**3. Heat transfer performance**

The heat transfer performance of the storage unit was numerically examined during the charging process. **Figure 7** shows the numerical results of the temperature and liquid fractions along the storage unit axis during the charging process when the elapsed time is 350 seconds (the inlet temperature of the HTF is 320 K, the mass flow rate is 0.0315 Kg/s).

It can be seen that the highest temperature of the PCM can be observed at the domain's top region close to the inlet of the HTF and can rise gradually in the regions closed to at the vicinity of the tube walls. Therefore, the top part of the domain converted into a liquid first and later on, melting expands to lower regions on the domain. **Figure 8**, illustrates the temperature variation in the PCM for three different radial locations. As expected, the higher temperatures are noticed in the regions close to the surface of the wall with the HTF, where the melting process takes place first. The temperature variations along the axis at the outer surface of the HTF tube are shown in **Figure 9**. It can be seen that higher temperatures exist at the domain's top. This is mainly because the HTF flows from the top to the bottom and thus the temperature rises faster at the vicinity of the tube walls close to the inlet. At this period, the effect of natural convection is not profound yet.

The effect of HTF inlet temperature on the melting process is demonstrated in **Figure 10**. It can be seen that the inlet temperature of the HTF considerably affects the rate of the melting and the PCM temperature distribution. The increase in the inlet temperature of the HTF leads to a rise in the temperature difference between the tube walls and the bulk of the PCM and thus enhance the heat transfer rate. It results in the faster rise of the liquid fraction and decreases the total melting time. **Figure 11** shows that the time for completion of melting for the HTF inlet temperature of 305 K is 2974 s, for 310 K this time is reduced to 1617s and finally, for 320 K, the time of melting is only 963 s. Therefore, the total melting time is reduced

**Figure 10.**

*Effect of the HTF inlet temperature on the melting process: (A) location T1 (h = 0.51 m, r = 0.002 m) and (B) location T2 (h = 0.95 m, r = 0.001 m),Tintial = 282.5 K, HTF mass flow rate = 0.0315Kg/s.*

**Figure 11.**

*The effect of the inlet temperature of the HTF on the formation of the liquid fraction of the PCM and on the melting time.*

**Figure 12.**

*Effect of the HTF flow rate on the melting process. The PCM temperature at locations (A) T1 (h = 0.51 m, r = 0.002 m) and (B) T2 (h = 0.95 m, r = 0.001 m),Tintial = 282.5 K, HTF mass flow rate = 0.0315Kg/s.*

#### **Figure 13.**

*The effect of the HTF flow rate on the liquid fraction formation and the melting time,Tintial = 282.5 K, Tinlet = 320 K.*

*CFD Model of Shell-and-Tube Latent Heat Thermal Storage Unit Using Paraffin as a PCM DOI: http://dx.doi.org/10.5772/intechopen.95847*

**Figure 14.**

*Temperature distribution and melting process at the bottom of the computational domain for the different HTF mass flow rates and inlet temperatures (elapsed time is 850 s).*

approximately by 68% when the inlet temperature is increased from 305 to 320 K and by 45.6% when the inlet temperature is increased from 305 to 310 K.

**Figure 12** demonstrates the effect of the HTF flow rate on the meting process. It can be seen that the flow rate accelerates the melting process due to the increased heat transfer rate. It can be seen from **Figure 13** that when the flow rate increases from 0.000315 to 0.00315 kg/s, the PCM melting time is reduced from 2781 to 1173 s (reduction by 57%). Also, the melting time is reduced by 17.8% when the mass flow rate increases from 0.00315 to 0.0315 kg/s. The effect of the HTF mass flow rate rise is less profound when compared with the effect of the rise in the HTF inlet temperature. This is demonstrated in **Figure 14** using temperature distribution counters and the PCM fluid fraction evolution diagrams.

### **4. Effect of natural convection**

To study the effect of the natural convection, the system needs to be installed in the horizontal position. Several points were created inside the computational domain to monitor the variation of the temperature inside the PCM during numerical CFD modelling. These monitoring points are placed in three planes, which are perpendicular to the axis of the system and located at distances of 0.07, 0.51 and 0.95 m from the front of the system (see **Figure 1**). **Figure 15** indicates the locations of monitoring points in the plane at a distance of 0.07 m from the front of the system.

**Figure 15.** *Locations of the monitoring points around the pipe.*

*CFD Model of Shell-and-Tube Latent Heat Thermal Storage Unit Using Paraffin as a PCM DOI: http://dx.doi.org/10.5772/intechopen.95847*

**Figure 16.** *Variation of the base PCM temperature with time.*

**Figure 16** shows the temperature variation at some of the monitoring points at the front plane of the system with the PCM for the case when the inlet temperature of the HTF is 320.7 K and the mass flow rate is 0.0315 kg/s. It can be seen that initially, the temperature increases rapidly from 280 to 299 K due to the heat transfer from the pipe walls to the solid PCM by conduction. The temperatures at monitoring points u1, r1, and b1 rise more rapidly due to their proximity to the pipe. Temperatures in monitoring points u3, r3, and b3 rise considerable slower since these points are located on the edge of the storage unit. During the melting process the temperature increases from 299 to 300.7 K and equalise at all monitoring points. Initially, during the melting process, a thin liquid layer is formed between the pipe and the solid PCM. Gradually, the solid–liquid interface expands in the axial and radial directions and the melting then is dominated by natural convection in the PCM's liquid regions. The melting process is intensified in the upper regions of the container, resulting in higher temperature recordings at the top of the computational domain (point u1).

The velocity vectors in the liquid pure PCM are shown in **Figure 17** for the elapsed time of 300, 400 and 550 s. It can be seen that the molten PCM ascends upwards from the top regions at the centre of the unit and then after cooling flows downwards to complete the natural convection circle. The convection is intensified as the liquid fraction volume increases. The velocity magnitude gradually decreases in time due to a reduction in the temperature difference in the molten PCM. These results are in good agreement with the results of several experimental and numerical investigations [35–37].

**Figure 17.** *Velocity vectors in pure PCM.*

### **5. Average heat transfer coefficient**

The local heat transfer coefficient could not be estimated accurately for the present thermal storage system as there is a temperature difference between the outer surface of the HTF pipe and the PCM along with both axial and radial directions. Consequently, the average heat transfer coefficient for the melting process is calculated instead using the following Equations [38] for the temporal heat transfer coefficient

$$h\_p = \frac{q}{A \times \Delta T (LMTD)}\tag{10}$$

where the surface area of HTFP is calculated using the following equation:

$$A = \pi D\_o l \tag{11}$$

The heat transfer rate (*q*) in the thermal storage can be calculated through the HTF's enthalpy reduction rate in the HTF [39]. This enthalpy reduction can be calculated through the following equation:

$$q = \dot{m}c\_p(T\_i - T\_o) \tag{12}$$

where, *m*\_ is the HTF mass flow,*Ti* and *To* are the HTF's inlet and outlet temperatures respectively.

The average heat transfer coefficient is:

$$\overline{h\_p} = \frac{Q\_{total}}{A \times \Delta T (LMTD) t\_n} \tag{13}$$

$$
\Delta T(LMTD) = \frac{(T\_{w1} - T\_{u1}) - (T\_{w2} - T\_{u2})}{[\ln \left(T\_{w1} - T\_{u1}\right) / (T\_{w2} - T\_{u2})]} \tag{14}
$$

where Tu1 and Tu2 are the PCM temperature at points u1 (first measurement plane) and u2 (last measurement plane); Tw1 and Tw2 are the pipe wall temperatures at the same corresponding measurement planes.

Finally, the temporal Nusselt number (*Nu*) is used to quantify heat transfer and this can be calculated using the following equation:

$$Nu = \frac{h\_p r\_{eq}}{k\_{pcm}}\tag{15}$$

The time-averaged Nusselt number *Nu* is defined as

$$\overline{Nu} = \frac{\overline{h\_p}r\_{eq}}{k\_{pcm}} \tag{16}$$

Different cases were analysed with various system's geometrical and thermophysical parameters. To generalise results it is vital to characterise them in the dimensionless form. More details about the dimensional parameters calculation and analysis can be found at [32].

The heat transfer coefficient values were calculated for the top, side and bottom regions of the storage unit for several cases during the melting processes.. **Figure 18** present results on the heat transfer coefficient changing as a function of time for the PCM during the case when the inlet temperature of the HTF is 320.7 K and the mass *CFD Model of Shell-and-Tube Latent Heat Thermal Storage Unit Using Paraffin as a PCM DOI: http://dx.doi.org/10.5772/intechopen.95847*

#### **Figure 18.**

*The heat transfer coefficient variation at the top, side, and bottom regions of the system.*

flow rate is 0.0315 kg/s. It can be seen that the heat transfer coefficients for both top and side regions increase with time. This agrees well with the initial solid PCM's temperature rise followed by the melting process. The higher heat transfer coefficient values can be observed in the top regions of the system. This can be attributed to the effect of natural convection. For the longer elapsed times, the heat transfer value stabilises. This is because the temperature distribution becomes more established within the PCM body when the system reaches the steady-state operation. At this stage, the majority of the PCM is melted and a very small part of it, which is close to the bottom regions of the unit, maybe in the solid-state.

The average heat transfer coefficient values for various HTF inlet temperature and flow rates were obtained and used to calculate the Nusselt number for the pure PCM. The flow rates of the HTF considered were 0.000315, 0.00315, 0.005, and 0.0063 Kg/s. The HTF inlet temperatures used in the modelling are 305, 310, and 320 K. To derive generic heat transfer correlations, the Nusselt numbers were calculated at the top, side and bottom regions of the storage unit. Thereafter, the average Nusselt number value for all regions is calculated. **Figures 19**-**21** show the results of these CFD modelling. As it can be seen in **Figure 19**, the Nusselt number increases with the rise in the Stefan number (see Eq. 18) which is proportional to the difference between the inlet temperature of the HTF and the melting temperature of the PCM (the increase in the inlet temperature of HTF increases the Stefan number). Similar observations can be made concerning the Rayleigh number (see Eq. 19), see **Figure 20**. The Rayleigh number is also proportional to the difference between the inlet temperature of the HTF and the melting temperature of the PCM.

$$\text{Ste} = \frac{\text{Cp}\_{pcm} \Delta T\_H}{\lambda\_{pcm}} \tag{17}$$

$$R\mathfrak{a} = \frac{\mathrm{g}\beta \mathrm{C}p\_{pcm}\rho\_{pcm}\Delta T\_H r\_{eq}^3}{\mu\_{pcm}k\_{pcm}}\tag{18}$$

$$Fo = \frac{\left(\alpha\_{pcm}t\_m\right)}{\left(r\_{eq}\right)^2} \tag{19}$$

*Variation in the Nusselt number as a function of Stefan number for (A) upper regions (B) side regions (C) bottom regions (D) the average Nusselt number (in the system).*

**Figure 20.**

*Variation in the Nusselt number as a function of Rayleigh number for (A) upper regions (B) side regions (C) bottom regions (D) the average Nusselt number (in the system).*

**Figure 21** show that the Nusselt number decreases with the rise in the Fourier number (see Eq. 20). The Fourier number is proportional to the melting time. However, an increase in the inlet temperature and flow rate of the HTF will lead to a decrease in the total melting time, and thus, reduce the Fourier number.

The average Nusselt numbers in the system are presented in **Figure 19D, 20D**, and **21D**. This data was used to derive the Nusselt number correlations for the pure *CFD Model of Shell-and-Tube Latent Heat Thermal Storage Unit Using Paraffin as a PCM DOI: http://dx.doi.org/10.5772/intechopen.95847*

**Figure 21.**

*Variation in the Nusselt number as a function of Fourier number for (A) upper regions (B) side regions (C) bottom regions (D) the average Nusselt number (in the system).*

PCM. The correlations are derived as a function of the Stefan, Fourier, and Rayleigh numbers.

The Nusselt number for the pure PCM is (*R*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*8524)

$$Nu = 2.9883 \left( \text{Ste} \right)^{0.0758} \left( Fo \right)^{-0.095} \left( Ra \right)^{0.0759} \tag{20}$$

The correlation between the numerically obtained and calculated using Eqs. (20) is shown in **Figure 22**. It can be seen that the Nusselt number varies between 2 and 4.3 for the system under investigations. The lower Nusselt numbers are for the low HTF inlet temperature, and the high Nusselt numbers values are obtained at the high HTF inlet temperatures. The total melting time of the PCM can then be calculated using the following formula (R<sup>2</sup> = 0.966)

**Figure 22.** *The correlation between numerically obtained and calculated using Eq. (20) Nusselt numbers - the pure PCM.*

$$t\_m = \frac{\rho D}{1.056 \overline{h}} \left[ \frac{c\_p (T\_m - T\_{\text{initial}}) + \lambda\_{pcm}}{(T\_{inlet} - T\_m)} \right] \tag{21}$$
