2. Mass transfer takes place in the two contacting phases as in extraction and absorption.

Transcription

1 PRT 11- CONVECTIVE MSS TRNSFER 2.1 Introdution 2.2 Convetive Mass Transfer oeffiient 2.3 Signifiant parameters in onvetive mass transfer 2.4 The appliation of dimensional analysis to Mass Transfer Transfer into a stream flowing under fored onvetion Transfer into a phase whose motion is due to natural onvetion 2.5 nalogies among mass, heat, and momentum transfer Reynolds analogy Chilton Colburn analogy 2.6 Convetive mass transfer orrelations For flow around flat plat For flow around single sphere For flow around single ylinder For flow through pipes 2.7 Mass transfer between phases 2.8 Simultaneous heat and mass transfer Condensation of vapour on old surfae Wet bulb thermometer 2.1 Introdution Our disussion of mass transfer in the previous hapter was limited to moleular diffusion, whih is a proess resulting from a onentration gradient. In system involving liquids or gases, however, it is very diffiult to eliminate onvetion from the overall mass-transfer proess. Mass transfer by onvetion involves the transport of material between a boundary surfae (suh as solid or liquid surfae) and a moving fluid or between two relatively immisible, moving fluids. There are two different ases of onvetive mass transfer: 1. Mass transfer taes plae only in a single phase either to or from a phase boundary, as in sublimation of naphthalene (solid form) into the moving air. 2. Mass transfer taes plae in the two ontating phases as in extration and absorption. In the first few setion we will see equation governing onvetive mass transfer in a single fluid phase. FE312-PRTII-CONVECTIVE MSS TRNSFER -1

2 2.2 Convetive Mass Transfer Coeffiient In the study of onvetive heat transfer, the heat flux is onneted to heat transfer oeffiient as ( t t ) Q q h (2.1) s m The analogous situation in mass transfer is handled by an equation of the form N ( C C ) s (2.2) The molar flux N is measured relative to a set of axes fixed in spae. The driving fore is the differene between the onentration at the phase boundary, C S (a solid surfae or a fluid interfae) and the onentration at some arbitrarily defined point in the fluid medium, C. The onvetive mass transfer oeffiient C is a funtion of geometry of the system and the veloity and properties of the fluid similar to the heat transfer oeffiient, h. 2.3 Signifiant Parameters in Convetive Mass Transfer imensionless parameters are often used to orrelate onvetive transfer data. In momentum transfer Reynolds number and frition fator play a major role. In the orrelation of onvetive heat transfer data, Prandtl and Nusselt numbers are important. Some of the same parameters, along with some newly defined dimensionless numbers, will be useful in the orrelation of onvetive mass-transfer data. The moleular diffusivities of the three transport proess (momentum, heat and mass) have been defined as: µ Momentum diffusivity ν (2.3) ρ Thermal diffusivity α (2.4) ρ C p and Mass diffusivity (2.5) It an be shown that eah of the diffusivities has the dimensions of 2 / t, hene, a ratio of any of the two of these must be dimensionless. FE312-PRTII-CONVECTIVE MSS TRNSFER -2

3 The ratio of the moleular diffusivity of momentum to the moleular diffusivity of heat (thermal diffusivity) is designated as the Prandtl Number Momentum diffusivity ν C p µ Pr (2.6) Thermal diffusivity α K The analogous number in mass transfer is Shmidt number given as Momentum diffusivity ν µ S (2.7) Mass diffusivity ρ The ratio of the moleular diffusivity of heat to the moleular diffusivity of mass is designated the ewis Number, and is given by Thermal diffusivity α e (2.8) Mass diffusivity ρ C p ewis number is enountered in proesses involving simultaneous onvetive transfer of mass and energy. et us onsider the mass transfer of solute from a solid to a fluid flowing past the surfae of the solid. The onentration and veloity profile is depited in Figure 2.1. Figure 2.1 The onentration and veloity profile of solid dissolving in fluid For suh a ase, the mass transfer between the solid surfae and the fluid may be written as ( C C ) s N (2.2 a) FE312-PRTII-CONVECTIVE MSS TRNSFER -3

4 Sine the mass transfer at the surfae is by moleular diffusion, the mass transfer may also desribed by N d C d y y (2.9) When the boundary onentration, C s is onstant, equation (2.9) may be written as ( C C ) d s N (2.10) d y y 0 Equation (2.2a) and (2.10) may be equated, sine they define the same flux of omponent leaving the surfae and entering the fluid d ( C C ) ( C C ) (2.11) s s d y y 0 This relation may be rearranged into the following form: d ( C C s ) d y ( C C ) y (2.12) Multiplying both sides of equation (2.12) by a harateristi length, we obtain the following dimensionless expression: d ( C C s ) d y ( C C ) S y (2.13) The right hand side of equation (2.13) is the ratio of the onentration gradient at the surfae to an overall or referene onentration gradient; aordingly, it may be onsidered as the ratio of moleular mass-transport resistane to the onvetive mass-transport resistane of the fluid. This ratio is generally nown as the Sherwood number, Sh and analogous to the Nusselt number Nu, in heat transfer. FE312-PRTII-CONVECTIVE MSS TRNSFER -4

5 2.4 ppliation of imensionless nalysis to Mass Transfer One of the method of obtaining equations for prediting mass-transfer oeffiients is the use of dimensionless analysis. imensional analysis predits the various dimensionless parameters whih are helpful in orrelating experimental data. There are two important mass transfer proesses, whih we shall onsider, the transfer of mass into a steam flowing under fored onvetion and the transfer of mass into a phase whih is moving as the result of natural onvetion assoiated with density gradients Transfer into a stream flowing under fored onvetion Consider the transfer of mass from the walls of a irular onduit to a fluid flowing through the onduit. The mass transfer is due to the onentration driving fore C s C. The result of the dimensional analysis of mass transfer by fored onvetion in a irular onduit indiates that a orrelating relation ould be of the form, Sh ( Re, S) ψ (2.14) where νρ Re and µ S C µ and Sherwood Number ρ Sh Whih is analogous to the heat transfer orrelation ψ ( Re, Pr) Nu (2.14) Transfer into a phase whose motion is due to Natural Convetion Natural onvetion urrents develop if there exists any variation in density within the fluid phase. The density variation may be due to temperature differenes or to relatively large onentration differenes. The result of the dimensional analysis of mass transfer by natural onvetion indiates that a orrelating relation ould be of the form, ( Gr S) Sh ψ, (2.15) where the Grashof number in heat transfer by natural onvetion 3 ρ g ρ Gr 2 µ FE312-PRTII-CONVECTIVE MSS TRNSFER -5

6 2.5 nalysis among Mass, Heat and Momentum Transfer nalogies among mass, heat and momentum transfer have their origin either in the mathematial desription of the effets or in the physial parameters used for quantitative desription. To explore those analogies, it ould be understood that the diffusion of mass and ondution of heat obey very similar equations. In partiular, diffusion in one dimension is desribed by the Fi s aw as J d C (2.16) d z Similarly, heat ondution is desribed by Fourier s law as d T q (2.17) d z where is the thermal ondutivity. The similar equation desribing momentum transfer as given by Newton s law is d ν τ µ (2.18) d z where τ is the momentum flux (or shear stress) and µ is the visosity of fluid. t this point it has beome onventional to draw an analogy among mass, heat and momentum transfer. Eah proess uses a simple law ombined with a mass or energy or momentum balane. In this setion, we shall onsider several analogies among transfer phenomenon whih has been proposed beause of the similarity in their mehanisms. The analogies are useful in understanding the transfer phenomena and as a satisfatory means for prediting behavior of systems for whih limited quantitative data are available. The similarity among the transfer phenomena and aordingly the existene of the analogies require that the following five onditions exist within the system 1. The physial properties are onstant 2. There is no mass or energy produed within the system. This implies that there is no hemial reation within the system FE312-PRTII-CONVECTIVE MSS TRNSFER -6

7 3. There is no emission or absorption of radiant energy. 4. There is no visous dissipation of energy. 5. The veloity profile is not affeted by the mass transfer. This implies there should be a low rate of mass transfer Reynolds nalogy The first reognition of the analogous behavior of mass, heat and momentum transfer was reported by Osborne Reynolds in lthough his analogy is limited in appliation, it served as the base for seeing better analogies. Reynolds postulated that the mehanisms for transfer of momentum, energy and mass are idential. ordingly, ν h ρ ν C p f (2.19) Here h is heat transfer oeffiient f is frition fator ν is veloity of free stream The Reynolds analogy is interesting beause it suggests a very simple relation between different transport phenomena. This relation is found to be aurate when Prandtl and Shmidt numbers are equal to one. This is appliable for mass transfer by means of turbulent eddies in gases. In this situation, we an estimate mass transfer oeffiients from heat transfer oeffiients or from frition fators Chilton Colburn nalogy Beause the Reynold s analogy was pratially useful, many authors tried to extend it to liquids. Chilton and Colburn, using experimental data, sought modifiations to the Reynold s analogy that would not have the restritions that Prandtl and Shmidt numbers must be equal to one. They defined for the j fator for mass transfer as j (2.20) ν 2 3 ( S) The analogous j fator for heat transfer is FE312-PRTII-CONVECTIVE MSS TRNSFER -7

8 2 3 j H St Pr (2.21) where St is Stanton number Nu Re Pr h ρ ϑ C p Based on data olleted in both laminar and turbulent flow regimes, they found j f j (2.21) H 2 This analogy is valid for gases and liquids within the range of 0.6 < S < 2500 and 0.6 < Pr < 100. The Chilton-Colburn analogy has been observed to hold for many different geometries for example, flow over flat plates, flow in pipes, and flow around ylinders. Example 2.1 stream of air at 100 Pa pressure and 300 K is flowing on the top surfae of a thin flat sheet of solid naphthalene of length 0.2 m with a veloity of 20 m/se. The other data are: Calulate: Mass diffusivity of naphthalene vapor in air 6 x 10 6 m 2 /se Kinemati visosity of air 1.5 x10 5 m 2.s Conentration of naphthalene at the air-solid naphthalene interfae 1x10 5 mol/m 3 (a) the overage mass transfer oeffiient over the flat plate (b) the rate of loss of naphthalene from the surfae per unit width Note: For heat transfer over a flat plate, onvetive heat transfer oeffiient for laminar flow an be alulated by the equation. Nu Re Pr you may use analogy between mass and heat transfer. FE312-PRTII-CONVECTIVE MSS TRNSFER -8

9 Solution: Given: Correlation for heat transfer Nu Re Pr The analogous relation for mass transfer is Sh Re S (1) where Sh Sherwood number / Re Reynolds number υρ/µ S Shmidt number µ / (ρ ) overall mass transfer oeffiient length of sheet diffusivity of in B υ veloity of air µ visosity of air ρ density of air, and µ/ρ inemati visosity of air. Substituting for the nown quantities in equation (1) ( 0.2) ( 0.2)( 20) 6x x x m/se 6x10 Rate of loss of naphthalene (C i C ) (1 x ) x 10 7 mol/m 2 se Rate of loss per meter width ( x 10 7 ) (0.2) x 10 8 mol/m.se gmol/m.hr. FE312-PRTII-CONVECTIVE MSS TRNSFER -9

10 2.5 Convetive Mass Transfer Correlations Extensive data have been obtained for the transfer of mass between a moving fluid and ertain shapes, suh as flat plates, spheres and ylinders. The tehniques inlude sublimation of a solid, vaporization of a liquid into a moving stream of air and the dissolution of a solid into water. These data have been orrelated in terms of dimensionless parameters and the equations obtained are used to estimate the mass transfer oeffiients in other moving fluids and geometrially similar surfaes Flat Plate From the experimental measurements of rate of evaporation from a liquid surfae or from the sublimation rate of a volatile solid surfae into a ontrolled air-stream, several orrelations are available. These orrelation have been found to satisfy the equations obtained by theoretial analysis on boundary layers, 5 ( laminar ) Re 3 * Sh Re S < (2.22) 5 ( turbulent) Re 3 * Sh Re S > (2.23) Using the definition of j fator for mass transfer on equation (2.22) and (2.23) we obtain ( laminar ) Re 3 * 10 j Re < (2.24) ( turbulent) Re 3 * 10 J Re > (2.25) These equations may be used if the Shmidt number in the range 0.6 < S < Example 2.2 If the loal Nusselt number for the laminar boundary layer that is formed over a flat plate is 1 2 1/ 3 Nu Re S x x Obtain an expression for the average film-transfer oeffiient, when the Reynolds number for the plate is a) Re b) Re The transition from laminar to turbulent flow ours at Re x 3 x FE312-PRTII-CONVECTIVE MSS TRNSFER -10

11 erivation: By definition : o dx o dx and Nu x x x v ρ µ ; Re x ; S ; µ ρ For Re ; (whih is less than the Reynolds number orresponding to Transition value of 3 x 10 5 ) 1 2 x v ρ o µ ( S) 1 3 x d x ( S) v ρ µ 1 2 d x 1 2 x o S 1 2 v ρ µ 1 2 x 1 2 [ ] o (i.e.) Re S [answer (a)] For Re (> 3 x 10 5 ) t o Re 1 2 x S d x x t Re x 4 5 S d x x where t is the distane from the leading edge of the plane to the transition point where Re x 3 x FE312-PRTII-CONVECTIVE MSS TRNSFER -11

12 0.332 S v ρ µ 1 2 t 4 5 d x v d x ρ S o x µ x t 4 5 [ x ] V ρ Ret Sx + S t 4 5 µ ( Re Re ) Re t S S t 1 2 t t Re S Re S Re S where Re t 3 x Single Sphere Correlations for mass transfer from single spheres are represented as addition of terms representing transfer by purely moleular diffusion and transfer by fored onvetion, in the form Sh o m n Sh + C Re S (2.26) where C, m and n are onstants, the value of n is normally taen as 1/3 For very low Reynold s number, the Sherwood number should approah a value of 2. This value has been derived in earlier setions by theoretial onsideration of moleular diffusion from a sphere into a large volume of stagnant fluid. Therefore the generalized equation beomes Sh m 2 + C Re S (2.27) For mass transfer into liquid streams, the equation given by Brain and Hales 2 3 ( 1.21 Pe ) 1 2 Sh (2.28) 4 orrelates the data that are obtained when the mass transfer Pelet number, Pe is less than 10,000. This Pelet number is equal to the produt of Reynolds and Shmidt numbers (i.e.) FE312-PRTII-CONVECTIVE MSS TRNSFER -12

13 Pe Re S (2.29) For Pelet numbers greater than 10,000, the relation given by evih is useful 1.01 Pe Sh (2.30) The relation given by Froessling 1 2 Sh Re S (2.31) orrelates the data for mass transfer into gases for at Reynold s numbers ranging from 2 to 800 and Shmidt number ranging 0.6 to 2.7. For natural onvetion mass transfer the relation given by Shutz Sh 1 4 ( Gr ) S (2.32) is useful over the range 2 x 10 8 < Gr S < 1.5 x10 10 Example 2.3 The mass flux from a 5 m diameter naphthalene ball plaed in stagnant air at 40 C and atmospheri pressure, is 1.47x10 3 mol/m 2. se. ssume the vapor pressure of naphthalene to be 0.15 atm at 40 C and negligible bul onentration of naphthalene in air. If air starts blowing aross the surfae of naphthalene ball at 3 m/s by what fator will the mass transfer rate inrease, all other onditions remaining the same? For spheres : Sh (Re) 0.5 (S) 0.33 where Sh is the Sherwood number and S is the Shmids number. The visosity and density of air are 1.8 x 10 5 g/m.s and g/m 3, respetively and the gas onstant is m 3. atm/mol.k. Calulations: Sh where is the harateristi dimension for sphere iameter. FE312-PRTII-CONVECTIVE MSS TRNSFER -13

14 S µ ρ R v ρ µ Mass flux, N K (1) Sh (Re) 0.5 (S) V ρ 0.5 µ ρ (2) also N K G p Therefore R T K G Given: K 3 mol N 1.47x10 2 m.se RT p RT (1.47x )10 4 mol 2 m.se 1.47x x82.06x ( ) m/se 2.517x 10 4 m/se (3) Estimation of : From (2), (2.517x10 4 )(5x10 2 ) 2 (sine v 0) FE312-PRTII-CONVECTIVE MSS TRNSFER -14

15 Therefore x 10 6 m 2 /se. nd (5x x10 ) (5x10 ) x3x x x (6.2925x10 ) [ 0.6 (96.74) (1.361)] m/se (4) (4) (3) N N x Therefore, rate of mass transfer inreases by 40.5 times the initial onditions Single Cylinder Several investigators have studied the rate of sublimation from a solid ylinder into air flowing normal to its axis. Bedingfield and rew orrelated the available data in the form G P S G m whih is valid for 400 < Re / < / 0. 4 ( Re ) (2.33) and 0.6 < S < 2.6 Where Re / is the Reynold s number in terms of the diameter of the ylinder, G m is the molar mass veloity of gas and P is the pressure Flow Through Pipes Mass transfer from the inner wall of a tube to a moving fluid has been studied extensively. Gilliland and Sherwood, based on the study of rate of vaporization of nine different liquids into air given the orrelation Sh p B, l m P Re S (2.34) FE312-PRTII-CONVECTIVE MSS TRNSFER -15

16 where p B, lm is the log mean omposition of the arrier gas, evaluated between the surfae and bul stream omposition. P is the total pressure. This expression has been found to be valid over the range 2000 < Re < < S < 2.5 inton and Sherwood modified the above relation maing it suitable for large ranges of Shmidt number. Their relation is given as 0.83 Sh Re S (2.35) and found to be valid for 2000 < Re < and 1000 < S < 2260 Example 2.4 solid dis of benzoi aid 3 m in diameter is spin at 20 rpm and 25 C. Calulate the rate of dissolution in a large volume of water. iffusivity of benzoi aid in water is 1.0x10 5 m 2 /se, and solubility is g/. The following mass transfer orrelation is appliable: Sh 0.62 Re ½ S 1/3 Where Re 2 ω ρ and ω is the angular speed in radians/time. µ Calulations: issolution rate N S (1) where N mass flux, and S surfae area for mass transfer N (C s C ) (2) where C s is the onentration of benzoi and at in water at the surfae of the dose. C is the onentration benzoi aid in wate for an from the surfae of the dis. Given: Sh 0.62 Re ½ S 1/3 FE312-PRTII-CONVECTIVE MSS TRNSFER -16

17 (i.e.) ω ρ µ (3) µ ρ 1 rotation 2 π radian Therefore 20 rotation per minute 20 x 2 π radian/min x 2π radian se For water ρ 1 g/m 3 µ 1 entipoise 0.01 g/m.se. From (3), From (2), 0.62 ω ρ µ 1 2 µ ρ (1.0x10 ) ( x x 10 4 m/se. 1 3 ( 40π 60) x From (1), N x 10 4 ( ) x 10 6 g/m 2.se N S N (2πr 2 ) (2.692 x 10 6 ) (2π x ) x 10 5 g/se g/hr. 2.7 Mass transfer between phases: Instead of a fluid in ontat with a solid, suppose we now onsider two immisible fluids, designated 1 and 2, in ontat with eah other. If fluid 1 has dissolved in it a substane that is also soluble in fluid 2, then as soon as the two fluids are brought together, substane will begin to diffuse into fluid 2. s long as the two phases remain in ontat, the transport of will ontinue until a ondition of equilibrium is reahed. FE312-PRTII-CONVECTIVE MSS TRNSFER -17

18 The situation disussed here ours in a variety of engineering proesses suh as gas absorption, stripping, and in liquid liquid extration. In all these separation proesses, two immisible fluids are brought into ontat and one or more omponents are transferred from one fluid phase to the other. In the system of fluids 1 and 2 with, the transported omponent, the onentration gradients in the region of the interfae between the two fluids are illustrated in Figure 2.1. Figure 2.1: Conentration gradients near the interfae between immisible fluids 1 and 2 Conentration C 1 and C 2 are the bul phase onentrations in fluids 1 and 2, respetively, C i is the onentration of at the interfae, and N is the molar flux of. For steady state onditions, we an define the flux of as N ( C C ) ( C C ) K ( C C ) (1) 1 1 i 2 i where individual mass transfer oeffiient defined in terms of the onentration differene in a single phase. K overall mass transfer oeffiient defined in terms of the overall differene in omposition. Equation (1) is analogous to that in heat transfer, where the individual oeffiients h are related to the overall oeffiient U. From equation (1), (2) K 2 In equation (1), the potential for mass transfer is exposed in terms of omposition. However, this is not always the most onvenient way to express it. For example, if fluid 1 is a gas and FE312-PRTII-CONVECTIVE MSS TRNSFER -18

19 fluid 2 a liquid, as in gas absorption, the potential in gas phase is often expressed in terms of partial pressures, while that in the liquid phase may be expressed in terms of onentrations. The expression for the molar flux is then written for the individual phases as: where N K ( P P ) K ( C C ) p G i i (3) p individual mass transfer oeffiient for the gas phase with the potential defined in terms of partial pressures. P G, C partial pressure and onentration of in the bul gas and liquid phases, respetively. P i, C i partial pressure and onentration of, respetively, at the interfae. t the interfae, it is usually assumed the two phases are in equilibrium. This means that P i and C i are related by an equilibrium relationship suh as Henry s law : P i H C i (4) where H is Henry s law onstant. The flux N an also be expressed in terms of overall mass transfer oeffiients as, ( P P ) K ( C C ) N K p G E E (5) where K p overall mass transfer oeffiient with the overall potential defined in terms of partial pressures. K overall mass transfer oeffiient with the overall potential defined in terms of onentrations. P E, C E equilibrium omposition. P E is related to the bul liquid omposition C S P E H C (6) similarly, C E P G (7) H The relationship between the individual and overall oeffiients is readily obtained through the use of equations (3) to (7) as 1 H H 1 + (8) K K p p FE312-PRTII-CONVECTIVE MSS TRNSFER -19

20 In many system, mass transfer resistane is mainly in one phase. For example, gases suh as nitrogen and oxygen do not dissolve muh in liquids. Their Henry s law onstant H is very large, thus K is a good approximation. In this ase, the liquid phase ontrols the mass transfer press sine mass transfer is slowest there. 2.8 Simultaneous Heat and Mass Transfer iffusional mass transfer is generally aompanied by the transport of energy, even with in an isothermal system. Sine eah diffusing onstituent arries its own individual enthalpy, the heat flux at a given plane is expressed as q N i H (1) i i where q is the heat flux due to diffusion of mass past the given plane, and H is the partial molar enthalpy of onstituent i in the mixture. i When there is a temperature differene, energy transfer also ours by one of the three heat transfer mehanisms (ondution, onvetion, radiation) ; for example, the equation for energy transport by onvetion and moleular diffusion beomes q h T + N i H i (2) i If the heat transfer is by ondution, the first term on the right hand side of equation (2) beomes T where is the thiness of the phase through whih ondution taes plae. The most ommon examples of proesses involving heat and mass transfer are ondensation of mist on a old surfae and in wet bulb thermometer. There are a number of suh proesses involving simultaneous heat and mass transfer suh as in formation of fog, and in ooling towers Condensation of vapor on old surfae: proess important in many engineering proesses as well as in day to day events involve the ondensation of a vapor upon a old surfae. Examples of this proess inlude sweating on old water pipes and the ondensation of moist vapor on a old surfae. FE312-PRTII-CONVECTIVE MSS TRNSFER -20

21 Figure illustrates the proess whih involves a film of ondensed liquid following down a old surfae and a film of gas through whih the ondensate is transferred by moleular diffusion. This proess involves the simultaneous transfer of mass and energy. The heat flux passing through the liquid film is given by ( ) q h l T 2 T 3 (1) This flux is also equal to the total energy transported by onvetion and moleular diffusion in the gas film.(i.e.,) q ( T T ) + N M ( H ) h H 2 (2) where M is the moleular weight of the diffusing onstituent. H 1 and H 2 are enthalpies of the vapor at plane 1 and liquid at plane 2. From equation (1) and (2) q ( T T ) h ( T T ) + N M ( H ) h (3) l H 2 The molar flux N is alulated by diffusion through stagnant gas model as C d y N 1 y d Z substituting the appropriate limits, the integral form of equation is N ( C ) ( y ) avg 1 y 2 ( Z 2 Z 1) yb, lm (4) The Wet bulb Thermometer nother example of simultaneous heat and mass transfer is that taing plae in wet-bulb thermometer. This onvenient devie for measuring relative humidity of air onsists of two onventional thermometers, one of whih is lad in a loth ni wet with water. The unlad dry-bulb thermometer measures the air s temperature. The lad wet-bulb thermometer measures the older temperature aused by evaporation of the water. We want to use this measured temperature differene to alulate the relative humidity in air. This relative humidity is defined as the amount of water atually in the air divided by the FE312-PRTII-CONVECTIVE MSS TRNSFER -21

22 amount at saturation at the dry-bulb temperature. To find this humidity, we an write equation for the mass and energy fluxes as: and Where the N ( C C ) ( y y ) i y i (1) ( T T ) q h i (2) C i and C are the onentrations of water vapor at the wet bulb s surfae and in bul of air, y i and y are the orresponding mole frations ; T i is the wet-bulb temperature, and T is the dry-bulb temperature. It an be noted that y i is the value at saturation at T i. In the air-film surroundings the wet-bulb, the mass and energy fluxes are oupled as Where Thus, N λ q (3) λ is the latent heat of vaporization of water. y Rearranging, ( y y ) λ h ( T T ) i i T i T λ h y ( y y ) i (4) From Chilton olbum analogy, j H j or h ( ) 2 2 Pr 3 ( S) 2 3 (5) ρ ν C ν α For gas Pr 1 and S 1. p Therefore equation (5) beomes, 1 (as y C y ρ ) h C Therefore equation (4) beomes p FE312-PRTII-CONVECTIVE MSS TRNSFER -22

23 T i T λ C ( y y ) i p where C p is the bumid heat of air. By similar method, the other industrial proesses of importane involving simultaneous heat and mass transfer suh as humidifiation and drying an be analysed. Example 2.5. ir at 1 atm is blown past the bulb of a merury thermometer. The bulb is overed with a wi. The wi is immersed in an organi liquid (moleular weight 58). The reading of the thermometer is 7.6 C. t this temperature, the vapor pressure of the liquid is 5 Pa. Find the air temperature, given that the ratio fo heat transfer oeffiient to the mass transfer oeffiient (psyhrometri ratio) is 2 J/g. ssume that the air, whih is blown, is free from the organi vapor. Solution: For simultaneous mass and heat transfer, heat flux q and mass flux N are related as q λ (1) N where λ is the latent heat of vaporization. Mass flux is given by N ' ( Y Y ') ω (2) Y where Y mass transfer oeffiient ' Y ω mass ratio of vapor in surrounding air at saturation; and Y mass ratio of vapor in surrounding air. Convetive heat flux is given by where q ( T ) h T ω (3) h heat transfer oeffiient; T ω wet bulb temperature of air; and T dry bulb temperature of air. Substituting for N and q from equation (2) and equation (3) in equation (1), ' ' ' ( T T ) ( T Y ) λ h y ω ω FE312-PRTII-CONVECTIVE MSS TRNSFER -23

24 T T ω λ ' ' ( Y Y ) ω Y h (4) Given: Y 0; λ 360 J/g; h/ Y 2 J/g.K; and T ω 7.6 C Y ' ω g organi vapor at g dry air saturation Substituting these in equation (4) T ( 360)( ) T C Temperature of air C FE312-PRTII-CONVECTIVE MSS TRNSFER -24