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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


                          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 ­

          So (\""n'~
                                       e2):- 0- (?fJ ~ If"...,,, G- (Tj]
                              ,             t..

                        Vi.       =                     	
     C-\va :           lG =- ei!.... :-
                                                   l... if:
                                                   "l.-       J&<   ~

                        b~            e:;lt2­
                       '\f"t -0-ft -= ])
                                              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
           , 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
\' ~                          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 
           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
                                                                                                                                                                                        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

            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
                                                  v oCA =   !?)AB   oZcA                                                  j                                               =1-          erf ~4!?)AaZ/Vmax
                                                   ~ 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
                                                                                        e.,- ~

                                                             ~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...
                                                                                                                                                                                                                                                    '::     I
   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
                   N Aiz)llI'=o = -9)AB            a;:             = CAO
                                                                                         1TZ          (17.5-16)

                                                                                                                                                                     -        CAO
                                                                                                                                                                                                                     '"' '"'

                                                                                                                       5l\4~           ..:~
The total moles of A transferred per unit time from the gas to the liquid                                                     ~f              »rrA                   -
film is

                                    IW    f:    NAlI'llI''''0   dz dy

                                                                                                                                                From this development it is seen that the mass transfer rate is directly
                                                                                                                                              proportional to the square root of the diffusivity and inversely proportional
                                            J9)    ABvrnax1L - H
                                                                               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 .•

                                                      7T           0
                                                                                                                  Jt- F-{",,4 f~f~w .                                                                                          .....
                                                           ABVrnax:                                   (17.5-17)   .' 5,~f Nz;vt',

       Integration of Concentration Profile over (Infinite) Film Thickness at z                            =L
                                                                                                                                          .,; A,)q ""
                                                                                                                                                             -                      dv.
                                                                                                                                                                                          = _ (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
                                                 (6 cAI.=L         dX)                                                                         The constant of integration is evaluated by using the boundary condition


                       WVrnax     Jo cAI.=L dx
                                                                                                                                               B.C. 2:
                                                                                                                                                                x                   at     =    a.
                                                                                                                                                                                                           V.   =0             v                    (2.2-15)

                   = WVrnaxCAO      1'"
                                          erfc ..I49) x L/Vrnax dx
                                                                                                                                               Substitution of this boundary condition into Eq. 2.2-14 shows that C2 ==
                                                                                                                                               (pg cos fJI2p)a 2• Therefore, the velocity distribution is

                    = WVmaxCAO'
                                                      (f'"             <t

                                                           v 4.!ii1AoL/vmax
                                                                                             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
 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

                                                                                                                                                                                      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

                                                v. dx                                                                       laminar flow without rippling
                                                                                                                            laminar flow with rippling
                                                                                                                                                                     4 to 25
                                                                                                                                                                               < 4 to 25
                                                                                                                                                                               < Re < lOOO to 2000
                                 = pgt5 cos
                                                                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

                                                    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.
           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).

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