PII: S0957-0233(99)96940-0. A simple method, the T -history method, of determining the heat of fusion, specific heat and thermal conductivity of phase-change.
Meas. Sci. Technol. 10 (1999) 201–205. Printed in the UK
PII: S0957-0233(99)96940-0
A simple method, the T -history method, of determining the heat of fusion, specific heat and thermal conductivity of phase-change materials Zhang Yinping, Jiang Yi and Jiang Yi Department of Thermal Engineering, Tsinghua University, Beijing 100084, People’s Republic of China Received 24 August 1998, in final form and accepted for publication 8 December 1998 Abstract. A simple method of determining the melting point, heat of fusion, specific heat
and thermal conductivity of phase-change materials (PCMs) is presented. Compared with other methods, such as conventional calorimetric methods, differential thermal analysis and differential scanning calorimetry methods, it has the following salient features: the experimental unit is simple, able to measure the heat of fusion, specific heat and thermal conductivity of several samples of PCMs simultaneously and allows one to observe the phase-change process of each PCM sample. Using the method, the thermophysical properties of various salt hydrates, paraffin and some PCMs developed by us were measured. For the PCMs whose thermophysical properties are available in the literature, our results gave fairly good agreement. The method is especially useful for the selection of lots of candidate PCMs used for the purpose of engineering and for preparing new PCMs. Keywords: phase-change materials, heat of fusion, specific heat, thermal conductivity
1. Introduction
In recent years latent heat storage systems have gained importance for many applications in space-based power plants, in some solar energy systems, in central airconditioning systems and in energy-conserving buildings, because of their high energy density and isothermal behaviour during charging and discharging. To some extent, the selection or preparation of a suitable phase-change material (PCM) for a latent-heat storage system is most difficult but most important. In any case, measurement of the thermal physical properties of the PCM is of importance. The methods available for determining the heat of fusion, specific heat and melting point in the field of latent-heat storage can be classified into three groups, conventional calorimetry methods, differential thermal analysis (DTA) and differential scanning calorimetry (DSC) methods [1]. Although DTA and DSC methods are well developed, their shortcomings are obvious: the samples tested by them are very small (1–10 mg) so that the thermophysical properties of samples are usually different from those of the bulk materials used in practical systems (for example, for liquid–solid phase-change processes of most composite PCMs, especially salt hydrates, if the phase-change material is put into a 0957-0233/99/030201+05$19.50
© 1999 IOP Publishing Ltd
small container, the degree of supercooling of the PCM is increased while the degree of phase segregation is decreased [2]); DTA and DSC measurement rigs are complicated and expensive; and they cannot measure heats of fusion, specific heats and thermal conductivities of several PCM samples simultaneously. Insofar as conventional calorimetry methods are concerned, almost all of them have the disadvantages mentioned above except the first one. Besides, the phasechange process of a PCM sample during a measurement is hard to observe clearly when using convectional methods. In view of these facts, we developed a new method; the T -history method, of determining the melting point, degree of supercooling, heat of fusion, specific heat and thermal conductivity of several PCM samples simultaneously. It is especially useful for the selection of lots of candidate PCMs or for the preparation of new PCMs for use in practical systems. 2. The principle of measurement
2.1. Determinations of the heat of fusion, specific heat and so on of PCMs If a tube containing a liquid PCM whose temperature is uniform and equal to T0 (T0 > Tm , Tm is the melting 201
Zhang Yingping et al
Figure 1. A typical T -history curve of a PCM during a cooling process (with supercooling).
Figure 3. A typical T -history curve of a PCM during a cooling process (without supercooling).
curve is as shown in figure 2. Considering that Bi < 0.1, similarly, we have
Figure 2. A typical T -history curve of water during a cooling
process.
temperature of the PCM) is suddenly exposed to an atmosphere whose temperature is T∞,a (which can be time dependent), the temperature versus time curve of the PCM, i.e. the T -history curve, is as shown in figure 1, where 1Tm (=Tm − Ts ) is the degree of supercooling. When Bi (= hR/(2k), i.e. the Biot number, where R is the radius of a tube, k the thermal conductivity of PCM and h the natural convective heat-transfer coefficient of air outside a tube) is less than < 0.1, the temperature distribution in the sample can be regarded as uniform and the lumped capacitance method can be used [3]. Hence, we have (mt cp,t + mp cp,1 )(T0 − Ts ) = hAc A1
(1)
where mp and mt are the masses of the PCM and tube, respectively, cp,l and cp,t are the mean specific heats of the liquid PCM and of the material of the tube, respectively, Ac is the convective heat-transfer area of a tube; A1 = Rand t1 (T − T∞,a ) dt. We have also 0 mp Hm = hAc A2
(2)
Hm is the heat of fusion of the PCM and A2 = Rwhere t2 t1 (T − T∞,a )dt (t1 → t2 is the time during which a phasechange process occurs) and (mt cp,t + mp cp,s )(Ts − Tr ) = hAc A3
(3)
where cp,s is the mean specific heat of the solid PCM, Rt A3 = t23 (T − T∞,a )dt and Tr is the reference temperature. If a tube containing pure water is suddenly exposed to the same atmosphere as that mentioned above, its cooling 202
(mt cp,t + mw cp,w )(T0 − Ts ) = hAc A01
(4)
(mt cp,t + mw cp,w )(Ts − Tr ) = hAc A02
(5)
where mw and cp,w are the mass and mean specific heat R t0 of water, respectively, A01 = 0 1 (T − T∞,a )dt and A02 = R t20 t1 (T − T∞,a )dt. In fact, using equation (4), the natural convective heattransfer coefficient of air outside a tube (h) can be obtained. It is about 5–6 W m−2 K−1 , so the condition about Bi 6 0.1 can be satisfied when ks > 0.2 W m−1 K−1 (ks of salt hydrates are all larger than 0.3 W m−1 K−1 ). From equations (1)–(5), we obtain cp,s =
mw cp,w + mt cp,t A3 mt cp,t − mp A02 mp
(6)
cp,1 =
mw cp,w + mt cp,t A1 mt cp,t 0 − mp A1 mp
(7)
Hm =
mw cp,w + mt cp,t A2 (T0 − Ts ). mt A01
(8)
For PCMs without supercooling (see figure 3, where the temperature range of the phase-change process is between Tm,1 and Tm,2 ), the expressions for cp,l and cp,s are the same as those above, but the heat of fusion should be rewritten as follows: mw cp,w + mt cp,t A2 Hm = (T0 − Tm,1 ) mp A01 mt cp,t (Tm,1 − Tm,2 ) . (9) − mp 2.2. Determination of the thermal conductivity of PCMs In order to measure the thermal conductivity of a PCM, the tube containing the melted PCM whose temperature is uniform and equal to T0 (T0 is a little higher than Tm ) has to be suddenly dipped into a cool water bath whose temperature is T∞,w (T∞,w is lower than Tm ). If the ratio of the length to the diameter of a tube is larger than 15, it is justifiable to assume that the heat transfer
Properties of phase-change materials Table 1. Relative errors of measurements.
Figure 4. An illustration of two phases of a PCM in a long tube.
is approximately one dimensional. For this case, the heatdiffusion equation for the cylinder shown in figure 4 is ∂T (r, t) 1 ∂T (r, t) 1 ∂ r = (ξ < r < R, t > 0) r ∂r ∂r αp ∂t (10) subject to the boundary condition ∂T ks = hw (T∞ − T ) t >0 ∂r r=R and the initial condition T (ξ = R) ∼ = Tm
t =0
where T (r, t) is the temperature of the PCM sample at radius r and instant t, αp is the thermal diffusivity of the PCM; ξ is the radius of the interface between the solid and liquid phases of the PCM and hw is the coefficient for convective heat transfer from the tube to the stirred cool water. For the interface between the two phases of the PCM, we have (11) T (r = ξ ) = Tm ∂T dξ (12) ks = ρp Hm . ∂r r=ξ dt Using the perturbation method [4, 5], neglecting the secondorder term of the perturbation expansion, we obtain cp (Tm − T∞,w ) tf (Tm − T∞,w ) 1 − ks = 1 + 4 Hm ρp R 2 Hm hw R (13) where ks is the effective thermal conductivity of the PCM in the solid state, ρp is the density of the PCM, which can be obtained easily, and tf is the time of full solidification of the molten PCM. For equation (13), the relative error in ks caused by neglecting the second-order term of the perturbation expansion is smaller than 5% for the condition that Bi > 0.1 and 0 < Ste < 0.5 where Ste is the Stefan number which equals cp |Tm − T∞ |/Hm [5]. Similarly, for a PCM in the solid state placed in a tube which is put into a hot water bath whose temperature is a little bit higher than Tm , the formula for calculating kl (the thermal conductivity of the liquid PCM) can be obtained. It should be mentioned that (i) equation (14) can be used for determining the heat conductivity only for a PCM
1T (◦ C)
0.5
0.1
0.01
1Hm /Hm (%) 1cp,s /cp,s (%) 1cp,l /cp,l (%) 1k/k (%)
30 30 20 80
6 6 4 16
0.6 0.6 0.4 1.6
in whose phase-change process there is one clear interface between two phases (in fact, for some salt hydrates, this condition cannot be satisfied) and (ii) hw can be obtained by using the T -history curve of a tube containing mercury and equation (4), where mw should be replaced by the mass of mercury (because the thermal conductivity of mercury is very large, the Bi number for this case is smaller than 0.1; therefore, the lumped capacitance method is applicable). From experiments, it was found that, for most cases, tf (Tm − T∞,w ) 1 . ρp R 2 Hm hw R Therefore, the term 1/(hw R) can be neglected for most cases. 2.3. Error analysis From equations (6)–(9) and (13), it can be seen that the errors in measurement of Hm , cp and k of PCMs mainly come from the errors in measurements of temperature. So, we can derive that 1Hm 41T 21T ≈ + (14) Hm Tm − T∞,a T0 − Tm 1cp,s 21T 21T ≈ + cp,s Tm − T∞,a Tr − Tm
(15)
1cp,l 41T 6 cp,l Tm − T∞,a
(16)
41T 21Hm 1ks ≈ + . ks Tm − T∞,w Hm
(17)
Besides this, by using equations (14)–(17), we also calculated the influences of errors in various parameters on the precision of measurement. The method is as follows. By using equations (6)–(9) and (13), for given errors of the related parameters, the deviations of the measured parameters can be calculated. The values of errors are almost the same as those estimated by using equations (14)–(17). The relative errors of the aforementioned parameters for various errors in measurement of the temperature 1T are listed in table 1. From table 1, it can be seen that, if 1T 6 0.1, the required precision of measurement of the thermophysical properties is satisfied. In brief, using equations (6)–(9), the specific heat of fusion of a PCM can be obtained. For a PCM in whose phase-change process there is one clear interface between two phases the thermal conductivity of the PCM can be measured by using equation (14). Besides this, the melting point and degree of supercooling can be readily found from the cooling curve of the PCM and the thermal diffusivity of the PCM (αp ) can be calculated by using its definition (αp = kp /ρp cp,p , where p denotes pcm). Therefore, by using the T -history curves of PCMs and water, almost 203
Zhang Yingping et al Table 2. Measured results of thermophysical properties of some PCMs (the number in the bracket after value 1 is the number of
measurements). Material
Item
Tm0 (◦ C)
58.7 wt% Mg(NO3 )2 · 6H2 O +41.3 wt% MgCl2 · 6H2 O 37.5 wt% NH4 NO3 +62.5 wt% Mg(NO3 )2 · 6H2 O Myristic acid
Value 1 (2) Value 2 Value 1(3) Value 2 Value 1(3) Value 2 Value 1(2) Value 2 Value 1(2) Value 2 Value 1(2) Value 2 Value 1(2) Value 2 Value 1(2) Value 2 Value 1(3) Value 1(2) Value 1(5) Value 1(3) Value 1(2) Value 1(2) Value 1(2) Value 2 Value 1(2) Value 2
58.3 58.0 48.7 51.0 49.7–52.7 52.1 30.1 30.1 52.5 52.1 54.60–57.69 56.3 58.0 58 48.0 48 27.91 52.30 38.92 29.28–46.35 34.7 32.4 20. 8 21.7 89. 3 89.9
Capric acid Lauric acid Hexacosane CH3 COONa · 3H2 O Na2 S2 O3 · 5H2 O KD-28a KD-52a KD-38a KD-46a KD-34a KD-32a Heptadecane Mg(NO3 )2 · 6H2 O a
1T (◦ C) 6.52 8.28
0.05 0.06 0.05 5.98 37.6 7.48 1.28 9.37 2.93 24.4 0 3.3
cp,l (kJ kg−1 K−1 )
cp,s (kJ kg−1 K−1 )
Hm (kJ kg−1 )
ks (W m−2 K−1 )
2.57 2.43 2.56
1.94 1.95 2.60
0.61 0.68 0.34
3.67
2.91
1.72 1.6 1.75 1.6 2.94
1.95
3.33
2.26 2.79
120 132 118 126 201 190 150 158 160 179 216 237 248 226 206 201 203 104 150 368 81 193 171 172 175 167
3.89
3.83 3.05 4.35 2.34 3.33 3.28 3.1 2.93
2.57
2.39 2.51
1.92 1.84
0.17 0.16 0.15 0.17 0.15 0.22
0.24 0.57 at 120 ◦ C
PCMs developed in our laboratory.
Figure 5. A schematic diagram of the experimental rig.
all thermophysical properties of PCMs can be obtained simultaneously, which is the reason that this method is named the ‘T -history method’.
3. The experimental system
An experimental rig based upon the principle was built (figure 5). In a test run seven samples and pure water are contained in eight glass tubes diameter is 10.4 mm and length 180.6 mm. A thermoprobe 108 mm in length and 0.7 mm in diameter is placed along the axis of each tube. Data acquisition is accomplished with a PC-based data-logger system. The advantages of this kind of experimental system are as follows. (i) Using conventional tubes as PCM sample containers makes measurement convenient and means that the phase-change process of each sample can be clearly observed. (ii) One is able to measure several samples during a test (the number of samples measured in a test run depends upon the number of channels of the data acquisition system). 204
Figure 6. Measured T -history curves of PCMs during cooling
processes.
4. Measured results
Figure 6 shows measured T -history curves of some PCMs. The thermophysical properties of the PCMs measured are
Properties of phase-change materials
listed in table 2. It can be seen that the relative deviations between the measured results (value 1) and those in the literature [4, 6, 7] (value 2) are less than 10%. 5. Conclusion
Compared with methods available such as conventional calorimetry methods and DSC methods, the method presented in this paper has the following salient features: the experimental system is simple; it is able to measure lots of samples and obtain several thermophysical properties of each PCM sample through a group of tests; the precision of measurement satisfies the need for engineering applications; and the phase-change process of each PCM sample can be observed clearly. Using the method, the thermophysical properties of several salt hydrates, paraffin and some PCMs developed in our laboratory (salt hydrates) were measured. The results agree well with those in the literature. This method is useful for rapid measurement of PCM candidate for use in engineering applications of latent thermal storage. The method is applicable to a larger range of temperature if an oil bath or other suitable liquid bath were used instead of a water bath.
Acknowledgment
This work was supported by the Nature Science Foundation of the People’s Republic of China. References [1] Chen Zeshao, Ge Xinshi and Gu Yuqing 1991 Measurement of Thermophysical Properties (Hefei: Publishing Company of USTC) pp 36–81 (in Chinese) [2] Abhat A 1981 Low temperature latent heat thermal energy storage Thermal Energy Storage ed C Beghi (Dordrecht: Reidel) [3] Incropera F P and Dewitt D P 1996 Fundamentals of Heat and Mass Transfer 4th edn (New York: Wiley) pp 212–17 [4] Zhang Yinping, Hu Hanping and Kong Xiangdong 1996 The Theory and Application of Heat Storage Phase-Change Materials (Hefei: Publishing Company of USTC) pp 339–41 (in Chinese) [5] Song Y W 1981 The solution of the solidification of PCMs in a cylindrical container by using the perturbation method J. Engng Thermal Phys. 2 211–6 (in Chinese, with English abstract) [6] George A L 1983 Solar Heat Storage: Latent Heat Material vol 2 (Boca Raton, FL: CRC) pp 4–67 [7] Garg H P 1985 Solar Thermal Energy Storage (Dordrecht: Reidel) pp 183–206
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