Maximization of heat transfer density rate from a single row of rhombic tubes cooled by forced convection based on constructal design

The heat transfer density rate from a row of rhombic tubes cooled by forced convection is maximized based on constructal design. A row of parallel rhombic tubes are placed in a fixed volume, the horizontal axis of the tubes is kept constant while the vertical axis of the tubes and the spacing between the tubes are changed to facilitate the heat flow from the tubes to the coolant. The tubes are kept at constant temperature and the incoming free‐stream flow is induced by constant pressure drop. For steady, two‐dimensional, incompressible, and laminar forced convection, the governing equations are solved numerically by finite volume method with SIMPLE algorithm. The dimensionless pressure drop (Bejan number, Be) ranging from 10 ³ to 10 ⁵, the range of the vertical axis of the tube is 0.2 ≤ B ≤ 2, and the working fluid is air ( Pr = 0.71). The results show that the optimal spacing decreases and the maximum heat transfer density increases as the Bejan number increases for all vertical axes of the tube. Bejan number and the bluntness of the tube have a significant effect of the flow structure (separation and vortex formation) around the tubes at the optimal spacings.     

Constructal design of longitudinally finned tubes cooled by forced convection

The constructal design method is used in the present study to find the configuration of longitudinally finned tubes cooled by forced convection. The finned tubes are arranged in parallel inside a fixed two‐dimensional domain. Two degrees of freedom inside the domain are considered for the design. The first degree of freedom is the tube‐to‐tube spacing, and the second is the length of the longitudinal fin. For both these degrees of freedom, a three‐fin position inside the domain is considered. The fin is placed in the front, back, and front and back of the tube in the first, second, and third positions, respectively. Maximization of the heat flow density (heat transfer/volume) from the finned tubes to the cold cross flow is the objective function of the present study. For the three fin positions, the constant pressure difference between the upstream and the downstream drives the cross flow. The dimensionless continuity, momentum, and energy equations for two dimensional, steady, and incompressible flows are solved by discretizing it according to the finite volume method. The thermal condition of the fins and the tubes is constant surface temperature. The dimensionless pressure drop known as Bejan number is varied in the range of 10³ ≤ Be ≤ 10⁵. The fin length is changed from L_f = 0 (unfinned tube) to L_f = 0.2, 0.4, and 0.4. The tubes are cooled by air (Prandtl number = 0.71). The results illustrated that for the considered Bejan numbers and fin positions, the spacing between the unfinned and the finned tubes can be adjusted to optimal spacing such that the heat flow from the tubes to the coolant is maximum.     

Maximization of heat transfer density from radially finned tubes in cross-flow using the constructal design method

The maximization of volumetric heat transfer density from radially finned tubes in cross-flow is investigated in this study based on the constructal design method. A row of radially finned tubes is placed in cross-air flow. The tubes and the radial fins are heated at uniform temperatures and cooled by the air cross-flow. The cross-air flow is generated by a finite pressure difference. Two dimensionless pressure differences (Bejan number) are considered (Be = 10³ and Be = 10⁵). The objective function, the degrees of freedom, and the constraints in the constructal design method should be identified. The objective function is the maximization of the heat transfer density from the finned tubes. The degrees of freedom are; the fin tip-to-fin tip spacing, the number of fins, the tube diameter, the fin thickness, and the angle between the fins. The constraints are the length and height of the space occupied by the finned tubes. The pressure-driven flow and energy equations (steady, two-dimensional, and incompressible) are solved by means of the finite volume method. The ranges of the dimensionless fin tip-to-fin tip spacing are (0.2 ≤ S ≤ 1 for Be = 10³ and 0.05≤ S ≤ 0.3 for Be = 10⁵). The number of fins is changed as (N = 2, 4, 6, 8, 10, and 12). The dimensionless tube diameter is changed as (D = 0.25, 0.5, and 0.75). The dimensionless fin thickness is changed as (T = 0.001, 0.01, and 0.05). The results showed that for both (Be = 10³) and (Be = 10⁵), the highest value of the maximum volumetric heat transfer density is for (N = 2) and decreases as the number of fins increases. In addition, the minimum values of the maximum volumetric heat transfer density occur when the vertical fins exist at (N = 4, 8, and 12).     

Maximization of heat transfer density from a single-row cross-flow heat exchanger with wing-shaped tubes using constructal design

A single-row cross-flow heat exchanger with wing-shaped tubes is designed in this paper using the constructal design method. The wing-to-wing spacing and the wing thickness are free to morph while the chord of the wings remains constant. Wing-to-wing spacing optimization or heat transfer density maximization from the wings is the objective function in this design. Two directions of the free-stream (incoming) flow are considered. The right- and left-flow directions are considered with a constant drop in pressure. All wings are heated at a uniform temperature, and the air is used to cool these wings. The two-dimensional conservation of mass, conservation of momentum, and the conservation of energy equations are solved by means of the finite volume method for steady and incompressible flow. The ratio of the wing thickness to the chord (dimensionless wing thickness) is changed from 0.2 to 1. For each wing thickness, three Bejan numbers (Be = 10³, 10⁴, and 10⁵) are used in the numerical simulation. The results revealed that in the case of the flow direction to the left, the maximum heat transfer density is higher than that in the case of the flow direction to the right for all wing thicknesses and all Bejan numbers.     

Constructal design of multiscale elliptic tubes in crossflow

The optimal configuration of two‐scale elliptic tubes in crossflow is found on the basis of the constructal design. The larger tubes are installed inside a domain of fixed length and height. In the same domain, smaller tubes are inserted between the larger tubes in the entrance region at the mid leading edge to leading edge distance of the larger tubes. The spacing between the larger tubes, the semiminor axis of the larger tubes, the major axis of the smaller tubes, and the semiminor axis of the smaller tubes are varied inside the domain freely to find the optimal configuration. There are two optimal configurations: one without the smaller tubes and the other with the presence of the smaller tubes. Both the larger and the smaller tubes are heated at a constant surface temperature. The flow is induced by a fixed pressure difference. The equations for steady, laminar, two‐dimensional, and incompressible flow are solved by finite volume method. In the absence of the smaller tubes, the range of Bejan number (dimensionless pressure drop) is ${10}^3<\mathit{Be}<{10}^5$, and in the presence of the smaller tubes, Bejan number is Be = 10⁵. The range of the dimensionless larger tubes semiminor axis is 0.1 ≤ B ≤ 0.4. Air is used to cool the row of the tubes with Prandtl number equal to 0.7. The results show that for different semiminor axes of the larger tubes, the heat transfer rate is enhanced when the smaller tubes are placed between the larger tubes.     

Constructal design of flat tubes cooled by natural convection

A set of vertical flat tubes cooled by natural convection and placed in a finite size space is designed based on the constructal law. The constraint in this design is the size of the space where the tubes are placed. The freedom inside the space is the distance between the tubes. When the constructal law is applied, the optimal distance between the tubes is determined. Rayleigh numbers are taken as (Ra = 10³, 10⁴, and 10⁵). The dimensionless tube diameter (tube diameter/tube height) is changed from (D* = 0.2) to (D* = 1) (circular tube). All the tubes are heated to the same wall temperature. The air used to cool the tubes has a Prandtl number (Pr = 0.72). The equations of conservation of mass, momentum, and energy for steady, two-dimensional, and incompressible flow are solved by the finite volume method. The result showed that the best or optimal distance at a given Rayleigh number remains constant for all tube diameters. The result also showed that the number of the small diameter tubes must be more than the number of the large-diameter tubes for the same Rayleigh number and the same size of the space to make the heat flow from the tubes to the coolant easier.     

MHD conjugate heat transfer and entropy generation analysis of MWCNT/water nanofluid in a partially heated divided medium

The aim of this article is to conduct the lattice Boltzmann simulation of the magnetohydrodynamic (MHD) natural conjugate heat transfer in an apportioned cavity loaded with a multiwalled carbon nanotube/water nanofluid. The divided cavity is, to some extent, heated and cooled at the upright walls, whereas the horizontal walls are adiabatic. The nanofluid properties are evaluated on the basis of experimental correlations. The parameters ranges in the study are as follows: nanoparticles' volume fraction (%): 0 ≤ ? ≤ 0.5, temperature (°C): T = 27, Rayleigh number (Ra): 10³ ≤ Ra ≤ 10⁵, Hartmann number (Ha): 0 ≤ Ha ≤ 90, and the magnetic field inclination angle (γ): 0 ≤ γ ≤ π/2. The current outcomes are observed to be in great concurrence with the numerical results introduced in the literature. The impacts of the aforesaid parameters on local and average heat transfer, entropy generation, and Bejan number (Be) are explored and discussed. Indeed, the transfer of heat increases linearly with ? for a low Ra. As Ra increases, the average Nusselt number decreases for a high value of ?. The increase of nanoparticles' volume fraction leads to a reduction in the entropy generation and an increase in the Bejan number for a high Ra, but at low Ra, these functions remain constant. As the Ha increases, the transfer of heat and the entropy generation decreases, whereas there is an increase in Be. The transfer of heat, total entropy generation, and the Be depends strongly on the direction of the magnetic field. The increase of heater and cooler size has a great influence on the transfer of heat, entropy generation, and Be.     

Effect of Darcy,Reynolds, and Prandtl Numbers on the Performance of Two‐Phase Flow Heat Exchanger Filled with Porous Media

Experimental investigation of two‐phase laminar forced convection in a single porous tube heat exchanger is presented. The effect of Darcy, Reynolds, and Prandtl numbers on the performance of this heat exchanger during the condensation process of carbon dioxide at different test conditions were investigated. Gravel sand with different porosities is used as a porous medium. The flow in the porous medium is modeled using the Brinkman–Forchheimer‐extended Darcy model. Parametric studies are also conducted to evaluate the effects of porosity and Reynolds and Prandtl numbers on the heat transfer coefficient and the friction factor. A dimensionless performance parameter is developed in order to be used in evaluating the porous tube heat exchanger based on both the heat transfer enhancement and the associated pressure drop. The study covers a wide range of inlet pressure (P_in), mass flow rate (), porosity of gravel sand (ε), and Darcy number (Da) which ranged: 34.5 ≤ P_in ≤ 43 bars, 8 * 10^{? 5} ≤ ≤ 16 * 10^{? 5} kg/s, 34.9% ≤ ε ≤ 44.5%, 1.6 * 10^{? 6} ≤ Da ≤ 5 * 10^{? 6}, respectively. The study predicted the combined effect of the Reynolds number, Darcy number, porosity, and Prandtl number on the heat transfer and pressure drop of carbon dioxide during the condensation process in a porous medium. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21117     

Numerical investigation of thermal performance of extended surfaces for a novel heat exchanger

The present paper provides a thorough numerical study of variation in geometrical parameters that affect the performance of the novel finned‐tube type heat exchanger design. The finite volume method was employed to discretize and solve the governing partial differential equations of heat conduction. A wide range of constant convective heat transfer coefficient (5 < h < 200 W/m² K) is chosen to reduce the computational time and power, which covers thermal applications of latent thermal energy storage, refrigeration & air‐conditioning, etc. The effects of the ratio of fin spacing of fins to the outer diameter of the tube (0.1 ≤ δ* ≤ 8), the material of fins (copper and stainless steel) and the ratio of fin thickness to the outer diameter of the tube (0.0333 ≤ t* ≤ 0.4) on the performance parameters namely efficiency (η) and effectiveness (ε) of the fins were studied. Temperature contours for a wide range of geometries were depicted. The maximum effectiveness of copper fins is 304.62, whereas that for steel fin is 219.33 with the optimum dimensionless fin thickness reported to be t* = 0.1666. Furthermore, the maximum overall efficiencies of fins were 99.98% and 99.62% for copper and steel fins, respectively.     

Experimental study on the effect of porous medium on performance of a single tube heat exchanger: A CO2 case study

An experimental study on single‐phase laminar forced convection in a single porous tube heat exchanger is presented. Parametric studies are conducted for different inlet pressures, different mass flow rates, and different porosities to evaluate the effects of particle diameter and Reynolds number on the heat transfer and friction factor. The Nusselt number and friction factor are developed for efficient design of a porous heat exchanger based on the present configuration. Heat is transferred to the walls of the heat exchanger by natural convection mode. Gravel sand with different porosities is used as a porous medium during the tests. The flow of carbon dioxide as a working fluid in the porous medium is modeled using the Brinkman–Forchheimer‐extended Darcy model. A dimensionless performance parameter is developed in order to be used in evaluating the porous tube heat exchanger based on both the heat transfer enhancement and the associated pressure drop. The study covers a wide range of inlet pressures (P_i), mass flow rates ( ), porosity of gravel sand (ε), and particle diameters (d_m) which ranged 34.5 ≤ P_i ≤ 43 bars, 8 ?? 10^?5 ≤ ≤ 16 ?? 10^?5 kg/s, 34.9% ≤ ε ≤ 44.5%, 1.25 ≤ d_m ≤ 5.15 mm, respectively. This study revealed that a smaller particle diameter can be used to achieve higher heat transfer enhancement, but a larger particle diameter leads to a more efficient performance based on heat transfer enhancement. The average heat transfer coefficient of carbon dioxide decreases when the porosity increases. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21059     

CFD analysis of a flat tube with semi-circular fins using graphene nanofluid and to compare performance with square type of fins

CFD analysis on a flat tube with semi-circular fins under laminar flow conditions was performed with graphene-based nanofluids considering the nanofluids as incompressible. Different simulations were performed at four different concentrations of nanofluids (0.01%, 0.1%, 0.2%, and 0.4%) and at different volume flow rates (4, 6, 8, and 10 LPM) and at four different forced convective heat transfer coefficients at different wind velocities at 300 K (50, 100, 150, and 200 W/m² K). It was observed that with an increase in the concentration of nanoparticles in nanofluids, the thermal conductivity of base fluid water was increased (at 353 K the nanofluid of 0.4% volume concentration, the thermal conductivity of nanofluid increased by 200% with respect to base fluid). Graphene-based nanofluids have higher effectiveness than most nanofluids hence it is considered for the analysis, at 0.4% concentration of nanofluid the effectiveness observed was 36.84% at 4 LPM, and for water, the effectiveness was 28.22% under similar conditions. The effect of flow rate on temperature drop was significant. At 4 LPM and at 0.4% of nanofluid, an outlet coolant temperature of 333 K was observed whereas the water outlet temperature at 10 LPM is 346.13 K. The effect of forced convective air heat transfer coefficient was significantly high. At h = 50 W/m² K the outlet temperature of 0.4% nanofluid at 4 LPM was 345.25 K and at h = 200 W/m² K, the outlet coolant temperature was 333.47 K. A single tube of the radiator was considered for the analysis whereas the original radiator consists of 50 tubes due to problems of Ansys in meshing.     

Entropy Generation of Natural Convection Heat Transfer in a Square Cavity Using Al2O3–Water Nanofluid

Entropy generation of an Al₂O₃–water nanofluid due to heat transfer and fluid friction irreversibility has been investigated in a square cavity subject to different side‐wall temperatures using a nanofluid for natural convection flow. This study has been carried out for the pertinent parameters in the following ranges: Rayleigh number between 10⁴ and 10⁷ and volume fraction between 0 and 0.05. Based on the obtained dimensionless velocity and temperature values, the distributions of local entropy generation, average entropy generation, and average Bejan number are determined. The results are compared for a pure fluid and a nanofluid. It is totally found that the heat transfer, and entropy generation of the nanofluid is more than the pure fluid and minimum entropy generation and Nusselt number occur in the pure fluid at any Rayleigh number. Results depict that the addition of nanoparticles to the pure fluid has more effect on the entropy generation as the Rayleigh number goes up.     

Constructal design of a PCM cylindrical heat sink with three-branched,two-stages dendritic tubes

Amputees who use prosthetic limbs suffer from the problem of high contact temperature between the socket of the prosthetic limb and the amputated part and lack of evaporation of sweat. These conditions lead to discomfort and failure to perform functions properly. In addition, these conditions help generate ulcers and accumulate harmful bacteria in this area. This paper presents a heatsink design to extract heat from the contact area. A cylindrical heat sink is designed for phase-changing materials with three branched tubes in two stages. The current heat sink is used to cool the contact area between the amputated part and the socket in the lower prostheses. Three distributions of pipe branches are proposed. The distribution and pipe lengths were obtained using a constructal design method. In the constructal design, the lengths of the branched tubes were the degrees of freedom, the objective function was the minimization of the inlet temperature to the heat sink, and the constraint was the volume of the cylindrical heat sink. The metabolic heat transfer during exercise was estimated and its value was used to calculate the size of the cylindrical heatsink and the selection of the phase change material by testing three of them: water, tridecane, and dodecane. It was found that water gives the highest latent heat of melting and the lowest volume in addition to its availability. On the other hand, two cooling fluids were tested: water and air. It was found that water as a cooling fluid gave the lowest flow and the largest heat capacity. Constructal theory was used to design a cylindrical heat sink using branched tubes for the coolant in two steps: the first with three branches, and the second with nine branches. The degree of freedom for constructal theory was the length of the branches through the choice of their end locations. It was found that the branches of the highest length led to a reduction in temperature from 40°C to 15.48°C compared with the single tube, which reduced the temperature to 23.87°C. All tests recorded a pressure drop within the acceptable range of 3.1–5.43 Pa for the branches examined. The research demonstrated that using constructal theory achieved the best thermal dissipation within a restricted volume.     

Correlations for Natural Convective Heat Transfer from Isothermal Surface of Octagonal and Hexagonal Shapes of Different Aspect Ratios

The effect of surface shape on laminar natural convective heat transfer from vertical isothermal hexagonal and octagonal flat plates embedded in a plane adiabatic surface, the adiabatic surface being in the same plane as the surface of the heated plate, has been numerically investigated. Results for the hexagonal and octagonal surface shapes with different aspect ratios have been obtained. It has been assumed that the fluid properties are constant except for the density change with temperature which gives rise to the buoyancy forces, this having been treated using the Boussinesq approach. The solution has been obtained by numerically solving the full three‐dimensional form of governing equations, these equations being written in dimensionless form. The solution was obtained using the commercial finite volume method based cfd code, FLUENT^©14.5. The solution has the surface shape, the Rayleigh number, the dimensionless plate width and the Prandtl number as parameters. Results have only been obtained for a Prandtl number of 0.7 for Rayleigh numbers between 10³ and 10⁸ for various surface shapes with width‐to‐height ratios between 0 and 0.6. The effect of these parameters on the mean Nusselt number has been studied and empirical correlation equations for the mean heat transfer rate have been derived.     

Performance predictions of laminar heat transfer and pressure drop in an in-line flat tube bundle using an adaptive neuro-fuzzy inference system (ANFIS) model

This paper shows how to predict the heat transfer and pressure drop for in-line flat tube configuration in a crossflow, using an adaptive neuro-fuzzy inference system (ANFIS). A numerical study of a 2D steady state and incompressible laminar flow for in-line flat tube configuration in a crossflow is also considered in this study. A finite volume technique and body-fitted coordinate system is used to solve the Navier–Stokes and energy equations. The Reynolds number varies from 10 to 320. Heat transfer and pressure drop results are presented for a tube configuration at transverse pitch and longitudinal pitch. The variation in velocity profile, isotherm contours and streamlines were compared for various configurations. The predicted results for average Nusselt number and dimensionless pressure show a good agreement with available previous work. The accuracy between numerical values and ANFIS model results were obtained with a mean relative error for average Nusselt number, pressure drop less than 1.9% and 2.97% respectively. Therefore, the ANFIS model is capable of predicting the performance of thermal systems in engineering applications, including the model of the tube bundle for heat transfer analysis and pressure drop.     

Numerical investigation for improved heat transfer characteristics in micro‐fin tubes

Generally, internal micro‐fin tubes are used for increasing the life and performance of electronic devices. The micro‐fins enhance the heat transfer rate by increasing the surface area with an increase of the pressure drop. In this study, heat transfer and pressure drop are analyzed by varying Reynolds number with the increase in the number of fins in tubes. Heat transfer and pressure drop, together with turbulence kinetic energy of micro‐fin tubes (helical and straight) and a smooth tube, have been evaluated for different Reynolds numbers (60 000, 40 000, 20 000, and 2000) at a constant temperature of 350 K, which clearly establishes laminar to turbulent flow. It is observed that the helical micro‐fin tube has a better result compared with the straight micro‐fin tube and smooth tube at Reynolds numbers 60 000, 40 000, and 20 000 at velocity 2, 1, and 0.5 m/s, respectively. This study is an attempt to establish a comparison of different micro‐fin geometries with varying Reynolds numbers, concluding that a high Reynolds number is suitable for the same.     

Combined Numerical Optimization and Constructal Theory for the Design of Microchannel Heat Sinks

This study deals with the geometric optimization of a silicon based microchannel heat sink using a combined numerical optimization and constructal theory. The objective is to minimize the wall peak temperature subject to various constraints. The numerical simulations are carried out with fixed volumes ranging from 0.7 mm³ to 0.9 mm³ and pressure drop between 10 kPa to 60 kPa. The effect of pressure drop on the optimized aspect ratio, solid volume fraction, hydraulic diameter, and the minimized peak temperature are reported. Results also show that as the dimensionless pressure drop increases the maximized global thermal conductance also increases.     

Heat transfer enhancement for combined convection flow of nanofluids in a vertical rectangular duct considering radiation effects

In this paper, combined convective heat transfer and nanofluids flow characteristics in a vertical rectangular duct are numerically investigated. This investigation covers Rayleigh numbers in the range of 2 × 10⁶ ≤ Ra ≤ 2 × 10⁷ and Reynolds numbers in the range of 200 ≤ Re ≤ 1000. Pure water and five different types of nanofluids such as Ag, Au, CuO, diamond, and SiO₂ with a volume fraction range of 0.5% ≤ φ ≤ 3% are used. The three‐dimensional steady, laminar flow, and heat transfer governing equations are solved using finite volume method (FVM). The effects of Rayleigh number, Reynolds number, nanofluids type, nanoparticle volume fraction of nano‐ fluids, and effect of radiation on the thermal and flow fields are examined. It is found that the heat transfer is enhanced using nanofluids by 47% when compared with water. The Nusselt number increases as the Reynolds number and Rayleigh number increase and aspect ratio decreases. A SiO₂ nanofluid has the highest Nusselt number and highest wall shear stress while the Au nanofluid has the lowest Nusselt number and lowest wall shear stress. The results also revealed that the wall shear stress increases as Reynolds number increases, aspect ratio decreases, and nanoparticle volume fraction increases. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20354     

Numerical Simulation of Natural Convection in a Triangle Cavity Filled with Nanofluids Using Tiwari and Das' Model: Effects of Heat Flux

In this article, we present a fully higher‐order compact (FHOC) finite difference method to investigate the effects of heat flux on natural convection of nanofluids in a right‐angle triangle cavity, where the left vertical side is heated with constant heat flux both partially and throughout the entire wall, the inclined wall is cooled, and the rest of walls are kept adiabatic. The Darcy flow and the Tiwari and Das’ nanofluid models are considered. Investigations with four types of nanofluids were made for different values of Rayleigh numbers with the range of 100 ≤ Ra ≤ 50,000, size of heat flux as 0.1 ≤ ε ≤ 1.0, enclosure aspect ratio as 0.5 ≤ AR ≤ 2.0, and solid volume fraction parameter of nanofluids with the range of 0% ≤ ? ≤ 20%. Results show that the average heat transfer rate increases significantly as particle volume fraction and Rayleigh numbers increase, and the maximum value of average Nusselt number is obtained by decreasing the enclosure aspect ratio. The results also show that the average heat transfer decreases with an increase in the length of the heater. Furthermore, multiple correlations in terms of the Rayleigh numbers and the solid volume fraction of four types of nanoparticles have been established in a general form.     

The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer

Experimental investigation is reported on natural convection heat transfer from the outer surface of a vertical array of horizontal square tubes in air. Five tubes equally spaced are used with cross section 0.02 × 0.02 m². The tubes are subject to constant heat flux boundary condition using internal constant heat flux heating elements in the range 46–510 W/m². Experiment is done for arrays of 2–5 square tubes and for four center-to-center separation distance to hydraulic diameter ratios. Study is concentrated on the effect of tube location in the array and on the geometry of the array. Results show that the downstream tubes exhibit reduced Nusselt numbers than that of a single tube for small center-to-center separation ratio of 2.5. This reduction depends on the location of the tube in the array and the number of tubes in each array. Results also show that as the ratio increases, enhancement in heat transfer over that of a single tube is observed and critical ratio is obtained at a specified value of the modified Rayleigh number for the upper (downward) tubes in each array. Local circumference averaged correlations are proposed for the upper tubes in each array and for any other individual tube in each array geometry. An overall general averaged correlation is also reported for each tube in the array.