Fundamentals and Semi-Analytical Analysis for Modeling the Capacity of Shell and Tube Heat Exchangers for Temperature Control in Hydrological Applications

Abstract

Currently, shell and tube heat exchangers comprise a group of equipment called profitable interest due to their usefulness in various industrial sectors. Its appropriate design is coupled to processes in a sustainable development scenario, mainly for use in laboratories or home services. The semi-analytical analysis of these complex systems currently allows for achieving temperature control objectives in the substances used, such as water, without detailed modeling of the hydrodynamics that develop due to the flow conditions. For this purpose, the development of computational tools that implement algorithms based on correlations for determining Nusselt numbers is proposed. Specifying key correlations that determine optimal operating conditions can be used to define process efficiency and contribute to the concept of net zero energy. Simultaneously, the transfer of thermal energy can continue in subprocesses, contributing to reuse and reducing the carbon footprint if the source of the energy comes from fuels or to efficiency if it comes from renewable energies.

1. Introduction

Heat exchangers are the focus of our discussion, and they are not only just essential equipment for energy transfer between fluids at different temperatures but also widely used in various industrial sectors. Their diverse construction and application in heat transfer processes have made them a staple in the petrochemical, chemical, food, and energy industries. The industry’s increasing energy needs have spurred the development of cutting-edge technologies for heat exchanger manufacturing, all adhering to the rigorous standards of the Tubular Exchanger Manufacturers Association (TEMA). These standards, tailored to the class of exchanger, meticulously detail materials, design techniques, dimensions, and construction methods, ensuring optimal mechanical and thermal performance and energy-saving processes.

Heat exchangers can be classified by direct or indirect transfer [1]. In general, in an indirect heat exchanger, the heat transfer process takes place through a conductive wall that separates two fluids. In a direct heat exchanger, the fluids interact directly with each other. In the development of this chapter, the indirect heat exchanger called the shell and tube type is of particular interest. Designing various models based on parallel or countercurrent flow in a perimeter shell structure over diverse tube banks is possible. If the fluid circulates at least twice about the axial axis of the exchanger shell, it is defined as a “step” design in which the installation of internal platforms establishes recirculation.

Shell and tube heat exchangers are considered one of the most common equipment due to their ability to adapt to various processes. Therefore, various references in thermal solutions such as Sterling in the UK, Komax-Systems in the USA, HRS in India, Goje in China, and Kelvion in Brazil, among others worldwide, offer custom design services to reduce the footprint of carbon from specific processes. Reduction in greenhouse gas emissions can be achieved by adopting cost-effective technologies such as shell and tube heat exchangers. Energy efficiency is estimated to represent more than 40% of emission reductions, according to the International Energy Agency (IEA). In this way, by optimizing the processes that implement heat exchangers and using the thermal load obtained in a subprocess, a means of use that contributes to the target percentage is established.

Regarding the concept of renewable energy, currently, exchangers provide heat obtained from ambient air and geothermal water to heating networks. They also transfer heat from solar energy transformation to heat drinking water for residences and the manufacturing industry. Erdogan et al. [2] proposed the design of a shell and tube heat exchanger in combination with a parabolic trough solar collector and a geothermal power plant based on an organic Rankine cycle. The researchers developed an algorithm using the logarithmic mean temperature difference method. They coded it using ESS to determine that the baffle spacing is the critical parameter for controlling the irradiated oil temperature. They also determined that the transfer surface area in the exchanger must increase depending on the increase in solar irradiation intensity. In another investigation, Rashidi et al. [3] discussed the decrease in entropy generation and exergy analysis in shell and tube heat exchangers, determining the relevance of mass flow rates, temperatures of streams in the process, and thermal properties of fluids in the entropy generation.

Looking ahead, the optimization and design of heat exchangers are paving the way for profitable technologies that harness thermal energy efficiently in a sustainable development scenario. As the world shifts toward renewable energy sources, shell and tube heat exchangers are at the forefront of this transition. They are being integrated into the concept of net zero energy, where thermal efficiency is hailed as the first fuel. This integration not only underscores the adaptability and versatility of these heat exchangers but also inspires a vision of a future powered by sustainable and efficient energy solutions. Therefore, in the context of this chapter, some systems that implement shell and tube heat exchangers will be analyzed, adopting a methodology similar to Erdogan et al. [2]. In the analysis, published and experimental data will be presented to implement algorithms that allow estimating the magnitudes of the temperatures of the fluids, where at least one is water, according to established processing conditions. The results would allow adjustments to the processes or the design of subprocesses for thermal energy using the fluid external to the primary process.

2. Shell and tube heat exchangers

Shell and tube heat exchangers are characterized by their design to obtain large heat transmission surfaces associated with the various tubes mounted in parallel inside a shell [4]. Due to the size of these components, the equipment can only operate with certain limitations, taking into account large pressure drops and their effect on the temperature of the fluids [5]. Figure 1 schematically shows a shell and straight-tube heat exchanger with countercurrent fluid flow.

Figure 1.

Shell-and-tube type heat exchanger with countercurrent fluid flow [6].

Shell and tube heat exchangers operate through the phenomenon of transferring thermal energy between fluids indirectly. Therefore, the greater the temperature gradient between the fluids, the greater the heat flow in transit. The primary heat transfer mechanism is convection. Therefore, since this is a complex phenomenon, the result of a specific convective condition is parameterized with data called heat transfer coefficients. In this way, by implementing these data to a thermal energy balance of the exchanger, it is possible to calculate the resulting temperature in the fluids. The energy balance can be proposed with microscopic or macroscopic models.

2.1 Microscopic analysis model

The variation equations for non-isothermal systems are complex in their argumentation and, therefore, in their solution. Eqs. (1) and (2) should be solved together to obtain a distribution field of temperature profiles linked to the velocity profile field of each fluid intervening in the heat exchanger and the temperature profiles in the solids with which they interact [7, 8]. Numerical methods are required to obtain a solution and a computational system with a processing unit that performs well. Even so, the continuity equation (Eq. (1)) represents the variation in density for a fixed point as a consequence of the variations of the mass velocity vector (ρv), and the energy equation as a function of the energy flux densities and momentum (Eq. (2)) is not complete. Implementing turbulent flow regime models is needed as a characteristic of the fluid circulation in the complex exchanger.

The flow regime of the fluid circulating on the shell side (exterior of the tubes) can be justified as turbulent due to the interaction characteristics with the tubes’ arrangement in its path. Furthermore, the flow direction changes concerning the location of the baffles and walls of the case, as schematically represented by the blue line in Figure 1. The laminar flow regime is more likely to develop in the circulation within the tubes. However, it will depend on the characteristics of the volumetric flow and the dimensions of the tubes. Under operating conditions, it would be desirable to maintain a turbulent regime to optimize the convection heat transfer mechanism and consequently increase the efficiency.

$$\frac{\partial \rho }{\partial t}=-\left(\nabla .\mathit{\rho v}\right)$$E1

$$\begin{array}{l}\rho C_v\left(\frac{\partial T}{\partial t}+v_x\frac{\partial T}{\partial x}+v_y\frac{\partial T}{\partial y}+v_z\frac{\partial T}{\partial z}\right)=-\left[\frac{\partial q_x}{\partial x}+\frac{\partial q_y}{\partial y}+\frac{\partial q_z}{\partial z}\right]\\ -T{\left(\frac{\partial p}{\partial T}\right)}_{\rho }\left(\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial x}+\frac{\partial v_z}{\partial x}\right)-\left\{{\tau }_{\mathit{xx}}\frac{{\partial v}_x}{\partial x}+{\tau }_{\mathit{yy}}\frac{{\partial v}_y}{\partial y}+{\tau }_{\mathit{zz}}\frac{{\partial v}_z}{\partial z}\right\}\\ -\left\{{\tau }_{\mathit{xy}}\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)+{\tau }_{\mathit{xz}}\left(\frac{\partial v_z}{\partial z}+\frac{\partial v_z}{\partial x}\right)+{\tau }_{\mathit{yz}}\left(\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right)\right\}\end{array}$$E2

Calculating the fluid flow pattern due to the interaction with solid sections determines the implementation of turbulence models. Turbulence or vortex generation starts with flow instabilities caused by average velocity gradients. Consequently, the initial vortices generate new instabilities, giving rise to smaller-scale vortices. The process continues until the vortices become small enough to be nullified by the effect of viscosity, and the turbulent energy dissipates as heat. In the ducts, the fluid may not be able to develop due to the dimensions and rough quality of its surfaces, so it circulates in a turbulent regime. The use of a turbulence model will depend on analyzing the system characteristics. The equations of the κ-ε model of turbulent kinetic energy (Eq. (3)) and the turbulent dissipation rate (Eq. (4)) for analyzing the velocity profile generate acceptable results in various applications. Eqs. (3) and (4) represent the corresponding general three-dimensional formulation.

$$\frac{\partial }{\partial t}\left(\mathit{\rho \kappa }\right)+\frac{\partial }{\partial x_i}\left(\mathit{\rho \kappa }{u}_i\right)=\frac{\partial }{\partial x_j}\left[\frac{{\mu }_t}{{\sigma }_{\kappa }}\frac{\partial \kappa }{\partial x_j}\right]+2{\mu }_t{E}_{\mathit{ij}}{E}_{\mathit{ij}}-\mathit{\rho \epsilon }$$E3

$$\frac{\partial \left(\mathit{\rho \epsilon }\right)}{\partial t}+\frac{\partial \left(\mathit{\rho \epsilon }{u}_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\frac{{\mu }_t}{{\sigma }_{\epsilon }}\frac{\partial \epsilon }{\partial x_j}\right]+C_{1\epsilon }\frac{\epsilon }{\kappa }2{\mu }_t{E}_{\mathit{ij}}{E}_{\mathit{ij}}-C_{2\epsilon }\rho \frac{{\epsilon }^2}{\kappa }$$E4

The term $E_{\mathit{ij}}$ represents the deformation rate components, $u_i$ the velocity component in the corresponding direction and ${\mu }_t$ the Eddy viscosity defined by Eq. (5).

$${\mu }_t=\rho C_{\mu }\frac{{\kappa }^2}{\epsilon }$$E5

Eqs. (3) and (4) contain predefined constants that determine conditions for the number of iterations in the fitting data of the wide turbulence intervals. Some suggested data are expressed in Eq. (6).

$$C_{\mu }=0.09,{\sigma }_{\kappa }=1.0,{\sigma }_{\epsilon }=1.3,C_{1\epsilon }=1.44,C_{2\epsilon }=1.92$$E6

According to the fluid-structure interaction conditions, Eqs. (3) and (4) depend on the relationship between the effects of laminar and turbulent flow in a small fraction of fluid on the surface of a solid wall. In other words, the effect will depend on the dimension of the space or control volume size chosen (cell) for microscopic analysis adjacent to the surface of the solid. The dimensionless variable Y-plus or $y^{+}$ is critical to this analysis. It is calculated as a dimensionless magnitude representing the normal distance from the solid surface’s wall to the cell’s centre. This variable, Y-plus, evaluates the proportionality of the mesh elements’ size over the solid’s wall. In this way, Eqs. (7)–(10) are required to define the wall law that complements the near-wall treatment of the turbulence model (refer to Table 1).

Turbulence model	Near wall treatment	Range of Y-plus
$\mathit{\kappa }\mathbf{-}\mathit{\epsilon }$	Standard wall functions	$30<y^{+}<300$
	Non-equilibrium fall functions	$30<y^{+}<300$
	Scallable wall functions	$y^{+}>11.225$
	Enhanced wall treatment	$y^{+}<5$

Table 1.

Y-plus specification according to the near-wall treatment of the turbulence model.

$$y=\frac{y^{+}{\mu }_f}{\rho u_{\tau }}$$E7

$$u_{\tau }=\sqrt{\frac{{\tau }_w}{\rho }}$$E8

$${\tau }_w=0.5C_f\rho U_{\infty }^2$$E9

$$C_f=\frac{0.26}{{\mathit{Re}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}$$E10

According to the above, not all engineering problems associated with heat exchangers can be solved by equations based on laws or balances of matter, energy, and momentum because they can be very complex and involve many variables. Consequently, empirical equations determined through controlled experiments in physical models were developed and adopted. The strategy consists of grouping the variables into a new dimensionless pseudo-variable to simplify the analysis. Buckingham’s Pi (π) Theorem allows us to obtain, through an appropriate approach, such relationships with dimensionless numbers from a set of variables associated with a particular problem. The dimensionless numbers are identified as π-groups and determine the structure of the correlations between the transcendent variables of a physical problem [9].

As a result of the dimensional analysis, semi-analytical formulations have been developed for the phenomenological characterization of heat exchange processes for various devices, including shell and tube devices. A certain degree of empiricism determines the formulations because the analysis is implemented on the mechanical behavior of the complete system (macroscopic analysis) and not a differential element as in the microscopic model. However, its development complies with mathematical formality as long as it is limited to the interval values defined for each variable or dimensionless number.

Conditions such as reducing thermal stress, preventing leaks, and making maintenance easy due to corrosion phenomena determine that the variables of most significant interest are operating temperatures and pressures. In predicting the exchanger’s performance, the heat transfer capacity is related to the overall transfer coefficient. The calculation of the global coefficient depends on the coefficients called internal and external transfer coefficients. The internal coefficient is estimated to determine the heat extraction or absorption capacity by the fluid circulating in the tubes. Respectively, the external coefficient is calculated to determine the same capacity of the fluid circulating on the shell side.

2.2 Macroscopic model: semi-analytical analysis

Even in such a structurally complex device, the heat exchange process can be defined macroscopically as a system of two substances, one giving off heat and the other absorbing it. Therefore, a global concept of the thermal phenomenon for modeling the exchanger is governed by a heat flow balance between the fluid on the shell side and that circulating on the tube side. The conditions under which this process develops can be diverse, and among the most relevant are the changes in the physical state of the substances and temporal continuity. Although it is not a general rule for applications, most heat exchangers are continuously operating systems, so they are said to reach a stable state of operation (unchanged once they reach their working configuration). Furthermore, in various systems in which water is used as the only or one of the substances, it is required to preserve its liquid state, so the formation of vapor or ice does not need to be considered. In this way, the heat flow balance is simplified and is determined by Eq. (11).

$$Q={\dot{m}}_s{C}_s\left({\overline{T}}_{\mathit{es}}-{\overline{T}}_{\mathit{\text{is}}}\right)={\dot{m}}_t{C}_t\left({\overline{T}}_{\mathit{et}}-{\overline{T}}_{\mathit{it}}\right)$$E11

Defining $\dot{m}$ as the mass flow of the substances, $C$ as their heat capacity, and $\overline{T}$ as average temperatures. The subscripts, $s$ and $t$, correspond to the specifications of the substances in the shell and tube sections, respectively. Additionally, the subscripts, $i$ and $e$, signify the entry and exit locations of the fluids in the exchanger.

Given the system’s complexity, considering significant mass flows of substances and turbulent flow regimes due to interaction with structures, the implementation and development of dimensional analysis and the concept of dimensionless numbers by physical-experimental modeling have represented a source of answers to acceptable control conditions.

Two methods in the analysis approach acquired relevance due to the results obtained in their application on various systems. Notably, the logarithmic mean temperature difference methods and the effectiveness-number of transfer units (ε-NTU) method can be coupled with correlations to determine convective heat transfer coefficients and thereby determine the average temperatures of the fluids in the outlet currents of the exchanger.

The following subsections address a brief description of each method’s thermal calculations, a model for determining the pressure drop in fluids, correlation information supported by dimensionless numbers, and a methodology for generating a computational graphical user interface for analysis assisted by the EES engineering application, which has an efficient system of equations solver.

2.2.1 Logarithmic mean temperature difference method

Because the fluids moving along the conduits in the heat exchanger lose or gain a temperature index locally, one way to obtain a representative temperature value is to implement the logarithmic mean temperature difference concept. Eq. (12) represents a relationship for fluid flow in a countercurrent system, where $T_h$ indicates the highest temperature of the fluid and $T_c$ is the lowest temperature. Indices 1 and 2 represent the data concerning the fluid on the shell and tube sides, respectively [1].

$$∆T_{\mathit{ml}}=\frac{\left(T_{\mathit{hs}}-T_{\mathit{ct}}\right)-\left(T_{\mathit{ht}}-T_{\mathit{cs}}\right)}{\mathit{Ln}\left[\frac{T_{\mathit{hs}}-T_{\mathit{ct}}}{T_{\mathit{ht}}-T_{\mathit{cs}}}\right]}$$E12

The above allows for analyzing or predicting the performance of a heat exchanger by quantifying the transferred energy $Q$ by estimating a heat transfer coefficient called global ($U_{\mathit{\text{global}}}$), as expressed in Eq. (13) [1].

$$Q=U_g{A}_T∆T_{\mathit{ml}}$$E13

The approximate calculation of the global heat transfer coefficient $U_g$ is obtained by posing the analogy with an electrical circuit in which various resistances are integrated. Thermal resistance originates from wall conduction and internal and external convection phenomena in tubes. The thermal resistance terms associated with the analogy are expressed as addends in the denominator of Eq. (14). The heat transfer coefficients associated with the fluid circulating inside the tubes ($h_i$) and outside ($h_o$) intervene in the equation, the conductivity of the material of the tube walls ($k_{\mathit{tm}}$), and the internal diameter ($D_{\mathit{it}}$) and external diameter ($D_{\mathit{ot}}$) of the tubes, as well as the fouling factor ($R_{\mathit{ic}}$) of the tube walls due to the accumulation of impurities.

$$U_g=\frac{1}{\frac{D_{\mathit{ot}}}{D_{\mathit{it}}{h}_i}+\frac{R_{\mathit{ic}}{D}_{\mathit{ot}}}{D_{\mathit{it}}}+\frac{D_{\mathit{ot}}\mathit{Ln}\left(\frac{D_{\mathit{ot}}}{D_{\mathit{it}}}\right)}{2k_{\mathit{tm}}}+\frac{1}{h_o}}$$E14

Eq. (15) defines the total area ($A_T$) associated with heat transfer to approximate its magnitude as the product of the external diameter of the tubes ($D_{\mathit{ot}}$), their length ($L_t$), and the total number of them ($N_t$) [1]. In Eq. (16), the term $\mathit{CL}$ acquires the value of 1.0 or 0.87 if the arrangement of the tubes allows establishing a geometric relationship between their centres of 90° and 45° (square), or 30° and 60° (triangular), respectively. On the other hand, $\mathit{CTP}$ acquires the values 0.93, 0.90, and 0.85, depending on whether the number of fluid passes in the equipment is defined as 1, 2, or 3, respectively. The vertical distance between the centres of the collinear tubes defines the $P_t$ (pitch) value.

$$A_T=\pi D_{\mathit{ot}}{L}_t{N}_t$$E15

$$N_t=0.785\sqrt{\frac{\mathit{CL}}{\mathit{CTP}}}\frac{D_s^2}{{\left(\mathit{PR}\right)}^2{D}_{\mathit{ot}}^2},\mathit{PR}=\frac{P_t}{D_{\mathit{ot}}}$$E16

2.2.2 The effectiveness and number of transfer units (ε-NTU) method

In the ε-NTU method, the determinations of two dimensionless numbers associated with the number of transfer units ($\mathit{NTU}$) and the heat capacity of the fluids in the exchanger ($C_r$) are used to calculate a third dimensionless number called effectiveness ($\epsilon$), which is defined as the quotient between the heat absorbed or delivered by the fluid of lower heat capacity and the maximum heat that could be exchanged. Eqs. (17) and (18) depend on the global heat transfer coefficient $U_g$ (Eq. (14)), the total exchange area $A_T$ (Eq. (15)) and the minor thermal capacity of the two substances in system $C_m$. Eq. (19), dependent on the previous equations, is defined for systems in which there is a single fluid passage in the casing, that is, a path in a non-return direction around the tube array.

$$\mathit{NTU}=\frac{U_g{A}_T}{C_m}$$E17

$$C_r=\frac{C_h}{C_m}$$E18

$$\epsilon =\frac{1-\mathit{exp}\left(-\mathit{NTU}\left(1-C_r\right)\right)}{1-C_r\mathit{exp}\left(-\mathit{NTU}\left(1-C_r\right)\right)}$$E19

If the exchanger has several steps, the effectiveness calculation is obtained from Eqs. (17), (18), (20), and (21) according to Eq. (22). The expression ${\left(\mathit{NTU}\right)}_1$ refers to the calculation of Eq. (6) considering the conditions of a step, which is subsequently magnified by multiplying by the number of steps over the arrangement of tubes $N_{s-s}$ in the exchanger.

$$\mathit{NTU}=N_{s-s}{\left(\mathit{NTU}\right)}_1$$E20

$${\epsilon }_1=2{\left[1+C_r-{\left(1-C_r^2\right)}^{0.5}\frac{1+\mathit{exp}\left(-{\left(\mathit{NTU}\right)}_1{\left(1-C_r^2\right)}^{0.5}\right)}{1-\mathit{exp}\left(-{\left(\mathit{NTU}\right)}_1{\left(1-C_r^2\right)}^{0.5}\right)}\right]}^{-1}$$E21

$$\epsilon =\left[{\left(\frac{1-{\epsilon }_1{C}_r}{1-{\epsilon }_1}\right)}^{N_{s-s}}-1\right]{\left[{\left(\frac{1-{\epsilon }_1{C}_r}{1-{\epsilon }_1}\right)}^{N_{s-s}}-C_r\right]}^{-1}$$E22

According to the above, the calculation of the total heat transferred is obtained with Eq. (23).

$$Q=\epsilon C_m\left({\overline{T}}_{\mathit{\text{is}}}-{\overline{T}}_{\mathit{it}}\right)$$E23

2.2.3 Pressure drop

The pressure drop $∆P_t$ in the fluid due to the circulation in the tube passages $N_{s-t}$ is obtained from Eqs. (24) and (25) [1]. The data $f$ is defined as the friction factor estimated from the Reynolds number considering the internal diameter of the tube as the characteristic length. The variables $v$ and $\rho$ represent the average velocity and density of the fluid, respectively.

$$∆P_t=\left(\frac{4f{L}_s{N}_{s-t}}{D_{\mathit{it}}}+4N_{s-t}\right)\left(\frac{\rho v^2}{2}\right)$$E24

$$f={\left(0.79\mathit{Ln}\left({\mathit{Re}}_D\right)-1.64\right)}^{-2},{\mathit{Re}}_D=\frac{\mathit{\rho v}{D}_{\mathit{it}}}{\mu }$$E25

On the other hand, the pressure drop in the fluid circulating in the casing (outside the tubes) is obtained with Eqs. (26)–(31). Again, f is defined as the friction coefficient that depends on the Reynolds number ${\mathit{Re}}_s$, which, in turn, depends on the mass velocity of the fluid $G_s$ and the equivalent diameter $D_e$ for a type of arrangement of tubes (triangular or square). Parameter $E$ is the vertical distance of clearance between tangents on the surface of the tubes. The number of deflectors is associated with parameter $N_b$, and the spacing between them defines the length $B$.

$$∆P_s=\frac{f{G}_s^2\left(N_b+1\right)D_s}{2\rho D_e{\left(\frac{\mu }{{\mu }_w}\right)}^{0.14}}$$E26

$$f=exp\left(0.576-0.19\mathit{Ln}\left({\mathit{Re}}_s\right)\right),{\mathit{Re}}_s=\frac{G_s{D}_e}{\mu }$$E27

$$G_s=\frac{{\dot{m}}_s}{A_s},A_s=\frac{D_s\mathit{EB}}{P_t}\equiv \left[1-\frac{D_{\mathit{ot}}}{P_t}\right]D_s\left[\frac{L_s}{N_b+1}\right]$$E28

$$D_s=0.637\sqrt{\frac{\mathit{CL}}{\mathit{CTP}}}{\left[\frac{A_T{\left(\mathit{PR}\right)}^2{D}_{\mathit{ot}}}{L_t}\right]}^{0.5}$$E29

$$D_e=\frac{4\left(\frac{P_t^2\sqrt{3}}{4}-\frac{\pi D_{\mathit{ot}}}{8}\right)}{\frac{\pi D_{\mathit{ot}}}{2}},\mathit{\text{triangular arrangement}}$$E30

$$D_e=\frac{4\left(P_t^2-\frac{\pi D_{\mathit{ot}}^2}{4}\right)}{\pi D_{\mathit{ot}}},\mathit{\text{square arrangement}}$$E31

However, the determination of the pressure drop is limited by the range of values of the Reynolds number (${\mathit{Re}}_s$) from 400 to 1.0E+06. The Reynolds number is dimensionless and represents the relationship between the inertial forces and the viscous forces determined by the flow of a fluid. Therefore, it is part of the π-group, and depending on its magnitude, it characterizes the movement of a fluid in laminar or turbulent regimes. It has been determined that the critical Reynolds number (Eq. (32)) is a function of the geometry of the fluid conduit [10]. Experimentally, it is suggested that, in the flow of the boundary layer (minimum layer of stable fluid on the surface of the conduit), with a value less than or equal to 2000, a laminar flow regime is preserved; therefore, when exceeding this magnitude, it is considered the domain of a turbulent flow [9].

$$\mathit{Re}=\frac{\mathit{\rho vD}}{\mu }$$E32

The dimensionless Prandtl number (Eq. (33)) determines the proportionality between the momentum quantity’s diffusion rate and the thermal diffusivity [11]. The magnitude of the Prandtl number determines the relationship between the thicknesses of the momentum and thermal boundary layers; small values define a rapid, effective diffusion of heat concerning the fluid velocity.

$$\mathit{Pr}=\frac{\raisebox{1ex}{$\mu $}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.}{\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$\rho C_p$}\right.}$$E33

2.2.4 Use of correlations to estimate internal and external flow heat transfer coefficients

The dimensionless Nusselt number (Nu) of the π-group represents the critical objective of the analysis of heat exchange processes since it determines the increase in heat transfer from a surface through which a fluid circulates concerning the process in which the transfer will be carried out only by heat conduction. The Nu numbers [12] can be obtained through correlations that depend on the Reynolds and Prandtl numbers. Eq. (34) expresses that estimating the heat transfer coefficient h from the dimensionless number Nu is possible. Analysis has been proposed in heat exchanger systems, such as the theories of the methods of Donohue, Tinker, Kern, Bell-Delaware, and Willis-Johnston, among others, which are based on correlations to obtain Nusselt numbers for the heat flow fluids inside tubes and on the outside or shell.

$$\mathit{Nu}=\frac{\mathit{hL}}{k}$$E34

The internal flow correlations referenced for their applicability in the analysis are those of the researchers Colburn [13], Dittus_Boelter [14], Gnielisnski [15], Sieder_Tate, Petukhov_Kirillov [16], and Notter_Sleicher. For external flow characterization, they are those of Zukauskas [17], Kern [8], Hilpert [18], Bell_Delaware [19], Taborek [20], Churchill_Berstein, and Hausen [21], for turbulent flow without phase change. Tables 2 and 3 list the corresponding correlations for determining internal and external heat transfer coefficients. Such correlations were used to analyze the subsequent cases presented in this chapter.

No.	Nusselt number correlation	Limitations
1	${\mathit{Nu}}_D=0.023{\mathit{Re}}_D^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}{\mathit{Pr}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	$\mathit{Re}\ge 10000$; $0.7\le \mathit{Pr}\le 160$; $\frac{L}{D}\ge 10$
2	${\mathit{Nu}}_D=0.023{\mathit{Re}}_D^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}{\mathit{Pr}}^{0.4}$
3	${\mathit{Nu}}_D=\frac{\left(\frac{f}{8}\right)\left[{\mathit{Re}}_D-1000\right]\mathit{Pr}}{1+12.7{\left(\frac{f}{8}\right)}^{0.5}\left({\mathit{Pr}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1\right)}$; $f={\left(0.69\mathit{ln}\left({\mathit{Re}}_D\right)-1.64\right)}^{-2}$	$3000\le \mathit{Re}\le 5E6$; $0.5\le \mathit{Pr}\le 2000$
4	${\mathit{Nu}}_D=0.027{\mathit{Re}}_D^{0.8}{\mathit{Pr}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(\frac{{\mu }_b}{{\mu }_w}\right)}^{0.14}$	$\mathit{Re}>10000$; $0.7\le \mathit{Pr}\le 16700$; $\frac{L}{D}>10$
5	${\mathit{Nu}}_D=\frac{\left(\frac{f}{2}\right){\mathit{Re}}_D\mathit{Pr}}{1.07+12.7{\left(\frac{f}{2}\right)}^{0.5}\left({\mathit{Pr}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1\right)}$; $f={\left(1.58\mathit{ln}\left({\mathit{Re}}_D\right)-3.8\right)}^{-2}$	$\mathit{Re}>2100$
6	${\mathit{Nu}}_D=5+0.015{\mathit{Re}}_D^m{\mathit{Pr}}^n$; $m=0.88-\frac{0.24}{4+\mathit{Pr}}$; $n=0.33+0.5\mathit{exp}\left(-0.6\mathit{Pr}\right)$	$1E4\le \mathit{Re}\le 1E6$; $0.1\le \mathit{Pr}\le 1E4$

Table 2.

Correlations for the estimation of the heat transfer coefficient for internal flow.

Authors: 1, Colburn; 2, Dittus_Boelter; 3, Gnielinski; 4, Sieder_Tate; 5, Petukhov_Kirillov; 6, Notter_Sleicher.

No.	Nusselt number correlation	Limitations
1	${\mathit{Nu}}_D=C{\mathit{Re}}_{D,\mathit{max}}^m{\mathit{Pr}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(\frac{\mathit{Pr}}{{\mathit{Pr}}_s}\right)}^{0.25}$	$10<\mathit{Re}\le 100,C=0.9,m=0.4$; $100<\mathit{Re}\le 1000,C=0.683,m=0.466$; $1000<\mathit{Re}\le 2E5,C=0.35,m=0.65$
2	${\mathit{Nu}}_D=0.36{\mathit{Re}}_D^{0.55}{\mathit{Pr}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(\frac{\mu }{{\mu }_b}\right)}^{0.14}$	$2E3<\mathit{Re}>1E6$; ${\mathit{Re}}_s\mathit{\text{defined}}\mathit{by}\mathit{Ecs}.17y18$
3	${\mathit{Nu}}_D=C{\mathit{Re}}_D^m{\mathit{Pr}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	$0.4<\mathit{Re}\le 4,C=0.989,m=0.33$; $4<\mathit{Re}\le 40,C=0.911,m=0.385$; $40<\mathit{Re}\le 4000,C=0.683,m=0.466$; $4000<\mathit{Re}\le 4E4,C=0.193,m=0.618$; $4E4<\mathit{Re}\le 4E5,C=0.027,m=0.805$
4	${\mathit{Nu}}_D=J_i{C}_p\left(\frac{\dot{m}}{A_s}\right){\left(\frac{k}{C_p\mu }\right)}^{0.66}{\left(\frac{\mu }{{\mu }_w}\right)}^{0.14}$; $J_i=0.37{\mathit{Re}}_s^{-0.395}$	$1E4<\mathit{Re}\le 1E5$
5	${\mathit{Nu}}_D=0.2{\mathit{Re}}_s^{0.6}{\mathit{Pr}}_s^{0.4}$	${\mathit{Re}}_s\mathit{\text{defined}}\mathit{by}\mathit{Ecs}.16y19o20$
6	${\mathit{Nu}}_D=0.3+\frac{0.62{\mathit{Re}}_D^{0.5}+{\mathit{Pr}}^{0.33}}{{\left[1+{\left(\frac{0.4}{\mathit{Pr}}\right)}^{0.66}\right]}^{0.25}}{\left[1+{\left(\frac{{\mathit{Re}}_D}{282000}\right)}^{0.625}\right]}^{0.8}$
7	${\mathit{Nu}}_D=0.35\mathit{Fa}{\mathit{Re}}_D^{0.57}{\mathit{Pr}}^{0.31}$; $\mathit{Fa}=1+0.1S_2+\frac{0.34}{S_1}$	Triangular arrangement of tubes

Table 3.

Correlations for the estimation of the heat transfer coefficient for external flow.

Authors: 1, Zukauskas; 2, Kern; 3, Hilpert; 4, Bell_Delaware; 5, Taborek; 6, Churchill_Bernstein; 7, Hausen.

2.2.5 Introduction to the EES (engineering equation solver) program for algorithm programming

The EES (Engineering Equation Solver) program is a valuable tool for solving problems formulated with systems of coupled nonlinear algebraic and differential equations [22].

The graphical user interface allows the creation of an interactive panel in which the user controls the data entry and, in turn, receives information due to the development of calculations. Therefore, this tool was used to create work files to analyze the methods in Sections 2.21 and 2.2.2 in conjunction with the correlations in Sections 2.2.3 and 2.2.4 to solve problems raised in the shell and tube heat exchangers field.

Figure 2 shows the graphical user interface of the EES program. Heat exchanger analysis algorithms require complex programming; however, each program is built from fundamental structures. A basic example with subsequent images will show the case study generation strategy. When starting the EES program in the equations window area, it will be demonstrated that introducing assignments of constants and operations between them is done relatively quickly. Previously, the unit system must be chosen by accessing it from the Options menu, as shown in Figure 3.

Figure 2.

The equations window is displayed initially by the graphical user interface of the EES program and access to the options of the Options menu.

Figure 3.

The pop-up window defines the system of units in a case study.

Figure 4 shows the simple process of assigning constant values and assigning functions, such as calculating thermal capacity that depends on temperature and pressure. In turn, Figure 5 shows the complete development in which it is now possible to estimate the magnitude of $Q$ by activating the Solve icon. Figure 6 shows the results window containing the information defined in the equations window and the expected result for both the Cpw and Q functions.

Figure 4.

Equation window with a definition of constants and a function for calculating the thermal capacity of water. Deployment of the pop-up window through the Options menu.

Figure 5.

The Equations window contains information about the complete calculation model that will be developed when the Solve icon is activated.

Figure 6.

Display of the results window, including the predefined data.

Alternatively, through the diagram window, it is possible to create interactive forms in which, through the toolbar, objects such as the calculation button and text boxes linkable to the predefined constants are introduced, with which it is possible to transform them into input or output variables. Figure 7 shows the complete process of converting the constants to variables the user can modify without accessing the equations window. Additionally, images can be integrated to understand better the analysis developed. The procedure consists of copying and pasting an edited image with the appropriate dimensions in the work area (see Figure 8).

Figure 7.

Convert to variables and insert the calculation activation button using the tools available in the diagram window.

Figure 8.

Pasting schematic figures to optimize the interactive form in the diagram window.

However, for the conversion to be operational, the constants defined in the equation window must be converted to comments. The conversion process is simple since only the text is selected, and the secondary mouse button accesses the options pop-up to select comment{}, as shown in Figure 9.

Figure 9.

Conversion of active constants to text comments through the pop-up menu displayed after selecting the text field in the equation window.

Other features can be explored in the program programming manuals. Therefore, the structure of more complex programs can be generated to solve the heat exchange process, as will be demonstrated in subsequent sections.

Figure 10 presents a general flow chart for the solution algorithms of each case study. The source quantities of the process variables are fed to each coded algorithm through each graphical user interface. Some data are processed to calculate thermal properties and subsequently transferred to a calculation cycle where the unknown dependent variables are determined. In the cycle, the systems of equations of the $\mathit{\text{LMTD}}$ and $\epsilon -\mathit{NTU}$ models and the correlation bank are implemented. The dependent variables are calculated with various data originating from combinations of correlations, and based on the target magnitudes, the appropriate set that characterizes the specific system is selected.

Figure 10.

Flow chart for the solution algorithms of each case study.

2.3 Analysis cases

2.3.1 Cylindrical exchanger for cooling water streams flowing in tubes and water streams flowing in the shell: case E1

The data on the operating conditions and geometric parameters presented in the analysis exercise were obtained from the work of Costa and Queiroz [23]. One of the objectives of the analysis of the heat exchange process is the determination of the outlet temperatures on the tube and shell sides. The comparison of data between the calculation and the source can determine that the calculation procedure proposed with the ε-NTU method constitutes a consistent calculation alternative. The operating parameters and characteristics of the heat exchanger obtained from [23] are provided in Tables 4 and 5. On the other hand, Table 6 establishes the magnitudes of surface area, amount of heat, and the velocity of the fluid in the tubes.

Section	Fluid	Volume flow	Mass flow	Inlet temp.	Pressure	Outlet temp.
		L min^−1	kg s^−1	°C	kPa	°C (target)
Shell	Water	50.18	8.33	17	12	40
Tubes	Water	84.72	13.88	67	12	—

Table 4.

Operating parameters in the cylindrical heat exchanger with water flowing on the tube and shell sides.

Section	Setting	Unit	Triangular arrangement
Shell	Length	m	0.762
	Inner diameter	m	0.438
	External diameter	m	—
	Number of steps	—	1
	Number of baffles	—	10
	Distance between baffles	m	—
Tube	Number of passages	—	4
	Distance between centers	m	0.025
	Length	m	3.048
	External diameter	m	0.03175
	Internal diameter	m	0.0301
	Amount	—	87
	Wall thermal conductivity	Wm^−1 K^−1	60

Table 5.

Characteristics of the cylindrical heat exchanger with water flowing on the tube and shell sides. Case E1.

Tubes arrangement	$A_T,m^2$	$Q,W$	$v,m{s}^{-1}$
Triangular	24.63	8.13E5	1.1

Table 6.

Transfer surface characteristics and total heat flow of the cylindrical heat exchanger with specification of the velocity of the water flowing on the tube side. Case E1.

In the analysis through the EES program, combinations of the correlations in Tables 2 and 3 were used to determine heat transfer coefficients for internal and external flows. Table 7 shows the magnitudes obtained from the Nutter_Sleicher and Bell_Delware correlations. With these data, the fluid temperature on the shell side closest to the target temperature was obtained with only a difference of 1.56°C. On the side of the tubes, the outlet temperature obtained was 52.27°C.

Internal flow correlations	$h_i,W{m}^{-2}{K}^{-1}$	External flow correlations	$h_0,W{m}^{-2}{K}^{-1}$
	Triangular arrangement		Triangular arrangement
Nutter_Sleicher	6303	Bell_Delaware	2098

Table 7.

The magnitudes of the heat transfer coefficients for internal and external flows gave rise to the best approximation of the shell-side outlet temperature, Case E1.

The transfer surface, the total heat flux, and the calculated fluid velocity differ slightly from the pressure drops. A high drop was determined on the tube side, and a shallow drop on the shell side. Table 8 shows the results calculated using the ε-NTU method (refer to Section 2.2.2) and the pressure drop equations (refer to Section 2.2.3). Figure 11 shows the interactive form created in the EES program for this exercise.

Tube arrangement	$A_T,m^2$	$Q,W$	$v$	$∆P,\mathit{Pa}$ (tubes)	$∆P,\mathit{Pa}$ (Shell)
Triangular	25.84	8.55E5	0.94	20,922	339.4

Table 8.

Transfer surface characteristics, total heat flow, average fluid velocity on the tube side, and pressure drops calculated for Case E1.

Figure 11.

An interactive form was created with the EES program for the analysis of the heat exchanger of Case E1.

2.3.2 Cylindrical exchanger for cooling methanol streams flowing in the shell with water streams flowing in the tubes: Case E2

In an industrial process, methanol is cooled through a heat exchanger using water as a secondary fluid. The average shell-side methanol inlet temperature is 90°C, and the mass flow is 31.22 kg s⁻¹. The average water inlet temperature on the side of the tubes is 30°C, and the mass flow is 102.9 kg s⁻¹. The analysis aims to promote both fluids to reach an average temperature of 40°C. The operating conditions and working pressure are detailed in Table 9.

Section	Fluid	Volume flow	Mass flow	Inlet temp.	Pressure	Outlet temp.
		L min^−1	kg s^−1	°C	kPa	°C (target)
Shell	Methanol	2485	31.22	90	1000	40
Tubes	Water	6210	102.9	30	1000	40

Table 9.

The operating parameters in the cylindrical heat exchanger are water flowing on the tube side and methanol on the shell side. Case E2.

Table 10 provides the exchanger’s operating parameters and characteristics, considering the triangular and square tube arrangement configuration.

Section	Setting	Unit	Arrangement
			Triangular	Square
Shell	Length	m	7	7
	Inner diameter	m	0.591	0.6697
	External diameter	m	0.6	0.6757
	Number of steps	—	1	1
	Number of baffles	—	8	8
Tube	Number of passages	—	1	1
	Distance between centers	m	0.018	0.018
	Length	m	6.025	6.025
	External diameter	m	0.016	0.016
	Internal diameter	m	0.01587	0.01587
	Amount	—	748	748
	Wall thermal conductivity	Wm^−1 K^−1	15.1	15.1

Table 10.

Characteristics of the cylindrical heat exchanger, with water flowing on the tube side and methanol on the shell side, with triangular and square tube arrangement configuration. Case E2.

According to the ε-NTU method (refer to Section 2.2.2) and the combinations of the correlations defined in Tables 2 and 3, the heat transfer coefficients for the internal and external flows presented in Table 11 were obtained to arrange tubes with triangular and square configurations.

Internal flow correlations	$h_i,W{m}^{-2}{K}^{-1}$		External flow correlations	$h_0,W{m}^{-2}{K}^{-1}$
	Tube arrangement			Tube arrangement
	Triangular	Square		Triangular	Square
Petukov	4122	4122	Taborek	711.9	717.9
Gnielinski	3966	3966	Zukauskas	1146	1178
Dittus	3739	3739	Hilpert	1274	1309
Colburn	3344	3344	Kern	1271	1438

Table 11.

Magnitudes of heat transfer coefficients for internal and external flows are calculated with various correlations for both a triangular and a square tube array configuration. Case E2.

The transfer area, heat flow data, and efficiency were determined constant for the corresponding triangular and square configurations. Also, the pressure drop on the tube side was steady for both configurations, and the pressure drop on the shell side was only 816 Pa different. Table 12 summarizes the corresponding information.

Tube arrangement	$A_T,m^2$	$Q,W$	$\epsilon$	$∆P,\mathit{Pa}$ (tubes)	$∆P,\mathit{Pa}$ (Shell)
Triangular	227.3	4.28E6	0.83	3592	4856
Square	227.3	4.28E6	0.83	3592	5672

Table 12.

Transfer surface characteristics and total heat flow, efficiency, and pressure drops calculated for Case E2.

An analysis of combinations of correlations for calculating heat transfer coefficients for the internal ($\mathit{\text{if}}$) and external ($\mathit{ef}$) flows resulted in the values obtained for the global heat transfer coefficient and the fluid outlet temperature on the tube and shell sides in Tables 13 and 14. For both cases of tube arrangement configurations, the temperatures are very close to the target value of 40°C. The furthest values are associated with combining the Petukhov_Kirillov and Taborek correlations.

Case	Correlation	$U_g$, Wm-2 K-1	${\overline{T}}_{\mathit{os}}$, °C	${\overline{T}}_{\mathit{ot}}$, °C
1	Gnielinski (if)	881.7	40.77	39.95
	Zukauskas (ef)
2	Dittus-Boelter (if)	869.8	40.92	39.92
	Zukauskas (ef)
3	Colburn (if)	977.2	39.69	40.17
	Kern (ef)
4	Petukhov (if)	591.6	46.42	38.81
	Taborek (ef)
5	Petukhov (if)	889.2	40.67	39.97
	Zukauskas (ef)
6	Gnielinski (if)	955.1	39.91	40.13
	Hilpert (ef)
7	Dittus-Boelter (if)	941.3	40.06	40.1
	Hilpert (ef)
8	Gnielinski (if)	1024	39.25	40.26
	Kern (ef)
9	Colburn (if)	846.4	41.24	39.86
	Zukauskas (ef)

Table 13.

Results of the overall heat transfer coefficient and fluid outlet temperatures on the tube and shell sides by combining correlations for a triangular configuration in the tubes. Case E2.

Case	Correlation	$U_g$, Wm-2 K-1	${\overline{T}}_{\mathit{os}}$, °C	${\overline{T}}_{\mathit{ot}}$, °C
1	Gnielinski (if)	942.3	40.1	40.05
	Zukauskas (ef)
2	Dittus-Boelter (if)	928.8	40.2	40.07
	Zukauskas (ef)
3	Colburn (if)	917	40.33	40.04
	Kern (ef)
4	Petukhov (if)	604.7	46.42	38.88
	Taborek (ef)
5	Petukhov (if)	950.9	39.96	40.12
	Zukauskas (ef)
6	Gnielinski (if)	1021	39.28	40.25
	Hilpert (ef)
7	Dittus-Boelter (if)	1005	39.28	40.25
	Hilpert (ef)
8	Gnielinski (if)	958.5	39.88	40.13
	Kern (ef)
9	Colburn (if)	902.2	40.51	40.01
	Zukauskas (ef)

Table 14.

Results of the global heat transfer coefficient and the fluid outlet temperatures on the tube and shell sides using a combination of correlations for a square configuration in the tubes. Case E2.

By obtaining results close to the target outlet temperatures with different combinations of correlations, it is verified that the exchanger operating conditions are within its predictive domain. From this perspective, the calculation procedure implemented for the form interface in the EES program, shown in Figure 12, can be used to generate data curves for various systems by modifying values in Table 9. However, the study could also be extended by modifying the values from Table 10.

Figure 12.

An interactive form was created with the EES program for the analysis of the heat exchanger of Case E2.

2.3.3 Cylindrical exchanger for cooling oil streams flowing in the tubes and water stream flowing in the casing: case E3

A heat exchanger designed to cool oil was instrumented with thermocouples to measure the inlet and outlet temperatures of the fluids on the tube and shell sides. Processing data for 30 tests are presented in Table 15.

Test	Oil (tube side)			Water (shell side)
	Volume flow, Ls^−1	Temperature, °C		Mass flow, gs^−1	Temperature, °C
		Inlet	Outlet		Inlet	Outlet
1	0.02	45	34.08	10.83	21.9	31.1
2	0.02	44	32.42	10.83	21.92	30.5
3	0.02	43	32.26	10.83	21.55	31.1
4	0.02	42	31.04	10.83	20.9	29.32
5	0.02	41	30.07	10.83	21.03	29.1
6	0.02	40	30.58	10.83	21.05	29.37
7	0.02	39	29.9	10.83	21.1	28.24
8	0.02	38	30	10.83	21.12	28.51
9	0.02	37	29.03	10.83	21.14	28
10	0.02	36	27.09	10.83	21.18	27.6
11	0.02	35	27.65	10.83	21.11	26.5
12	0.02	45	34.19	10.83	22.08	32.36
13	0.019	45	35.07	10.83	22.1	32.59
14	0.0181	45	34.02	10.83	22.13	32.83
15	0.0171	45	33.22	10.83	22.13	33
16	0.0162	45	33.48	10.83	22.18	33.3
17	0.0152	45	33.08	10.83	22.28	33
18	0.0143	45	33	10.83	22.28	29.9
19	0.0133	45	32.78	10.83	22.37	29.5
20	0.0123	45	32.6	10.83	22.34	31.2
21	0.02	45	34.02	10.83	22.38	31.1
22	0.02	45	34.03	10.5	22.41	31.5
23	0.02	45	34.35	10.1	22.43	31.8
24	0.02	45	34.46	9.8	22.46	32
25	0.02	45	34.56	9.5	22.47	32
26	0.02	45	34.82	9	22.61	32.1
27	0.02	45	34.94	8.8	22.62	32.3
28	0.02	45	35.14	8.5	22.68	32.7
29	0.02	45	35.32	8.1	22.7	33
30	0.02	45	35.42	7.8	22.73	33.6

Table 15.

Test data on the heat exchanger for oil cooling. Case E3.

The characteristics of the heat exchanger are presented in Table 16.

Table 16.

Characteristics of the cylindrical heat exchanger, with oil flowing on the tube side and water on the shell side, with triangular tube arrangement configuration. Case E3.

The graph in Figure 13(a) shows the measured and calculated oil outlet temperature data for each test in Table 15. As noted, the Sider_Tate and Hilpert correlations generated results close to the data recorded by the thermocouples in the tests on the exchanger. The results of the Dittus_Boelter-Churchill_Bernstein and Dittus_Boelter-Kern combinations were eliminated because they significantly exceeded the desired temperature. In turn, the graph in Figure 13(b) shows the data for water. In this case, the calculations with the combination of the Sieder_Tate-Hilper, Gnielinski-Zukauskas and Sieder_Tate-Churchill_Berstein correlations adequately approximate the target temperature.

Figure 13.

Measured outlet temperatures (listed in Table 15) and calculated with internal and external flow correlations for fluids: (a) oil and (b) water.

According to the above, the analysis through development in the ESS program allowed obtaining a control tool on the behavior of the heat exchanger for oil cooling. It is natural for the physical properties of the oil to change due to the recirculatory use of the oil and contamination of the oil. However, the information can be adjusted to the data required by the calculation procedure until the results of the calculated outlet temperatures converge. This condition turns the computational tool into a simulator with prediction capacity.

The form was created in the EES program for analysis using the ε-NTU method in Figure 14.

Figure 14.

An interactive form was created with the EES program for the analysis of the heat exchanger of Case E3.

2.3.4 Cylindrical exchanger for cooling water streams flowing in tubes and R134a refrigerant stream flowing in the casing: case E4

An exchanger is used for zone cooling and water, and R134a refrigerant is involved. The objective of the analysis is to determine the appropriate correlation information to validate the exchanger’s behavior using the ε-NTU method. Table 17 summarizes the operation data recorded in seven processes.

Case	Section	Volume flow	Mass flow	Inlet temp.	Pressure	Outlet temp.
		m3 h^−1	kg s^−1	°C	Bar	°C
1	Shell	—	0.055	56.5	11.36	42.68
	Tubes	1.021	—	23.83	—	32.02
2	Shell	—	0.0541	68.65	14.95	52.34
	Tubes	0.581	—	26.61	—	39.87
3	Shell	—	0.0603	66.76	13.3	47.84
	Tubes	0.588	—	15.87	—	31.04
4	Shell	—	0.0466	51.2	10.73	40.83
	Tubes	1.027	—	24.72	—	31.6
5	Shell	—	0.0551	67.98	14.58	51.18
	Tubes	0.638	—	26.06	—	38.5
6	Shell	—	0.036	45.67	9.54	36.54
	Tubes	1.042	—	23.69	—	29.17
7	Shell	—	0.0576	72.57	16.02	27.07
	Tubes	0.563	—	27.07	—	41.83

Table 17.

Measurement data in the heat exchanger for oil cooling. Case E4.

The behavior of the exchanger is characterized by three regions associated with the refrigerant’s aggregation state. The first zone (Z1) is associated with the decrease in the temperature of the refrigerant in the vapor state from the maximum temperature to the change of state temperature, that is, to the condensation temperature. The second zone (Z2) is associated with the transformation process by condensation to the liquid state, and the third zone (Z3) with cooling until the refrigerant’s minimum temperature or outlet temperature on the shell side is reached. Table 18 summarizes the characteristics of the exchanger with a triangular tube arrangement configuration.

Section	Setting	Unit	Arrangement Triangular
Shell	Length	m	0.87
	Inner diameter	m	0.183
	External diameter	m	0.195
	Number of steps	—	1
	Number of baffles	—	4
Tube	Number of passages	—	2
	Distance between centers	m	0.0195
	Length	m	0.8
	External diameter	m	0.016
	Internal diameter	m	0.013
	Amount	—	10
	Wall thermal conductivity	Wm^−1 K^−1	60

Table 18.

Characteristics of the cylindrical heat exchanger, with water flowing on the tube side and R134a refrigerant on the shell side, with triangular tube arrangement configuration. Case E4.

The thermal processes in the three zones will be characterized according to the correlations that allow the estimation of the convective coefficients that determine the amount of heat transferred. Therefore, the approach to the energy balance and the identification of the heat exchange surfaces will support the description of the thermal phenomenon in each zone.

In Z1, the cooling of the superheated vapor that enters the condenser from the discharge of the compressor of the system in which the exchanger operates originates. At Z2, the refrigerant goes from the saturated vapor to the saturated liquid state. In this zone, condensation will be considered to occur at a constant temperature (such as a pure fluid or azeotropic mixture), so it is easy to define that the water stream has the minimum thermal capacity. Finally, in Z3, the temperature is decreased below the saturation condition. In this regard, the minimum capacitance is determined in Z1 and Z3, and the efficiency of the exchanger in Z2 is defined by Eq. (35).

$${\eta }_{Z2}=1-\mathit{exp}\left(-{\mathit{NTU}}_{Z2}\right)$$E35

${\mathit{NTU}}_{Z1}$ and the global heat transfer coefficient $U_g$ are determined with Eqs. (14) and (17), respectively. However, Eq. (14) is modified to introduce a heat transfer coefficient associated with the condensation phenomenon derived from the extension of Nusselt’s analysis [10] defined by Eq. (36).

$$h_{Z2}=0.729{\left[\frac{g{\rho }_l\left({\rho }_l-{\rho }_v\right)k_l^3{H}_{\mathit{fg}}^{\prime }}{N_t{\mu }_l\left(T_{\mathit{sat}}-T_{\mathit{\text{wall}}}\right)D_{\mathit{os}}}\right]}^{0.25}$$E36

The term $H_{\mathit{fg}}^{\prime }$ represents the latent heat modified by thermal advection effects that can be obtained by Eq. (37).

$$H_{\mathit{fg}}^{\prime }=H_{\mathit{fg}}\left(1+0.68J_a\right),J_a={\mathit{Cp}}_l\left(\frac{T_{\mathit{sat}}-T_{\mathit{\text{wall}}}}{H_{\mathit{fg}}}\right)$$E37

Eq. (38) expresses the heat exchange surfaces and the total heat transfer in the total balance.

$$A_T=A_{Z1}+A_{Z2}+A_{Z3},Q=Q_{Z1}+Q_{Z2}+Q_{Z3}$$E38

In the analysis procedure, different correlations were combined to calculate the heat transfer coefficients for internal and external flow. In summary, the association of the Dittus_Boelter-Hilpert correlations were the most appropriate to characterize the thermal phenomena in zones Z1 and Z3. For Z2, the Dittus_Boelter-Nusselt combination was used. Table 19 summarizes the information on the transfer surface and the amount of heat transferred calculated for each exchanger zone. Also, the corresponding effectiveness is listed in the table, which indicates a relatively regular behavior for the seven processes analyzed.

Case	Cooling in Z1			Condensation			Cooling Z2
	$A_{Z1},m^2$	$\epsilon$	$Q_{Z1},W$	$A_{Z2},m^2$	$\epsilon$	$Q_{Z2},W$	$A_{Z3},m^2$	$\epsilon$	$Q_{Z3},W$
1	0.0817	0.103	125.2	1.048	0.355	8716	1.74	0.752	842.7
2	0.124	0.136	235.9	0.903	0.403	7875	1.84	0.747	1123
3	0.093	0.1	235.1	0.902	0.383	9129	1.874	0.732	14.92
4	0.066	0.09	82.63	1.034	0.357	7496	1.77	0.783	623.1
5	0.13	0.14	250.6	0.898	0.384	8092	1.84	0.749	1161
6	0.065	0.11	58.1	0.977	0.346	5957	1.83	0.83	412.6
7	0.163	0.165	328.3	0.887	0.4	8171	1.82	0.73	1301

Table 19.

Transfer surface data and the amount of heat transferred were calculated for each zone, and the effectiveness obtained for each process. Case E4.

Regarding the outlet temperatures of the fluids, measured and calculated on the tube and casing sides, it is observed that there is a good approximation, and the maximum differences do not exceed 3 °C. Therefore, the model can be considered very useful for predicting exchanger performance. Table 20 summarizes the corresponding data and allows a comparison of the information.

Process	Outlet temperature results, °C
	Refrigerant R134a (side shell)		Water (side tubes)
	Measured	Calculated	Measured	Calculated
1	56.5	54.65	32.02	31.99
2	71.11	68.65	39.87	40.3
3	69.76	66.76	31.04	31.76
4	53.6	51.2	31.6	31.59
5	69.9	67.98	38.5	38.88
6	48.3	45.67	29.17	29
7	75.2	72.57	41.83	42.05

Table 20.

Measured and calculated fluid outlet temperatures for the R134a refrigerant cooling heat exchanger processes on the shell side with water on the tube side. Case E4.

2.3.5 Cylindrical exchanger for cooling water streams flowing in tubes and water streams flowing in the shell: case E5

Some shell and tube-type heat exchangers are designed to heat water in swimming pools with a maximum volume of 450 m³ through a simple and flexible installation. Hot water can come from a boiler, heat pump, and solar panel systems, among others, toward the shell side with a maximum volumetric flow of 200 L min⁻¹. Cold water enters from the side of the tubes with a maximum volumetric flow of 25 L min⁻¹. Table 21 includes additional information on the exchanger operation and suggests three analysis processes.

Section	Fluid	Volume flow	Pressure	Inlet temperature
		L min^−1	kPa	°C (proc. 1)	°C (proc. 2)	°C (proc. 3)
Shell	Water	200	31.22	60	70	45
Tubes	Water	25	102.9	10	20	10

Table 21.

Operating parameters in the cylindrical heat exchanger include water flowing on the tube side and water on the shell side. Case E5.

Table 22 lists the characteristics of the heat exchanger models with differences in the heat transfer surface. Model 1 defines an area of 2.65 m², while model 2 defines 2.96 m².

Section	Setting	Unit	Triangular arrangement
			Model 1	Model 2
Shell	Length	m	1.2	1.6
	Inner diameter	m	—	—
	External diameter	m	—	—
	Number of steps	—	1	1
	Number of baffles	—	6	8
Tube	Number of passages	—	1	1
	Distance between centers	m	0.04	0.045
	Length	m	1.2	1.6
	External diameter	m	0.02	0.031
	Internal diameter	m	0.019	0.03
	Amount	—	37	19
	Wall thermal conductivity	Wm^−1 K^−1	60	60

Table 22.

Characteristics of the cylindrical heat exchanger, with water flowing on the tube side and water on the shell side, triangular tube arrangement configuration. Case E5.

In this case, in addition to the macroscopic semi-analytical analysis essentially referred to as the ε-NTU method, the results of the microscopic analysis obtained using the Ansys Fluent program are presented. Tables 23–25 summarize the results of the water outlet temperature on the tube and shell sides for the two heat exchanger models and the three processes, both for the macroscopic analysis (ε-NTU) and for the microscopic analysis (Ansys Fluent).

Model	Side	Water outlet temperature, °C (process 1)
		Fluent	Petukhov-Taborek	Gnielinski-Zukauskas	Dittus-Hausen	Colburn-Kern
1	Shell	31.64	31.06	13.92	31.25	25.29
	Tubes	57.29	57.32	59.5	57.3	58.05
2	Shell	29.31	30.56	17.63	30.96	27.02
	Tubes	57.26	57.38	59.03	57.33	57.83

Table 23.

Water outlet temperatures on the tube and shell sides for the two models of the exchanger with conditions for process 1. Case 5.

Model	Side	Fluid outlet temperature, °C (process 2)
		Fluent	Petukhov-Taborek	Gnielinski-Zukauskas	Dittus-Hausen	Colburn-Kern
M1	Shell	41	42.72	27.96	42.9	36.67
	Tubes	67.3	67.1	68.3	67.02	67.87
M2	Shell	39.5	41.42	32.72	41.6	38.8
	Tubes	67	67.14	68.38	67.08	67.6

Table 24.

Water outlet temperatures on the tube and shell sides for the two models of the exchanger with conditions for process 2. Case 5.

Model	Side	Fluid outlet temperature, °C (process 3)
		Fluent	Petukov-Taborek	Gnielinski-Zukauskas	Dittus-Hausen	Colburn-Kern
M1	Shell	26.7	25.6	12.66	25.74	20.53
	Tubes	42.3	43.16	44.66	43.14	43.67
M2	Shell	25.64	24.31	15.3	24.66	21.85
	Tubes	43.06	43.2	44.33	43.15	43.5

Table 25.

Water outlet temperatures on the tube and shell sides for the two models of the exchanger with conditions for process 3. Case 5.

When analyzing the information in Tables 23–/bold>25, it is observed that the results obtained with the combinations of the Petukhov_Kirillov-Taborek and Dittus_Boelter-Hausen coefficients present the most minor difference concerning the results obtained with Ansys Fluent. Due to the increase in the heat transfer surface in model 2, the temperature is consistently lower than that obtained for model 1 in all processes.

Tables 26 and 27 summarize the results of the heat transfer coefficient calculations for the internal and external flow of each process and model, respectively. The combination of Petukhov_Kirillov-Taborek and Dittus_Boelter-Hausen coefficients generates similar results for the overall heat transfer coefficient (refer to Table 28).

Internal flow correlations			$h_i,W{m}^{-2}{K}^{-1}$
	Process 1		Process 2		Process 3
	M1	M2	M1	M2	M1	M2
Petukhov	535.4	375.5	611.2	428.6	535.4	375
Gnielinski	72.83	104.6	208.3	193.2	73	105
Dittus	479.3	338.4	542.5	383	479.3	338.4
Colburn	409	288.8	472.7	333.8	409	288.8

Table 26.

Estimated heat transfer coefficients for the internal flow of the two exchanger models with conditions for the three processes. Case 5.

External flow correlations			$h_o,W{m}^{-2}{K}^{-1}$
	Process 1		Process 2		Process 3
	M1	M2	M1	M2	M1	M2
Taborek	1475	2336	1525	2417	1388	2199
Zukauskas	248.8	1754	277.2	1958	216	1517
Hausen	2410	17,820	2517	18,615	2229	16,485
Kern	686.6	1975	717.1	2064	649.3	1869

Table 27.

Estimated heat transfer coefficients for the external flow of the two exchanger models with conditions for the three processes. Case 5.

Correlation association			$U_g,W{m}^{-2}{K}^{-1}$
	Process 1		Process 2		Process 3
	M1	M2	M1	M2	M1	M2
Petukhov-Taborek	373.4	323.4	414.6	364	367.6	320.6
Gnielinski-Zukauskas	54.03	98.67	115.1	175.9	53	98
Dittus-Hausen	378	332.1	421.6	375.3	373.3	331.6
Colburn-Kern	246	252	273.6	287.3	241.1	250

Table 28.

Estimated global heat transfer coefficients of the two exchanger models with conditions for the three processes. Case 5.

Although the magnitudes of the heat transfer coefficients for the internal flow vary, they balance with the determined magnitudes of the heat transfer coefficients for the external flow. Therefore, the calculated efficiency presents the same pattern as the overall heat transfer coefficient (Table 29). The above indicates that macroscopic balances are certainly not precise, although they represent a solution for the system in approximation to the microscopic balance. The form was created in the EES program for analysis using the LMTD and ε-NTU methods in Figure 15.

Correlation association			$\epsilon$
	Process 1		Process 2		Process 3
	M1	M2	M1	M2	M1	M2
Petukov-Taborek	042	0.41	0.45	0.44	0.41	0.4
Gnielinski-Zukauskas	0.1	0.15	0.15	0.25	0.1	0.15
Dittus-Hausen	0.41	0.4	0.45	0.44	0.41	0.4
Colburn-Kern	0.33	0.34	0.33	0.37	0.3	0.33

Table 29.

Estimated efficiency of the two exchanger models with conditions for the three processes. Case 5.

Figure 15.

An interactive form was created with the EES program for the analysis of the heat exchanger of Case E5.

On the other hand, the simulation of the process based on the microscopic balance will depend significantly on the precise definition of the system’s geometric characteristics. Therefore, if the details of the shapes and the amount of information about the materials involved in each section increase, obtaining a solution by solving numerically Eqs. (1)–(6) becomes complex. The system must be subdivided or meshed into several elements to get the numerical solution and avoid variation in the results. The excessive meshing and its quality imply a high computing time cost. Therefore, mesh sensitivity studies are necessary to determine the optimal analysis time. Consequently, this process can be time-consuming and impractical, unlike macroscopic analysis. Figure 16(a) and (b) show the water flow pathlines on the shell side obtained by solving the microscopic model in the geometric representation of the heat exchanger through Ansys Fluent®. Fortunately, the potential of computing systems has increased enormously in recent decades, reducing analysis times. Therefore, computational fluid dynamics (CFD) analysis is more accessible but still expensive.

Figure 16.

The water flow pathlines on the shell side were obtained through Ansys Fluent for (a) model 1 and (b) model 2 of the heat exchanger. Case 5.

3. Conclusions

Heat exchangers are cost-effective technology due to their adaptability in process optimization. This chapter analyzed use cases of exchangers, showing that with established operating parameters and correlations, the amount of heat transferred, and the fluid temperatures on the shell and tube sides are precisely quantified. Calculations in the opposite direction were also obtained using the algorithms proposed and codified in ESS. The functionality of this program for solving systems of equations facilitates the ability to create a predictive tool. Therefore, the tool applied to various operating conditions produced satisfactory results by having a database of correlations for estimating heat transfer coefficients for internal and external flows. Although the correlations are limited with the numbers of Re and Pr, the determination of the Nusselt number that allows calculating the global transfer coefficients is defined with the results of fluid temperatures on the shell and tube side. In this sense, the measurement of experimental data is required to validate the calculations and define applicable margins of the specific correlations to characterize the process of interest.

According to the above, macroscopic balances constitute an efficient methodology for the design objective of heat exchangers. However, simulations using microscopic balances represent a fundamental activity in determining a design fully adapted to a process. Even though it requires a relatively long time and expensive software resources, determining the velocity fields and temperature distribution clarifies the definition of dimensions, geometries, and materials for the construction of the exchanger structure. In this sense, fuel from non-renewable resources is used efficiently. Consequently, the carbon footprint of the process is reduced, especially if part of the thermal energy in transit continues to be used in alternate subprocesses. The above determines the concept of net zero energy to reduce costs and generate more significant benefits and is complementary to the use of renewable energies. In the case of using renewable resources as a source of thermal energy, efficiency is still an essential factor since generation is complex, and the objective is its maximum use.

Acknowledgments

This work was supported by the Universidad Politécnica de Juventino Rosas.

The authors thank the Universidad de Guanajuato for the facilities and for using the EES program to generate the graphical user interface examples for the chapter’s analysis.

We also thank the Tecnológico Nacional de México, which provided access to the Ansys Fluent program through the Instituto Tecnológico de Morelia.

Conflict of interest

The authors declare no conflict of interest.

Thanks

Thanks are due to the access to the technical information provided by Dr. Juan Gregorio Hortelano Capetillo, the material discussed in this chapter was possible to concentrate on and present.

Nomenclature

$∆P_t$	pressure drop in the fluid
$\epsilon$	effectiveness/viscous dissipation
${\epsilon }_1$	effectiveness for one-step conditions
κ	turbulence kinetics
$\mu$	viscosity
${\mu }_t$	Eddy viscosity
$\eta$	efficiency of the exchanger
$\rho$	density
${\sigma }_{\epsilon }$	constant
${\sigma }_{\kappa }$	constant
$\tau$	shear strength
$A_T$	total exchange area
$B$	spacing between deflectors
$C_{\mu }$	constant
$C_{1\epsilon }$	constant
$C_{2\epsilon }$	constant
$C_m$	minor thermal capacity of the two substances in the system
$C_p$	thermal capacity at constant pressure
$C_r$	heat capacity of the fluids in the exchanger
$C_s$	heat capacity of shell material in the exchanger
$C_t$	heat capacity of tubes material in the exchanger
$C_v$	thermal capacity at constant volume
$D_e$	equivalent diameter
$D_{\mathit{it}}$	tube internal diameter
$D_{\mathit{ot}}$	tube external diameter
$e$	output location subscript
$E$	vertical distance of clearance between tangents on the surface of the tubes
$E_{\mathit{ij}}$	deformation rate components
$f$	friction factor estimated from the Reynolds
$G_s$	the mass velocity of the fluid
$h_i$	heat transfer coefficient associated with internal flow
$h_o$	heat transfer coefficient associated with external flow
$H_{\mathit{fg}}^{\prime }$	latent heat modified by thermal advection effects
$H_{\mathit{fg}}$	latent heat of transformation gas liquid
$i$	input location subscript
$k_{\mathit{tm}}$	the conductivity of the material of the tube walls
$k$	thermal conductivity
$L$	length
$L_t$	tube length
$L_s$	shell length
$\dot{m}$	mass flow
$m$	exponent
$n$	exponent
$N_b$	number of deflectors
$N_{s-s}$	number of steps over the arrangement of tubes
$N_{s-t}$	number of tube passages
$\mathit{NTU}$	number of transfer units
$\mathit{Nu}$	Nusselt number
$p$	pressure
$\mathit{Pr}$	Prandtl number
$q$	heat flux
$Q$	transferred energy
${\mathit{Re}}_s$	Reynolds number depends on the mass velocity of the fluid
${\mathit{Re}}_D$	diameter-dependent Reynolds
$R_{\mathit{ic}}$	Fouling factor
$t$	time variable
$∆T_{\mathit{ml}}$	logarithmic mean temperature difference
$T$	temperature variable
$\overline{T}$	average temperature
$T_c$	lowest temperature of the fluid
$T_h$	highest temperature of the fluid
$T_{\mathit{sat}}$	saturation temperature
$T_{\mathit{\text{wall}}}$	wall temperature
$U_g$	global heat transfer coefficient
$v$	average velocity
$x$	coordinate system direction
$y$	coordinate system direction
$y^{+}$	Y-plus is the normal distance from the solid surface’s wall to the cell’s centre
$z$	coordinate system direction
Z1	cooling of the superheated vapor zone
Z2	condensation zone
Z3	cooling below the saturation condition