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1 80 CHAPTER 5 1965 Guide And'Data Book fig. 7 .... Mass Transfer from a fiat Plate used for this forced convection mass transfer process. At least,' at low mass transfer rates, where a,- is small,* the factor should be adequate. Experiments have proved the as sumption to be' justified as will be shown below*. Therefore, the'total mass transfer rate from the interface may-be written O.IO --------------T WETTED - WALL COLUMNS I------H----- 0X8 0.04 *9 0.01 0.008 0.002 KEY. : - , .. . M 00V O - GILLILAND * - GILLILAND 0.60-(WATER) 1.60 - - ~ GILLILAND 2.20' BARNET AND KOBE 0.60 (WATER) A - CHILTON AND COLStRN 0.60 (WATER) A - CHAMBERS 0.60 .(WATER) 1 1 III 1 : -- JOHNSTONE AND PIGFORIi , binary rect FICATION 1 1 1 1 1 u_____:____ - TA' 1 . .1 FT- 3g 1: -McADAMS, HEAT TRANSTER^ , TO GASES (Jm) ;1 IOOO| , , ,, '] 2 4 6 : IOOOO, . 2 REYNOLDS NUMBER 4 fig. 8 .... Vaporization and Absorption in a Wetted Wall Column m.JA - DJ*, / dp,\ R.Tp,, V dy )i (20) The concentration gradient at the interface (dp/dy)< must be.evaluated experimentally. Rather than work with, this gradient,, it is the common practice to define a naa transfer coefficient, as.in Equation 2. It followsthat . " (*o/ft.r)(P,, ---(). . By some mathematical rearrangement this becomes f DJ-, L Hs/L) X . I2,1' where Ngk is the dimensionless Sherwood number and L is some characteristic dimension of the mass transfer surface such as the length of a plate or diameter of a cylinder. Equa tion'21 gives a simple, but valuable, physicsd interpretation df ' the ' dimensionless mass transfer' coefficient, i.e., the Sherwood number.' It' is seen to be the dimensionless concern tration* gradient at the interface or mass' transfer boundary. The'gradieht is illustrated'in Fig. 6. '* ' `Mass transfer coefficients have been established experimen tally for a number of flow geometries, and are presented in the literature on this subject.1However, data on'mass transfer are relatively sparse as compared to the wealth.of data on heat transfer coefficients. Recognizing that.there isah analogy between these transfer processes, it should be possible to use these heat transfer data.in predicting'mass transfer, coeffi cients. Such similarity relations do exist, and their reliability has been'well established at low nuw transfer rates.. / Similarity Relations for Convective Mass Transfer Heat transfer from the wetted surface of Fig. 6 to the air* stream'ean be expressed in'an exactly analogous form to that of the imm transfer. Thus, the heat flux per unit area is - MA -*-J or, by mathematical rearrangement : The familiar dimensionless Nosselt number (ATy) of heat transfer is now recognized as the dimensionless temperature gradient'at the heat transfer boundary, (interface); Further more, the Sherwood and Nusselt numbers are recognized as analogous expressions for dimensionless mass transfer and heat transfer coefficients, respectively. - " - Again,' in the of momentum-transfer,'it:is the usual practice to define a friction factor / such that the momentum flux at the interface (pro per unit area is* - f (pV-L\ rdCT./F,)-], . , V, p, / "L d(y/L) X (23) when' -*.-' ' Vm -- mean,or free stream, velocity of gas, feet per.hour. ' Equation 23 states that one-half tire friction factor'times the Reynolds number (Nat--pV*%L/p) equals the dimensionless velocity gradient at the stationary surface or interface.' Equations'21, 22, and 23 show'the-similarities between mass, heat, and momentum transport in a turbulent boundary layer (where both eddy and molecular diffusion are opera tive), in tire same way'-that Equations 8, 9, and 10'demon strated' the similarities for a laminar boundary layer (where only moleculardiffusionia present).-Again assume a gaseous system in'which^ by chance, ),=a:=> (thatis, the molecular diffusion - constants for itmwh, energy, and 'momentum* are identical). From 'a purely theoretical consideration of the ! s Mass Transfer s>b< 8); physical process, of eddy diffusion, the .eddy diffusivities for these three'rate processes must.be identical, since all rely on the same particle mixing action for the transfer op-, eration. Experiments verify this conclusion..Therefore, under the previous assumption,.the dimensionless partial pressure,, temperature,.and .velocity.- profiles must be identical.;.This permits equating the left-hand side of Equations 21, 22, and 23 to obtain Vsk=Viir=(//2) (NaJor. ., -t. ': hpLpam Dft hL f VmL k "2 * *( * As pointed out in the discussion ;of molecular.diffusion, the postulate made here is.equivalent;to-the statement,that the' Prahdtl humber(AVr"<^7i) and Schmidt number (Vsi=p/p),),are unity. Using this statement Equation'24 can be rearranged to give '>* < ' ^"'JJLSL - . (24a) or hDp^ h . '' /' . ` VP. " pc,Vm m 2 The two right-hand terms of Equation 24 are'known' as:the Reynolds .analogy. The left-hand-term-is an extension; of Reynolds analogy to include mass transfer. Investigators of convective heat transfer have long recog nized that Reynolds analogy gave a reliable correlation be tween friction factors and heat' transfer "coefficients for common gases (where JV/r=1), and where.the temperature potential is moderate. Since, the time-of. Reynolds, not less * modified analogies have, been suggested to account for the effect of Prandti number. One of the simplest and most often.used among these is that suggested by Chilton arid Colburn1 who showed that the correlation, of heat, transfer date with friction data could be greatly imprqved by substi- fating (ATpr)1 for NPt in Equation 24a which leads to ' * 1' 3* " \pc,vj\ k ) ~ 2 * By anally, Chilton and Colburn suggested ,that it shouli possible to represent irrrr transfer data with good ac f^cy by a similar change in the Schmidt number parameter Equations 25 and 26 are known as the j-fadirr analogy,and are widely used today.for plotting and predicting heat and mass transfer data. * The power of the Chilton-Colburn /-factor analogy is repre sented in Figs. 7, 8,; 9, and 10.. Fig. 7 is a plot of experimental values of jo by a number of investigators of mass transfer from a flat plate with flow parallel to the plate surface. The solid line,'.which represents the data' to .near .perfection,' is actually //2 from Blasius solution.of laminar flow on a flat plate (left hand portion of the solid line) -and Goldstein's solution for a turbulent boundary layer (right.hand portion). The right hand ..portion of the solid line also represents McAdams* correlation of turbulent flow heat transfer coeffi cients for a fiat' plate. A wetted-wall column is a vertical tube in which a thin liquid film adheres to the tube surface.and exchanges maaa by evaporation or absorption with a gas flowing through the tube. It is a device of practical interest and provides one of the best tests of transport theories. Fig. 8 illustrates typical data on vaporization'in' wetted-wall ` columns, 'plotted- as '/*> vs Nsr The spread of the points with variation in /i/pd is due to the. fact that'Gilliland found an. exponent of 0.56, not 2/3, as representing the effect of the Schmidt number.*. This is shown by the fact that Gilliland's equation may be written jD - 0.023Nx4~*-i',ip/pD,)a i1 '' (27) - Similarly McAdams* equation for heat transfer in pipes may be expressed as **, ' -- >. *' in .- 0.023(Ar J:'. .(28) This is represented by the dashed curve, shown;onjFig.-8, which falls somewhat below the mass transfer datel The curve //2 representing friction in smooth' tubes is the upper 'solid curved* _'*: ''*`! : * Data for evaporation of liquids from single* cylinders' into gas streams flowing transversely to the cylinder axis are shown in Fig. 9. Although the dash-dot line on Fig. 9 represent the data well, it is.actuaUy taken from McAdams* as representa tive of a large collection .of date on heat transfer to single cylinders placed transverse to air streams. In order to com pare these date with friction,- it is necessary to distinguish between total drag and skin friction. The analogies are based on skin friction, hence the normal pressure drag must lib subtracted from the measured total drag. At-ATb;= 1000 the skin ' friction is 12.6 percent-of-the total dragand! at?N&