Equation of Continuity II

1. Equation of Continuity II

2. Summary of Equations of Change

3. Summary of Equations of Change

4. Summary of Equations of Change The momentum molecular flux, molecular stresses = pressure + viscous stresses *

5. Summary of Equations of Change The energy molecular flux is the partial molar enthalpy of species α

6. Summary of Equations of Change Recall: the combined energy flux vector e

7. Combined Energy Flux Vector We introduce something new to replace q: Combined Energy Flux Vector: Heat Rate from Molecular Motion Convective Energy Flux Work Rate from Molecular Motion

8. Combined Energy Flux Vector We introduce something new to replace q: Combined Energy Flux Vector: Recall the molecular stress tensor: When dotted with v: Substituting into e:

9. Summary of Equations of Change Recall: Substituting the equation for q into e

10. Summary of Equations of Change Recall: Substituting the equation for q into e per unit mass partial molar

11. Summary of Equations of Change Recall: Substituting the equation for q into e

12. Summary of Equations of Change

13. Summary of Equations of Change

14. Simultaneous Heat and Mass Transfer Example 1. Hot condensable vapor, A, diffusing through a stagnant film of non-condensable gas, B, to a cold surface at y=0, where A condenses Find:

15. Simultaneous Heat and Mass Transfer Assumptions: Steady-state Ideal gas behavior Total c is constant Uniform pressure Physical properties are constant, evaluated at mean T and x. Neglect radiative heat transfer

16. Simultaneous Heat and Mass Transfer Equations of Change: Continuity (A)

17. Simultaneous Heat and Mass Transfer Equations of Change: Energy * Both NAy and ey are constant throughout the film

18. Simultaneous Heat and Mass Transfer To determine the mole fraction profile: Recall: The molar flux for diffusion of A through stagnant B

19. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Since B is stagnant,

20. Simultaneous Heat and Mass Transfer To determine the mole fraction profile: Recall: The molar flux for diffusion of A through stagnant B Recall: Integration of the above equation

21. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Let C1 = -ln K1 and C2 = -lnK2, B.C. at z = z1, xA= xA1 at z = z2, xA= xA2

22. Simultaneous Heat and Mass Transfer To determine the mole fraction profile: Recall: The molar flux for diffusion of A through stagnant B Using the appropriate B.C.s at y= 0, xA= xA0 at y = δ,xA= xAδ

23. Simultaneous Heat and Mass Transfer To determine the mole fraction profile: Evaluating NAy from the equations above Note that:

24. Simultaneous Heat and Mass Transfer BUT… ]

25. Simultaneous Heat and Mass Transfer ]

26. Simultaneous Heat and Mass Transfer Rearranging and combining

27. Simultaneous Heat and Mass Transfer @ y = y, xA = xA

28. Simultaneous Heat and Mass Transfer @ y = y, xA = xA @ y = δ, xA = xAδ Taking the ratios of the two equations

29. Simultaneous Heat and Mass Transfer To determine the temperature profile: Note: where the enthalpy of mixing is often neglected for gases at low to moderate pressures

30. Simultaneous Heat and Mass Transfer To determine the temperature profile: The general solution is where

31. Simultaneous Heat and Mass Transfer where At y = 0, T = T0 At y = δ, T = Tδ Subtracting the two equations

32. Simultaneous Heat and Mass Transfer Since

33. Simultaneous Heat and Mass Transfer where

34. Simultaneous Heat and Mass Transfer If we did not consider mass transfer

35. Simultaneous Heat and Mass Transfer With mass transfer

36. Simultaneous Heat and Mass Transfer Comparison of the energy flux with & without the presence of mass transfer: Rate of heat transfer is directly affected by simultaneous mass transfer BUT mass flux is not directly affected by simultaneous heat transfer