Convective Mass Transfer
PART 11- CONVECTIVE MASS TRANSFER2.1 Introduction2.2 Convective Mass Transfer coefficient2.3 Significant parameters in convective mass transfer2.4 The application of dimensional analysis to Mass Transfer2.4.1 Transfer into a stream flowing under forced convection2.4.2 Transfer into a phase whose motion is due to natural convection2.5 Analogies among mass, heat, and momentum transfer2.5.1 Reynolds analogy2.5.2 Chilton – Colburn analogy2.6 Convective mass transfer correlations2.6.1 For flow around flat plat2.6.2 For flow around single sphere2.6.3 For flow around single cylinder2.6.4 For flow through pipes2.7 Mass transfer between phases2.8 Simultaneous heat and mass transfer2.8.1 Condensation of vapour on cold surface2.8.2 Wet bulb thermometer2.1 IntroductionOur discussion of mass transfer in the previous chapter was limited to molecular diffusion,which is a process resulting from a concentration gradient. In system involving liquids orgases, however, it is very difficult to eliminate convection from the overall mass-transferprocess.Mass transfer by convection involves the transport of material between a boundary surface(such as solid or liquid surface) and a moving fluid or between two relatively immiscible,moving fluids.There are two different cases of convective mass transfer:1. Mass transfer takes place 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 takes place in the two contacting phases as in extraction andabsorption.In the first few section we will see equation governing convective mass transfer in a singlefluid phase.FDE312-PARTII-CONVECTIVE MASS TRANSFER -1
2.2 Convective Mass Transfer CoefficientIn the study of convective heat transfer, the heat flux is connected to heat transfercoefficient as()Q A q h t s t m -------------------- (2.1)The analogous situation in mass transfer is handled by an equation of the form()N A k c C As C A -------------------- (2.2)The molar flux N A is measured relative to a set of axes fixed in space. The driving force isthe difference between the concentration at the phase boundary, C AS (a solid surface or afluid interface) and the concentration at some arbitrarily defined point in the fluid medium,C A . The convective mass transfer coefficient k C is a function of geometry of the system andthe velocity and properties of the fluid similar to the heat transfer coefficient, h.2.3 Significant Parameters in Convective Mass TransferDimensionless parameters are often used to correlate convective transfer data. Inmomentum transfer Reynolds number and friction factor play a major role. In the correlationof convective heat transfer data, Prandtl and Nusselt numbers are important. Some of thesame parameters, along with some newly defined dimensionless numbers, will be useful inthe correlation of convective mass-transfer data.The molecular diffusivities of the three transport process (momentum, heat and mass) havebeen defined as:andMomentum diffusivityν µ ----------------------------- (2.3)ρThermal diffusivityα k --------------------------- (2.4)ρCpMass diffusivity D AB --------------------------- (2.5)It can be shown that each of the diffusivities has the dimensions of L2 / t, hence, a ratio ofany of the two of these must be dimensionless.FDE312-PARTII-CONVECTIVE MASS TRANSFER -2
The ratio of the molecular diffusivity of momentum to the molecular diffusivity of heat(thermal diffusivity) is designated as the Prandtl NumberCp µMomentum diffusivityν Pr KThermal diffusivityα------------------------ (2.6)The analogous number in mass transfer is Schmidt number given asνµMomentum diffusivity-------------- (2.7) Sc D ABρ D ABMass diffusivityThe ratio of the molecular diffusivity of heat to the molecular diffusivity of mass isdesignated the Lewis Number, and is given bykαThermal diffusivity------------- (2.8) Le D ABρ C p D ABMass diffusivityLewis number is encountered in processes involving simultaneous convective transfer ofmass and energy.Let us consider the mass transfer of solute A from a solid to a fluid flowing past the surfaceof the solid. The concentration and velocity profile is depicted in Figure 2.1.Figure 2.1 The concentration and velocity profile of solid dissolving in fluidFor such a case, the mass transfer between the solid surface and the fluid may be written as()N A k c C As C A ---------------------- (2.2 a)FDE312-PARTII-CONVECTIVE MASS TRANSFER -3
Since the mass transfer at the surface is by molecular diffusion, the mass transfer may alsodescribed byN A D AB------------------------- (2.9)dCAdyy 0When the boundary concentration, C As is constant, equation (2.9) may be written asN A D AB(d C A C As)dy---------------------- (2.10)y 0Equation (2.2a) and (2.10) may be equated, since they define the same flux of component Aleaving the surface and entering the fluid()k c C A s C A D AB(dC A C Asdy)--------------- (2.11)y 0This relation may be rearranged into the following form:kcD AB ()d C A C As d y(C A C A )-------------------- (2.12)y 0Multiplying both sides of equation (2.12) by a characteristic length, L we obtain the followingdimensionless expression:kc LD AB ()d C A C As d y(C A S C A ) Ly 0----------------- (2.13)The right hand side of equation (2.13) is the ratio of the concentration gradient at thesurface to an overall or reference concentration gradient; accordingly, it may be consideredas the ratio of molecular mass-transport resistance to the convective mass-transportresistance of the fluid. This ratio is generally known as the Sherwood number, Sh andanalogous to the Nusselt number Nu, in heat transfer.FDE312-PARTII-CONVECTIVE MASS TRANSFER -4
2.4 Application of Dimensionless Analysis to Mass TransferOne of the method of obtaining equations for predicting mass-transfer coefficients is theuse of dimensionless analysis. Dimensional analysis predicts the various dimensionlessparameters which are helpful in correlating experimental data.There are two important mass transfer processes, which we shall consider, the transfer ofmass into a steam flowing under forced convection and the transfer of mass into a phasewhich is moving as the result of natural convection associated with density gradients.2.4.1 Transfer into a stream flowing under forced convectionConsider the transfer of mass from the walls of a circular conduit to a fluid flowing throughthe conduit. The mass transfer is due to the concentration driving force C As – C A .The result of the dimensional analysis of mass transfer by forced convection in a circularconduit indicates that a correlating relation could be of the form,Sh ψ (Re, Sc ) --------------------------- (2.14)whereRe Dνρµand S C µρ D ABand Sherwood Number Sh kc DD ABWhich is analogous to the heat transfer correlationNu ψ (Re, Pr ) ---------------------------- (2.14)2.4.2 Transfer into a phase whose motion is due to Natural ConvectionNatural convection currents develop if there exists any variation in density within the fluidphase. The density variation may be due to temperature differences or to relatively largeconcentration differences.The result of the dimensional analysis of mass transfer by natural convection indicates that acorrelating relation could be of the form,()Sh ψ Gr AB , Sc ---------------------------- (2.15)where the Grashof number in heat transfer by natural convectionL3 ρ g ρ AGrAB 2µFDE312-PARTII-CONVECTIVE MASS TRANSFER -5
2.5 Analysis among Mass, Heat and Momentum TransferAnalogies among mass, heat and momentum transfer have their origin either in themathematical description of the effects or in the physical parameters used for quantitativedescription.To explore those analogies, it could be understood that the diffusion of mass andconduction of heat obey very similar equations. In particular, diffusion in one dimension isdescribed by the Fick’s Law asJA D ABdCAdz------------------------------ (2.16)Similarly, heat conduction is described by Fourier’s law asq kd T --------------------------------- (2.17)dzwhere k is the thermal conductivity.The similar equation describing momentum transfer as given by Newton’s law isτ µdν----------------------------- (2.18)dzwhere τ is the momentum flux (or shear stress) and µ is the viscosity of fluid.At this point it has become conventional to draw an analogy among mass, heat andmomentum transfer. Each process uses a simple law combined with a mass or energy ormomentum balance.In this section, we shall consider several analogies among transfer phenomenon which hasbeen proposed because of the similarity in their mechanisms. The analogies are useful inunderstanding the transfer phenomena and as a satisfactory means for predicting behaviorof systems for which limited quantitative data are available.The similarity among the transfer phenomena and accordingly the existence of the analogiesrequire that the following five conditions exist within the system1. The physical properties are constant2. There is no mass or energy produced within the system. This implies that there is nochemical reaction within the systemFDE312-PARTII-CONVECTIVE MASS TRANSFER -6
3. There is no emission or absorption of radiant energy.4. There is no viscous dissipation of energy.5. The velocity profile is not affected by the mass transfer. This implies there should bea low rate of mass transfer.2.5.1 Reynolds AnalogyThe first recognition of the analogous behavior of mass, heat and momentum transfer wasreported by Osborne Reynolds in 1874. Although his analogy is limited in application, itserved as the base for seeking better analogies.Reynolds postulated that the mechanisms for transfer of momentum, energy and mass areidentical. Accordingly,kcνh ρν Cp f -------------------------------- (2.19)2Here h is heat transfer coefficientf is friction factorν is velocity of free streamThe Reynolds analogy is interesting because it suggests a very simple relation betweendifferent transport phenomena. This relation is found to be accurate when Prandtl andSchmidt numbers are equal to one. This is applicable for mass transfer by means of turbulenteddies in gases. In this situation, we can estimate mass transfer coefficients from heattransfer coefficients or from friction factors.2.5.2 Chilton – Colburn AnalogyBecause the Reynold’s analogy was practically useful, many authors tried to extend it toliquids. Chilton and Colburn, using experimental data, sought modifications to the Reynold’sanalogy that would not have the restrictions that Prandtl and Schmidt numbers must beequal to one. They defined for the j factor for mass transfer asjD kcν (Sc ) 2 3--------------------------- (2.20)The analogous j factor for heat transfer isFDE312-PARTII-CONVECTIVE MASS TRANSFER -7
j H St Pr23----------------------------- (2.21)where St is Stanton number Nuh Re Prρϑ C pBased on data collected in both laminar and turbulent flow regimes, they foundjD jH f ----------------------------- (2.21)2This analogy is valid for gases and liquids within the range of 0.6 Sc 2500 and0.6 Pr 100.The Chilton-Colburn analogy has been observed to hold for many different geometries forexample, flow over flat plates, flow in pipes, and flow around cylinders.Example 2.1 A stream of air at 100 kPa pressure and 300 K is flowing on the top surface of athin flat sheet of solid naphthalene of length 0.2 m with a velocity of 20 m/sec. The otherdata are:Mass diffusivity of naphthalene vapor in air 6 x 10 –6 m 2/secKinematic viscosity of air 1.5 x10 –5 m 2.scConcentration of naphthalene at the air-solid naphthalene interface 1x10 –5 kmol/m3Calculate:(a) the overage mass transfer coefficient over the flat plate(b) the rate of loss of naphthalene from the surface per unit widthNote: For heat transfer over a flat plate, convective heat transfer coefficient for laminar flowcan be calculated by the equation.Nu 0.664 Re1L 2 Pr 1 3you may use analogy between mass and heat transfer.FDE312-PARTII-CONVECTIVE MASS TRANSFER -8
Solution:Given: Correlation for heat transferNu 0.664 Re1L 2 Pr 1 3The analogous relation for mass transfer isSh 0.664 Re1L 2 Sc 1 3 h Sherwood number kL/D ABRe L Reynolds number Lυρ/µSc Schmidt number µ / (ρ D AB )k overall mass transfer coefficientL length of sheetD AB diffusivity of A in Bυ velocity of airµ viscosity of airρ density of air, andµ/ρ kinematic viscosity of air.Substituting for the known quantities in equation (1)k (0.2 ) (0.2 )(20 ) 0.664 6 5 6 x10 1.5 x10 k 0.014 m/sec12 1.5 x10 5 6 6 x10 13Rate of loss of naphthalene k (C Ai – C A ) 0.014 (1 x 10 –5 – 0) 1.4024 x 10 –7 kmol/m 2 secRate of loss per meter width (1.4024 x 10 –7) (0.2) 2.8048 x 10 –8 kmol/m.sec 0.101 gmol/m.hr.FDE312-PARTII-CONVECTIVE MASS TRANSFER -9
2.5 Convective Mass Transfer CorrelationsExtensive data have been obtained for the transfer of mass between a moving fluid andcertain shapes, such as flat plates, spheres and cylinders. The techniques include sublimationof a solid, vaporization of a liquid into a moving stream of air and the dissolution of a solidinto water.These data have been correlated in terms of dimensionless parameters and the equationsobtained are used to estimate the mass transfer coefficients in other moving fluids andgeometrically similar surfaces.2.5.1Flat PlateFrom the experimental measurements of rate of evaporation from a liquid surface or fromthe sublimation rate of a volatile solid surface into a controlled air-stream, severalcorrelations are available. These correlation have been found to satisfy the equationsobtained by theoretical analysis on boundary layers,Sh 0.664 ReL1 2Sc 1 3 (laminar ) Re L 3 * 10 5 ------------- (2.22)Sh 0.036 ReL0.8 Sc 1 3 (turbulent ) Re L 3 * 10 5 ----------- (2.23)Using the definition of j factor for mass transfer on equation (2.22) and (2.23) we obtainj D 0.664 ReL 1 2 (laminar ) Re L 3 * 10 5 ------------- (2.24)J D 0.037 ReL 0.2 (turbulent ) Re L 3 * 10 5 ----------- (2.25)These equations may be used if the Schmidt number in the range 0.6 Sc 2500.Example 2.2 If the local Nusselt number for the laminar boundary layer that is formed overa flat plate isNu x 0.332 Re 1x 2 Sc 1 / 3Obtain an expression for the average film-transfer coefficient k c , when the Reynoldsnumber for the plate isa) Re L 100 000b) Re L 1500 000The transition from laminar to turbulent flow occurs at Re x 3 x 10 5.FDE312-PARTII-CONVECTIVE MASS TRANSFER -10
Derivation:L k c dxBy definition :kc oL dxoand Nu x kc xRe x ;D ABxv ρµ;Sc µρ D AB;For Re L 100 000 ; (which is less than the Reynolds number corresponding to Transitionvalue of 3 x 10 5) xv ρ 0.332 µ oLkc 121(Sc )3D ABxdxL120.332 (Sc )13 v ρ µ Ldxox1 2D AB 12 v ρ 0.332 Sc 1 3 1µ L2L[ ]D AB x 1 2(i.e.) k c L 0.664 Re 1 2 Sc 1 3LD ABLo[answer (a)]For Re L 1500 000 ( 3 x 10 5)kc D AB LL t 0.332 Re 1 2 Sc 1 3 d x 0.0292 Re xx xLt oL45Sc13d x x where L t is the distance from the leading edge of the plane to the transition point whereRe x 3 x 10 5.FDE312-PARTII-CONVECTIVE MASS TRANSFER -11
kc D AB12 Lt dx 0.332 Sc 1 3 v ρ 0.0292 Sc 1 3 12 µ o x L[ ]kcL0.0292 0.664 Re1t 2 Sx 1 3 Sc 1 3 x45D AB45 LLt( v ρ µ V ρ µ 0.664 Re 1t 2 Sc 1 3 0.0365 Sc 1 3 Re L4 5 Re t4 5d x 15 Lt x 45 L45)kcL 0.664 Re 1t 2 Sc 1 3 0.0365 Re L4 5 Sc 1 3 0.0365 Ret4 5 Sc 1 3D ABwhere Re t 3 x 10 52.5.2Single SphereCorrelations for mass transfer from single spheres are represented as addition of termsrepresenting transfer by purely molecular diffusion and transfer by forced convection, in theformSh Sh o C Re m Scn---------------------- (2.26)where C, m and n are constants, the value of n is normally taken as 1/3For very low Reynold’s number, the Sherwood number should approach a value of 2. Thisvalue has been derived in earlier sections by theoretical consideration of molecular diffusionfrom a sphere into a large volume of stagnant fluid. Therefore the generalized equationbecomesSh 2 C Re m Sc 1 3 -------------------------- (2.27)For mass transfer into liquid streams, the equation given by Brain and Hales(23Sh 4 1.21 Pe AB)12-------------------------- (2.28)correlates the data that are obtained when the mass transfer Peclet number, Pe AB is lessthan 10,000. This Peclet number is equal to the product of Reynolds and Schmidt numbers(i.e.)FDE312-PARTII-CONVECTIVE MASS TRANSFER -12
Pe AB Re Sc ---------------------------------- (2.29)For Peclet numbers greater than 10,000, the relation given by Levich is usefulSh 1.01 Pe1AB3 --------------------------- (2.30)The relation given by FroesslingSh 2 0.552 Re 1 2 Sc 1 3 ----------------------- (2.31)correlates the data for mass transfer into gases for at Reynold’s numbers ranging from 2 to800 and Schmidt number ranging 0.6 to 2.7.For natural convection mass transfer the relation given by Schutz(Sh 2 0.59 Gr AB Scis useful over the range)1 4----------------------- (2.32)2 x 10 8 Gr AB Sc 1.5 x10 10Example 2.3 The mass flux from a 5 cm diameter naphthalene ball placed in stagnant air at40 C and atmospheric pressure, is 1.47x10 –3 mol/m 2. sec. Assume the vapor pressure ofnaphthalene to be 0.15 atm at 40 C and negligible bulk concentration of naphthalene in air.If air starts blowing across the surface of naphthalene ball at 3 m/s by what factor will themass transfer rate increase, all other conditions remaining the same?For spheres :Sh 2.0 0.6 (Re) 0.5 (Sc)0.33where Sh is the Sherwood number and Sc is the Schmids number. The viscosity and densityof air are 1.8 x 10 –5 kg/m.s and 1.123 kg/m 3, respectively and the gas constant is82.06 cm 3 . atm/mol.K.Calculations:Sh kc LD ABwhere L is the characteristic dimension for sphere L Diameter.FDE312-PARTII-CONVECTIVE MASS TRANSFER -13
µSc Rc ρ D ABDv ρµMass flux, N A K c c ------------------------------(1)Sh 2.0 0.6 (Re) 0.5 (Sc) 0.33kc DD AB DV ρ 2.0 0.6 0 .5 µ ρ D AB 0.33----------------------- (2)also N K G p AThereforekcRT KGGiven:N 1.47 x10 3Kcmol p2m . sec RTAk c 0.15mol 0 (1.47 x10 3 )10 4 RT 1cm 2 . sec 1.47 x10 7kc x82.06 x(273 40 )0.15 0.0252 cm/seck c 2.517x 10 –4 m/sec ------------------------------(3)Estimation of D AB :From (2),(2.517 x10 4 )(5 x10 2 ) 2 (since v 0)D ABFDE312-PARTII-CONVECTIVE MASS TRANSFER -14
Therefore D AB 6.2925 x 10 –6 m2/sec.Andk c (5 x10 2 )6.2925 x10 6 (5 x10 2 ) x3 x1.123 2 0.6 1.8 x10 5 0. 5 1.8 x10 5 6 1.123(6.2925 x10 ) 0.337946 k c 2 [ 0.6 (96.74) (1.361)]k c 0.0102 m/sec. --- (4)N A2( 4)0.0102 40.5(3)N A1 2.517 x10 4Therefore, rate of mass transfer increases by 40.5 times the initial conditions.2.5.3Single CylinderSeveral investigators have studied the rate of sublimation from a solid cylinder into airflowing normal to its axis. Bedingfield and Drew correlated the available data in the formk G P Sc0.56( ) 0.4 ------------------------ (2.33) 0.281 Re /Gmwhich is valid for 400 Re / 25000and0.6 Sc 2.6Where Re / is the Reynold’s number in terms of the diameter of the cylinder, G m is the molarmass velocity of gas and P is the pressure.2.5.4Flow Through PipesMass 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 liquidsinto air given the correlationShp B, l mP 0.023 Re 0.83 Sc 0.44 -------------- (2.34)FDE312-PARTII-CONVECTIVE MASS TRANSFER -15
where p B, lm is the log mean composition of the carrier gas, evaluated between the surfaceand bulk stream composition. P is the total pressure. This expression has be
2.4.2 Transfer into a phase whose motion is due to natural convection . 2.5 Analogies among mass, heat, and momentum transfer . In the correlation of convective heat transfer data, Prandtl and Nusselt numbers are important. Some of the . Let us consider the mass transfer of solute
Basic Heat and Mass Transfer complements Heat Transfer,whichispublished concurrently. Basic Heat and Mass Transfer was developed by omitting some of the more advanced heat transfer material fromHeat Transfer and adding a chapter on mass transfer. As a result, Basic Heat and Mass Transfer contains the following chapters and appendixes: 1.
23.10 Radiant Heat Transfer Between Gray Surfaces 381 23.11 Radiation from Gases 388 23.12 The Radiation Heat-Transfer Coefficient 392 23.13 Closure 393 24. Fundamentals of Mass Transfer 398 24.1 Molecular Mass Transfer 399 24.2 The Diffusion Coefficient 407 24.3 Convective Mass Transfer 428 24.4 Closure 429 25.
Abstract—Recently, heat and mass transfer simulation is more and more important in various engineering fields. In order to analyze how heat and mass transfer in a thermal environment, heat and mass transfer simulation is needed. However, it is too much time-consuming to obtain numerical solutions to heat and mass transfer equations.
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1 THE COUPLING BETWEEN CONVECTIVE VARIABILITY AND LARGE-SCALE FLOW 2 PATTERNS OBSERVED DURING PISTON 2018-2019 3 . Convective rain rates and 30-dBZ echo top heights increased with feature size and . 55 board the R/V Thomas G. Thompson (2018, hereafter referred to as TGT) .
wind shear likely to produce ordinary, convective storms. Weisman and Klemp (1986) describe the short-lived single cell storm as the most basic convective storm. Therefore, this ordinary convective storm type will be the focus of this study. 1.2 THUNDERSTORM CHARGE STR
The mass-transfer coefficient is related to: (1).The properties of the fluid, (2).The dynamic characteristics of the flowing fluid, and (3).The geometry of the specific system of interest Evaluation of the mass transfer coefficient There are four methods of ev
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