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 thermalhydraulic 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 thermalhydraulic 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.
I. INTRODUCTION
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 heattransferareatovolume 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 MOQOJaya 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 MOQOJaya algorithm, which demonstrated superior optimization precision and efficiency compared to the NSGAII 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 MOQOJaya algorithm outperformed the NSGAII algorithm in DCHE optimization. The maximum VHTC optimized using MOQOJaya was 3.4 times higher than that with NSGAII under the same entransy dissipation and population number. Moreover, the running efficiency of MOQOJaya was 2.24 times that of NSGAII under the same conditions, indicating a shorter computational time in favor of MOQOJaya.
Serageldin et al.^{27} had proposed a novel spiraldouble ground heat exchanger (GHX) that reduced costs, facilitated installation, and improved thermal performance. The performance of the GHX was evaluated using threedimensional (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 Utube and spiral GHXs, with higher thermal effectiveness, heat transfer rates, and lower thermal resistance.
Dagdevir^{28} presented an investigation into the geometrical parameters of dimpled heat exchanger tubes, optimized using the Taguchi method and Gray Relational Analysis (GRA). Singleobjective 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. Multiobjective 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 multiobjective optimization analyses using a nondominated sorting genetic algorithm (NSGAII) 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 = ( Nu / Nu 0 ) ( f / f 0 ) \u2212 1 / 3$. Here, $ 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 Rao^{30} explored the use of particle swarm optimization (PSO) for designing shellandtube 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 nonNewtonian 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 MOQOJaya 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 highdimensional spaces effectively while keeping computational costs under control.
Authors .  Optimization technique .  Heat exchanger type .  Major findings . 

Li et al.^{26}  MOQOJaya algorithm  Direct contact heat exchanger (DCHE)  The MOQOJaya algorithm demonstrated superior optimization precision and efficiency compared to the NSGAII algorithm. 
Serageldin et al.^{27}  Response surface methodology and sensitivity analyses  Spiraldouble ground heat exchanger (GHX)  The proposed GHX outperformed traditional single Utube and spiral GHXs, with higher thermal effectiveness, heat transfer rates, and lower thermal resistance. 
Dagdevir^{28}  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 NSGAII  Corrugated surface straight tubes  Corrugated surfaces are more effective in laminar flow conditions. 
Patel and Rao^{30}  Particle swarm optimization (PSO)  Shellandtube heat exchangers  PSO showed potential for optimizing heat exchanger design from an economic perspective. 
Bahiraei et al.^{31}  Artificial neural network (ANN)  NonNewtonian nanofluid containing Cu nanoparticles in annuli  ANN model accurately predicted convective heat transfer coefficients and pressure drop. 
Ali et al.^{24}  Discrete adjointbased 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}  MOQOJaya algorithm  Direct contact heat exchanger (DCHE)  The MOQOJaya algorithm demonstrated superior optimization precision and efficiency compared to the NSGAII algorithm. 
Serageldin et al.^{27}  Response surface methodology and sensitivity analyses  Spiraldouble ground heat exchanger (GHX)  The proposed GHX outperformed traditional single Utube and spiral GHXs, with higher thermal effectiveness, heat transfer rates, and lower thermal resistance. 
Dagdevir^{28}  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 NSGAII  Corrugated surface straight tubes  Corrugated surfaces are more effective in laminar flow conditions. 
Patel and Rao^{30}  Particle swarm optimization (PSO)  Shellandtube heat exchangers  PSO showed potential for optimizing heat exchanger design from an economic perspective. 
Bahiraei et al.^{31}  Artificial neural network (ANN)  NonNewtonian nanofluid containing Cu nanoparticles in annuli  ANN model accurately predicted convective heat transfer coefficients and pressure drop. 
Ali et al.^{24}  Discrete adjointbased optimization (ABO)  Vortex generators in a rectangular channel  ABO obtained an optimal design at a reduced computational cost. 
In this context, adjointbased 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 fuelefficient 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 powerefficient.^{36,37} Adjointbased 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 adjointbased 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 wellsuited 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: Adjointbased 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 Adjointbased 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 deltawinglet pair of vortex generators. This study demonstrated the efficiency of adjointbased 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 adjointbased 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 costeffective 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.
II. METHODOLOGY
The computational methodology, including governing equations, ABO, and numerical implementation details, is presented in the present section (Sec. II).
A. Computational domain
The computational domain consists of a CTHE with a countercurrent flow arrangement (Fig. 1). Hot water flows inward through the inner tube (D_{i} = 20 mm), while cold water flows outward through the outer annulus (D_{o} = 40 mm). The heat exchanger's length is L = 2000 mm. The adjoint control volume (CV) (Fig. 1) consists of a $ 40 \xd7 40 \xd7 1800 \u2009 mm 3$ rectangular paralleled, located at a distance of $ \u2113 = 100$ mm measured from both the cold and hot inlets, thus enveloping the majority of the inner wall surface to be morphed by the adjoint algorithm. Moreover, a restriction is imposed on the deformation of the inner wall such that the latter stays symmetric concerning the yz plane.
B. Governing equations and boundary conditions
The Reynolds Averaged Navier–Stokes (RANS) equations were employed to model the fluid flow in three dimensions (3D). Note other methods, such as highresolution and highorder 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 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 $ C p$, and the Prandtl and turbulent Prandtl numbers by $ Pr$ and $ Pr t$, respectively. The hot and cold fluids are water, with similar thermophysical properties.
The $ k \u2212 \omega \u2212 SST$ (shearstress transport) model^{41} 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 \u2212 \omega $ model and a $ k \u2212 \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 \u2212 \omega \u2212 SST$ turbulence model. For incompressible, steadystate flows, these equations are presented as follows.
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.
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 $ m \u0307 h = 0.126$ kg/s.

Cold inlet: A constant temperature $ T c , i = 283$ K, and a mass flow rate at the inlet $ m \u0307 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 inner \u2009 wall , h \u2033 = q inner \u2009 wall , c \u2033$, and $ T inner \u2009 wall , h = T inner \u2009 wall , c$.

Outer wall: An adiabatic wall with zero heat flux.

Hot and cold outlets: A zeropressure 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 watertowater 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 solarheated 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, secondorder upwind scheme is utilized to methodically solve the flow equations, specifically for spatial discretization of the convective terms.^{42} For the diffusion terms, a centraldifference technique with secondorder 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 underrelaxation for steadystate scenarios. This approach accelerates the solution acquisition and enhances the solutions' robustness, providing a more reliable outcome in steadystate 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 \u2212 5$. A more stringent convergence criterion of $ 10 \u2212 7$ is employed for the energy equation to ensure accurate convergence.
C. Mesh sensitivity and numerical validation
Three polyhedral meshes were chosen and refined at the walls using carefully selected inflation layers to ensure a $ y + \u2264 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.
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 \u2265 1.3$. This approach led to a final fine mesh 3, composed of approximately 1.7 × 10^{6} polyhedral cells, as displayed in Table II.
.  Mesh 1 .  Mesh 2 .  Mesh 3 . 

Number of cells (N)  279 497  687 234  1 700 000 
Grid size $ s = [ 1 N \u2211 i = 1 i = N ( \Delta V i ) ] 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 = [ 1 N \u2211 i = 1 i = N ( \Delta V i ) ] 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$ 
.  $ \varphi 1$ .  $ \varphi 2$ .  $ \varphi 3$ .  $ \varphi ext$ .  $ e a 23$ (%) .  p_{c} .  $ GCI fine$ (%) . 

Heat transfer q (W)  8764.88  7721.68  7682.94  7681.45  0.50  10.91  0.024 
Pumping power $ P \u0307 total$ (W)  0.1609  0.1460  0.1442  0.1440  1.25  7.00  0.215 
.  $ \varphi 1$ .  $ \varphi 2$ .  $ \varphi 3$ .  $ \varphi ext$ .  $ e a 23$ (%) .  p_{c} .  $ GCI fine$ (%) . 

Heat transfer q (W)  8764.88  7721.68  7682.94  7681.45  0.50  10.91  0.024 
Pumping power $ P \u0307 total$ (W)  0.1609  0.1460  0.1442  0.1440  1.25  7.00  0.215 
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 $ P \u0307 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 doublepipe 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 $ P \u0307 total$. Based on these results, mesh 3 can be employed as a reference mesh for all studies conducted in this article.
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 \xb7 q 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 $ P \u0307 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.
D. Discrete adjointbased optimization in CFD
We have employed the optimization process in the framework of ANSYS Fluent^{46} (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.
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 methods^{24} 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 adjointbased methods is more accurate in computing the sensitivity field, particularly in turbulent flow conditions, than the continuous adjoint method.^{48,49}
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 remeshed during each design iteration whenever its quality deteriorates. This maintains a highquality 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.
III. RESULTS AND DISCUSSION
A. Adjoint design evolution and flow patterns
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 (DI11) exhibits a wavy structure where the heat transfer rate can no longer increase with further deformation. Consequently, DI11 is considered the optimal converged case, demonstrating the maximum heat transfer rate compared to the initial straight nondeformed tube case, DI0. It has been observed that the amplitude of deformation increases with adjoint iterations, causing the crosssectional area of the inner wall to successively expand and contract along the heat exchanger (HE) length.
The variation of streamwise dimensionless heat rate $ q z / q 0$ for DI11 is shown in Fig. 6 for the dimensionless position z/L taken between 0.45 and 0.5, where q_{0} corresponds to the total heat transfer rate of the nondeformed initial design DI0. 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 q_{0} for the nondeformed case DI0. The minimum oscillatory value is approximately close to 1, which shows a performance close to DI0.
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 crosssectional 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 crosssectional plane.
At $ z / L = 0.452$, the contraction of the inner tube creates two lowpressure 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 counterrotating 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 lowpressure 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.
Figure 9 presents crosssectional contour plots of the dot product between the velocity vector U and the temperature gradient $ \u2207 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.
B. Quantitative analysis
A quantitative analysis of the heat transfer enhancement in the optimized geometry is presented and compared with the initial design. Figure 10 displays $ r \u2212 \theta $ plots of the tangential Nusselt number variation, $ Nu \theta $, for both the inner and annular flows at different axial positions. The global average Nusselt number normalizes the tangential Nusselt number, $ Nu 0$, corresponding to the initial design DI0.
The results at different positions reveal that the $ Nu \theta / 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 $ Nu \theta / Nu 0$ is greater than one at angles that span from $ 0 \xb0$ to $ 180 \xb0$, 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 nondeformed case.
The radially dominant flow significantly influences the ratio of $ Nu \theta / Nu 0$ at $ z / L = 0.463$ and $ z / L = 0.478$. An important observation from the $ r \u2212 \theta $ plots is that the shape of the variation of $ Nu \theta / 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 $ Nu \theta / 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 , \u2009 U / U 0$, and $ HTA / 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 DI11, representing a 54% increase in the heat transfer rate compared to the initial nondeformed concentric tube DI0. 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 nondeformed design (DI0) 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 (DI0).
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 nondeformed 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 $ HTA / 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 $ HTA / 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 DI0 to a maximum value of 0.32 at the final design iteration DI11. The optimized case DI11 can transfer heat at a rate q that is 32% of the maximum heat transfer rate $ q max$ compared to 21% for the initial case DI0. 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.
Once $ Re 0$ is found, we can calculate the heat transfer rate q_{0} using numerical simulations on the undeformed geometry at the corresponding Reynolds number $ Re 0$. Finally, the thermalhydraulic performance factor η can be determined using Eq. (29). As depicted in Fig. 11(d), the variation of the thermalhydraulic performance factor η concerning design iterations (DI) shows that η attains values greater than 1 for each DI, reaching a maximum of $ \eta \u2248 1.2$ at the final optimal case DI11. This illustrates that the benefits of heat transfer enhancement during the morphing process outweigh the associated pressure drop penalty.
IV. CONCLUSION
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 (DI11) achieved a 54% increase in the heat transfer rate compared to the initial nondeformed concentric tube (DI0), and the overall heat transfer coefficient was improved by 47%. Moreover, the final converged design exhibits a thermalhydraulic factor of 1.2, reflecting a 20% enhancement in heat transfer performance when compared with the initial undeformed design (DI0) 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 crosssectional area of the inner wall to expand and contract along the CTHE length successively. The wavy structure observed in DI11 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. Crosssectional 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 $ Nu \theta / 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 nondeformed case. It is worth noting that the different flow structures responsible for this enhancement, including the counterrotating 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 watertowater 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 energyefficient industrial processes.
ACKNOWLEDGMENTS
The authors would like to thank the EditorinChief and Physics of Fluids staff for their assistance during the peerreview and publication of the manuscript.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Samer Ali: Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (lead); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Talib Dbouk: Conceptualization (lead); Formal analysis (equal); Investigation (equal); Methodology (lead); Project administration (lead); Software (supporting); Supervision (lead); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal). Mahmoud Khaled: Formal analysis (equal); Investigation (equal); Methodology (supporting); Software (supporting); Supervision (supporting); Writing – original draft (equal); Writing – review & editing (equal). Jalal Faraj: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (supporting); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal). Dimitris Drikakis: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Project administration (supporting); Supervision (equal); Writing – original draft (equal); Writing – review & editing (lead).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.