Determination of the Reference Temperature for a Convective Heat Transfer Coefficient in a Heated Tube Bank

Abstract

The paper presents a theoretical analysis of heat transfer in a heated tube bank, based on the Nusselt number computation as one of the basic dimensionless criteria. To compute the Nusselt number based on the heat transfer coefficient, the reference temperature must be determined. Despite the value significance, the quantity has several different formulations, which leads to discrepancies in results. This paper investigates the heat transfer of the inline and staggered tube banks, made up of 20 rows, at a constant tube diameter and longitudinal and transverse pitch. Both laminar and turbulent flows up to Re = 10,000 are considered, and the effect of gravity is included as well. Several locations for the reference temperature are taken into consideration on the basis of the heretofore published research, and the results in terms of the overall Nusselt number are compared with those obtained by the experimental correlations. This paper provides the most suitable variant for a unique reference temperature, in terms of a constant value for all tube angles, and the Reynolds number ranges of 100–1000 and 1000–10,000 which are in good agreement with the most frequently used correlating equations.

1. Introduction

Heat exchangers consisting of a heated tube bank are widely used in practice to transfer heat from one medium to another. Typically, one fluid flows over the tube bank, while another fluid of a different temperature passes through the tubes. The tubes are arranged as a bank in an inline or staggered manner to heat the surroundings, dissipate excessive heat, or recover waste heat. Tube banks in crossflow are used in many fields, such as industry, biomedicine, mechanisms for heat transfer increase, and nanofluid applications. They can be found in the economizer and evaporator of an air conditioning system [1]. Moreover, they are parts of processes such as biological systems, filtration flow, fibrous media encountered in polymer processing, and in insulation materials [2].

Amatachaya and Krittacom [3] discussed the inline tube bank heat exchanger installed on a gas porous burner, using the air as a working fluid and LPG as a fuel. Gu and Min [4] investigated the thermal-hydraulic characteristics of the bare tube bank and plain finned tube heat exchangers used to cool the compressor bleed air in an aero turbine engine. Wang et al. [5] performed an analysis and optimization of metal-foam tube banks for waste heat recovery from the engine’s exhaust gas. Kang et al. [6] investigated the heat transfer and pressure drop of sodium-to-air heat exchanger tube banks on a sodium-cooled fast reactor. A steam reactor powered by a propane fuel consisting of a shell and tube heat exchanger was investigated by Barnoon et al. [7]. Some authors use the nanofluid to improve the heat transfer parameters [8,9,10].

To evaluate the heat transfer in a tube bank, the Nusselt number is used as one of the basic criteria. It is defined as a ratio of convective to conductive heat transfer across a boundary. In other words, the Nusselt number is a dimensionless heat transfer coefficient. The purpose of using dimensionless numbers is to compare the results of the used quantities independently. There are two well-known ways to obtain the Nusselt number. The first one is based on the heat transfer coefficient, characteristic length, and thermal conductivity; the other one uses the empirical correlations based on the Reynolds and Prandtl numbers. The latter approach is usually mentioned in literature dealing with experimental results. Some authors have created more difficult and complex formulas where other variables are used as well. They are related to the manner in which fluid properties are evaluated. The base is either the free-stream temperature, the film temperature, which is the average of the free-stream and surface temperatures, or the base is the average of the inlet and outlet temperatures, which is not easy to determine without a calculation or experiment.

Due to the lack of computational technology, the first studies of heat transfer were conducted only experimentally and reported the tube spacing effect for banks in crossflow. Grimison [11] and Pierson [12] created the correlations for various tube bank arrangements in the Reynolds number range of 2 × 10³ and 4 × 10⁴, while Huge [13] tested different tube diameters at Reynolds numbers ranging from 2 × 10³ to 7 × 10⁴. He confirmed the similarity principle applied to the tube banks, despite some departure from the true geometric similarity in the ratio of length to the diameter or to the intertube space. Brevoort and Tifford [14] detailed the flow conditions in a bank of 20 staggered circular tubes with a Reynolds numbers of over 2 × 10⁴. McAdams [15] discussed the effect of the Prandtl number and large differences in temperature-dependent properties in a wide range of the Reynolds numbers from laminar to turbulent flow. The most extensive and precise measurement of the local heat transfer coefficient around a cylinder was carried out by Schmidt and Werner [16]. Kays and Lo [17] extended the available data for a normal gas flow to the small tube diameter banks of various staggered arrangements to lower the Reynolds numbers from 10³ to 10⁴. In the paper by Welch and Fairchild [18], heat transfer coefficients are obtained for the individual rows of the ten-row inline tube banks, under various pitch ratio arrangements.

To study heat transfer rates in the crossflow over the smooth tube banks, the correlating equations obtained by Žukauskas and his co-authors are widely used. One of these investigations, by Samoshka et al. [19], was of a closely spaced staggered tube bank of large smooth tubes in water streams within 21 turbulent regions. They found out that the efficiency of the banks from an energetic point of view increased as the tube spacing decreased. Žukauskas [20] reported the pressure and hydrodynamic resistance of single tubes and tube banks of various arrangements in flows of gases and v (viscous liquids) at higher Reynolds numbers and various Prandtl numbers from 0.7 to 500. The effect of the fluid properties was considered by Žukauskas and Ulinskas [21,22] for the inline and staggered tube banks in a water crossflow at Reynolds numbers ranging from 5 × 10⁴ to 2 × 10⁶ and Prandtl numbers from 3 to 7. They determined the optimal arrangements and geometries of the tube banks in an oil crossflow at Reynolds numbers ranging from 1 to 2 × 10⁴. The former Grimison correlations were modified by Hausen [23], where the empirical formulas were created instead of a graphical representation.

Ramezanpour et al. [24] used the Ansys Fluent software and the RNG k-ε turbulence model with a modified dissipation term in the ε equation, where the Reynolds numbers (based on the maximum mean velocity inside a tube bank and the tube hydraulic diameter) of 10³, 5 × 10³, 10⁴, and 10⁵ were used. According to the results, the optimal tube spacing for a staggered tube array in crossflow was found for S_T/D = 1.5, 1.3 and S_L/D = 1.15, 1.05, respectively (Figure 1 and Figure 2). Khan et al. [25,26] presented the models for the inline and staggered arrangement of tube banks applicable over a wide range of Reynolds and Prandtl numbers, as well as the longitudinal and transverse pitch ratios. It was determined that the average heat transfer coefficient for tube banks in crossflow depends on the number of longitudinal rows, longitudinal and transverse pitch ratios, and the Reynolds and Prandtl numbers. Moreover, it was proved that the staggered arrangement gave higher heat transfer rates than the inline one. The same findings were observed by Haider et al. [27].

Kim [28] indicated that the heat transfer coefficient might be reduced by 37.1% from the prediction of a well-known correlation by Žukauskas, as the longitudinal pitch decreased. The heat transfer degradation can be estimated by using an empirical correction factor, and the Žukauskas correlation can predict existing experimental data when the correlation is combined with the empirical factor developed in the study. Yilmaz et al. [29,30] extended and developed the formulas of Gnielinski, Gaddis, and Žukauskas for the calculation of the Nusselt number in tube banks. The main contribution is a unique formula for a wide range of the Reynolds numbers, while the Žukauskas correlations are divided into the Reynolds number subintervals. Niemelä et al. [31] created the CFD model by comparing different boundary conditions, domain dimensions, and turbulence models for unsteady simulations of the inline tube banks. The correlation of Gnielinski is recommended for the tube banks with large transverse spacing as it agrees, within ±13%, with the numerically obtained values.

The former way to compute the Nusselt number based on the heat transfer coefficient is derived from the Fourier and Newton law. As a part of the formula, the reference temperature should be expressed. Depending on the solved problem, the reference temperature is taken at different locations. In a heat exchanger tube, it could be defined as an average bulk fluid temperature at the axial position, according to Córcoles et al. [32]. In some cases, it is taken far from the surface and defined as the inlet temperature. In other cases, it could be defined as a bulk temperature or film temperature. According to Beale [33], some authors, such as Žukauskas, used the temperature in a free stream at the beginning of the subdomain. In the same way, the bulk temperature of the fluid entering the tube is used in the paper by Ge et al. [34], who optimized the shape of the staggered tube bank. Alternatively, authors such as Gnielinski [35] or Bergelin [36] used the log-mean temperature difference, while Le Feuvre [37] and Massey [38] use the midway subdomain temperature. Furthermore, the mass weighted average temperature of the inlet and outlet, with subsequent averaging, was used in the investigation by Castro et al. [39], while Mangrulkar et al. [40] used the temperature of the fluid past the corresponding cylinder. Muzaffar et al. [41] used the logarithmic mean temperature difference when studying the heat transfer of a half-cycle air condition system, using liquefied petroleum gas. The same approach was used to calculate the average air-side heat transfer coefficient for a tube bundle heat exchanger with a novel fin design in the paper by Unger et al. [42]. Xu et al. [43] used a linear interpolation of the average temperature at the inlet and outlet of the channel when evaluating the mean Nusselt number of a staggered array of Kagome lattice structures. Wang et al. [44] proved that an incorrect reference temperature of the fin side surface heat transfer coefficient leads to discrepancies between the experimental and numerical results.

The manuscript points out a need for the correct determination of the reference temperature to calculate the heat transfer coefficient and Nusselt number in a heated tube bank. When considering high Reynolds numbers, it is stated that either of the former methods converge to the unique value. The problem appears at low Reynolds numbers, where no single reference temperature is assumed. In this paper, some of the mentioned locations will be tested to determine the reference temperature, and the resulting Nusselt number will be compared with those obtained by the correlating equations. As a part of the reverse investigation, several representative temperatures for a subdomain were created and compared. Finally, the most suitable reference value with the lowest discrepancy to the experimental measurements is provided in terms of a constant for all tube angles and the Reynolds number ranges of 100–1000 and 1000–10,000.

2. Numerical Investigation

The numerical analyses were performed on the inline and staggered tube banks, which are made up of 20 rows with tube diameters of D = 20 mm, and a longitudinal and transverse pitch S_L = S_T = 40 mm. The wall and inlet temperatures were constantly set to T_w = 333.15 K and T_∞ = 293.15 K, respectively, while the inlet velocities U_∞ corresponded to the Reynolds number in the range of 10²–10⁴. Due to a symmetrical character of the task, the domains shown in Figure 1 and Figure 2 are considered.

The flow is considered to be two-dimensional, steady, and incompressible. The governing equations of mass, momentum, and energy conservation are expressed in terms of the velocity vector v, density ρ, static pressure p, gravity g, dynamic viscosity μ, specific heat capacity c_p, temperature T, and thermal conductivity of fluid k_f as follows:

$$\nabla ·\overrightarrow{v}=0$$

(1)

$$\rho \overrightarrow{v}·\nabla \overrightarrow{v}=-\nabla p+\rho g+\mu \Delta \overrightarrow{v}$$

(2)

$$\rho c_p\overrightarrow{v}·\nabla T=k_f\Delta T$$

(3)

The boundary conditions in terms of the velocities U, V (x and y direction) are applied:

$$\mathrm{inlet:} U=U_{\infty }, V=0, T=T_{\infty }$$

(4)

$$\mathrm{outlet}: P=P_{atm}, T=T_{\infty }$$

(5)

$$\mathrm{tubes}: U=0, V=0, T=T_w$$

(6)

$$\mathrm{symmetry}: \frac{ \partial U}{\partial y}=0, V=0, \frac{\partial T}{\partial y}=0$$

(7)

Governing Equations (1)–(3) considering the boundary conditions (4)–(7) were solved by the finite volume method using the Ansys Fluent software. The steady pressure-based analyses with the variable air properties and incompressible ideal gas density model were performed. Regarding the variable properties, specific heat c_p (Jkg⁻¹K⁻¹), thermal conductivity k_f (Wm⁻¹K⁻¹), and dynamic viscosity µ (kgm⁻¹s⁻¹) are considered as polynomial functions of temperature according to the standard [45]:

$$c_p\left(T\right)=3.34·{10}^{-4}{T}^2-0.156T+1023.53$$

(8)

$$k_f\left(T\right)=-2.48·{10}^{-8}{T}^2+8.92·{10}^{-5}T+1.12·{10}^{-3}$$

(9)

$$\mu \left(T\right)=-3.76·{10}^{-11}{T}^2+6.95·{10}^{-8}T+1.12·{10}^{-6}$$

(10)

To perform the numerical analyses, the Direct Numerical Simulation solver was used for the laminar flow, while the turbulent one was solved by the k-ω SST model, using a low inlet turbulent intensity. The pressure-velocity coupling was handled by the coupled scheme with the pseudo transient formulation. As a spatial discretization, the QUICK schemes were used. The solution was considered to be fully converged when the residuals of continuity, x-velocity, y-velocity, energy, k, and ω parameters met the convergence criterion 10⁻⁶. A preliminary study of the mesh grid size was carried out where quadrilateral elements were used. As it is shown in

Table 1. Mesh independence test for the tube-adjacent area.

Elements in Adjacent Area	y+	Average Nu	Error (%)
960	6.153	70.012	-
1280	3.608	57.382	18.04
1600	2.178	56.582	1.39
1920	2.107	56.245	0.60
2240	1.396	56.029	0.39

, five different numbers of elements were created in the wall-adjacent area, and the average Nusselt number for the tube was evaluated. Each mesh variant has a different number and bias of elements in a normal direction to the tube wall in a distance of 1.3 mm, where a boundary layer is predicted. On the basis of this study, the last mesh variant with 14 perpendicular elements was chosen from the accuracy point of view. Furthermore, the y⁺ values are included in this study, with the worst values are shown in Table 1. The maximum element height out of the adjacent area is 0.3 mm. Considering the element quality (aspect ratio, angles, etc.), the element widths were appropriately adjusted (Figure 3).

On the basis of the Fourier and Newton law, the local heat transfer coefficient is defined as:

$$h_{\theta }=-k_w{\frac{dT}{dr}|}_{r=0}\frac{1}{T_w-T_{ref}}$$

(11)

The mean or average heat transfer coefficient over the investigated domain is given as:

$$h_{avg}=\frac{1}{A}{\int }^{ }{h}_{\theta }dA$$

(12)

The Nusselt number over the investigated domain is defined as [46]:

$$Nu=\frac{h_{avg}D}{k_f}$$

(13)

where k_w is the thermal conductivity at the tube wall and A is the tube area.

Performing our search for the reference temperature location, several variants were tested according to Figure 4, as described in

Table 2. The reference temperature characteristics.

Denotation	Description
$T_{ef}^{\left(i\right)}$	Temperature for ith row in the free stream of the subdomain inlet.
$T_{ec}^{\left(i\right)}$	Temperature for ith row in the half of the subdomain inlet.
$T_{em}^{\left(i\right)}$	Temperature for ith row on the bound of the subdomain inlet.
$T_{mf}^{\left(i\right)}$	Temperature for ith row in the free stream of the subdomain centre.
$T_{mc}^{\left(i\right)}$	Temperature for ith row in the half of the subdomain centre.
$T_w^{\left(i\right)}$	Temperature for ith row at the tube wall.
$T_{e,mass}^{\left(i\right)}$	Mass weighted average temperature for ith row at the subdomain inlet.
$T_{m,mass}^{\left(i\right)}$	Mass weighted average temperature for ith row at the subdomain centre.
$\Delta T_{ef}^{\left(i\right)}$	LMTD for ith row based on $T_{ef}$
$\Delta T_{ec}^{\left(i\right)}$	LMTD for ith row based on $T_{ec}$
$\Delta T_{em}^{\left(i\right)}$	LMTD for ith row based on $T_{em}$
$\Delta T_{e,mass}^{\left(i\right)}$	LMTD for ith row based on $T_{e,mass}$

. Except the mentioned variants, some combination trials were built, and the most suitable ones are described later.

The logarithmic mean temperature difference (LMTD) is computed based on the subdomain inlet temperature T_in, outlet temperature T_out, and wall temperature T_w as follows:

$$\Delta T=\frac{\left(T_{in}-T_w\right)-\left(T_{out}-T_w\right)}{\ln \left(\frac{T_{in}-T_w}{T_{out}-T_w}\right)}$$

(14)

To provide the reference temperature with a better agreement compared to the other variants, the mean of the selected temperature points on the subdomain bound was considered where the formula can be expressed as follows:

$$T_{ref}^{\left(i\right)}=\frac{T_{em}^{\left(i\right)}+T_{em}^{\left(i+1\right)}+T_{mf}^{\left(i\right)}+T_w^{\left(i\right)}}{4}$$

(15)

When considering the correlating equations, the Reynolds number is defined with the velocity in the minimum cross-section, while the Prandtl number is defined based on the thermal conductivity (k) location:

$$R{e}_{max}=\frac{\rho U_{max}D}{\mu }$$

(16)

$$Pr=\frac{c_p\mu }{k}$$

(17)

The following correlating equations will be compared with the computations:

$$\mathrm{Grimison}: Nu=1.13C_{1}R{e}_{max}^{m}P{r}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$

(18)

$$Ž\mathrm{ukauskas}: Nu=C_{2}R{e}_{max}^{m}P{r}^{0.36}{\left(\frac{Pr}{P{r}_w}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$$

(19)

$$\mathrm{Yilmaz}: Nu=C_{2}R{e}_{max}^{0.4}{\left[1+{\left(\frac{R{e}_{max}}{p}\right)}^m\right]}^{n}P{r}^{0.36}$$

(20)

$$\mathrm{Hausen}: Nu=C_1{C}_{2}R{e}_{max}^{m}P{r}^{0.31}$$

(21)

$$\mathrm{Khan}: Nu=C_2\left(C_{1}R{e}_{max}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}P{r}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+0.001R{e}_{max}\right)$$

(22)

$$\mathrm{ESDU}: Nu=C_{2}R{e}_{max}^{m}P{r}^{0.34}{\left(\frac{P{r}_w }{Pr}\right)}^{0.26}$$

(23)

The arrangement factor C₁ for S_T/D = S_L/D = 2 and the correction factors C₂ and m, which are dependent on Re, are noted in

Table 3. The correlation factors for the inline arrangement.

Author	C1	C2	m	n	p
Grimison [11]	0.229	-	0.632	-	-
Žukauskas [20,21,22]	-	0.52 * 0.27 ×	0.5 * 0.63 ×	-	-
Yilmaz [29,30]	-	0.9	5	0.5	290
Hausen [23]	$1-0.0605\sqrt{\frac{1000}{R{e}_{max}}}$	0.34	0.61	-	-
Khan [25,26]	0.542	1.43	-	-	-
ESDU [47]	-	0.742 Δ 0.211 ◊	0.431 Δ 0.651 ◊	-	-

* for Re = 10²–10³; ^× for Re = 10³–10⁴; ^Δ for Re = 10–3·10²; ^◊ for Re = 3·10²–2·10⁵.

and

Table 4. The correlation factors for the staggered arrangement.

Author	C1	C2	m	n	p
Grimison [11]	0.482	-	0.556	-	-
Žukauskas [20,21,22]	-	1.04 * 0.71 × 0.35 ꜝ	0.4 * 0.5 × 0.6 ꜝ	-	-
Yilmaz [29,30]	-	1.04	1.84	0.125	500
Hausen [23]	1.18	0.35	0.57	-	-
Khan [25,26]	0.567	1.61	-	-	-
ESDU [47]	-	1.309 Δ 0.273 ◊	0.36 Δ 0.635 ◊	-	-

* for Re = 0–5·10²; ^× for Re = 5·10²–10³; ꜝ for Re = 10³–2·10⁵; ^Δ for Re = 10–3·10²; ^◊ for Re = 3·10²–2·10⁵.

.

3. Results and Discussion

A dependence of the Nusselt number on the Reynolds number for the inline tube bank is shown in Figure 5, where the mentioned variants of the reference temperatures were used to compare the computed Nusselt numbers with the correlating equations of Žukauskas, Grimison, Yilmaz, Hausen, Khan, and ESDU standards. Except the variants T_em, T_ec, ΔT_em, and ΔT_ec, the other methods, shown hatched, underestimate the experimental results in the investigated area. Although the former methods meet the value order, the course slope is not in agreement with the correlating equations for a significant part of the Re range.

When considering the variant with the lowest discrepancies to any correlating equation, the temperature T_ec provides a suitable reference temperature for the Nusselt numbers that are in a good agreement with the Žukauskas correlating equation in the range of Re = 10²–10³ (Figure 6). The maximum discrepancies appear on the boundary Reynolds numbers (4.5%) due to the Žukauskas discontinuous correlations. The average and maximum discrepancies, within the correlation interval, are 0.8% and 1.9%, respectively.

For the turbulent flow over Re = 10³, the approach according to the Equation (15) was tested. As is shown in Figure 7, the results are in good agreement with the Žukauskas correlating equation, where the average and maximum discrepancies are 1.25% and 2.86%, respectively.

In the range of 0 < Re < 10², any constant reference temperature was found to meet the value order and course slope of the mentioned correlating equations, which confirms the hypothesis of the listed authors: that there is no reference temperature which is both representative and easy to compute or measure at low Reynolds numbers.

A dependence of the Nusselt number on the Reynolds number for the staggered tube bank is shown in Figure 8, in the same way as for the inline tube bank. Almost all of the investigated reference temperature variants meet the error range and curve slope of any of the correlating equations. From the accuracy point of view, the temperature T_mf is the most suitable to use where the maximum discrepancy for 10³ < Re < 10⁴ is 2.24% compared to the Žukauskas correlating equation. Due to its range discontinuity, there is no smooth course and therefore the correlation according to the ESDU is taken into consideration for 10² < Re < 10³. The lowest discrepancies for this variant are obtained when using the temperature T_{e, mass}, where the highest one is 6.23%. A detailed comparison is shown in Figure 9.

The temperature fields (Figure 10) show a thicker boundary layer for lower Reynolds numbers, which causes the diminishing of the temperature gradient on the surface and the reduction in the local and average Nusselt numbers. From the reference temperature point of view, the flow at the staggered tube bank is mixed even for higher Reynolds numbers; therefore, it is simpler to find a unique reference temperature. On the other hand, the inline tube bank has a subdomain divided into the free stream and stream between the tubes, and therefore, there is a need to compile the reference temperatures from several locations for higher Reynolds numbers.

The local heat transfer coefficients for the inline and staggered tube banks at Re = 500 and 5000 are shown in Figure 11 and Figure 12. As investigated by Žukauskas, the maximum value of the heat transfer for the inline tube bank was observed at φ = 50°, which is closed to the obtained numerical results. Considering the staggered tube bank, the heat transfer in the first row is similar to that of a single tube. In subsequent rows, an increase in heat transfer is achieved. On the other hand, the heat transfer at low Reynolds numbers is similar for all the tubes. For both the inline and staggered tube banks, the heat transfer becomes similar from the third or fourth row.

This paper deals with an approach where a constant tube wall temperature is considered. According to Žukauskas [20], when a constant heat flux is set on the tube wall, the temperature is variable and the average Nusselt number is generally higher. The difference increases with the increase in the Reynolds number.

When considering gravity, it is proved that at lower Reynolds numbers, gravity has a significant effect on the local Nusselt numbers for a single tube. The higher the Reynolds number, the lower the Nusselt number difference noticed. Since gravity is ubiquitous in real life, it was considered in the paper as well.

4. Conclusions

The reference temperature determination is required when computing the Nusselt number based on the heat transfer coefficient. For some geometries, the reference temperature is unambiguously determined. The tube bank does not have a unique location and authors use different ways to determine the temperature, whereby a comparison becomes difficult. This paper considers several temperature locations in a tube bank subdomain and compares the Nusselt numbers with the most used correlating equations. On the basis of the Žukauskas correlating equations, a unique reference temperature for both inline and staggered tube bank arrangements is determined with two Reynolds number ranges.

Concerning the inline tube bank, the temperature in the half of the subdomain inlet is the most appropriate reference temperature for the Reynolds number in the range of 10² and 10³. Over this interval, a combination of the temperature in the bound of the subdomain inlet, the temperature in the free stream of the subdomain center, and the wall temperature should be used. When the staggered tube bank is considered, the mass weighted average temperature at the subdomain inlet is recommended for Re = 10²–10³, while the temperature in the free stream of the subdomain center should be used over this interval. The mentioned approach provides more accurate numerical results that are comparable with those obtained by the experiments.