**3. Experimental methods of determining the air-side heat transfer coefficient in fin-and-tube heat exchanger**

The experimental-numerical method for determining the average air-side heat transfer coefficient was described in details in ref. [14, 17]. Moreover, in ref. [17], the detailed list of measurement points, used in this work, is presented. The experimental-numerical method involves the performance tests of a car radiator (Fig. 1) and allows to obtain the formulas for the Nusselt number for the air and water flows. During the measurements the inlet and outlet air temperatures (*T'a* and *T'''a*), the inlet and outlet water temperatures (*T'w* and *T'"w*), the volumetric mass flow rate of water *V*˙ *<sup>w</sup>*, and the inlet velocity of the air *w*0, are determined. The following change ranges of *T'a*, *T'''a*, *V*˙ *<sup>w</sup>*, *T'w*, *T'"w* and *w0* were examined:


The value of the experimental heat transfer coefficient *h <sup>a</sup>*, *<sup>i</sup> <sup>e</sup>* for the air flow is determined based on the condition that the calculated outlet temperature *T <sup>w</sup>*, *<sup>i</sup>* ''' (*h <sup>a</sup>*, *<sup>i</sup> <sup>e</sup>* ) of water must be equal to the measured temperature (*T <sup>w</sup>*, *<sup>i</sup>* ''' )*<sup>e</sup>* , where *i*=1,..., *n* is the dataset number. The following nonlinear algebraic equation must be solved for each dataset to determine *h <sup>a</sup>*, *<sup>i</sup> <sup>e</sup>* :

$$\left(\left(T\_{w,i}^{\cdots}\right)^{\epsilon} - T\_{w,i}^{\cdots}\right)\left(\left.h\_{a,i}^{\epsilon}\right) = 0\ , i = 1 , \dots \,\mathrm{n} \tag{1}$$

where *n* is the number of datasets. This study employs the mathematical model of the heat exchanger developed in [11] to calculate the water outlet temperature *T <sup>w</sup>*, *<sup>i</sup>* ' '' as a function of the heat transfer coefficient *h <sup>a</sup>*, *<sup>i</sup> <sup>e</sup>* . The heat transfer coefficient for the air flow *<sup>h</sup> <sup>a</sup>*, *<sup>i</sup> <sup>e</sup>* is determined by searching for such a preset interval that makes the measured outlet temperature of water (*T <sup>w</sup>*, *<sup>i</sup>* ''' )*<sup>e</sup>* and the computed outlet temperature *T <sup>w</sup>*, *<sup>i</sup>* ' '' the same. The outlet water temperature *T <sup>w</sup>*, *<sup>i</sup>* '' (*h <sup>a</sup>*, *<sup>i</sup> <sup>e</sup>* ) is calculated at each search step. Next, a specific form is adopted for the formula on the air-side Colburn factor *ja=ja*(Re*a*), with *m* = 2 unknown coefficients. The least squares method allows to determine the coefficients *x*1, *x*2 under the condition:

$$\mathbf{S}\_{m\mathbf{m}} = \sum\_{l=1}^{n} \left[ \mathbf{j}\_{a,l}^{\epsilon} - \mathbf{j}\_{a,l} \left( \mathbf{x}\_{1}, \mathbf{x}\_{2} \right) \right]^{2} = \min \mathbf{m}\_{\prime} \ m \le n \tag{2}$$

where:

intermediate header (2), the mixing of the water streams from the first (4) and second (5) row occurs. The intermediate temperature of the water is equal to *T"w*. Next, the water reverses and flows into the two rows of the tubes located in the second (lower) pass. Finally, the liquid, cooled down to temperature *T'''w* flows out of the heat exchanger through the outlet manifold (3). The air with inlet temperature *T'a* flows in the normal direction to the both rows of the finned tubes. After the first and second row, air temperature is *T"a* and *T'''a*, respectively (Fig.

For the CFD calculations presented in this paper (section 4), the flow in a narrow passage

The experimental-numerical method for determining the average air-side heat transfer coefficient was described in details in ref. [14, 17]. Moreover, in ref. [17], the detailed list of measurement points, used in this work, is presented. The experimental-numerical method involves the performance tests of a car radiator (Fig. 1) and allows to obtain the formulas for the Nusselt number for the air and water flows. During the measurements the inlet and outlet air temperatures (*T'a* and *T'''a*), the inlet and outlet water temperatures (*T'w* and *T'"w*), the volumetric mass flow rate of water *V*˙ *<sup>w</sup>*, and the inlet velocity of the air *w*0, are determined. The

*<sup>e</sup>* for the air flow is determined based

*<sup>e</sup>* :

*<sup>e</sup>* ) of water must be equal to

' '' as a function of the

'''(*h <sup>a</sup>*, *<sup>i</sup>*

, where *i*=1,..., *n* is the dataset number. The following non-

, ,, - = 0 , 1,.., = (1)

1). The plate fins (6) are used to enhance the heat transfer from the air side.

following change ranges of *T'a*, *T'''a*, *V*˙ *<sup>w</sup>*, *T'w*, *T'"w* and *w0* were examined:

/h,

**3. Experimental methods of determining the air-side heat transfer**

formed between two consecutive fins is considered.

**coefficient in fin-and-tube heat exchanger**

/h – 2186.40 dm3

The value of the experimental heat transfer coefficient *h <sup>a</sup>*, *<sup>i</sup>*

on the condition that the calculated outlet temperature *T <sup>w</sup>*, *<sup>i</sup>*

''')*<sup>e</sup>*

linear algebraic equation must be solved for each dataset to determine *h <sup>a</sup>*, *<sup>i</sup>*

() ( ) *<sup>e</sup> <sup>e</sup>*

exchanger developed in [11] to calculate the water outlet temperature *T <sup>w</sup>*, *<sup>i</sup>*

*wi wi ai T Th i n* ''' '''

where *n* is the number of datasets. This study employs the mathematical model of the heat

**•** *T'a* = 12.5 ºC – 15 ºC,

262 Heat Transfer Studies and Applications

**•** *<sup>V</sup>*˙ *<sup>w</sup>* = 865.8 dm3

**•** *T'''a* = 38.51 ºC – 57.66 ºC,

**•** *T'w* = 61.0 ºC – 71.08 ºC, **•** *T'''w* = 49.58 ºC – 63.83 ºC,

**•** *w0* = 1 m/s – 2.2 m/s.

the measured temperature (*T <sup>w</sup>*, *<sup>i</sup>*

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

is the air Colburn factor and Pra *= μa cpa / ka* is the air Prandtl number. The Nua *= ha dh / ka* and Rea *= wmax dh /νa* are the air Nusselt and Reynolds numbers, respectively. The velocity *wmax* is the air velocity in the narrowest free flow cross-section *Amin*. The symbol *j a*, *i <sup>e</sup>* is the experimen‐ tally determined Colburn factor, and *ja, i* is the *j*-factor calculated with the approximating function for the set value of the Reynolds number Rea*, i*. The Colburn factor *ja* is approximated by a power-law function:

$$j\_a = \propto\_1 \operatorname{Re}\_{\text{a}}^{\text{x}\_2}.\tag{4}$$

The unknown coefficients *x1* and *x2* are determined by the Levenberg-Marquardt method [35], using the Table-Curve program [36]. Combining Equations (3) and (4) one gets:

$$\mathbf{Nu\_{a}} = \mathbf{x\_{1}Re\_{a}^{\left(1\ast x\_{2}\right)}Pr\_{a}^{1/3}}.\tag{5}$$

The *wmax* air velocity in the narrowest cross-section of flow *Amin* is defined as:

$$w\_{\text{max}} = \frac{sp\_1}{A\_{\text{min}}} \left( \frac{\overline{T}\_a + 273.15}{T'\_a + 273.15} \right) w\_{0'} \tag{6}$$

where *Amin* is

$$A\_{\
u \mu \nu} = \left(\mathbf{s} - \mathcal{S}\_f\right) \left(p\_1 - d\_{\
u \mu \nu}\right). \tag{7}$$

The equivalent diameter for the air flow passage *dh* is [17, 18-19]:

$$d\_h = \frac{4A\_{\min}L\_t}{A\_f + A\_e} \tag{8}$$

where the fin surface of a single passage *Af* is:

$$A\_f = \mathcal{Z} \cdot \mathcal{Z} \left( p\_1 p\_2 - A\_{\text{out}} \right) = \left( 4p\_1 p\_2 - \pi d\_{\text{min}} d\_{\text{max}} \right) \prime \tag{9}$$

the tube external surface between two fins *Ae* is:

$$A\_{\iota} = \mathbf{2} \cdot P\_o \left(\mathbf{s} - \boldsymbol{\delta}\_f\right). \tag{10}$$

For the given parameters of the air-flow passage, the equivalent hydraulic diameter is *dh* = 0.00141 m. The arithmetic average air temperature *T*¯ *<sup>a</sup>* taken from the inlet air temperature *Ta* ' and the outlet air temperature *Ta* ''' is used to evaluate the thermal properties.

Air-side heat transfer correlations found in this chapter will be compared with the correlations of Kröger [37, 38].

The air-flow Nusselt number correlations, determined via the measurements, are listed in Table 1 [19, 20]. These correlations are paired with the water-flow heat transfer formulas, given in the literature [39- 41]. The correlations presented in Table 1 were employed to determine the outlet temperature of water *T <sup>w</sup>*, *<sup>i</sup>* ' '' using the heat exchanger model [11].

The water flow criteria numbers are: Nuw *= hin dt /kw* and Rew *= ww dt /νw*. The friction factor *ξ* is defined as:

$$\zeta = \frac{1}{\left(1.82 \,\mathrm{log}\,\mathrm{Re}\_{\mathrm{w}} - 1.64\right)^{2}} = \frac{1}{\left(0.79 \,\mathrm{ln}\,\mathrm{Re}\_{\mathrm{w}} - 1.64\right)^{2}}\tag{11}$$

The mean water velocity in a single tube – *ww* is calculated using the total volumetric flow rate *<sup>V</sup>*˙ *<sup>w</sup>* as follows:

$$
\Delta w\_w = \dot{V}\_w \,/\left(n\_{tp} A\_{w,in}\right) \, \tag{12}
$$

where *ntp* is the number of tubes in a single pass of the heat exchanger and *Aw, in* is the crosssectional area of the flow related to one tube.

Computer-Aided Determination of the Air-Side Heat Transfer Coefficient and Thermal Contact Resistance for… http://dx.doi.org/10.5772/60647 265


**Table 1.** Nusselt number formulas for the air flow Nua obtained from the measurements

The water-flow equivalent hydraulic diameter *dt* is calculated as

The equivalent diameter for the air flow passage *dh* is [17, 18-19]:

where the fin surface of a single passage *Af*

264 Heat Transfer Studies and Applications

the tube external surface between two fins *Ae* is:

0.00141 m. The arithmetic average air temperature *T*¯

and the outlet air temperature *Ta*

the outlet temperature of water *T <sup>w</sup>*, *<sup>i</sup>*

x

sectional area of the flow related to one tube.

of Kröger [37, 38].

defined as:

*<sup>V</sup>*˙ *<sup>w</sup>* as follows:

*h*

*t*

p

''' is used to evaluate the thermal properties.

' '' using the heat exchanger model [11].

2 2

= = - - (11)

*w V nA w w tp w in* ( ) , <sup>=</sup> / , & (12)

), (9)

(10)

*<sup>a</sup>* taken from the inlet air temperature *Ta*

(8)

'

*f e*

is:

*A L <sup>d</sup> A A* <sup>4</sup> <sup>=</sup> , <sup>+</sup> *min*

*A pp A pp d d <sup>f</sup>* =× - = - 2 2( )( 1 2 *oval* 4 1 2 min max

*A Ps e of* =× - 2 . ( )

d

For the given parameters of the air-flow passage, the equivalent hydraulic diameter is *dh* =

Air-side heat transfer correlations found in this chapter will be compared with the correlations

The air-flow Nusselt number correlations, determined via the measurements, are listed in Table 1 [19, 20]. These correlations are paired with the water-flow heat transfer formulas, given in the literature [39- 41]. The correlations presented in Table 1 were employed to determine

The water flow criteria numbers are: Nuw *= hin dt /kw* and Rew *= ww dt /νw*. The friction factor *ξ* is

w w 1 1 ( 1.82 log Re 1.64 ) ( 0.79 lnRe 1.64 )

The mean water velocity in a single tube – *ww* is calculated using the total volumetric flow rate

where *ntp* is the number of tubes in a single pass of the heat exchanger and *Aw, in* is the cross-

$$d\_t = \frac{4A\_{w,in}}{P\_{in}} \,\prime\tag{13}$$

where *Pi* denotes the oval perimeter (refered to inner tube wall). In this study, the water side hydraulic diameter *dt* is 0.00706 m.
