# Experimental Verification of an Analytical Mathematical Model of a Round or Oval Tube Two-Row Car Radiator

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## Abstract

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## 1. Introduction

## 2. Mathematical Model of the Two-Pass PFTHE accounting for Different HTCs in Every Tube Row

#### 2.1. Analytical Model for the First Row of Pipes

#### 2.2. Analytical Model for the Second Row of Pipes

## 3. An Analytical Model of an Internal Combustion Engine Radiator

## 4. The Air-Side Heat Transfer Correlations

- oval pipe radiator

- round pipe radiator

## 5. The Liquid-Side Heat Transfer Correlations

## 6. Experimental Results

#### 6.1. Engine Cooler Made of Oval Tubes

#### 6.2. Engine Cooler Made of Round Tubes

## 7. Analysis and Discussion of Heat Flow Rates from Water to Air Transferred in the Entire Heat Exchanger and Individual Rows of Pipes

#### 7.1. Engine Cooler Made of Oval Tubes

_{0}varied between 0.71 m/s and 2.2 m/s. Figure 3b presents heat flow rates obtained for the following data: ${\overline{\dot{V}}}_{w}=1273.37$ liters/h, ${\overline{T}}_{am}^{\prime}=14.28\xb0\mathrm{C}$, and ${\overline{T}}_{w}^{\prime}=60.51\xb0\mathrm{C}$. The water flow in the pipes was turbulent as the Reynolds number in the first pass of the cooler varied from 5238 to 5440, and in the second pass ranged from 5820 to 5883. The air velocity in front of the heat exchanger changed from 0.71 m/s to 2.2 m/s.

#### 7.2. Engine Cooler Made of Circular Tubes

## 8. Conclusions

- An exact analytical model of a single-pass double-row heat exchanger with different heat transfer coefficients on the first and second rows of pipes.
- Exact analytical model of a two-row, two-pass car radiator (plate-fin and tube heat exchanger (PFTHE)) with different heat transfer coefficients for the first and second tube rows.
- Calculation method of PFTHE without using empirical heat transfer correlations on the water and air sides.
- The results of extensive experimental research on two car radiators, one of which is made of oval tubes and the other of round tubes.
- The water flow in the pipes can be laminar, transient or turbulent while maintaining the continuity of the heat transfer coefficient when changing the flow regime. CFD modeling of all three flow ranges would be difficult.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Flow arrangement of the car radiator analysed in the study: the two-pass car radiator with two rows of tubes consists of the first tube row in the first pass (1), second tube row in the first pass (2), first tube row in the second pass (3), and second tube row in the second pass (4).

**Figure 3.**The heat transfer rate ${Q}_{t,1HTC}$ from water to air in an engine radiator made of oval tubes with an equal HTC ${h}_{a}$ on the air side of the entire radiator; (

**a**) ${\overline{\dot{V}}}_{w}=326.06$ litres/h, ${\overline{T}}_{am}^{\prime}=13.62\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=59.61\xb0\mathrm{C}$; (

**b**) ${\overline{\dot{V}}}_{w}=1273.37$ litres/h, ${\overline{T}}_{am}^{\prime}=14.28\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=60.51\xb0\mathrm{C}$; 1—air side equation for ${\mathrm{Nu}}_{a}$ obtained from the CFD modelling, 2—empirical air side equation for ${\mathrm{Nu}}_{a}$, 3–relative difference e.

**Figure 4.**Heat flow rate $Q$ transferred from hot water to cool air in the car radiator; 1: ${Q}_{w,\mathrm{exp}}$ values determined using the measurement results, 2: heat flow rate ${Q}_{w,calc}$ obtained for the air-side Nusselt number ${\mathrm{Nu}}_{a}$ estimated by CFD simulation (

**a**) or empirical correlation (

**b**), 3: heat flow rate ${Q}_{w,calc}$ calculated using the CFD-based Equation (a) or empirical Equation (b) and the average input data from seven measurement series: ${\overline{\dot{V}}}_{w}=326.06$ liters/h, ${\overline{T}}_{am}^{\prime}=13.62\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=59.61\xb0\mathrm{C}$, 4: the relative difference between ${Q}_{w,\mathrm{exp}}$ and ${Q}_{w,calc}$ (between 1 and 2).

**Figure 5.**The thermal output $Q$ of the car radiator; 1: ${Q}_{w,\mathrm{exp}}$ values determined using the measurement results, 2: thermal output ${Q}_{w,calc}$ calculated with the air-side Nusselt number ${\mathrm{Nu}}_{a}$ obtained by CFD simulation (

**a**) or empirical Equation (

**b**), 3: thermal output ${Q}_{w,calc}$ obtained using the CFD-based correlation (

**a**) or empirical correlation (

**b**) and the average input data from seven measurement series: ${\overline{\dot{V}}}_{w}=1273.37$ litres/h, ${\overline{T}}_{am}^{\prime}=14.28\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=60.51\xb0\mathrm{C}$, 4: the relative difference between ${Q}_{w,\mathrm{exp}}$ and ${Q}_{w,calc}$.

**Figure 6.**Comparison of thermal outputs of the specific pipe rows and whole-car radiator considering that the air-side HTC is constant in both pipe rows with the corresponding thermal outputs for different HTCs in the first and second pipe rows; (

**a**) ${\overline{\dot{V}}}_{w}=326.06$ litres/h, ${\overline{T}}_{am}^{\prime}=13.62\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=59.61\xb0\mathrm{C}$, (

**b**) ${\overline{\dot{V}}}_{w}=1273.37$ litres/h, ${\overline{T}}_{am}^{\prime}=14.28\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=60.51\xb0\mathrm{C}$; ${Q}_{i,1HTC}$, $i=1,\dots ,4$: thermal output of the specific pipe row (Figure 1) adopting the same correlation (32) for the Nusselt number on the air side, ${Q}_{i,2HTC}$, $i=1,\dots ,4$: thermal outputof a specific pipe row for different correlations for the air side Nusselt number (Figure 1); Equation (30) was used for the first row and Equation (31) for the second row of tubes, ${Q}_{t,1HTC}$: car radiator output for constant HTC calculated from Equation (32) for all rows of pipes, ${Q}_{t,2HTC}$: car radiator output for different HTCs; Equation (30) was used for the first tube row and Equation (31) for the second tube row.

**Figure 7.**The relative differences between thermal outputs of specific tube rows for different and constant HTCs; (

**a**) ${\overline{\dot{V}}}_{w}=326.06$ litres/h, ${\overline{T}}_{am}^{\prime}=13.62\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=59.61\xb0\mathrm{C}$, (

**b**) ${\overline{\dot{V}}}_{w}=1273.37$ litres/h, ${\overline{T}}_{am}^{\prime}=14.28\xb0\mathrm{C}$ and ${\overline{T}}_{w}^{\prime}=60.51\xb0\mathrm{C}$.

**Figure 8.**The heat flow rate ${Q}_{t,1HTC}$ from water to air in an engine cooler constructed from round tubes with an equal HTC on the air side of the entire radiator versus the first-pass Reynolds number ${\mathrm{Re}}_{w,u}$ on the water side; 1: air-side correlation for ${\mathrm{Nu}}_{a}$ based on CFD modelling, 2: air-side relationship for ${\mathrm{Nu}}_{a}$ based on experimental data, 3: relative difference e.

**Figure 9.**The comparison of radiator output $Q$ versus the water-side Reynolds number ${\mathrm{Re}}_{w,u}$ obtained experimentally and using the mathematical model of the engine cooler with the uniform air-side Nusselt number ${\mathrm{Nu}}_{a}$ determined by (

**a**) CFD modelling and (

**b**) empirical correlation; $e$: relative difference.

**Figure 10.**Comparison of thermal capacities of specific pipe rows and the whole-car radiator supposing that HTC on the air-side is uniform in both pipe rows with the corresponding thermal capacities obtained for different HTCs in the first and second pipe rows; ${\overline{w}}_{0}=2.27$ m/s, ${\overline{T}}_{am}^{\prime}=8.24\xb0\mathrm{C}$, and ${\overline{T}}_{w}^{\prime}=70.56\xb0\mathrm{C}$; ${Q}_{i,1HTC}$, $i=1,\dots ,4$: thermal output of the specific pipe rows (Figure 1) supposing the same Equation (32) for the Nusselt number on the air-side throughout the radiator, ${Q}_{i,2HTC}$, $i=1,\dots ,4$: thermal outputs of individual pipe rows (Figure 1) for different equations for the air-side Nusselt in both pipe rows; Equation (30) was used for the first row of tubes and Equation (31) for the second row, ${Q}_{t,1HTC}$: car radiator output calculated using Equation (32) for all rows of pipes, ${Q}_{t,2HTC}$: car radiator output calculated using different heat transfer correlations for the first and second tube rows.

**Figure 11.**The relative differences ${e}_{i}$ and ${e}_{t}$ between thermal outputs of individual tube rows and the entire car radiator for different and uniform HTCs calculated using Equations (55) and (56); ${\overline{w}}_{0}=2.27$ m/s, ${\overline{T}}_{am}^{\prime}=8.24\xb0\mathrm{C}$, and ${\overline{T}}_{w}^{\prime}=70.56\xb0\mathrm{C}$.

Geometric Data | The Heat Exchanger of Oval Pipes | The Heat Exchanger of Round Pipes |
---|---|---|

$\mathrm{Outer}\mathrm{tube}\mathrm{diameter}{d}_{o}$, mm | $\mathrm{Maximum}\mathrm{tube}\mathrm{diameter}{d}_{o,\mathrm{max}}=11.82$ mm $\mathrm{Minimum}\mathrm{tube}\mathrm{diameter}{d}_{o,\mathrm{min}}=6.35$ mm | ${d}_{o}$ = 7.2 mm |

$\mathrm{Tube}\mathrm{wall}\mathrm{thickness}{\delta}_{w}$, mm | ${\delta}_{w}$ = 0.4 mm | ${\delta}_{w}$ = 0.5 mm |

$\mathrm{Height}{p}_{1}$$\mathrm{and}\mathrm{width}{p}_{2}$ of the apparent fin associated with one tube, mm | ${p}_{1}=18.5$$\mathrm{mm},{p}_{2}=17$ mm | ${p}_{1}=18.5$$\mathrm{mm},{p}_{2}=12$ mm |

$\mathrm{Fin}\mathrm{thickness}{\delta}_{f}$, mm | ${\delta}_{f}$ = 0.08 mm | ${\delta}_{f}$ = 0.08 mm |

$\mathrm{Air}-\mathrm{side}\mathrm{hydraulic}\mathrm{diameter}{d}_{ha}$, mm | ${d}_{ha}$ = 1.41 mm | ${d}_{ha}$ = 1.95 mm |

$\mathrm{Water}-\mathrm{side}\mathrm{hydraulic}\mathrm{diameter}{d}_{hw}$, mm | ${d}_{hw}$ = 7.06 mm | ${d}_{hw}={d}_{o}-2{\delta}_{w}$ = 6.2 mm |

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**MDPI and ACS Style**

Taler, D.; Taler, J.; Trojan, M.
Experimental Verification of an Analytical Mathematical Model of a Round or Oval Tube Two-Row Car Radiator. *Energies* **2020**, *13*, 3399.
https://doi.org/10.3390/en13133399

**AMA Style**

Taler D, Taler J, Trojan M.
Experimental Verification of an Analytical Mathematical Model of a Round or Oval Tube Two-Row Car Radiator. *Energies*. 2020; 13(13):3399.
https://doi.org/10.3390/en13133399

**Chicago/Turabian Style**

Taler, Dawid, Jan Taler, and Marcin Trojan.
2020. "Experimental Verification of an Analytical Mathematical Model of a Round or Oval Tube Two-Row Car Radiator" *Energies* 13, no. 13: 3399.
https://doi.org/10.3390/en13133399