*4.3.4 Heat lost (Qloss)*

It is the amount of heat lost from the external surface of the storage tank during the charging process. It is evaluated using Eq. (9).

$$Q\_l = U\_{overall} \, A \, dT \qquad W \tag{9}$$

*dT = Tavg - T*<sup>∞</sup> *Tavg* = Average temperature of all the layers of water in the storage tank, °C *Uoveral l* = Overall heat loss coefficient of the storage tank, W/m<sup>2</sup> K *A* = Area of the storage tank, m2

#### *4.3.5 Heat stored (Qs)*

It is the actual amount of heat retained in the storage tank during the charging or discharging process after subtracting the heat lost to the surrounding, from the

storage tank. It gives an idea as to what amount of heat is stored and is calculated using Eq. (10).

$$Q\_{stored} = Q\_{cum} - Q\_{loss} \tag{10}$$

#### *4.3.6 Charging efficiency*

The charging efficiency of the storage tank is the ratio of the instantaneous heat transfer to the maximum heat transfer at a given inlet temperature at any instant, keeping constant the flowrate of the HTF at the inlet. It illustrates how the efficiency varies with respect to time for the charging done to the storage tanks without and with PCM. It is determined by the following Eq. (11)

$$\eta\_{ch} = \frac{[T\_{H-in} - T L b]\_t}{T\_{H-in} - T\_{ini}} \tag{11}$$

Tini = Initial temperature of the tank

#### *4.3.7 Discharging efficiency*

The discharging efficiency of the storage tank is the ratio of the instantaneous heat transfer to the maximum heat transfer at a given inlet temperature at any instant keeping constant flowrate of the HTF at the inlet as given by Eq. (12). The cold water inlet at 30°C (TC-in) is made to enter the storage tank till the end of the experiment.

$$\eta\_{\rm disk}(\mathbf{t}) = \frac{(T\_{\rm C-out} - T\_{\rm C-in})\_t}{(T\_{\rm C-out})\_{ini} - T\_{\rm C-in}} \tag{12}$$

*TCin* = Cold inlet water temperature to the tank at the bottom, °C *TCout* = Outgoing water temperature at the top, °C

#### *4.3.8 Stratification number (Str)*

It is the ratio of the average temperature gradients at each interval of time to the temperature gradient at the initial charging process (t = 0). Eq. (13) shows the evaluation of stratification number, depending on the temperatures acquired at equidistant points in the thermal energy storage tanks

$$\text{Str} = \frac{(\partial T/\partial \mathbf{y})t}{(\partial T/\partial \mathbf{y})t = \mathbf{0}} \tag{13}$$

where

$$
\left[\frac{\partial T}{\partial \mathbf{y}}\right] = \frac{1}{N - 1} \left[ \sum\_{k=1}^{N-1} \frac{(T\_k - T\_{k+1})}{\Delta Z} \right]
$$

$$
\left[\frac{\partial T}{\partial \mathbf{y}}\right]\_{t=0} = \frac{T\_{H-in} - T\_{ini}}{(N - 1)\Delta Z}
$$

where k is the nodal points where the temperature measurements are made, *N* is the number of nodal points and *ΔZ* is the distance between the nodal points *T\_(H-i*n) and *T\_ini* which shows the hot water inlet and initial temperatures of the HTF, respectively.

#### *4.3.9 Richardson number (Ri)*

It is an effective indicator of stratification performance that considers the various aspects of storage systems such as the geometry of the storage tank, specific velocity, and the top and bottom temperature of the layers within the tank. It is given by the ratio of the buoyancy forces to the mixing forces and is estimated by Eq. (14).

$$\text{Ri} = \frac{\text{g}\beta H[(T\_{H-in}) - (T\_{H-out})]\_{\text{l}}}{v\_{\text{g}'}^2} \tag{14}$$

$$\nu\_{\text{f}'} = \frac{V}{\pi r^2}$$

where *TH-in* and *TH-out* represent the top and bottom temperature of the HTF measured in the TES tank and H is the distance between these locations. *V* and *νsf* represent the volumetric flow rate and superficial velocity of HTF entering the TES tank, respectively and r is the radius of the inlet pipe which is ¼th of an inch in the present experiment.

#### *4.3.10 Mix number*

Mix number is the level of energy in the TES tank weighted by its vertical distance from the bottom of the storage tank. The vertical moment of energy in a storage tank was shown in Eq. (15).

$$M\_E = \int\_0^H \mathcal{y} dE \tag{15}$$

This equation was further deciphered as shown in Eq. (16), as follows

$$\mathcal{M}\_{\rm E} = \sum\_{i=1}^{N} \mathcal{y}\_{i} E\_{i} \tag{16}$$

where *N* = the number of layers used

*yi* = the distance measured from the bottom of the tank.

*Ei* ¼ *ρCpViTi* � � *water* þ *ρCpViTi* � � *pcm*, which is the energy stored at a particular level i On calculating the energy stored for PCM, the sensible heat is accounted as *mCpViTi* � � *pcm* and the latent heat is accounted as ð Þ *ρViL pcm* cumulatively based on the temperature which the PCM attained on heating or cooling and when the PCM reached the phase change temperature. The term *Vi* represents the volume of the PCM or the water at a particular level i and *L* here represents the latent heat of the PCM when the PCM reaches the phase transition temperature.

When considering mixing as a function of time of the day, the moment of energy increases with increase in temperature of the heated water and decreases when hot water is withdrawn from the tank. The temperature profiles were determined experimentally in the fully mixed and the unmixed storage tank. Hence the moments of

energy for the unmixed or fully stratified (*Mstr*) and the fully mixed (*Mmix*) are determined and substituted in the *MIX* number as shown in Eq. (17) given as follows:

$$\text{MIX} = \frac{(\mathbf{M}\_{tr} - \mathbf{M}\_{actual})}{(\mathbf{M}\_{str} - \mathbf{M}\_{mix})} \tag{17}$$

For a given set of inlet conditions, the Mstr and Mmix represent the largest and the smallest values of the moment of energy, respectively.

#### **4.4 Error analysis**

The error associated with various primary experimental measurements and the calculation of estimated uncertainties for the performance parameters are given in **Table 4**.

#### **4.5 Results**

The temperatures being measured across the HEM tank along with the PCM while carrying out the experiment are plotted under various dimension with respect to time and analysed in the following section. The temperature variation is initially noted with respect to time and later on various types of data like the amount of heat transferred is quantified along with the different stratification analysis.


#### **Table 4.**

*Summary of the estimated uncertainties.*

*Technology in Design of Heat Exchangers for Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.108462*

*4.5.1 Comparing the charging and discharging characteristics of the storage tanks filled with PCM for all the HEM*

**Figure 9** shows the temperature variations of the heat transfer fluid in the top and bottom layers of the storage tank along with the variation of the HTF inlet temperature during the charging process. **Figures 9a** and **b** represent the temperature variations of the HTF in the storage tank with PCM balls while charging and discharging respectively. In both the figures, the shaded portion represents the temperature variations of the later of HTF along the height of the storage tank at any instant of time. The thick line on the top for **Figures 9a** and **b** represents the incoming and outgoing HTF temperature of the storage tank on charging and discharging, respectively.

In the case of storage tank with PCM balls, while charging as shown in **Figure 9a**, the incoming hot water referred by the dark line on the top of the storage tank is slowly heated and the temperature of it rises until it reaches around 95 °C after which the thermostat stops heating the incoming water temperature to rise, as it forms steam at 100 °C. As the cold water is circulated from the bottom back to the heater tank, the HTF temperature keeps rising due to the mixing of incoming hot water and the layers temperature slowly rises. Hence the stratification or the temperature difference for the HEM continuously increases to 16.68°C till the end of charging upto 270 minutes. This temperature difference of more than 15°C is maintained for more than 125 minutes at the end of charging.

However in discharging we find that a maximum temperature difference above 20°C is available for 25 minutes and it diminishes to 7°C at the end of discharging upto 66 min. The Tc-out temperature of water is the outgoing hot water temperature and more than 80°C of water can be retrieved for around 24 minutes and later on the temperature of the outgoing water reduces gradually. The outgoing water temperature runs through the center of the shaded portion as mixing of the HTF layers are predominant. Substantial number of balls present with high heat capacity is facing the

#### **Figure 9.**

*Temperature variation in the top and bottom layers of the storage tank with PCM balls in the HEM while a) charging b) discharging.*

**Figure 10.** *Instantaneous and cumulative heat transfer during the charging and discharging process.*

incoming hot water in the layers of HEM, and due to this arrangement of the PCM in the storage tank, it helps to achieve the required type of stratification.

**Figure 10** shows the instantaneous and cumulative heat transfer during the charging and discharging process which are calculated using the temperature data. It is seen from the figure that the instantaneous heat transfer continuously rises and the cumulative heat stored in the HEM at the end of the charging process is around 14,000 kJ. After deducting the heat lost during the charging process from the cumulative heat transfer, the heat stored is also calculated. From the graph, the heat lost is found to be significant at the end of the charging process.

During the discharging process, the instantaneous heat transfer is initially very high in the range of 4 to 4.5 kW. This higher instantaneous heat transfer is maintained for the first 10 min duration. Even at the end of the discharging, the instantaneous heat transfer is maintained consistently higher till the heat is discharged in the HEM. The gradual fall in the instantaneous heat transfer could be due to the slow retrieval of heat in the PCM at the bottom of the storage tank, due to high stratification created as the cold water enters from the bottom and cannot move up quickly due to high density. Hence in this case of the HEM, almost all the heat available in the storage tank of 14000 kJ is removed within the duration of 65 min. Since the instantaneous heat transfer rate decreases, further removal of heat from the storage tank appreciably will be a prolonged process.

Charging efficiency and discharging efficiency have been plotted in **Figure 11** for the HEM considered in the present investigation. It is seen from **Figure 11a** that charging efficiency is high initially as the temperature difference between the incoming hot water and the room temperature is high and hence stratification effect increases. Also during the charging process, the presence of reasonable quantity of PCM balls in the top layer retains more amount of heat at the top layer of HEM and hence the charging efficiency remains steady up to 38 % till the end and does not drop further.

During the discharging also the discharging efficiencies gradually drops to 20% for a time span of 66 min as the convection heat gradually flow from the top and bottom

*Technology in Design of Heat Exchangers for Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.108462*

**Figure 11.** *a) Charging efficiency and b) Discharging efficiency for the HEM.*

layers. Initially, the heat transfer is arrested as the PCM balls melt upto the first 23 min and the temperatures also remain high. This temperature pattern of discharging provides favourable heat transfer during the entire discharge process.

**Figure 12** shows the variation of the stratification number in the HEM during the charging and discharging process. It is observed that in this module there is a gradual increase in the stratification that favoured the rate of charging and hence the charging efficiency is upto 40%. The stratification increases upto 160 min as there is continuous difference of temperature which causes the convection of heat from top to bottom

**Figure 12.** *The variation of stratification number for a) charging and b) discharging of HEM.*

**Figure 13.** *The variation of the Richardson number in the storage tank during the charging and discharging process.*

#### **Figure 14.**

*Variation of MIX number during the a) charging process and the b) discharging process.*

layers of HTF and later on stabilises once the temperature of HTF is around the melting point of PCM 89°C.

The stratification behaviour during the discharging process shows a steep increase after 10 minutes and stratification number reaches its maximum of 5.58 till 38 min. Then it decreases slowly and reaches a level around 2 after a period of 63 min. This infers that the maximum amount of heat can be discharged upto 40 min due to better stratification and also though stratification is reduced but significant amount of heat can be retrieved in this HEM.

*Technology in Design of Heat Exchangers for Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.108462*

**Figure 13** shows the variation of the Richardson number in the storage tank during the charging and discharging process. Richardson number is a more accurate method to calculate the stratification as it includes the buoyant forces also. When the Richardson number increases it shows greater stratification whereas when the Richardson number decreases, it represents mixing forces dominate. Continuous rise in the Richardson number shows stratification found till the end and while discharging also it supports the retrieval of heat by rising gradually and then decreasing gradually after 40 minutes. The gradual increase and decrease of stratification give us an idea as to how the increase of heat charging can effectively retain and discharge the heat effectively.

**Figure 14** illustrates the MIX number variation while charging and discharging. The stratification remains long as the MIX number ends reduce from 1 to 0 while charging. The MIX = 1 represents a completely mixed condition and reaches MIX = 0 for a totally unmixed condition due to charging. The reduction of MIX number is gradual due to gradual increase in stratification.

On discharging, it is observed that there is a slow and steady increase and the mix number only increases after 10 minutes so the stratification initiates. The gradual rise in MIX number shows that there is a gradual increase in stratification and represents consistent discharging of uniform heat which is very vital for any discharging application, to achieve maximum heat retrieval.

#### **5. Conclusions**

Understanding the heat exchanger design in a thermal storage helps to design a module better. To enhance this design various parameters can be utilised as discussed above. A heat exchanger with the results is being analysed and it can conclude that the amount of heat charged is easily discharged when the temperature of melting is 89°C. The average charging and discharging efficiencies are maintained high for the HEM tank as the PCM balls present retain high heat capacity and also help to maintain better stratification along with high Richardson number.

#### **Nomenclature**


