**4. Determining of the air-side heat transfer coefficient using CFD simulations**

The CFD simulations [32] were performed to model the heat and fluid flow processes in the air-flow passage, shown in Fig. 2. As a result, the air temperature and velocity are determined. Moreover, with the application of a conjugate heat transfer treatment, the wall temperature of fin and tube are calculated. A similar modeling approach for the gas flow in fin-and-tube heat exchangers was used in papers [15, 18-20]. The approach allows to simplify the computational domain and reduce the computational costs. In this study, the CFD software ANSYS CFX release 13.0 [31] was used. The phenomenon of air flow across the passage is complex e.g. flow is turbulent at the heat exchanger inlet and laminar between the fins. Hence, the SST turbulence model with Gamma-theta transitional turbulence formulation [33, 34] is used in computations. The model allows to study at the same time both the laminar and turbulent flows.

The element based finite volume method is used to discretize the differential governing equations. The coupled solver is used for the momentum and continuity equations. The Rhie-Chow interpolation scheme with the co-located grid is applied for pressure. The so-called "high resolution" scheme is used to discretize the convective terms [31].

Fig. 3 shows the discrete model and the applied boundary conditions. The model consists of three heat transfer domains: air (1), fin (2), and tube (3). The inlet boundary condition, where the values of air velocity *w*<sup>0</sup> and temperature *T'a* are prescribed, is denoted as (I). At the outlet boundary (II) the pressure level was held constant at 1 bar. At the inner tube surface (III) the convective boundary condition is applied to model the heat transfer from the water to the tube wall. The water-side heat transfer coefficient *hin* was determined from the experimental correlation for Nu*<sup>w</sup>* given in Table 1. The bulk temperature of the water *T*¯ *<sup>w</sup>* flowing through the tube is calculated as the arithmetic average of the measured temperatures: *T* '*w* and *T* '''*w*.

The thermal resistance between between external tube surface and fin *Rtc* was set at location (IV). The symmetry boundary condition is applied at the location of (V) in Fig. 3.

**Figure 3.** Flow passage studied during the computations: 1 –air, 2 – fin, 3 – tube; boundary conditions: I – inlet, II – outlet, III – convective surface, IV – solid\solid interface (thermal contact resistance), V – symmetry.

The numerical mesh, shown in Fig. 3 was used in the computation (the number of nodes: 452917, the number of elements: 404560). The grid independence tests were performed for the mass averaged outlet air temperature. Refining this numerical model does not lead to the relative change in the obtained results more than 0.1 %. The global imbalance of mass, momentum and energy equations were less than 0.1%. The boundary flow region computa‐ tional accuracy was controlled by the so-called *y*<sup>+</sup> value which was less than 3 in the present computations.

domain and reduce the computational costs. In this study, the CFD software ANSYS CFX release 13.0 [31] was used. The phenomenon of air flow across the passage is complex e.g. flow is turbulent at the heat exchanger inlet and laminar between the fins. Hence, the SST turbulence model with Gamma-theta transitional turbulence formulation [33, 34] is used in computations.

The element based finite volume method is used to discretize the differential governing equations. The coupled solver is used for the momentum and continuity equations. The Rhie-Chow interpolation scheme with the co-located grid is applied for pressure. The so-called

Fig. 3 shows the discrete model and the applied boundary conditions. The model consists of three heat transfer domains: air (1), fin (2), and tube (3). The inlet boundary condition, where the values of air velocity *w*<sup>0</sup> and temperature *T'a* are prescribed, is denoted as (I). At the outlet boundary (II) the pressure level was held constant at 1 bar. At the inner tube surface (III) the convective boundary condition is applied to model the heat transfer from the water to the tube wall. The water-side heat transfer coefficient *hin* was determined from the experimental

the tube is calculated as the arithmetic average of the measured temperatures: *T* '*w* and *T* '''*w*.

The thermal resistance between between external tube surface and fin *Rtc* was set at location

**Figure 3.** Flow passage studied during the computations: 1 –air, 2 – fin, 3 – tube; boundary conditions: I – inlet, II –

The numerical mesh, shown in Fig. 3 was used in the computation (the number of nodes: 452917, the number of elements: 404560). The grid independence tests were performed for the mass averaged outlet air temperature. Refining this numerical model does not lead to the relative change in the obtained results more than 0.1 %. The global imbalance of mass,

outlet, III – convective surface, IV – solid\solid interface (thermal contact resistance), V – symmetry.

*<sup>w</sup>* flowing through

The model allows to study at the same time both the laminar and turbulent flows.

"high resolution" scheme is used to discretize the convective terms [31].

266 Heat Transfer Studies and Applications

correlation for Nu*<sup>w</sup>* given in Table 1. The bulk temperature of the water *T*¯

(IV). The symmetry boundary condition is applied at the location of (V) in Fig. 3.

The CFD simulation results, obtained for the following parameters: *w*<sup>0</sup> = 0.8 m/s, *T'a* = 14.98 ºC, *hw* = 1512 W/(m2 ·K), *Tw* =73.85 ºC, *Rtc* = 0 (m2 ·K)/W are presented in Fig. 4 [19].

**Figure 4.** The results of test CFD simulation: a) air temperature distribution at the symmetry plane between two neigh‐ boring fins b) fin temperature c) air velocity distribution at the symmetry plane between two neighboring fins.

The temperature variations for the air and tube are shown in Fig. 4a. The air temperature is determined at the middle plane between fins. Figure 4b shows fin surface temperature while Figure 4c plots the air velocity distribution. The considerable increase of air temperature can be observed in the first tube row. The increase is larger compared to the second tube row (Fig. 4a). Also, the temperature difference between the fin surface and air is larger in the first row than in the second. Fig 4a and 4b reveals that the temperature difference between the fin surface and fluid is large in the entrance region, what in turn increases the heat flow rate. The efficient heat transfer at the inlet section is the main reason of the significant heat flow rate transferred from water to air in the first row of tubes.

In the existence of the low velocity region between the tubes along the symmetry plane, where the wake behind the upstream tube is bounded by the stagnation on the downstream one (Fig. 4c), the fin temperature (Fig. 4b) in the second tube row is high. Due to the recirculation zones the air entrapped in the vortices is heated almost up to the fin temperature (Fig. 4a). In this region the heat flow rate is close to zero, since the temperature difference between the fin surface and recirculating air is close to zero [19].

The presence of two dead-air zones near the tubes located in the second row decrease the heat flow rate from the second tube row to air. The average heat flux *q* at the outer tube surface on the length of one pitch *s* between two *y* coordinates: *y*¯ *<sup>n</sup>* and *y*¯ *<sup>n</sup>*+1 (Fig. 5) can be calculated as [19]:

$$q\left(\overline{y}\_{n+1/2}\right) = q\left(\frac{\overline{y}\_n + \overline{y}\_{n+1}}{2}\right) = \frac{\int\_{A\_o} q\_o dA\_o + \int\_{A\_c} q\_c dA\_c}{A\_o + A\_c} \Bigg|\_{\overline{y} = \overline{y}\_n} \tag{14}$$

with:

*dAo* -the elemental surface area on the outer surface of the oval tube,

*dAc* - the elemental surface area on the contact surface between fin and tube,

*qo* - the heat flux from the outer tube surface to the air across the elemental surface *dAo*,

*qc* - the heat flux from the outer tube surface to the fin base across the elemental surface *dAc*,

*y*¯ - the vertical distance from horizontal plane passing through the center of the oval tube to the elevation of the point situated on the tube outer surface.

**Figure 5.** The outer surface of oval tube (grey elements) and the contact surface between fin and tube (red elements).

The variation of outer surface heat flux with the direction of air flow, is presented in form of dimensionless coordinate *ξ*,

$$
\xi = -\frac{2\overline{y}}{d\_{\text{max}}} \tag{15}
$$

The symbol *y*¯ denotes a distance in the vertical direction between the horizontal plane passing through the oval gravity center '0' and the point located at the outer surface of the tube wall. Figure 6 [19] shows the variation of the heat flux *q* with the dimensionless major radius *ξ* of the oval tube for the first and second tube rows.

( )

+

*n*

268 Heat Transfer Studies and Applications

with:

*n*

<sup>+</sup> =

1

(14)

*y y*

*n*

*y y*

=

*o c*

ò ò

*n n A A*

*y y qy q A A*

*dAc* - the elemental surface area on the contact surface between fin and tube,

æ ö <sup>+</sup> = = ç ÷

*dAo* -the elemental surface area on the outer surface of the oval tube,

the elevation of the point situated on the tube outer surface.

dimensionless coordinate *ξ*,

1/2 2

1

è ø +

*qo* - the heat flux from the outer tube surface to the air across the elemental surface *dAo*,

*qc* - the heat flux from the outer tube surface to the fin base across the elemental surface *dAc*, *y*¯ - the vertical distance from horizontal plane passing through the center of the oval tube to

**Figure 5.** The outer surface of oval tube (grey elements) and the contact surface between fin and tube (red elements).

The variation of outer surface heat flux with the direction of air flow, is presented in form of

*y d*max 2

The symbol *y*¯ denotes a distance in the vertical direction between the horizontal plane passing through the oval gravity center '0' and the point located at the outer surface of the tube wall.

= - (15)

x

+

*oo cc*

*q dA q dA*

+

*o c*

**Figure 6.** The variations of heat flux *q* on the outer surface of tube wall for the first and second row.

The heat flux *q* reaches its highest value equal to *q* = 4.72 104 W/m2 in the first row at the inflow surface of the oval profile (*ξ* = -1), i.e. front stagnation point. In the area of the rear stagnation point (*ξ* = 1), the considerable heat flux decrease can be observed in both the first and second tube row. In the rear stagnation point on the tube in the first row, the heat flux is only *q* = 2.04 103 W/m2 .

The heat transfer is more efficient in the first row of tubes, than in the second. The mean (areaweighted) values of heat flux in the first and second tube row are: *q*¯*I* = 2.19 104 W/m2 and *q*¯*II* = 5.62 103 W/m2 , respectively. Thus the average value falls almost four times.

In subsections, 4.1 and 4.2 two methods of determining the air-side heat transfer coefficient are presented. The first considers the application of the analytical model of fin-and-tube heat exchanger while the second allows determining the air-side heat transfer coefficient directly from CFD simulations.

#### **4.1. Determination of the gas-side heat transfer coefficient using the analytical model of finand-tube heat exchanger and CFD simulation results**

The CFD calculations allow to determine the temperature and heat flux distributions in heat transfer domains. It should be noted that the local and average heat transfer coefficients are difficult to determine due to the unclear definition of fluid bulk temperature. From the definition the local heat transfer coefficient is calculated as a ratio of the local heat flux and difference between the fin surface temperature and air temperature (averaged in the reffered flow cross-section). In the case that the average temperature of the air is calculated as the arithmetic mean of the inlet and outlet temperature, the fin surface temperature at the inlet section of a channel formed by the fins is lower than the air mean temperature and then the calculated local heat transfer coefficient can be negative. This is due to a large change in air temperature with the flow direction. Another possibility of determining the average heat transfer coefficient is to calculate first the local distribution of the heat transfer coefficient and then its average value. Nevertheless, this method encounters difficulties in evaluating the local mass-averaged temperature of the air (air bulk temperature) due to the different directions of air flow in the duct between the fins (in vicinity of flow stagnation zones).

A method for determining heat transfer coefficient [18], presented in this study, aims to avoid defining the bulk temperature of air, local or average for the entire flow passage. The method is appropriate for determining the average heat transfer coefficient using the analytical solution for the temperature distribution of air flowing through the two row fin-and-tube heat exchanger. The method is compatible with experimental predictions of heat transfer correla‐ tions.

The mean heat transfer coefficient on the air side is determined from the condition that the air temperature increase over two rows of tubes, is the same for the analytical method and for the CFD calculations (Fig. 7a) [19]. To compare the air temperature difference in the heat exchang‐ er, the inlet and outlet air temperatures obtained from the CFD simmulations should be mass weighted over the inlet and outlet cross-sections. From the comparison of the difference of the air mass averaged temperatures between the inlet and outlet cross-sections with analytical temperature difference, the average heat transfer coefficient on the air side is computed. The analytical model assumes that the air side heat transfer coefficient is constant. Fig. 7b depicts the positions of evaluation planes used in the CFD simulations to determine the mass-weighted air temperatures.

The average heat transfer coefficient *ha* on the tube and fin surface is determined from the condition that the total mass average air temperature difference *ΔT*¯ *to*,*CFD* computed using ANSYS CFX program is equal to the air temperature difference *∆Tto* (*Rtc*, *ha*) calculated from an analytical model

$$
\Delta T\_{\rm to} \left( R\_{\rm tc} = 0, h\_{\rm a, CFD} \right) - \Delta \overline{T}\_{\rm to, CFD} = 0 \tag{16}
$$

The total air temperature difference *∆Tto* is

$$
\Delta T\_{\text{to}} = T\_a^{\text{"}} - T\_a^{\text{"}} = \Delta T\_{\text{l}} + \Delta T\_{\text{ll}} \tag{17}
$$

where *ΔTI* =*T* ''*<sup>a</sup>* −*T 'a* and *ΔTII* =*T* '''*<sup>a</sup>* −*T* ''*a* is the air temperature increase over the first and second tube row, respectively (Fig. 7a).

average heat transfer coefficient on the air side is computed. The analytical model assumes that the air side heat transfer coefficient is constant.. Fig. 7b depicts the positions of evaluation planes used in the Computer-Aided Determination of the Air-Side Heat Transfer Coefficient and Thermal Contact Resistance for… http://dx.doi.org/10.5772/60647 271

to a large change in air temperature with the flow direction. Another possibility of determining the average heat transfer coefficient is to calculate first the local distribution of the heat transfer coefficient and then its average value. Nevertheless, this method encounters difficulties in evaluating the local mass-averaged temperature of the air (air bulk temperature) due to the different directions of

 A method for heat transfer coefficient [18], presented in this study, aims to avoid defining the bulk temperature of air, local or average for the entire flow passage.. The method is appropriate for determining the average heat transfer coefficient using the analytical solution for the temperature distribution of air flowing through the two row fin-and-tube heat exchanger. The method of determining the heat transfer coefficient is the same as in experimental prediction of heat transfer

 The mean heat transfer coefficient on the air side is determined from the condition that the air temperature increase over two rows of tubes, is the same for the analytical method and for the CFD calculations (Fig. 7a) [19]. To compare the air temperature difference in the heat exchanger, the inlet and outlet air temperatures obtained from the CFD simmulations should be mass weighted over the inlet and outlet cross-sections. From the comparison of the difference of the air mass averaged temperatures between the inlet and outlet cross-sections with analytical temperature difference, the

air flow in the duct between the fins (in vicinity of flow stagnation zones).

CFD simulations to determine the mass-weighted air temperatures.

correlations..

difference between the fin surface temperature and air temperature (averaged in the reffered flow cross-section). In the case that the average temperature of the air is calculated as the arithmetic mean of the inlet and outlet temperature, the fin surface temperature at the inlet section of a channel formed by the fins is lower than the air mean temperature and then the calculated local heat transfer coefficient can be negative. This is due to a large change in air temperature with the flow direction. Another possibility of determining the average heat transfer coefficient is to calculate first the local distribution of the heat transfer coefficient and then its average value. Nevertheless, this method encounters difficulties in evaluating the local mass-averaged temperature of the air (air bulk temperature) due to the different directions of

A method for determining heat transfer coefficient [18], presented in this study, aims to avoid defining the bulk temperature of air, local or average for the entire flow passage. The method is appropriate for determining the average heat transfer coefficient using the analytical solution for the temperature distribution of air flowing through the two row fin-and-tube heat exchanger. The method is compatible with experimental predictions of heat transfer correla‐

The mean heat transfer coefficient on the air side is determined from the condition that the air temperature increase over two rows of tubes, is the same for the analytical method and for the CFD calculations (Fig. 7a) [19]. To compare the air temperature difference in the heat exchang‐ er, the inlet and outlet air temperatures obtained from the CFD simmulations should be mass weighted over the inlet and outlet cross-sections. From the comparison of the difference of the air mass averaged temperatures between the inlet and outlet cross-sections with analytical temperature difference, the average heat transfer coefficient on the air side is computed. The analytical model assumes that the air side heat transfer coefficient is constant. Fig. 7b depicts the positions of evaluation planes used in the CFD simulations to determine the mass-weighted

The average heat transfer coefficient *ha* on the tube and fin surface is determined from the

ANSYS CFX program is equal to the air temperature difference *∆Tto* (*Rtc*, *ha*) calculated from

where *ΔTI* =*T* ''*<sup>a</sup>* −*T 'a* and *ΔTII* =*T* '''*<sup>a</sup>* −*T* ''*a* is the air temperature increase over the first and

D = -D = *TR h T to tc a CFD to CFD* ( 0, , , ) 0 (16)

*to a a I II TTT T T* ''' ' D = - =D +D (17)

*to*,*CFD* computed using

condition that the total mass average air temperature difference *ΔT*¯

air flow in the duct between the fins (in vicinity of flow stagnation zones).

tions.

270 Heat Transfer Studies and Applications

air temperatures.

an analytical model

The total air temperature difference *∆Tto* is

second tube row, respectively (Fig. 7a).

**Fig. 7.** Cross flow heat exchanger with two rows of tubes: a) air flow passage used in analytical model, b) evaluation planes for mass averaged temperatures *T'a* , *T''a,* and *T'''a* used in CFD simulations. **Figure 7.** Cross flow heat exchanger with two rows of tubes: a) air flow passage used in analytical model, b) evaluation planes for mass averaged temperatures *T'a*, *T''a,* and *T'''a* used in CFD simulations.

The average heat transfer coefficient *ha* over two rows of tubes is calculated by solving equation (16). This study assumes the same water temperature *Tw* in the first and the second tube. This small temperature difference has insignificant influence on the average heat transfer coefficient *ha*. Furthermore, the water temperatures are assumed as constant along the tube length. Under these assumptions, the following differential equations with appropriate boundary conditions describe the air temperature [19] The average heat transfer coefficient *ha* on the tube and fin surface is determined from the condition that the total mass average air temperature difference *Tto CFD* , computed using ANSYS CFX program is equal to the air temperature difference *Tto* (*Rtc*, *ha*) calculated from an analytical model

$$\frac{d\left(T\_a\left(y\_I^{\*}\right)\right)}{dy\_I^{\*}} = \mathbf{N}\_a^I \left[\left.T\_w - T\_a\left(y\_I^{\*}\right)\right]\right] \tag{18}$$

$$T\_a \Big|\_{y\_1^\* = 0} = T\_a' \tag{19}$$

$$\frac{d\left(T\_a\left(\boldsymbol{y}\_{ll}^{\*}\right)\right)}{d\left(\boldsymbol{y}\_{ll}^{\*}\right)} = \mathbf{N}\_a^{\mathrm{II}} \left[\boldsymbol{T}\_w - \boldsymbol{T}\_a\left(\boldsymbol{y}\_{ll}^{\*}\right)\right] \tag{20}$$

$$T\_{\
u} \Big|\_{y\_H^\* = 0} = T\_{\
u}^{\prime \prime} \tag{21}$$

Solving the initial-boundary problems (18-19) and (20-21) yields

$$T\_a \left( y\_I^\* \right) = T\_w + \left( T\_a' - T\_w \right) e^{-N\_a^{\bar{\imath}} y\_I^\*} \tag{22}$$

$$T\_a\left(y\_{ll}^{+}\right) = T\_w + \left(T\_a' - T\_w\right)e^{-\left(N\_a^{I} + N\_a^{\overline{w}}\left|y\_{ll}^{+}\right.\right)}\tag{23}$$

where

$$\mathbf{N}\_a^I = \mathbf{U}\_o^I \ A\_o \big/ \left(\dot{m}\_a \ c\_{pa}\right) \quad \mathbf{N}\_a^{\mathrm{II}} = \mathbf{U}\_o^{\mathrm{II}} \ A\_o \big/ \left(\dot{m}\_a \ c\_{pa}\right) \big|$$

The symbols *m*˙ *<sup>a</sup>* and *A* denote the air mass flow rate and the outer surface area of the bare tube, respectively. The overall heat transfer coefficient referred to surface area *Ao* can be expressed as [15, 19-22]:

$$\mathcal{U}\_o = \frac{1}{\frac{A\_o}{A\_{in}}\frac{1}{h\_{in}} + \frac{2A\_o}{A\_{in} + A\_o}\frac{\delta\_t}{k\_t} + \frac{1}{h\_o}}\tag{24}$$

with: *Ain* – area of the inner tube surface, *δ<sup>t</sup>* - the thickness of tube wall, *kt* - the thermal conductivity of the tube, *hin* - the water side heat transfer coefficient. The equivalent air-side heat transfer coefficient *h* ¯ *<sup>a</sup>* referred to the tube outer surface area *Ao* is defined as:

$$\overline{h}\_a = \frac{A\_f \eta\_f \left(R\_{\text{ac}}, h\_a\right) + A\_e}{A\_g} h\_a \tag{25}$$

where [19]

$$\ln \eta\_f \left( R\_{\text{ic}}, h\_a \right) = \frac{\left( c\_1 + c\_3 R\_{\text{ic}} + c\_5 h\_a + c\_7 R\_{\text{ic}}^2 + c\_8 h\_a^2 + c\_{11} R\_{\text{ic}} h\_a \right)}{\left( 1 + c\_2 R\_{\text{ic}} + c\_4 h\_a + c\_6 R\_{\text{ic}}^2 + c\_8 h\_a^2 + c\_{10} R\_{\text{ic}} h\_a \right)} \tag{26}$$

The unknown coefficients in the function (26) were estimated by the Levenberg – Marquardt method using a commercial software Table Curve 3d version 4.0 [36]. The coefficients appear‐ ing in the function *η<sup>f</sup>* (*Rtc*, *ha*) are shown in Table 2 [19].

The differences of air temperature over the first and second tube row can be calculated as follows

Computer-Aided Determination of the Air-Side Heat Transfer Coefficient and Thermal Contact Resistance for… http://dx.doi.org/10.5772/60647 273


**Table 2.** The coefficients of function *ηf,* (*Rtc*, *ha*) given by expression (26) [19].

( ) ( ) *<sup>I</sup> N y a I aI w w Ty T T T e* <sup>+</sup> <sup>+</sup> - =+ -*<sup>a</sup> '*

( ) ( ) ( ) *I II N Ny a a II a II w <sup>w</sup> Ty T T Te* <sup>+</sup> - + <sup>+</sup> =+ -*<sup>a</sup> '*

( ) ( ) *I I II II N U A mc N U A mc a o o a pa a o o a pa* = = & & ,

The symbols *m*˙ *<sup>a</sup>* and *A* denote the air mass flow rate and the outer surface area of the bare tube, respectively. The overall heat transfer coefficient referred to surface area *Ao* can be expressed

> *o o t in in in o t a*

*A h A Ak h*

+ + +

conductivity of the tube, *hin* - the water side heat transfer coefficient. The equivalent air-side

*f f tc a e* ( ) *a a g*

*A Rh A h h A*

( ) ( )

+ ++ ++ , 1+ + + + +

(*Rtc*, *ha*) are shown in Table 2 [19].

( ) *tc a tc a tc a*

2 4 6 8 10

*cR ch cR ch c Rh*

*c cR ch cR ch c Rh*

1 3 5 7 9 11

The unknown coefficients in the function (26) were estimated by the Levenberg – Marquardt method using a commercial software Table Curve 3d version 4.0 [36]. The coefficients appear‐

The differences of air temperature over the first and second tube row can be calculated as

*tc a tc a tc a*

2 2

= (26)

2 2

h

*<sup>a</sup>* referred to the tube outer surface area *Ao* is defined as:

1 1 1 2 d

*<sup>U</sup> A A*

<sup>=</sup>

*o*

with: *Ain* – area of the inner tube surface, *δ<sup>t</sup>*

¯

*f tc a*

h

*R h*

heat transfer coefficient *h*

where [19]

ing in the function *η<sup>f</sup>*

follows

where

272 Heat Transfer Studies and Applications

as [15, 19-22]:

(22)

(23)


, <sup>+</sup> <sup>=</sup> (25)

(24)

$$
\Delta T\_l = T\_a \Big|\_{y\_l^\* = 1} - T\_a \Big|\_{y\_l^\* = 0} = \left( T\_w - T\_a' \right) \left( 1 - e^{-N\_s'} \right) \tag{27}
$$

$$
\Delta T\_{ll} = T\_a \left|\_{y\_H^\*-1} - T\_a \right|\_{y\_H^\*-0} = \left( T\_w - T\_a' \right) e^{-N\_s^I} \left( 1 - e^{-N\_s^H} \right) \tag{28}
$$

Assuming that the heat transfer coefficients in the first and second tube row are equal, i.e. *ha I* = *ha II* = *ha* and the water side heat transfer coefficient *hin* is the same in both tubes results in the equality of the numbers of heat transfer units across the first and second row, i.e. *Na* = *Na <sup>I</sup>* <sup>=</sup> *Na II* . Hence, the total temperature difference *ΔTto* over two rows can be defined as

$$
\Delta T\_{\rm to} = \Delta T\_{\rm I} + \Delta T\_{\rm II} = \left(T\_w - T\_a'\right)\left(1 - e^{-2N\_s}\right) \tag{29}
$$

The temperature difference *ΔTto* given by expression (29) and Eq. (17) are nonlinear functions of the heat transfer coefficient *ha*. Also, the overall heat transfer coefficient *Uo* =*Uo I* = *Uo II* is a nonlinear function of *h* ¯ *<sup>a</sup>*, which in turn depends on *ha*. The expression (29) is used in Equation (16) to evaluate the heat transfer coefficient *ha* while the temperature difference *ΔT*¯ *to*,*CFD* obtained from the CFD simulations is assumed as a measured temperature difference.

#### **4.2. Determination of the gas-side heat transfer coefficient directly from CFD simulations of fin-and-tube heat exchanger**

The method of determining the average heat transfer coefficient directly from CFD simulation was presented in [20]. The average heat transfer coefficients can be calculated, based on the following relationship:

$$\mathcal{H}\_{\text{avg}, \mathcal{CFD}} = \frac{\mathcal{Q}}{A\_t \left(\overline{T}\_{\text{null}} - T\_{\text{oc}}\right)} \tag{30}$$

where the heat transfer rate, referenced to a single pitch, is:

$$\mathcal{Q} = \dot{m} \left( \dot{\mathbf{i}}\_{0,outet} - \dot{\mathbf{i}}\_{0,inlet} \right) \tag{31}$$

where *m*˙ denotes the mass flow rate of the air, *i*0, *outlet* and *i*0, *inlet* are the air static enthalpy calculated at the outlet and inlet of the flow passage, respectively. The total heat transfer area is calculated as:

$$A\_t = A\_f + A\_{\iota^{\prime}} \tag{32}$$

the area averaged wall temperature is defined as:

$$\overline{T}\_{\text{null}} = \frac{1}{A\_t} \int T\_{\text{null}} dA\_\prime \tag{33}$$

the air bulk temperature *T∞* is calculated as the arithmetic mean temperature from the air inlet and outlet temperatures:

$$T\_{\ll} = \overline{T}\_{a} = 0.5 \left( T\_{a}^{\cdot} + T\_{a}^{\cdot \cdot} \right). \tag{34}$$

Correlations for air-side heat transfer coefficient will be determined using both methods presented in this chapter. If the air temperature increase (*Ta* ''' −*Ta* ' ) is small then both procedures described in the sections 4.1 and 4.2 give the same results.

#### **5. Results and discussion**

#### **5.1. The correlation on gas-side heat transfer coefficient obtained directly from CFD simulations**

Table 3 lists the flow and heat transfer parameters studied during the performed computa‐ tional cases [20]. Moreover the values of the computed outlet air temperature *T'''a* are given in Table 3.


**4.2. Determination of the gas-side heat transfer coefficient directly from CFD simulations**

The method of determining the average heat transfer coefficient directly from CFD simulation was presented in [20]. The average heat transfer coefficients can be calculated, based on the

> ( ) *avg CFD t wall*

*AT T* , ,

where *m*˙ denotes the mass flow rate of the air, *i*0, *outlet* and *i*0, *inlet* are the air static enthalpy calculated at the outlet and inlet of the flow passage, respectively. The total heat transfer area

> *t wall wall t A T T dA A*

the air bulk temperature *T∞* is calculated as the arithmetic mean temperature from the air inlet

Correlations for air-side heat transfer coefficient will be determined using both methods

**5.1. The correlation on gas-side heat transfer coefficient obtained directly from CFD**

Table 3 lists the flow and heat transfer parameters studied during the performed computa‐ tional cases [20]. Moreover the values of the computed outlet air temperature *T'''a* are given in

¥

<sup>=</sup> - (30)

*Q mi i* = - & ( 0, 0, *outet inlet* ), (31)

*AAA t fe* = + , (32)

<sup>1</sup> <sup>=</sup> , ò (33)

*T T TT a aa* ( ) ' ''' 0.5 . ¥ == + (34)

'''

−*Ta* '

) is small then both procedures

*<sup>Q</sup> <sup>h</sup>*

where the heat transfer rate, referenced to a single pitch, is:

the area averaged wall temperature is defined as:

presented in this chapter. If the air temperature increase (*Ta*

described in the sections 4.1 and 4.2 give the same results.

**of fin-and-tube heat exchanger**

274 Heat Transfer Studies and Applications

following relationship:

is calculated as:

and outlet temperatures:

**5. Results and discussion**

**simulations**

Table 3.


**Table 3.** The list of the computational cases used in the CFD simulations and the values of inlet air velocity *w*0, inlet air temperature *T'a*, the average heat transfer coefficient for water flow *hin*, average temperature of water *<sup>T</sup>*¯ *<sup>w</sup>* and outlet temperature of the air *T'''a*

The determined values of the average heat transfer coefficients *havg*.*CFD* are listed in Table 4 [20]. The computations were carried out for the mean water temperatures: *T*¯ *w* = 30 ºC and *T*¯ *<sup>w</sup>* = 65 ºC, respectively, to demonstrate that the influence of the tube wall temperature on the determined air side heat transfer coefficients is insignificant. The maximum relative difference between the heat transfer coefficients for *T*¯ *w* = 30 ºC and *T*¯ *<sup>w</sup>* = 65 ºC does not exceed 2.9 %. These discrepancies are due to different temperature in the boundary layer, which in turn affects the value of thermal conductivity and kinematic viscosity of air, although the air side Prandtl number is 0.7 in both cases. A similar effect of wall temperature on the value of heat transfer coefficient on the air side can be expected in experimental studies [20].


**Table 4.** The values of the heat transfer rate *Q* referenced to a single pitch, the area averaged wall temperature *<sup>T</sup>*¯ *wall* , the bulk temperature of the air *T∞* and the average heat transfer coefficient *havg*,*CFD* for the air flow, obtained for the computational cases listed in Table 1

The values of *havg*,*CFD* obtained when *T*¯ *w* = 30 ºC and *T*¯ *<sup>w</sup>* = 65 ºC do not differ significantly for the same air velocity. Table 5 [20] lists the Nusselt number correlation obtained from CFD simulations.


**Table 5.** Nusselt number formulas for the air flow Nua obtained from the CFD simulations based on the mean arithmetic temperatures of the air: *T<sup>∞</sup>* =65<sup>ο</sup> C and *T<sup>∞</sup>* =30<sup>ο</sup> C

The air-flow Nusselt number correlations obtained from CFD simulations are compared with the experimental correlations listed in Table 1. Fig. 8 reveals that the correlations for the airflow Nusselt number, determined via the CFD simulations, predicts slightly lower values than the one obtained via the measurements. The maximum percentage differences can be observed for Rea = 150, where the values of the Nusselt number, obtained using the CFD simulations are from 10.1 % to 13.7% lower than those obtained from the measurements. For the largest value of Rea (Rea = 400) these differences are smaller: from 0.5 % to 8.4 % [20].

**Case no.** *w0, m/s <sup>Q</sup>* **, W** *<sup>T</sup>***¯**

276 Heat Transfer Studies and Applications

computational cases listed in Table 1

simulations.

1

2

The values of *havg*,*CFD* obtained when *T*¯

arithmetic temperatures of the air: *T<sup>∞</sup>* =65<sup>ο</sup>

No. Correlation – CFD simulations

NUa

NUa

(*T<sup>∞</sup>* =65<sup>ο</sup>

(*T<sup>∞</sup>* =30<sup>ο</sup>

C)= *x*1Rea *x*2 Pra 1/3

C)= *x*1Rea *x*2 Pra 1/3

**Table 5.** Nusselt number formulas for the air flow Nua obtained from the CFD simulations based on the mean

C

C and *T<sup>∞</sup>* =30<sup>ο</sup>

150 < Rea < 400 Pra = 0.7

150 < Rea < 400 Pra = 0.7

*wall* **, ºC** *<sup>T</sup>***¯**

 1 0.8609 59.049 37.014 39.385 1.2 1.0089 58.059 36.521 47.121 1.4 1.1445 57.152 36.066 54.155 1.6 1.2678 56.321 35.651 60.347 1.8 1.3804 55.569 35.275 65.849 2 1.4806 54.865 34.922 70.589 2.2 1.575 54.247 34.614 74.774 2.4 1.6608 53.672 34.326 78.506 2.5 1.7007 53.403 34.191 80.204 1 0.2570 28.228 21.604 38.913 1.2 0.3010 27.938 21.459 46.399 1.4 0.3405 27.661 21.321 53.069 1.6 0.3765 27.416 21.198 58.935 1.8 0.4091 27.186 21.083 64.106 2 0.4392 26.989 20.985 68.628 2.2 0.4662 26.798 20.889 72.563 2.4 0.4913 26.625 20.803 76.083 2.5 0.5039 26.551 20.765 77.804

**Table 4.** The values of the heat transfer rate *Q* referenced to a single pitch, the area averaged wall temperature *<sup>T</sup>*¯

the bulk temperature of the air *T∞* and the average heat transfer coefficient *havg*,*CFD* for the air flow, obtained for the

*w* = 30 ºC and *T*¯

the same air velocity. Table 5 [20] lists the Nusselt number correlation obtained from CFD

*<sup>a</sup>* **=***T∞* **, ºC** *havg***,***CFD* **, W/(m2**

 **K)**

*wall* ,

*<sup>w</sup>* = 65 ºC do not differ significantly for

Estimated parameters

*x*1 = 0.0674±0.00621 *x* 2 = 0.7152±0.0612

*x*1 = 0.0623±0.00574 *x* 2 = 0.7336±0.0703

**Figure 8.** The values of the Nusselt number of the air Nua obtained for the Reynolds numbers Rea = 150 – 400 and the Prandtl number Pra = 0.7, using the correlations listed in Table 1 (experimental correlations: Cor. 1 – Cor. 3) and in Table 5 (correlations based on CFD: Cor. CFD 1, Cor. CFD 2).

The values of the Prandtl numbers for the air and water: Pra = 0.7 and Prw = 2.75 are typical for air temperatures *T*¯ *<sup>a</sup>* from 10 ºC to 40 ºC and for water temperature *T*¯ *<sup>w</sup>* = 65 ºC. Fig. 8 and Fig. 9 reveal that the experimental correlation 1 (see Table 1) predicts the largest values of the Nusselt number for the air flow if Rea > 150 and for water flow if Rew > 10364. Experimental correlation 2 predicts the lowest values of the Nusselt number for the air flow if Rea > 150 and for water flow if Rew > 4000. Experimental correlation 3 predicts slightly larger values of Nua if Rea > 150 and the largest values of Nuw if Rew < 10364.

During the CFD simulations the idealistic heat transfer conditions were assumed: the constant inlet velocity and the perfect contact between the fin and the outer surface of tube wall. In a real fin-and-tube heat exchanger the maldistribution of air flow as well as the thermal contact resistance between the fin and tube [18, 19] can significantly influence the heat and momentum transfer. Furthermore, the maldistributions of water flow to the tubes of heat exchanger exists for these devices [21-23].The circumstances, mentioned above, explain why the Nusselt

**Figure 9.** The values of the Nusselt number of water Nuw obtained for the Reynolds numbers Rew = 4000 – 12000 and the Prandtl number Prw = 2.75 using the correlations presented in Table 1.

number correlations obtained using CFD simulation differ slightly from the experimental correlations. The analytical-numerical approach for calculating the average thermal contact resistance for a studied fin-and-tube heat exchanger is presented in section 6.

#### **5.2. The correlation on gas-side heat transfer coefficient obtained using fin-and-tube heat exchanger model and CFD simulations**

Application of the proposed method is illustrated by the following data set[19]:


The temperatures *T'a*, *Tw*, and the heat transfer coefficient *hin* were held constant, while the inlet air velocity *w*0 was varied from *w0* = 1 m/s to *w0* = 2.5 m/s (Table 6). First, the CFD simulations were performed without including thermal contact resistance (*Rtc* = 0). Table 6 [19] lists the air temperature differences obtained from the CFD simulations, for the first and second tube rows (*ΔT*¯ *<sup>I</sup>* ,*CFD* and *ΔT*¯ *II* ,*CFD*) as well as the total air temperature difference *ΔT*¯ *to*,*CFD*. The secant method was employed to solve the nonlinear algebraic equation (16) for the air-side heat transfer coefficient *ha, CFD*. The values of *ha, CFD* and heat transfer coefficients *ha, me* obtained based on the experimental data (correlation 4 in Table 1), are shown in Table 7 [19].

Computer-Aided Determination of the Air-Side Heat Transfer Coefficient and Thermal Contact Resistance for… http://dx.doi.org/10.5772/60647 279


**Table 6.** Temperature differences for the first and second row of tubes *ΔT*¯ *t*,*CFD* and *ΔT*¯ *II* ,*CFD*) and the total temperature difference *ΔT*¯ *to*,*CFD* obtained using CFD simulations for different air inlet velocities *w*<sup>0</sup>


**Table 7.** Air-side heat transfer coefficient for entire heat exchanger obtained from CFD simulation: *ha,CFD* and experimental correlation *ha, me* (correlation 4 in Table 1) for different air inlet velocities *w*0.

The air-side Reynolds and Prandtl numbers (Re*a* and Pr*a*) were calculated as presented in section 3 for the experimental method. For the determined heat transfer coefficients *ha, CFD* the heat transfer correlations are derived as follows. First, the Colburn factor *ja* is approximated using the power law function [20]

$$j\_a = \propto\_1 \text{Re}\_a^{x\_1} \tag{35}$$

where the Colburn factor *ja* is defined as [19, 20]

number correlations obtained using CFD simulation differ slightly from the experimental correlations. The analytical-numerical approach for calculating the average thermal contact

**Figure 9.** The values of the Nusselt number of water Nuw obtained for the Reynolds numbers Rew = 4000 – 12000 and

**5.2. The correlation on gas-side heat transfer coefficient obtained using fin-and-tube heat**

The temperatures *T'a*, *Tw*, and the heat transfer coefficient *hin* were held constant, while the inlet air velocity *w*0 was varied from *w0* = 1 m/s to *w0* = 2.5 m/s (Table 6). First, the CFD simulations were performed without including thermal contact resistance (*Rtc* = 0). Table 6 [19] lists the air temperature differences obtained from the CFD simulations, for the first and second tube rows

*II* ,*CFD*) as well as the total air temperature difference *ΔT*¯

method was employed to solve the nonlinear algebraic equation (16) for the air-side heat transfer coefficient *ha, CFD*. The values of *ha, CFD* and heat transfer coefficients *ha, me* obtained based

on the experimental data (correlation 4 in Table 1), are shown in Table 7 [19].

·K).

*to*,*CFD*. The secant

resistance for a studied fin-and-tube heat exchanger is presented in section 6.

Application of the proposed method is illustrated by the following data set[19]:

**•** air velocity *w*0 in front of heat exchanger: 1 m/s – 2.5 m/s,

the Prandtl number Prw = 2.75 using the correlations presented in Table 1.

**•** air temperature before the heat exchanger *T'a* = 14.98 ºC,

**•** water side heat transfer coefficient *hin* = 4793.95 W/(m2

**•** mean water temperature in the tubes *Tw* = 68.3 ºC,

**exchanger model and CFD simulations**

278 Heat Transfer Studies and Applications

(*ΔT*¯

*<sup>I</sup>* ,*CFD* and *ΔT*¯

$$j\_a = \text{Nu}\_a \,/\,\text{(Re}\_a \text{Pr}\_a^{1/3}\text{)}\tag{36}$$

Based on the heat transfer coefficients *ha, CFD* obtained from the solution of Equation (16), the Colburn factors (Table 7) *j a*,*i CFD* =Nu*a*,*<sup>i</sup> CFD* / (Re*a*,*<sup>i</sup>* Pr*a*,*<sup>i</sup>* 1/3), *i* = 1,.., 8, were calculated. The symbol Nu*a*,*<sup>i</sup> CFD* <sup>=</sup>*ha*,*CFD dh ka* is the Nusselt number for i*th* data set CFD. The unknown coefficients *x*1 and *x*2 in the function (35) were determined using the least squares method. The coefficients *x*<sup>1</sup> and *x*2 were selected to minimize the following sum of squares:

$$\mathbf{S} = \sum\_{l=1}^{n=8} \left( f\_{a,l}^{\text{CFD}} - \mathbf{x}\_1 \, \text{Re}\_{a,l}^{\text{x}\_2} \right)^2 \tag{37}$$

The symbol *n* is the number of data sets shown in Table 7.

The coefficients *x*1 and *x*2 obtained using the least squares method for the data sets listed in Table 5 are: *x*1 = 0.188 and *x*<sup>2</sup> = - 0.382. To find the optimum values of *x*1 and *x*2 the Levenberg-Marquardt method was used [35]. The MATLAB R2012 curve fitting toolbox [42] was used for this purpose. Figure 10 [19] depicts the obtained correlation *j a CFD*(Re*a*), also the prediction bounds set at 95 % confidence level are presented.

**Figure 10.** Correlation *j a CFD*(Re*a*)=0.188*Rea* <sup>−</sup>0.382 - continuous line, and prediction bounds set at 95% confidence lev‐ el – dashed line. The correlation was based on the CFD data set.

Fig. 10 reveals that the correlation *j <sup>a</sup>*(Re*a*)=0.1878*Rea* <sup>−</sup>0.382 predicts the values of Colburn factor *j a CFD* well for Re*<sup>a</sup>* ∈ (170, 390). The expression on the air side Nusselt number is obtained after rearranging Eq. (36)

Computer-Aided Determination of the Air-Side Heat Transfer Coefficient and Thermal Contact Resistance for… http://dx.doi.org/10.5772/60647 281

$$\mathbf{Nu}\_a = \mathbf{x}\_1 \mathbf{Re}\_a^{(1 \times \mathbf{x}\_2)} \mathbf{Pr}^{1/3} \tag{38}$$

The following formula for the air-side heat transfer coefficient was obtained after substituting the estimated coefficients *x*1 and *x*2 into the correlation (38),

$$\mathcal{H}\_{a, \text{CFD}} = \frac{k\_a}{d\_h} \mathcal{N} \boldsymbol{\mu}\_a = \frac{k\_a}{d\_h} 0.188 \,\text{Re}\_a^{0.618} \,\text{Pr}\_a^{1/3} \tag{39}$$

In ref. [37] similar correlations for continuous-fin and tube heat exchangers can be found. The correlation

$$h\_a = \frac{k\_a}{d\_h} 0.174 \,\mathrm{Re}\_a^{0.613} \,\mathrm{Pr}\_a^{1/3} \tag{40}$$

obtained by Kröger [38] is similar to the correlation (39).

Based on the heat transfer coefficients *ha, CFD* obtained from the solution of Equation (16), the

Pr*a*,*<sup>i</sup>*

*x*2 in the function (35) were determined using the least squares method. The coefficients *x*<sup>1</sup> and

( ) *<sup>n</sup> CFD <sup>x</sup> a i a i*

Re

The coefficients *x*1 and *x*2 obtained using the least squares method for the data sets listed in Table 5 are: *x*1 = 0.188 and *x*<sup>2</sup> = - 0.382. To find the optimum values of *x*1 and *x*2 the Levenberg-Marquardt method was used [35]. The MATLAB R2012 curve fitting toolbox [42] was used for

*S jx* <sup>2</sup> <sup>2</sup> <sup>8</sup> , 1,

is the Nusselt number for i*th* data set CFD. The unknown coefficients *x*1 and

= - å (37)

*a*

<sup>−</sup>0.382 - continuous line, and prediction bounds set at 95% confidence lev‐

<sup>−</sup>0.382 predicts the values of Colburn factor

*CFD*(Re*a*), also the prediction

1/3), *i* = 1,.., 8, were calculated. The symbol

*CFD* / (Re*a*,*<sup>i</sup>*

Colburn factors (Table 7) *j*

280 Heat Transfer Studies and Applications

*dh ka*

Nu*a*,*<sup>i</sup>*

*CFD* <sup>=</sup>*ha*,*CFD*

**Figure 10.** Correlation *j*

rearranging Eq. (36)

*j a* *a*

Fig. 10 reveals that the correlation *j*

*CFD*(Re*a*)=0.188*Rea*

*<sup>a</sup>*(Re*a*)=0.1878*Rea*

*CFD* well for Re*<sup>a</sup>* ∈ (170, 390). The expression on the air side Nusselt number is obtained after

el – dashed line. The correlation was based on the CFD data set.

*a*,*i CFD* =Nu*a*,*<sup>i</sup>*

*x*2 were selected to minimize the following sum of squares:

The symbol *n* is the number of data sets shown in Table 7.

bounds set at 95 % confidence level are presented.

this purpose. Figure 10 [19] depicts the obtained correlation *j*

*i*

1

= =

> The thermal contact resistance exists between the tube and fin for some methods of attaching the fins on the tubes. It reduces the heat transfer rate between the fluids in the heat exchanger.

> The correlation (39) leads to over-prediction of the heat transfer rate from the hot to the cold fluid, when the contact resistance occurs. The thermal contact resistance between the tube and the fin base will be determined by using the correlation (39) and the experimental results.
