**1. Introduction**

Plate and fin heat exchangers (PFTHEs) are complicated devices. Their construction, which involves cross thin metal sheets and tubes with different numbers of rows, different fin pitches, and tube sizes, causes complicated phenomena on the air- and/or water-side [1]. Each row in PFTHEs operates in a different way. Differences are caused by variations in air and water temperatures, airflow turbulence, or even air velocity. Those elements cause different HTCs in each row [2].

Performance characteristics of tube heat exchangers are most often determined experimentally. This is usually due to the high complexity of the systems studied. An example is the experimental testing of photovoltaic (PV) cells cooled by phase-change materials (PCM) [3]. Fouling phenomena in exchangers are also investigated experimentally. Ali et al. [4] investigated the influence of dust deposited on the surface of two types of PVs, monocrystalline silicon, and polycrystalline silicon modules. Experimental studies are also widely used to determine the flow and heat characteristics of finned tube heat exchangers. Heat transfer correlations are determined on the side of the flowing gas, which is usually air, in a wide range of Reynolds number variations [5]. Additionally, the effectiveness of various types of improvements in the design of heat exchangers, such as oval tubes [6], new

**Citation:** Marcinkowski, M.; Taler, D.; Taler, J.; W˛eglarz, K. Thermal Calculations of Four-Row Plate-Fin and Tube Heat Exchanger Taking into Account Different Air-Side Correlations on Individual Rows of Tubes for Low Reynold Numbers. *Energies* **2021**, *14*, 6978. https:// doi.org/10.3390/en14216978

Academic Editor: Artur Bartosik

Received: 27 September 2021 Accepted: 20 October 2021 Published: 25 October 2021

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fin shapes, or guide vanes forming the air flow in the exchanger [5], is usually evaluated experimentally. Despite the high reliability of the obtained results, a disadvantage is the cost of experimental studies. To determine the experimental characteristics of an exchanger with a different number of rows, tubes, or different construction, it is necessary to build an experimental test facility equipped with a measuring apparatus and a computer data acquisition system. Moreover, the time and therefore costs of experimental investigations are high. Therefore, CFD modelling is increasingly used in the development of tubular cross-flow heat exchangers [7]. The influence of various innovations in the design of the exchanger is modelled in different ranges of Reynolds numbers on the air-side and on the fluid-side flowing inside the tube. Four round-convex strips were placed around the tube to enhance air-side heat transfer [8].

If the experimental testing cannot be fully replaced, then the experiment comes down to verification of a certain part of the results [9]. The CFD modelling gives us increased flexibility in the research and industry. This study shows the newly determined correlations of the air-side Nusselt number for the four-row PFTHE. Those correlations illustrate new functions to determine HTC in a particular row of PFTHE. Individual correlations can also change the way we look at the heat flow inside PFTHE.

Until recently, plenty of Nusselt number, Colburn factor, or HTC correlations were determined. However, the majority are referred to as average correlations. Some studies presented local HTCs and only a few of them showed results or determined Nusselt number correlations for a particular row in multi-row PFTHEs.

The first group of research overviews is about average HTCs in PFTHEs. Khan et al. [10] investigated twisted oval tube HEXs and designated the average Nusselt numbers and pressure drop correlations. Lindqvist et al. [11] conducted CFD research taking into account different tube bundle array angles. They also presented diagrams with the Colburn factors for low Reynold numbers. Ł ˛ecki et al. [12] presented a comparison between HTCs obtained using CFD simulation to the one calculated with VDI correlation for three-row inline PFTHE. Sadeghianjahromi et al. [13] determined HTCs and pressure drop factors of PFTHEs with different fin types and circular and flat tubes under dry and wet conditions. Elmekawy et al. [14] showed that attaching the splitter plates to the tubes could increase the Nusselt number and reduce the pressure drop. Okbaz et al. [15] presented different Colburn factors correlations for PFTHEs with different numbers of rows. However, those correlations were averaged for entire PFTHEs. Petrik and Szepesi [16] determined Nusselt number correlations for one and two-row U-shape HEXs. Additionally in another piece of research, Petrik et al. [17] presented Nusselt number correlations but for standard PFTHEs. González et al. [18] presented the average Nusselt numbers as a function of fin material and Reynolds number for two-row PFTHE with inline tube arrangement. Awais and Bhuiyan [19] collected plenty of Colburn factor correlations in their state-of-the art paper. It is possible to present many more average correlations. However, it is not necessary for this research. Adam et al. [20], in a state-of-the-art paper, shows studies of local HTCs and local heat/mass transfers coefficients.

All of the above research is related to the average Nusselt numbers, Colburn factors, or HTC correlations, or presents local HTCs to predict heat transfer correlations for the entire PFTHE. There are only a couple instances of research referring to heat transfer correlations for a particular row in PFTHEs. During the 1970s, Rich et al. [21] showed experimental research where a Colburn factor in a particular row for multi-row PFTHEs, with one-row to eight-rows, was presented. This was the first time when someone showed variation in the HTC inside PFTHEs. Taler et al. [2] determined individual Nusselt number correlations for each row in the case of two-row PFTHE. Despite its high practical importance and current marginal existence in the literature, there is a lack of heat transfer correlations on individual tube rows for heat exchangers with different number of tube rows. Additionally, the literature is full of average Nusselt number, Colburn factor, or HTC correlations for different fin geometries, tube geometries, fin pitches, tube sizes, air velocities, etc.

Only Rich et al. [21] showed that further tube rows in multi-row PFTHEs are inefficient when the air velocity is low and the airflow in the exchanger is laminar. However, correlations on Nusselt number for individual tube rows have not been developed. Taler et al. [22] showed that in a two-row automotive radiator, made with round or oval tubes, the first row of tubes is more efficient than the second row when the air velocity is less than 2.5 m/s. However, coolers with larger tube numbers have not been modelled by CFD or studied experimentally. There are many problems to solve, such as: whether a larger cross-section of a PFTHE, but with a smaller number of rows, would be more efficient than a three-, four-, or five-row PFTHE. Which row is the least effective? Should we consider individual correlations in the case of multi-row PFTHEs?

The present research refers to four-row PFTHE with the air velocity before the HEX changing in the range from 0.5 m/s to 2.5 m/s. This paper covers:

