The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 55
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In the flame the vapor reacts with oxygen toform gaseous products, primarily CO2 and HzO. When a fuel droplet ignites, there is a short initialtransient during which the droplet heats up, until further conduction into the droplet is negligible andthe droplet attains a steady temperature (approximately the wet-bulb temperature, which is very closeto the boiling point for a typical hydrocarbon fuel).
The reaction in the flame can be modeled as a singlestep reaction with a constant stoichiometric ratio, r, and heat of combustion Dhc J/kg of fuel.The burning (mass transfer) rate of the droplet is given by the Godsave–Spalding formula,m˙ ¢¢ =k cpRln[1 + B ] kg m 2 sec(4.7.55)whereB=mox ,e Dhc r + c p (Te - Ts )h fgis the mass transfer driving force (or transfer number). The droplet lifetime is thent=rl DO2( )8 k c p ln(1 + B )sec(4.7.56)Based on experimental data for alkane droplets burning in air, Law and Williams (1972) recommendthat properties be evaluated at a reference temperature Tr = (1/2)(TBP + Tflame) where Tflame is the adiabaticflame temperature.
The reference specific heat is cpr = cpfu, and the reference thermal conductivity is kr© 1999 by CRC Press LLC4-226Section 4FIGURE 4.7.10 Combustion of a volatile fuel droplet burning in air: (a) schematic showing the flame, (b) concentration and temperature profiles.= 0.4kfu + 0.6kair . Radiation has been ignored in the analysis leading to Equation (4.7.55) but is accountedfor in using the Law and Williams reference-property scheme.For example, consider a 1-mm-diameter n-octane droplet burning in air at 1 atm and 300 K, at nearzero gravity. For n-octane (n-C8H18), rl = 611 kg/m3, hfg = 3.03 ´ 105 J/kg, Dhc = 4.44 ´ 107 J/kg, andTBP = 399 K.
The flame temperature is Tflame = 2320 K. At a reference temperature of (1/2) (Tflame + TBP)= 1360 K, property values of n-octane vapor include k = 0.113 W/m K, cp = 4280 J/kg K. The reaction isC 8 H18 + 12.5O 2 ® 8CO 2 + 9H 2 OHence, the stoichiometric ratio r = 400/114.2 = 3.50. Also mox,e = 0.231 and Ts @ TBP = 399 K.
Thus,the transfer number isB=(0.231) (4.44 ´ 10 7 ) (3.50) + 4280(300 - 399)3.03 ´ 10 5= 8.27At Tr = 1360 K, kair = 0.085 W/m K. Hence,kr = 0.4k fu + 0.6kair = (0.4)(0.113) + (0.6) (0.085) = 0.096 W m Kand the droplet lifetime is2(611) (1 ´ 10 -3 )t== 1.53 sec(8) (0.096 4280) ln(1 + 8.27)Mass ConvectionThe terms mass convection or convective mass transfer are generally used to describe the process ofmass transfer between a surface and a moving fluid, as shown in Figure 4.7.11. The surface may be that© 1999 by CRC Press LLC4-227Heat and Mass TransferFIGURE 4.7.11 Notation for convective mass transfer into an external flow.of a falling water film in an air humidifier, of a coke particle in a gasifier, or of a silica-phenolic heatshield protecting a reentry vehicle. As is the case for heat convection, the flow can be forced or natural,internal or external, and laminar or turbulent.
In addition, the concept of whether the mass transfer rateis low or high plays an important role: when mass transfer rates are low, there is a simple analogybetween heat transfer and mass transfer that can be efficiently exploited in the solution of engineeringproblems.Mass and Mole Transfer ConductancesAnalogous to convective heat transfer, the rate of mass transfer by convection is usually a complicatedfunction of surface geometry and s-surface composition, the fluid composition and velocity, and fluidphysical properties.
For simplicity, we will restrict our attention to fluids that are either binary mixturesor solutions, or situations in which, although more than two species are present, diffusion can beadequately described using effective binary diffusion coefficients, as was discussed in the section onordinary diffusion.
Referring to Figure 4.7.11, we define the mass transfer conductance of species 1,gm1, by the relationj1,s = g m1 Dm1 ;Dm1 = m1,s - m1,e(4.7.57)and the units of gm1 are seen to be the same as for mass flux (kg/m2sec). Equation (4.7.57) is of a similarform to Newton’s law of cooling, which defines the heat transfer coefficient hc.
Why we should not usea similar name and notation (e.g., mass transfer coefficient and hm) will become clear later. On a molarbasis, we define the mole transfer conductance of species 1, Gm1, by a corresponding relation,J1,s = G m1 Dx1 ;Dx1 = x1,s - x1,e(4.7.58)where Gm1 has units (kmol/m2sec).Low Mass Transfer Rate TheoryConsider, as an example, the evaporation of water into air, as shown in Figure 4.7.12. The water–airinterface might be the surface of a water reservoir, or the surface of a falling water film in a coolingtower or humidifier. In such situations the mass fraction of water vapor in the air is relatively small; thehighest value is at the s-surface, but even if the water temperature is as high as 50°C, the correspondingvalue of mH2O,s at 1 atm total pressure is only 0.077. From Equation 4.7.54 the driving potential fordiffusion of water vapor away from the interface is Dm1 = m1,s – m1,e, and is small compared to unity,even if the free-stream air is very dry such that m1,e .
0. We then say that the mass transfer rate is lowand the rate of evaporation of the water can be approximated as j1,s; for a surface area A,()m˙ 1 = m1,s ns + j1,s A . j1,s A kg sec© 1999 by CRC Press LLC(4.7.59)4-228Section 4FIGURE 4.7.12 Evaporation of water into an air flow.In contrast, if the water temperature approaches its boiling point, m1,s is no longer small, and of course,in the limit of Ts = TBP , m1,s = 1.
The resulting driving potential for diffusion Dm1 is then large, and wesay that the mass transfer rate is high. Then, the evaporation rate cannot be calculated from Equation4.7.59, as will be explained in the section on high mass transfer rate theory. For water evaporation intoair, the error incurred in using low mass transfer rate theory is approximately (1/2) Dm1, and a suitablecriterion for application of the theory to engineering problems is Dm1 < 0.1 or 0.2.A large range of engineering problems can be adequately analyzed assuming low mass transfer rates.These problems include cooling towers and humidifiers as mentioned above, gas absorbers for sparinglysoluble gases, and catalysis. In the case of catalysis, the net mass transfer rate is actually zero. Reactantsdiffuse toward the catalyst surface and the products diffuse away, but the catalyst only promotes thereaction and is not consumed.
On the other hand, problems that are characterized by high mass transferrates include condensation of steam containing a small amount of noncondensable gas, as occurs in mostpower plant condensers; combustion of volatile liquid hydrocarbon fuel droplets in diesel engines andoil-fired power plants, and ablation of phenolic-based heat shields on reentry vehicles.Dimensionless GroupsDimensional analysis of convective mass transfer yields a number of pertinent dimensionless groupsthat are, in general, analogous to dimensionless groups for convective heat transfer. The most importantgroups are as follows.1.
The Schmidt number, Sc12 = m/rD12, which is a properties group analogous to the Prandtl number.For gas mixtures, Sc12 = O(1), and for liquid solutions, Sc12 = O(100) to O(1000). There are notfluids for which Sc12 ! 1, as is the case of Prandtl number for liquid metals.2. The Sherwood number (or mass transfer Nusselt number).
Sh = gm1L/rD12 (= Gm1L/cD12) is adimensionless conductance.3. The mass transfer Stanton number Stm = gm1/rV (= Gm1/cV ) is an alternative dimensionlessconductance.As for convective heat transfer, forced convection flows are characterized by a Reynolds number, andnatural convection flows are characterized by a Grashof or Rayleigh number. In the case of Gr or Ra itis not possible to replace Dr by bDT since density differences can result from concentration differences(and both concentration and temperature differences for simultaneous heat and mass transfer problems).© 1999 by CRC Press LLC4-229Heat and Mass TransferAnalogy between Convective Heat and Mass TransferA close analogy exists between convective heat and convective mass transfer owing to the fact thatconduction and diffusion in a fluid are governed by physical laws of identical form, that is, Fourier’sand Fick’s laws, respectively.
As a result, in many circumstances the Sherwood or mass transfer Stantonnumber can be obtained in a simple manner from the Nusselt number or heat transfer Stanton numberfor the same flow conditions. Indeed, in most gas mixtures Sh and Stm are nearly equal to their heattransfer counterparts. For dilute mixtures and solutions and low mass transfer rates, the rule for exploitingthe analogy is simple: The Sherwood or Stanton number is obtained by replacing the Prandtl numberby the Schmidt number in the appropriate heat transfer correlation. For example, in the case of fullydeveloped turbulent flow in a smooth pipeNu D = 0.023Re 0D.8 Pr 0.4 ;Pr > 0.5(4.7.60a)Sh D = 0.023Re 0D.8 Sc 0.4 ;Sc > 0.5(4.7.60b)which for mass transfer becomesAlso, for natural convection from a heated horizontal surface facing upward,14Nu = 0.54(GrL Pr ) ;10 5 < GrL Pr < 2 ´ 10 713Nu = 0.14(GrL Pr ) ;2 ´ 10 7 < GrL Pr < 3 ´ 1010(laminar)(turbulent)(4.7.61a)(4.7.61b)which for isothermal mass transfer with rs < re become14Sh = 0.54(GrL Sc) ;10 5 < GrL Sc < 2 ´ 10 713Sh = 0.14(GrL Sc) ;2 ´ 10 7 < GrL Sc < 3 ´ 1010(laminar)(turbulent)(4.7.62a)(4.7.62b)With evaporation, the condition, rs < re will be met when the evaporating species has a smaller molecularweight than the ambient species, for example, water evaporating into air.
Mass transfer correlations canbe written down in a similar manner for almost all the heat transfer correlations given in Section 4.2.There are some exceptions: for example, there are no fluids with a Schmidt number much less thanunity, and thus there are no mass transfer correlations corresponding to those given for heat transfer toliquid metals with Pr ! 1. In most cases it is important for the wall boundary conditions to be ofanalogous form, for example, laminar flow in ducts. A uniform wall temperature corresponds to a uniformconcentration m1,s along the s-surface, whereas a uniform heat flux corresponds to a uniform diffusiveflux j1,s. In chemical engineering practice, the analogy between convective heat and mass transfer iswidely used in a form recommended by Chilton and Colburn in 1934, namely, Stm/St = (Sc/Pr)–2/3.
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