*3.2. Development of New Empirical Relations for Fanning f and Colburn j-Factor*

For the complex geometries used in the present investigation, the experimental data obtained in the present investigation were used for developing a set of novel design equations for the prediction of Fanning *f-* and the Colburn *j*-factors. As previously stated, the *f*- and *j*-factors quantify the pressure drop and heat transfer characteristics of the heat exchanger units. Therefore, it is imperative to develop correlations that relate them with the flow, fluid, and geometrical parameters. The correlation was developed using multivariate regression analysis using the curve fitting tools in Microsoft Excel's Solver® which are based on the least squares' method. The dimensionless parameters used to develop the predictive correlation are, as mentioned earlier, the Reynolds number, *ReD* and the ratio between total fin surface area to the total heat transfer surface area *Af* /*At* of the heat exchanger which is a design parameter used by Palmer et al. [26] that can be used for estimating heat transfer areas needed for a given amount of heat to be transmitted. Other authors have used similar dimensionless groupings to correlate the heat transfer properties of fin and tube heat exchangers [13,17,18]. The newly derived equations are as follows:

$$j = 10^{4.595} \left(\frac{A\_f}{A\_t}\right)^{29.918} Re\_D^{-0.374} \tag{9}$$

$$f = 10^{1.203} \left(\frac{A\_f}{A\_t}\right)^{12.811} Re\_D^{-0.139} \tag{10}$$

where *ReD* is the Reynolds number calculated using the hydraulic diameter of the heat exchanger face's cross-section; *Dc* is the outside diameter of the fin's collar (m): *Af* is the total surface area of the fins (m2); *At* is the heat exchanger's overall heat transfer surface area (m2). The Equations (9) and (10) show that the Colburn and Fanning factors are inversely proportional to the Reynolds numbers which are consistent with experimental observations (in Figure 7a,b). Additionally, the relatively large indices (29.218 and 12.811) for the *Af* /*At* parameter reflects the small magnitude of the *Af* /*At* ratio. It is advised that the equations are valid within the Reynolds number range: 6 <sup>×</sup> 103 <sup>≤</sup> *ReD* <sup>≤</sup> <sup>30</sup> <sup>×</sup> <sup>10</sup><sup>3</sup> and for the heating cycle in forced convection heat transfer.

In order to estimate the accuracy of predictions from the developed equations Figure 8a,b have been prepared to depict the relationship between the experimentally evaluated values and the predicted values of Colburn factor (*j*) and Fanning friction factor (*f*), respectively. From the Figure 7a,b, it can be concluded that there is a good match between the experimental and the computed values and almost 100% of the data lie within ±15% error band. Furthermore, the correlation coefficients which compare the goodness of fit between the experimentally evaluated and predicted data for Equations (9) and (10) are 0.853 and 0.811, respectively. As such, the newly developed correlations can be used with confidence.

In order to account for geometric variations of the fins (their spacing *Fp*, longitudinal pitch *Lp* and transverse pitch *Tp*), validated CFD data (previously reported in ref. [18]) was combined with the experimental data obtained in this study to generate new empirical correlations for j and f for the plain fin model using nonlinear least squares regression. The longitudinal pitches considered were 20, 22 and 24 mm; transverse pitches were 23.5, 25 and 26.5 mm; while the fin spacings investigated were 3.7, 4.2 and 4.7 mm at air Reynolds numbers of 6000 to 32,000. There were a combined total of 21 points (15 CFD and 6 experimental). Figure 7c,d show that the correlations have very high goodness of fit R<sup>2</sup> values of 0.931 and 0.979 for the j and f factors, respectively, and the predictions of the new correlations are well within ±15% of the underlying CFD and experimental data. The two new correlations are given below as follows:

$$j = 0.173 \ Re\_D{}^{-0.388} \left(\frac{F\_p}{D\_c}\right)^{-0.199} \left(\frac{L\_p}{F\_w}\right)^{-0.297} \left(\frac{T\_p}{F\_H}\right)^{-0.089} \tag{11}$$

$$f = 0.084 \ Re\_D \, ^{-0.213} \left( \frac{F\_p}{D\_c} \right)^{-0.334} \left( \frac{L\_p}{F\_w} \right)^{-0.151} \left( \frac{T\_p}{F\_H} \right)^{-0.262} \tag{12}$$

To summarise, it can be concluded that the developed equations are very much capable of predicting the Fanning *f-* and Colburn *j*-factors of these heat exchangers having the stated fin geometries with sufficient accuracy. Consequently, the equations can be used during the design and evaluation of existing multi-tube multi-fin heat exchanger with plain, perforated or louvred fins.

#### **4. Conclusions**

This study has presented novel geometric configurations for multi-tube multi-fin heat exchanger. The configurations were designed in order to conduct a robust experimental investigation with three heat exchanger geometries namely plain, perforated plain and louvred fin heat exchangers. Some important observations were made during the experiments and analysis of the pressure drop and heat transfer data. It was found that for all inlet air and water flow rates and hence velocities, the louvred fins produced the highest heat transfer rate. This was attributed to increased surface area available for heat transfer. Conversely, it also produced the highest pressure losses when compared to the other two designs. Also, while the new perforated design produced a slightly higher pressure drop than the plain fin design, due to the vortices generated by the perforations, an enhancement in its heat transfer characteristics was observed when comparing with the plain and louvred fin models. This enhancement is relatively high at a small water flow rate. The experimental results were subsequently used to generate a set of novel empirical equations for design optimisation which can be used to predict the heat transfer and pressure drop characteristics of the heat exchangers represented by the Colburn and Fanning factors. The empirical equations were developed as functions of the heat exchangers' geometrical parameters, and we have shown that the performance of the equations are well within acceptable ±15% error margins in relation to the experimental data.

**Author Contributions:** M.A.: Experimentation, Data curation, formal analysis, Investigation, methodology, validation, visualization, writing—original draft; R.M.: Conceptualisation, supervision, writing—review and editing; A.M.A.: Writing—review & editing, formal analysis; K.J.K.: supervision, formal analysis, writing—review and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research did not receive any external funding.

**Data Availability Statement:** Data is available upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

