1. Gas Absorbtion in a Falling Liquid Film: Gas absorbers are im- portant technologies for flue-gas...

 

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1. Gas Absorbtion in a Falling Liquid Film: Gas absorbers are im- portant technologies for flue-gas cleanup, and have potential for direct air capture of CO2. Both applications can mitigate climate change. We discuss a primitive model for this operation. A gas A sparingly soluble in a non-volatile liquid B contacts a falling film of the pure liquid in the system depicted in the attached figure. For x > 0 the solute A dissolves into the liquid at the gas/liquid interface, located at y = 0, and diffuses into the film towards the impermeable wall located at y = H. We want to know the local rate of absorption of A into the film as expressed by a local Sherwood number Shloc Shloc kloc H DAB where kloc is a local mass transfer coefficient, defined here by NAyly=0 kloc(CA(x, y =0) - CA,ave (x)) [=] where c (y = 0) means the liquid phase molar density of A at the gas/liquid interface (usually taken to be the solubility of A in B CAO; note A0 = 0 < < 1 for a sparingly soluble gas), while CA,ave means the average molar density of A in the film at the downstream location x CA,ave (x) Here v (y) is the steady laminar downstream velocity profile in the film, determined by fluid mechanics as Gas CA = Co X = Vx We want to determine Shloc for a falling film absorber from the FCMT model discussed in class in the case of large Pem= Umax H >> 1 far DAB enough downstream such that x = y SCA(x, y)v. (y)dy vz(y)dy H Liquid VT vx (y) = Umax [-6] 1 X Pem H Wall 2 ~ mole LT > 1. Assert a DAB Hn(y) with Go ~ where the An and Hn(y) are solutions to the eigenvalue (Sturm- Liouville) problem 1 ~1- H (0) e -XH(y) for = 2 1 12 (0) - H + X [1 y] Hn = 0 Hn (0) = 0; H(1) = 0 (d) Far downstream where x = O(1) the first term in the Pe H expansion for dominates giving the very good approximation X Pem H Show that for this region downstream, the Sherwood number is given by 2. Shloc Hint: Section 10.5 of Deen is helpful. 1. Gas Absorbtion in a Falling Liquid Film: Gas absorbers are im- portant technologies for flue-gas cleanup, and have potential for direct air capture of CO2. Both applications can mitigate climate change. We discuss a primitive model for this operation. A gas A sparingly soluble in a non-volatile liquid B contacts a falling film of the pure liquid in the system depicted in the attached figure. For x > 0 the solute A dissolves into the liquid at the gas/liquid interface, located at y = 0, and diffuses into the film towards the impermeable wall located at y = H. We want to know the local rate of absorption of A into the film as expressed by a local Sherwood number Shloc Shloc kloc H DAB where kloc is a local mass transfer coefficient, defined here by NAyly=0 kloc(CA(x, y =0) - CA,ave (x)) [=] where c (y = 0) means the liquid phase molar density of A at the gas/liquid interface (usually taken to be the solubility of A in B CAO; note A0 = 0 < < 1 for a sparingly soluble gas), while CA,ave means the average molar density of A in the film at the downstream location x CA,ave (x) Here v (y) is the steady laminar downstream velocity profile in the film, determined by fluid mechanics as Gas CA = Co X = Vx We want to determine Shloc for a falling film absorber from the FCMT model discussed in class in the case of large Pem= Umax H >> 1 far DAB enough downstream such that x = y SCA(x, y)v. (y)dy vz(y)dy H Liquid VT vx (y) = Umax [-6] 1 X Pem H Wall 2 ~ mole LT > 1. Assert a DAB Hn(y) with Go ~ where the An and Hn(y) are solutions to the eigenvalue (Sturm- Liouville) problem 1 ~1- H (0) e -XH(y) for = 2 1 12 (0) - H + X [1 y] Hn = 0 Hn (0) = 0; H(1) = 0 (d) Far downstream where x = O(1) the first term in the Pe H expansion for dominates giving the very good approximation X Pem H Show that for this region downstream, the Sherwood number is given by 2. Shloc Hint: Section 10.5 of Deen is helpful.

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Part 1 To determine the local Sherwood number Shloc for the gas absorption in a falling liquid film we can use the PCMT Penetration and Creeping Motio... View the full answer