Dept of Chemical and Biomolecular Engineering CN2125E Heat and Mass Transfer

1. Dept of Chemical and Biomolecular EngineeringCN2125E Heat and Mass Transfer Dr. Tong Yen Wah, E5-03-15, 6516-8467 chetyw@nus.edu.sg (Mass Transfer, Radiation)

2. Course Outline • Week 9-12: Mass Transfer • Week 9: Steady-state Diffusion (WWWR Ch 26) • Week 10: Unsteady-state Diffusion (WWWR Ch 27) • Week 11: Convective Mass Transfer (WWWR Ch 28) • Week 13: Radiation Heat Transfer (WWWR Ch 23, ID Ch 12-13)

3. Tutorial #9 WWWR # 26.17 & 26.27 To be discussed on March 25, 2014. By either volunteer or class list. HW/Tutorial Week #9WWWR Chapters 26, ID Chapter 14

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

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

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

7. 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

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

9. 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

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

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

12. Example 1

17. 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

18. 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

19. Example 2

27. 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

28. 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,

29. 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:

30. 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

31. 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

32. 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,

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

34. 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:

35. 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

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

37. 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,

38. Combining diffusion with reaction process, we get

39. Example 3

43. 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

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

45. 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,

46. 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

47. 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.

48. Example 4