Ahmed H. Ahmed*
| Maki H. Zaidan | Manar S.M. Al-Jethelah
© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
The present study considers two-dimensional laminar incompressible flow over a longitudinally finned flat tube bank heat exchanger (LFFTBHE) based on the constructal design method and the Fuzzy logic control. The impact of Reynolds number (Re) and dimensionless upstream fin length (Lu/DT) on the LFFTBHE performance was studied. The finite volume method using a commercial CFD package (ANSYS 2022/R2) was used to solve the governing equations. Fuzzy logic control was used to optimize the results. The results showed that the upstream fin length insignificantly impacted the Nusselt and Bejan numbers. Increasing the dimensionless upstream fin length, i.e., from 0.4 to 1, decreased the Nusselt number up to 10.3% at Re=1200. However, the dimensionless upstream fin length increase increased the Bejan number up to 28.8% at Re=1200.
constructal design, laminar forced convection, longitudinally finned flat tubes bank heat exchanger, upstream fin length
A heat exchanger's performance can be evaluated by estimating the largest amount of energy transmitted and the least amount of power expended. The performance of the air side can be improved by changing the design of the fin or tube, or in some cases both due to the resistance to heat transfer from the air side is dominant. This resistance results from the low specific heat of the air and the high heat transfer coefficient on the water or the internal fluid side [1-4]. It is necessary to control the tube shape, vertical distance, horizontal distance, fin height, and the distance between the fins to reach the optimal design. All the above variables directly impact on thermal conductance or the largest value of the heat transfer coefficient, as well as the area density and the consequent cost, size, and weight, considering the metal type. The above considerations were adopted in studying forced and natural convection heat transfer to analyze the performance of tube bank heat exchangers (TBHE) [5-9]. Previous research in the literature suggests that the tube's form and design can improve heat exchanger design criteria, including the exterior tube heat transfer coefficient and the external flow pressure drop [10-16]. The width of the flow channel is not equal to the convective body that forms on the channel walls as a result of boundary layers because the dimensions of the flow body and the channel are different. A different strategy is to equalize the size of the flow body and flow channel. This methodology's cornerstone is the constructal design technique, based on the constructal law.
The optimal flow channel size is determined by the ease with which heat can be transferred from the walls to the air streams [17, 18]. The constructal design procedure for rhombic tubes in crossflow and three dimensionless pressure differences (Be=103, 104, and 105) was investigated. It was reported that at any dimensionless pressure difference, the maximal heat transfer density was higher for narrower rhombic tubes [19]. To enhance the density of heat transfer, a vertical array of flat tubes in cross-flow with a fixed pressure drop was investigated. The maximum heat transfer densities of naturally chilled tubes with varying cross-sectional areas were reported [20, 21]. Other studies were investigated the maximum heat transfer densities from multiscale tubes and fins in forced and natural convection [22-26]. The thermal fluid properties in two dimensions over two staggered flat tubes and the thermal fluid characteristics across a tube series between two parallel plates were studied. As was already indicated, studies suggested a proportional relationship between the Re and Nu [27-32]. A semicircular segment of a tube placed in a crossflow at different angles was numerically and experimentally investigated. the studied Re ranged from 2200 and 45000. The greatest Nu number was at the arc's surface, which was normal to the flow direction (zero angle of attack) [33]. The airflow turbulence in an elliptic tube bundle has been studied using experimental and computational fluid dynamics models. The Reynolds number range examined was $5600 \leq R e \leq 40000$ . Tests were conducted with an air angle of attack ranging from 0° to150°, and the four-axis ratios of the tube were 0.25, 0.33, 0.5, and 1. The lowest thermal performance was at 90°, and the highest was at 0° [34]. Heat transfer and pressure drop through finned flat plate banks was investigated for Pr=0.71 and $124 \leq R e \leq 1238$ . As Re increased from 124 to 1238, Nu increased from 113.6% to 150.3%, and the friction factor decreased 57.7% [35]. A numerical investigation studied the impact of Re and the fin pitch on the thermal and hydraulic performance of finned-tube heat exchanger. Increasing Re from 750 to 2350 increased the heat transfer rate by 84% at fin pitch of 20 mm. At Re=2350, increasing the fin pitch from 7.5 mm to 20 mm increased the heat rate by 100%. It was found that the j factor decreased by 34% when Re increased from 750 to 2350 at fin pitch of 20mm. At Re=2350, the j factor increased 12% as the fin pitch increased from 7.5 mm to 20 mm [36]. The impact of a wavy fin on the thermal behavior of a heat exchanger was numerically studied [37]. It was found that the heat transfer coefficient decreased by 23% at fin pitch of 1.2mm and 11.5% at 2.0 mm.
Heat transfer and crossflow air in a staggered configuration of flat tube heat exchangers were experimentally studied [38]. The air attack angles and tube aspect ratios effect were investigated. The analysis covers a range of air velocities from 2 to 6 m/s and includes 19 heat exchanger cases with six attack angles from 0 to 150° with a 30° step and four aspect ratios: 0.18, 0.39, 0.66, and 1. The results showed that the pressure drop and thermal performance were controlled by the tube aspect ratios and attack angles. The Neuro-fuzzy inference system model was employed to predict the heat transfer coefficient and pressure drop values [39]. Three longitudinal pitches (3, 4, and 6) and transverse pitches from 1.5 to 4.5 with a step of 1.0 were investigated for Reynolds numbers ranged from 10 to 320. The average discrepancy between the numerical and ANFIS model values for the average Nu was 1.9%, and the dimensionless pressure drop was 2.97%.
The previous investigations conducted numerical and experimental studies on identically shaped tubes, i.e., cylindrical and elliptic, for inline and staggered flattened tube configurations. In the studies mentioned above, the constructal design for LFFTBHE was not considered. Also, the present study introduces a fuzzy logic control of LFFTBHE in crossflow. The present work studied the impact of upstream length at $200 \leq R e \leq 1200$ on Nu and Be numbers. The studied fluid was air, Pr=0.7108. The tube was maintained at a constant surface temperature. Four dimensionless upstream fin lengths (Lu/DT) (0.4, 0.6, 0.8, and 1) were considered. The objective function of the present investigation is to maximize heat transfer rate, i.e., Nu number, and minimize the pressure drop, i.e., Be number.
The air-side physical model of LFFTBHE, shown in Figure 1, was considered to study its heat transfer performance. Usually, this kind of heat exchanger has a large number of LFFT. The following assumptions were applied: a steady state, laminar flow, constant surface temperature, incompressible fluid, no viscous dissipation, and constant thermophysical properties of the working fluid. The airflow was heated by convection heat transfer caused by the horizontal x-y plane. The two-dimensional governing equations that describe the flow and heat transfer fields are given below [40, 41]:
Continuity equation:
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ （1）
Momentums equations in the x- and y- directions:
$\rho\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)$ (2)
$\rho\left(u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+\mu\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}\right)$ (3)
Energy equation:
$\left(u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}\right)=\alpha\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}\right)$ (4)
Figure 1. Rows of LFFTBHE in crossflow
Eqs. (1) to (4) can be non-dimensionalized using the following dimensionless forms:
$X, Y=\frac{x, y}{D t}, U, V=\frac{u, v}{(\Delta p / \rho)^{\frac{1}{2}}}, P=\frac{p}{p_{\text {in }}-p_{\text {out }}}$,$\theta=\frac{T-T_o}{T_w-T_o}, P r=\frac{\mu C p}{K}$ (5)
Now, with the above dimensionless variables in Eq. (5), Eqs. (1)-(4) become:
$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (6)
$U \frac{\partial U}{\partial X}+V \frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\sqrt{\frac{P r}{B e}}\left(\frac{\partial^2 U}{\partial X^2}+\frac{\partial^2 U}{\partial Y^2}\right)$ (7)
$U \frac{\partial V}{\partial X}+V \frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial X}+\sqrt{\frac{P r}{B e}}\left(\frac{\partial^2 V}{\partial X^2}+\frac{\partial^2 V}{\partial Y^2}\right)$ (8)
$U \frac{\partial \theta}{\partial X}+V \frac{\partial \theta}{\partial Y}=\frac{1}{\sqrt{(\operatorname{BePr})}}\left(\frac{\partial^2 \theta}{\partial X^2}+\frac{\partial^2 \theta}{\partial Y^2}\right)$ (9)
The key dimensionless numbers, i.e., Re, Be, and Nu, are defined as:
$\mathrm{Re}=\frac{\rho \mathrm{VD}}{\mu}, B e=\frac{\Delta p D T^2}{\mu \alpha}, \mathrm{Nu}=\frac{\mathrm{h} \mathrm{D}_h}{k}$ (10)
The average Nu can be defined as:
$\mathrm{Nu}=\frac{1}{s} \int-\left.\frac{\partial \theta}{\partial n}\right|_s d s$ (11)
Other dimensionless parameters are normalized by the transverse tube diameter (DT), as the upstream fin length=(Lu/DT), transverse pitch ratio=(ST/DT), the longitudinal pitch=(SL/DT), downstream fin length (Ld/DT), dimensionless fin angle $\theta ds=\theta \pi / 180$.
Figure 2 displays the computing domain and the concept of staggered tube banks. The applied boundary conditions are as follows:
A: $\quad \frac{\partial U}{\partial Y}=0, V=0, \frac{\partial \theta}{\partial Y}=0$ (12)
B: $\quad U=1, \frac{\partial V}{\partial X}=0, \theta=0$ (13)
C: $\quad \frac{\partial U}{\partial X}=\frac{\partial V}{\partial X}=0, \frac{\partial \theta}{\partial X}=0$ (14)
D: $\quad U=V=0, \theta=1$ (15)
Figure 2. Boundary conditions around and on LFFTBHE in crossflow
The present problem was numerically solved using a control-volume-based finite volume approach (Fluent-CFD) software solver was used to solve the governing equations under the assumed boundary conditions. To interpolate the convection fluxes, upwind differencing scheme, hybrid scheme, and central differencing Scheme were employed. SIMPLE algorithm was used in pressure-velocity coupling. The maximum residue permitted for convergence check was 108. Figure 3 presents the model flowchart.
Figure 3. Model flowchart
The entire studied computational domain was discretized with more than 45000 unstructured mesh elements. Quadratic element order was selected. The solver has a size of 0.01mm for the entire computational domain and a smooth inflation transition near tube and fin surfaces.
Code validation is an essential part of any numerical investigation [42, 43]. The validation of the computational code (ANSYS FLUENT 2022/R2) for test problems and predictions was performed using the developed code with exact solutions and experimental data, as well as the standard issues available in previous studies. The numerical predictions established in this study were compared with those in the literature under similar tube conditions. The Re numbers of $10 \leq R e \leq 100$ were adopted to predict the average Nu number. Comparison with previous experimental results was performed to validate the present numerical results obtained through the finite volume codes for the flat shape in the staggered arrangement. The completion of the code validation was based on the overall heat transfer rate. Figure 4 presents a comparison between the present simulation results and [44]. The deviation increased with Re. The maximum deviation values were approximately 2.62% and 2.48% for Re 10 and 100, respectively. These deviations appeared due to ignoring the dimensionless temperature at the inlet and outlet of heat exchanger modules, as well as the influence of the changes in thermal properties with temperature and the approximate thermal properties.
Figure 4. Comparison between the present study with [44] for average Nusselt Number with Reynolds number
Since the 1960s, fuzzy logic has been utilized more and more in control research. The extension of well-known Boolean logic methods serves as the basis for the fuzzy logic [45]. Analyzing the mathematical model dynamics of a physical process using fuzzy logic and arbitrary order has been extensively studied in the last few decades [46]. Applications of fuzzy logic aid in solving issues about measuring uncertainty in mathematical disease modeling. A model with a triangle membership function based on a specific population was calculated, and the fuzzy reproduction number and fuzzy equilibrium points were found. The primary LFFTBHE parameters are suggested to be controlled by the fuzzy logic in this work. First, the system's mathematical model was used to calculate the pressure drop, Bejan Number, Nusselt Number, and heat transfer rate. For simple and complicated systems, the fuzzy logic serves as monitor and processing tools. The membership functions were used to translate the steps of fuzzification and defuzzification. The input and output values were first transformed from crisp to fuzzy, i.e., a process known as fuzzification, and then from Fuzzy to crisp once again to finish this translation, i.e., a process known as defuzzification.
Table 1. The input/ output ranges of data
|
I/P |
Min |
Max |
O/P |
Min. |
Max. |
|
$\widehat{L u}$ |
0.4 |
1 |
$B e$ |
4220281.79 |
118042599.50 |
|
$R e$ |
200 |
1200 |
$N u$ |
7.22 |
15.65 |
|
|
|
|
$P_{d s}$ |
8.04 |
9.29 |
|
|
|
$\hat{Q}$ | 134.04 |
271.28 |
(a)
(b)
(c)
Figure 5. Fuzzy model and member ships
A summary of the input/output data is provided in Table 1. Figure 5(a) illustrates how the input parameter (fuzzification) directly affects the output (defuzzification). A particular range of data for each input is anticipated during the mathematical module, which has already been completed in the present work. The data type represented in the Fuzzification process, which turns numerical values into linguistic variables, is the membership function (MF), which is 3D. For every input, three preset values represent the overall levels of any range, i.e., low, medium, and high. The transition form used in this work is of the Gaussian type, which is close to the actual input and output parameters shown in Figure 5(b)-(c).
Because it illustrates the time shift between the two levels of the variables, the overlap between the MFs is crucial. The y-intercept is set at zero. However, the y-delta is low, mid, and high. There is a nonlinear transition between the parameter values. For a nonlinear operation closest to reality, the MF is fixed at three for each input variable and of Gaussian type [47]. The primary processor for the present work is the Mamdani inference system. If the parameters in the LFFTBHE have a certain value, which results in a particular value for the answers. Since responses are the primary regulated parameter in this project, the rules coded into the inference system should consider the ideal level of outputs.
The numerical solutions for laminar forced convection heat transfer across LFFTBHE, based on fuzzy logic control and the constructal design method, are presented in this section. The optimum of the staggered finned flat tubes at a fixed volume was considered concerning the upstream fin length variable, and the other parameters were constant. The dimensions of the fixed volume are presented in Figure 6.
Figure 6. Geometric parameters of LFFTBHE arrangements
Figure 7 shows the upstream fin length effect on the temperature and velocity contours $0.4 \leq L u / D T \leq 1$ with $S L / D T=4, S T / D T=3, L d / D T=0.8, \theta=0.53$, and $200 \leq R e \leq 1200$. The flow separation occurred when the fluid collides with the upstream fin length. The fluid angle begins to expand relatively with the upstream fin length. The fluid moves away from the front of the LFFTBHE. In other words, the adjacent layer grows continuously along the front interface, even at a high Reynolds number. A distinct region is established between the body and the fluid stream when a fluid detaches from a body. The segregated region, a low-pressure zone located behind the body, is the site of recirculation and backflows. The pressure drag is directly affected by the size of the separation zone. A bigger separated region results in a higher pressure drag. The downstream effects of the flow separation are characterized by a decrease in velocity in comparison to the upstream velocity. The wake is the term used to describe the trailing region of flow, where the body's influence on velocity is readily apparent. The region separated is concluded upon the reconnection of the two flow streams.
Figure 7. Temperature and velocity contours for $0.4 \leq \frac{L u}{D T} \leq 1$ with $\frac{S L}{D T}=4, \frac{S T}{D T}=3, \frac{L d}{D T}=0.8, \theta=0.53$, and $200 \leq R e \leq 1200$
Figure 8. Nusselt number against dimensionless upstream fin length for various Reynolds numbers
Figure 9. Reynolds number and dimensionless upstream fin length against Nusselt number
The divided area is therefore an enclosed volume. Simultaneously, the wake keeps growing behind the body until the wake region fluid returns to its original velocity, and the velocity profile almost flattens out once more. Viscous and rotational effects play a crucial role in the boundary layer, the separated region, and the wake [48, 49].
Figure 10. Bejan number against dimensionless upstream fin length for various Reynolds numbers
Figure 11. Reynolds number and dimensionless upstream fin length against Bejan number
Figures 8 and 9 show the impact of the upstream fin length on the Nusselt number for various Reynolds numbers. The Nusselt is inversely proportional to the upstream fin length. At Re=200, Nu was 7.7 and 7.2 at 0.4 and 1 dimensionless upstream fin length. At Re=1200, Nu was 15.6 and 14.0 at 0.4 and 1 dimensionless upstream fin length. The main reason for this phenomenon is the separation of the fluid in the first stages, which prevents it from colliding with the upstream fin. The adjacent layer grows due to the absence of the effect of fluid momentum, which prevents it from sweeping the adjoining layer. This process is accompanied by an increase in thermal insulation between the surface of the pipe and the fluid. As a result, the heat transfer between the fluid and the surface decreases. This result is consistent with references [33, 50]. Nu increased 101% with Re from 200 to 1200 at 0.4 dimensionless upstream fin length. While Nu 92.8% with Re from 200 to 1200 at 1 dimensionless upstream fin length. The increase in Nu value with Re is due to increasing the flow mixing. In other words, the dimensionless heat transfer coefficient increases with mass flow rate, which agrees with [51]. Figures 10 and 11 show the impact of the dimensionless upstream fin length on the Bejan number for various Reynolds numbers. Be increased from 0.45×107 to 0.56×107 as the dimensionless upstream fin length increased from 0.4 to 1 at Re=200. Be increased from 9.24×107 to 11.9×107 as the dimensionless upstream fin length increased from 0.4 to 1 at Re=1200. Referring to the velocity contour and observing the up-stream fin length fluid flow lines thoroughly explains the turbulence process or not depending on the up-stream fin length. In addition, the fluid colliding with the front of the fin almost completely cancels the momentum resulting from the collision, or rather, it disappears relatively as the upstream fin length increases. The fluid tends to fill the space between the tubes, so friction begins to decrease relatively, accompanied by a clear decrease in the pressure drop. This process is accompanied by an increase in the speed of the fluid displaced between the rows of the heat exchanger to collide with the upstream fins. Therefore, the pressure drop gained can be considered a result of this procedure.
The findings of the present study in terms of increasing the heat transfer and reduce the pressure drop could be valuable for heat exchangers in several applications, such as automobiles and generators.
The construction and validation of a two-dimensional numerical model of a LFFTBHE have been completed successfully utilizing the constructal design approach. The results obtained showed the behavior of the heat transfer density as a function various dimensionless upstream fin length for 0.4≤Lu/DT≤1 and 200≤Re≤1200. The following conclusions have been attained as an outcome of the present study.
The authors thank the Faculty of Mechanical Engineering at Tikrit University and Northern Technical University, Technical Institute, Renewable Energy Research Unit, Hawija, Iraq.
|
Be |
Bejan number |
|
Dh |
hydraulic diameter, m |
|
DT |
transverse diameter, m |
|
DL |
longitudinal diameter, m |
|
H |
domain height, m |
|
h |
convection heat transfer coefficient |
|
k |
thermal conductivity, W/m K |
|
Re |
Reynolds number |
|
Nu |
Nusselt number |
|
P |
dimensionless pressure, Pa |
|
p |
pressure, Pa |
|
Pr |
Prandtl number |
|
SL |
longitudinal pitch, mm |
|
ST |
transverse pitch, mm |
|
Ld |
downstream fin length |
|
Lu |
upstream fin length |
|
T |
temperature, °C |
|
u, v |
velocities in x and y- directions, m/s |
|
U, V |
dimensionless velocities in X and Y direction |
|
x, y |
cartesian coordinates, m |
|
X, Y |
dimensionless cartesian coordinates |
|
W |
width |
|
Greek symbols |
|
|
$\alpha$ |
thermal diffusivity, m2/s |
|
$\varphi$ |
angle between the fins |
|
$\theta$ |
dimensionless temperature |
|
$\mu$ |
dynamic viscosity, Kg/ms |
|
$\rho$ |
density, Kg/m3 |
|
$\infty$ |
ambient condition |
|
Subscripts |
|
|
n |
normal |
|
s |
surface |
|
Abbreviation |
|
|
CFD |
computational fluid dynamic |
|
ANFIS |
adaptive neuro-fuzzy model |
|
MF |
membership function |
|
LFFTB |
longitudinally finned flat tube bank |
|
LFFTBHE |
longitudinally finned flat tube bank heat exchanger |
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