Understanding Mass Flows in Thermodynamic Systems

1. MASS BALANCE: WHY MASS FLOWS? SURROUNDING Phase a Ta, Pa, mia Phase b Tb, Pb, mib NO NO ISOLATED SYSTEM WORK HEAT Ideal fixed permeable membrane ai = ith species activity Ta = TbPa= Pb mia = mia EQUILIBRIUM CONDITIONS dU=dUa+dUb = 0

2. MASS BALANCE: EQ.CON. ALTERATION 1 Pa increase (>Pb) [Ta = Tb; mia = mib] Phase a T, Pa, mi Phase b T, Pb, mi MASS TRASPORT (CONVECTION) permeable membrane Mass transport represents a possible way the system has to get new equilibrium conditions once the original ones have been altered.

3. MASS FLOW (Kg/s) =V*S*Ci V = velocity (m/s) S = cross section (m2) Ci = “i” concentration (Kg/m3) S V Ci MASS FLUX (Kg/sm2) =V*Ci V = velocity (m/s) Ci = “i” concentration (Kg/m3) S = 1 m2 V Ci

4. mia>mib [Ta = Tb; Pa = Pb] 2 Phase a T, P, mia Phase b T, P, mib permeable membrane MASS TRASPORT (DIFFUSION) Mass transport represents a possible way the system has to get new equilibrium conditions once the original ones have been altered.

5. FICK LAW DX CiX+DX S CiX MASS FLOW (Kg/s) =-S*D*DCi/DX S = cross section (m2) D = diffusion coefficient (m2/s) Ci = “i” concentration (Kg/m3) DCi/DX = gradient concentration (Kg/m2s) DCi/DX = (CiX+DX-CiX)/DX MASS FLUX (Kg/sm2) =-D*DCi/DX

6. MASS BALANCE Z X Y DZ DX (X+DX, Y+ DY, Z+ DZ) G (X, Y, Z) DY (X, Y+ DY, Z) (X+DX, Y+ DY, Z)

7. MASS BALANCE: EXPRESSION DX DZ (X, Y, Z) G DY Ci = f(X, Y, Z, t)

8. DX DZ (X, Y, Z) G DY DIVIDING FOR DV

10. FLUXES EXPRESSIONS Ideal solution Diffusion Diffusion Diffusion Convection Convection Convection Bi = mobility of the diffusing components

11. Remembering that:

12. MASS BALANCE: CYLINDRICAL COORDINATES Jir+dr Jir+dr Jiz Jiz+dz Jir Jir Jir Jiz Jiz+dz Jir+dr r dr dz TWO DIMENSIONS r z

13. Ci = f(z, r, t) r dr dz Dividing by 2prdrdz

14. Remembering the derivativedefinition