# Water Heating and Circulating Heating System with Energy-Saving Optimization Control
## 3 The LPPT Method
### Water Heating and Circulating System and LPP  
Figure 4 shows the water heating and circulating system architecture, which includes a heat generation module and a thermal load module. The heat generation module (Sous-Vide Cooker) is composed of a heater and a pump, and the inputs are heater input power \(P_{h}\) and pump speed \(\omega\), respectively, to generate heat \(Q_{g}\). The heat load includes the heat content of the water in the pot, the heat loss \(Q_{o}\) from the external environment, and the heat absorption \(Q_{d}\) of the internal heating foods. The heat balance equation for the system is established  
by Formula (1), where \(T_{w}\) is the water temperature, \(\dot{T}_{w}\) represents the derivative of water temperature with respect to time, and \(C_{w}\) is the heat content of the water in the pot. \[\dot{T}_{w}=\frac{1}{C_{w}}\Big{(}Q_{g}-\ Q_{fd}-Q_{o}\Big{)} \tag{1}\]  
The heat \(Q_{g}\) output by the Sous-Vide Cooker is shown in Formula (2), and it is obtained by multiplying the difference between the heater temperature and the tank water temperature by the pump flow rate. Here, \(\dot{m}_{w}\) is the output flow of the pump, \(S_{w}\) is the specific heat of water, and \(T_{h}\) is the output temperature of the heater. The heater temperature \(T_{h}\) is controlled by the input power \(P_{h}\), and the pump flow \(\dot{m}_{w}\) is controlled by the pump motor speed \(\omega\). \[Q_{g}=\dot{m}_{w}S_{w}\ (T_{h}-T_{w}) \tag{2}\]  
The heater has three parameters: \(C_{h}\) is its heat capacity, \(P_{h}\) is its input power, and \(k_{h}\) is its heat transfer coefficient, and the calculation of the output temperature \(T_{h}\) of the heater is shown in Formula (3). \[\dot{T}_{w}=\frac{1}{C_{h}}(P_{h}-\ k_{h}T_{h}) \tag{3}\]  
For the calculation of the pump flow rate, it is a function of the pump motor speed \(\omega\), as shown in Formula (4), where \(\rho\) is the water density, \(A_{g}\) is the area of the pump outlet, and \(r\) is the radius of the pump fan blade radius. In this example, the transient change in pump flow can be ignored because the time constant of pump flow is much smaller than the time constant of water temperature heating. \[\dot{m}_{w}=\ \rho\ A_{g}\ r\ \omega \tag{4}\]  
The above is the establishment of the heating model of the Sous-Vide Cooker. Next, we will explain the model of the pot and heating foods. The heat lost \(Q_{o}\) by the pot to the outside includes the convective heat \(Q_{cv}\) to the outside, which is expressed as in (5), and the heat conduction heat \(Q_{cd}\) between the pot and the outside is shown in Equation (6). \[Q_{cv}=A_{cv}h_{cv}(T_{w}-T_{air}) \tag{5}\]  
\[Q_{cd}=A_{cd}k_{cd}(T_{w}-T_{o}) \tag{6}\]  
where \(A_{cv}\) is heat convection area, \(h_{cv}\) is heat convection coefficient, \(T_{air}\) is ambient temperature for heat convection, \(A_{cd}\) is heat transfer area, \(k_{cd}\) is heat transfer coefficient, and \(T_{o}\) is ambient temperature for heat conduction. In order to define the heat absorption capacity of the tabletop and air surrounding the pot shell, Equation (6) assumes a conduction distance  
Figure 4: The water heating and circulating system architecture. of 1 m and multiplies it by \(k_{cd}\) to simplify the calculation model. Finally, calculate the heat absorption \(Q_{fd}\) of the heating foods, as shown in (7) and (8), where \(R_{fd}\) is the thermal resistance of the ingredients, and \(T_{f}\) and \(C_{fd}\) are the temperature and heat capacity of the foods, respectively. \[Q_{fd}=\frac{1}{R_{fd}}\Big{(}T_{w}-\ T_{f}\Big{)} \tag{7}\]  
\[\dot{T}_{f}=\frac{1}{C_{fd}}Q_{fd} \tag{8}\]  
By organizing the interconnections between Formulas (1)-(8), the block diagram of the water heating and circulating system can be derived, as shown in Figure 5.  
Food has a long maturation period when using the water heating and circulating method. In this way, the overall operating efficiency can be effectively improved if it can be controlled in a steady state at the lowest input power point.