Morphing optimization of flow and heat transfer in concentric tube ... - American Institute of Physics

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Concentric tube heat exchangers are vital in various industrial applications, including chemical, process, energy, mechanical, and aeronautical engineering. Advancements in heat transfer efficiency present a significant challenge in contemporary research and development. This study concerns optimizing flow and heat transfer in concentric tube heat exchangers by morphing the tube's walls. The adjoint shape optimization approach is implemented in a fully turbulent flow regime. The effect of inner tube deformation on flow physics and heat transfer is examined. The results show that morphing can lead to a 54% increase in the heat transfer rate and a 47% improvement in the overall heat transfer coefficient compared to straight concentric tube designs. Moreover, the thermal-hydraulic performance factor is calculated to account for the relative increase in heat transfer when the optimal and initial designs are operated under the same pumping power. A thermal-hydraulic performance factor of 1.2 is obtained for the new design, showing that the heat transfer enhancement caused by morphing the tube's walls outweighs the increase in pumping power. The physics of a radial flow, resulting from an adverse pressure gradient in an annular region caused by the successive inner tube deformation, significantly augments heat transfer. This study shows morphing can lead to higher thermal efficiencies, and numerical optimization can assist in achieving this goal.

Heat exchangers (HEs) are essential for transferring heat between two fluids at different temperatures. They are used in several industrial applications, such as solar systems,^1,2 thermal energy storage,^3,4 automotive systems,^5,6 power generation,^7,8 heating, ventilation and air conditioning (HVAC),^9,10 and waste heat recovery systems.^11,12 There is a growing need to increase energy efficiency and reduce energy costs, thus leading to a more sustainable society.^13,14

The development and implementation of heat exchangers encompass energy and space utilization challenges. The goal is to optimize flow and improve the heat exchangers' thermal performance while reducing their size or increasing compactness, i.e., achieving a high heat-transfer-area-to-volume ratio. Several studies were conducted to improve heat exchangers' performance; these can be categorized into passive, active, and compound methods. Active methods require external power input, including mechanical, electric, magnetic, and pulsating force.^15,16 The passive method does not require external power and usually involves changes in equipment structure or working fluid to intensify heat transfer.^17,18 The compound method combines two or more changes, including alterations in the working medium, equipment structure, and external power. Since passive methods enhance heat transfer without additional equipment, they have become a practical and convenient choice for many applications.

Furthermore, passive methods can improve heat transfer efficiency without ongoing maintenance or monitoring. Research has been conducted on passive heat enhancement, primarily involving changes to the equipment structure or working fluid. Changes to the structure typically include the use of fin inserts,^19,20 deformation of the pipe structure,^21,22 and the use of vortex generators,^23–25 among other techniques.

Optimizing heat exchangers improves heat transfer efficiency while minimizing pressure drop, size, and weight. Several optimization techniques were proposed, and indicative examples are discussed below.

Li et al.²⁶ utilized the MOQO-Jaya algorithm to optimize the direct contact heat exchanger (DCHE), a crucial component in energy conversion systems. This study stands out for its innovative application of the MOQO-Jaya algorithm, which demonstrated superior optimization precision and efficiency compared to the NSGA-II algorithm, a more commonly used method. The researchers focused on optimizing several key parameters of the DCHE, including the inlet temperature of the thermal oil, R245fa flow rate, thermal oil flow rate, evaporator height, nozzle diameter, and number of jets. They aimed to maximize the Volumetric Heat Transfer Coefficient (VHTC) while minimizing entransy dissipation and entropy production. The results revealed a negative correlation between the heat transfer capacity and entransy dissipation and entropy production. This suggests that reducing these two factors can enhance the heat transfer performance of the DCHE. Specifically, the VHTC increased by 33.89% and entransy dissipation decreased by 44% compared to traditional experiments. Further optimization of the initial temperature of the continuous phase led to an additional 10.7% increase in the VHTC and a 12.3% reduction in entransy dissipation. Regarding algorithm efficiency, the MOQO-Jaya algorithm outperformed the NSGA-II algorithm in DCHE optimization. The maximum VHTC optimized using MOQO-Jaya was 3.4 times higher than that with NSGA-II under the same entransy dissipation and population number. Moreover, the running efficiency of MOQO-Jaya was 2.24 times that of NSGA-II under the same conditions, indicating a shorter computational time in favor of MOQO-Jaya.

Serageldin et al.²⁷ had proposed a novel spiral-double ground heat exchanger (GHX) that reduced costs, facilitated installation, and improved thermal performance. The performance of the GHX was evaluated using three-dimensional (3D), transient, and conjugated finite volume simulations. A response surface methodology and sensitivity analyses optimized its design parameters. The results showed that the proposed GHX outperformed traditional single U-tube and spiral GHXs, with higher thermal effectiveness, heat transfer rates, and lower thermal resistance.

Dagdevir²⁸ presented an investigation into the geometrical parameters of dimpled heat exchanger tubes, optimized using the Taguchi method and Gray Relational Analysis (GRA). Single-objective optimization revealed optimal configurations for the highest heat transfer coefficient, h, and the lowest pressure drop, $\Delta P$⁠. The most effective configuration for h was a dimple diameter, D_d, of 7 mm, a pitch length, PL, of 10 mm, and a dimple depth, H, of 1.4 mm. Conversely, the optimal configuration for $\Delta P$ was D_d of 3 mm, PL of 30 mm, and H of 1.0 mm. Parameter contributions to h were 10.88% (D_d), 57.78% (PL), and 23.82% (H). For $\Delta P$⁠, contributions were 8.63% (D_d), 51.80% (PL), and 31.64% (H). The pitch length, PL, was the most impactful parameter on both h and $\Delta P$⁠, with increased PL improving heat transfer but increasing the $\Delta P$ penalty due to heightened flow resistance. Multi-objective optimization highlighted the configuration with D_d of 7 mm, PL of 30 mm, and H of 1.0 mm, providing superior thermal and hydraulic performance.

Kirkar et al.²⁹ investigated corrugated surfaces on straight tubes for passive heat transfer improvement. Genetic Aggregation Response Surface Methodology and multi-objective optimization analyses using a non-dominated sorting genetic algorithm (NSGA-II) were employed to maximize heat transfer and minimize pressure drops. To account for both heat transfer and pressure drop, the thermal performance factor was used, defined as $\eta =(\text{Nu}/{\text{Nu}}_0){(f/f_0)}^{-1/3}$⁠. Here, ${\text{Nu}}_0$ and $f_0$ represent the Nusselt number and friction factor of a smooth pipe, respectively. The results indicated that corrugated surfaces are more effective in laminar flow conditions, achieving a maximum thermal performance factor of 2.48.

Patel and Rao³⁰ explored the use of particle swarm optimization (PSO) for designing shell-and-tube heat exchangers from an economic perspective, minimizing total annual cost by optimizing three design variables (internal shell diameter, outer tube diameter, and baffle spacing) and two tube layouts (triangle and square) showing the potential for PSO to optimize heat exchanger design. Finally, Bahiraei et al.³¹ developed an artificial neural network (ANN) model to predict convective heat transfer coefficients and pressure drop of a non-Newtonian nanofluid containing Cu nanoparticles in annuli. Their study suggested optimal states with maximum heat transfer and minimum pressure drop from the designer's viewpoint, considering various conditions.

To better illustrate the variety of optimization methods used in heat exchanger research, Table I provides a summary of the aforementioned studies, the techniques they employed, the type of heat exchanger they focused on, and their key findings. As illustrated in Table I, a wide range of optimization methods, including the MOQO-Jaya algorithm,²⁶ Response Surface Methodology,²⁷ and Gray Relational Analysis,²⁸ have been employed in heat exchanger research. These studies show the broad applicability of these techniques to diverse types of heat exchangers and the significant improvements they can bring to heat exchanger performance. Despite the demonstrated potential of these methods, they often face significant challenges when applied to large parametric spaces, a common scenario in heat exchanger optimization. The challenge lies in the high computational cost of exploring the large parametric spaces typical in heat exchanger design. This necessitates extensive computational fluid dynamics (CFD) simulations, leading to high computational overhead even for minor parameter variations. As such, there is a pressing need for optimization methods to navigate these high-dimensional spaces effectively while keeping computational costs under control.

TABLE I.

Summary of optimization techniques used in heat exchanger research.

Authors .	Optimization technique .	Heat exchanger type .	Major findings .
Li et al.26	MOQO-Jaya algorithm	Direct contact heat exchanger (DCHE)	The MOQO-Jaya algorithm demonstrated superior optimization precision and efficiency compared to the NSGA-II algorithm.
Serageldin et al.27	Response surface methodology and sensitivity analyses	Spiral-double ground heat exchanger (GHX)	The proposed GHX outperformed traditional single U-tube and spiral GHXs, with higher thermal effectiveness, heat transfer rates, and lower thermal resistance.
Dagdevir28	Taguchi method and gray relational analysis (GRA)	Dimpled heat exchanger tubes	Pitch length was the most impactful parameter on both heat transfer coefficient and pressure drop.
Kirkar et al.29	Genetic aggregation response surface methodology and NSGA-II	Corrugated surface straight tubes	Corrugated surfaces are more effective in laminar flow conditions.
Patel and Rao30	Particle swarm optimization (PSO)	Shell-and-tube heat exchangers	PSO showed potential for optimizing heat exchanger design from an economic perspective.
Bahiraei et al.31	Artificial neural network (ANN)	Non-Newtonian nanofluid containing Cu nanoparticles in annuli	ANN model accurately predicted convective heat transfer coefficients and pressure drop.
Ali et al.24	Discrete adjoint-based optimization (ABO)	Vortex generators in a rectangular channel	ABO obtained an optimal design at a reduced computational cost.

Authors .	Optimization technique .	Heat exchanger type .	Major findings .
Li et al.26	MOQO-Jaya algorithm	Direct contact heat exchanger (DCHE)	The MOQO-Jaya algorithm demonstrated superior optimization precision and efficiency compared to the NSGA-II algorithm.
Serageldin et al.27	Response surface methodology and sensitivity analyses	Spiral-double ground heat exchanger (GHX)	The proposed GHX outperformed traditional single U-tube and spiral GHXs, with higher thermal effectiveness, heat transfer rates, and lower thermal resistance.
Dagdevir28	Taguchi method and gray relational analysis (GRA)	Dimpled heat exchanger tubes	Pitch length was the most impactful parameter on both heat transfer coefficient and pressure drop.
Kirkar et al.29	Genetic aggregation response surface methodology and NSGA-II	Corrugated surface straight tubes	Corrugated surfaces are more effective in laminar flow conditions.
Patel and Rao30	Particle swarm optimization (PSO)	Shell-and-tube heat exchangers	PSO showed potential for optimizing heat exchanger design from an economic perspective.
Bahiraei et al.31	Artificial neural network (ANN)	Non-Newtonian nanofluid containing Cu nanoparticles in annuli	ANN model accurately predicted convective heat transfer coefficients and pressure drop.
Ali et al.24	Discrete adjoint-based optimization (ABO)	Vortex generators in a rectangular channel	ABO obtained an optimal design at a reduced computational cost.

In this context, adjoint-based optimization (ABO) emerges as a promising approach. It has already proven its worth in various fields outside of heat exchanger design. They have been employed successfully in diverse fields, such as aerodynamics, structural mechanics, and electrical engineering, thus demonstrating the method's versatility. For example, the adjoint method was used in aerodynamics to optimize aircraft wing shapes to minimize drag and maximize lift, leading to a more fuel-efficient flight.^32,33 In structural mechanics, it was used to optimize the shapes of structures for stress minimization and load maximization, contributing to safer and more efficient structures.^34,35 In electrical engineering, the adjoint method is often used in designing and optimizing electrical circuits, leading to circuits that are more power-efficient.^36,37 Adjoint-based optimization methods (ABO) can provide sensitivity gradients from the flow field solution,^38,39 which can help identify how performance would change by morphing a given geometry. More specifically, when compared to other optimization techniques presented earlier, such as the response surface method, artificial neural networks, particle swarm optimization, and genetic algorithms, the adjoint-based optimization offers several distinct advantages:

• Efficiency: Unlike other optimization methods that require a high number of iterations or sampling to obtain a satisfactory solution, adjoint methods typically require fewer calculations to converge to an optimal solution. This is because they simultaneously compute sensitivity information (gradients) for all design parameters, which can accelerate the optimization process.

• Scalability: Adjoint methods are well-suited for problems with many design variables because the computational cost of the gradient computation is essentially independent of the number of variables. This contrasts with methods like the response surface method, which require an increased number of samples, thus computational cost as the number of variables increases.

• Deterministic: Unlike stochastic methods like particle swarm optimization and genetic algorithms, adjoint methods are deterministic. This means they will produce the same result if run multiple times with the same initial conditions.

• Incorporation of constraints: Adjoint-based optimization handles constraints within the optimization process, making it easier to include realistic engineering constraints.

This broad applicability and the specific advantages of the adjoint method served as the primary motivation for its application in the context of heat exchanger design. In our work, we aim to leverage these benefits to optimize the design of a concentric tube heat exchanger (CTHE).

In a recent article, Ali et al.²⁴ used discrete Adjoint-based optimization (ABO) to determine the optimal shape for rigid vortex generators (RVG) that maximize their thermal performance in a rectangular channel under turbulent flow conditions. The adjoint morphing technique was employed to obtain three vortex generator shapes depending on the specific definition of the objective function used in the optimization process. Specifically, the third shape achieved the highest thermal performance factor of 1.28 compared to an original thermal performance factor of 1.16 for a traditional delta-winglet pair of vortex generators. This study demonstrated the efficiency of adjoint-based optimization (ABO) to obtain an optimal design at a reduced computational cost.

This study makes several contributions to heat exchanger design and optimization that distinguish our work from existing literature. First, while adjoint-based optimization (ABO) has been applied in various fields, such as aerodynamics, structural mechanics, and electrical engineering, its application to the design of a concentric tube heat exchanger (CTHE) is a novel contribution. This approach allows us to optimize the heat exchanger design efficiently, overcoming the computational challenges associated with traditional optimization methods. Second, our focus on maximizing the heat transfer rate between the hot and cold fluids in turbulent–turbulent flow regimes addresses a complex task that has not been extensively explored in previous studies. More specifically, we tackle a unique adjoint optimization problem that involves heat transfer between two fluids via a shared heat exchange wall. Our approach departs from the previous study by Ali et al.,²⁴ in which heat exchanger performance was optimized using vortex generators embedded within a channel. In that study, the channel wall was assumed to maintain a constant high temperature, thus simulating the hot side of the heat exchanger and simplifying the heat transfer dynamics by not requiring interaction with another hot fluid in motion. In contrast, our current study focuses on manipulating the heat exchange wall separating hot and cold fluids. Any changes to this wall directly influence the flow and thermal dynamics on both the hot and cold sides of the wall. This approach provides novel insights into optimizing heat exchangers by accounting for the complex interplay between structural deformation, fluid dynamics, and heat transfer. Finally, unlike many studies that enhance heat exchanger performance through specific inserts or other modifications, we achieve improved efficiency solely through pipe deformation. This approach offers a practical and cost-effective solution for enhancing heat exchanger performance. Collectively, these contributions advance the field of heat exchanger design and offer new avenues for future research and development.

The Reynolds Averaged Navier–Stokes (RANS) equations were employed to model the fluid flow in three dimensions (3D). Note other methods, such as high-resolution and high-order large eddy simulation (LES),⁴⁰ could be used in modeling and simulating the flow; however, the cost would make the application of adjoint optimization impossible due to the excessive computational cost.

The mass flow rate at the hot and cold inlets is set to produce Reynolds numbers 8000, resulting in a turbulent flow for both hot and cold fluids. The equations of continuity, momentum, and energy for an incompressible Newtonian fluid are given by

$\nabla ·\mathbf{U}=0,$

(1)

$\rho \frac{\partial \mathbf{U}}{\partial t}+\rho \nabla ·\left[\mathbf{UU}\right]=\nabla ·[(\mu +{\mu }_t)\nabla \mathbf{U}],$

(2)

$\rho C_p\frac{\partial T}{\partial t}+\rho C_p\nabla ·\left[\mathbf{U}T\right]=C_p\nabla ·\left[\right(\frac{\mu }{\text{Pr}}+\frac{{\mu }_t}{{\text{Pr}}_{\mathrm{t}}})\nabla T].$

(3)

The velocity vector is denoted by U, the temperature by T, the fluid density by ρ, the fluid's dynamic viscosity by μ, the turbulent dynamic viscosity by μ_t, the specific heat by ${\mathrm{C}}_{\mathrm{p}}$⁠, and the Prandtl and turbulent Prandtl numbers by $\text{Pr}$ and ${\text{Pr}}_{\mathrm{t}}$⁠, respectively. The hot and cold fluids are water, with similar thermophysical properties.

The $k-\omega -\text{SST}$ (shear-stress transport) model⁴¹ is employed to solve the RANS equations for turbulent conditions. The SST model uses two additional partial differential equations coupled with Eqs. (1)–(3) to provide a more accurate representation of the turbulent kinetic energy k and the turbulent dissipation rate ω. The SST model seamlessly transitions between a $k-\omega$ model and a $k-\epsilon$ model in the boundary layer and freestream regions.

The transport equations for the turbulent kinetic energy, k, and the specific dissipation rate, ω, are essential components of the $k-\omega -\text{SST}$ turbulence model. For incompressible, steady-state flows, these equations are presented as follows.

For the turbulent kinetic energy, k, we have

$\rho \nabla ·(k\mathbf{U})=\nabla ·\left[(\mu +{\sigma }_k{\mu }_t)\nabla k\right]+P_k-C_k\rho \omega k.$

(4)

In this equation, P_k is the production of turbulent kinetic energy, σ_k is the turbulent Prandtl number for k, and C_k is a model constant.

Similarly, for the specific dissipation rate, ω, we find

$\rho \nabla ·(\omega \mathbf{U})=\nabla ·\left[(\mu +{\sigma }_{\omega }{\mu }_t)\nabla \omega \right]+\frac{\alpha \omega }{k}{P}_k-C_{\omega }\rho {\omega }^2.$

(5)

In this equation, α and $C_{\omega }$ are model constants, and ${\sigma }_{\omega }$ is the turbulent Prandtl number for ω. These equations capture the advection, diffusion, production, and dissipation of k and ω. Combined with the continuity, momentum, and energy equations, they form a complete system of equations that can provide detailed insights into the behavior of turbulent flows.

The boundary conditions (Fig. 1) are set as follows:

• Hot inlet: A constant temperature $T_{h,i}=353$ K, and a mass flow rate at the inlet ${\dot{m}}_h=0.126$ kg/s.

• Cold inlet: A constant temperature $T_{c,i}=283$ K, and a mass flow rate at the inlet ${\dot{m}}_c=0.378$ kg/s.

• Inner wall: Zero thickness coupled wall, where the heat flux and temperature are conserved on both the hot and cold faces across the inner wall, i.e., $q_{\text{inner}\hspace{0.17em}\text{wall},h}^{″}=q_{\text{inner}\hspace{0.17em}\text{wall},c}^{″}$⁠, and $T_{\text{inner}\hspace{0.17em}\text{wall},h}=T_{\text{inner}\hspace{0.17em}\text{wall},c}$⁠.

• Outer wall: An adiabatic wall with zero heat flux.

• Hot and cold outlets: A zero-pressure Dirichlet boundary condition.

It is important to note that the chosen temperatures and mass flow rates are not arbitrary. They are, in fact, reflective of conditions commonly found in practical applications of water-to-water heat transfer in concentric tube heat exchangers. For example, heat exchangers transfer heat from solar collectors to a water storage tank in solar water heating systems. The hot inlet temperature of 353 K (80 °C) could represent the temperature of the solar-heated fluid. In comparison, the cold inlet temperature of 283 K (10 °C) might correspond to the initial temperature of the water in the storage tank. Moreover, drain water heat recovery (DWHR) systems recover waste heat from warm drain water to preheat a cold supply. The hot inlet temperature can be viewed as an upper estimate for the drain water temperature (from hot showers or dishwashers, for instance), and the cold inlet temperature of 283 K (10 °C) could represent the cold city water supply. In addition, the selected mass flow rates (0.126 kg/s for the hot fluid and 0.378 kg/s for the cold fluid) are designed to yield a Reynolds number of approximately 8000, ensuring turbulent flow. Turbulent flow is generally sought in heat exchanger operations due to its enhanced heat transfer efficiency.

A double precision, second-order upwind scheme is utilized to methodically solve the flow equations, specifically for spatial discretization of the convective terms.⁴² For the diffusion terms, a central-difference technique with second-order precision is chosen. The COUPLED algorithm and the pseudo transient option are implemented for managing pressure–velocity coupling.

The pseudo transient option is a form of implicit under-relaxation for steady-state scenarios. This approach accelerates the solution acquisition and enhances the solutions' robustness, providing a more reliable outcome in steady-state cases. We solve the governing equations iteratively until the convergence criteria are met. These criteria are defined by the point at which the scaled residuals for momentum, continuity, turbulent kinetic energy (k), and specific dissipation rate (ω) reach values less than ${10}^{-5}$⁠. A more stringent convergence criterion of ${10}^{-7}$ is employed for the energy equation to ensure accurate convergence.

Three polyhedral meshes were chosen and refined at the walls using carefully selected inflation layers to ensure a $y^{+}\le 1$ throughout the investigation, as shown in Fig. 2. The Grid Convergence Method, described by Celik et al.,⁴³ was adopted to provide a quantitative measure of numerical uncertainty in CFD simulations due to discretization errors.

FIG. 2.

Cross-sectional view accompanied by related magnified streamwise view illustrating the three meshes used in the mesh study.

FIG. 2.

Cross-sectional view accompanied by related magnified streamwise view illustrating the three meshes used in the mesh study.

Close modal

The sensitivity analysis was conducted for hot and cold fluid streams at a Reynolds 8000. The three grids were refined systematically, following the recommendation of Celik et al.⁴³ of using a refinement ratio $r\ge 1.3$⁠. This approach led to a final fine mesh 3, composed of approximately 1.7 × 10⁶ polyhedral cells, as displayed in Table II.

TABLE II.

Mesh characteristics used for the sensitivity study.

.	Mesh 1 .	Mesh 2 .	Mesh 3 .
Number of cells (N)	279 497	687 234	1 700 000
Grid size $s={\left[\frac{1}{N}\sum _{i=1}^{i=N}(\Delta V_i)\right]}^{\frac{1}{3}}$	0.009 652	0.007 151	0.005 288
Refinement factor $r=s_i/s_j$	⋯	$s_1/s_2=1.35$	$s_2/s_3=1.35$

.	Mesh 1 .	Mesh 2 .	Mesh 3 .
Number of cells (N)	279 497	687 234	1 700 000
Grid size $s={\left[\frac{1}{N}\sum _{i=1}^{i=N}(\Delta V_i)\right]}^{\frac{1}{3}}$	0.009 652	0.007 151	0.005 288
Refinement factor $r=s_i/s_j$	⋯	$s_1/s_2=1.35$	$s_2/s_3=1.35$

Table III presents the calculation of the variables ${\varphi }_1,\hspace{0.17em}{\varphi }_2$⁠, and ${\varphi }_3$⁠. These variables are calculated based on the heat transfer rate q and the total pumping power ${\dot{P}}_{\mathit{\text{total}}}$ for the hot and cold streams, as observed in meshes 1, 2, and 3. The total pumping power is evaluated using the following equation:

${\dot{P}}_{\mathit{\text{total}}}=\frac{{\dot{m}}_h}{{\rho }_h}\Delta p_h+\frac{{\dot{m}}_c}{{\rho }_c}\Delta p_c.$

(6)

TABLE III.

Calculation of discretization error for the mesh study.

.	${\varphi }_1$ .	${\varphi }_2$ .	${\varphi }_3$ .	${\varphi }_{\text{ext}}$ .	$e_a^{23}$ (%) .	pc .	${\text{GCI}}_{\text{fine}}$ (%) .
Heat transfer q (W)	8764.88	7721.68	7682.94	7681.45	0.50	10.91	0.024
Pumping power ${\dot{P}}_{\text{total}}$ (W)	0.1609	0.1460	0.1442	0.1440	1.25	7.00	0.215

.	${\varphi }_1$ .	${\varphi }_2$ .	${\varphi }_3$ .	${\varphi }_{\text{ext}}$ .	$e_a^{23}$ (%) .	pc .	${\text{GCI}}_{\text{fine}}$ (%) .
Heat transfer q (W)	8764.88	7721.68	7682.94	7681.45	0.50	10.91	0.024
Pumping power ${\dot{P}}_{\text{total}}$ (W)	0.1609	0.1460	0.1442	0.1440	1.25	7.00	0.215

Here, $\Delta p_h$ and $\Delta p_c$ represent the pressure drops across the hot and cold streams. In Table III, ${\varphi }_{\mathit{\text{ext}}}$ denotes the extrapolated variable, which is calculated from the variable values on meshes 2 and 3, denoted as ${\varphi }_2$ and ${\varphi }_3$⁠, respectively, the refinement factor r, and the apparent order of convergence p_c. The equation for ${\varphi }_{\mathit{\text{ext}}}$ is given by

${\varphi }_{\mathit{\text{ext}}}=\frac{(r_{23}^{p_c}{\varphi }_3-{\varphi }_2)}{(r_{23}^{p_c}-1)}.$

(7)

The relative error $e_a^{23}$ between mesh 2 and mesh 3 is calculated using the absolute difference between ${\varphi }_3$ and ${\varphi }_2$⁠, divided by ${\varphi }_3$ as follows:

$e_a^{23}=\frac{|{\varphi }_3-{\varphi }_2|}{{\varphi }_3}.$

(8)

The apparent order of convergence p_c is solved using fixed-point iteration on the following three equations:

$p_c=\frac{1}{\hspace{0.17em}\text{ln}\hspace{0.17em}(r_{23})}|\hspace{0.17em}\text{ln}\hspace{0.17em}|{\epsilon }_{12}/{\epsilon }_{23}|+g(p_c)|,$

(9)

$g(p_c)=\text{ln}\hspace{0.17em}\left(\frac{r_{23}^{p_c}-n}{r_{12}^{p_c}-n}\right),$

(10)

$n=\text{sgn}({\epsilon }_{12}/{\epsilon }_{23}),$

(11)

where ${\epsilon }_{12}={\varphi }_1-{\varphi }_2$ and ${\epsilon }_{23}={\varphi }_2-{\varphi }_3$⁠. Finally, the Grid Convergence Index (GCI) measures the solution's uncertainty due to grid discretization. For fine grid solutions, the GCI, denoted as ${\text{GCI}}_{\text{fine}}$⁠, is calculated as

${\text{GCI}}_{\text{fine}}=\frac{1.25e_a^{23}}{r_{23}^{p_c}-1}.$

(12)

This method allows for a rigorous grid convergence study that estimates the discretization error in the numerical solution and ensures that the solution is independent of the grid. For a more comprehensive understanding of these calculations and their implications, the reader is referred to the work of Celik et al.⁴³ 

As demonstrated in Table III, the uncertainty in the fine grid solution is 0.024% for the heat transfer rate q and 0.215% for the total pumping power ${\dot{P}}_{\mathit{\text{total}}}$⁠. Furthermore, the order of convergence p_c is remarkably high for both variables, indicating rapid convergence, particularly for a straightforward CFD case involving an empty double-pipe heat exchanger. The relative error percentage between meshes 2 and 3 is $e_a^{23}=0.5\%$ for the heat transfer rate q and $e_a^{23}=1.25\%$ for the total pumping power ${\dot{P}}_{\mathit{\text{total}}}$⁠. Based on these results, mesh 3 can be employed as a reference mesh for all studies conducted in this article.

To validate the numerical approach, the simulation results at $\text{Re}=8000$ for both the hot and cold sides are compared based on the heat transfer rate q and total pumping power ${\dot{P}}_{\mathit{\text{total}}}$ with empirical correlations found in the existing literature. To calculate the heat transfer rate q, the $\epsilon -\text{NTU}$ method is used for a counter flow CTHE as follows:

$\epsilon =\frac{1-\text{exp}[-\text{NTU}(1-C_r)]}{1-C_r\hspace{0.17em}\text{exp}[-\text{NTU}(1-C_r)]},$

(13)

where ε is the effectiveness of the heat exchanger, defined as the ratio of the actual heat transfer rate q to the maximum possible heat transfer rate $q_{\text{max}}$⁠. NTU, or the Number of Transfer Units, is given by $UA/C_{\text{min}}$⁠, where UA is the product of the overall heat transfer coefficient U and the heat exchanger surface area A, and $C_{\text{min}}$ is the minimum heat capacity rate between the hot and cold fluids. C_r, the heat capacity rate ratio, is defined as the ratio of the minimum to the maximum heat capacity rate, i.e., $C_{\text{min}}/C_{\text{max}}$⁠. The overall heat transfer coefficient U is evaluated as

$U=\left(\frac{1}{h_h}+\frac{1}{h_c}\right).$

(14)

The thermal resistance due to conduction is assumed to be negligible in the simulations since the heat transfer area is usually made of high thermal conductivity materials. To evaluate the convection heat transfer coefficients, the empirical correlation proposed by Petukhov and Kirilov⁴⁴ is adopted,

$\begin{array}{c}\text{Nu}=\frac{h{D}_h}{\lambda }=\frac{(f/8)·\text{Re}·\text{Pr}}{1.07+900/\text{Re}-0.36/(1+10\text{Pr})+12.7·{(f/8)}^{1/2}·({\text{Pr}}^{2/3}-1)},\hfill \end{array}$

(15)

where D_h is the hydraulic diameter and is equal to the inner diameter D_i for the hot water flow inside the inner tube and the difference $(D_o-D_i)$ for the cold water flow inside the annulus. The Darcy friction factor f is related to the Reynolds number and is obtained from the following Blasius equation for turbulent flow inside a smooth pipe,⁴⁵ 

$f\hspace{0.17em}\approx \hspace{0.17em}0.316{\text{Re}}^{-1/4}.$

(16)

Moreover, the friction factor in Eq. (16) is used to empirically estimate the pressure drop on the hot and cold sides as follows:

$\Delta p=\frac{1}{2}\rho V_{\mathit{\text{inlet}}}^{2}f\frac{L}{D_h},$

(17)

where V_inlet represents the fluid velocity at the inlet.

To determine the empirical heat transfer rate q, the hot and cold convection coefficients h_h and h_c are evaluated using the Nusselt number correlation of Eq. (15). Once the convection coefficients are obtained, the overall heat transfer coefficient U can be calculated using Eq. (14). Finally, the effectiveness ε of the heat exchanger can be assessed using Eq. (13), which is employed to find the heat transfer rate as $q=\epsilon ·q_{\mathit{\text{max}}}$⁠. However, to determine the empirical pumping power, the pressure drop for both the hot and cold fluids must first be estimated using Eq. (17), which is based on the friction factor correlation provided by Eq. (16). The pressure drop is then utilized to determine the empirical total pumping power according to Eq. (6). Table IV compares the results for the total heat transfer rate q and the total pumping power ${\dot{P}}_{\mathit{\text{total}}}$ between the empirical correlations and the CFD simulations. The error percentage is acceptable, with a 2.33% discrepancy for the heat transfer rate and an approximate 6% deviation for the total pumping power.

TABLE IV.

Comparison of the total heat transfer rate q and total pumping power ${\dot{P}}_{\mathit{\text{total}}}$ is made between the numerical simulations using mesh 3 and the empirical correlations found in the literature.

.	Correlations .	CFD .	Error (%) .
Heat transfer q (W)	7866.02	7682.94	2.33
Pumping power ${\dot{P}}_{\mathit{\text{total}}}$ (W)	0.1361	0.1442	5.98

.	Correlations .	CFD .	Error (%) .
Heat transfer q (W)	7866.02	7682.94	2.33
Pumping power ${\dot{P}}_{\mathit{\text{total}}}$ (W)	0.1361	0.1442	5.98

We have employed the optimization process in the framework of ANSYS Fluent⁴⁶ (version 2022 R1), which involves a series of steps to determine the optimum solution. A general overview of the procedure is shown in Fig. 3. The discrete adjoint optimization approach maximizes the heat transfer rate q between the hot and cold fluids. The process begins by minimizing a cost function F and maximizing the heat transfer rate q. The computational domain is discretized on the mesh, and the boundary conditions are specified. A flow simulation using the RANS equations is then performed to determine the flow properties and temperature fields.

FIG. 3.

The Computational Algorithm of the Adjoint and CFD Solvers with Mesh Morphing technique. The optimization convergence criterion is based on the steepest descent gradient-based optimization sub-algorithm.

FIG. 3.

The Computational Algorithm of the Adjoint and CFD Solvers with Mesh Morphing technique. The optimization convergence criterion is based on the steepest descent gradient-based optimization sub-algorithm.

Close modal

Once the flow simulation is fully converged, an adjoint simulation is performed to calculate the sensitivity of the cost function F concerning changes in the geometry of the inner wall. The sensitivity information is used to guide an iterative optimization process, where the geometry of the inner wall is updated to minimize the cost function or, in other words, to improve the heat transfer rate q.

The morphing tool modifies the boundary and interior mesh during each iteration based on the calculated sensitivity field to produce a new geometry. The flow simulation is then repeated, followed by another adjoint simulation until the optimum solution is reached, where the heat transfer rate q is maximized. The optimization loop continues until the heat transfer rate q no longer improves with further modifications to the inner wall. The final solution represents the optimized geometry that maximizes the heat transfer rate q.

Adjoint optimization methods²⁴ have an advantage of larger space design exploration, unlike other restricted methods such as parametric optimization techniques in CFD.⁴⁷ Adjoint optimization can lead to more optimal engineering designs at a reduced computational cost. The discrete approach of adjoint-based methods is more accurate in computing the sensitivity field, particularly in turbulent flow conditions, than the continuous adjoint method.^48,49

The mathematical problem of the ABO is solved by minimizing the cost function F where a minimum value of F represents a maximum heat transfer rate q between the hot and cold fluid as follows:

$F=-q=-q_{\text{inner}\hspace{0.17em}\text{wall}}^{″}\times \text{HTA},$

(18)

where $q_{\text{inner}\hspace{0.17em}\text{wall}}^{″}$ is the heat flux evaluated at the inner wall, and $\text{HTA}$ is the heat transfer area, which is equivalent to the inner wall area in this case.

The goal of the adjoint optimization is to compute the shape sensitivity of the heat transfer area, which is the inner wall, concerning an objective function F. In this process, c is introduced as the Cartesian coordinates (x, y, and z) for every cell or node in the computational domain, which is the adjoint control volume encompassing the inner wall surface, as shown in Fig. 1. The objective function F not only depends on the design parameters c but also on the flow solution vector β,

$\begin{array}{c}\beta (c)=\left[u_x,u_y,u_z,p,T\right],\\ F=F(\beta (c),c).\end{array}$

(19)

The equations representing heat and mass transfer can be expressed in a simpler form

$R(\beta (c),c)=0,$

(20)

where R is the residual vector/matrix of the fluid flow solution. With any modification in the flow state $\beta (c)$ and the control variables c, an observable variation in the chosen optimization objective also occurs

$\delta F=\frac{\partial F}{\partial \beta }\delta \beta +\frac{\partial F}{\partial c}\delta c.$

(21)

The equation's first term pertains to the change in a flow state, which the steady-state RANS equations control, in reaction to alterations in geometry or other boundary conditions. The adjoint method offers a method for substituting this term with an expression solely dependent on c. This substitution is realized by leveraging the linearization of the equations defining heat and mass transfer. The optimization must function within specified flow field boundaries: flow and design variables. Therefore, when the flow field changes, the residual variable $(\delta R)$ must remain at zero,

$\delta R=\frac{\partial R}{\partial \beta }\delta \beta +\frac{\partial R}{\partial c}\delta c=0.$

(22)

Following this, it is essential to determine the Lagrange Multipliers to convert the constrained optimization problem into an unconstrained one,

$L(f,c,\lambda )=F(\beta ,c)-{\lambda }_m^{T}R(\beta ,c).$

(23)

In Eq. (23), the parameter λ_m is identified as the vector of adjoint variables

$\lambda =\left[u_x^{*},u_y^{*},u_z^{*},p^{*},T^{*}\right].$

(24)

Merging Eqs. (21) and (23) yields

$\begin{array}{c}\delta F=\frac{\partial F^T}{\partial \beta }\delta \beta +\frac{\partial F^T}{\partial c}\delta c-{\lambda }_m^T\left(\frac{\partial R}{\partial \beta }\delta \beta +\frac{\partial R}{\partial c}\delta c\right)\hfill \\ =\left(\frac{\partial F^T}{\partial \beta }-{\lambda }_m^T\frac{\partial R}{\partial \beta }\right)\delta \beta +\left(\frac{\partial F^T}{\partial c}-{\lambda }_m^T\frac{\partial R}{\partial c}\delta c\right)\delta c.\hfill \end{array}$

(25)

The value of λ_m is set to eliminate the impact of flow variables. This gives rise to the adjoint equation to be solved,

$\frac{\partial F^T}{\partial c}={\lambda }_m^T\frac{\partial R}{\partial c}\delta c.$

(26)

By substituting Eq. (26) into (25), the dependency between the cost function and design variables can be attained

$\delta F=\left(\frac{\partial F^T}{\partial \beta }-{\lambda }_m^T\frac{\partial R}{\partial \beta }\right)\delta \beta .$

(27)

In the next step, the direct interpolation mesh morphing algorithm modifies the inner wall surface inward or outward to minimize the objective function F, and the steepest descent algorithm ensures the optimization problem converges toward a local minimum of F. The direct interpolation method in ANSYS Fluent projects the deformations from the boundary into the interior of the mesh. The displacement of the interior mesh nodes is calculated as a weighted average of all boundary node displacements as follows:

$\Delta x_v^i=\frac{\sum _{j=0}^{N_b}{w}_j(r_{ij})\Delta x_b^j}{\sum _{j=0}^{N_b}{w}_j(r_{ij})},$

(28)

where $\Delta x_v^i$ and $\Delta x_b^j$ are the interior and boundary node displacements, respectively, w_j is the weighting function, and r_ij is the distance between the ith and jth nodes. This approach allows for a smooth deformation of both the boundary and the interior mesh while satisfying the design conditions. Subsequently, by altering the cell nodes, the new shape of the geometric model is obtained. The flow analysis is then repeated. Once the assumed value of the objective function is reached, the optimization process is completed.

The computational domain is generated by polyhedral cells using a local mesh refinement technique and volume control in the inner wall design space zone. The evolution of the mesh, relative to the design iterations (DI), is depicted in Fig. 4. The mesh is carefully examined and re-meshed during each design iteration whenever its quality deteriorates. This maintains a high-quality mesh, emphasizing a minimal dimensionless wall distance, y⁺, in both the inner tube and outer annulus flow. By doing so, we ensure that y⁺ remains consistently low throughout the iterative process of mesh morphing.

FIG. 4.

Cross-sectional view at the streamwise position of $z/L=0.49$⁠, together with a magnified streamwise view showcasing the evolution of the computational mesh across different design iterations (DI).

FIG. 4.

Cross-sectional view at the streamwise position of $z/L=0.49$⁠, together with a magnified streamwise view showcasing the evolution of the computational mesh across different design iterations (DI).

Close modal

The effect of inner tube deformation on heat transfer performance and its relation to flow structure were examined. Figure 5 presents the evolution of the inner wall deformation regarding adjoint design iterations. The last iteration (DI-11) exhibits a wavy structure where the heat transfer rate can no longer increase with further deformation. Consequently, DI-11 is considered the optimal converged case, demonstrating the maximum heat transfer rate compared to the initial straight non-deformed tube case, DI-0. It has been observed that the amplitude of deformation increases with adjoint iterations, causing the cross-sectional area of the inner wall to successively expand and contract along the heat exchanger (HE) length.

FIG. 5.

Inner wall morphing after 11 adjoint design iterations (DI-0 to 11). The figures to the right are a zoomed-in view of a region located at z = 1 m measured from the hot inlet.

FIG. 5.

Inner wall morphing after 11 adjoint design iterations (DI-0 to 11). The figures to the right are a zoomed-in view of a region located at z = 1 m measured from the hot inlet.

Close modal

The variation of streamwise dimensionless heat rate $q_z/q_0$ for DI-11 is shown in Fig. 6 for the dimensionless position z/L taken between 0.45 and 0.5, where q₀ corresponds to the total heat transfer rate of the non-deformed initial design DI-0. The behavior of the dimensionless heat rate is oscillatory. It is influenced by the converging–diverging repeating pattern of the inner tube, where the heat rate q_z is reaching values as high as 2.5 times the average heat rate q₀ for the non-deformed case DI-0. The minimum oscillatory value is approximately close to 1, which shows a performance close to DI-0.

FIG. 6.

Streamwise variation of the heat rate q_z normalized for the undeformed inner tube total heat rate q₀.

FIG. 6.

Streamwise variation of the heat rate q_z normalized for the undeformed inner tube total heat rate q₀.

Close modal

To further investigate the reasons behind the maximum and minimum streamwise heat ratios and relate them to important flow structures, we take several positions highlighted in blue on Fig. 6 and show cross-sectional contours of the velocity and temperature fields in Fig. 7 at the associated dimensionless positions z/L. The velocity magnitudes displayed here are based on the velocity field's x and y components, which are tangential to the shown cross-sectional plane.

FIG. 7.

Cross-sectional velocity and temperature fields at different dimensionless positions. Velocity vectors are also shown in black on the velocity field contour plots.

FIG. 7.

Cross-sectional velocity and temperature fields at different dimensionless positions. Velocity vectors are also shown in black on the velocity field contour plots.

Close modal

At $z/L=0.452$⁠, the contraction of the inner tube creates two low-pressure regions in the annular region, causing the fluid to displace from the upper part of the annular region to the adjacent right and left parts, creating a counter-rotating vortex pair that is also detected in the streamlines in Fig. 8 at $z/L=0.452$ and $z/L=0.459$⁠. Examining the inner tube flow, the expansion of the tube from $z/L=0.452$ to $z/L=0.459$ generates low-pressure zones in the inner tube region, which are responsible for the rotating vortices inside the inner tube, as seen in the red streamlines at $z/L=0.459$⁠. An important key feature is that the flow diverges radially in the annular region when the inner tube contracts at $z/L=0.463$⁠, shown in the radially oriented velocity vectors in Fig. 7. This radial flow, caused by the strong adverse pressure gradient in the annular region, greatly enhances heat transfer. The heated region near the inner tube wall mixes with the cold annular flow, as evident in the related temperature contours. The resulting enhanced heat transfer is observed with a peak in the normalized heat transfer rate $q_z/q_0$ at $z/L=0.463$ in Fig. 6. This peak is repeated at $z/L=0.478$⁠, where the radial flow dominates at this position, as seen in the related velocity vectors. It is worth noting that the strong radial flow is present after each contraction of the inner tube following an expansion.

FIG. 8.

Streamlines and pressure contour plots for the optimal case DI-11 at different dimensionless positions. The red contours for the pressure indicate regions of maximum pressure, while the blue contours indicate regions of minimum pressure. The min/max pressure contour scale is adjusted locally at every position to identify red and blue zones.

FIG. 8.

Streamlines and pressure contour plots for the optimal case DI-11 at different dimensionless positions. The red contours for the pressure indicate regions of maximum pressure, while the blue contours indicate regions of minimum pressure. The min/max pressure contour scale is adjusted locally at every position to identify red and blue zones.

Close modal

Figure 9 presents cross-sectional contour plots of the dot product between the velocity vector U and the temperature gradient $\mathbf{\nabla }T$⁠. The analysis of the resulting scalar field is based on the idea that the synergy between the velocity and temperature fields can be used to determine areas of high heat transfer. By analyzing the field synergies, it is possible to identify regions where the flow enhances the temperature gradients, and the heat transfer rate is maximum.⁵⁰ To further increase heat transfer performance, the field synergy principle implies that the temperature gradients and velocity fields are more closely aligned and work together to transfer heat. The radial flow is aligned with the temperature gradient and is responsible for higher heat transfer rates. This is observed especially at $z/L=0.463$ and 0.478 in Fig. 9, where high synergy is present near the inner wall.

FIG. 9.

Contours of the dot product between the velocity vector field and the temperature gradient for the optimal case DI-11 at different dimensionless axial positions. The blue velocity vectors correspond to the annular/cold water flow; the red vectors correspond to the inner/hot water tube flow.

FIG. 9.

Contours of the dot product between the velocity vector field and the temperature gradient for the optimal case DI-11 at different dimensionless axial positions. The blue velocity vectors correspond to the annular/cold water flow; the red vectors correspond to the inner/hot water tube flow.

Close modal

A quantitative analysis of the heat transfer enhancement in the optimized geometry is presented and compared with the initial design. Figure 10 displays $r-\theta$ plots of the tangential Nusselt number variation, ${\text{Nu}}_{\theta }$⁠, for both the inner and annular flows at different axial positions. The global average Nusselt number normalizes the tangential Nusselt number, ${\text{Nu}}_0$⁠, corresponding to the initial design DI-0.

FIG. 10.

$r-\theta$ plots of the tangential Nusselt number ${\text{Nu}}_{\theta }$ for the optimal case DI-11 normalized by the global average Nusselt number ${\text{Nu}}_0$ for the undeformed initial case (DI-0).

FIG. 10.

$r-\theta$ plots of the tangential Nusselt number ${\text{Nu}}_{\theta }$ for the optimal case DI-11 normalized by the global average Nusselt number ${\text{Nu}}_0$ for the undeformed initial case (DI-0).

Close modal

The results at different positions reveal that the ${\text{Nu}}_{\theta }/{\text{Nu}}_0$ ratio is greater than or equal to one, indicating superior convection coefficients for hot and cold water flows. Starting from $z/L=0.452$⁠, the ratio of ${\text{Nu}}_{\theta }/{\text{Nu}}_0$ is greater than one at angles that span from $0°$ to $180°$⁠, covering the location of the rotating vortex pairs observed in the streamlines of the annular and inner tube regions in Fig. 8. Furthermore, the ratio becomes three at $z/L=0.463$⁠, 0.467, 0.478, and 0.484, indicating a threefold enhancement of convection coefficients compared to the non-deformed case.

The radially dominant flow significantly influences the ratio of ${\text{Nu}}_{\theta }/{\text{Nu}}_0$ at $z/L=0.463$ and $z/L=0.478$⁠. An important observation from the $r-\theta$ plots is that the shape of the variation of ${\text{Nu}}_{\theta }/{\text{Nu}}_0$ precisely follows the observed flow patterns. For instance, when comparing positions $z/L=0.452$ to $z/L=0.472$⁠, the ratio of ${\text{Nu}}_{\theta }/{\text{Nu}}_0$ is higher than one in the upper part of the annular region at $z/L=0.452$ and shifts to higher values in the lower part of the annular region at $z/L=0.472$⁠. The same trend is observed when comparing positions $z/L=0.475$ to $z/L=0.49$⁠. This behavior can be explained by examining Fig. 9, where the velocity vectors are more aligned with the temperature gradient in the upper part of the annular flow at $z/L=0.452$ and more aligned with the lower part of the annular flow at $z/L=0.472$⁠. The same applies to positions $z/L=0.475$ and 0.490.

To evaluate the thermal performance of the optimization process globally, the ratios of $q/q_0,\hspace{0.17em}U/U_0$⁠, and $\text{HTA}/{\text{HTA}}_0$ are plotted in Fig. 11(a) concerning the adjoint design iterations. The results show that the adjoint algorithm can advance the thermal performance of the CTHE by enhancing the heat transfer rate between hot and cold water. The ratio $q/q_0$ achieves a value of 1.54 for the converged final case DI-11, representing a 54% increase in the heat transfer rate compared to the initial non-deformed concentric tube DI-0. Moreover, Fig. 11(b) shows the evolution of the hot and cold water outlet temperatures concerning the adjoint optimization process. The optimized CTHE geometry significantly improved over the initial non-deformed design (DI-0) for hot and cold water outlet temperatures. There is a 22.5 K drop in the hot water temperature between the inlet and the outlet for the optimal design compared to a temperature drop of 14.6 K for the initial undeformed design (DI-0).

FIG. 11.

This figure shows the variation of (a) the ratios of heat rate (⁠ $q/q_0$⁠), U-value (⁠ $U/U_0$⁠), and heat transfer area (⁠ ${\text{HTA}/\text{HTA}}_0$⁠) for adjoint design iterations, (b) the outlet temperatures of hot and cold water, (c) the effectiveness ε, and (d) the thermal-hydraulic performance factor η. The plotted data illustrate how these quantities evolve throughout the adjoint design optimization process.

FIG. 11.

This figure shows the variation of (a) the ratios of heat rate (⁠ $q/q_0$⁠), U-value (⁠ $U/U_0$⁠), and heat transfer area (⁠ ${\text{HTA}/\text{HTA}}_0$⁠) for adjoint design iterations, (b) the outlet temperatures of hot and cold water, (c) the effectiveness ε, and (d) the thermal-hydraulic performance factor η. The plotted data illustrate how these quantities evolve throughout the adjoint design optimization process.

Close modal

Furthermore, the cold water temperature can be increased by 7.5 K for the optimal geometry compared to a 4.9 K temperature increase for the non-deformed geometry. These findings demonstrate the potential of the optimized CTHE for applications such as preheating and precooling as well as waste heat recovery. The optimized design has the potential to enhance the performance of thermal systems, reduce costs, and lower greenhouse gas emissions.

Analysis of the ratios of overall heat transfer coefficient $U/U_0$ and the heat transfer area ${\text{HTA}/\text{HTA}}_0$ reveals that the increase in the heat transfer rate ratio $q/q_0$ is mainly due to the reduction of thermal resistances by convection between the hot and cold water and is slightly influenced by the increase in the heat transfer area. The ratio of heat transfer area $\text{HTA}/{\text{HTA}}_0$ reaches a value of 1.15 at the end of design iterations, representing a 15% increase for the heat transfer area of the initial CTHE geometry. In contrast, the overall heat transfer coefficient $U/U_0$ ratio reaches a value of 1.47, representing an increase in 47%. The geometry morphing leads to significant flow rearrangement, particularly vortical structures, which greatly reduces the thermal resistances and improves the overall heat transfer coefficient with a small increase in the heat transfer area at the end of the design iterations.

The variation of the effectiveness ε concerning the adjoint design iterations is shown in Fig. 11(c). An increase in the effectiveness throughout the optimization iterations is obtained, ranging from a value of 0.21 at DI-0 to a maximum value of 0.32 at the final design iteration DI-11. The optimized case DI-11 can transfer heat at a rate q that is 32% of the maximum heat transfer rate $q_{\text{max}}$ compared to 21% for the initial case DI-0. This improvement in the heat exchanger's effectiveness highlights the adjoint algorithm's efficacy in providing morphing solutions that enhance the heat exchanger's thermal performance.

Finally, we calculate the overall thermal-hydraulic performance factor η, as described in Ref. 51. This factor accounts for the ratio of heat transfer in the enhanced geometry q_e to the heat transfer in the reference, undeformed geometry (denoted as “0”) at equal pumping powers ${\dot{P}}_{\text{total}}$⁠. In simpler terms, this factor measures the efficacy of the heat transfer performance in an enhanced heat exchanger compared to a reference one, with both operating under the same pumping power conditions. The thermal-hydraulic performance factor can be expressed as follows:

$\eta ={\frac{q_e}{q_0}|}_{{\dot{P}}_{\text{total}}}.$

(29)

To achieve the same pumping power in the undeformed case as in the deformed case at any design iteration DI, the required Reynolds number ${\text{Re}}_0$ can be determined by simultaneously solving the following two equations for both the hot and cold sides of the heat exchanger:

$f_0·{\text{Re}}_0^3=f_e·{\text{Re}}_e^3,$

(30)

$f_0\approx 0.316\hspace{0.17em}{\text{Re}}_0^{-1/4}.$

(31)

Once ${\text{Re}}_0$ is found, we can calculate the heat transfer rate q₀ using numerical simulations on the undeformed geometry at the corresponding Reynolds number ${\text{Re}}_0$⁠. Finally, the thermal-hydraulic performance factor η can be determined using Eq. (29). As depicted in Fig. 11(d), the variation of the thermal-hydraulic performance factor η concerning design iterations (DI) shows that η attains values greater than 1 for each DI, reaching a maximum of $\eta \approx 1.2$ at the final optimal case DI-11. This illustrates that the benefits of heat transfer enhancement during the morphing process outweigh the associated pressure drop penalty.

In this study, we investigated the effect of inner tube deformation on heat transfer performance and its relation to flow structure in a CTHE. The results presented in this paper show that the adjoint algorithm can significantly enhance the thermal performance of the CTHE by improving the heat transfer rate between hot and cold water. The final converged design iteration (DI-11) achieved a 54% increase in the heat transfer rate compared to the initial non-deformed concentric tube (DI-0), and the overall heat transfer coefficient was improved by 47%. Moreover, the final converged design exhibits a thermal-hydraulic factor of 1.2, reflecting a 20% enhancement in heat transfer performance when compared with the initial undeformed design (DI-0) under identical pumping power conditions.

The results of our study revealed that the amplitude of deformation of the inner tube increased with adjoint iterations, causing the cross-sectional area of the inner wall to expand and contract along the CTHE length successively. The wavy structure observed in DI-11 could no longer increase the heat transfer rate with further deformation, making it the optimal converged case.

Our analysis also showed that the radial flow, caused by the strong adverse pressure gradient in the annular region, greatly enhanced heat transfer. The radial flow was present after each contraction of the inner tube directly following an expansion. Cross-sectional contour plots of the dot product between the velocity vector and the temperature gradient were used to determine high heat transfer areas. High synergy was present near the inner wall.

The tangential Nusselt number was used to present a quantitative analysis of the heat transfer enhancement in the optimized geometry and compare it with the initial design. The results showed that the ${\text{Nu}}_{\theta }/{\text{Nu}}_0$ ratio was greater than or equal to one, indicating superior convection coefficients for both the hot and cold water flows. The ratio reached three at several axial positions, indicating a threefold enhancement of convection coefficients compared to the non-deformed case. It is worth noting that the different flow structures responsible for this enhancement, including the counter-rotating vortices, are created solely with the inner surface deformation and without vortex generators, as it is traditionally used in the literature by many researchers.

Optimizing CTHE using the adjoint method has significant potential for improving the efficiency of waste heat recovery systems that involve water-to-water heat transfer. Waste heat recovery applications have gained increasing importance recently due to the need to reduce energy consumption and greenhouse gas emissions. Optimized heat exchangers can help recover waste heat, reduce energy consumption, and increase process efficiency. The results presented in this study demonstrate that the adjoint method can be used to design highly efficient heat exchangers that significantly increase the heat transfer rate and overall heat transfer coefficient. The optimized design can also reduce the thermal resistances between the hot and cold fluids, leading to increased heat exchanger effectiveness. Therefore, optimizing heat exchangers can be critical in achieving more sustainable and energy-efficient industrial processes.

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