. with boundary conditions: at z = z1, yA = yA1 at z = z2, yA = yA2 So, the concentration profile is:

    11. Example 1

    16. Pseudo-Steady-State Diffusion When there is a slow depletion of source or sink for mass transfer Consider the Arnold diffusion cell, when liquid is evaporated, the surface moves, at any instant, molar flux is

    17. Molar flux is also the amount of A leaving Under pseudo-steady-state conditions, which integrated from t=0 to t=t, z=zt0 to z=zt becomes

    18. Example 2

    26. Equimolar Counterdiffusion Flux of one gaseous component is equal to but in the opposite direction of the second gaseous component Again, for steady-state, no reaction, in the z-direction, the molar flux is

    27. In equimolar counterdiffusion, NA,z = -NB,z Integrated at z = z1, cA = cA1 and at z = z2, cA = cA2 to: Or in terms of partial pressure,

    28. The concentration profile is described by Integrated twice to With boundary conditions at z = z1, cA = cA1and at z = z2, cA = cA2 becomes a linear concentration profile:

    29. Systems with Reaction When there is diffusion of a species together with its disappearance/appearance through a chemical reaction Homogeneous reaction occurs throughout a phase uniformly Heterogeneous reaction occurs at the boundary or in a restricted region of a phase

    30. Diffusion with heterogeneous first order reaction with varying area: With both diffusion and reaction, the process can be diffusion controlled or reaction controlled. Example: burning of coal particles steady state, one-dimensional, heterogeneous

    31. 3C (s) + 2.5 O2 (g) ? 2 CO2 (g) + CO (g) Along diffusion path, RO2 = 0, then the general mass transfer equation reduces from to For oxygen,

    32. From the stoichiometry of the reaction, We simplify Fick’s equation in terms of oxygen only, which reduces to

    33. The boundary conditions are: at r = R, yO2 = 0 and at r = ?, yO2 = 0.21, Integrating the equation to: The oxygen transferred across the cross-sectional area is then:

    34. Using a pseudo-steady-state approach to calculate carbon mass-transfer output rate of carbon: accumulation rate of carbon: input rate of carbon = 0 Thus, the carbon balance is

    35. Rearranging and integrating from t = 0 to t = ?, R = Ri to R = Rf, we get For heterogeneous reactions, the reaction rate is

    36. If the reaction is only C (s) + O2 (g) ? CO2 (g) and if the reaction is not instantaneous, then for a first-order reaction, at the surface, then,

    37. Combining diffusion with reaction process, we get

    38. Example 3

    42. Diffusion with homogeneous first-order reaction: Example: a layer of absorbing liquid, with surface film of composition A and thickness ?, assume concentration of A is small in the film, and the reaction of A is

    43. Assuming one-direction, steady-state, the mass transfer equation reduces from to with the general solution

    44. With the boundary conditions: at z = 0, cA = cA0 and at z = ?, cA = 0, At the liquid surface, flux is calculated by differentiating the above and evaluating at z=0,

    45. Thus, Comparing to absorption without reaction, the second term is called the Hatta number. As reaction rate increases, the bottom term approaches 1.0, thus

    46. Comparing with we see that kc is proportional to DAB to ½ power. This is the Penetration Theory model, where a molecule will disappear by reaction after absorption of a short distance.

    47. Example 4

    55. 2- and 3-Dimensional Systems Most real systems are two- and three-dimensional Analytical solution to general differential equation with the boundary conditions Requires partial differential equations and complex variable theories.

    56. Unsteady-State Diffusion Transient diffusion, when concentration at a given point changes with time Partial differential equations, complex processes and solutions Solutions for simple geometries and boundary conditions

    57. Fick’s second law of diffusion 1-dimensional, no bulk contribution, no reaction Solution has 2 standard forms, by Laplace transforms or by separation of variables

    58. Transient diffusion in semi-infinite medium uniform initial concentration CAo constant surface concentration CAs Initial condition, t = 0, CA(z,0) = CAo for all z First boundary condition: at z = 0, cA(0,t) = CAs for t > 0 Second boundary condition: at z = ?, cA(?,t) = CAo for all t Using Laplace transform, making the boundary conditions homogeneous

    59. Thus, the P.D.E. becomes: with ?(z,0) = 0 ?(0,t) = cAs – cAo ?(?,t) = 0 Laplace transformation yields which becomes an O.D.E.

    60. Transformed boundary conditions: General analytical solution: With the boundary conditions, reduces to The inverse Laplace transform is then

    61. As dimensionless concentration change, With respect to initial concentration With respect to surface concentration The error function is generally defined by

    62. The error is approximated by If ? ? 0.5 If ? ? 1 For the diffusive flux into semi-infinite medium, differentiating with chain rule to the error function and finally,

    63. Transient diffusion in a finite medium, with negligible surface resistance Initial concentration cAo subjected to sudden change which brings the surface concentration cAs For example, diffusion of molecules through a solid slab of uniform thickness As diffusion is slow, the concentration profile satisfy the P.D.E.

    64. Initial and boundary conditions of cA = cAo at t = 0 for 0 ? z ? L cA = cAs at z = 0 for t > 0 cA = cAs at z = L for t > 0 Simplify by dimensionless concentration change Changing the P.D.E. to Y = Yo at t = 0 for 0 ? z ? L Y = 0 at z = 0 for t > 0 Y = 0 at z = L for t > 0

    65. Assuming a product solution, Y(z,t) = T(t) Z(z) The partial derivatives will be Substitute into P.D.E. divide by DAB, T, Z to

    66. Separating the variables to equal -?2, the general solutions are Thus, the product solution is: For n = 1, 2, 3…,

    67. The complete solution is: where L = sheet thickness and If the sheet has uniform initial concentration, for n = 1, 3, 5… And the flux at z and t is

    68. Example 1

    75. Example 2

    83. Concentration-Time charts

    86. Example 3

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