2. Convection Coefficient: h=150 Wm2∙K

3. Specific Heat of Water: CP=4.184 kJkg∙K

5. Node C2 (Top Right Corner)

6. Node C3 (Bottom Left Corner)

7. Node C4 (Bottom Right Corner)

8. Node Group 1 (Heater Wall Nodes)

9. Node Group 2 (Water Surface Nodes)

10. Node Group 3 (Axis of Symmetry Nodes)

11. Node Group 4 (Pool Floor Nodes)

12. Interior NodesResults and Discussion:<br />Based on the finite difference approximation, the pool takes roughly 5 days and 11 hours to heat up to an average temperature of 30 ºC. Figure 4 illustrates an approximate linear increase in temperature over time. In contrast, when using a lumped system approximation, the average temperature of the pool appears to oscillate in a sinusoidal manner as shown in Figure 5. In the lumped system analysis, it is important to note that the average temperature never reaches 30 ºC. It converges to a steady state condition in which the convection from the ambient air and radiation from the sun contribute to a larger heat flux that the heaters can provide.<br />Figure 6 shows a lumped system approximation over 20 days. The heater was turned off after 10 days to show effects of not having the heater on. The steady state temperature average with no heater is only about 2 °C lower than the system with the heaters on.<br />The main difference between finite difference analysis and the lumped system approximation is the assumption that in finite difference analysis, the system acts as a ‘solid’ in conduction with the heat sources, with a relatively low thermal conductivity. From Figure 7, we can see that because of the low thermal conductivity of water, the lower middle portion of the pool was not affected by the temperature gradient and heat sources, even after 7 days of heat transfer. This shows that the pool’s absorption of heat is not even throughout the fluid. The sides can reach around 60 °C while the lower middle part stays around 10 °C. Because of low thermal conductivity, there is not a significant change in average temperature during the night when the temperature approaches freezing temperature. This is because only the surface of the pool is exposed to the convection, and since the thermal conductivity of water is so low, the rate of heat transfer between water molecules is low relative to that of the convection to the ambient air (Figure 8). <br />The lumped capacitance method would assume perfect mixing of the water, and essentially a negligible temperature gradient between the nodes in the pool. Although this method would eliminate the need to consider natural convection as a source for heat dissipation, it is not a valid approach, since the rate of convective heat transfer is much higher than the rate of conductive heat transfer within the water.<br />We would be able to more accurately represent the true heat distribution throughout the pool if we considered the effects of mixing, and natural convection. Since the outer edges of the pool experienced a high heat transfer, their densities would have changed, causing natural convection, and a driving force for fluid motion. We assumed, however, that the densities remained constant, and because of this minor detail, we did not attain a perfect model of the heat distribution inside the pool.<br />-25908015240<br />Figure 4: Average pool temperature per day using a finite difference approximation over 7 days.<br />-167640193040<br />Figure 5: Average pool temperature per day using a lumped system approximation over 7 days.<br />-106680-289560<br />Figure 6: Average pool temperature per day using Lumped system approximation with heaters turned off after 10 days. <br />9525024765<br />Figure 7: Temperature distribution at 30 °C average Temperature: lack of uniformity of distribution<br />Figure 8: Temperature of pool over time: the convection at the surface cools the top most layer of water, but because of the low thermal conductivity of water, heat isn’t transferred well to the top layer, keeping it colder than the layer under it.<br />