**1. Introduction**

During design, operation, and troubleshooting of various process and power equipment-containing tube bundles, it often is important to know the velocity and temperature fields in both the tube and the shell sides. These are obtained predominantly using Computational Fluid Dynamics (CFD) models and, therefore, articles covering a wide range of such applications are available. For example, Wei et al. [1] discussed a coupled CFD-Lagrange multipliers optimization method for flow distribution adjustments to prevent freezing of power generation natural draft cooling systems during winter operation. Chien et al. [2], on the other hand, presented a coupled CFD-surrogate-based optimization of flow distribution in a heat exchanger inlet header. Zhou et al. [3] focused on CFD investigation and optimization of a compact heat exchanger comprising a single row of tubes, and Łopata et al. [4] published an article covering the experimental investigation of flow distribution in a similar cross-flow heat exchanger, but with a tube bank consisting of elliptical tubes. CFD evaluation and optimization of solar collectors, commonly also using a single row of risers, were discussed, for instance, by García-Guendulain et al. [5], who aimed to improve the collector performance by changes of riser-to-header cross-sectional area

and diameter ratios. Karvounis et al. [6] carried out a numerical and experimental study of the flow field in a forced circulation Z-type flat plate solar collector. Articles focusing on two-phase flow are also common. Li et al. [7] studied flow reversal in vertically inverted U-tube steam generators used in marine nuclear power plants, whereas Klenov and Noskov [8] investigated the effect of two-phase flow pattern in an inlet duct on flow distribution in the upper part of a trickle bed reactor. As for dispersion headers, which are often used in flue gas cleaning equipment, a CFD investigation of the impacts of inlet flow rate, hole diameter, and downstream distance on the flow distribution in an annular multi-hole header was presented by He et al. [9]. Other frequent research areas where the knowledge of flow distribution is critical are fuel cells and micro-channel heat exchangers. One might mention, e.g., the CFD and laser Doppler velocimetry investigation of flow distribution in a polymer electrolyte membrane fuel cell stack by Bürkle et al. [10], the CFD evaluation of pressure and flow distribution effects on the performance of polymer electrolyte membrane fuel cells by Heck et al. [11], or the CFD optimization of a liquid cooling system of a power inverter in an electric vehicle presented by Hur et al. [12]. Various studies involving liquid-cooled micro-channel heat sinks for electronics are quite common as well. See, e.g., the article by Li et al. [13] discussing the optimization of the micro-channel topology.

Studies not utilizing CFD are much less common and, typically, focus on evaluations of the respective equipment via physical experiments. To name just a few, one could mention the experimental investigation of flow distribution and its effect on the performance of a plate-fin heat exchanger by Zhu et al. [14], the study of two-phase refrigerant distribution in a finned-tube evaporator by Tang et al. [15], or the article by Quintanar et al. [16] covering natural circulation flow distribution in a multi-branch manifold. Micro-channel equipment was discussed, e.g., by Yih and Wang [17], who carried out an experimental investigation of the thermal-hydraulic performance of a micro-channel heat exchanger for waste heat recovery, or by Lugarini et al. [18], who focused on the evaluation of flow distribution uniformity in a comb-like micro-channel network. On the other end of the size spectrum are heat exchanger networks, in which Ishiyama and Pugh [19] studied thermo-hydraulic channeling in the individual parallel branches. In their paper, they also presented a model for prediction of flow distribution in the branches for the case when no flow control was implemented. Similarly, Novitsky et al. [20] discussed multilevel modeling and optimization of large-scale pipeline systems using specialized software tools. In these two studies, however, modelling of pressure drop was only done in a simplified manner. Korelstein [21], on the other hand, discussed mathematical properties of classical hydraulic network flow distribution problems which included pressure-dependent closure relations. An essentially identical problem can also be encountered when it comes to the design of water distribution networks. Still, proper inclusion of pressure drop in the respective models is rare because they generally focus on layout optimization while meeting the local water demands (see, for instance, the article by Cassiolato et al. [22], who proposed a Mixed-Integer Nonlinear Programming (MINLP) model for this purpose), identification of sources of contamination (as done, e.g., using a convolutional neural network by Sun et al. [23]), detection of leakage points (see, for example, the article by Fang et al. [24]), evaluation of the network performance and reliability (as discussed, e.g., in the Hanoi case study by Jeong and Kang [25]), etc.

To the best of the authors' knowledge, there currently is no semi-accurate but fast Finite Element Analysis-based model of fluid flow other than [26] that would properly include the pressure drop. This model, however, does not consider heat transfer and, thus, is of limited practical value to the designers of process and power equipment. Consequently, CFD models, because of their significant computational cost, are being employed for evaluations of fluid flow distribution and heat transfer only if absolutely necessary. As a result, the corresponding temperature fields, which, to a large degree, depend on mass flow rates through individual tubes, are also unknown. This may not pose significant problems if thermal stress is relatively even throughout the tube bundle in question. Nonetheless, mechanical failures of bundles, stemming from improper design procedures which a priori assume uniform flow distribution, are not uncommon when it comes to heat exchangers featuring large changes

in stream temperatures. The present paper, therefore, introduces a significantly extended version of the flow-only, adiabatic model discussed in [26]. This includes heat transfer between the fluids in the tube and the shell sides of a cross-flow tube bank heat exchanger (e.g., an economizer) as well as various other improvements. Shell-side flow was modelled as unidirectional. As test cases, a simple cross-flow tube bundle heat exchanger from the literature and an existing heat recovery hot water boiler, for which the necessary data had been provided by its operator, were considered. These were compared to the results obtained using the present model and, to gain further insight, also to the data from an industry-standard heat exchanger design software package. A good agreement among the data sets was observed.
