**2.1 Determination of enthalpy as a function of temperature**

In order to obtain the most suitable method to determine enthalpy as a function of temperature during the solid-liquid phase change, two main thermal analysis methods were studied: differential scanning calorimetry (DSC) and adiabatic calorimetry. In addition, a customized method was studied: the T-history method. A complex review of the work on thermophysical properties was carried out with some conclusions being (Lazaro, 2009):


DSC, adiabatic calorimetry and T-history method were studied and compared. Factors considered in the method selection are: sample size, heating and cooling rate, obtainability of the h-T curve, introduction to the market, easiness to build, cost, use, maintenance. The Thistory method was selected as it provides the enthalpy vs. temperature curves and also uses sample sizes and heating/cooling rates similar to those used in real applications.

### **2.1.1 The T-history method**

Zhang et al., 1999, developed a method to analyze PCM enthalpy. The T-history method is based on an air enclosure where the temperature is constant and two samples are introduced at a different temperature from the temperature in the air enclosure. During cooling processes, three temperatures are registered: the ambient (air enclosure) and those of the two samples. The two samples are one reference substance whose thermal properties are known (frequently water) and one PCM whose thermal properties will be determined with the results of the test. Figure 1 shows the basic scheme of the T-history method.

Fig. 1. Scheme of T-history installation.

PCM-Air Heat Exchangers: Slab Geometry 431

The set of equations 1 summarize the calculations considering the improvements. There *At* denotes the tube lateral area, *mp* the PCM mass, *mt* the tube mass, *cpt* the specific heat of the tube, *cpw* the specific heat of water, *h* the convection coefficient whereas *h(T)* denotes enthalpy. The little temperature steps, *ΔTi*, varies in accord to the corresponding time intervals for the PCM (*Δti*=ti+1-ti) and for water (*Δt'i*=t'i+1-t'i). The integral of the temperature difference against time, is the area under the curve in Figure 2 for the PCM (*Ai*) and for

When analyzing errors with T-history, the most important factor is the precision in the temperature measurement. Thermal sensors used in previous implementations have been thermocouples, while Pt-100 was chosen for this new installation due to the higher precision: ±0.05ºC with a 4 threads assembly. However, Pt-100 has a longer response time, but will not affect the results provided that the response time is the same for all temperature measurements. This objective is achieved by using Pt-100 of the same manufacture set, and characteristics will be identical. Enthalpy is expressed in a mass unit basis; therefore the precision in mass measurements is as important as the precision in temperature measurements. A 0.1 mg precision scale is used to measure the mass of samples. The sample containers have been designed so that the method standards are fulfilled (Bi<0.1). Churchill-Chu (Marin & Monne, 1998) natural convection correlations for cylinders were used to calculate the suitable radius/length rate of the tubes. The chosen material was glass, since it allows the observation of the phase change process. Cylinders of 13 cm in length and 1 cm in diameter were used. A data logger was used with a RS-232 connection with 22 bits and 6 ½ resolution. A thermostatic bath (0.1 K precision) was used to fix the initial temperature of the samples. A calculation software, especially developed in Labview, was used to obtain

**2.1.2 Design of a new installation to implement the T-history method** 

the h-T curves. The new T-history implementation based its improvements on:

A guarantee that there is no contribution of heat transfer by radiation.

Horizontal position of samples in the air enclosure, minimizing convective movements.

A program designed (Labview) for calculations and real time view of the

Examples of T-history analysis applied to two typical PCM (organic and inorganic) are shown in Figure 3. Typical phenomena as hysteresis or sub-cooling can also be observed.

The objective of analyzing organic and inorganic substances is to confirm the expected differences in behaviour: the inorganic PCM presents the sub-cooling phenomenon that occurs during cooling, presenting more hysteresis and quite higher stored energy density

The procedure used was: mass measurements of the samples and sample containers using a precision scale, then the Pt-100 were introduced into the samples (one into the PCM and one into the water), and the tubes were inserted into the thermostatic bath at the desired initial temperature. The initial temperature depends on the PCM to be tested as well as if it is for a heating or a cooling test. For a heating test, the initial temperature must be lower than the phase change temperature. For a cooling test, it must be higher. Once the temperature inside

Obtainability of the h-T curves during cooling and heating.

Utilization of more precise instrumentation.

measurements.

when compared to organic PCM.

water (*Ai'*).

The basic aspects of the T-history method are (Zhang et al., 1999):


To evaluate the temperature vs. time evolution, Zhang proposed three stages: liquid, phase change, and solid. Therefore, with this method it is possible to obtain *cp,liquid*, *cp,solid* and *hsl*. Marin et al., 2003, made improvements, based on the finite increments method, in order to obtain the h-T curves. Figure 2 shows how the calculations were carried out.

$$m\_p \Delta h\_p \left( T\_i \right) + m\_t c\_{pt} \left( T\_i \right) \left( T\_i - T\_{i+1} \right) = hA\_t \int\_{t\_i}^{t\_i + A t\_i} \left( T - T\_{\alpha, a} \right) dt = hA\_t A\_i \tag{1a}$$

$$\left[m\_t c\_{pt}\left(T\right) + m\_w c\_{pw}\left(T\right)\right] \left(T\_i - T\_{i+1}\right) = hA\_t \int\_{t^\*}^{t^\* + At^\*} \left(T - T\_{w,a}\right) dt = hA\_t A\_i \tag{1b}$$

$$
\Delta\hbar \left( T\_i \right) = \left( \frac{m\_w c\_{pw} \left( T\_i \right) + m\_t c\_{pt} \left( T\_i \right)}{m\_p} \right) \frac{A\_i}{A\_i} \Delta T\_i' \frac{m\_t}{m\_p} c\_{pt} \left( T\_i \right) \Delta T\_i \tag{1c}
$$

$$\mathcal{M}\_p\left(T\right) = \sum\_{i=1}^N \mathcal{A}h\_{pi} + h\_{p0} \tag{1d}$$

$$
\sigma\_p = \partial \mathbb{M} / \partial \mathbb{T} \tag{1e}
$$

Fig. 2. Calculation of the improvements achieved by Marin et al., 2003.

Heat transfer is one-dimensional in the radial direction since the samples containers are

Containers with water and PCM samples are designed with Bi<0.1 and therefore are

 Heat transfer occurs by free convection between the samples and air. Containers must be identical in order to have a very low and almost the same free convection

To evaluate the temperature vs. time evolution, Zhang proposed three stages: liquid, phase change, and solid. Therefore, with this method it is possible to obtain *cp,liquid*, *cp,solid* and *hsl*. Marin et al., 2003, made improvements, based on the finite increments method, in order to

1 , *i i*

' '

 

*p p i t pti i i t a ti*

*t pt w pw ii t a ti*

*m c T m c T T T hA T T dt hA A*

'

*N p pi p i hT h h* 

Fig. 2. Calculation of the improvements achieved by Marin et al., 2003.

*w pw i t pt i i t i i pti i p p i*

> <sup>0</sup> 1

*m c T mc T A m h T T cTT m m A*

*m h T m c T T T hA T T dt hA A*

*i*

1 , '

(1b)

*t*

*t t*

*t*

*t t*

(1a)

'

(1c)

 

(1d)

*<sup>p</sup> c hT* (1e)

obtain the h-T curves. Figure 2 shows how the calculations were carried out.

The basic aspects of the T-history method are (Zhang et al., 1999):

long cylinders.

coefficient.

considered capacity systems.

 The set of equations 1 summarize the calculations considering the improvements. There *At* denotes the tube lateral area, *mp* the PCM mass, *mt* the tube mass, *cpt* the specific heat of the tube, *cpw* the specific heat of water, *h* the convection coefficient whereas *h(T)* denotes enthalpy. The little temperature steps, *ΔTi*, varies in accord to the corresponding time intervals for the PCM (*Δti*=ti+1-ti) and for water (*Δt'i*=t'i+1-t'i). The integral of the temperature difference against time, is the area under the curve in Figure 2 for the PCM (*Ai*) and for water (*Ai'*).
