MODULE 3: MASS TRANSFER COEFFICIENTS

Transcription

1 MOULE 3: MASS TRASFER COEFFICIETS LECTURE O Penetration theory Most of the industrial proesses of mass transfer is unsteady state proess. In suh ases, the ontat time between phases is too short to ahieve a stationary state. This non stationary phenomenon is not generally taen into aount by the film model. In the absorption of gases from bubbles or absorption by wetted-wall olumns, the mass transfer surfae is formed instantaneously and transient diffusion of the material taes plae. Figure 3.4 demonstrates the shemati of penetration model. Basi assumptions of the penetration theory are as follows: 1) Unsteady state mass transfer ours to a liquid element so long it is in ontat with the bubbles or other phase ) Equilibrium eists at gas-liquid interfae 3) Eah of liquid elements stays in ontat with the gas for same period of time Liquid elements are sliding down Rising gas bubble Figure 3.4: Shemati of penetration model. Joint initiative of IITs and IIS Funded by MHR Page 1 of 5

2 Under these irumstanes, the onvetive terms in the diffusion an be negleted and the unsteady state mass transfer of gas (penetration) to the liquid element an be written as: t Z (3.60) The boundary onditions are: t = 0, Z > 0 : = Ab and t > 0, Z = 0 : = Ai. The term Ab is the onentration of solute at infinite distane from the surfae and Ai is the onentration of solute at the surfae. The solution of the partial differential equation for the above boundary onditions is given by the following equation: Ai Ai Ab erf Z t Where erf() is the error funtion defined by (3.61) erf ( ) 0 ep( Z ) dz (3.6) If the proess of mass transfer is a unidiretional diffusion and the surfae onentration is very low ( Ab ~0), the mass flu of omponent A, A [g m s 1 ], an be estimated by the following equation: A 1 Ab Z Z0 Z Z0 (3.63) Substituting Equation (3.61) into Equation (3.63), the rate of mass transfer at time t is given by the following equation: t) ( Ai ) (3.64) t A( Ab Then the mass transfer oeffiient is given by L( t) (3.65) t The average mass transfer oeffiient during a time interval t is then obtained by integrating Equation (3.61) as Joint initiative of IITs and IIS Funded by MHR Page of 5

3 t 1 L, av ( t) dt t 0 t (3.66) So from the above equation, the mass transfer oeffiient is proportional to the square root of the diffusivity. This was first proposed by R. Higbie in 1935 and the theory is alled Higbie s penetration theory Surfae Renewal Theory For the mass transfer in liquid phase, anwert (1951) modified the Higbie s penetration theory. He stated that a portion of the mass transfer surfae is replaed with a new surfae by the motion of eddies near the surfae and proposed the following assumptions: 1) The liquid elements at the interfae are being randomly swapped by fresh elements from bul ) At any moment, eah of the liquid elements at the surfae has the same probability of being substituted by fresh element 3) Unsteady state mass transfer taes plae to an element during its stay at the interfae. Hene, average molar flu, A,av ( C C ) s A, av Ai Ab (3.67) Comparing Equation (3.67) with Equation (3.8) we get s L, av (3.68) where s is fration of the surfae renewed in unit time, i.e., the rate of surfae renewal [s -1 ] Boundary Layer Theory Boundary layer theory taes into aount the hydrodynamis/flow field that haraterizes a system and gives a realisti piture of the way mass transfer at a phase boundary. A shemati of onentration boundary layer is shown in Figure 3.5. Joint initiative of IITs and IIS Funded by MHR Page 3 of 5

4 y U C Ab () m() Flat plate u U C A =C Ai -0.99(C Ai -C Ab ) Figure 3.5: Shemati of onentration boundary layer. When ()u=u and when m () u=0.99u distane over whih solute onentration drops by 99% of (C Ai -C Ab ). Sh. L, (Re) ( S) (3.69) where, is the distane of a point from the leading edge of the plate; L, is the loal mass transfer oeffiient. Sh l. L, av 0.664(Re) ( S) (3.70) where, l is the length of the plate. Eample problem 3.: Zaausas (Adv. Heat Transfer, 8, 93, 197) proposed the following orrelation for the heat-transfer oeffiient in a staggered tube ban arrangement: u 0.453Re Pr, where 10 Re( ugd / ) 10 6 and 0.7 Pr 500 Estimate the mass-transfer oeffiient by using the mass and heat transfer analogy if to be epeted for evaporation of n-propyl alohol into arbon dioide for the same geometrial arrangement of tube diameter (d) of 38 mm when the arbon dioide flows at a maimum veloity (u g ) of 10 m/s at 300 K and 1 atm. Properties of dilute mitures of propyl alohol in arbon dioide at 300 K and 1 atm are: Moleular weight (M) = 44 gm/mole, density (ρ) = 1.8 g/m 3, Visosity (μ) = g/m.s, diffusivity ( ) = m /s and universal gas Joint initiative of IITs and IIS Funded by MHR Page 4 of 5

5 onstant (R) = J/mole.K. Sherwood number is defined based on diameter of the tube as RTd / G Solution 3.: Re = 1.8*10*0.038/ = S = ( )/(1.8* ) = 1.08 Analogy to heat transfer Sh 0.453Re S = So, RTd / = G Therefore G = (07.68* )/(RTd) = mole/m.s.pa Joint initiative of IITs and IIS Funded by MHR Page 5 of 5