As the result of the theoretical consideration concerning with the analogy among fluid friction, heat transfer and mass transfer in common gaseous systems (σ, φ=0.5～2.5), the following relation between the coefficients of heat and mass transfer was obtained : h/κ' = (φ/σ)[1/2]・Cp and this was ascertained experimentally, using the wet and dry bulb thermometer

MIZUSHINA, T.

NAKAJIMA, M.

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Simultaneous Heat and MassTransfer : The Relation betweenHeat transfer and Mass transferCoefficientsMIZUSHINA, T.; NAKAJIMA, M.MIZUSHINA, T. ...[et al]. Simultaneous Heat and Mass Transfer : The Relation betweenHeat transfer and Mass transfer Coefficients. Memoirs of the Faculty of Engineering, KyotoUniversity 1951, 13(1): 40-481951-01-31http://hdl.handle.net/2433/280226Simultaneous Heat and Mass Transfer The Relation between Heat transfer and .Mass transfer Coefficients Dept. of Chemical Engineering K¥oto Univ. Japan T. MizUSHINA and M. NAKAJIMA (Received Dec., 1950) General Simultaneous heat and mass transfer is an important problem in studying the unit operations in chemical engineering such as drying, humidification, de-humidification and so on. In order to solve the problem of heat and mass transfer, the analogy among fluid friction, heat transfer and mass transfer is very useful. With the development of the theory of fluid flow, this has been improved too. Mizushina has slso offered a new one 5). Using this the authors found the relation between coefficients of heat and mass transfer in common gaseous systems (a, p=0.5~2.5) as follows and the equation was found to fit expe_rimental data by means of a wet and dry-bulb hygrometer using water and several kind of organic solvents. Theoretical consideration The analogy between heat transfer and fluid friction in turbulent flow was originally offered by Reynolds, and developed by Taylor, Prandtl, von Karman, Chilton and Colburn, assuming that fluid friction is produced by exchange of momentum, heat transfer by exchange of heat, and both are similar phenomena. Similarly this analogy may be applied to the relation between fluid friction and mass transfer. Consequently, fluid friction, heat transfer and mass transfer can be considered to be quite analogous as Sherwood l) suggested. Reynolds' analogy on heat transfer for fluid flow in a round pipe is shown as follows: h (1) Simultaneous Heat and Mass Transfer 41 Taylor and Prandtl 2) tried to expand this theory introducing : h = l+(a-1)6.2J// (2) where Karman 3) gave the following equation : h (3) Chilton and Colburn 4 ) reduced the relation as follows : 1 2 /cppllo h = a 213 (4) Similarly concerning mass transfer where (5) The power index 2/3 in Eq. (5) is the mean value in the range of a, </>= 0.5~100. )Vhile within the limit of a, </>=0.5~2.5 as in the case of gas humidi-fier and dehumidifier, Eq. (5) deviates slightly from experimental values. Now, Mizushina 5) reexamined the above-mentioned analogy of Tayler and Prandtl and proposed the following equation : (6) Similarly ,;,' 1 I, 2 Pllo l+C,,·-1) 6.3J/T 'fl <p 0.2 l+(cf>-l)6.3J/TN, 0-2 1 +(a-1) 6.3J/ I J a 0•2 Cp (7) 42 2.5 2.0 .. .. ~ 1.5 ~ I<) '° -IN ,.---. 1.0 ?::, I ~ + 0.5 T. MIZUSHINA and M. NAKAJIMA Fig. 1 shows relations be• tween [1 +(11-l) 6.3J///11°-2]. the denominator of Eq. (6), and Prandtl number 11 within the limits of 11=0.5~2.5, taking 6.3J/T =0.5. This line can be substituted by a straight line which represents 11 112• Then the denominator of Eq. (6) becomes 11 112 • The same thing 0.4 "----'------'------'---L-.....1---l may be said about <j, in case of 0.4 0.5 1.0 1.5 2.0 2.5 30 mass transfer. Then in the er= Cp)l/A range of 11, <j,=0.5~2.5 Eq. (7) Fig. 1. Comparison of denominator of equ. (6) with ,,½ becomes ~ = (: )l • Cp ( 8 ) This relation is for the stream in a round pipe. But Eq. (8) may be applic• able to the case of fluid flow at right angle to a simple rod as in the case of wet-bulb thermometer. Wet and dry bulb hygrometer If a drop of liquid is left in the unsaturated gas, it rapidly assumes the wet-bulb temperature tw, then the heat required for evaporation will be supplied from gas since the temperature of gas t is higher than that of the drop tw i. e. where Q = hA(t-tw) A : Surface area of the drop. • Under adiabatic conditions, taking the saturated humidity at the temperature of tw as H '° and the humidity of gas at the temperature of t as H, (10) where r,o: Letent heat of liquid at tw, (~ t:, = rw(I[:_ ~ H) (11) Now it is possible to calculate the value of (h/i;.')oba when t and tw are measured. But Eq. (10) can not be applied actually with perfect accuracy be-cause some factors influence the reading of wet-bulb temperature. * (Note) Many investigators have calculated (h/,r,') by Eq. (11). The authors will use, therefore, the notation (h/ i;.')oba for the value of (h/ K-1 ) given by Eq. (11 ). Simultaneous Heat and Mass Transfer 43 One of these factors is the radiation. In the usual case the temperature of sourroundings is near the dry-bulb temperature t higher than that. of the wet-bulb. So repr~senting the coefficient of heat transfer by radiation as hr, the heat radiated~'to the wet-bulb is to be hrA(t-tw). , Secondly, we must consider the heat input by conduction through the wet-bulb thermometer stem. Take h0 as the coefficient of heat transfer by conduc-tion, then the heat conducted is to be h 0 A(t-t,,,). Usually a capillary phenomenon is used in order to wet the surface of wet-bulb. Arnold asserts himself in the discussion of Sherwood's paper 6) that capil-larity is not operative unless a partial drying out of the wick occurs. In th~se cases we must consider that the surface area of evaporation of the wet-bulb is a little smaller than that of heat transfer. Let us take the ratio of both area as a, then a~l according to the conditions of wick and the. method of liquid supply. In the experiments of investigators referred here and that of authors, the pre-cooling of supplied liquid has been devised so as to limit the sensible heat input with it to be only 1 ~2¾ ot total quantity of heat transfer. Therefore, this influence can be neglected. Considering what is mentioned above, the heat balance on the wet-bulb sur-face becomes (h+hr+h0 )(t-t,,,) = rw,c'a(H.,,-H) From Eqs. (11) and (12) (~)· . - rw(Hw-H) h+hr+ho h h+hr+ho "' ob, - t-t.,, = ,c'a = "f? ha. (12) Since the velocity of air flow is considered constant during one experiment, h is constant, and hr and he are also almost constant as to one apparatus. Fur-thermore, a may be considered as a constant irrespective of the kind of liquid in one experiment. Therefore h+hr+ho -n-;;:-is almost constant in one experiment. Then (~) = rw(Hw-H) = 04 = 0(i..)~cp " . ob, t - tw " . d I (13) (14) As 0>1, the value of (h/i.')o&, obtained by wet-and dry-bulb thermometer is a little larger than the theoretical value of hji.'. In our experiment the value of 0 is 1.26. As to the experiments of Sherwood 44 T. MIZUSHINA and M. NAKAJIMA and Comings 6 \ Mark 7l and Arnold 8 l it is 1.31, 1,16, 1.31 respectively. In all cases {3 is calculated from data for toluene. Apparatus and operations The air is sent into a glass tube of 1. 7 cm inner diameter from a compressor through several calcium dichloride tubes and an active carbon tube to remove the water and organic vapors. (Fig. 2) The amount of air flow is controlled at the orifice. While the experiment is being done, the velocity of air flow is kept constant (about 4 m/sec.). Larger velocity was desirable in order to minimize the influence by radiation and other factors but it could not be realized. At the end of the glass tube two copper-constantan thermo-couples protected with small glass tubes of about 1~2 mm inner diameter were fixed as shown in Fig. 3. B Fig. 2. A: Compressor B: Valve C: CaCl2 Tube C ': Active Carbon Tube D: Orifice E: Manometer F: Electric Heater G: Transformer H: Glass Tube I : Wet-and Dry-bulb Ther-mometer Schematic drawing of apparatus Fig. 3. Dry and wet bulb thermometer The wet-bulb thermo-couple is covered with cotton and filter paper, which are kept wet with liquid drops from a burette. The outer diameter of the wet-bulb is about 4 mm. The liquid flows down along the cotton, and a part of it evaporates through the surface of the filter paper. The liquid is cooled gradually losing the latent heat of vaporization, and at the end of the wet-bulb it will take the wet-bulb temperature. This considerably long wick is also useful to prevent the heat input caused by conduction through the thermometer stem. The rate of liquid dripping is controlled so that drops may be held at the lower end of the wick and not fall down. If it drops, by accident, it will soon be driven out from the Simultaneous Heat and Mass Transfer 45 bottom hole and its vapor will be blown off by the downward air flow. There-fore, the main flow of air will not contain the vapor of liquid. Wet and dry bulb temperatures can be measured with a potentiometer. With a electric heater the temperature of air flow is varied and (h/1r,')u1>, is obtained for each temperature using Eq. (11). Thus experiments were repeated by using water and several kind of liquids, obtaining the mean value of (h/r.' )ob, for each liquid. Result of Experiment Table 1 shows the mean values of (hf ,.')ob, for each liquid obtained by Sher-wood and Comings 6), Mark 7), Arnold B) and the auth,c;,~s:. Prandtl number a was taken as 0.74 for air. Schmidt number can be calculated us~ng µ/ p=0.132cm2/sec for pure air and the diffusion coefficient. All of these values are at 0°C and 1 atm. The diffusion coefficient for water, Methyl alcohol, ethyl alcohol, n-propyl alcohol, benzene, ethyl acetate, toluene, ethylpropionate and xylene of 0.220, 0.133, 0.102, 0.085, 0.077, 0.072, 0.071, 0.065 and 0.059cm2/sec respectively were taken from the International Critical Tables. As there were no proper data for propyl acetate, chlorobenzene, carbon tetrachloride and ethylen tetrachloride, they were calculated by Arnold's equation 9) and the following values were obtained respec-tively, 0.065, 0.065, 0.065 and 0.064. Table 1 also shows the calculated values of (,(hjr.')=(,(<J>Ja)½-c,, which are in fair agreement with the values of ( h Jr.' )ob,. 20 ,--- o Sherwood &Comings v - • Mo.tie o Arnold J - .... -::: ·,--- • Mizushi11c1itNakq_Jim11 · .......... t -,,,.... I) ~/ ~ • - ¼--(fjJ;,.)\ .... - ~ / to / / Ett,<S) ........... V ' 08 OS" D f.5 ... o '5 50 Fig. 4. Correlation of data for dry and wet bulb thermometers by theoretical equation (8) 46 T. MIZUSHINA and M. NAKAJIMA N ( h ) {rw( H w - H )} l d ( / ) · ow, K! c;, = -( t _ r,;,)13c-;; are P otte vs. </J t1 usmg the logarithmic c.oordinate3 in Fig. 4. Eq. (8) is represented by a straight line. From the above the theoretical relation expressed by Eq. (8) is found to fit experimental data quite satisfactorily. Summary As the result of the theoretical consideration concerning with the analogy among fluid friction, heat transfer and mass tranfer in common gaseous systems (t1, </J=0.5~2.5), the following relation between the coefficients of heat and mass transfer was obtained : and this was ascertained experimentally, using the wet and dry bulb thermometer. Acknowledgment The authors wish to express their thanks to Professor S. Kamei for his attentive instructions, and also to the Education Ministry for the subsidy of the Science Research Funds which helped this research. (Nomenclature) cp specific heat of fluid D diffusion coefficient f friction factor 0.079 Re-¼ h coefficient of heat transfer he coefficient of heat transfer by conduction through the thermometer stem h., coefficient of heat transfer by radiation H absolute humidity of gas Hw : saturated absolute humidity at fw K. 1 coefficient of mass transfer rw latent heat of vaporization of liquid at tw t temperature of gas tw wet-bulb temperature u0 m,ean velocity of fluid "' ratio of evaporation surface area to heat transfer surface area of wet-bulb [:I constant defined by Eq. (13) A heat conductivity of fluid µ absolute viscocity of fluid p density of fluid " Prandtl number Cp µ/ A ,t, Schmidt number µ/ pD Table 1 Liquid used I (! )ob. 1 ~ = ( h/ t.1 )ob,, D I 5b=L I ,;, I fj(f )lcp I cm2/sec I -IC1 Cp fj•Cp pD " Water 0.29 0.92 0.220 0.60 0.81 0.28 Sherwood & Benzene 0.40 1.27 0.077 1.72 2.32 0.48 Comingss) Toluene 0.50 1.59 0.071 1.86 2.51 0.50 fj=l.31 n-Propyl acetate 0.52 1.65 0.065 2.03 2.75 0.52 Carbon tetrachloride 0.50 1.59 o'.065 2.03 2.75 0.52 Water 0.26 0.94 0.220 I o.60 0.81 0.25 Mark7) Benzene 0.41 1.48 0.077 1.72 2.32 0.42 fj=I.16 Ethyl acetate 0.42 1.51 0.072 1.83 2.48 I 0.44 Toluene 0.44 1.59 0.071 1.86 2.51 0.44 Chlorobenzene 0.44 1.59 0.065 2.03 2.75 0.46 Carbon tetrachloride 0.44 1.59 0.065 2.03 2.75 0.46 Ethylen tetrachloride 0.50 1.80 0.064 2.06 2.79 0.46 Water 0.27 0.86 0.220 0.60 0.81 0.28 ArnoldB) Methyl alcohol 0.35 1.11 0.133 0.99 1.34 0.36 fj=l.31 n-Propyl alcohol 0.43 1.37 0.085 1.55 2.10 0.46 Toluene 0.50 1.59 0.071 1.86 2.51 0.50 Ethyl propionate 0.50 1.59 0.065 2.03 2.75 0.52 Water 0.26 0.88 0.220 0.60 0.81 0.27 Mizushina Ethyl alcohol 0.41 1.35 0.102 1.29 1.75 0.40 & Nakajima Benzene 0.44 1.44 0.077 1.72 2.32 0.46 fj=l.26 Ethyl acetate 0.46 1.51 0.072 1.83 2.48 0.47 Toluene 0.4B 1.59 0.071 1.86 2.51 0.48 Carbon tetrachloride 0.47 1.54 0.065 2.03 2.75 0.50 Xylene o.55 1.83 0.059 2.24 3.02 0.52 48 T. MIZUSHINA and M. NAKAJIMA Literature cited 1) Th. K. Sherwood: Absorption and Extraction 31, (1937) 2) L. Prandtl : Physikal. Zft., 29, 487 (1916) 3) Th. von Karman: Trans. A. S. M. E., 61, 705 (1939) 4) T. H. Chilton and A. P. Colburn: Ind. Eng. Chem, 26, 1183 (1934) 5) T. Mizushina: Chem. Eng. Japan, 13, 22 (1949) 6) Th. K. Sherwood and E. W. Comings: Trans. Am. Inst. Chem. Eng., 28, 83 (1932) 7) J. G. Mark: Data presented by Sherwood: Trans. Am. Inst. Chem. Eng., 28, 107 (1932) 8) J. H. Arnold: Physics, 4, 255 (1933) 9) J. H. Arnold: Ind. Eng. Chem., 22, 1091 (1930) 

Simultaneous Heat and Mass Transfer : The Relation between Heat transfer and Mass transfer Coefficients

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Simultaneous Heat and Mass Transfer : The Relation between Heat transfer and Mass transfer Coefficients

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