**2. Mathematical model of the two-pass PFTHE with different heat transfer coefficients on each tube row**

An analytical mathematical model of a two-pass radiator with two rows of pipes was developed. The first and second-row HTCs were calculated using heat transfer correlations, which were determined based on CFD (Computational Fluid Dynamics) simulations. The radiator flow diagram is shown in **Figure 1**.

The mass flow rate *m*\_ *<sup>w</sup>* to the car radiator is divided into two equal parts *m*\_ *<sup>w</sup>=*2 flowing through the first and second row of pipes in the first pass. The starting point for building a mathematical model of a car radiator is a single-pass double-row heat exchanger (first pass of two-pass heat exchanger with two rows of tubes) (**Figure 2**).

#### **Figure 1.**

*Flow arrangement of the analyzed two-pass car radiator with two rows of tubes; 1 – First tube row in the first pass, 2 - second tube row in the first pass, 3 - first tube row in the second pass, 4 - second tube row in the second pass.*

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

## **2.1 Analytical model for the first row of pipes**

The governing energy conservation equation equations for the water and air in the first tube row are

$$\frac{d\,T\_{w,1}(\boldsymbol{\chi}^{+})}{d\boldsymbol{\chi}^{+}} = -N\_{w}^{I} \left[ T\_{w,1}(\boldsymbol{\chi}^{+}) - T\_{ma}^{I}(\boldsymbol{\chi}^{+}) \right] \qquad \mathbf{0} \leq \mathbf{x}^{+} \leq \mathbf{1} \tag{1}$$

$$\frac{\partial T\_a^l(\mathbf{x}^+, \mathbf{y}\_1^+)}{\partial \mathbf{y}\_1^+} = \mathbf{N}\_a^l \left[ T\_{w, \mathbf{1}}(\mathbf{x}^+) - T\_a^l(\mathbf{x}^+, \mathbf{y}\_1^+) \right] \qquad \mathbf{0} \le \mathbf{x}^+ \le \mathbf{1} \qquad \mathbf{0} \le \mathbf{y}\_1^+ \le \mathbf{1} \tag{2}$$

where: *x*<sup>þ</sup> ¼ *x=Lc*, *y*<sup>þ</sup> <sup>1</sup> ¼ *y*1*=p*<sup>2</sup> dimensionless coordinates; *x*, *y*<sup>1</sup> Cartesian coordinates (**Figure 2**) *Tw*,1 the temperature of the water in the first row of pipes, *T<sup>I</sup> <sup>a</sup>* the temperature of the air in the first row of pipes.

Two local dimensionless coordinate systems were introduced *x*þ, *y*<sup>þ</sup> 1 � � and *x*þ, *y*<sup>þ</sup> 2 � � (**Figure 2**). The average air temperature *T<sup>I</sup> m a x*<sup>þ</sup> ð Þ over the first-row width *p*<sup>2</sup> is determined by the expression

$$T\_{m,a}^{I}(\boldsymbol{\varkappa}^{+}) = \int\_{\boldsymbol{\varkappa}\_{1}^{+} = 0}^{1} T\_{a}^{I}(\boldsymbol{\varkappa}^{+}, \boldsymbol{\jmath}\_{1}^{+}) \, d\boldsymbol{\jmath}\_{1}^{+} \tag{3}$$

The number of heat transfer units on the water-side *N<sup>I</sup> <sup>w</sup>* and air-side *N<sup>I</sup> <sup>a</sup>* are defined as follows

$$\mathbf{N}\_w^I = \mathbf{2} \mathbf{U}^I \mathbf{A}\_{\rm{bout}}^I / \left( \dot{m}\_w \ \overline{c}\_{pw} \right) \qquad \mathbf{N}\_a^I = \mathbf{U}^I \mathbf{A}\_{\rm{bout}}^I / \left( \dot{m}\_a \ \overline{c}\_{pa} \right) \tag{4}$$

where: *A<sup>I</sup> bout* <sup>¼</sup> *nI uPoutLc*, *nI <sup>u</sup>* total number of pipes in the first row of the first pass, *AI bout* outer surface area of the bare pipes in the first row of the first pass, *Pout* outer circumference of the plain tube, *Lc* length of a tube in the PFTHE, The symbol*cpw* means the average specific heat of water in the temperature range from *T*<sup>00</sup> *<sup>w</sup>* to *T*<sup>0</sup> *<sup>w</sup>*, and the average specific heat of air*cpa* in the temperature range from *T*<sup>0</sup> *am* to *T*<sup>00</sup> *am* (**Figure 2**).

The overall HTC *U<sup>I</sup>* is calculated using the expression

$$\frac{\mathbf{1}}{\mathbf{U}^{I}} = \frac{A\_{bout}^{I}}{A\_{bin}^{I}} \frac{\mathbf{1}}{h\_{in}^{I}} + \frac{A\_{bout}^{I}}{A\_{wm}^{I}} \frac{\delta\_{w}}{k\_{w}} + \frac{\mathbf{1}}{h\_{oc}^{I}} \tag{5}$$

where: *AI bin* <sup>¼</sup> *<sup>n</sup><sup>I</sup> uPinLc* inner surface area of the pipes in the first row of the first pass, *Pin* inner circumference of the tube, *hI in* HTC on the tube inner surface in the

$$\begin{array}{c|c} \chi^{\*} = \mathsf{O} & \begin{array}{c|c} \chi^{\*} = \mathsf{O} & \begin{array}{c|c} \chi^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} \end{array} \\ \hline \chi^{\*} = \mathsf{O} & \begin{array}{c|c} \chi^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} \neq \mathsf{O} \end{array} & \begin{array}{c|c} \chi^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} \end{array} \\ \hline \chi^{\*} = \mathsf{O} & \begin{array}{c|c} \chi^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} \\ \chi^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} \\ \chi^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} & \text{if } \mathsf{w}^{\*} = \mathsf{O} \\ \end{array} \end{array}$$

**Figure 2.** *Diagram of a single-pass two-row heat exchanger.* first row of the first pass, *Awm* <sup>¼</sup> *AI bout* <sup>þ</sup> *AI bin* � �*=*2 mean heat transfer area of the plain tubes in the first row of the first pass, *δ<sup>w</sup>* tube wall thickness, *kw* thermal conductivity of the tube material. The equivalent HTC *hI oe*, taking into account the heat transfer through the fins fixed to the outer surface of a plain pipe for the first row of pipes, is given by

$$h\_{o\epsilon}^{I} = h\_a^{I} \left[ \frac{A\_{bf}^{I}}{A\_{bout}^{I}} + \frac{A\_{f}^{I}}{A\_{bout}^{I}} \eta\_f \left( h\_a^{I} \right) \right] \tag{6}$$

where: *h<sup>I</sup> <sup>a</sup>* air-side HTC in the first row of tubes in the first pass, *A<sup>I</sup> bf* tube surface area between fins, *η <sup>f</sup>* fin efficiency.

The boundary conditions for Eqs. (1) and (2) are as follows

$$T\_{w,1}|\_{\mathbf{x}^+=\mathbf{0}} = T\_{w,\mathbf{1}}' \tag{7}$$

*dTw*,2 *x*<sup>þ</sup> ð Þ

*∂TII*

*TII*

*<sup>a</sup> x*þ, *y*<sup>þ</sup> 2 � � *∂ y*<sup>þ</sup> 2

where

*d x*<sup>þ</sup> ¼ �*NII*

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

*<sup>w</sup>* <sup>¼</sup> <sup>2</sup>*UII <sup>A</sup>II*

<sup>¼</sup> *<sup>N</sup>II*

*NII*

should then be replaced by *hII*

*m a x*<sup>þ</sup> ð Þ is determined as follows

appropriate boundary conditions are

using Eq. (6). HTC *h<sup>I</sup>*

row of pipes is *T*<sup>0</sup>

*Tw*,2 *x*<sup>þ</sup> ð Þ¼

0≤*x*<sup>þ</sup> ≤1

*TII <sup>a</sup> x*þ, *y*<sup>þ</sup> 2

where

*<sup>D</sup>* <sup>¼</sup> *<sup>N</sup>II w NII a*

**35**

*E*

0≤*x*<sup>þ</sup> ≤1 0 ≤*y*<sup>þ</sup>

<sup>1</sup> � exp �*NII*

behind the second row of pipes *T*‴

account in Eq. (21) that *y*<sup>þ</sup>

for the air temperature *T*‴

*T*0 *am* þ *<sup>w</sup> Tw*,2 *<sup>x</sup>*<sup>þ</sup> ð Þ� *<sup>T</sup>II*

*<sup>a</sup> x*þ, *y*<sup>þ</sup> 2 � � � � 0≤ *x*<sup>þ</sup> ≤1 0 ≤*y*<sup>þ</sup>

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

Eq. (5) allows to determine the total HTC *UII*. Heat transfer coefficient *hI*

*<sup>a</sup>* should then be substituted by HTC *<sup>h</sup>II*

tubes. The average air temperature over the width of the second row of pipes

1

*TII <sup>a</sup> x*þ, *y*<sup>þ</sup> 2 � � *dy*<sup>þ</sup>

After taking into account that the water temperature at the inlet to the second

*<sup>w</sup>*,2, and the temperature of the air at the inlet to the second row of

*<sup>w</sup>*,2 � *T*<sup>0</sup>

<sup>2</sup> <sup>≤</sup><sup>1</sup> (21)

*am* � � <sup>1</sup> � exp �*N<sup>I</sup>*

*<sup>a</sup> x*<sup>þ</sup> ð Þ (**Figure 2**), it is necessary to take into

<sup>2</sup> ¼ 1. Then Eq. (20) should be substituted into Eq. (21).

*y*þ <sup>2</sup> ¼0

pipes is equal to the outlet temperature of the air from the first row of pipes the

*Tw*,2 *<sup>x</sup>*j þ¼<sup>0</sup> ¼ *T*<sup>0</sup>

� <sup>¼</sup> *<sup>T</sup>*<sup>00</sup>

The solution to Eqs. (14) and (15) under boundary conditions (18) and (19) is

*<sup>a</sup> <sup>x</sup>*<sup>þ</sup> ð Þ � � exp �*NII*

*<sup>w</sup>*,1 � *T*<sup>0</sup>

Eq. (13) allows to calculate the coefficient B. To determine the air temperature

The resulting equation allows, after simple transformations, to write an expression

*<sup>a</sup> x*<sup>þ</sup> ð Þ behind the second row of pipes

*oe*. The effective air-side HTC *hII*

*bout<sup>=</sup> <sup>m</sup>*\_ *<sup>w</sup> cp w* � � *<sup>N</sup>II*

*<sup>a</sup> Tw*,2 *<sup>x</sup>*<sup>þ</sup> ð Þ� *<sup>T</sup>II*

*TII*

*m a <sup>x</sup>*<sup>þ</sup> ð Þ¼ <sup>ð</sup>

*TII a y*<sup>þ</sup> <sup>2</sup> ¼0 � �

*<sup>D</sup>* � *<sup>B</sup>* exp �*B x*<sup>þ</sup> ð Þ� exp �*D x*<sup>þ</sup> <sup>½</sup> ð Þ� þ *<sup>T</sup>*<sup>0</sup>

� � <sup>¼</sup> *Tw*,2 *<sup>x</sup>*<sup>þ</sup> ð Þ� *Tw*,2 *<sup>x</sup>*<sup>þ</sup> ð Þ� *<sup>T</sup>*<sup>00</sup>

*a* � � � � *<sup>E</sup>* <sup>¼</sup> *D T*<sup>0</sup>

*m a <sup>x</sup>*<sup>þ</sup> ð Þ � � <sup>0</sup>≤*x*<sup>þ</sup> <sup>≤</sup>1 (14)

*<sup>a</sup>* <sup>¼</sup> *<sup>U</sup>II AII*

<sup>2</sup> ≤1 (15)

*oe*

(20)

*bout<sup>=</sup> <sup>m</sup>*\_ *<sup>a</sup> cp a* � � (16)

*oe* can be calculated

<sup>2</sup> (17)

*<sup>w</sup>*,2 (18)

*<sup>a</sup> x*<sup>þ</sup> ð Þ (19)

*am* � � exp �*D x*<sup>þ</sup> ð Þ

*<sup>a</sup> y*<sup>þ</sup> 2 � �

*a* � � � � (22)

*<sup>a</sup>* for the second row of

$$T\_{\
u}^{I}\Big|\_{\mathcal{Y}\_1^+=\mathbf{0}} = T\_{\
u m}^{\prime}\tag{8}$$

The solution to Eqs. (1) and (2) with boundary conditions (7) and (8) is

$$T\_{w,1}(\mathbf{x}^+) = T\_{am}' + \left(T\_{w,1}' - T\_{am}'\right) \exp\left\{-\frac{N\_w^l}{N\_a^l} \left[1 - \exp\left(-N\_a^l\right)\right] \mathbf{x}^+\right\} \qquad \mathbf{0} \le \mathbf{x}^+ \le \mathbf{1} \tag{9}$$

$$T\_a^l \left(\mathbf{x}^+, y\_1^+\right) = T\_{w,1}(\mathbf{x}^+) - \left[T\_{w,1}(\mathbf{x}^+) - T\_{am}^l\right] \exp\left(-N\_a^l y\_1^+\right) \quad \mathbf{0} \le \mathbf{x}^+ \le \mathbf{1} \quad \mathbf{0} \le y\_1^+ \le \mathbf{1} \tag{10}$$

By substituting *y*<sup>þ</sup> <sup>1</sup> ¼ 1 into Eq. (10) and performing appropriate transformations the air temperature *T*<sup>00</sup> *<sup>a</sup> x*<sup>þ</sup> ð Þ behind the first row of pipes can be obtained

$$T\_a''(\mathbf{x}^+) = T\_a^l \left(\mathbf{x}^+, \mathbf{y}\_1^+ = \mathbf{1}\right) = T\_{w,1}(\mathbf{x}^+) - \left[T\_{w,1}(\mathbf{x}^+) - T\_{am}^{\prime}\right] \exp\left(-N\_a^l\right) \tag{11}$$

Expression (9) describes the temperature *Tw*,1 *x*<sup>þ</sup> ð Þ. The temperature *T*<sup>00</sup> *am* denotes the average air temperature after the first row of tubes. The temperature *T*<sup>00</sup> *am* can be calculated as follows

$$T\_{am}^{\prime\prime} = \int\_0^1 T\_a^{\prime\prime}(\mathbf{x}^+) \, d\mathbf{x} = T\_{am}^{\prime} + \frac{N\_a^I}{N\_w^I} \left( T\_{w,1}^{\prime} - T\_{am}^{\prime} \right) \left[ 1 - \exp \left( -B \right) \right] \tag{12}$$

where *B* stands for

$$B = \frac{N\_w^I}{N\_a^I} \left[ \mathbf{1} - \exp\left(-N\_a^I\right) \right] \tag{13}$$

The temperature distribution of water and air in the first row of pipes of the first pass can be determined using Eqs. (9)–(13).

#### **2.2 Analytical model for the second row of pipes**

The relevant differential equations for the second row of pipes are

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

$$\frac{d\,T\_{w,2}(\mathbf{x}^+)}{d\mathbf{x}^+} = -N\_w^{\text{II}} \left[ T\_{w,2}(\mathbf{x}^+) - T\_{ma}^{\text{II}}(\mathbf{x}^+) \right] \qquad \mathbf{0} \le \mathbf{x}^+ \le \mathbf{1} \tag{14}$$

$$\frac{\partial T\_a^{\text{II}}(\mathbf{x}^+, \mathbf{y}\_2^+)}{\partial \mathbf{y}\_2^+} = \mathbf{N}\_a^{\text{II}} \left[ T\_{w,2}(\mathbf{x}^+) - T\_a^{\text{II}}(\mathbf{x}^+, \mathbf{y}\_2^+) \right] \qquad \mathbf{0} \le \mathbf{x}^+ \le \mathbf{1} \qquad \mathbf{0} \le \mathbf{y}\_2^+ \le \mathbf{1} \tag{15}$$

where

$$\mathbf{N}\_{w}^{\mathrm{II}} = 2\mathbf{U}^{\mathrm{II}}\mathbf{A}\_{\mathrm{bout}}^{\mathrm{II}} / \begin{pmatrix} \dot{m}\_{w} \ \overline{c}\_{pw} \end{pmatrix} \qquad \mathbf{N}\_{a}^{\mathrm{II}} = \mathbf{U}^{\mathrm{II}}\mathbf{A}\_{\mathrm{bout}}^{\mathrm{II}} / \begin{pmatrix} \dot{m}\_{a} \ \overline{c}\_{pa} \end{pmatrix} \tag{16}$$

Eq. (5) allows to determine the total HTC *UII*. Heat transfer coefficient *hI oe* should then be replaced by *hII oe*. The effective air-side HTC *hII oe* can be calculated using Eq. (6). HTC *h<sup>I</sup> <sup>a</sup>* should then be substituted by HTC *<sup>h</sup>II <sup>a</sup>* for the second row of tubes. The average air temperature over the width of the second row of pipes *TII m a x*<sup>þ</sup> ð Þ is determined as follows

$$T\_{ma}^{\text{II}}(\mathbf{x}^+) = \int\_{y\_2^+=0}^1 T\_a^{\text{II}}(\mathbf{x}^+, y\_2^+) \, dy\_2^+ \tag{17}$$

After taking into account that the water temperature at the inlet to the second row of pipes is *T*<sup>0</sup> *<sup>w</sup>*,2, and the temperature of the air at the inlet to the second row of pipes is equal to the outlet temperature of the air from the first row of pipes the appropriate boundary conditions are

$$T\_{w,2}|\_{\mathbf{x}^+=\mathbf{0}} = T\_{w,2}' \tag{18}$$

$$T\_{a}^{II}\Big|\_{\mathbb{Y}\_{2}^{+}=0} = T\_{a}^{\prime\prime}(\mathbb{x}^{+})\tag{19}$$

The solution to Eqs. (14) and (15) under boundary conditions (18) and (19) is

$$\begin{aligned} T\_{w,2}(\mathbf{x}^+) &= \\ T\_{am}' + \frac{E}{D-B} \left[ \exp\left(-B\mathbf{x}^+\right) - \exp\left(-D\mathbf{x}^+\right) \right] + \left(T\_{w,2}' - T\_{am}'\right) \exp\left(-D\mathbf{x}^+\right) \\ \mathbf{0} &\leq \mathbf{x}^+ \leq \mathbf{1} \end{aligned} \tag{20}$$

$$\begin{aligned} T\_a^{ll}(\mathbf{x}^+, \mathbf{y}\_2^+) &= T\_{w,2}(\mathbf{x}^+) - \left[ T\_{w,2}(\mathbf{x}^+) - T\_a''(\mathbf{x}^+) \right] \exp\left( -\mathbf{N}\_a^{ll} \mathbf{y}\_2^+ \right) \\ \mathbf{0} \le \mathbf{x}^+ \le \mathbf{1} & \mathbf{0} \le \mathbf{y}\_2^+ \le \mathbf{1} \end{aligned} \tag{21}$$

where

$$D = \frac{N\_w^{\mathrm{II}}}{N\_a^{\mathrm{II}}} \left[ 1 - \exp\left(-N\_a^{\mathrm{II}}\right) \right] \qquad \qquad E = D \left(T\_{w,1}' - T\_{am}'\right) \left[1 - \exp\left(-N\_a^{\mathrm{I}}\right) \right] \tag{22}$$

Eq. (13) allows to calculate the coefficient B. To determine the air temperature behind the second row of pipes *T*‴ *<sup>a</sup> x*<sup>þ</sup> ð Þ (**Figure 2**), it is necessary to take into account in Eq. (21) that *y*<sup>þ</sup> <sup>2</sup> ¼ 1. Then Eq. (20) should be substituted into Eq. (21). The resulting equation allows, after simple transformations, to write an expression for the air temperature *T*‴ *<sup>a</sup> x*<sup>þ</sup> ð Þ behind the second row of pipes

*Heat Transfer - Design, Experimentation and Applications*

$$\begin{aligned} T\_{\boldsymbol{a}}^{\prime}\left(\mathbf{x}^{+}\right) &= T\_{\boldsymbol{a}}^{\mathrm{II}}\left(\mathbf{x}^{+}, \mathcal{Y}\_{2}^{+} = \mathbf{1}\right) = \\ T\_{\boldsymbol{a}m}^{\prime} &+ \left[\mathbf{1} - \exp\left(-N\_{\boldsymbol{a}}^{\mathrm{II}}\right)\right] \left\{ \frac{E}{D - B} \left[\exp\left(-B\mathbf{x}^{+}\right) - \exp\left(-D\mathbf{x}^{+}\right)\right] + \\ &+ \left(T\_{\boldsymbol{w},2}^{\prime} - T\_{\boldsymbol{a}m}^{\prime}\right) \exp\left(-D\mathbf{x}^{+}\right) \right\} + \left(T\_{\boldsymbol{w},1}^{\prime} - T\_{\boldsymbol{a}m}^{\prime}\right) \left[\mathbf{1} - \exp\left(-N\_{\boldsymbol{a}}^{\prime}\right)\right] \exp\left(-B\mathbf{x}^{+} - N\_{\boldsymbol{a}}^{\prime\prime}\right) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mathbf{0} \leq \mathbf{x}^{+} \leq \mathbf{1} \end{aligned} \tag{23}$$

*T*‴

*<sup>a</sup> <sup>x</sup>*<sup>þ</sup> ð Þ¼ *<sup>T</sup>II*

� *C x*<sup>þ</sup> þ *T*<sup>0</sup>

0≤*x*<sup>þ</sup> ≤1

*<sup>a</sup> x*þ, *y*<sup>þ</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

where the symbol *C* means

capacity of 1600 cubic centimeters.

*kw* = *kf* = 207 W/(m�K).

• oval pipe radiator

below:

**37**

*<sup>w</sup>*,1 � *T*<sup>0</sup>

<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> *<sup>T</sup>*<sup>0</sup>

*am* exp �*B x*<sup>þ</sup> ð Þþ *<sup>T</sup>*<sup>0</sup>

*C* ¼ *B T*<sup>0</sup>

**3. An analytical model of a car engine radiator**

detail in articles [9, 10], respectively. First, the temperatures *T*<sup>00</sup>

the first (upper) pass is *m*\_ *anu=*ð Þ *nu* þ *nl* , and the second (lower) pass is *m*\_ *anl=*ð Þ *nu* þ *nl* , respectively. The symbol *m*\_ *<sup>a</sup>* denotes the mass flow rate of air flowing through the radiator. The symbols *nu* and *nl* stand respectively for the number of pipes in the first and second pass in the first row of tubes. The numbers *nu* and *nl* are equal ten and nine, respectively. Thermal calculations of the first and second row of pipes in the second (lower) pass have been carried out analogously to

those of the first and second row in the first (upper) pass.

**4. Heat transfer correlations on the air-side**

and second row of pipes and for the entire cooler.

*<sup>a</sup>* <sup>¼</sup> <sup>30</sup>*:*7105 Re ‐0*:*<sup>24</sup> *<sup>a</sup>* Pr<sup>1</sup>*=*<sup>3</sup>

*<sup>a</sup>* <sup>¼</sup> <sup>0</sup>*:*0744 Re <sup>0</sup>*:*<sup>7069</sup>

Nu*<sup>a</sup>* <sup>¼</sup> <sup>1</sup>*:*0605 Re <sup>0</sup>*:*<sup>2974</sup>

Nu*<sup>I</sup>*

Nu*II*

*<sup>w</sup>*,1 � *T*<sup>0</sup>

*am* <sup>þ</sup> <sup>1</sup> � exp �*N<sup>I</sup>*

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

*am* <sup>1</sup> � exp �*N<sup>I</sup>*

The relationships derived for the single-pass double-row PFTHE, taking into account the different HTCs on the first and second row of pipes, were used to develop an analytical mathematical model of the entire radiator, the flow diagram of which is shown in **Figure 1**. This is an engine radiator with a displacement

The constructions of coolers made of round and oval tubes are described in

liquid at the outlet from the first pass of the cooler are calculated. Then the temperature *Twm* of the liquid is calculated after mixing the liquid streams from the first and second pipe row (**Figure 1**). Water at the *Twm* temperature feeds the second pass of the automobile radiator with the water mass flow rate *m*\_ *<sup>w</sup>* divided into two equal parts *m*\_ *<sup>w</sup>=*2 flowing through the first and second row of pipes. The air stream flows through the entire cross-section of the radiator. The mass flow rate through

The thermal conductivity of aluminum tubes and fins was assumed to be

Based on CFD modeling performed in ANSY-CFX, the air-side heat transfer coefficient values were determined. The paper [19] provides a description of the method for determining the correlations for the average Nusselt number in the first

Nusselt number correlations for PFTHE from oval and round pipes are given

*<sup>a</sup>* Pr<sup>1</sup>*=*<sup>3</sup>

*<sup>a</sup>* Pr<sup>1</sup>*=*<sup>3</sup>

�

*<sup>w</sup>*,1 � *T*<sup>0</sup>

*a*

*am* exp � *B x*<sup>þ</sup> <sup>þ</sup> *<sup>N</sup><sup>I</sup>*

*a*

(29)

*<sup>w</sup>*,1 and *T*<sup>00</sup>

*<sup>a</sup>* 150 ≤ Re *<sup>a</sup>* ≤330 (30)

*<sup>a</sup>* 150 ≤ Re *<sup>a</sup>* ≤330 (31)

*<sup>a</sup>* 150 ≤ Re *<sup>a</sup>* ≤ 330 (32)

*<sup>w</sup>*,2 of the

*a*

(28)

The average air temperature after the second row of pipes *T*‴ *am* is determined by the formula

$$T\_{am}^{''} = \begin{bmatrix} T\_a^{'} \left( \mathbf{x}^{+} \right) d\mathbf{x}^{+} = T\_{am}^{'} + \left[ \mathbf{1} - \ \exp \left( -N\_a^{\mathrm{II}} \right) \right] \left\{ \frac{E}{(D-B)B} \left[ \mathbf{1} - \ \exp \left( -B \right) \right] - \right\} \tag{24}$$

$$- \frac{E}{(D-B)D} \left[ \mathbf{1} - \ \exp \left( -D \right) \right] + \frac{\left( T\_{w,2}^{'} - T\_{am}^{'} \right)}{D} \left[ \mathbf{1} - \ \exp \left( -D \right) \right] \right\} -$$

$$- \frac{\left( T\_{w,1}^{'} - T\_{am}^{'} \right)}{B} \left[ \mathbf{1} - \ \exp \left( -N\_a^{\mathrm{I}} \right) \right] \left[ \exp \left( -B - N\_a^{\mathrm{II}} \right) - \exp \left( -N\_a^{\mathrm{II}} \right) \right] \tag{25}$$

When the air-side HTC *hI <sup>a</sup>* on the first pipe row is equal to the HTC *<sup>h</sup>II <sup>a</sup>* on the second pipe row, then *D* = *B* and the denominator (*D-B*) is going to be zero. When *D=B* the formula (20) requires transformation. Writing Eq. (20) as

$$T\_{w,2}(\mathbf{x}^{+}) = T\_{am}' + \frac{E \exp\left(-D\mathbf{x}^{+}\right)}{D - B} \left\{ \exp\left[ (D - B)\mathbf{x}^{+} \right] - \mathbf{1} \right\}$$

$$+ \left(T\_{w,2}' - T\_{am}' \right) \exp\left(-D\mathbf{x}^{+}\right) \tag{25}$$

and applying the L'Hôpital rule to an expression where, at *D=B*, the numerator and the denominator are equal to zero, one obtains

$$\lim\_{D \to B} \frac{\exp\left[ (D - B)x^{+} \right] - 1}{D - B} = \lim\_{x \to 0} \frac{\exp\left( zx^{+} \right) - 1}{z} = \lim\_{x \to 0} \frac{\frac{d}{dz} \left[ \exp\left( zx^{+} \right) - 1 \right]}{\frac{d}{dz}} = 1$$
  $\lim\_{x \to 0} \frac{x^{+} \exp\left( zx^{+} \right)}{1} = x^{+} \tag{26}$ 

where *z=D – B*.

Taking into account the relation (26) for *D=B* in the formula (25), i.e. for *hI <sup>a</sup>* ¼ *hII <sup>a</sup>* ¼ *ha*, it takes the following form

$$\begin{aligned} T\_{w,2}(\mathbf{x}^+) &= T\_{am}^{\prime} + \left\{ B \left( T\_{w,1}^{\prime} - T\_{am}^{\prime} \right) \left[ \mathbf{1} - \exp \left( -N\_a^{I} \right) \right] \mathbf{x}^+ + \left( T\_{w,2}^{\prime} - T\_{am}^{\prime} \right) \right\} \exp \left( -B \mathbf{x}^+ \right) \\ \mathbf{0} &\leq \mathbf{x}^+ \leq \mathbf{1} \end{aligned} \tag{27}$$

where *N<sup>I</sup> <sup>a</sup>* <sup>¼</sup> *<sup>N</sup>II <sup>a</sup>* ¼ *Na*.

If *h<sup>I</sup> <sup>a</sup>* <sup>¼</sup> *hII <sup>a</sup>* ¼ *ha*, then the air temperature *T*‴ *<sup>a</sup> x*<sup>þ</sup> ð Þ after the second row of pipes is obtained with Eq. (23) after taking into account the relation (26)

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

$$\begin{aligned} T\_a'' \left( \mathbf{x}^+ \right) &= T\_a^{\text{II}} \left( \mathbf{x}^+, \mathbf{y}\_2^+ = \mathbf{1} \right) = T\_{am}' + \left[ \mathbf{1} - \exp \left( -N\_a^I \right) \right] \times \\ &\times \left\{ \left[ \mathbf{C} \mathbf{x}^+ + \left( T\_{w,1}' - T\_{am}' \right) \right] \exp \left( -B \mathbf{x}^+ \right) + \left( T\_{w,1}' - T\_{am}' \right) \exp \left[ -\left( B \mathbf{x}^+ + N\_a^I \right) \right] \right\} \\ &\mathbf{0} \le \mathbf{x}^+ \le \mathbf{1} \end{aligned} \tag{28}$$

where the symbol *C* means

$$\mathbf{C} = B \left( T\_{w,1}' - T\_{am}' \right) \left[ \mathbf{1} - \exp \left( -\mathbf{N}\_a^I \right) \right] \tag{29}$$

#### **3. An analytical model of a car engine radiator**

The relationships derived for the single-pass double-row PFTHE, taking into account the different HTCs on the first and second row of pipes, were used to develop an analytical mathematical model of the entire radiator, the flow diagram of which is shown in **Figure 1**. This is an engine radiator with a displacement capacity of 1600 cubic centimeters.

The constructions of coolers made of round and oval tubes are described in detail in articles [9, 10], respectively. First, the temperatures *T*<sup>00</sup> *<sup>w</sup>*,1 and *T*<sup>00</sup> *<sup>w</sup>*,2 of the liquid at the outlet from the first pass of the cooler are calculated. Then the temperature *Twm* of the liquid is calculated after mixing the liquid streams from the first and second pipe row (**Figure 1**). Water at the *Twm* temperature feeds the second pass of the automobile radiator with the water mass flow rate *m*\_ *<sup>w</sup>* divided into two equal parts *m*\_ *<sup>w</sup>=*2 flowing through the first and second row of pipes. The air stream flows through the entire cross-section of the radiator. The mass flow rate through the first (upper) pass is *m*\_ *anu=*ð Þ *nu* þ *nl* , and the second (lower) pass is *m*\_ *anl=*ð Þ *nu* þ *nl* , respectively. The symbol *m*\_ *<sup>a</sup>* denotes the mass flow rate of air flowing through the radiator. The symbols *nu* and *nl* stand respectively for the number of pipes in the first and second pass in the first row of tubes. The numbers *nu* and *nl* are equal ten and nine, respectively. Thermal calculations of the first and second row of pipes in the second (lower) pass have been carried out analogously to those of the first and second row in the first (upper) pass.

The thermal conductivity of aluminum tubes and fins was assumed to be *kw* = *kf* = 207 W/(m�K).

#### **4. Heat transfer correlations on the air-side**

Based on CFD modeling performed in ANSY-CFX, the air-side heat transfer coefficient values were determined. The paper [19] provides a description of the method for determining the correlations for the average Nusselt number in the first and second row of pipes and for the entire cooler.

Nusselt number correlations for PFTHE from oval and round pipes are given below:

• oval pipe radiator

$$\mathrm{Nu}\_{a}^{I} = \mathbf{30.7105} \,\mathrm{Re}\_{a}^{\cdot 0.24} \mathrm{Pr}\_{a}^{1/3} \qquad \mathbf{150} \le \mathrm{Re}\_{a} \le \mathbf{330} \tag{30}$$

$$\mathrm{Nu}\_{a}^{H} = \mathbf{0.0744} \,\mathrm{Re}\_{a}^{0.7069} \mathrm{Pr}\_{a}^{1/3} \qquad \mathbf{150} \le \mathrm{Re}\_{a} \le \mathbf{330} \tag{31}$$

$$\text{Nu}\_d = \text{1.0605} \,\text{Re}\_d^{0.2974} \text{Pr}\_d^{1/3} \qquad \text{150} \le \text{Re}\_d \le \text{330} \tag{32}$$

The Reynolds number on the air-side can be determined by knowing the hydraulic diameter and the air velocity in the smallest flow area. The hydraulic diameter for the oval tube cooler analyzed was *dh a* ¼ 1*:*42 mm. The average Nusselt number on the first row of tubes varies very little (the exponent at the Reynolds number takes small values). Hydraulically and thermally developed laminar flow is characterized by a constant Nusselt number. Because the exponent at the Reynolds number for the second row of pipes takes larger values, the phenomenon of vortices forming near the front and rear stagnation points is more visible there. The heat transfer relation (32) for the average Nusselt number in a two-row bundle is similar to the expression for the average Nusselt number on a plane surface for thermally and hydraulically developing laminar flow, for which the exponent at the Reynolds number is 1/3. Formulas for determining the air-side Nusselt number for an engine radiator made of circular tubes are presented below.

	- Nu*<sup>I</sup> <sup>a</sup>* <sup>¼</sup> <sup>1</sup>*:*6502 Re <sup>0</sup>*:*<sup>2414</sup> *<sup>a</sup>* Pr<sup>1</sup>*=*<sup>3</sup> *<sup>a</sup>* 100≤ Re *<sup>a</sup>* ≤ 525 (33)

Nu*<sup>m</sup>*,*<sup>q</sup>* <sup>¼</sup> Nu3

laminar flow

*<sup>m</sup>*,*q*,1 <sup>þ</sup> <sup>0</sup>*:*63 <sup>þ</sup> Nu*<sup>m</sup>*,*q*,2 � <sup>0</sup>*:*<sup>6</sup> � �<sup>3</sup> <sup>þ</sup> Nu3 *<sup>m</sup>*,*q*,3 h i<sup>1</sup>*=*<sup>3</sup>

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

Nu*<sup>m</sup>*,*q*,1 <sup>¼</sup> <sup>48</sup>

Γð Þ 2*=*3 Re *<sup>w</sup>*Pr*<sup>w</sup>*

Nu*<sup>m</sup>*,*q*,3 ¼ 0*:*924 Re *<sup>w</sup>*

Re *<sup>w</sup>* ¼ 2300 and *ξw*ð Re *<sup>w</sup>* ¼ 3000Þ ¼ 0*:*04355

were evaluated at the mean temperature *Tw* ¼ *T*<sup>0</sup>

**6. Experimental results**

**39**

stant liquid velocity at the tube inlet

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

Nu*<sup>m</sup>*,*q*,2 <sup>¼</sup> <sup>3</sup><sup>1</sup>*=*<sup>3</sup>

Taler's relatonship [28].

Taler formula [29]

Nu*<sup>w</sup>* ¼ Nu*<sup>m</sup>*,*<sup>q</sup>*j Re *<sup>w</sup>*¼<sup>2300</sup> þ

The symbol Nu*<sup>m</sup>*,*q*,1 designates the average Nusselt number for the developed

The second Nusselt number Nu*<sup>m</sup>*,*q*,2 represents the Lévêque solution [4] for developing flow over planar surface and Nu*<sup>m</sup>*,*q*,3 was derived numerically for con-

> *dhw Lt*

*dhw Lt* � �<sup>1</sup>*=*<sup>2</sup>

� �<sup>1</sup>*=*<sup>3</sup>

*ξw*

1*:*084 þ 12*:*4

<sup>11</sup> <sup>¼</sup> <sup>4</sup>*:*364 Re *<sup>w</sup>* <sup>≤</sup><sup>2300</sup> (37)

¼ 1*:*9530 Re *<sup>w</sup>*Pr*<sup>w</sup>*

Pr<sup>1</sup>*=*<sup>3</sup>

*w*

Pr<sup>2</sup>*=*<sup>3</sup> *<sup>w</sup>* � <sup>1</sup> � �

The Nusselt number in the transition and turbulent flow range was evaluated by

<sup>8</sup> ð Þ Re *<sup>w</sup>* � <sup>2300</sup> Pr<sup>1</sup>*:*<sup>008</sup>

The Darcy-Weisbach friction factor for turbulent flow, was calculated by the

The relationship for the friction factor in the transition region was obtained using the linear interpolation [29] between the value *ξ<sup>w</sup>* ¼ 64*=*2300 ¼ 0*:*02783 for

*<sup>ξ</sup><sup>w</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>02783</sup> <sup>þ</sup> <sup>2</sup>*:*<sup>2457</sup> � <sup>10</sup>�<sup>5</sup> ð Þ Re *<sup>w</sup>* � 2 300 <sup>2300</sup><sup>≤</sup> Re *<sup>w</sup>* <sup>≤</sup>3 000 (42)

Based on the experimental testing of a car radiator made of oval and round tubes

(**Table 1**), heat transfer correlations on the air side were determined.

The Reynolds number on the water side Re *<sup>w</sup>* ¼ *ww dh w=ν<sup>w</sup>* is based on the hydraulic diameter *dhw*. The hydraulic diameter was *dhw* ¼ 7*:*06 mm for oval tubes and *dhw* ¼ *din* ¼ 6*:*2 mm for circular tubes. The physical properties of the water

ffiffiffiffiffi *ξw* 8 r

<sup>2300</sup> <sup>≤</sup> Re *<sup>w</sup>* <sup>≤</sup> <sup>10</sup><sup>6</sup> <sup>0</sup>*:*1≤Pr*<sup>w</sup>* <sup>≤</sup>1000, *dhw*

*<sup>ξ</sup><sup>w</sup>* <sup>¼</sup> ð Þ <sup>1</sup>*:*2776 log Re *<sup>w</sup>* � <sup>0</sup>*:*<sup>406</sup> �2*:*<sup>246</sup> <sup>3000</sup> <sup>≤</sup> Re *<sup>w</sup>* (41)

*<sup>w</sup>* þ *T*<sup>00</sup> *w* � �*=*2. *dhw Lt*

� �<sup>1</sup>*=*<sup>3</sup>

Re *<sup>w</sup>* ≤2300

*<sup>w</sup>* Re *<sup>w</sup>* ≤2 300 (39)

*dhw Lt* � �<sup>2</sup>*=*<sup>3</sup> " #

> *Lt* ≤1

1 þ

Re *<sup>w</sup>* ≤2300

(36)

(38)

(40)

$$\mathrm{Nu}\_{a}^{H} = \mathbf{0.1569} \,\mathrm{Re}\_{a}^{0.5499} \mathrm{Pr}\_{a}^{1/3} \qquad \mathbf{100} \le \mathrm{Re}\_{a} \le \mathbf{525} \tag{34}$$

$$\mathrm{Nu}\_{d} = \mathbf{0}.6070 \,\mathrm{Re}\_{a}^{0.3678} \mathrm{Pr}\_{a}^{1/3} \qquad \mathbf{100} \le \mathbf{Re}\_{a} \le \mathbf{525} \tag{35}$$

Analyzing formulas (33) and (34), it can be observed that the exponent at Reynolds number for the first row of circular pipes is low and equals 0.2414. This indicates the dominance of laminar flow in the first row of pipes on the air-side. In the second row of tubes, the exponent at Reynolds number is higher and equals 0.5499. There are vortexes at the front and rear of the tube.

However, the average Nusselt number in the first row is higher than in the second row. This is because the inlet section is located in the gap formed by two adjacent fins, which have very high values of heat transfer coefficients at the fin surfaces. This results in a greater heat flow rate absorbed by the air in the first row than in the second row. The vortexes formed near the front and rear stagnation points on the surface of the second-row tube reduce the heat transfer to a large extent. The rotating air has a temperature close to that of the fin and pipe surfaces. From the heat transfer point of view, these are dead zones.

The comparison of the heat correlations determined by CFD modeling with the correlations determined based on experimental tests was carried out for two different car radiators. The flow arrangement of both radiators is the same (**Figure 1**). The first radiator is made of round pipes and the second one of the oval pipes.

#### **5. Heat transfer correlations on the liquid-side**

The present chapter demonstrates that by using theoretical relationships to calculate the heat transfer coefficient on the inner surfaces of the heat exchanger tubes and the use of the air-side correlations determined by CFD modeling, very similar results to experimental HTCs can be obtained.

The proposed methodology for determining the heat-transfer coefficient will reduce the cost of experimental tests for designed heat exchangers or even eliminate them. In determining the HTC for laminar flow, Gnielinski's formulas were used [27]. For the transition and turbulent flow, the formulas proposed in [28] were used.

In the range of laminar flow regime, the average HTC along the entire length of pipe *Lt*, assuming that the velocity profile at the pipe inlet is flat and the liquid flow is hydraulically and thermally developing, is determined by the following relationship [27]. *The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

$$\mathbf{Nu}\_{m,q} = \left[ \mathbf{Nu}\_{m,q,1}^3 + \mathbf{0.6}^3 + \left( \mathbf{Nu}\_{m,q,2} - \mathbf{0.6} \right)^3 + \mathbf{Nu}\_{m,q,3}^3 \right]^{1/3} \qquad \text{Re}\_w \le 2300 \tag{36}$$

The symbol Nu*<sup>m</sup>*,*q*,1 designates the average Nusselt number for the developed laminar flow

$$\text{Nu}\_{m,q,1} = \frac{48}{11} = 4.364 \qquad \qquad \text{Re}\_w \le 2300 \tag{37}$$

The second Nusselt number Nu*<sup>m</sup>*,*q*,2 represents the Lévêque solution [4] for developing flow over planar surface and Nu*<sup>m</sup>*,*q*,3 was derived numerically for constant liquid velocity at the tube inlet

$$\mathrm{Nu}\_{m,q,2} = \mathfrak{Z}^{1/3} \Gamma(2/3) \left( \mathrm{Re}\_w \mathrm{Pr}\_w \frac{d\_{hw}}{L\_t} \right)^{1/3} = \mathbf{1.9530} \left( \mathrm{Re}\_w \mathrm{Pr}\_w \frac{d\_{hw}}{L\_t} \right)^{1/3} \tag{38}$$
 
$$\mathrm{Re}\_w \le 2 \, 300$$

$$\mathrm{Nu}\_{m,q,3} = 0.924 \left( \mathrm{Re}\_w \, \frac{d\_{hw}}{L\_t} \right)^{1/2} \mathrm{Pr}\_w^{1/3} \qquad \mathrm{Re}\_w \le 2300 \tag{39}$$

The Nusselt number in the transition and turbulent flow range was evaluated by Taler's relatonship [28].

$$\mathrm{Nu}\_{w} = \mathrm{Nu}\_{m,q}|\_{\mathrm{Re}\_{w} = 2300} + \frac{\frac{\tilde{\xi}\_{w}}{8} (\mathrm{Re}\_{w} - 2300) \, \mathrm{Pr}\_{w}^{1.008}}{\mathrm{1.084} + 12.4 \sqrt{\frac{\tilde{\xi}\_{w}}{8}} (\mathrm{Pr}\_{w}^{2/3} - 1)} \left[ 1 + \left( \frac{d\_{hw}}{L\_{t}} \right)^{2/3} \right] \tag{40}$$

$$2300 \le \mathrm{Re}\_{w} \le 10^{6} \quad 0.1 \le \mathrm{Pr}\_{w} \le 1000, \ \frac{d\_{hw}}{L\_{t}} \le 1$$

The Darcy-Weisbach friction factor for turbulent flow, was calculated by the Taler formula [29]

$$
\xi\_w = (\text{1.2776 } \log \text{Re}\_w - 0.406)^{-2.246} \qquad \text{ 3000} \\
\leq \text{Re}\_w \qquad \text{ (41)}
$$

The relationship for the friction factor in the transition region was obtained using the linear interpolation [29] between the value *ξ<sup>w</sup>* ¼ 64*=*2300 ¼ 0*:*02783 for Re *<sup>w</sup>* ¼ 2300 and *ξw*ð Re *<sup>w</sup>* ¼ 3000Þ ¼ 0*:*04355

$$
\xi\_w = 0.02783 + 2.2457 \cdot 10^{-5} (\text{Re}\_w - 2.300) \qquad 2300 \le \text{Re}\_w \le 3000 \tag{42}
$$

The Reynolds number on the water side Re *<sup>w</sup>* ¼ *ww dh w=ν<sup>w</sup>* is based on the hydraulic diameter *dhw*. The hydraulic diameter was *dhw* ¼ 7*:*06 mm for oval tubes and *dhw* ¼ *din* ¼ 6*:*2 mm for circular tubes. The physical properties of the water were evaluated at the mean temperature *Tw* ¼ *T*<sup>0</sup> *<sup>w</sup>* þ *T*<sup>00</sup> *w* � �*=*2.

#### **6. Experimental results**

Based on the experimental testing of a car radiator made of oval and round tubes (**Table 1**), heat transfer correlations on the air side were determined.


• Velocity: 0.74 m/s - 2.27 m/s

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

• Temperature: 3.6°C - 15.5°C

• Volume flow rate: 300 l/h - 2410 l/h

• Temperature: 50.0°C - 72.0 °C.

found

The water parameters at the cooler inlet were as follows:

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

Nusselt number Nu*<sup>a</sup>* was obtained using the least-squares method

*<sup>a</sup>* Pr<sup>1</sup>*=*<sup>3</sup>

**7. Analysis and discussion of heat flow rates from water to air**

1273*:*4 liters/h. For a car radiator built of round tubes, the results of tests and calculations are presented as a function of the Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* on the water side for the first pass of the heat exchanger for the preset air velocity *w*<sup>0</sup> in front of

Nu*<sup>a</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>1</sup> Re *<sup>x</sup>*<sup>2</sup>

the air side for two selected volume flows of water \_

the radiator equal to 2.27 m/s.

inlet temperature *T*<sup>0</sup>

60*:*51o C.

**41**

**7.1 Engine cooler made of oval tubes**

326*:*06 liters/h, average air inlet temperature *T*<sup>0</sup>

*<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61<sup>o</sup>

obtained for the following data: *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*37 liters/h, *<sup>T</sup>*<sup>0</sup>

The water flow regime changed from laminar to transitional and then to turbulent during testing. The air-side Reynolds Re *<sup>a</sup>* number ranged from 150 to 560. The Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* on the water side in the first pass of the heat exchanger varied from 1450 to 16,100 during the test. The water-side HTC in round tubes was evaluated using the relationships (36)–(42). The following Equation for the air-side

Based on the 70 data series, the following values of parameters *x*<sup>1</sup> and *x*<sup>2</sup> were

Numbers with a sign � in Eq. (46) represent half of the 95% confidence interval.

**transferred in the entire heat exchanger and individual rows of pipes**

In the case of a radiator made of oval pipes, the results of mathematical modeling and measurements are presented as a function of the Reynolds Re *<sup>a</sup>* number on

The thermal capacity of the radiator was calculated by using an analytical mathematical model of the radiator proposed in this paper, assuming the uniform airside HTC on both rows of pipes. **Figure 3a** shows heat flow rates calculated using the mathematical model of the PFTHE for average volumetric flow rate *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup>

C. The average values of *V*\_ *<sup>w</sup>*, *T*<sup>0</sup>

calculated based on seven measured data sets. **Figure 3b** presents heat flow rates

The water flow in the pipes was laminar. The Reynolds number on the waterside in the first pass was between 1222 and 1287, and in the second pass between

*am* <sup>¼</sup> <sup>13</sup>*:*62o

*x*<sup>1</sup> ¼ 0*:*5248 � 0*:*1572 *x*<sup>2</sup> ¼ 0*:*4189 � 0*:*0501 (46)

*<sup>a</sup>* 225 ≤ Re *<sup>a</sup>* ≤ 560 (45)

*Vw* <sup>¼</sup> <sup>326</sup>*:*1 liters/h and *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup>

C, and average water

*<sup>w</sup>* were

*<sup>w</sup>* ¼

*am*, and *T*<sup>0</sup>

*am* <sup>¼</sup> <sup>14</sup>*:*28oC and *<sup>T</sup>*<sup>0</sup>

#### **Table 1.**

*Geometrical data of finned tubes in the heat exchanger made of oval and round tubes.*

#### **6.1 Engine cooler made of oval tubes**

A detailed description of the test stand for experimental research of the engine radiator constructed from oval tubes can be found in [10]. The necessary parameters, including the exponent at Reynolds number, were determined from measured data. The air parameters during the radiator studies were as follows:


The water parameters at the cooler inlet were as follows:


The Reynolds number on the air side ranged from 150 to 330, and the water-side Reynolds number in the first (upper) pass ranged from 2500 to 11,850. When determining the air-side heat transfer correlations, the water-side HTC was calculated using (36)–(42). The following Equation for the air-side Nusselt number Nu*<sup>a</sup>* was found using the experimental data

$$\mathbf{Nu}\_d = \boldsymbol{\varkappa}\_1 \, \mathbf{Re}\_d^{\times 1} \mathbf{Pr}\_d^{1/3} \qquad \qquad \mathbf{150} \le \mathbf{Re}\_d \le \mathbf{330} \tag{43}$$

where the parameters *x*<sup>1</sup> and *x*<sup>2</sup> are

$$\varkappa\_1 = 0.5162 \pm 0.0116 \qquad \varkappa\_2 = 0.4100 \pm 0.0041 \tag{44}$$

Numbers with a sign � in Eq. (44) represent half of the 95% confidence interval.

#### **6.2 Engine cooler made of round tubes**

A test stand description of a car radiator made of round tubes can be found in [10]. Flow and heat measurements allowed to determine the air-side heat transfer correlations. The air parameters during the radiator tests were as follows:

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*


The water parameters at the cooler inlet were as follows:


The water flow regime changed from laminar to transitional and then to turbulent during testing. The air-side Reynolds Re *<sup>a</sup>* number ranged from 150 to 560. The Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* on the water side in the first pass of the heat exchanger varied from 1450 to 16,100 during the test. The water-side HTC in round tubes was evaluated using the relationships (36)–(42). The following Equation for the air-side Nusselt number Nu*<sup>a</sup>* was obtained using the least-squares method

$$\mathrm{Nu}\_{d} = \varkappa\_{1} \mathrm{Re}\_{d}^{\times\_{1}} \mathrm{Pr}\_{d}^{1/3} \qquad \qquad \mathtt{225} \leq \mathrm{Re}\_{d} \leq \mathrm{560} \tag{45}$$

Based on the 70 data series, the following values of parameters *x*<sup>1</sup> and *x*<sup>2</sup> were found

$$\varkappa\_1 = 0.5248 \pm 0.1572 \qquad \qquad \varkappa\_2 = 0.4189 \pm 0.0501 \tag{46}$$

Numbers with a sign � in Eq. (46) represent half of the 95% confidence interval.

## **7. Analysis and discussion of heat flow rates from water to air transferred in the entire heat exchanger and individual rows of pipes**

In the case of a radiator made of oval pipes, the results of mathematical modeling and measurements are presented as a function of the Reynolds Re *<sup>a</sup>* number on the air side for two selected volume flows of water \_ *Vw* <sup>¼</sup> <sup>326</sup>*:*1 liters/h and *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> 1273*:*4 liters/h. For a car radiator built of round tubes, the results of tests and calculations are presented as a function of the Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* on the water side for the first pass of the heat exchanger for the preset air velocity *w*<sup>0</sup> in front of the radiator equal to 2.27 m/s.

#### **7.1 Engine cooler made of oval tubes**

The thermal capacity of the radiator was calculated by using an analytical mathematical model of the radiator proposed in this paper, assuming the uniform airside HTC on both rows of pipes. **Figure 3a** shows heat flow rates calculated using the mathematical model of the PFTHE for average volumetric flow rate *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> 326*:*06 liters/h, average air inlet temperature *T*<sup>0</sup> *am* <sup>¼</sup> <sup>13</sup>*:*62o C, and average water inlet temperature *T*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61<sup>o</sup> C. The average values of *V*\_ *<sup>w</sup>*, *T*<sup>0</sup> *am*, and *T*<sup>0</sup> *<sup>w</sup>* were calculated based on seven measured data sets. **Figure 3b** presents heat flow rates obtained for the following data: *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*37 liters/h, *<sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>14</sup>*:*28oC and *<sup>T</sup>*<sup>0</sup> *<sup>w</sup>* ¼ 60*:*51o C.

The water flow in the pipes was laminar. The Reynolds number on the waterside in the first pass was between 1222 and 1287, and in the second pass between

#### **Figure 3.**

*The heat flow rate Qt*,1*HTC from water to air in an engine cooler built with oval pipes with uniform heat transfer coefficient ha on the air side of the whole cooler; a) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*<sup>06</sup> *liters/h, <sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>13</sup>*:*62o C *and T*<sup>0</sup> *<sup>w</sup>* ¼ 59*:*61<sup>o</sup> <sup>C</sup>*; b) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> *liters/h, <sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>14</sup>*:*28oC *and <sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51oC*; 1 - air-side equation for* Nu*<sup>a</sup> based on the CFD modeling, 2 - air-side equation for* Nu*<sup>a</sup> based on the experimental data, 3 - relative difference.*

1358 and 1430. The velocity of the flowing air was *w0*, and its values varied from 0.71 m/s to 2.2 m/s. The heat flow rates shown in **Figure 3b** were determined for the following data: *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*37 liters/h, *<sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>14</sup>*:*28oC and *<sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51o C. The water flow in the pipes was turbulent. The Reynolds number in the first pass of the heat exchanger on the water-side ranged from 5238 to 5440 and in the second pass from 5820 to 5883. The air velocity before the cooler varied from 0.71 m/s to 2.2 m/s. The heat transfer coefficient on the water-side was calculated using the radiator mathematical model based on Eqs. (36)–(42). The air-side HTC was calculated using Eq. (32) based on CFD simulation (Eq. 1 in **Figure 3**) or empirical Eq. (43) (Eq. 2 in **Figure 3**).

The relative difference was calculated using the expression *e* ¼ 100 *Q*<sup>2</sup> � *Q*<sup>1</sup> ð Þ*=Q*2, where *Q*<sup>1</sup> is the radiator thermal output calculated using the CFD based air-side Eq. (1) and *Q*<sup>2</sup> is the radiator thermal output calculated using the empirical Eq. (2) (**Figure 3**). The analysis of the results presented in **Figure 3a** and **b** shows that the absolute value of the difference e does not exceed 2.75%. The heat flow rates based on the measurement data *Qw*, exp and CFD simulations *Qw*,*calc* were calculated as follows

$$Q\_{w, \text{exp}} = \dot{V}\_w \rho\_w \left( T\_{w, \text{exp}}' \right) \overline{c}\_w \left( T\_{w, \text{exp}}' - T\_{w, \text{exp}}' \right) \tag{47}$$

Calculations were performed for averaged measurement data: *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*06 liters/h,

used to determine the heat flow rate *Q* (Entries 1 and 2). The water-side heat transfer coefficient in the radiator mathematical model used was determined with Eqs. (36–42). The water flow in the cooler tubes was laminar because the water flow rate was low. The air-side HTC was determined by Eq. (32) based on CFD modeling (**Figure 4a**) or by empirical correlation (43) (**Figure 4b**). **Figure 4** also shows the relative difference values *e* calculated using Eq. (49). The results presented in **Figure 4a** and **b** show an excellent agreement between the calculated and measured heat flow rates when the air-side HTC is determined using Eq. (32) based on CFD

*The heat transfer rate Q from hot water to air in a car radiator; 1 - Qw*, exp *values determined using the measurement results, 2 - heat flow rate Qw*,*calc calculated with the air-side Nusselt number* Nua *based on CFD modeling (a) or empirical correlation (b), 3 – Heat flow rate Qw*,*calc calculated with the CFD based correlation (a) or empirical correlation (b) and the average input data from 7 measurement series: <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> *liters/h,*

C*, 4 - the relative difference between Qw*, exp *and Qw*,*calc (between 1 and 2).*

*Heat transfer rate Q from hot water to air in a car radiator; 1 - Qw*, exp *values determined using the measurement results, 2 - heat flow rate Qw*,*calc calculated with the air-side Nusselt number* Nua *based on CFD modeling (a) or empirical correlation (b), 3 – Heat flow rate Qw*,*calc calculated with the CFD based correlation (a) or empirical correlation (b) and the average input data from 7 measurement series: <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*<sup>06</sup> *liters/h,*

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

C. The mathematical model of the car radiator was

C*, 4 - the relative difference between Qw*, exp *and Qw*,*calc (between 1 and 2).*

*T*0

*T*0

**Figure 4.**

*am* <sup>¼</sup> <sup>13</sup>*:*62o

*am* <sup>¼</sup> <sup>13</sup>*:*62o

modeling.

**Figure 5.**

*am* <sup>¼</sup> <sup>14</sup>*:*28oC *and <sup>T</sup>*<sup>0</sup>

*<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51o

*T*0

**43**

C and *T*<sup>0</sup>

*<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61o

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

C *and T*<sup>0</sup>

*<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61<sup>o</sup>

$$Q\_{w,calc} = \dot{V}\_w \rho\_w \left( T\_{w,exp}' \right) \overline{c}\_w \left( T\_{w,exp}' - T\_{w,calc}' \right) \tag{48}$$

where: *cw* - the mean specific heat of the water in the range between the outlet *T*<sup>00</sup> *<sup>w</sup>*, exp (Eq. (47)) or *T*<sup>00</sup> *<sup>w</sup>*,*calc* (Eq. (48)) and inlet water temperature *T*<sup>0</sup> *<sup>w</sup>*, exp , *<sup>V</sup>*\_ *<sup>w</sup>* water volume flow rate measured at the car radiator inlet.

The relative difference e was calculated from the following expression

$$e = \frac{Q\_{w, \text{exp}} - Q\_{w, \text{calc}}}{Q\_{w, \text{exp}}} \mathbf{100}, \text{@ }\tag{49}$$

The results of measurements and calculations of the heat transfer rate *Q* are shown in **Figure 4**. The results are presented for seven measurement series.

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

#### **Figure 4.**

*Heat transfer rate Q from hot water to air in a car radiator; 1 - Qw*, exp *values determined using the measurement results, 2 - heat flow rate Qw*,*calc calculated with the air-side Nusselt number* Nua *based on CFD modeling (a) or empirical correlation (b), 3 – Heat flow rate Qw*,*calc calculated with the CFD based correlation (a) or empirical correlation (b) and the average input data from 7 measurement series: <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*<sup>06</sup> *liters/h, T*0 *am* <sup>¼</sup> <sup>13</sup>*:*62o C *and T*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61o C*, 4 - the relative difference between Qw*, exp *and Qw*,*calc (between 1 and 2).*

Calculations were performed for averaged measurement data: *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*06 liters/h, *T*0 *am* <sup>¼</sup> <sup>13</sup>*:*62o C and *T*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61<sup>o</sup> C. The mathematical model of the car radiator was used to determine the heat flow rate *Q* (Entries 1 and 2). The water-side heat transfer coefficient in the radiator mathematical model used was determined with Eqs. (36–42). The water flow in the cooler tubes was laminar because the water flow rate was low. The air-side HTC was determined by Eq. (32) based on CFD modeling (**Figure 4a**) or by empirical correlation (43) (**Figure 4b**). **Figure 4** also shows the relative difference values *e* calculated using Eq. (49). The results presented in **Figure 4a** and **b** show an excellent agreement between the calculated and measured heat flow rates when the air-side HTC is determined using Eq. (32) based on CFD modeling.

#### **Figure 5.**

*The heat transfer rate Q from hot water to air in a car radiator; 1 - Qw*, exp *values determined using the measurement results, 2 - heat flow rate Qw*,*calc calculated with the air-side Nusselt number* Nua *based on CFD modeling (a) or empirical correlation (b), 3 – Heat flow rate Qw*,*calc calculated with the CFD based correlation (a) or empirical correlation (b) and the average input data from 7 measurement series: <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> *liters/h, T*0 *am* <sup>¼</sup> <sup>14</sup>*:*28oC *and <sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51o C*, 4 - the relative difference between Qw*, exp *and Qw*,*calc (between 1 and 2).*

Both when using heat transfer correlation (32) based on CFD modeling and empirical correlation (43), the relative differences *e* range from about (�1.5%) to about 5.2%.

**Figure 5** shows a similar comparison as in **Figure 4**, but for other averaged measurement data: *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*37 liters/h, *<sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>14</sup>*:*28oC and *<sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51o C. Due to the higher flow rate of water flow through a car radiator equal to *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> liters/h, the flow regime of water in pipes was turbulent. The Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* on the water side of the first pass pipes was between 5238 and 5440. In the lower pass, the Reynolds number Re *<sup>w</sup>*,*<sup>l</sup>* was higher due to the smaller number of pipes and varied between 5820 and 5883.

It was found that the correlation for the air-side Nusselt number Nu*<sup>a</sup>* determined for the entire double-row radiator based on CFD modeling provides a very good match between the calculated and measured heat flow rates transferred from hot water to air (**Figures 4** and **5**). Then, the heat flow rates transferred in the first and second row of pipes in both radiator passes were calculated taking into account different first and second row HTCs. The following relationships were applied to calculate heat flow rates in each row of pipes in the first and second pass

$$Q\_{1jHTC} = \frac{\dot{m}\_w}{2} \overline{c}\_w \left( T\_w' - T\_{w,1}'' \right) \quad j = 1, 2 \tag{50}$$

A comparison of the first and second pipe row heat flow rates in the first and second pass of the radiator and the radiator heat output for the uniform HTC throughout the radiator and different HTCs in the first and second pipe row are shown in **Figure 6a** for *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*06 liters/h and in **Figure 6b** for *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup>

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

The results presented in **Figure 6a** and **b** show that the radiator thermal output is almost identical for the uniform HTC throughout the radiator and for different HTCs in the first and second row of tubes. As shown in **Figures 5** and **6**, the radiator capacity determined by applying a mathematical model of the car cooler using the CFD based air-side correlation for the Nusselt number is very well in line with the measurements. It can also be seen that the heat flow rates exchanged in the first row of the heat exchanger, both in first and second pass, are significantly higher for different correlations for the air-side Nusselt compared to the heat flow rates

The heat capacity of the heat exchanger is almost identical with the same and different HTCs on the first and second pipe row (**Figures 5** and **6**). CFD modeling and analytical formula [19] were used to determine the air temperature increase in the entire heat exchanger. A uniform HTC for the two pipe rows was then determined from the equality condition of the calculated air temperature increases. The thermal output of the entire heat exchanger will be nearly identical with uniform and different HTCs on the first and second rows of tubes. This is because the air temperature increase on both tube rows is equal to the sum of temperature increases

*Comparison of heat flow rates transferred from hot water to air in specific pipe row and entire PFTHE assuming that the air-side HTC is the same in both pipe rows with the corresponding heat transfer rates calculated using different HTCs in the first and second pipe rows; a) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*<sup>06</sup> *liters/h, <sup>T</sup>*<sup>0</sup>

*heat flow rates transferred in specific pipe rows (Figure 1) assuming the same correlation (32) for the air-side Nusselt number, Qi*,2*HTC, i* ¼ 1, … , 4 *- heat flow rate transferred in individual pipe rows (Figure 1) considering different correlations for the air-side Nusselt number, Eq. (30) for the first tube row and Eq. (31) for the second tube row, Qt*,1*HTC - car radiator output for the same heat transfer correlation (32) for all rows of pipes, Qt*,2*HTC - car radiator output for different heat transfer correlations for the first and second tube rows,*

*am* <sup>¼</sup> <sup>14</sup>*:*28oC *and <sup>T</sup>*<sup>0</sup>

*<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51o

<sup>C</sup>*, b) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> *liters/h, <sup>T</sup>*<sup>0</sup>

*Eq. (30) for the first row and Eq. (31) for the second tube row.*

*am* <sup>¼</sup> <sup>13</sup>*:*62o

C*; Qi*,1*HTC, i* ¼ 1, … , 4 *-*

C

obtained assuming the same HTC in both pipe rows.

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

on the first and second tube rows.

liters/h.

**Figure 6.**

*and T*<sup>0</sup>

**45**

*<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61o

$$Q\_{2\bar{y}HTC} = \frac{\dot{m}\_w}{2} \overline{c}\_w \left( T\_w' - T\_{w,2}'' \right) \quad j = 1, 2 \tag{51}$$

$$Q\_{3\text{-}jHTC} = \frac{\dot{m}\_w}{2} \overline{c}\_w \left( T\_{wm} - T\_{w,3}' \right) \quad j = 1,2\tag{52}$$

$$Q\_{\mathcal{A}\_jHTC} = \frac{\dot{m}\_w}{2} \overline{c}\_w \left( T\_{wm} - T\_{w,4}^{\prime} \right) \quad j = 1,2 \tag{53}$$

The total heat transfer rate from the air to the water was calculated using the following Equation

$$Q\_{t\_{\vec{y}}\vec{H}\text{TC}} = \dot{m}\_w \overline{c}\_w \left( T\_w' - T\_w' \right) \quad j = \mathbf{1}, 2 \tag{54}$$

The symbols *Qi*,1*HTC i* ¼ 1, … , 4 designate the heat flow rate transferred in the specific pipe row (**Figure 1**) with the uniform HTC throughout the heat exchanger given by Eq. (32). The symbol *Qi*,2*HTC i* ¼ 1, … , 4 stands for the heat flow rate transferred in the particular row, with the different HTCs in the first and second pipe rows. The HTC in the first row was evaluated using Eq. (30) and in the second row by applying Eq. (31). If the air-side HTC is uniform in the whole heat exchanger, then in Eqs. (50)–(54) *j* ¼ 1 is assumed, and if the HTCs in the first and second tube row are not identical, then *j* ¼ 2 should be taken. The relative differences *ei* between the heat flow rate with different HTCs in the first and second row of tubes and the heat flow rate with uniform HTC in the entire car radiator were also calculated

$$e\_i = \frac{Q\_{i,2HTC} - Q\_{i,1HTC}}{Q\_{i,2HTC}} \cdot 100 \quad i = 1, \ldots, 4 \tag{55}$$

The relative *et* difference for the whole heat exchanger was calculated with a similar formula

$$\mathbf{e}\_{l} = \frac{\mathbf{Q}\_{\rm t,2HTC} - \mathbf{Q}\_{\rm t,1HTC}}{\mathbf{Q}\_{\rm t,2HTC}} \cdot \mathbf{100} \tag{56}$$

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

A comparison of the first and second pipe row heat flow rates in the first and second pass of the radiator and the radiator heat output for the uniform HTC throughout the radiator and different HTCs in the first and second pipe row are shown in **Figure 6a** for *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*06 liters/h and in **Figure 6b** for *<sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> liters/h.

The results presented in **Figure 6a** and **b** show that the radiator thermal output is almost identical for the uniform HTC throughout the radiator and for different HTCs in the first and second row of tubes. As shown in **Figures 5** and **6**, the radiator capacity determined by applying a mathematical model of the car cooler using the CFD based air-side correlation for the Nusselt number is very well in line with the measurements. It can also be seen that the heat flow rates exchanged in the first row of the heat exchanger, both in first and second pass, are significantly higher for different correlations for the air-side Nusselt compared to the heat flow rates obtained assuming the same HTC in both pipe rows.

The heat capacity of the heat exchanger is almost identical with the same and different HTCs on the first and second pipe row (**Figures 5** and **6**). CFD modeling and analytical formula [19] were used to determine the air temperature increase in the entire heat exchanger. A uniform HTC for the two pipe rows was then determined from the equality condition of the calculated air temperature increases. The thermal output of the entire heat exchanger will be nearly identical with uniform and different HTCs on the first and second rows of tubes. This is because the air temperature increase on both tube rows is equal to the sum of temperature increases on the first and second tube rows.

#### **Figure 6.**

*Comparison of heat flow rates transferred from hot water to air in specific pipe row and entire PFTHE assuming that the air-side HTC is the same in both pipe rows with the corresponding heat transfer rates calculated using different HTCs in the first and second pipe rows; a) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*<sup>06</sup> *liters/h, <sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>13</sup>*:*62o C *and T*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61o <sup>C</sup>*, b) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> *liters/h, <sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>14</sup>*:*28oC *and <sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51o C*; Qi*,1*HTC, i* ¼ 1, … , 4  *heat flow rates transferred in specific pipe rows (Figure 1) assuming the same correlation (32) for the air-side Nusselt number, Qi*,2*HTC, i* ¼ 1, … , 4 *- heat flow rate transferred in individual pipe rows (Figure 1) considering different correlations for the air-side Nusselt number, Eq. (30) for the first tube row and Eq. (31) for the second tube row, Qt*,1*HTC - car radiator output for the same heat transfer correlation (32) for all rows of pipes, Qt*,2*HTC - car radiator output for different heat transfer correlations for the first and second tube rows, Eq. (30) for the first row and Eq. (31) for the second tube row.*

data: *<sup>w</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*:*27 m/s, *<sup>T</sup>*<sup>0</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

to 16084.

(**Figure 8**).

(**Figure 9b**).

(**Figures 10** and **11**).

tubes.

**Figure 9.**

*difference.*

**47**

*am* <sup>¼</sup> <sup>8</sup>*:*24oC and *<sup>T</sup>*<sup>0</sup>

rate of water *V*\_ *<sup>w</sup>* at the radiator inlet varied from 309 to 2406 liters/h. The air-side Reynolds number Re *<sup>a</sup>* changed from 482.5 to 489.5 and the

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

The results presented in **Figure 8** show that the capacity of the cooler constructed with round tubes calculated for the empirical correlation to the air-side Nusselt number exceeds the capacity of the cooler calculated using the correlation to the air-side Nusselt number determined by CFD modeling. The relative difference *e* calculated from Eq. (49) is about 5% for a water-side

The relative difference *e* between the cooler output determined

of the radiator and different HTCs in the first and second tube rows.

water-side Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* in the first pass of the cooler varied from 1834

Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* of 1800 and about 10% for a Reynolds number of 16,000

**Figure 10** shows a comparison of the heat flow rates transferred in each pipe rows assuming the same air-side heat transfer coefficient in the mathematical model

The air in the first pipe row takes up a significantly higher heat flow rate assuming different HTCs in the first and second pipe row compared to the heat flow rate determined assuming an even HTC throughout the heat exchanger

The relative differences *e*<sup>1</sup> and *e*<sup>3</sup> are approximately 7.5% for Re *<sup>w</sup>*,*<sup>u</sup>* equal to 1000 and increase with Re *<sup>w</sup>*,*<sup>u</sup>* to about 15% for the Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* equal to 18,000. The relative differences of *e*<sup>2</sup> and *e*<sup>4</sup> are approximately �8% for Re *<sup>w</sup>*,*<sup>u</sup>* equal to 1000 and decrease with Re *<sup>w</sup>*,*<sup>u</sup>* to approximately �18% for the Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* equal to 18,000. The results depicted in **Figures 10** and **11** show that the radiator thermal output is almost identical with an even HTC and different HTCs on both rows of pipes, despite the various capacities of the first and second rows of

*The comparison of heat transfer rate Q from hot water to air in a car radiator as a function of the water-side Reynols number* Re *<sup>w</sup>*,*<sup>u</sup> determined experimentally and using the mathematical model of the car radiator with the uniform air-side Nusselt number* Nua *based on CFD modeling (a) and empirical correlation (b); e- relative*

experimentally and that calculated using the mathematical model of the cooler with the air-side Nusselt number correlation obtained from CFD modeling ranges from 3% to about 8.3% (**Figure 9**). When the air-side correlation is determined experimentally, the relative difference *e* is smaller and ranges from �4–0%

*<sup>w</sup>* <sup>¼</sup> <sup>70</sup>*:*56<sup>o</sup>

C. The volume flow

#### **Figure 7.**

*The relative differences between heat flow rates for different and uniform HTCs; a) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>326</sup>*:*<sup>06</sup> *liters/h, T*0 *am* <sup>¼</sup> <sup>13</sup>*:*62o C *and T*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>59</sup>*:*61o <sup>C</sup>*, b) <sup>V</sup>*\_ *<sup>w</sup>* <sup>¼</sup> <sup>1273</sup>*:*<sup>37</sup> *liters/h, <sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>14</sup>*:*28oC *and <sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>60</sup>*:*51<sup>o</sup> C*.*

**Figure 7a** and **b** illustrate the higher heat absorption by the air in the first pipe row and the significant reduction of heat transfer in the second pipe row if different correlations for the Nusselt numbers in the first and second pipe row are taken into account compared to the respective outputs of the individual rows with a uniform HTC.

**Figure 7a** and **b** show that the more considerable differences in the exchanged heat flow through the first and second row of pipes for uniform and different HTCs are higher for the larger volume flow rate passing the cooler. The differences in the heat flow rates exchanged by the individual pipe rows with equal and different HTCs on the first and second pipe row become smaller as the air-side Reynolds number increases.

#### **7.2 Engine cooler made of circular tubes**

**Figure 8** depicts a comparison of the heat flow rates for the CFD based air-side correlation (35) and empirical correlation (45) for averaged measurement

#### **Figure 8.**

*The heat transfer rate Qt*,1*HTC from water to air in a car radiator made of round tubes with an equal HTC on the air side of the entire radiator as a function of the water-side Reynolds number* Re *<sup>w</sup>*,*u; 1 - air-side correlation for* Nu*<sup>a</sup> based on the CFD modeling, 2 - air-side relationship for* Nu*<sup>a</sup> based on the experimental data, 3 - relative difference* e*.*

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

data: *<sup>w</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*:*27 m/s, *<sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>8</sup>*:*24oC and *<sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>70</sup>*:*56<sup>o</sup> C. The volume flow rate of water *V*\_ *<sup>w</sup>* at the radiator inlet varied from 309 to 2406 liters/h. The air-side Reynolds number Re *<sup>a</sup>* changed from 482.5 to 489.5 and the water-side Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* in the first pass of the cooler varied from 1834 to 16084.

The results presented in **Figure 8** show that the capacity of the cooler constructed with round tubes calculated for the empirical correlation to the air-side Nusselt number exceeds the capacity of the cooler calculated using the correlation to the air-side Nusselt number determined by CFD modeling. The relative difference *e* calculated from Eq. (49) is about 5% for a water-side Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* of 1800 and about 10% for a Reynolds number of 16,000 (**Figure 8**).

The relative difference *e* between the cooler output determined experimentally and that calculated using the mathematical model of the cooler with the air-side Nusselt number correlation obtained from CFD modeling ranges from 3% to about 8.3% (**Figure 9**). When the air-side correlation is determined experimentally, the relative difference *e* is smaller and ranges from �4–0% (**Figure 9b**).

**Figure 10** shows a comparison of the heat flow rates transferred in each pipe rows assuming the same air-side heat transfer coefficient in the mathematical model of the radiator and different HTCs in the first and second tube rows.

The air in the first pipe row takes up a significantly higher heat flow rate assuming different HTCs in the first and second pipe row compared to the heat flow rate determined assuming an even HTC throughout the heat exchanger (**Figures 10** and **11**).

The relative differences *e*<sup>1</sup> and *e*<sup>3</sup> are approximately 7.5% for Re *<sup>w</sup>*,*<sup>u</sup>* equal to 1000 and increase with Re *<sup>w</sup>*,*<sup>u</sup>* to about 15% for the Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* equal to 18,000. The relative differences of *e*<sup>2</sup> and *e*<sup>4</sup> are approximately �8% for Re *<sup>w</sup>*,*<sup>u</sup>* equal to 1000 and decrease with Re *<sup>w</sup>*,*<sup>u</sup>* to approximately �18% for the Reynolds number Re *<sup>w</sup>*,*<sup>u</sup>* equal to 18,000. The results depicted in **Figures 10** and **11** show that the radiator thermal output is almost identical with an even HTC and different HTCs on both rows of pipes, despite the various capacities of the first and second rows of tubes.

#### **Figure 9.**

*The comparison of heat transfer rate Q from hot water to air in a car radiator as a function of the water-side Reynols number* Re *<sup>w</sup>*,*<sup>u</sup> determined experimentally and using the mathematical model of the car radiator with the uniform air-side Nusselt number* Nua *based on CFD modeling (a) and empirical correlation (b); e- relative difference.*

heat transfer coefficient was proposed. Radiators constructed from round and oval pipes were investigated. Mathematical models of car radiators were developed, in which only one average heat transfer coefficient for the entire exchanger was used, as well as different heat transfer correlations for evaluating the air-side heat transfer coefficient on the first and second row of pipes. The use of the proposed heat exchanger calculation method will allow designing a heat exchanger with the optimum number of tube rows. Taking into account different heat transfer coefficients on each row of tubes in the calculations also allows for a more accurate calculation

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows…*

The water flow regime in the tubes may be laminar, transitional, or turbulent. The results of the calculations of both tested car radiators were compared with experimental data. The compatibility of calculation and measurement results is very good. The presented method of calculating plate-fin-and-tube-heat exchangers is attractive because of eliminating costly experimental research necessary for the experimental determination of the heat transfer correlations on the air and

Department of Energy, Faculty of Environmental Engineering, Cracow University

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

of the thermal output of individual rows of tubes.

*DOI: http://dx.doi.org/10.5772/intechopen.97402*

The authors declare no conflict of interest.

water side.

**Conflict of interest**

**Author details**

**49**

Dawid Taler, Jan Taler\* and Marcin Trojan

provided the original work is properly cited.

\*Address all correspondence to: jan.taler@pk.edu.pl

of Technology, Cracow, Poland

#### **Figure 10.**

*Heat flow rates transferred from hot water to air in specific pipe row and entire car radiator assuming that the air-side HTC is the same in both pipe rows with the corresponding heat transfer rates calculated using different HTCs in the first and second pipe rows; <sup>w</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>27</sup> *m/s, <sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>8</sup>*:*24oC *and <sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>70</sup>*:*56oC*; Qi*,1*HTC, i* <sup>¼</sup> 1, … , 4 *- heat flow rates transferred in specific pipe rows (Figure 1) assuming the same Eq. (32) for the airside Nusselt number throughout the radiator, Qi*,2*HTC, i* ¼ 1, … , 4 *- heat flow rate transferred in individual pipe rows (Figure 1) assuming different equations for the air-side Nusselt number in the first and second tube row, Eq. (30) was used for the first tube row and Eq. (31) for the second tube row, Qt*,1*HTC - car radiator output for the same heat transfer Eq. (32) for all rows of pipes, Qt*,2*HTC - car radiator output for different heat transfer equations for the first and second tube rows.*

#### **Figure 11.**

*The relative differences ei and et between heat flow rates for different and uniform HTCs calculated using Eq. (55) and (56); <sup>w</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>27</sup> *m/s, <sup>T</sup>*<sup>0</sup> *am* <sup>¼</sup> <sup>8</sup>*:*24oC *and <sup>T</sup>*<sup>0</sup> *<sup>w</sup>* <sup>¼</sup> <sup>70</sup>*:*56o C*.*

#### **8. Conclusions**

An analytical model of a two-pass car radiator with two rows of pipes was built. The use of the correlations determined by CFD modeling to calculate the air-side

*The CFD Based Method for Determining Heat Transfer Correlations on Individual Rows… DOI: http://dx.doi.org/10.5772/intechopen.97402*

heat transfer coefficient was proposed. Radiators constructed from round and oval pipes were investigated. Mathematical models of car radiators were developed, in which only one average heat transfer coefficient for the entire exchanger was used, as well as different heat transfer correlations for evaluating the air-side heat transfer coefficient on the first and second row of pipes. The use of the proposed heat exchanger calculation method will allow designing a heat exchanger with the optimum number of tube rows. Taking into account different heat transfer coefficients on each row of tubes in the calculations also allows for a more accurate calculation of the thermal output of individual rows of tubes.

The water flow regime in the tubes may be laminar, transitional, or turbulent. The results of the calculations of both tested car radiators were compared with experimental data. The compatibility of calculation and measurement results is very good. The presented method of calculating plate-fin-and-tube-heat exchangers is attractive because of eliminating costly experimental research necessary for the experimental determination of the heat transfer correlations on the air and water side.
