# An Economical and Precise Cooling Model and Its Application in a Single-Cylinder Diesel Engine

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## Abstract

**:**

## 1. Introduction

## 2. Model of the Cooling System

#### 2.1. Heat Rejection Model of Engine

#### 2.2. Target Temperature of the Water Jacket

#### 2.3. Model of the Radiator

#### 2.4. Model of the Fan and Pump

## 3. Model Solutions

#### 3.1. Electric Cooling Model Solution

- Matching the fan and water pump to the cooling system: $\Delta {p}_{a}$ in Equation (4) should be equal to $\delta {p}_{a}$ in Equation (7) to ensure that the fan works normally. This step is the same for the water pump. We combine these equations to obtain:$$\begin{array}{ccc}\hfill {q}_{a}& =& {q}_{a}\left({n}_{a}\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{a}={P}_{a}\left({n}_{a}\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {q}_{w}& =& {q}_{w}\left({n}_{w}\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{w}={P}_{w}\left({n}_{w}\right)\hfill \end{array}$$
- Calculating the heat dissipation of the radiator: Plug ${q}_{w},{q}_{a}$ in Equations (8) and (9) into Equation (3) to obtain,$${Q}_{c}={Q}_{c}(\phantom{\rule{3.33333pt}{0ex}}\Delta T,\phantom{\rule{3.33333pt}{0ex}}{q}_{a}\left({n}_{a}\right),\phantom{\rule{3.33333pt}{0ex}}{q}_{w}\left({n}_{w}\right)\phantom{\rule{3.33333pt}{0ex}}),$$By assuming that the heat rejection in Equation (1) is equal to the heat dissipation of the radiator, one obtains:$$Q(n,\varphi )={Q}_{c}(\phantom{\rule{3.33333pt}{0ex}}\Delta T,\phantom{\rule{3.33333pt}{0ex}}{q}_{a}\left({n}_{a}\right),\phantom{\rule{3.33333pt}{0ex}}{q}_{w}\left({n}_{w}\right)\phantom{\rule{3.33333pt}{0ex}}).$$Recall that $\Delta T=T(n,\varphi )-{T}_{atm}$, where $T(n,\varphi )$ is the target temperature of the water jacket and ${T}_{atm}$ is the atmospheric temperature. In Equation (11), only ${n}_{a},{n}_{w}$ are unknown; thus, the fan speed ${n}_{a}$ can be expressed as a function of the water pump speed ${n}_{w}$,$${n}_{a}={n}_{a}\left({n}_{w}\right).$$
- Obtaining $({n}_{a},{n}_{w})$ and the total power consumption P: P can be written as: $P={P}_{a}\left({n}_{a}\right)+{P}_{w}\left({n}_{w}\right)$ and combined Equation (12), P can be rewritten as a single-variable function of ${n}_{w}$, thus minimizing the total power consumption; therefore, P can be written as:$$\frac{\mathrm{d}P}{\mathrm{d}{n}_{w}}=0,$$Using this equation, ${n}_{w}$ and then ${n}_{a}$ can be solved, where P has a minimum value. At this point, the solution of the electric cooling model is complete.

#### 3.2. Mechanical Cooling Model Solution

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 10.**The plot ensures that the cooling system works well in all cases. The gray surface representing ${Q}_{c}$ in Equation (14) is just above the colorful surface, which represents the heat rejection Q of the engine.

PM | CO | CH + NOx | |
---|---|---|---|

China-IV | 0.6 | 5.5 | 7.5 |

EPA 3 | 0.4 | 5.0 | 4.7 |

Electric cooling system | 0.374 | 1.34 | 6.5 |

Mechanical cooling system | 0.436 | 4.19 | 9.2 |

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**MDPI and ACS Style**

Xie, Z.; Wang, A.; Liu, Z.
An Economical and Precise Cooling Model and Its Application in a Single-Cylinder Diesel Engine. *Appl. Sci.* **2021**, *11*, 6749.
https://doi.org/10.3390/app11156749

**AMA Style**

Xie Z, Wang A, Liu Z.
An Economical and Precise Cooling Model and Its Application in a Single-Cylinder Diesel Engine. *Applied Sciences*. 2021; 11(15):6749.
https://doi.org/10.3390/app11156749

**Chicago/Turabian Style**

Xie, Zhifeng, Ao Wang, and Zhuoran Liu.
2021. "An Economical and Precise Cooling Model and Its Application in a Single-Cylinder Diesel Engine" *Applied Sciences* 11, no. 15: 6749.
https://doi.org/10.3390/app11156749