15
cm9 2509/91 $3.00 + 0.00 0 1991 Pergamon Press plc Chemical Engineering Scwnce, Vol. 46, No. 7. pp. 1573 1587,199l Pnnted in Great Britain. CYCLIC MELTING AND FREEZING M. HASAN’, A. S. MUJUMDAR and M. E. WEBER* Department of Chemical Engineering, McGill University, 3480 University Street, Montreal, Canada H3A 2A7 (First receiued 26 October 1989; accepted for publication in revised form 3 December 1990) Abstract-This paper reports the results of a numerical and experimental investigation of one-dimensional conduction-controlled cyclic melting and freezing. The phase change material (PCM) was a plane slab, one surface of which was subjected to alternating periods of constant temperature above and below the fusion temperature while the other surface was insulated. Due to the cyclic change of temperature, multiple solid and liquid regions develop within the material. A front-tracking, self-starting finite-difference scheme was used to solve the resulting multiple moving boundary problem. Numerical computations were carried out from a uniform initial condition until a periodic steady state was attained. In this periodic steady state all temperatures and solid/liquid interface movements were identical from one cycle to the next. Experiments were performed on n-octadecane in a thin horizontal cell with heat transfer from above. The numerical results were in good agreement with these conduction-controlled experiments. 1. INTRODUCTION Latent heat storage systems have attracted con- siderable attention, particularly for the utilization of solar energy for low-temperature applications. The phase change commonly utilized for thermal energy storage involves the liquid and solid phases and the energy is stored in the form of the latent heat of fusion. Energy is stored during melting and recovered during freezing. The mathematical description of a phase change storage module requires the analysis of cycles caused by alternate melting and freezing of the phase change material (PCM). Due to cyclic temperature changes around the fusion temperature the PCM may contain multiple moving boundaries which separate alternating liquid and solid phases. During cycling, liquid and solid phases grow and shrink and the tracking of these offers challenges compared to single front problems. Despite the significance of cyclic melting and freezing, only three relevant studies have been published. Bransier (1979) considered two PCM capsules: (1) a slab module insulated on one side and (2) a concentric cylindrical module with the PCM between two cylin- ders and the heat transfer fluid (HTF) Bowing through the inner cylinder, while the outer cylinder was in- sulated. The temperature of the HTF varied sinus- oidally about the fusion temperature. Heat transfer between the HTF and PCM took place with a con- stant heat transfer coefficient. Bransier found that a maximum of two interfaces could co-exist during cyclic melting and freezing. It is not clear whether a periodic steady-state (PSS) condition was achieved which was independent of the initial thermal condi- tion of the PCM. Transient solutions prior to the ‘Present address: Department of Mining & Metallurgical Engineering. ‘Author to whom correspondence should be addressed. attainment of the PSS condition were not given and no experimental results were reported. Kalhori and Ramadhyani (1985), while studying the melting characteristics of vertical cylinders embedded in a 99% pure n-ecosane PCM, carried out a single cyclic melting/freezing experiment They used cooling water with a temperature 123°C below’the fusion temperature during each freezing period of 15.25 h. During each 7.25 h long melting period, hot water with a temperature ll.l”C higher than the fusion temperature was used. Identical temperature distribu- tions recurred from cycle to cycle at corresponding times after 12 cycles. They observed multiple layers of solid and liquid during the initial transient period consisting of the first three or four cycles. At the PSS only a narrow annular region of PCM around the cylindrical heat source/sink underwent phase change. This annular region was surrounded by a mass of solid that never melted. Jariwala et al. (1987) carried out an experimental study of the cyclic thermal performance of a latent heat storage unit which consisted of a cylindrical helically coiled copper tube immersed in a cylindrical tank of a commercial paraffin wax. Hot and cold water was circulated alternately inside the coiled tube. Irrespective of the durations of the individual melting or freezing periods, a PSS, which was independent of the initial thermal condition of the PCM, was estab- lished by the third cycle. Jariwala et al. also developed a quasi-steady one-dimensional model by assuming conductive heat transfer during the freezing cycle and convective heat transfer during the melting cycle. This study considers cyclic phase change heat transfer for a plane PCM slab of finite thickness, one side of which acts as the heat exchanging surface while the other side is insulated. The heat transfer surface is exposed in a cyclic fashion to temperatures above and below the fusion temperature. The results of com- putations for a conduction model are presented along 1573

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Page 1: Cyclic melting and freezing

cm9 2509/91 $3.00 + 0.00 0 1991 Pergamon Press plc

Chemical Engineering Scwnce, Vol. 46, No. 7. pp. 1573 1587,199l Pnnted in Great Britain.

CYCLIC MELTING AND FREEZING

M. HASAN’, A. S. MUJUMDAR and M. E. WEBER* Department of Chemical Engineering, McGill University, 3480 University Street, Montreal,

Canada H3A 2A7

(First receiued 26 October 1989; accepted for publication in revised form 3 December 1990)

Abstract-This paper reports the results of a numerical and experimental investigation of one-dimensional conduction-controlled cyclic melting and freezing. The phase change material (PCM) was a plane slab, one surface of which was subjected to alternating periods of constant temperature above and below the fusion temperature while the other surface was insulated. Due to the cyclic change of temperature, multiple solid

and liquid regions develop within the material. A front-tracking, self-starting finite-difference scheme was used to solve the resulting multiple moving boundary problem. Numerical computations were carried out from a uniform initial condition until a periodic steady state was attained. In this periodic steady state all temperatures and solid/liquid interface movements were identical from one cycle to the next. Experiments were performed on n-octadecane in a thin horizontal cell with heat transfer from above. The numerical results were in good agreement with these conduction-controlled experiments.

1. INTRODUCTION

Latent heat storage systems have attracted con-

siderable attention, particularly for the utilization of solar energy for low-temperature applications. The phase change commonly utilized for thermal energy storage involves the liquid and solid phases and the energy is stored in the form of the latent heat of fusion. Energy is stored during melting and recovered during freezing.

The mathematical description of a phase change storage module requires the analysis of cycles caused by alternate melting and freezing of the phase change material (PCM). Due to cyclic temperature changes around the fusion temperature the PCM may contain multiple moving boundaries which separate alternating liquid and solid phases. During cycling, liquid and solid phases grow and shrink and the tracking of these offers challenges compared to single front problems. Despite the significance of cyclic melting and freezing, only three relevant studies have been published.

Bransier (1979) considered two PCM capsules: (1) a slab module insulated on one side and (2) a concentric cylindrical module with the PCM between two cylin- ders and the heat transfer fluid (HTF) Bowing through the inner cylinder, while the outer cylinder was in- sulated. The temperature of the HTF varied sinus- oidally about the fusion temperature. Heat transfer between the HTF and PCM took place with a con- stant heat transfer coefficient. Bransier found that a maximum of two interfaces could co-exist during cyclic melting and freezing. It is not clear whether a periodic steady-state (PSS) condition was achieved which was independent of the initial thermal condi- tion of the PCM. Transient solutions prior to the

‘Present address: Department of Mining & Metallurgical Engineering.

‘Author to whom correspondence should be addressed.

attainment of the PSS condition were not given and no experimental results were reported.

Kalhori and Ramadhyani (1985), while studying the melting characteristics of vertical cylinders embedded in a 99% pure n-ecosane PCM, carried out a single cyclic melting/freezing experiment They used cooling water with a temperature 123°C below’the fusion temperature during each freezing period of 15.25 h. During each 7.25 h long melting period, hot water with a temperature ll.l”C higher than the fusion temperature was used. Identical temperature distribu- tions recurred from cycle to cycle at corresponding times after 12 cycles. They observed multiple layers of solid and liquid during the initial transient period

consisting of the first three or four cycles. At the PSS only a narrow annular region of PCM around the cylindrical heat source/sink underwent phase change. This annular region was surrounded by a mass of solid that never melted.

Jariwala et al. (1987) carried out an experimental study of the cyclic thermal performance of a latent heat storage unit which consisted of a cylindrical helically coiled copper tube immersed in a cylindrical tank of a commercial paraffin wax. Hot and cold water was circulated alternately inside the coiled tube. Irrespective of the durations of the individual melting or freezing periods, a PSS, which was independent of the initial thermal condition of the PCM, was estab- lished by the third cycle. Jariwala et al. also developed a quasi-steady one-dimensional model by assuming conductive heat transfer during the freezing cycle and convective heat transfer during the melting cycle.

This study considers cyclic phase change heat transfer for a plane PCM slab of finite thickness, one side of which acts as the heat exchanging surface while the other side is insulated. The heat transfer surface is exposed in a cyclic fashion to temperatures above and below the fusion temperature. The results of com- putations for a conduction model are presented along

1573

Page 2: Cyclic melting and freezing

1574 M. HASAN rt al.

with experimental data for cyclic melting and freezing of n-octadecane.

2. MODEL FORMULATION

The physical system is a large plane slab of phase change material; one surface (x = 0) exchanges heat while the opposite surface (x = L) is insulated (see Fig. 1). The temperature of the surface at x = 0 is cycled above and below the fusion temperature of the PCM. The positions of the interfaces are denoted by yi. The following assumptions are made:

(1) (2)

(3)

(4)

(5)

The PCM is isotropic and homogeneous. Heat transfer in the PCM is by one-dimensional conduction. The densities of the solid and liquid are equal and constant. Other physical properties of the solid and liquid phases are constant, but not necessarily equal. The phase change takes place at a discrete fusion temperature, r,. The interfaces between solid and liquid are planar.

Initially the sIab is a single phase liquid or solid with a uniform temperature of T,,. The heat transfer surface is exposed to a rectangular wave temperature variation around the fusion temperature, 7’,,, (see Fig. 2). The surface temperature is Twr for the freezing period of duration tl. and T,,,, for the melting period of duration t,. A cycle consists of a melting period followed by a freezing period. The thermal swing is defined as the difference between the surface temper- ature and the fusion temperature, i.e. T,, ~ T, dur- ing the melting period and Tws - T,,, during the freezing period.

If the PCM is initially liquid, the temperature

variation at x = 0 is

T,= 7,,, t<O (1)

Tw = Twr> for nt, < t < nt, + t/ n = 0, 1, 2, .

Tw = Tw,,, for nt, + t, < t -c fn + l)t, (2)

where the cycle time, t,, is

t, = t, + Cf. (3)

After a large number of identical cycles, a PSS will be achieved. In this state the time variation of the tem- perature at every point and the time variations of the positions of the interfaces are repeated during each cycle.

The following development considers the first freezing period of an initially liquid PCM. The energy equations for the solid and liquid layers are:

a27-11 3T” 0~ 2 = 2 s ax2 at

for 0 < x G yr(t), t > 0 (4)

for yr(t) < x G L, t > 0 (5)

the superscript 1 refers to the shrinking region (liquid) and the superscript Ii refers the growing (solid) re- gion, while the subscripts “s” and “I” denote the solid and liquid phases, respectively. The thermal di&siv- ities of the solid and liquid are E, and c+, respectively. The position of the first interface separating the solid and liquid is denoted by y,(t). The following are the initial, boundary and moving boundary conditions:

Initial conditions The initial temperature in the liquid is uniform, i.e.

at t = 0, T:(x. 0) = T;.. 0 i x G 15. (6)

THERMAL ENERGY

STORAGE AND

DISCHARGE

tl

Fig. 1. Physical model and coordinate system.

Page 3: Cyclic melting and freezing

(a)

Cyclic melting and freezing 1575

Fig. 2. Thermal boundary condition at the heat transfer surface; (a) equal periods and equal swings; (b) equal periods and unequal swings; (c) equal swings and unequal periods.

(7)

The initial position of the first-freezing front is

att=O, y,(r)=O.

Boundary conditions Heat transfer surface: For the first-freezing period

there is a step change in temperature.

atx=O, T,“(O,t)=T,,, t>O. (8)

Insulated boundary condition:

at x = L, ar: ~ = 0, ax t > 0. (9)

Moving boundary conditions From the continuity of the temperature at the

interface:

atx=y,(t),T:‘(x,r)=T{(x,t)=T,,t>O. (10)

The Stefan condition, which results from the energy balance at the interface, is:

at x = yr(t), k,s - k,Z= @%, t > 0 (11)

where ), is the latent heat of fusion and k, and k, are the thermal conductivities of the solid and liquid, respectively.

In terms of the following dimensionless variables

X = x/L, r = a,c/L2, Qjl = c, (Tf’ - T,J/A

0: = c,(T.! - T,,,)/A, S, = y,(t)/L (12)

the equations are:

Solid region:

Liquid region:

Initial conditions:

e:(X, 0) = c,(T;:, - T,)/A

S,(O) = 0.

Boundary conditions:

0f’(O, i) = c,(T,, - T,,,)/A = 8,

as: -=0 atX=l, t>O. 8X

Moving boundary conditions:

Of’(X, T) = 0:(X, r) = 0, = 0, at X = S,(T), 5 > 0

(19)

g _ (k,,k,) !!& = 2, at X = S,(r), r > 0.

(20)

Solution methodology A front-tracking, self-starting explicit difference

scheme was developed to solve eqs (13)-(20). A finite- difference grid was constructed with spatial incre- ments of equal size and time steps of equal size. Since

Page 4: Cyclic melting and freezing

1576 M. HASAN et a!.

were not generally located at nodes. To illustrate briefly the solution methodology, a solidification period is de- scribed below.

Figure 3 illustrates the finite-difference (FD) grid with “N + 1” grid points in the X-coordinate. Let S,(r) be the location of the solid/liquid interface measured from the heat transfer surface (X = 0). The node in the solid region closest to the interface is “r” and node ‘T + 1’ is in the liquid region. The nodal spacing AX is ljiy while 6X,(z) is the time-varying distance between node “r” and S, (7). At this particular time, 7, the location of the solid/liquid interface, S,(z), is known, as well as the nodal temperatures, 0:’ (i = 1, . , r) and Qf(i = T + 1, . . , N + 1). The following finite-difference expression is applicable to the entire grid except at the nodes, 1, P, Y + 1, and N + 1.

where

CC = 1 when 0 = 0:’ (solid region, i < r)

a = (a,/a,) when 19 = Si (liquid region, i > r).

The wall temperature Q1 is known from the boundary condition. The temperature of the insulated boundary is determined by putting i = N + 1 in eq. (21) and setting ON+Z,j = ON,j (from &?/aXl,+,,j = 0): thus

6 ..,,j*,=e,+,;j[1-2~]+2~e,.j.

(22)

Near the interface The finite-difference expressions for the first and

second derivatives of 0.:’ and 0: at the solid/liquid interface, written by using Taylor’s series expansion, are

aeil ,I=_5 = I [ (1 + X1)% 5,C’,

s ax s, 510 + i’l) + (1 + 5,)

(24)

ae: m:==W = 1 [ (2 -w,, (3 - 25,)

(l-e,) 1+’ - (1 - 5,)(2 - 5i)Hm

(1 + 51),, 1 L (25) (2 - 51) ‘+’ AX

and

cm: -I _[ f3

8x2 s, ,20:‘;,, + (1 - si2 - 5,)

e:+, 2 (l-tT,) ihx)z 1

(261

where

so that

51 (z) = 6x1 WAX (27)

0 < Cl(t) < 1. (28)

Equations (23) and (24) are singular at (I = 0 as eqs (25) and (26) are at <I = 1. Since rr(r) can take any value from 0 to 1, the following procedures were adopted to avoid singularity during the computation.

When tl(z) lies in the region 0 < cl(z) d 0.5, the temperature distribution of the solid phase near the interface S, is represented by a quadratic profile with time-dependent coefficients. In the vicinity of x = S,(T):

edr=e,+A,(X-sS,)+B,(X--S,)z (29)

where B, = 0 and

A,=m:=g 1 F20”

and B,=-‘” 2 2x2 s,’

(30) s,

In order to find ~9~‘. ,,,+l andOf+,,jc, twomorebound- ary conditions at the interface S1 are derived by using the fact that when the interface temperature is con- stant throughout the process, the substantial derivat- ive of temperature with time at the interface, dQ/dt Is,, equals zero. Thus at X = S,(t)

5x2 6%

Fig. 3. Location of interfaces and nodes.

Page 5: Cyclic melting and freezing

Cyclic melting and freezing 1577

and

Now applying the field equations, eqs (13) and (14) at X = S,(z) and using eqs (33) and (34), the following equations are derived:

(33)

(34)

Using eqs (20) and (301, eq. (33) can be written as:

2B, + A, [A, - (k,/k,)m:] = 0. (35)

Employing eq. (29) to find the temperature at the node “r - 1” in the solid phase, gives

8” r l,j+l = 0, - A,(1 + <lWX + B,U + t,)‘AX*.

(34)

Note that 0,” 1, j+ 1 is known after solving eq. (21) for the solid phase. The values of A, and B, are obtained by solving eqs (35) and (36).

Using eq. (29) to find the temperature at the node “r” in the solid phase gives

@r. r,j+1 = 8, + [ - A, + B,6X,]6X,. (37)

Since 6X, as well as A, and B, are known from the previous time step, eq. (37) permits evaluation of 8r at the advanced time step. From eqs (25), (26) and (34), the temperature of the grid point close to the liquid side of the interface, f$+ 1, j+ 1 can be obtained

The interface motion, dS,/dr, which is determined a posteriori, is calculated from

Si+ 1 = Sl; + Az(A, - (k,/k,)m;). (39)

When tl(~) lies in the region 0.5 < tl(z) < 1, the singularities in eqs (25) and (26) are avoided by representing the temperature in the liquid phase near the interface by a quadratic profile with time-depend- ent coefficients similar to that used in the solid phase when tl(~) < 0.5.

Starting solution

PCM, a similar strategy to the one described above was used. Details of the numerical algorithm, the validation of the code and the grid independency check are available elsewhere [Hasan, 19881. The numerical results presented here are for a time step (Ar) of 5.0 x 10m4 and spatial step size (AX) of 0.05. All calculations were carried out on an AMDAHL 5850 computer in double precision. A typical simu- lation run of 6 melting/freezing cycles required about 60 s CPU time. The numerical results showed no instabilities and were confirmed to be independent of the step sizes selected.

The phase change problem is indeterminate at From this numerical scheme the instantaneous tem- z = 0. To initiate computation, S,(T) should have a perature distribution in the PCM and the positions of finite value, i.e. a solid and a liquid region should exist the interfaces are known. The magnitude of the cumu- initially. Consider solidification with the boundary at lative energy stored or recovered per unit area, Q, was X = 0 subjected to a constant temperature 8,. At the calculated as a function of time for each period. This end of the first time step AZ, the solid/liquid interface calculation was made by computing the energy con- moves to a distance SX, from the boundary. By tent of the PCM at each time increment using the choosing an appropriately small time step, 6X, solid PCM at its fusion temperature as the reference < AX (see Fig. 3). Assuming that the temperature state. The energy content at the beginning of the

distribution within the solid is linear yields

aoIl *I=S = UL - Rn) s ax s1 6s

(40)

Equation (41) is obtained by setting r = 1 in eq. (25). At z = 0, &$/dXI,, is approximated by

(3 - 25,)

(l - 51)e0 (42) (2 - 511 11 1

where S: and 0: are the initial temperatures of nodes 2 and 3, respectively. Substituting, the expressions for #‘/ax Is, and t?lI~/aX Is, into the energy balance at the interface [eq. (20)] gives

dS, 6X, ’ _=_ dz AT =E

(‘,; Orn) _ (k,/k,) 1

k&; 1

(3 - 2<,) (1 - <I),,

-(1--51)(2--51)8”-(2--~) 3 >I 01

6X, = AZ (0, - &,,I (k,lk,)

e, AX i

(2AX - 6X,Ie0 -~ -(AX-6X,) *

(3AX - 26X,)

- (AX - SX,) (2AX - aX,)em

(AX - SX,)

-(ZAX-SX1)@ )I

The latter equation is solved iteratively by Newton-Raphson method to find S, = 6X,.

When multiple interfaces developed within

(43)

(44)

the

the

Page 6: Cyclic melting and freezing

1578 M. HASAN et al.

period was subtracted from the instantaneous energy content to give the instantaneous energy change. The absolute value 4 the latter quantity is Q. The value of Q is zero at the beginning of a period and it increases during the period to Qr, the cumulative energy stored or recovered in the period. The quantity QT is ident- ical for each period in the PSS.

The maximum amount of energy which can be stored or recovered (per unit area of the heat transfer surface) in a melting or freezing period is

Qw = PUN + c,V,n - Tw,-) + Dow, - Tm)l (45)

or, in dimensionless form,

3. COMPUTED RESULTS

Numerical computations were made for the three modes of cyclic operation shown in Fig. 2. The solu- tions are functions of dimensionless time and seven dimensionless parameters: thermal conductivity ratio, k,/k,; thermal diffusivity ratio, cr,/cr,; temperature swing during melting, O,,; temperature swing during freezing, 1 Bwf 1; melting period, 7,; freezing period, ‘5p, and initial temperature, 8,“. The values of the para- meters used in the computations are Iisted in Table 1.

The dimensionless time and the dimensionless tem- perature are defined with the properties of the solid PCM. The temperature swing parameters are related to the Stefan numbers for freezing and melting by

I&, I = Ste, (47)

and

0 wm =

When the dimensionless temperature swings are equal, B,, = 1 O,, 1, the temperature differences (at X

= 0) above and below the fusion temperature are equal, T,, - T,=T,,-Tw,.

In presenting the results for equal dimensionless swings, i.e. for j 0,,1 = B,,, a single abbreviation, / 8,I, is used. Similarly, for equal periods, i.e. when To

= T.&f. zp is used. Interfaces are described as solid/ liquid if the solid phase is nearer the heat transfer surface at X = 0 and liquid/solid if the liquid phase is nearer X = 0. During freezing a solid/liquid interface moves outward from X = 0. During the subsequent melting period a liquid/solid interface moves outward.

Figure 4 shows the movement of the interfaces during the transient cycles leading to the periodic steady state (PSS) for a PCM which was initially a saturated liquid, i.e. ein = O(I). The thermal swing If?,1 is 0.2 and the period zp is 1.0. In this figure Si = 0 is the heat transfer surface and .I$ = 1 is the insulated boundary. Early in the first-freezing period a solid/ liquid interface (curve 1F) advances into the liquid in

proportion to ,,6 in accord with the solution to the classical “Stefan problem”. At the end of this freezing process about 60% of the slab is solid. The liquid/solid front (curve 1M) at the end of the tirst- melting period did not merge with the solid/liquid front from the initial freezing period. At the end of one freezing and one melting period (r = 2.0) three layers are present: a liquid layer (thickness z 0.4), a solid layer (thickness z 0.2) and another liquid layer (thick-

ness z 0.4). The first solid/liquid front is almost motionless during the first-melting period. During the

Table 1. Parameter values for numerical solutions

Parameters

k/k, and ~,/a, k, and I 64 T,,, and TV 0,”

Values

0.1, 0.2,0.4, 0.7, 1.0 0.1, 0.2,0.3,0.4 OS, 1.0, 1.5,2.s - 0.4, - 0.2, qs), @I), 0.2, 0.4

1.0 ’

ClOtlID SDLIO 0.8 -

0 1 2 3 rr 5 6 7 8 9 10 11 Z

Fig. 4. Interface positions vs time during transient periods [k,/k, = 0.4, ~,/a, = 1.0, ip = 1.0, 1 fJw 1 = 0.2,

Bi, = O(I)].

Page 7: Cyclic melting and freezing

Cyclic malting and freezing 1579

second-freezing period (curve 2F) the second solid/ liquid interface merges with the first liquid/solid inter- face thus eliminating the melt layer which was sandwiched between the two solid layers at z = 2.0.

The first solid/liquid front continues to move toward the insulated boundary during the second-freezing period. The liquid/solid interface which moves into the solid from the heat transfer surface during the second-melting period (curve 2M) is eliminated by the solid/liquid interface developed during the third-freez- ing period. A maximum of three phase fronts segre- gating the PCM into alternate liquid and solid re- gions existed within the PCM slab. With continued cyclic melting and freezing the first solid/liquid front reaches the insulated boundary at 7 a 6.3. For longer times only the PCM near the heat transfer surface (Si < 0.35) melts or freezes. One cycle after the origin- ally liquid PCM was frozen to the insulated surface, the movement of the solid/liquid and liquid/solid interfaces becomes repetitive. The lower thermal con- ductivity of the Iiquid causes slower movement of the liquid/solid front during melting compared to the movement of the solid/liquid front during freezing. Additional computations were made with the same parameters except that the PCM was initially a satur- ated solid, i.e. Bi, = O(s). In this case the movement of the interfaces becomes repetitive after t = 3. The PSS was the same independent of the initial condition, as expected.

Figure 5 shows the normalized energy stored or recovered, Qr/QM, in each melting or freezing period for the PCM used in Fig. 4. Results are shown for four initial conditions: subcooled solid (curve a), saturated solid (curve b), saturated liquid (curve c) and super- heated liquid (curve d). The interface positions shown in Fig. 4 are for the conditions of curve c. In Fig. 5 values of QT/QI, each corresponding to the end of either a melting or a freezing period, are joined by straight lines. A cycle constitutes two consecutive values of QT/QM. A PSS is attained with

QT/QM= 0.42 independent of the initial condition of the PCM: however, the number of cycles or the time required to approach the PSS depends upon this initial condition. If the PCM is initially a solid, the PSS is reached quickly: however, many more cycles are required if the PCM is initially a liquid. In the initially liquid PCM large oscillations of QT/QM persist until the solid/liquid interface from the first- freezing period reaches the insulated boundary (see Fig. 4).

Figure 6 displays transient temperatures corres- ponding to the PCM of Fig. 4 at the dimensionless distances of 0.2 (curve a), 0.5 (curve b) and 0.8 (curve c) from the heat transfer surface. In this figure a positive value of e indicates a superheated liquid, a negative value of 0 signifies a subcooled solid, while 0 = 0 is the fusion temperature. Since IfI, 1 = 0.2, the maximum value of 0 which the superheated melt can attain is 0.2 and the minimum value of 0 the subcooled solid can attain is - 0.2. At the fifth cycle (T = 8 to 10) the profiles attain the PSS. Figure 6 also shows that the

L

0.60 .

0.55 _

0.50 .

d d

0.45 .

0.40 .

0.35 .

0.30’ . . B ’ ’ ’ ’ m ’ ’ ’ 0 2 4 6 8 10 12

K

Fig. 5. Dimensionless cumulative energy transferred in melting/freezing periods [k,/k, = 0.4, or,/a, = 1.0, TV = 1.0,

I8,I = 0.23.

maximum temperature at X = 0.2 during melting gradually decreases with continued cycling and at- tains a constant value of 0.08 at PSS. During freezing the minimum temperature at X = 0.2 decreases with continued cycling and attains a constant value of - 0.19 at PSS. In each melting period the temper-

ature at X = 0.2 shows an inflexion point near 8 = 0 because of the slower movement of the liquid/solid interface due to the lower thermal conductivity of the liquid. The temperatures at X = 0.5 and X = 0.8 nearly reach the saturation temperature during melting since the liquid/solid interface penetrates only to about X z 0.4 during each melting period. The inflection point in each of the curves, a, b and c, at 5 z 6.7, is due to the complete solidification of the liquid PCM (see Fig. 4).

Figure 7 shows QT/Qnr vs T for a PCM having equal solid/liquid thermophysical properties with B,, = 0.2 and two values of 0,. Starting with a solid reduces the time needed to reach the PSS as shown in Fig. 5. As R,,,, increases, the normalized energy stored or recovered increases. In all cases the PSS was independent of the initial condition of the PCM.

The time variation of the interface position at the PSS is presented in Figs 8 and 9 in a two panel format. The first panel represents a melting period while the second represents the following freezing period. The PCM has equal thermo-physical properties. The cycle has equal thermal swings and equal freezing and

melting periods of 2.5.

Figure 8 shows the positions of the interfaces for

three values of the thermal swing, / .G,l = 0.1, 0.2 and

0.4. The single-dotted broken curves (curve a) show

that for IQ,,1 = 0.1 the melt layer reaches only Si x 0.7. The freezing starts in the presence of the liquid/solid interface remaining from the preceding

Page 8: Cyclic melting and freezing

0.65

U.60

0.55

0.50

d d

0.45

0.40

0.35

M. HASAN et al.

a 0.2

b 0.5

Fig. 6. Transient temperatures at X = 0.2, 0.5 and 0.8 [k,/k, = 0.4, CL,/OI~ = 1.0, zP = 1.0, IB,,, = 0.2, tin = O(l)].

n zn 1 Y,,”

0 12 3 4 5 6 7 8 9 10 I1 .I2

z

Fig. 7. Dimensionless cumulative energy transferred in melting/freezing periods (k,/k, = 1.0, a,/a, = 1.0, T,, = 1.0,

I Bw, ( = 0.2).

melting period. As freezing begins a solid/liquid inter- face moves from S, = 0 outward into the liquid and at 7 z 4.5 it meets the original liquid/solid front and the entire slab is solid. After T > 4.5 heat extraction causes

subcooling of the solid throughout. The b curves, which portray the results for IB,,I = 0.2, are similar to the a curves. The solid curves (curve c) show that for 1 (il.+, / = 0.4, the liquid/solid interface reaches the in- sulated boundary before the melting period ends, i.e. the entire slab melts. Similarly, during the freezing period the entire slab freezes.

Figure 9 displays the effect of k,/k, on the interface movement for 1 @,,,I = 0.2. At any instant during the melting period, the higher the k-ratio, i.e. the larger k,, the thicker is the melt layer. During melting the liquid/solid interface does not reach the insulated boundary for any of the parametric values. The freezing panel of Fig. 9 shows that freezing starts in the presence of a liquid/solid interface remaining from the preceding melting period. The rate of propagation of the freezing front is essentially independent of the value of k, for a fixed k,. Note that r is defined with the solid thermal diffusivity, a,. Calculations were also made with (k,/k,) = 1.0 and various ~,/a, ranging from 0.2 to 1.0. On the coordinates of Fig. 9 there was little effect of the a-ratio. The curve for (~,/a,) = 0.2 was only about 5% below the topmost curve in the figure. For a fixed k,/k,, changes in q/a, are due to

changes in the ratio of the heat capacities. Since most of the energy is stored or removed as latent heat, the heat capacity ratio has little effect on interface movement.

Page 9: Cyclic melting and freezing

Cyclic melting and freezing 1581

0.6

6‘

0.4

0.2

0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

z

Fig. 8. Interface positions vs time at PSS (k,/k, = 1.0, a,/a, = 1.0, T, = 2.5).

Fig. 9. Interface positions YS time at PSS (a,/~, = 1.0, fp = 2.5, I@,1 = 0.2).

In the PSS the PCM behaves in one of the following ways.

(1) The entire PCM melts and freezes in each period.

(2) The PCM melts and freezes only in an “active zone” adjacent to the heat transfer surface at X = 0. The PCM between this active zone and the insulated surface at X = 1 does not change phase.

The first behaviour is illustrated by curve c in Fig. 8 while the second is illustrated by curves a and b of this figure, by all of the curves in Fig. 9 and by Fig. 4 for z > 7. Our calculations indicate that if the PCM melts and freezes completely in each period at the PSS, the energy stored or released is within 5% of the PSS value in 2 cycles starting from a uniform temperature for’liquid or solid. On the other hand, if there is an active zone at the PSS, the number of cycles required to approach the PSS depends upon the initial phase of a uniform temperature PCM. If the initial phase of the PCM is opposite to that of the PCM beyond the active zone at the PSS, 4 to 6 cycles are required for the energy stored or released to approach within 5% of the PSS value. Curves c and d in Fig. 5 illustrate

this phenomena. For the parameters of this figure there is an active zone located at X < 0.35 and the region beyond this is solid (see Fig. 4) for 7 > 7. If the PCM is initially liquid, 5 cycles are required to ap preach the PSS (curves c and d in Fig. 5); if the PCM is initially solid, 2 cycles are required (curves a and b in Fig. 5).

Figures 10 and 11 display dimensionless temper- atures as functions of dimensionless time in the PSS at distances of 0.2 (curve a), 0.5 (curve b) and 0.8 (curve c) from the heat transfer surface. The thermal swing is fixed at 0.2 and the period at 2.5. Figure 10 presents the temperature variation for equal thermo-physical properties corresponding to the topmost curve in Fig. 9. The thermal response curve at X = 0.2 (curve a) does not show an inflection point whereas curves b and c show inflection points at 0 = 0. A sharp change in slope is expected as the phase change interface passes a given position. Since the interface moves rapidly past the plane X = 0.2, the sharp change in slope is not visible in the scale of the figure. As the liquid/solid interface approaches X = 0.5, the temper- ature reaches the fusion temperature. The interface passes X = 0.5 at 7 x 0.8. For larger 5 the liquid superheats and the temperature rises more rapidly.

Page 10: Cyclic melting and freezing

1582 M. HASAN er al.

Fig. 10. Temperature variation at PSS at X = 0.2,0.5,0.8 fk,/k, = 1.0, q/as = 1.0, 1p = 2.5, 10, I = 0.2).

0

RLTING

Fig. 11. Temperature variation at PSS at X = 0.2,0.5,0.8 (k,/k, = 0.4, q/n, = 1.0, rp = 2.5, IfI, = 0.2).

Since the interface moves more slowly at larger values of X, the temperature remains near the fusion temper- ature for a considerable length of time. When the interface passes, liquid phase superheating begins. The freezing curves in Fig. 10 are nearly mirror images of the melting curves except for z > 4.5 where the sharp drop of 0 is a reflection of complete freezing and subsequent subcooling throughout (see Fig. 9). Figure 11 displays the dimensionless temperature variations for (~+/a,) = 1.0 and (k,/k,) = 0.4, which corresponds to the middle curve in Fig. 9. Curve a in Fig. 11 shows an inflection point since the interface moves more slowly than in Fig. 10 due to the lower thermal con- ductivity of the liquid. The temperature-remains near zero at X = 0.5 and X = 0.8 because the liquid/solid interface reaches Si = 0.58 only at the end of the melting period. The subcooling of the solid is re-

moved at X = 0.8 by r z 1 .O and the solid remains at

the fusion temperature during remainder of the melting period. During the freezing period the melt layer ahead of the solid/liquid front releases sensible heat very rapidly and the .temperature there reaches the fusion temperature. The sharp drop in B for z > 3.2 is again due to complete freezing (see Fig. 9) and the rapid removal of sensible heat.

Figure 12 shows dimensionless instantaneous heat fluxes at the heat transfer surface in the PSS for (a,/~,) = 1.0, 1 B, 1 = 0.2 and rp = 2.5 and for three k-ratios.

The heat flux results are plotted in a two panel format, one for the melting Period and the other for the freezing period. During melting the instantaneous heat flux falls rapidly as the conduction resistance increases with the thickening melt layer. At a fixed T the largest flux during melting occurs for the highest k-ratio. Immediately after freezing begins the heat Rux is higher for lower k-ratios. The change in the curva-

Page 11: Cyclic melting and freezing

Cyclic melting and freezing I583

Fig. 12. Instantaneous heat flux at X = 0 at PSS (a,/or, = 1.0, rp = 2.5, 1 f3,I = 0.2).

ture of the heat flux curves during freezing occurs when the advancing solid/liquid interface meets the pre-existing liquid/solid interface and the entire slab becomes solid. This occurs at .= = 2.8, 3.3 and 4.5 for (k,/k,) = 0.2, 0.4 and 1.0, respectively (see Fig. 9). Subsequently, only sensible heat is removed.

Figure 13 is a plot of QT/QM at the PSS against (k,/kJ with (al/as) = 1.0. The two sets of curves, differ- entiated by solid and dotted lines, correspond to equal melting and freezing periods of 1.0 and 2.5, respect- ively. Each set contains curves a, b and c for IQ,,,1 =O.l, 0.2, and 0.4, respectively. The value of QT/QM increases with (k,/k,) but at a decreasing rate with increasing k-ratio. Each curve attains its highest value when the thermal conductivities of the liquid and solid are equal. For fixed (k,/k,) and 10 w 1 the value of Qr/Q,,, increases with z,, but at a decreasing rate as rp is increased. Similarly, for a fixed (k$kJ and ~~ the value of Qr/QM increases with 10, I but at a rate which diminishes as ) flw I increases.

Calculations made with different cc-ratios at (k,/k,) = 1.0 indicated that the dimensionless instantaneous heat flux was nearly the same for a-ratios from 0.2 to 1.0. On the coordinates of Fig. 12, the heat flux for (x,/a,) = 0.2 and was only about 5% below the curve for (k,/k,) = 1.0 in this figure. Figure 14 shows the effect of (=,/a,) on QT/QM at the PSS for (k,/k,) = 1.0. There is little effect of (a!/~(~) for ratios larger than about 0.3. The decrease of QT/QM for small a-ratio is due to the large increase of QY as (a,/~,) -B 0. For equal thermal swings eq. (46) becomes

(49)

With a k-ratio of unity, QM + co as (X,/C(,) + 0, thus the decrease of Qr/QM reflects an increase of QM

rather than a decrease of QT.

4. EXPERKMENTS

Cycling melting and freezing cxperimcnts wcrc per- formed on a slab of n-octadecane to test the numerical

1.0

0.8

s:‘: 0.2

0.1 2.1 -- -

.

b 0.2

c 0.4 I

__--;

0 0 0.2 0.4 0.6 0.8 1.0

*I’%

-i

Fig. 13. Dimensionless energy transferred per period at PSS vs k-ratio @,/a, = 1.0).

0.8

0.6

s 0.4 0.2

I 0 0.2 0.4 0.6 0.8 1.0

9%

Fig. 14. Dimensionless energy transferred per period at PSS vs a-ratio (k,/k, = 1.0).

solution. The experimental set-up consisted of a test

cell and a heat transfer fluid (HTF) loop for circu- lating alternately hot and cold water to induce cyclic melting and freezing. The cell allowed observation

Page 12: Cyclic melting and freezing

1584 M. HASAN et al.

and photographic recording of the interface shapes and positions.

AppLWUt#S Figure 15 is a longitudinal section of the cell

through its central axis. The cell was a rectangular trough with inside dimensions of 29 cm length, 6 cm depth and 6 cm height. The walls of the cell were made from 0.64 cm thick Plexiglass. For better insutation, the front, back and base were double walls con- structed from two Plexiglass sheets with a 0.32 cm gap between them. The heat source/sink was a 0.67 cm thick copper plate measuring 15 cm by 6 cm in the streamwise and transverse directions, respectively. Grooves were milled into the plate on the water side to form an inline strip-fin heat transfer surface.

The copper heat exchange surface occupied a length of 15 cm out of the total 29 cm trough length. A 7 cm x 6 cm x 6 cm piece of Styrofoam blocked one end of the trough while a 6 cm x 6 cm x 5 cm piece blocked the other end, leaving a gap of 2 cm x 6 cm x 5 cm to accommodate the volume change due to expansion during melting. The PCM containment space was 15 cm x 6 cm x 0.276 cm. A filling tube was inserted through a 0.635 cm diameter hole drilled through the top cover. The tube extended down about

1 HTF out

1 cm into the expansion space. Temperature measure-

ments were made with precalibrated, teflon-insulated 20 gauge copper-constantan (T-type) thermocouples (TC). Two thermocouples (TC, and TC,) were de- ployed at the inlet and two (TC, and TC,) at the outlet of the heat exchanger (HE) to measure the inlet and outlet temperatures of the HTF. Five thermo- couples (TC,-TCB) were deployed inside the PCM, with three (TC,, TC, and TC,) in the same horizontal plane, one (TC,) near the HE surface and the other (TC,) near the base.

The test cell, which was insulated in all sides except the front with LO cm thick fibreglass insulation, was placed inside a 50 cm x 60 cm x 55 cm plywood box. The sides of the box were insulated with 6 cm thick Styrofoam insulation. After the cell was placed inside the box the empty space inside the box was filled with fibreglass insulation. A glass window at the front of the box was provided to permit observation.

The HTF was water supplied from one of two constant temperature baths. One bath was main- tained above the fusion temperature of the PCM while the second was maintained below the fusion temperature. The temperature deviation within each bath was at most 0.2”C. Duplicate valves and rotameters were used to circulate water to the cell

I HTF in

EXPANSION CHAMBER

Fig. 15. The test cell.

Page 13: Cyclic melting and freezing

Cyclic melting and freezing 1585

from the appropriate bath. Water flow rates were Haven, CT, USA). This material has a fusion temper- measured to within 0.5% by collection and weighing. ature of 27.4”C. The thermophysical properties of n- Details are given by Hasan (1988). octadecane were taken from Ho (1982).

Procedure The preparation for a cyclic melting/freezing ex-

periment began with the filling of the cell with the PCM. The solid was melted and poured into the cell through the filling port until there was excess liquid to compensate for contraction during freezing. Repeated melting and freezing of the PCM with occasional shaking of the test cell eliminated gas bubbles. The initial condition of the PCM was a solid at uniform temperature.

5. EXPERIMENTAL RESULTS AND DISCUSSION

During an experiment, the temperature difference (AT) between the inlet and outlet streams of water as well as the temperature of the inlet of the HE (TC,) were recorded continuously. The water Aow rate was measured at least three times during a cycle. From the average flow rate, the heat capacity of water and the area under the AT vs. time curve for a cycle, the energy stored or recovered was obtained. In addition, the temperatures from the five TCs inside the PCM (TC,-TC,) as well as the one in the outlet water stream (TC,) were recorded every 15 min. During the first 15 min of a cycle the latter TC readings were recorded every 5 min. Throughout the cyclic melting/ freezing experiments the liquid/solid as well as the solid/liquid interfaces were essentially planar and horizontal. The positions of the interfaces were meas- ured with a cathetometer having a minimum resolu- tion of 0.05 mm. A typical experiment ran for approx- imately 6 h. Additional details are available in Hasan (1988).

The initial experiments involved melting the PCM from above to simulate the classical “Stefan problem” of melting a semi-infinite solid at its fusion temper- ature. This arrangement provided a stable stratifica- tion during melting. The solid PCM was initially at its fusion temperature. The experimental interface loca- tions and the energy transferred were within 0.5% of the Stefan solution (Ozisik, 1980) until the liquid/ solid interface travelled 90% of the thickness of the cell. This agreement confirms the accuracy of the experimental results. Our results were similar to those of Hale and Viskanta (1980) and Gau and Viskanta (1985) who melted materials from above and found that the interface remained planar and that its motion was in agreement with the analytical solution.

The phase change material was 99% pure n-octa- decane (Humphrey Chemical Company, North

Figure 16 shows the dimensionless position of the interface as a function of dimensionless time from the first-melting period until well into the PSS. The re- sults in this figure are for equal melting and freezing temperature swings of 19°C (top) and 12°C (bottom). The liquid/solid interface during melting and the solid/liquid interface during freezing remained planar. During melting the liquid/solid interface was planar because the temperature gradient in the melt was stable. During freezing the temperatures registered by the thermocouples within the PCM indicated that the superheat of the melt underneath the growing solid layer was lost within 5 min of the onset of freezing. In this initial period of freezing the advancing solid/ liquid interface showed slight deviations from a plane.

Fig. 16. Interface position YS time (k,/k, = 0.41, q/c(, = 0.40, tp = 1.5 h, r, = 27°C).

Page 14: Cyclic melting and freezing

1586

0.6

Fig. 1

M. HASAN et al.

I I I I r I I I EFiF+Im 0

CONDUCTION MODEL -- -

/

/ / ,<-,---i- _ -m--m %- _-< - ~ b

IT, - T,,,I PERIODS

t”cl (hr)

0 19 1.5 b 19 1.0

C I2 1.5

L 1 1 1 I L ..I

1M 1F 2M 2F 3M 3F 4M 4Y 5M

MLTINdFREEZI NG CYCLES

‘. Q,/Q, vs number of melting/freezing periods for t, = t, (k,/k, = 0.41, al/cc, = 0.39, 8, = - 0.0034).

Later in the freezing process these deviations disap- peared. The experimental data are in excellent agree- ment with the numerical calculations of the conduc- tion model. Figure 16 shows that the PSS was reached after 3 cycles starting from an initially solid PCM. This figure also shows that the active layer, the region of the PCM which melts and freezes, is shallower for the smaller thermal swing.

tion model with complementary experiments. From the theoretical and experimental parts of this work the following conclusions are drawn.

(1) At the PSS one or two interfaces are present. (2) The number of cycles required to reach the PSS

Figure 17 shows the dimensionless energy stored or recovered in a melting or a freezing period, Qr/Qnr, from the beginning of cycling until well into the PSS, for the two cases in Fig. 16. In addition, Fig. 17 shows QT/QM for a swing of 19°C and for a melting and freezing period of 1 h. The notation of the abscissa indicates the period number, i.e. 1M denotes the end of the first-melting period, 2M denotes the end of the second-melting period, etc. The value of Qw, the maximum energy transferred, was 63.1 kJ for a ther- mal swing of 19°C and 57.2 kJ for 12°C. Latent heat comprises 74% of Q,, for the larger swing and 82% of QM for the smaller swing. In this figure the nu- merically-computed discrete values of QT/QM are joined by straight lines. The energy stored or re- covered comes within 3% of the PSS-value after the second-melting period. The experimental data agree well with the numerical predictions of the conduction model.

depends upon the initial condition of the PCM. More than three interfaces may be present during the transient leading to the PSS.

(3) In the periodic steady state the time variation of the position of the interface(s) is a strong function of the temperature swing and the k-ratio. There is little effect of the a-ratio.

(4) The dimensionless energy stored or recovered in the periodic steady state increases with increasing temperature swing and increasing (k,/k,). There is relatively little effect of (at/a,) except for (a,/a,) < 0.3.

(5) Experiments involving periodic melting and freezing of the PCM from the top resulted in essen- tially planar liquid/solid and solid/liquid interfaces. The experimental interface positions and the energy transferred agreed well with the conduction model calculations.

Acknowledgement-This work was supported by the Natural Sciences and Engineering Research Council of Canada. M. Hasan gratefully acknowledges financial support in the form of’ the Bindra Fellowship from McGill University.

6. CONCLUSION NOTATION

Periodic phase change heat transfer for a PCM slab C specific heat, kJ/kg K was investigated through a one-dimensional conduc- k thermal conductivity, W/m K

Page 15: Cyclic melting and freezing

QM

QT si

S,

S*

Ste,

Ste,

Cyclic melting and freezing 1587

thickness of slab, m e WI?2 thermal swing for a melting period, dimen- number of space increments, dimensionless sionless cumulative energy transferred in a period, I latent heat of fusion, kJ/kg kJ/m’ P density, kg/m3 maximum possible energy stored or released T dimensionless time, ( = a,t/L’) in a period, kJ/m’ tc dimensionless time for a cycle ( = TV + TV) energy stored or released in a period, kJ/m’ z,, dimensionless time for a freezing or a melt- dimensionless distance of an interface from ing period (when 5M = TV) x=0 dimensionless distance of the solid/liquid Subscripts

interface from X = 0 F freezing dimensionless distance of the liquid/solid i general nodal point interface from X = 0 in initial condition Stefan number for freezing [ = c, (r,,, j general time step

- ~w,Y4 I liquid Stefan number for melting [ = c,(T,, M melting

- ~,)/4 n refers to the number of cycles time, min 4 nodal point identification freezing period, min r nodal point identification melting period, min s solid time for a cycle ( = TF + tM), min W heat transfer surface temperature, “C fusion temperature, “C Superscripts temperature at the heat transfer surface, “C 1 first liquid region temperature at the heat transfer surface dur- II first solid region ing freezing, “C III second liquid region temperature at the heat transfer surface dur- 0 initial value ing melting, “C A? first dimensionless time step distance, m dimensionless space variable ( = x/L) distance of an interface from x = 0 REFERENCES

. I Greek letters

Bransier, J., 1979, Stockage periodique par chaleur latente: aspets fondamentaux lies 2 la cinetique des transferts. Int. J. Heat Mass Transfer 22,875-883.

Gau, C. and Viskanta, R., 1985, Effect of crystal anisotropy on heat transfer during melting and solidification of a pure metal. J. Hear Transfer 107, 706-709.

Hale, N. W. and Viskanta, R., 1980, Solid-liquid phase change heat transfer and interface motion in materials cooled or heated from above or below. Int. J. Heat Mass Transfer 23, 283-292.

ix,

thermal diffusivity, m2/s

dimensionless distance of the solid/liquid

interface (S,) from the nearest grid point to the left

3x2 dimensionless distance of the liquid/solid interface (S,) from the nearest grid point to the left

AZ dimensionless time step I9 dimensionless temperature [ = c,(T - T,,,)/

4 4.j temperature at node ‘3” at “j” time step,

dimensionless

e, fusion temperature of the PCM, dimen- sionless

e NN+I temperature of the last grid point, dimen- sionless

&V thermal swing at the HTS, dimensionless I3 wf thermal swing for a freezing period, dimen-

sionless

Hasan, M., 1988, Cyclic phase change: energy storage and recovery. Ph.D. thesis, Department of Chemical Engineer- ing, McGill University, Montreal, Canada.

Ho, C. J., 1982, Solid-liquid phase change heat transfer in enclosures. Ph.D. thesis, Department of Mechanical Engineering, Purdue University, Wast Lafayette, Indiana.

Jariwala, V., Mujumdar, A. S. and Weber, M. E., 1987, The periodic steady state for cyclic energy storage in paraffin wax. Can. J. Chem. Engng 65, 899-906.

Kalhorui, B. and Ramadhyani, S., 1985, Studies on heat transfer from a vertical cylinder, with or without lim

embedded in a solid phase change medium. J. Heat Trans- fer 107, 44-51.

Ozisik, M. N., 1980, Heat Conduction, pp. 399435. Wiley, New York.