The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 25
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The processmust then be modeled by an appropriate set of continuity, momentum, energy, mass conservation, andstate equations, which need to be solved simultaneously.More-detailed information about melting and freezing can be found in the monograph by Alexiadesand Solomon (1993) and in the comprehensive reviews by Fukusako and Seki (1987) and Yao and Prusa(1989).Melting and Freezing of Pure MaterialsThorough mathematical treatment of melting and freezing is beyond the scope of this section, butexamination of the simplified one-dimensional case for a pure material and without flow effects providesimportant insights into the phenomena, identifies the key parameters, and allows analytical solutionsand thus qualitative predictive capability for at least this class of problems.In the freezing model, described in Figure 4.4.15, a liquid of infinite extent is to the right (x > 0)of the infinite surface at x = 0, initially at a temperature Ti higher than the fusion temperature Tf .
Attime t = 0 the liquid surface temperature at x = 0 is suddenly lowered to a temperature T0 < Tf , andmaintained at that temperature for t > 0. Consequently, the liquid starts to freeze at x = 0, and thefreezing interface (separating in Figure 4.4.15 the solid to its left from the liquid on its right) located© 1999 by CRC Press LLC4-107Heat and Mass TransferFIGURE 4.4.15 Freezing of semi-inifinite liquid with heat conduction in both phases.at the position x = X(t) moves gradually to the right (in the positive x direction). We note that in thisproblem heat is conducted in both phases.Assuming for simplification that heat transfer is by conduction only — although at least naturalconvection (Incropera and Viskanta, 1992) and sometimes forced convection and radiation also takeplace — the governing equations areIn the liquid: The transient heat conduction equation is¶Tl ( x, t )¶ 2 Tl ( x, t )= alin X (t ) < x < ¥ for t > 0¶t¶x 2Tl ( x, t ) = Tiin x > 0, at t = 0(4.4.29)(4.4.30)where al is the thermal diffusivity of the liquid, with the initial condition and the boundary conditionTl ( x ® ¥, t ) ® Tifor t > 0(4.4.31)In the solid: The transient heat conduction equation is¶Ts ( x, t )¶ 2 Ts ( x, t )= asin 0 < x < X (t ) for t > 0¶x 2¶t(4.4.32)where as is the thermal diffusivity of the solid, with the boundary conditionTs (0, t ) = T0for t > 0(4.4.33)The remaining boundary conditions are those of temperature continuity and heat balance at the solid–liquid phase-change interface X(t),Tl [ X (t )] = Ts [ X (t )] = T f© 1999 by CRC Press LLCfor t > 0(4.4.34)4-108Section 4dX (t )æ ¶T öæ ¶T öks ç s ÷for t > 0- kl ç l ÷= rhslè ¶x ø [ X (t )]è ¶x ø [ X (t )]dt(4.4.35)where ks and kl are the thermal conductivities of the solid and liquid, respectively, r is the density (hereit is assumed for simplicity to be the same for the liquid and solid), and hsl is the latent heat of fusion.The two terms on the left-hand side of Equation (4.4.35) thus represent the conductive heat flux awayfrom the phase-change interface, into the solid at left and the liquid at right, respectively.
Energyconservation at the interface requires that the sum of these fluxes leaving the interface be equal to theamount of heat generated due to the latent heat released there, represented by the term on the right-handside of the equation.The analytical solution of Equations (4.4.29) to (4.4.35) yields the temperature distributions in theliquid and solid phases,(Tl ( x, t ) = Ti - Ti - Tfæ x öerfcç÷è 2 a lt ø) erfc(las al)æ x öerfcç÷è 2 a st øTs ( x, t ) = T0 + Tf - T0erfcl()(4.4.36)(4.4.37)where erf and erfc are the error function and the complementary error function, respectively, and l isa constant, obtained from the solution of the equation2kel- lerf l ksa a l2a s Ti - Tfe( s l )l p=a l Tf - T0 erfc l a s a lSte s()(4.4.38)where Ste is the Stefan number (dimensionless), here defined for the solid asSte s º(cs Tf - T0)hsl(4.4.39)and cs is the specific heat of the solid.
Solutions of Equation (4.4.38) are available for some specificcases in several of the references, and can be obtained relatively easily by a variety of commonly usedsoftware packages.The solution of Equations (4.4.29) to (4.4.35) also gives an expression for the transient position ofthe freezing interface,12X (t ) = 2l(a s t )(4.4.40)where l is the solution of Equation 4.4.38, and thus the expression for the rate of freezing, i.e., thevelocity of the motion of the solid liquid interface, is© 1999 by CRC Press LLC4-109Heat and Mass Transfer12dX (t )= la s t 1 2dt(4.4.41)FIGURE 4.4.16 Melting of semi-infinite solid with conduction in the liquid phase only.For a simple one-dimensional melting example of an analytical solution for melting, consider the semiinfinite solid described in Figure 4.4.16, initially at the fusion temperature Tf .
For time t > 0 thetemperature of the surface (at x = 0) is raised to T0 > Tf , and the solid consequently starts to melt there.In this case the temperature in the solid remains constant, Ts = Tf , so the temperature distribution needsto be calculated only in the liquid phase. It is assumed that the liquid formed by melting remainsmotionless and in place. Very similarly to the above-described freezing case, the equations describingthis problem are the heat conduction equation¶ 2 Tl ( x, t )¶Tl ( x, t )= alin 0 < x < X (t ) for t > 0¶x 2¶t(4.4.42)with the initial conditionTl ( x, t ) = Tfin x > 0, at t = 0(4.4.43)the boundary conditionTl (0, t ) = T0for t > 0(4.4.44)and the liquid–solid interfacial temperature and heat flux continuity conditionsTl [ X (t )] = T ffor t > 0dX (t )æ ¶T ö- kl ç l ÷= rhlsfor t > 0è ¶x ø [ X (t )]dt© 1999 by CRC Press LLC(4.4.45)(4.4.46)4-110Section 4The analytical solution of this problem yields the temperature distribution in the liquid,æ x öerf ç÷è 2 a lt øTl ( x, t ) = T0 - T0 - Tferf l ¢()for t > 0(4.4.47)where l¢ is the solution of the equation2l ¢ e l ¢ erf (l ¢) =Ste lp(4.4.48)with Stel here defined for the liquid asSte l º(cl T0 - T f)hsl(4.4.49)l¢ as a function of Ste, for 0 £ Ste £ 5, is given in Figure 4.4.17.
The interface position is12X (t ) = 2l ¢(a l t )(4.4.50)FIGURE 4.4.17 The root l¢ of Equation 4.4.48.The solution of the freezing problem under similar conditions, i.e., of a semi-infinite liquid initiallyat temperature Tf where T(0,t) is abruptly reduced to T0 < Tf for t > 0, is identical to the above if everysubscript l is replaced by s and the latent heat hsl is replaced by –hsl.Example: The temperature of the vertical surface of a large volume of solid paraffin wax used for heatstorage, initially at the fusion temperature, Ti = Tf = 28°C, is suddenly raised to 58°C.
Any motion inthe melt may be neglected. How long would it take for the paraffin to solidify to a depth of 0.1 m?Given properties: al = (1.09) 10–7 m2/sec, rs = rl = 814 kg/m3, hsl = 241 kJ/kg, cl = 2.14 kJ/kg°C. Tofind the required time we use Equation (4.4.50), in which the value of l¢ needs to be determined. l¢ iscalculated from Equation (4.4.48), which requires the knowledge of Stel. From Equation (4.4.49)© 1999 by CRC Press LLC4-111Heat and Mass TransferSte l =(2.14kJ kg°C) (58°C - 28°C)= 0.266241.2 kJ kgThe solution of Equation (4.4.48) as a function of Stel is given in Figure 4.4.17, yielding l » 0.4.
Byusing Equation (4.4.50), the time of interest is calculated by2[ X(t )]t=4l2 a l=(0.1 m)2= (1.43)10 5 sec = 39.8 hr4(0.4) [(1.09)10 7 m 2 sec]2The axisymmetric energy equation in cylindrical coordinates, applicable to both the solid phase andimmobile liquid phase (with appropriate assignment of the properties) is¶T (r, t )1 ¶ æ k ¶T (r, t ) ö=ç÷ for t > 0¶trc ¶r è r ¶r ø(4.4.51)and the temperature and heat balance conditions at the solid–liquid phase-change interface r = R(t) areTl [ R(t )] = Ts [ R(t )] for t > 0(4.4.52)dR(t )æ ¶T öæ ¶T ö- kl ç l ÷= hslks ç s ÷è ¶r ø R(t )è ¶r ø R(t )dt(4.4.53)Because of the nature of the differential equations describing nonplanar and multidimensional geometries, analytical solutions are available for only a few cases, such as line heat sources in cylindricalcoordinate systems or point heat sources in spherical ones, which have very limited practical application.Other phase-change problems in nonplanar geometries, and in cases when the melt flows during phasechange, are solved by approximate and numerical methods (Yao and Prusa, 1989; Alexiades and Solomon,1993).Some Approximate SolutionsTwo prominent approximate methods used for the solution of melting and freezing problems are theintegral method and the quasi-static approximation.
The integral method is described in Goodman (1964),and only the quasi-static approximation is described here.To obtain rough estimates of melting and freezing processes quickly, in cases where heat transfertakes place in only one phase, it is assumed in this approximation that effects of sensible heat arenegligible relative to those of latent heat (Ste ® 0), thus eliminating the sensible-heat left-hand side ofthe energy equations (such as (4.4.29), (4.4.32), and (4.4.51)). This is a significant simplification, sincethe energy equation then becomes independent of time, and solutions to the steady-state heat conductionproblem are much easier to obtain. At the same time, the transient phase-change interface condition(such as Equations (4.4.35) and (4.4.53)) 5 is retained, allowing the estimation of the transient interfaceposition and velocity. This is hence a quasi-static approximation, and its use is shown below.We emphasize that these are just approximations, without full information on the effect of specificproblem conditions on the magnitude of the error incurred when using them.
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