Heat Source Length Research Articles

Characterization of the acoustic pressure waveforms in Rijke tubes with spatially varying heat sources and temperature distributions

In this work, we implement an asymptotic expansion technique that leverages a small perturbation parameter that arises in the context of one-dimensional tubes with open-open end point configurations and spatially varying heat sources. This approach, when paired with a spectral collocation eigensolver, enables us to produce accurate predictions of the pressure mode shapes and frequencies over a wide range of parameters. These include the temperature gain across the heat source, the heat source length and location, and the overall thermal profile. The latter is intended to reproduce different flow heating configurations that emulate Rijke tube characteristics. Specifically, this investigation begins by considering three piecewise representations of the heat source by juxtaposing constant–constant temperatures before and after a heating element whose temperature is prescribed locally using three analytical functions: linear, exponential, and power-law profiles. This is followed by a logistic distribution that can be globally applied to provide a uniformly valid, continuous, and differentiable thermal profile spanning the entire tube, including the heat source element. Our fundamental formulation relies on Green's functions and an integral formulation that enables us to extract all acoustic frequencies analytically. These are found to increase monotonically with successive elevations in the temperature gain across the heat source, retractions of the heat source, length reductions in the heat source, and smoothing of the temperature gain. Along similar lines, the pressure mode shapes are found to exhibit blunter and often linear variations for higher temperature gains, longer heat sources, and upstream displacements of the heat source toward the inlet.

Opposing Mixed Convection in an Open Parallelogram Cavity with the Horizontal Channel: Effects of the Heat Source Length and Location

In this work, a numerical simulation was used to investigate the impact that the length and positioning of the heat source had on the mixed convection opposing flow that occurred within the horizontal channel that included an open parallelogram cavity. When the wall being heated is facing away from the direction of the incoming force. Different placements of the heat source along the cavity's sidewalls were explored, with the length of the heat source set at (ε) (0.25<ε<1). The cool, steady-speed air came in through the sides of the canal. All other walls are adiabatic, while the vertical walls on the inflow and outflow sides are isothermal. The governing equations were solved using the finite element technique. For several different values of the Richardson number (Ri=0.1-100), we estimated the flow and heat fields. While the Prandtl number is held at 0.71 and the Reynolds number is maintained at 100. The average Nusselt values, as well as the findings of the flow and temperature fields, were reported. The findings demonstrate that both the Richardson number (Ri) and the distance from the heat source (ε) positively affect the heat transfer rate. It was also determined that for all Richardson numbers, the highest average Nusselt number is attained at the higher portion of the right wall of the hollow.

Numerical Computation of Natural Convection of Nanofluid in an Open Wavy Porous Cavity Heated Partially

In this work, we emphasise the heat transfer and fluid flow due to buoyancy force in a wavy open porous cavity, placed horizontally having filled with porous media and, its top wall is being kept open, whereas the right cold wall is wavy and, its left vertical wall is heated partially, keeping all other walls at thermally insulated. The non-dimensional ψ − θ formulation of mass, momentum, and energy conservation laws for porous media are solved by the standard finite difference scheme for a wide range of pertinent parameters such as nanoparticle volume fraction (0.05 ≤ Φ ≤ 0.2), Rayleigh-Darcy number (10 ≤ Ra ≤ 103), length of heat source (0.25 ≤ ε ≤ 1), and parameters controlling waviness of right wall (1 ≤ N ≤ 5) and amplitude (0.05 ≤ a ≤ 0.25). The simulated results are presented in the form of streamlines and isotherms; global and local Nusselt numbers are computed. Obtained results are analyzed and it is observed that the convection process is augmented at the presence of nanoparticle for low Ra but decreases at high Ra for all pertinent parameters; also, the wall waviness augments convection low Ra.

Laminar Natural Convection of Water-Based Alumina Nanofluids in a Square Enclosure

Steady-state, laminar, natural convection of water-based alumina nanofluids in a square enclosure with partial heating from the bottom and symmetrical cooling from the sides has been investigated based on numerical simulations. The effects of Rayleigh number, normalized heat source length and nanoparticle volume fraction on the heat transfer behavior have been investigated. An increase in Rayleigh number, for given values of normalized heat source length and nanoparticle volume fraction, leads to an increase in Nusselt number. For a given set of values of Rayleigh number based on base-fluid properties and normalized heat source length, the Nusselt number does not vary significantly with increasing nanoparticle volume fraction because the strengthening viscous effects with increasing volume fraction for water-based alumina nanofluids counteracts the enhanced thermal diffusion in nanofluids. However, when effective Rayleigh number is based on nanofluid properties, Nusselt number decreases with increasing nanoparticle volume fraction. For a given set of values of Rayleigh number and nanoparticle volume fraction, the Nusselt number increases with increasing normalized heat source length due to the strengthening of advective transport. A correlation for mean Nusselt number is proposed which adequately captures qualitative and quantitative behavior obtained from the simulations across the considered parameter range.

Mixed Convection in a Double Lid-Driven Wavy Shaped Cavity Filled with Nanofluid Subject to Magnetic Field and Internal Heat Source

A numerical investigation is carried out to analyze the impacts of internal heat source size, solid concentration of nanoparticles, magnetic field, and Richardson number on flow characteristics in an oppositely directed lid-driven wavy-shaped enclosure. The left and right vertical walls of the enclosure are cooled isothermally and moving with fixed velocity in upward and downward directions, respectively. The bottom wall is wavy shaped and isothermally cooled as the vertical walls while the top wall is kept adiabatic. A rectangular heater is placed horizontally in the center of the cavity. The physical problems are characterized by 2D governing partial differential equations accompanying proper boundary conditions and are discretized using Galerkin’s finite element formulation. The study is executed by analyzing different ranges of geometrical and physical parameters, namely, internal heat source length 0.2 ≤ CH ≤ 0.6 , solid concentration of nanoparticles 0 ≤ φ ≤ 0.09 , Hartmann’s number 0 ≤ Ha ≤ 70 , and Richardson’s number 0.1 ≤ Ri ≤ 10 . The results indicate that the overall heat transfer rate declines with the increasing length of internal heat source. The presence and rising values of solid concentration of nanoparticles cause the augmentation of heat transfer whereas the magnetic field has a negative influence and the Richardson number has a positive influence on heat transfer.

Radiative effects on unsteady MHD natural convection flow in an inclined wavy porous cavity using hybrid nanofluid containing a square obstacle

The introduction of hybrid nanofluids is a significant notion in several engineering and industrial applications. It is utilized prominently in diverse engineering applications, such as low-pressure drop, wider absorption range, generator cooling, good thermal conductivity, nuclear system cooling, heat exchangers. The present model describes the unsteady MHD free convection flow in an inclined wavy permeable enclosure using hybrid nanofluid (Aluminum oxide/Copper-water) containing a square obstacle with the effects of radiation and heat generation/ absorption. The flow is buoyancy-driven and is subjected to heat radiation (Rd) and an angled magnetic field that is constant (B0). The left wall of the cavity is partially heated and cooled by the wavy-right wall, while the remaining sides are adiabatic and contain a square solid block. A numerical Finite difference technique is utilized to resolve the governing PDEs formulated in stream function, temperature and nanoparticle volume fraction. The acquired results are validated with previous numerical studies. The parameters number in this study are discussed as the inclination angle 0°⩽α⩽π/2, Dimensionless heat source length (0.2 ≤ B ≤ 0.8), porosity of the porous medium 0.1⩽ε⩽0.9 dimensionless location of the left heater (0.3 ≤ D ≤ 0.7), Heat generation -2⩽Q⩽2, nanoparticles volume fraction (0.01 ≤ φ ≤ 0.02), thermal radiation 0⩽Rd⩽10 and the Hartmann number (0 ≤ Ha ≤ 100). The results show that improving the porosity raises the average Nusselt number; however, the rate of increase is greater at higher heat flux locations. The average Nusselt number grows when the nanoparticle volume fraction raises in Rd. The distribution of isothermal lines is slightly reduced for the higher value of the Hartman number. Natural convective hybrid nanofluid flow in a wavy porous square cavity combined with heat source/sink has a vital role in various engineering and industrial processes such as solar collector electric machinery, creation of constructing shelters and attics.

Numerical simulation of natural convection MHD flow in a wavy L‐shaped cavity with line heat source using hybrid EFGM/FEM

AbstractIn the current paper, the natural convection of power‐law nanofluid within a wavy l‐shaped cavity under the magnetic field has been simulated by a hybrid (EFGM/FEM) technique with parallel implementation. A line heat source with length Lhs and height Hhs is applied on the right vertical wall of the cavity. The cavity is filled with power‐law non‐Newtonian nanofluid. Parallel implementation with hybrid EFGM/FEM is used to solve the governing partial differential equations. Effect of Rayleigh number, Hartmann number, power‐law index, aspect ratio, nanoparticle volume fraction, location, and length of the heat source has been studied on the fluid flow, isotherms, and heat transfer. It is observed that the power‐law index, Hartmann number, and height of heat source have a decreasing effect on heat transfer rate while the Rayleigh number, length of heat source, and volume fraction of nanoparticle increases the heat transfer rate. Heat transfer rate improves by 51% and 73% with increasing the volume fraction of nanoparticles and Rayleigh number while heat transfer rate reduces by 28% with increasing the Hartmann number. The main application of such problems is found in the building corners, lubricating machines, and medical and electronic equipment (thermometer device and PCB). The hybrid method is one of the novel contributions of the author. Parallel implementation of the meshfree method (EFGM) is one of the novel features of this paper.

Magneto-Nanofluid Flow via Mixed Convection Inside E-Shaped Square Chamber

Nanofluids play a crucial role in the augmentation of heat transfer in several energy systems. They exhibit better thermal conductivity and physical strength compared to normal fluids. Here, we conduct an evaluative investigation of the magnetized flow of water–copper nanofluid and its heat transport inside a symmetrical E-shaped square chamber via mixed convective impact with a heated corner. The chamber was constructed symmetrically with an inclined magnetic field strength, and the upper surface of the chamber was isolated and set to move at a fixed velocity. The heated corner was set at a fixed hot temperature in both the left and lower directions. The right side was maintained at a fixed cold temperature, while the remaining portions of the left and lower parts were isolated. The investigation was implemented computationally, solving each of the energy and Navier–Stokes models via the application of a symmetrical finite volume method. The following topics have been addressed in this study: the consequences of the magnetic field, the volumetric fraction of nanoparticles, the heat generation–absorption parameters, and the effects of heat-source length and Richardson number on the fluid comportment and heat transport. The outputs of this symmetric study enabled us to arrive at the following derivation: the magnetic field reduces the fluid circulation inside the E-shaped square chamber. The augmentation of the Richardson number leads to an increase in the heat transfer. Moreover, the decrease in heat generation coefficient lowers the nanofluid temperature and weakens the flow fields.

Optimization of cooling of three radiant heat source inside an inclined porous medium using particle swarm optimization

In this study, an optimization of the maximum cooling of three radiant heat sources inside a closed square porous environment is performed. Three radiant-heat sources are cooled by airflow in a porous medium with an inclination angle. It is applied in the cooling of electronic equipment inside laptop computers. The cooling phenomenon is a function of five independent variables, such as the length of heat source, radiation parameter, Darcy number, the angle of inclination of a porous medium, and Rayleigh number. Differential equations are solved using the finite-difference approach. And those are solved by the Gauss-Seidel approach. Simulation is a numerical solution of the vorticity–stream equations. Coding is done in MATLAB software. And it is obtained the maximum objective function using the algorithm of particle swarm optimization. Cooling simulations are performed by drawing flow lines, constant temperature lines, dimensionless velocity component profiles, local Nusselt numbers, and mean Nusselt numbers. The main goal is to find the highest cooling rate in different amounts of five independent variables. The flow is in the laminar flow regime, and inclination angles are between 0 to 360°. The present study is the first study on the numerical solution of the cooling of three radiant-heat sources with an inclination angle by the porous media-filled square environment. And it is developed from the cooling of hot radiation walls problems in an inclined enclosure with constant temperature local radiant heat sources. This investigation showed the flow lines and their direction are influenced by the inclination angle of the porous medium strongly, and the maximum heat transfer is achieved at an angle of inclination of zero degrees. This investigation showed at the Darcy number, 0.001, and the radiation parameter, 1. The maximum and minimum Nusselt number is obtained for the inclination angle 60 and 180°, respectively.

Effectiveness of magnetize flow on nanofluid via unsteady natural convection inside an inclined U-shaped cavity with discrete heating

This analysis investigates the unsteady MHD free convection flow and heat transfer in an inclined U-shaped cavity filled with Cu-water nanofluid. Both the right and left walls of the cavity and the internal walls of the cavity are supposed to be adiabatic, while the upper horizontal walls of the cavity are kept at a lower temperature. A heat source is located on the bottom horizontal wall with its position that alters the left vertical wall. The dimensionless governing equations are solved using the Successive Under-Relaxation technique. The investigation is achieved by controlling the impact of a set of pertinent parameters, namely; the size and position of the heat element (B = 0.3–0.6, D = 0.3–0.6), Hartman number (Ha = 0–50), nanoparticle volume fraction (ϕ = 0.0–0.07), heat source generation/ absorption (Q = −4.0–2.0), and aspect ratio (AR = 0.2–0.6). The results show that the mean Nu declines with heat source length and Hartmann number, whereas it augments with dimensionless heat source location. Furthermore, the mean Nu rises in parallel with the solid volume fraction of nanoparticles and decreases with aspect ratio. In addition, the increasing of aspect ratio reduces the convection mode compared to conduction heat transfer.

Heat and mass source effect on MHD double-diffusive mixed convection and entropy generation in a curved enclosure filled with nanofluid

This paper examines the two-dimensional laminar steady magnetohydrodynamic doublediffusive mixed convection in a curved enclosure filled with different types of nanofluids. The enclosure is differentially heated and concentrated, and the heat and mass source are embedded in a part of the left wall having temperature Th (>Tc) and concentration ch (>cc). The right vertical wall is allowed to move with constant velocity in a vertically upward direction to cause a shear-driven flow. The governing equations along with the boundary conditions are transformed into a nondimensional form and are written in stream function-velocity formulation, which is then solved numerically using the Bi-CGStab method. Based on the numerical results, the effects of the dominant parameters such as Richardson number (1 ≤ Ri ≤ 50), Hartmann number (0 ≤ Ha ≤ 60), solid volume fraction of nanoparticles (0.0 ≤ ϕ ≤ 0.02), location and length of the heat and mass source are examined. Results indicate that the augmentation of Richardson number, heat and mass source length and location cause heat and mass transfer to increase, while it decreases when Hartmann number and volume fraction of the nanoparticles increase. The total entropy generation rises by 1.32 times with the growing Richardson number, decreases by 1.21 times and 1.02 times with the rise in Hartmann number and nanoparticles volume fraction, respectively.

Understanding the Thermal Time Constants of GaN HEMTs through Model Order Reduction Technique

This paper described a comparison between a numerical Finite Element Analysis (FEA) and an analytical approach in order to extract the thermal time constants and the thermal resistances of simple but realistic structures. Understanding the complex contribution of multidimensional thermal spreading, the effect of multiple layers, and the correlation with the heat source length is mandatory due to the severe mismatch of thermal expansion in different epitaxial layers and high operating temperatures. This is especially true on GaN HEMT (High Electron Mobility Transistor) with the continuous decrease of the gate length and the increase of the power density. Moreover, in this paper, we extracted the time constants with a Model Order Reduction (MOR) technique based on the Ritz vector approach with inputs coming from the FE matrices. It was found that the time constants obtained by an analytical solution and a model order extraction from FEA were exactly the same. This result validated the idea that our MOR technique provides the real time constants and resistances for our device structures and in this case unified the analytical world with the numerical one.

An analytical solution of temperature near nuclear waste canister under three-dimensional heat transfer condition

The temperature distribution is an important issue for the design and safety assessment of the repository for disposing high-level radioactive nuclear waste (HLW). In order to analyze the heat conduction property near the HLW canister under three-dimensional heat transfer condition, a solution to the temperature distribution is obtained by combining the following two parts: (1) The ratio of temperature between the finite length heat source (FLHS) and infinite length heat source (ILHS), which are obtained by integrating the solution of the point source heat transfer model. (2) An analytical solution to the temperature distribution near HLW canister with two-dimensional heat transfer for double-layered media in the lateral direction. Then, the solution of temperature near the HLW canister under three-dimensional heat transfer condition is verified by checking against the existing ones for a special case in the literature. Finally, a parametric study is performed to investigate the temperature performance in the buffer layer and surrounding rock with different parameters.

Numerical study of natural convection nanofluids flow in tilted cavities with a partially thermally heat source

Numerically, natural convection heat transfer of nanofluids in a two-dimensional tilt square enclosure was investigated, with a partial heat source embedded on the bottom wall subject to a fixed heat flux. The remaining portions of the horizontal bottom wall are assumed to be adiabatic, while the upper horizontal wall and the vertical ones are supposed to be at a relatively low temperature. Using the finite volume method and the SIMPLER algorithm, the governing equations have been discretized and solved. Simulations have been carried out for more than one nanoparticle and base fluid, a range of Rayleigh numbers ([Formula: see text] Ra [Formula: see text]), various values of heat source length and location (0.2 [Formula: see text] B [Formula: see text] 0.8 and 0.2 [Formula: see text] D [Formula: see text] 0.5, respectively), solid volume fraction ([Formula: see text]) as well as tilt angle ([Formula: see text]). The results indicate that the heat transfer performance increases by adding nanoparticles into the base fluid. An optimum solid volume fraction raises and reduces the heat transfer rate and maximum temperature of the surface heat source. respectively. Moreover, the results show a significant impact of the tilt angle on the flow, temperature patterns, and the heat transfer rate with a specific tilt angle depending to the pertinent parameters.

Finite Volume Simulation of Natural Convection for Power-Law Fluids with Temperature-Dependent Viscosity in a Square Cavity with a Localized Heat Source

In this paper, numerical study on natural convection heat transfer for confined thermo-dependent power-law fluids is conducted. The geometry of interest is a fluid-filled square enclosure where a uniform flux heating element embedded on its lower wall is cooled from the vertical walls while the remaining parts of the cavity are insulated, without slipping conditions at all the solid boundaries. The governing partial differential equations written in terms of non-dimensional velocities, pressure and temperature formulation with the corresponding boundary conditions are discretized using a finite volume method in a staggered grid system. Coupled equations of conservation are solved through iterative Semi Implicit Method for Pressure Linked Equation (SIMPLE) algorithm. The effects of pertinent parameters, which are Rayleigh number (103 ≤ Ra ≤ 106), power-law index (0.6 ≤ n ≤ 1.4), Pearson number (0 ≤ m ≤ 20) and length of the heat source (0.2 ≤ W ≤ 0.8) on the cooling performance are investigated. The results indicate that the cooling performance of the enclosure is improved with increasing Pearson and Rayleigh numbers as well as with decreasing power-law index and heat source length.

Electroosmosis-modulated bio-flow of nanofluid through a rectangular peristaltic pump induced by complex traveling wave with zeta potential and heat source.

Electrokineticmicroperistaltic pumps are important biomechanical devices that help in targeted drugging of sick body parts. This article is focused on mathematical modeling and analysis of some important aspect of such flows in a rectangular duct with wall properties. Effects of zeta potential, heat source, and deby length are also studied. Carbon nanotubes (CNTs) in the Newtonian base fluid are assumed as drugging material. A comparison of single-walled CNTs and multiwalled CNTs is also presented. It is considered that the walls are flexible and encapsulating the region with limited permeability. The defined flow problem is modeled and analyzed analytically for the transport of CNT-water nanofluid. It is accepted that the flow is steady, nonturbulent, and propagating waves do have a considerably longer wavelength when compared to amplitude. The conditions and assumptions lead to a model of coupled partial differential equations of order two. The exact results using the eigenfunction expansion method are procured and shown accordingly. The predictions about the behavior of important parameters are displayed for single-walled CNT and multiwalled CNT-water nanofluidic behavior-using figures. The impact of sundry parametersis are analyzed. The application of the current study involved a transporting/targeted drug delivery system using peristaltic micropumps and magnetic fields in pharmacological engineering.

Numerical Simulation of Entropy Generation of Conjugate Heat Transfer in A Porous Cavity with Finite Walls and Localized Heat Source

In this research, the entropy production of the conjugate heat transfer in a tilted porous cavity in respect to heat source and solid walls locations has been studied numerically. Three different cases of the cavity with finite walls thickness and heat source locations are considered in the present study. For both cases one and two, the cavity considered has a vertical finite walls thickness, while the cavity with the horizontal finite walls thickness is considered for case three. For cases one and two, the left sidewall of the cavity is exposed to heat source, whereas the rest of this wall as well as the right sidewall are adiabatic. The upper and lower cavity walls are adiabatic. For case three, the lower wall is exposed to a localized heat source, while the rest of it is assumed adiabatic. The upper wall is cold, whereas the left and right sidewalls are adiabatic. The flow and thermal fields properties along with the entropy production are computed for the modified Rayleigh number (150 ? Ram ? 1000), thermal conductivity ratio (1 ? Kr ? 10), heat source length (0.2 ? B ? 0.6), aspect ratio (0.5 ? AR ? 2) and walls thickness (0.1 ? D1 ? 0.2 and 0.1 ? D2 ? 0.2) respectively. The results show that, the maximum values of the entropy generated from fluid friction develop close to the cavity wall-fluid interfacial, while the maximum values of the entropy generated from heat transfer develop nearby the heat source region. The average Bejan number (Beav) is higher than (0.5) for cases one and two. While for case three, it was found to be less than (0.5). Also, the results show that as the modified Rayleigh number, thermal conductivity ratio, heat source length and aspect ratio increased, the fluid flow intensity in the cavity increased. While, it decreased when the walls thickness increased. From the results, it is concluded that case three gives a higher heat transfer enhancement. The obtained results are compared against another published results and a good agreement is found between them.

Design of ultrathin thermal meta-substrate for uniform cooling

Thermal metamaterials have been demonstrated to be powerful on heat manipulation and promising for thermal management in recent years, but most thermal functionalities remain in theory, and thus practical applications of thermal metamaterials are extremely urgent and worth exploring. Here, by drawing inspiration from the uniform top temperature distribution of vapor chamber that enables efficient electronics cooling, we design a thermal meta-substrate for achieving uniform top temperature profile under the design framework of the illusion thermotics theory. Further, the key parameter for better uniform cooling is identified as the length of illusion heat source by parameter optimization. Such an all-solid substrate not only combines the advantages of metamaterials and the uniform top temperature distribution of vapor chamber, but also breaks the thickness limit of the conventional liquid-based vapor chamber toward ultrathin thickness. It is expected that the present substrate concept can be used for electronics cooling and trigger further exploration on the practical application of thermal metamaterials.

Effect of aspect ratio on natural convection in a wavy porous cavity submitted to a partial heat source

In this numerical study effect of aspect ratio on natural convection in a wavy porous cavity is investigated. The cavity is placed vertically, and its right wavy wall is kept at fixed low temperature, whereas a partial heat source is embedded at the left vertical wall keeping top and bottom walls as adiabatic. The non-dimensional governing equations describing Darcy flow are solved by the finite difference method. The parameters involving the investigation are aspect ratio (Ar=0.2, 0.5, 2.0, 5.0), Rayleigh-Darcy number (Ra=10, 102, 103), dimensionless length of heat source (ε=0.25, 0.50, 1.0) along with number of undulations (N=1, 3, 5) per unit length and dimensionless amplitude of the wave (a=0.05, 0.15, 0.25) which controls waviness of the wall. The increment in aspect ratio increases the cooling efficiency of the heat source.

The effects of L-shaped heat source in a quarter-tube enclosure filled with MHD nanofluid on heat transfer and irreversibilities, using LBM: numerical data, optimization using neural network algorithm (ANN)

In this paper, the process of natural convection heat transfer (NCHT) and production of entropy (POF) in a portion of a tube has been modeled two-dimensionally. The examined problem is a quarter-tubular enclosure (quarter of a tube) which is filled with water-alumina nanofluid and subjected to a magnetic field (MF) of strength B0 at angle ω relative to horizon. Lattice Boltzmann Method (LBM) is used to simulate this problem. The ranges of parameters used in this investigation are: 0 < ω < 90, 0 < Ha < 60, and 0.1 < L, H < 0.5, and the obtained results include the Nusselt number (Nu), generated entropy, and Bejan number (Be). The results of thermal and dynamic analyses indicate that by growing the Hartmann number (Ha), the NCHT and POF values go up and the Bediminishes. Heat transfer is also improved by increasing the length of the enclosure’s hot walls. The highest amount of heat transfer occurs at the MF angle of 60º, and it is 10.3% greater than the amount of heat transfer occurring at horizontal MF. Finally, an artificial neural network was used to simulate the cavity performance based on these parameters. An optimization is performed on the parameters of heat source length and Ha. The optimization is aimed at finding suitable parameter values that lead to the highest heat transfer rate and lowest POF. A table listing a number of optimal points has been presented at the end of the paper. The optimization results indicate a desirability of about 0.56, which is achieved under the best conditions at defined parameter ranges. In this case, the values of Nu and POF are 7.26 and 3.54, respectively. The optimal state occurs at the non-dimensional hot wall length of about 0.5 and Ha of 37.7.