# Water Heating and Circulating Heating System with Energy-Saving Optimization Control

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. The History of Development—Water Heating and Circulating System

#### 2.2. The Current Technical Features and Capabilities

- Vacuum-sealed foods;
- Precise and long-term temperature control;
- The control of water flow;
- The combination of heating and water temperature.

#### 2.3. Temperature Control

#### 2.4. The Water Flow Control

#### 2.5. The Heating Efficiency of Water Heating and Circulating

## 3. The LPPT Method

#### 3.1. Water Heating and Circulating System and LPP

_{h}and pump speed ω, 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

_{fd}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.

_{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, ṁ

_{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 ṁ

_{w}is controlled by the pump motor speed ω.

_{h}is its heat transfer coefficient, and the calculation of the output temperature ${T}_{h}$ of the heater is shown in Formula (3).

_{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.

_{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).

_{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 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.

_{g}for the water temperature to be quickly heated to the desired level. As the water temperature reaches its target, the demand for Q

_{g}decreases. Heater input power drops to less than about four times pump motor-rated power. It is possible to optimize the system’s efficiency by controlling the pump motor speed as the proportion of pump motor power increases in relation to the total input power.

_{h}of the heater can be computed using Formulas (2)–(4), as depicted in Formula (9). It can be observed that P

_{h}is inversely proportional to the pump motor speed ω.

_{m}is directly proportional to the cube of the pump motor speed ω [28]. The pump power P

_{m}is shown in (10), where Tm is the output torque of the pump motor, T

_{m}= ṁ

_{w}*ω.

_{in}of the system can be expressed as shown in Formula (11), which clearly reveals that P

_{in}is a function of the pump speed ω. By taking the differentiating of the input power with respect to the pump speed, the minimum input power point can be obtained when the condition of dP

_{in}/dω equals zero.

_{LPP}, corresponding to the minimum input power point, is represented in Formula (12). Upon reaching a state of steady temperature control, the output power Q

_{g}of the Sous-Vide Cooker only has to be responsible for compensating for heat loss due to conduction and convection within the heating pan. In this scenario, Formula (12) can be applied as Q

_{g}= Q

_{cd}+ Q

_{cv}.

_{h}of the heater is inversely proportional to the rotational speed ω of the pump motor. In general, pump power P

_{m}is proportional to pump motor speed ω

^{3}, and the relationship is shown in Figure 6. The solid line in Figure 6 shows the total input power P

_{in}, which is calculated by summing the two parameters. With the change in pump motor speed, there will be a minimum input power point.

_{in}, heater power P

_{h}, and pump power P

_{m}have been simulated and calculated in a steady state. The results are shown in Figure 7 and Figure 8, and the system parameters utilized can be found in Table 1. Figure 7 shows the relationship between power and pump speed under different temperature settings and a constant volume of 20 L. As the temperature decreases, the speed at the lowest input power point decreases correspondingly. By comparing the relationship between power and pump speed under different volumes with a constant temperature setting of 65 °C, it can be observed that the speed at the lowest input power point decreases as the volume decreases, as illustrated in Figure 8.

#### 3.2. The Methodology of Control and Lowest Power Point Tracking

_{g}. However, Formula (2) shows that the heating capacity Q

_{g}of the Sous-Vide Cooker is a function of the pump flow rate ṁ

_{w}and the heater temperature T

_{h}, and the two are inversely proportional to each other. That is, in the steady state, T

_{h}can be changed through the adjustment of the flow ṁ

_{w}, and the input power of the Sous-Vide Cooker can be further reduced.

_{in}and Δω, are calculated. As the positive and negative relationship between the two changes, gradually decrease the pump speed command until the system finds the lowest point of power input.

## 4. System Experiment Simulation

#### 4.1. Simulation in Water Heating and Circulating System

#### 4.2. Same Pot Volume, Different Target Temperature

#### 4.3. Same Target Temperature, Different Pot Volume

#### 4.4. Simulation Result

## 5. Following the Experimental Study

#### 5.1. Experimental System

#### 5.2. Experimental Results

#### 5.3. LPPT Control Results

#### 5.4. Experimental Data Comparison

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Thermostats for controlling the temperature of food and water (from seattlefoodgeek.com (1 February 2020).

**Figure 7.**Power–speed relationship with constant volume (20 liters) and varying temperatures, denoted by minimum input power point ‘o’.

**Figure 8.**Power–speed relationship with constant temperature (65 °C) and varying volumes, denoted by minimum input power point ‘o’.

**Figure 13.**Traditional water heating and circulating control results. (

**a**) Pump speed; (

**b**) Water temperature; (

**c**) Input power.

**Figure 15.**Same target temperature, different pot volume. (

**a**) Pump speed; (

**b**) Water temperature; (

**c**) Input power—traditional control; (

**d**) LPPT control.

**Figure 20.**Experimental results of traditional temperature control. (

**a**) Temperature control; (

**b**) Pump speed.

Symbol | Quantity | Value | Units |
---|---|---|---|

S_{w} | Water-Specific Heat | 4.184 | J/g⋅K |

k_{h} | Heater Heat Transfer Coefficient | 1.3 | W/m⋅K |

k_{cd} | Pot Heat Transfer Coefficient | 60 | W/m^{2}.K |

h_{cv} | Air Heat Convection Coefficient | 5 | W/m^{2}.K |

ρ | Water Density | 1000 | kg/m^{3} |

r | Pump Blade Radius | 0.021 | m |

A_{g} | Pump Outlet Area | 0.0002 | m^{2} |

A_{cd} | Heat Transfer Area (20 L) | 0.292 | m^{2} |

A_{cv} | Heat Convection Area (20 L) | 0.146 | m^{2} |

T_{o} | Ambient Temperature for Heat Conduction | 30 | °C |

T_{air} | Ambient Temperature for Heat Convection | 30 | °C |

**Table 2.**Input wattage and energy-saving ratio for temperature of 55 °C, 65 °C, and 75 °C (pot volume 5 L).

5 L | Traditional Control | LPPT Control | Improvement Percentage |
---|---|---|---|

55 °C | 206 | 170 | 17.5% |

65 °C | 220 | 195 | 11.4% |

75 °C | 235 | 220 | 6.4% |

**Table 3.**Input wattage and energy-saving ratio for temperature of 55 °C, 65 °C, and 75 °C (pot volume 10 L).

10 L | Traditional Control | LPPT Control | Improvement Percentage |
---|---|---|---|

55 °C | 222 | 204 | 8.3% |

65 °C | 247 | 232 | 6.1% |

75 °C | 275 | 267 | 2.9% |

**Table 4.**Input wattage and energy-saving ratio for temperature of 55 °C, 65 °C, and 75 °C (pot volume 15 L).

15 L | Traditional Control | LPPT Control | Improvement Percentage |
---|---|---|---|

55 °C | 243 | 230 | 5.5% |

65 °C | 280 | 272 | 2.9% |

75 °C | 320 | 317 | 1.0% |

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**MDPI and ACS Style**

Lin, F.-C.; Chen, C.-S.; Lin, C.-J.
Water Heating and Circulating Heating System with Energy-Saving Optimization Control. *Appl. Sci.* **2023**, *13*, 5542.
https://doi.org/10.3390/app13095542

**AMA Style**

Lin F-C, Chen C-S, Lin C-J.
Water Heating and Circulating Heating System with Energy-Saving Optimization Control. *Applied Sciences*. 2023; 13(9):5542.
https://doi.org/10.3390/app13095542

**Chicago/Turabian Style**

Lin, Feng-Chieh, Chin-Sheng Chen, and Chia-Jen Lin.
2023. "Water Heating and Circulating Heating System with Energy-Saving Optimization Control" *Applied Sciences* 13, no. 9: 5542.
https://doi.org/10.3390/app13095542