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SURROUNDING

MASS BALANCE: WHY MASS FLOWS?. SURROUNDING. Phase a T a , P a , m i a. Phase b T b , P b , m i b. NO. NO. ISOLATED. SYSTEM. WORK. HEAT. Ideal fixed permeable membrane. a i = i th species activity. T a = T b P a = P b m i a = m i a. EQUILIBRIUM CONDITIONS

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SURROUNDING

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  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

  3. 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.

  4. 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)

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

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

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

  8. Remembering that:

  9. MASS BALANCE EQUATION FOR ith SPECIES where: Remembering the definition of the NABLA operator:

  10. As: … the summation of ith mass balance over all the r components yields to the well known continuity equation:

  11. 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

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

  13. Constant diffusion coefficient D Remembering the derivativedefinition

  14. MOMENTUM BALANCE Incoming sand Escaping sand Sand containing vessel Sliding vessel motion direction Inclinate plane Friction Gravity NO Yes

  15. DZ DX (X+DX, Y+ DY, Z+ DZ) (X, Y, Z) DY (X, Y+ DY, Z) (X+DX, Y+ DY, Z) Body forces: gravitational, electro-magnetic fields Surface forces: viscous drag and pressure

  16. Surface forces gravitational field Incoming and exiting mass

  17. ENERGY BALANCE Conduction Expansion Viscous heating

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