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" '1.(!e )tJ Y'5fy.·b c...~'-- " ~f . , \ : • ."1 fairly widerange - e ratio N* is indeed very near be used to describe absorptlo ere! b .. g r - l nity, and... . A-S may ... O. Similar ~;.;':, lification t11J.(~) J~IIi<W~_~ is possible Cor more realistic mass- ,' / • P$ ,. .1 'T C>' tA ~1.- ) /4 J.!Z.5 DIFFUSION INTO A FALLING LIQUID FILM: "R5ittED-CONVECTION MASS TRANSFER B. c. ~;: b Vi = 0 In this section we present an illustration of forced-convection mass transfer in which viscous flow and diffusion occur under such conditions that the ~-;:o 'A1(" - iGC 0 So (\""n'~ e2):- 0- (?fJ ~ If"...,,, G- (Tj] . , t.. Vi. = C-\va : lG =- ei!.... :- '3j'- l... if: "l.- J&< ~ b~ e:;lt2 '\f"t -0-ft -= ]) ~t- 'b:'-.A 148 ~x:a.. v:: [,_ (L1~) ~=- 1>ItS d~ bJ J e>t: )..u.yc 0)(2. FI,. 17.5-1. Absorption Into a falling film. velocity field can be considered virtually unaffected by the diffusion. Specifi cally. we consider the absorption of gas A by a laminar falling 1.ilrn of liquid B. The material A is only slightly soluble in B, so that the viscosjty of the liquids is not changed appreciably. We shall make the further restriction L. that the diffusion take place so slowly in the liquid film that A will not "penetrate" very far into B-that the· pen~tration distance be small in comparison with the film thickness. The system is sketched in Fig. 17.5-1. . Let us now set up the differential equations describing the process. First of all we have to solve the momentum transfer problem to obtain the velocity profile v.(x) for the film; this has already been worked out in §2.2 in the < 538 ) Concentration Distributions In Solids and In Laminar Flow "'''''', Diffusion Into a Fallin, Liquid Film ~J ) 539 , absence of ",..55 transfer at the fluid surface, and we know that the result is This partial differential equation is to be solved with the following boundary conditions: \' ~ I': V~(X) == vmax[l - (~)] (17.5-1) B. C. 1: at z == 0, CA =0 (17.5-8) ( provided that "end effects" are not considered. B. C. 2: at x = 0, cA = c AO (17.5-9) Next we have to establish a mass balance on component A. We note that C.,{ will be changing both with x and with z. Hence as the element of volume B. C. 3: at x = 15, oCA == 0 (17.5-10) over which we set up the mass balance we select the volume formed by the ax intersection of a slab of thickness az with a slab of thickness ax. Then the when the fluid film begins as pure B, and the interface concentration in the mass balance on A is simply liquid is taken to be the solubility _of A in B, here designated as CAo. This problem has been solved by Pigford,l but we do not give that solution here. NAzlzW Llx - NA.I.+t..Wax + N~o:IIIIW az - N&o:lo:+t.o:Waz == 0 (17.5-2) Instead, we solve to obtain only a limiting expression valid for "short in which W is the width of the film. By dividing by wax az and performing contact times"-that is, small values of L/Vmax. the usual limiting process as the volume element becomes infinitesimally If, as is indicated in Fig. 17.5-1, the substance A has penetrated only a we get short distance into the film, then the A species for the most part has the aNAl + oNAo: = 0 impression that the film is moving throughout with a velocity equal to Vmax. (17.5-3) Furthermore, if A does not penetrate very far, it does not "feel" the presence oz ax of the solid wall at x = <5. Hence, if the film were of infinite thickness moving Into this equation we now have to insert the expressions for NA • and NAIYJ with the velocity Vmax. the diffusing material would not know the difference. by making appropriate simplifications of Eq. 17.0-1. For the molar flux in This physical argument suggests then that we replace Eq. 17.5-7 and its the z-direction, we write, assuming constant C . boundary conditions by OC A A oZc NAt = -f!#A1J]!.c.A-+'XA(NA~ + N B .) ~ cAv.(x) (17.5-4) Vmax a; = !?)AB oxt (17.5-11) _-, oz i. B. C. 1: at z== 0, ,CA =0 (17.5-12) That is, A moves in the z-direction primarily because of the flow of the film, the diffusive contribution being negligible. The molar flux in the x-direction B. C. 2: at x == 0, C.t = CAO (17,5;-13) will be oC ...-t'~/'~ oC B. C. 3: at x = 00, cA == 0 (17.5-14) = -f!#AB-;;- + xA (N4-a>-.+·').VBIII)*:S -f!#AB-;;- (17.5-5) A A oX .,- oX The solution to Eq. 17.5-11 with these boundary conditions is2 That is, in the x-direction A is transported primarily by diffusion, there ~ IYJ r~~ being almost no convective transport because of the very slight solubility - C& 1 2 IV'4lJu',vmn e-"dt: == - - 10 of A in B. Substitution of these expressions for NAill and NA• into Eq. 17.5-3 CAO V; 0 gives x v oCA = !?)AB oZcA j =1- erf ~4!?)AaZ/Vmax (17.5-6) ~ 01, ox2 or which is the differential equation describing cA(x, z). C,A. = erfc x (17.5-15) The forced-convection mass transfer is then described by Eqs. 17.5-1 CAO .j4!?)AaZlVmax and 17.5~6. When these two equations are combined, we get finally 1 R. L. Pigford, doctoral dissertation, University of Illinois (1941). S'u Vma.x X)2JOCA _ (d a; - m. oZcA ;:;gAB aX" (17.5-7) I The solution is worked out in detail in Example 4.1-1 by the "method of combination of variables." i * , 'L ---J f e.,- ~ t rJs ' '" " ~A (f "f1\' j o ~: _ j.C{'-'I'J V~. , 5'10 )' '::oncentration Distributions In Solids and in Laminar Flow y'l Diffusion Into a Failing Liquid Film ')} 541 Here erf y is the "error function" defined in §4.1 and erfc y = 1 - erfy is At this juncture we change the' order of integration 3 in the double integral, ,the "complementary error function"; both are standard tabulated functions. which procedure enables us to evaluate the double integral analytically and Once the concentration profiles are known, the total mass-transfer rate may to obtain Eq. 17.5-17 thus: be found by either of the following two methods: 1, -t.,f.H~\' Integration of Mass Flux over Length of Film P'!J..v, .. v I.f'I LCftJ \ !!,y A = WLCAoj49)-:;max. 2 L'" e-et(fdU) d~ J... ,ox ':: I j..'<tt.t The local mass flux at the surface x = 0 at a position z down the plate is , J;l \l-,j , ~ \ ~ ~ ,',.- l : '+' ' ''-~',:-'t- - WL j49)ABv 21'"e J9) dl:! N Aiz)llI'=o = -9)AB a;: = CAO ABvmax 1TZ (17.5-16) 1 "1\ t;r - CAO 7TL rnax • 0 -ill:! '"' '"' 5l\4~ ..:~ J49) The total moles of A transferred per unit time from the gas to the liquid ~f »rrA - - WL CAO ABVmax (17.5-19) film is IW f: NAlI'llI''''0 dz dy 7TL From this development it is seen that the mass transfer rate is directly proportional to the square root of the diffusivity and inversely proportional - - IX! rrCAO J9) ABvrnax1L - H Z dZ to the square root of the "exposure time," t exp == Llvm!).x. This approach for studying gas absorption was apparently first proposed by Higbie .• j49) 7T 0 Jt- F-{",,4 f~f~w . ..... - - WL CAO ABVrnax: (17.5-17) .' 5,~f Nz;vt', 7TL Integration of Concentration Profile over (Infinite) Film Thickness at z =L ~:r' .,; A,)q "" "u - dv. dx = _ (pgJ;M1J\ x pJ (2.2-13) According to an over-all mass balance on the film, the total moles of A i This equation is easily integrated to give transferred per unit time across the gas-liquid interface must be the same as the total molar rate of flow of A across the plane z = L, which may be \- calculated by multiplying the volume rate of flow across the plane z = L V. - - -(pg~x2 + C2 ----:;;- r (2.2-14) by the average concentration at that plane: = lim (W bVrnax) (! o aJ (6 cAI.=L dX) The constant of integration is evaluated by using the boundary condition = 6-", WVrnax Jo cAI.=L dx ('" B.C. 2: ----------------.. x at = a. -.-~~,~.," V. =0 v (2.2-15) = WVrnaxCAO 1'" 0 erfc ..I49) x L/Vrnax dx --;==::=~~_ AB Substitution of this boundary condition into Eq. 2.2-14 shows that C2 == (pg cos fJI2p)a 2• Therefore, the velocity distribution is = WVmaxCAO' 2 (f'" <t v 4.!ii1AoL/vmax e- fl d~) dx J v. = pg~;;~1 - (~rJ (2.2-16) = WVmaxCAO j 49) AB. 2 L ,_. Vrnax '" 7T 1"'(f'"e-f') du 0 IS d~ (17.5-18) Hence the velocity profile is parabolic. (See Fig. 2.2-2.) Once the velocity profile has been found, a number of quantities may be calculated: In the last line the new variable u = x/V4!?)ABL/vrnax has been introduced. (i) The maximum velocity vz,max is clearly the velocity at x = 0; that is pg{)2~ VZ,max == ~ (2.2-17) Velocity Distributions In Laminar FloW' ) -10 Flow of a failing Film \/,. (ii) The average velocity (v~) over a cross section of the film is obtained by the kinematic viscosity 11 == t-tl p decreases, the nature of the flow the following calculation: 7 changes; in this gradual change three more or less distinct stable types of fl, wp can be observed: (a) laminar flow with straight streamlines, (b) laminar n, 10 z dx dy V with rippling. and (c) turbulent flow. Quantitative information concenll v >= "0 0 Z fW f6 the type of flow that can be expected under a given set of physical condittc Jo Jo dx dy seems to be only fragmentary. For vertical walls, the following informali may be given :1,2 =-11 t5 6 0 v. dx laminar flow without rippling laminar flow with rippling Re 4 to 25 < 4 to 25 < Re < lOOO to 2000 = pgt5 cos 2 P f[l - (~rJ d(~) turbulent flow Re > lOOO to 2000 in which Re = 4t5(v.)plt-t = 4r/t-t is the Reynolds number for this == pg'd2 cos f3 (2.2-18) Why this dimensionless group is used as a criterion for flow patterns discussed further in subsequent chapters. ,j 3t-t (iii) The Q.o/ume rate ot.,fJ.ow. Q is obtained from the average velocity or by \ Exam'ple 2.2-1. Calculation of Film Velocity integration of the velocity distribution: \. fW fd , = pgW ~3 cos f3 An oil has a kinel1i'll.tic viscosity of 2 x 10- 4 m2 sec-1 and a density of 0.8 A Q Jo Jo v.. dx dy = 3t-t , (2.2-19) kg m- 3 • What should he mass rate of flow of this film down a vertical wall be order to have a film thl kness of 2.5 mm? (iv) The film thickness d may be given in terms of the average velocity. Solution. According 0 Eq. 2.2-20, the mass rate of flow per unit width of \\ the volume rate of fiow, or the mass rate of flow per unit width of wall is (all numerical values gi en in mks units) (~(V,)\~~ J"--3;(~J=-:T~F7 31 {)3 (2.5 x 10-3)3(0.8 x 1<fl)(9.80) 3t-t r (2.2-20) r pgcosp ~~::.; p2gcosp::.; pgW (v) The z-compo~;~tof the fficeiFiTiiie- flEjd..on tbe...!J!.iface is given by integrating the momentum flux over the fluid-solid interface: -. This is the desired result if and o,ly if the flow is indeed laminar. To ascertain l nature of the flow, we calculate a ~eynolds number based on the mass rate of fLfW just found: . F. == Jo Jo T".I,,=6 dy dz Re = 4r = 4r _ 4(0.204) p, pO' - (0. x 103)(2 x 10-4) 5.1 (dimensionk 51 L W == dv. o 0 -t-t dx I ""'0 dy dz This Reynolds number is below the ob~rved upper limit for laminar flow stal above, and therefore the calculated valu~ of r is valid. == (LW)(-t-t)( pgt5 ~os P) Example 2.2-2. Falling Film \yith Variable Viscosity = pg bLW cos f3 (2.2-21) Rework the falling film problem for the situa..tion in which the viscosity depcn upon position in the following manner: This is clearly just the z-component of the weight of the entire fluid in the film. (2.2-2 The foregoing analytical results are ~nly when the film is fa~g in 1~ with straight streamlines. For the slow flow of thm viscous 1 T. K. Sherwood and R. L. Pigford, Absorption and Ex films, these conditions are satisfied. It has been found experimentally that as (1952), p. 265. the film velocity (v z ) increases, as the thickness ofthe film t5 increases, and as • S. S. Grimley, Trans. Inst. Chern. Engrs. (London), 23, ~8-235 (1948).