6.9 Applications of SolidLiquid Phase Change

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1 6.9 Applications of Solid-Liquid Phase Change Latent Heat Thermal Energy Storage The theoretical model employed in this study is shown in Fig At the very beginning of the process (t =0), the tube, which has a radius of R0, is surrounded by a liquid phase change material with uniform temperature Tf>Tm. The temperature of the working fluid inside the tube is Ti and the convective heat transfer coefficient between the working fluid and the internal tube wall is hi. Both hi and Ti are kept constant throughout the process. The thickness of the tube is assumed to be very thin, so the thermal resistance of the tube wall can be neglected. 1

2 Figure 6.31 Solidification around a horizontal tube 2

3 The phase change material can be treated as if it were directly in contact with the coolant inside the tube. Liquid adjacent to the cooled surface will be solidified, and the temperature difference between the solid-liquid interface and the otherwise quiescent liquid will drive natural convection in the liquid region. 3

4 The liquid is Newtonian and Boussinesq as well as incompressible. The Prandtl number of the liquid phase change material is larger. The solid is homogeneous and isotropic. The liquid motion induced by volumetric variation during solidification is neglected, i.e., the density of the solid is equal to the density of the liquid. In addition, the phase change material s properties are constant in the liquid and solid regions. The solid-liquid interface is assumed to be a smooth cylinder concentric with the cooled tube. The effect of natural convection is restricted within the boundary layer and the bulk liquid has a uniform temperature, Tf. 4

5 The boundary layer equations of the problem can be written as follows: u v + = 0 x y (6.358) 2u ν + g β (T Tm )sin ϕ = 0 2 y (6.359) T (ut ) (vt ) + + =α t x y (6.360) f 2T y2 5

6 Since the Prandtl number of the liquid is much larger than unity, the inertia terms in the momentum equation have been neglected. These equations were solved by an integral method. Assuming a polynomial profile, the temperature and velocity profiles inside the boundary layer of thickness are expressed as y T = T f (T f Tm ) 1 δ y y u = U 1 δ δ 2 (6.361) 2 (6.362) 6

7 g β sin ϕ U= (T f Tm )δ 3ν 1 δ dr 1 g β sin ϕ 3δ 2α + (T f Tm ) t dt 90 ν R ϕ δ (6.363) 2 f = 0 (6.364) Heat transfer in the solidified layer is dominated by conduction. The governing equation of the solid layer and corresponding boundary conditions are as follows: 1 T 1 T R0 < r < R t > 0 (6.365) r = r r r α s t 7

8 T ks = hi (T Ti ) r = R0 r Ts (r, t ) = Tm Ts ks r T kf y r= R y= 0 r= R t> 0 dr = ρ hsl dt t> 0 (6.366) (6.367) r = R t > 0 (6.368) Equations (6.364) (6.368) can be nondimensionalized and solved using integral approximate method. 8

9 When Bi, it corresponds to boundary conditions of the first kind, i.e., the tube wall temperature is and is kept steady throughout the process. Wang et al. (1991) experimentally investigated the solidification process around a horizontal cooled tube. A comparison of the predicted solidification rate, V / V0 = ( R / R0 ) 2 and the experimental results is shown in Fig When Ra = 0, i.e., no superheat exists in the liquid region or the solidification process is dominated by conduction, the predicted value is 18% lower than the experimental data. 9

10 Figure 6.32 Comparison of predicted solidification rate with experiments. 10

11 During the conduction dominated freezing process, the front of the freezing layer is a dendritic layer, as indicated by Wang et al. (1991). Therefore, it is believed that the solid-liquid interface is extended by the dentric layer. As the Rayleigh number increases, natural convection occurs in the liquid region, and the solid-liquid interface becomes smooth because of the natural motion of the liquid. When Ra = 1.8x105, the predicted value is only 8% lower than the experimental data, so the agreement is satisfactory. The effect of Biot number on the wall temperature and solidification rate was also studied by Zhang et al. (1997). 11

12 6.9.2 Heat Pipe Startup from Frozen State When a high-temperature heat pipe starts from room temperature, the working fluid within the wick structure is in the frozen state. During startup from the frozen state, large thermal gradients generate significant internal stresses within the pipe wall. These stresses may severely shorten the life of the heat pipe. As a result, information on the stresses is needed for design purposes, and it is necessary to determine the temperature distribution during the transient frozen startup period. 12

13 Figure 6.33 Evolution of heat pipe startup process from the frozen state (not to scale; Cao and Faghri, 1993a). 13

14 The startup process of a liquid metal heat pipe from the frozen state may be divided into several periods for convenience of analysis: The heat pipe wick is initially in the frozen state. The heat pipe core can be considered to be completely evacuated before startup. The heat pipe in its initial state is illustrated in Fig (period 1). Power is applied to the evaporator section of the heat pipe. Heat conduction through the wall and melting of the working fluid in the porous wick take place. However, the liquid-solid melting interface has not yet reached the wick vapor boundary, and the heat pipe core is still evacuated because no evaporation has taken place in the evaporator sec tion (Fig. 6.33, period 2). 14

15 In the evaporator section, the working fluid in the wick is completely melted, and evaporation takes place at the liquid vapor interface. The vapor pressure is so low that the vapor flow is in a rarefied or free molecular condition. Also, some working fluid in the wick is still in the solid state in the adiabatic and condenser sections of the heat pipe (Fig. 6.33, period 3). As evaporation continues, the amount of vapor accumulated in the core is large enough in the evaporator that a continuum flow is established there. Near the condenser end of the heat pipe, however, the vapor flow is still in the rarefied or free molecular regime. This period may be called the intermediate period (Fig. 6.33, period 4). The working fluid in the wick is completely melted and continuum flow is established over the entire heat pipe length. The heat pipe continues to operate and gradually reaches steady state (Fig. 6.33, period 5). 15

16 For the porous wick structure, the change of phase of the frozen working fluid must be considered. The wick structure is assumed to be isotropic and homogeneous. The temperature transforming model (Cao and Faghri, 1990a) is employed to study melting of working fluid during the heat pipe startup. The energy equation in the wick structure (a porous medium) is T 1 T ( ρ eff ct ) = keff + keff r t z z r r r ( ρ eff b) (6.369) t 16

17 The coefficients c, b, and keff in eq. (6.369) are cse c(t ) = cm + hsl /(2 T ) c le T < Tm T Tm T < T < Tm + T (6.370) T > Tm + T T < Tm T cse ( T Tm ) b(t ) = cme T + hsl / 2 ( cme + hsl /(2 T ) ) Tm Tm T < T < Tm + T c T + h c T T > Tm + T sl le m se (6.371) 17

18 keff kse = kse + (kle kse )(T Tm + T ) /(2 T ) k le T < Tm T Tm T < T < Tm + T T > Tm + T where ρ eff = ϕ ρ l + (1 ϕ ) ρ ws, cse = ϕ cs + (1 ϕ )cws, cle = ϕ cl + (1 ϕ )cws and cme = 0.5(cse + cle ). The values of k se and kle depend on the particular heat pipe wick structure. 18

19 (6.372) Figure 6.34 Outer wall temperature compared with experimental data for case 11a-11f 19

20 6.9.3 Thermal Protection from Intense Localized Heating Using PCM In some applications, a surface is hit by a moving high intensity heat source, as shown in Fig The major concern here is how to protect the surface from being burned out by the moving heat flux. This is indeed a concern in laser thermal threats and reentry situations. Ablation and heat pipe technologies are usually sources of protection for surfaces in danger of being burned out by a high heat flux. 20

21 Figure 6.35 Schematic of localized heating 21

22 Phase-change materials (PCMs) have a large melting heat, so then offer an efficient means of absorbing the heat energy while the materials are subjected to heat input, and releasing it afterward at a relatively constant temperature. Another alternative, as shown in Fig. 6.36, advantageously incorporates the merits of the above technologies as protection for surfaces attacked by a high heat flux. There, the incoming heat input moves along the surface with speed U and the heat is conducted through the outside wall to the PCM. The PCM beneath the surface melts and absorbs a large amount of the incoming heat. Because of the large melting latent heat of the PCM and the constant melting temperature Tm, the peak wall temperature will be maintained at a temperature moderately higher than Tm. With a low or moderate Tm, the reduction of the peak wall temperature is evident. 22

23 using PCM 23

24 The dividing sheet, the soft insulating material and the supporting plate may be used to prevent the PCM from separating from the surface wall during the melting process, and to prevent the incoming heat from being conducted into the cabin. An analysis in a fixed coordinate system is difficult for this problem. It is convenient to study it in a moving coordinate system where the origin is fixed at the heated spot. For an imaginary observer riding along with the incoming heat beam, the outside wall and PCM will travel by at the same speed, U. The energy equation in the Cartesian coordinate system for this problem is (Cao and Faghri, 1990b) h h T ρ ρu = k t x x x T k + y y T + k z z (6.373) where the second term on the left-hand side is a convection term, while the terms on the right-hand side are diffusive terms. 24

25 25

26 ( ρ h) 1 (rvr ρ h) 1 (vθ ρ h) + + t r r r θ 1 (Γ h) 1 1 (Γ h ) (Γ h ) = r + + r x r r θ r θ z z 1 S 1 1 S 2S + r r x r r θ r θ z (6.374) where vr = U cosθ and vθ = U sin θ are the velocity components in the cylindrical coordinate system, and Γ and S for the PCM can be obtained from eqs. (6.224) and (6.225). For the wall, no phase change occurs, and therefore h = cwt, Γ = k w / cw and S=0. Equation (6.375) with corresponding boundary conditions is solved using a finite volume method. The resulting algebraic equations are solved with the Gauss-Siedel method. 26

27 6.9.4 Microwave Thawing of Food and Biological Materials Food thawing processes are becoming more important because the demand for frozen food products continuously increases. The interest in microwave thawing is stimulated by its ability to avoid some of the disadvantages associated with conventional thawing: long processing times, large space requirements, microbial problems, chemical deterioration, drip loss, high fresh water consumption. 27

28 During microwave heating, the transient temperature distribution within the material is determined by the internal heat generation attributed to dissipation of electrical energy from microwave radiation, and the heat transfer by conduction and convection. The moisture transfer within the material and evaporation at the surface can also influence the temperature profile. Modeling microwave thawing heat transfer is difficult due to the effects of highly nonlinear phenomena, such as the rate of energy dissipation and the energy distribution within the material. These phenomena are governed by the thermal, electrical, and physical properties of the material, and vary with temperature during the microwave thawing process. 28

29 Microwaves are electromagnetic waves with wavelengths ranging from 1 cm to 1 m. The heat generation within the microwave-irradiated frozen materials results from dipole excitation and ion migration. There may also be other mechanisms of interaction between the material and the electromagnetic field which cause the dissipation and heating effects of microwave energy. The equations governing the absorption of microwave radiation by a conducting material are the Maxwell equations of electromagnetic waves. In the time-varying case, the different field equations are coupled, since a changing magnetic flux induces an electric field and a time-varying electric flux induces a magnetic field. 29

30 B + E= 0 t ε (6.375) E + σ EΗ = t (6.376) (ε Η ) = 0 (6.377) B= 0 (6.378) B= µh (6.379) 30

31 where E is the electric field strength, B is the magnetic flux density, H is the magnetic field intensity, and are magnetic permeability, electric permissivity and conductivity of the material, respectively. Maxwell s first equation is Faraday s law of induction, which states that a time variation of flux density is accompanied by the curl of the electrical field. Maxwell s second equation refers to Ampere s law, whose integral form states that the magnetic field over a closed path is equal to the enclosed current. Equations (6.377) and (6.378) are known respectively as Gauss magnetic law and Gauss electric law, while eq. (6.379) is the constitutive relation for a simple medium. 31

32 For the microwave thawing process, the enthalpy formulation including a heat generation term is given by h = (k T ) + q t where h is enthalpy and k is the thermal conductivity. The heat generation term accounts for the conversion of microwave energy to heat energy. Its relationship to the electrical field intensity E at that location can be derived from eqs. (6.375) (6.379): q = 2π f ε 0ε E (6.380) 2 (6.381) where the magnetic losses in the food material have been ignored. 32

33 The microwave frequency f is 2450 MHz for most domestic ovens and 915 MHz for some industrial equipment. ε0 is the dielectric constant of free space, and ε is the loss factor for the dielectric being heated. The basic assumptions concerning the heat generation term include the following: a) b) c) d) The microwaves are planar and propagate perpendicularly to the material. The microwave field at the material surface is uniform. Microwave energy entering from different sides is considered separately, and decreases exponentially from the surface into the material. The total amount of heat absorbed by a sample during a heating cycle is constant. 33

34 Generally, it is difficult to either measure or calculate the strength of the electrical field inside the material from Maxwell s equations, so an exponential variation in the field intensity from the surface of the material into its interior is generally assumed (Datta, 1990): 2 Ex = E02 exp( 2α x) Substituting eq. (6.382) into eq. (6.831), we have q x = q x 0 exp( 2α x) (6.382) (6.383) where q x 0 is the rate of heat generation corresponding to an electric field E0 at the surface of the material, and x is the distance from the surface into the material. 34

35 The heat generation term can be expressed in general as q = q x 0 exp( 2α x) + q y 0 exp( 2α y ) + q z 0 exp( 2α z ) (6.384) where q x 0, q y 0 and q z 0 are the rates of heat generation corresponding to an electric field E0 at the surfaces of the material in the different coordinate directions. The attenuation coefficient α is calculated from the dielectric constant ε, the dielectric loss factor ε, and the wavelength λ0 in vacuum, which determines the energy distribution within the material: 2π ε {[1 + (ε / ε ) 2 ]1/ 2 1} α = (6.385) λ0 2 35

36 In the microwave thawing experiments conducted by Zeng (1991), frozen cylinders (D H = 2.0 cm 2.0 cm) of Tylose were thawed. Aluminum foil covered both ends, so microwave energy entered only in the radial direction. Symmetry about the two central axes of the cylinder is assumed. Figure 6.38 shows the comparison of the predicted and experimental temperature histories for a D H = 2.0 cm 2.0 cm cylinder (Case 3a). For the time interval of 0 t < 25 s, predicted temperature curve is rather flat since there is no microwave radiation during off' period. For 25 s t < 30 s, the microwave oven radiated the Tylose sample. The incident microwave energy was dissipated into heat energy within the sample, which caused the sharp temperature rise. For 30 s t < 170 s, the temperature history indicates that mushy region phase change was occurring. At t 170 s, the thawing process was completed, and the temperature rose sharply. 36

37 history during microwave thawing of a Tylose cylinder (P=64 W). 37

38 6.9.5 Laser Drilling Melting and vaporization occur Assuming vaporization has started and the liquid-vapor interface is formed, the geometric shape of the liquidvapor interface and solid-liquid interface are respectively expressed as f1 ( z, r, t ) = z s1 (r, t ) = 0 (6.386) f 2 ( z, r, t ) = z s2 ( r, t ) = 0 (6.387) The temperatures at the two interfaces satisfy T = Tsat, z = s1 (r, t ) (6.388) (6.389) f 2 ( z, r, t ) = z s2 ( r, t ) = 0 38

39 Figure 6.39 Physical model of laser drilling process. 39

40 The energy balance at the liquid-vapor and solid-liquid interfaces can be used to obtain the locations of the two interfaces 2 r T T s1 s m 1 (6.390) ρ hl v = α a I 0 e R kl sat 1 +, z = s1 (r, t ) 2 2 t s2 s1 T Tm s T ρ hsl 2 = ks s + kl sat 1+ t z s2 s1 (6.391) Ganesh et al derived an equation of saturation temperature by a using gas dynamic model hlv 1 Tl γ +1 1 (6.392) γ Rg Tsat I abs kl = p0 exp γ hlv r 2 s2, z = s2 (r, t ) r n1 Rg Tsat,0 Tsat The average material removal rate in the laser drilling process, which was obtained by the following integration 2π ρ (6.393) MR = s ( r, t ) rdr tp 0 1 p 40

41 6.9.6 Selective Last Sintering (SLS) of Metal Powders SLS of metal powder involves fabrication of near full density objects from powder via melting induced by a directed laser beam (generally CO2 or YAG) and resolidification In order to overcome balling phenomena caused by surface tension, a powder mixture containing two powders with significantly different melting points can be used in the SLS process 41

42 Figure 6.41 Physical model of 3-D sintering of two-component metal powder 42

43 The liquid velocity, vl, must satisfy the continuity equation ϕ l &0 (6.394) The solid low melting point powder vanishes at the same rate ϕ s (ϕ s ws ) + = Φ& 0L (6.395) t z The continuity equation for the high melting point ϕ H (ϕ H ws ) material is + = 0 (6.396) t z The solid velocity can be determined by integrating eq. (3) z> s 0 (6.367) ws = ϕ s,i s t + (ϕ l v l ) = Φ 1 ε t L z< s 43

44 The liquid flow occurs in three directions, and the velocities can be determined using the Darcy's law KK rl (6.368) v wk= ( p + ρ gk ) l s ϕ lµ c l The temperature transforming model using a fixed grid method is employed to describe melting and resolidification in the powder bed and the energy equation is { ϕ ρ c + (ϕ + ϕ ) ρ c T } + (ϕ v ρ c T ) (6.369) H H ph l s L pl l l L t ws (ϕ H ρ H c ph + ϕ s ρ L c pl )T = (k T ) + z ρ L [ (ϕ l + ϕ s )b ] + (ϕ l v l b) + ϕ w b ( s s ) t z pl 44

45 Figure 6.42 Comparison of cross-section area obtained by numerical simulation and experiment (Ni=0.0749, Ub=0.124) 45

Analysis of Forced Convection Heat Transfer in Microencapsulated Phase Change Material Suspensions

JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER Vol. 9, No. 4, October-December 1995 Analysis of Forced Convection Heat Transfer in Microencapsulated Phase Change Material Suspensions Yuwen Zhangh and Amir