**6. Estimation of the thermal contact resistance between the tube outer surface and fin base usingCFD simulations and experimental data**

The correlation for the air-side Nusselt number was derived based on: the experimental data and the CFD simulation. The values of the heat transfer coefficients obtained from the CFD simulation *ha, CFD* and from the experiment *ha, me* differ from each other (compare Table1 and Table 7). The method based on the CFD simulation gives larger values of *ha* in comparison to the experimental-numerical method (Table 7). The reason for this discrepancy is the thermal contact resistance between the fin and tube in the tested car radiator.

The air temperature increase across two tube rows *ΔT*¯ *to*,*CFD* calculated using the heat transfer coefficient *ha, CFD* obtained from the CFD based method, is greater than the calculated temper‐ ature rise *ΔT*¯ *to*,*me* obtained with the heat transfer coefficient *ha, me*. The temperature differences *ΔT*¯ *to*,*CFD* and *ΔT*¯ *to*,*me* can be equal if a thermal contact resistance is included in the CFD simulations.

The air temperature difference *ΔT*¯ *to*,*CFD* through the entire heat exchanger depends on the thermal contact resistance *Rtc* and air-side heat transfer coefficient *ha*. To determine the thermal contact resistance *Rtc*, the nonlinear algebraic equation

$$
\Delta \overline{T}\_{\text{to,CFD}} \left( \mathcal{R}\_{\text{tc}}, h\_{\text{a,CFD}} \right) - \Delta \overline{T}\_{\text{to,mc}} = 0 \tag{41}
$$

was solved, for the given values of *ha, CFD*, listed in Table 7. The value of the thermal contact resistance *Rtc* was so adjusted that Eq. (41) is satisfied. Equation (41) was solved using the Secant method. Note that the predicted value of total air temperature difference *ΔTto, CFD* determined from Eq. (29) depends on fin-efficiency *η<sup>f</sup>* , which in turns depends on *Rtc*. Heat transfer coefficient *ha, CFD* is a function of air velocity *w*0 and is independent of the thermal contact resistance *Rtc*. The heat transfer coefficient *ha, CFD* was calculated using the correlation (39).

Table 8 [19] lists the measurement data sets and the obtained values of thermal contact resistance.


**Table 8.** Thermal contact resistance *Rtc* determined using experimental data sets and the heat transfer coefficient *ha, CFD* obtained from the CFD simulations

The mean value of thermal contact resistance, obtained for data set given in Table 8, is *R*¯ *tc* = 3.16 10-5 (m2 K)/W. To calculate the total air temperature differences *ΔT*¯ *to*, *CFD* the *R*¯ *tc* was included in the CFD model of heat exchanger.

Figure 11 presents the results of CFD simulations for computational cases listed in Table 8.

Equation (14) was used to determine the heat flux *q* variations at the outer surface of tube wall with dimensionless coordinate *ξ*. Fig. 12 presents the results for the first tube row and Fig 13 for the second tube row [19]. Additionally, the computed values of heat flux *q* for the thermal contact resistance *R*¯ *tc* = 0 (m2 K)/W are compared with that obtained for *R*¯ *tc* = 3.16 10-5 (m2 K)/W.

Fig 12 reveals that the thermal contact resistance significantly reduces heat flux through the finned outer surface of the tube. The influence of contact resistance on the average heat flux in the second row of tubes (Fig. 13) is smaller than in the first row of tubes (Fig. 12). The overall heat transfer rate decreases significantly if the thermal contact resistance exists because the largest amount of heat is transferred across the first row of tubes.

Table 9 [19] compares the temperature differences across the two rows of tubes computed using ANSYS CFX for the average thermal contact resistance *R*¯ *tc* = 3.16 10-5 (m2 K)/W with the

Computer-Aided Determination of the Air-Side Heat Transfer Coefficient and Thermal Contact Resistance for… http://dx.doi.org/10.5772/60647 283

D*T Rh T to CFD tc a CFD to me* ,, , ( , 0 ) -D = (41)

**/h** *hin ,* **W/(m2 ·K)**

, which in turns depends on *Rtc*. Heat transfer

*ha, CFD ,* **W/(m2 ·K)**

*R*¯

*tc* = 3.16 10-5 (m2

*tc* 3.16·10-5

*to*, *CFD* the *R*¯

*tc* = 3.16 10-5 (m2

K)/W with the

*tc* =

*tc* was

*Rtc ,* **(m2 ·K)/W**

was solved, for the given values of *ha, CFD*, listed in Table 7. The value of the thermal contact resistance *Rtc* was so adjusted that Eq. (41) is satisfied. Equation (41) was solved using the Secant method. Note that the predicted value of total air temperature difference *ΔTto, CFD* determined

coefficient *ha, CFD* is a function of air velocity *w*0 and is independent of the thermal contact resistance *Rtc*. The heat transfer coefficient *ha, CFD* was calculated using the correlation (39).

Table 8 [19] lists the measurement data sets and the obtained values of thermal contact

**I** 1.00 14.98 42.67 68.35 1, 892.40 4, 793.95 71.14 4.45·10-5 **II** 1.27 13.49 39.74 65.02 1, 882.20 4, 813.42 82.45 3.27·10-5 **III** 1.77 13.03 35.83 63.14 1, 789.80 4, 743.65 101.03 2.42·10-5 **IV** 2.20 12.69 31.83 61.24 1, 788.00 4, 739.78 115.34 2.42·10-5

**Table 8.** Thermal contact resistance *Rtc* determined using experimental data sets and the heat transfer coefficient *ha, CFD*

The mean value of thermal contact resistance, obtained for data set given in Table 8, is *R*¯

Figure 11 presents the results of CFD simulations for computational cases listed in Table 8. Equation (14) was used to determine the heat flux *q* variations at the outer surface of tube wall with dimensionless coordinate *ξ*. Fig. 12 presents the results for the first tube row and Fig 13 for the second tube row [19]. Additionally, the computed values of heat flux *q* for the thermal

Fig 12 reveals that the thermal contact resistance significantly reduces heat flux through the finned outer surface of the tube. The influence of contact resistance on the average heat flux in the second row of tubes (Fig. 13) is smaller than in the first row of tubes (Fig. 12). The overall heat transfer rate decreases significantly if the thermal contact resistance exists because the

Table 9 [19] compares the temperature differences across the two rows of tubes computed

K)/W are compared with that obtained for *R*¯

K)/W. To calculate the total air temperature differences *ΔT*¯

*to***,***me* **, ºC** *Tw, ºC <sup>V</sup>***˙** *<sup>w</sup>* **, dm3**

from Eq. (29) depends on fin-efficiency *η<sup>f</sup>*

**Case** *w0 , m/s T'a, ºC <sup>Δ</sup>T***¯**

282 Heat Transfer Studies and Applications

obtained from the CFD simulations

included in the CFD model of heat exchanger.

*tc* = 0 (m2

largest amount of heat is transferred across the first row of tubes.

using ANSYS CFX for the average thermal contact resistance *R*¯

3.16 10-5 (m2

contact resistance *R*¯

K)/W.

resistance.

**Figure 11.** The results of CFD simulation for data sets I - IV listed in Table 8: a) temperature distribution in the air domain at the middle of flow passage, b) fin surface temperature, c) air velocity distribution at the middle of flow pas‐ sage [19].

temperature differences obtained from the expression (29) for the experimentally determined heat transfer coefficient *ha, me* (correlation 4, Table 1)

**Figure 12.** The distribution of heat flux *q* on the outer surface of tube wall for the first tube row, for computational cases I - IV listed in Table 8.

**Figure 13.** The distribution of heat flux *q* on the outer surface of tube wall for the second tube row, for computational cases I - IV listed in Table 8.

The relative temperature difference |*εa*| between the obtained results, is calculated as:

$$\left| \mathcal{E}\_a \right| = \left| \frac{\Delta \overline{T}\_t - \Delta \overline{T}\_{t,mc}}{\Delta \overline{T}\_{t,mc}} \right| \cdot 100\%. \tag{42}$$


**Table 9.** Air temperature differences *ΔT*¯ *to*,*CFD* over two rows of tubes obtained using the CFD simulations with the thermal contact resistance *<sup>R</sup>*¯ *tc* = 3.16 10-5 (m2 K)/W and the temperature difference *ΔT*¯ *to*,*me* obtained from Eq. (29) for the experimentally detemined heat transfer coefficients *ha, me*

The largest value of this difference was obtainedfor the case I - |*εa*| = 3.98 % (Table 9). For the other computational test cases, the value of |*εa*| is less than 3 %. The performed calculations demonstrate the effectiveness of the method developed. The estimated contact resistance can be used in the calculation of equivalent heat transfer coefficient using (Eq. (25)) and in the analytical calculations of the heat transfer rate in the heat exchanger:

$$
\dot{Q} = F A\_o \mathcal{U}\_o \Delta T\_{lm} \tag{43}
$$

where the symbol *F* denotes the correction factor based on the logarithmic mean temperature difference ∆*Tlm* for a counter-current flow arrangement.

The method proposed for determining the air side heat transfer correlations based on the CFD computations, can easily account for the thermal contact resistance between the tube outer surface and fin bases. The method can also be used for heat exchangers with various tube shapes and other types of the fin to tube attachment as well as for different tube arrangements.

#### **7. Conclusions**

The relative temperature difference |*εa*| between the obtained results, is calculated as:

*t t me*

100%.

, ,

**Figure 13.** The distribution of heat flux *q* on the outer surface of tube wall for the second tube row, for computational

**Figure 12.** The distribution of heat flux *q* on the outer surface of tube wall for the first tube row, for computational

cases I - IV listed in Table 8.

284 Heat Transfer Studies and Applications

cases I - IV listed in Table 8.

D -D <sup>=</sup> <sup>×</sup> <sup>D</sup> (42)

*T T T*

*t me*

*a*

e

The experimental and CFD based methods for determining the air-side heat transfer coeffi‐ cient, for fin-and-tube heat exchanger, are presented in this study. Two types of CFD based methods were described. The first one allows determining the air-side heat transfer coefficient directly from CFD simulations while the second employs the analytical model of fin-and-tube heat exchanger to determine the air-side heat transfer coefficient. The results obtained using these two methods were compared with the experimental data.

Moreover, the method for determination of the thermal contact resistance between the fin and tube was presented. The CFD simulations are appropriate for predicting heat transfer corre‐ lations for the plate fin and tube heat exchanger with tubes of various shapes and flow arrangements. Using the experimental data and CFD simulations, the thermal contact resist‐ ance between the fin base and tube was estimated. The fin efficiency appearing in the formula for the equivalent air side heat transfer coefficient is a function of the air side heat transfer coefficient and the thermal contact resistance. The air-side heat transfer correlations are determined based on the CFD simulations. The heat transfer coefficients predicted from the CFD simulations were larger than those obtained experimentally, because in the CFD model‐ ing the thermal contact resistance between the fin and tube was neglected. A new procedure for estimating the thermal contact resistance was developed to improve the accuracy of the heat exchanger calculation. When the value of mean thermal contact resistance, determined by the proposed method, is included in the CFD model, then the computed air temperature distributions show better agreement with measurements.

The computations presented in this study allows to draw the following conclusions. CFD modeling is an effective tool for flow and thermal design of plate fin-and-tube heat exchangers. and is an effective tool for finding heat transfer correlations in the newly designed heat exchangers. However, to obtain good agreement between the CFD modeling and experimental data, it is necessary to adjust some parameters of the CFD model using the experimental results. An example of such a parameter may be thermal contact resistance between the tube and the fin base.
