Surface-averaged Nusselt Number Research Articles

Effects of Flow Pulsation and Surface Geometry On Heat Transfer Performance in a Channel with Teardrop-Shaped Dimples Measured by Transient Technique

Abstract This study focuses on the heat transfer performance of a pulsating flow over a channel surface with teardrop-shaped dimples. Heat transfer measurements were performed by a transient technique with compensation of three-dimensional heat conduction under a bulk Reynolds number of 25,000. Seven types of surfaces with the teardrop-shaped dimples were examined, where dimple arrangement (in-line/staggered) and inclination angle (0-60 deg) were varied. A pulsating flow with the Strouhal number of 0.15 was generated by vibrating a rubber film section on the channel wall using a vibration generator. The pulsation amplitude was evaluated by calculating the root-mean-square value of the phase averaged velocity. Two conditions of the pulsation amplitudes were examined (0.09 and 0.12 of mean velocity). The results showed that the surface-averaged Nusselt number and friction factor for the pulsating flow increased from those for the steady flow. The highest increases of the surface-averaged Nusselt number and heat transfer efficiency index appeared in the 30deg in-line arrangement, and those were 16.1% and 9.8%, respectively, at most as compared with the steady case. Due to the flow pulsation, the local Nusselt number was enhanced at the leading-edge region of the dimples, and supplementary RANS/URANS results showed that the flow separation size was shrunk by the flow pulsation there.

Effects of Flow Pulsation and Surface Geometry on Heat Transfer Performance in a Channel With Teardrop-Shaped Dimples Investigated by Large Eddy Simulation

Abstract Large eddy simulation was performed to investigate heat transfer performance of a pulsating flow over teardrop-shaped dimples. A total of six geometries of dimpled surfaces were examined for dimple arrangements of in-line/staggered/original and dimple inclination angle of 0–60 deg. Pulsating flows were generated by sinusoidally varying the volume-averaged velocity. The pulsation frequency and amplitude were changed for the Strouhal number of 0–0.60 and the root-mean-square velocity amplitude normalized by the bulk flow velocity of 0–0.14. The results showed that the surface-averaged Nusselt number and friction factor were larger for the pulsating flow case than for the steady flow case. The surface-averaged Nusselt number ratio and the friction factor increased with the Strouhal number up to the Strouhal number of 0.30. For the Strouhal number larger than 0.30, they decreased with the Strouhal number or stayed almost constant. Consequently, the heat transfer efficiency index increased with the Strouhal number. The increase in the local Nusselt number ratio due to the flow pulsation was observed at the leading-edge region of the dimples. The results of the streamlines near the dimple showed that the swirling separation bubble was located closer to the leading-edge region due to the pulsation, which resulted in the increase of the absolute values of the turbulent heat flux and the local Nusselt number ratio.

Experimental investigation of dual jet flow past a heated surface: Effect of Reynolds number

A dual jet flow comprises a wall jet, discharged immediately parallel to a solid boundary, flowing alongside a parallel jet whose exit is offset from the solid boundary by some distance. Despite the widespread industrial application of dual jet flows, experimental analysis of their flow and heat transfer characteristics is severely underrepresented in the published literature. As a result, the heat transfer capabilities and driving flow mechanisms associated with dual jet flows are not yet fully understood, meaning they cannot be optimised towards specific applications. The present study is thus concerned with characterising the transfer of heat to a dual jet flow for varying Reynolds number through experimental means. Using air as the working fluid, each jet Reynolds number is varied in the range 5500≤Re≤12,000 and the offset ratio and velocity ratio are each maintained at a constant value of 1. A constant heat flux generation of 1670W/m2 is maintained in the bounding wall and the surface Nusselt number distribution is measured using infrared thermography techniques. A distinct Nusselt number profile is observed with respect to downstream distance, which remains consistent regardless of the Reynolds number. The results indicate that increasing Re elevates the Nusselt number proportionally at all downstream locations, where the surface-averaged Nusselt number rises linearly with increasing jet Reynolds number.

Numerical investigation on vortex cooling flow and heat transfer characteristics for gas turbine blade with variable coolant chamber cross-sectional shapes

In this paper, according to the actual size of gas turbine blade interior space, eight vortex cooling models with different cross-sectional shapes of coolant chambers are established. Rectangular, Circle, Trapezoidal up, Trapezoidal down, Ellipse up, Ellipse down, Sine up and Sine down are the eight coolant chamber shapes. Functions of variable coolant chamber structures on vortex cooling flow and heat exchange characteristics are numerically analyzed by verifying the Standard k- ω turbulence model and conducting grid independence analysis. Research results show that the heat exchange ability of Sine up coolant chamber is the best compared with the other coolant chambers. The cool air enters the vortex chamber at a faster speed and impacts on the target surface to form a high-pressure area and a low-temperature swirl. The more the cool air moves downstream, the greater the turbulent kinetic energy becomes, which is more conducive to the heat transfer. Therefore, Sine up coolant chambers have the highest thermal performance factor. Due to the low drag coefficient and high target surface average Nusselt number of Ellipse up, its thermal exchange performance is second only to that of Sine up. Circle coolant chamber will not bring favorable factors to engineering application because of the slowest flow speed and the worst heat exchange capacity. For Trapezoidal up and Trapezoidal down, the flow velocity, pressure distribution, turbulent kinetic energy intensity and heat transfer uniformity are almost the same as Rectangle coolant chamber. The models of Ellipse down and Sine down coolant chambers exhibit less turbulent kinetic energy strength to heat transfer and higher temperature in the vortex chamber, and it is not suggested for cooling system optimal design.

Effects of the Wavelength for the Sinusoidal Cylinder and Reynolds Number for Three-Dimensional Mixed Convection

A numerical study estimated the mixed convection around a sinusoidal cylinder inside a rectangular cavity. The key parameters for sinusoidal circular cylinders were the wavelength and Reynolds number (Re = 100, 500, and 1000) with a fixed Prandtl number of 0.7. The flow regimes were greatly affected by these parameters. The unsteadiness of the flow pattern was also affected by λ and Re. The results were analyzed by the thermal structure, vortical structure, and Nusselt number. Furthermore, the orthogonal enstrophy was used to quantify the three-dimensional flow characteristics, and the frequency of the surface-averaged Nusselt number for the heated bottom wall was used to analyze the unsteady characteristics. The heat transfer performance was enhanced by 6.6% at Re = 100 for λ = 3 compared to the circular cylinder. In addition, the heat transfer performance was enhanced by 6.6% and 2.6% at Re = 500 for λ = 2 and Re = 1000 for λ = 1, respectively, compared to the circular cylinder.

Natural-convection heat transfer from regularly ribbed vertical surfaces: Homogenization-based simulations towards a correlation for the Nusselt number

Free-convection heat transfer from vertical surfaces is widely encountered in engineering applications, yet the role played by surface alterations in the heat transfer process and their practical effectiveness are still points of confusion. In this work, buoyancy-driven flows over periodically ribbed vertical plates of different surface micro-textures are investigated, mainly based on an asymptotic homogenization model through which the expensive resolution of the velocity and thermal fields within the inter-rib regions is bypassed, by imposing equivalent effective boundary conditions at a virtual plane surface. Efficiency of the homogenized simulations in detecting macroscopic behavior of the Nusselt number is first assessed, compared with full feature-resolving simulations in which the effects of complex flow patterns, near and within wall corrugations, on the local Nusselt number are captured. Second, the validated model is used to construct a database of numerical results describing deviations of the average Nusselt number over different ribbed surfaces, relative to a corresponding smooth surface. Under the conditions investigated, it is found that surface roughening generally deteriorates heat transfer from vertical surfaces, with slight enhancement for geometries characterized by low thermal slip, for example, rectangular ribs of narrow inter-rib spaces. Finally, a multiple-regression analysis is conducted to formulate a correlation describing effects of the thermal-slip coefficient, the number of ribs, and the Grashof number on the surface-averaged Nusselt number; accuracy of the proposed correlation is attested via further validation. This paper aims to call attention of the heat transfer society to the ability of the homogenization approach to considerably alleviate the computational requirements for relevant simulations and, thus, to significantly accelerate parametric optimization studies.

Heat transfer of pulsating water flow through aluminum-foam channel under asymmetric constant heat flux boundary condition

In the present study, the effect of pulsating inlet velocity condition on heat transfer in metal foam was investigated experimentally. Square wave shaped water velocity profiles with different amplitudes and frequencies were applied to the inlet of a rectangular channel filled with aluminum foam having 20 pores per inch and a porosity of 91.8%. A constant heat flux (12787 W/m2) was applied to one side of the porous channel. The non-heated surfaces were insulated. The surface temperatures, water inlet and outlet temperatures and volumetric flow rates were measured and recorded. The dimensionless flow frequency parameter (M) and dimensionless flow amplitude (Rem) parameter were changed in the ranges 4.8–7.3 and 492–1388, respectively. The results showed that there was a strong relationship between average Nusselt number and flow velocity amplitude. Generally, average Nusselt number was seen to increase with increasing frequency parameter and/or amplitude parameter. Correlations for the local time-averaged and surface-averaged Nusselt numbers are proposed. A meaningful comparison between steady-state and pulsating heat transfer is provided. It showed that heat transfer advantages of pulsating flow were only present for a certain low range of Reynolds number, beyond which pulsating flow heat transfer approached that of steady state.

Heat Transfer Due to Annular Jets Impinging on a Moving Surface

Abstract This work investigates flow and heat transfer under an array of annular jets impinging on a heated moving surface. Numerical solutions of the full Navier–Stokes equation were attempted with a highly refined mesh. This study reports results for Reynolds numbers up to 500. In the surface movement direction, a periodic element from a jet-bank configuration was chosen, and the nondimensional surface velocity was considered from zero (i.e., a stationary plate) to two times the jet velocity. The impact of annular jet impingement over a moving surface on flow and heat transfer characteristics, including the development of the flow field, velocity profiles, skin friction coefficient and topology of skin friction lines, and local as well as surface averaged Nusselt number distribution are presented. It is observed that both the flow field and thermal performance are strongly affected by the surface motion. Heat transfer from the surface initially increases with the increasing surface motion, and after attainment of the highest value, heat transfer reduces with a further increase in surface velocity. However, higher surface velocity leads to higher uniformity in heat transfer, which may be beneficial for situations demanding uniformity in heat transfer.

Flow and thermal characteristics produced by a curved rectangular winglet vortex generator in a channel

Three-dimensional numerical computations are performed to investigate the performance of curved winglets mounted on a rectangular channel. The concave and convex shapes of curved rectangular winglet vortex generators (RWVGs) are named according to flow facing surface. Curvature of winglet mainly depends on arc angle, radius of curvature, arc length and chord length. Present study investigates the influence of arc angle variation of curved RWVG on flow and thermal characteristics for fixed chord length. Arc length and radius of curvature of curved RWVG vary according to arc angle. With the help of streaklines plots and iso-Q surfaces, flow behavior around the curved RWVG has been investigated. Heat transfer enhancement of curved RWVGs has been examined by using the dimensionless parameters surface-averaged Nusselt number, while pressure loss is estimated by friction factor. The overall thermohydraulic performance of curved RWVGs is inspected by thermal performance factor. Investigations of secondary flow intensity are carried out to understand its impact on heat transfer mechanism. From the investigation of secondary flow, it is observed that secondary flow provides a supplement to the traditional method of analyzing heat transfer performance.

Leading edge impingement cooling analysis with separators of a real gas turbine blade

• Experiments of impingement cooling at real temperature conditions are conducted. • A new cooling design containing separators is suggested to prevent cross flow. • Deviation by using low temperature conditions of impingement cooling is presented. • Heat transfer data of impingement cooling are compared with published studies. This paper reports an analysis on leading edge impingement cooling of a turbine blade under real operation conditions of gas turbine. In the experiments, both mainstream and cooling air are heated to real high temperatures, and overall cooling effectiveness on the external blade mid-span surface is measured. Numerical strategy including RNG k-ε turbulence model is validated by the experimental data. Impingement cooling models with and without separators (Model A and Model B) are used as specimens, and the effects of high temperature conditions, separators and temperature ratios are discussed. The results indicate that: 1) When jet Reynolds number, mass flow ratio and temperature ratio are matched to the real conditions, the Nusselt numbers predicted by low temperature conditions can be generally matched, exclude the forepart of the internal surface at the blade leading edge. But the error of overall cooling effectiveness on the blade external surface is not neglected. 2) The application of separators can prevent the heat transfer deterioration caused by cross flow with extending the heat exchange area, thus the leading edge with separators can be further cooled over 30 K compared to the traditional design. 3) Under the real operating conditions of gas turbine, two new correlations between the surface averaged Nusselt number with jet Reynolds number and temperature ratio are suggested.

Applying Bayesian Markov chain Monte Carlo (MCMC) modeling to predict the melting behavior of phase change materials

A key practical application of PCM-based spherical containers is in packed bed thermal energy storage (TES) devices utilized for air conditioning in large buildings. Proposing high-precision models for the melting characteristics in spherical containers can provide deep insights into the design of TES systems. Herein, a methodology based on Bayesian inference was adopted to provide reliable models that predict the melting rate (f) and surface-averaged Nusselt number (Nu) as two significant factors in PCM melting process in spherical heat storage units. Five proposed models for f and Nu prediction were applied to available datasets. The input of Bayesian-based predictive models included three important dimensionless parameters of the melting problem, namely Ste, Gr, and Fo. Ste (Stefan number) describes the driving force of melting, Gr (Grashof number) reflects the natural convection intensity, and Fo (Fourier number) represents the dimensionless time. For accurate inference of model parameters using the Bayesian Markov chain Monte Carlo (MCMC) technique, an analysis was performed by coding in the WinBUGS language. Several statistical performance criteria (R2, RMSE, and MSE) were utilized to evaluate the efficiency of Bayesian-based predictive models. The results showed that model #4 recorded R2 = 0.980 in the prediction of melting rate. Furthermore, model #5 with R2 = 0.966, had the best performance in predicting the values of surface-averaged Nusselt number. The models recommended in this research yield superior results compared to the previous study, which suggested R2 = 0.975 and R2 = 0. 925 for f and Nu, respectively.

Heat transfer characteristics in a narrow confined channel with discrete impingement cooling

Heat transfer characteristics in a narrow confined channel with discrete impingement cooling were investigated using thermal infrared camera. Detailed heat transfer distributions and comparisons on three surfaces with three impact diameters were experimentally studied in the range of Reynolds number of 3000 to 30000. The experimental results indicated that the strong impingement jet leaded to a high strength heat transfer zone in the ΔX=±2.5Dj range of the impact center, which was 1.3–2.5 times of the average heat transfer value of the impingement wall. With the same coolant mass flow rate, small diameter case had lower heat transfer coefficient on both inner wall and outside wall, while the impingement wall was insensitive to the impact diameter. The surface averaged Nusselt number of inner wall was only 43%–57% of impingement wall, while the outside wall can reach up to 80%–90%. The larger the diameter, the higher heat transfer enhancement and the smaller the channel flow resistance was observed in term of Reynolds number. The surface averaged Nusselt numbers were developed as the function of Reynolds number and the impingement height-to-diameter for further engineering applications.

Augmentation of tunnel-air convective heat transfer forced by duct ventilation in a geothermal construction tunnel

This work conducts numerical simulations and parameter analyses to investigate the augmentation performance of tunnel-air convective heat transfer (CHT) forced by duct ventilation in a dead-end tunnel concerned in tunnel space cooling inside a geothermal construction tunnel. Attributed to duct nozzle jet-flow, the CHT performance varies longitudinally and augments in the region close to duct nozzle, whereas holds a flat trend in the tunnel entrance region far away. In this work, the CHT is represented by surface-averaged Nusselt number (Nu) at the tunnel-air interface with a spatial resolution to the tunnel length, and the CHT augmentation region is highlighted by the bulge pattern of the Nu distribution curve. The three-dimensional Computational Fluid Dynamics is based on to simulate the actual CHT concerning both airflow turbulence and near-wall conditions, and the influences of parameters considering different (ⅰ) shapes of tunnel cross-section, (ⅱ) outlet airflow Reynolds number (Re), and (ⅲ) ventilation duct position and (ⅳ) diameter on the CHT augmentation are respectively identified. The results show the CHT augmentation region is less influenced by the shape of tunnel cross-section and Reynolds number but notably affected by ventilation duct diameter and position. A duct with a smaller diameter and position at the tunnel side is optimal to the CHT augmentation, and hence should be considered in the duct ventilation design for tunnel space cooling in a geothermal construction tunnel. Furthermore, a semi-empirical equation about the local Nu-Re correlation for the CHT augmentation region is proposed based on numerical results.

Forced convection from a sphere to power-law fluids in a tapered tube

The Poiseuille flow of power-law fluids past a heated sphere in a tapered tube is studied over the following ranges: Blockage ratio, BR (0.1 to 0.5), Separation ratio, SR (0.1 to 0.7), taper angle, α (1o to 20o), Reynolds number, Re (1 to 100), Prandtl number, Pr (10 to 100), and the power law index, n (0.2 to 1). The hydrodynamic force exerted on the sphere is expressed using the total drag coefficient and its pressure component. Both exhibit the expected inverse dependence on Re while it bears a positive dependence on n, SR and BR. The normalized drag for a confined sphere also exhibits a complex functional relationship with each of these parameters. The normalized drag is significantly influenced by the taper angle. In general, SR, BR and α delay the boundary layer separation and hence, stabilize the flow. Similarly, the heat transfer characteristics are described in terms of isotherms, local and surface average Nusselt number. The Nusselt number shows a positive relationship both with SR and BR. The taper angle exerts only a weak effect on the Nusselt number. The heat transfer coefficient is augmented up to 59% in shear-thinning fluids above that in a Newtonian fluid.

Effect of variable pitch on forced convection heat transfer around a helically twisted elliptic cylinder

The present study numerically evaluated the performance of the variable pitch helically twisted elliptical (VPHTE) cylinder on the forced convection heat transfer at the Reynolds number of 3000 by employing a large eddy simulation (LES). A parametric study on the variable pitch ratio (VP) and the simulations of the smooth and the helically twisted elliptical (HTE) cylinders for the purpose of the comparison were performed. The present results validated the VPHTE disturbance as the passive flow control to achieve the drag reduction and the suppression of the lift fluctuation, which is consistent with the findings of previous research. The VPHTE cylinders provided smaller values of total surface-averaged Nusselt number over the time than the smooth and HTE cylinders. In addition, the VPHTE cylinders considerably stabilized the time series of total surface-averaged Nusselt number. The VPHTE and HTE cylinders formed the regions containing the local peaks of time -averaged Nusselt number build the inclined elliptic formation with the center at the front stagnation point of the node. The distribution of the temperature contours clearly reflects the stably elongated shear layers and the delay of the vortex shedding due to the geometric disturbances of the HTE and VPHTE cylinders.

On heat transfer and flow characteristics of jets impinging onto concave surface with varying bleeding arrangements

Purpose The purpose of this study is to investigate the effects of film holes’ arrangements and jet Reynolds number on flow structure and heat transfer characteristics of jet impingement conjugated with film cooling in a semicylinder double wall channel. Design/methodology/approach Numerical simulations are used in this research. Streamlines on different sections, skin-friction lines, velocity, wall shear stress and turbulent kinetic energy contours near the concave target wall and vortices in the double channel are presented. Local Nusselt number contours and surface averaged Nusselt numbers are also obtained. Topology analysis is applied to further understand the fluid flow and is used in analyzing the heat transfer characteristics. Findings It is found that the arrangement of side films positioned far from the center jets helps to enhance the flow disturbance and heat transfer behind the film holes. The heat transfer uniformity for the case of 55° films arrangement angle is most improved and the thermal performance is the highest in this study. Originality/value The film holes’ arrangements effects on fluid flow and heat transfer in an impingement cooled concave channel are conducted. The flow structures in the channel and flow characteristics near target by topology pictures are first obtained for the confined film cooled impingement cases. The heat transfer distributions are analyzed with the flow characteristics. The highest heat transfer uniformity and thermal performance situation is obtained in present work.

Numerical simulation of the drag and heat-transfer characteristics around and through a porous particle based on the lattice Boltzmann method

Porous particle flow is universal in nature and industry. However, in previous numerical simulations, porous particles have usually been assumed to be solid. It is necessary to study the flow and heat-transfer characteristics around porous particles because they are greatly different from those of impermeable particles. In this study, two-dimensional steady flow and heat transfer around and through a porous particle with a constant temperature placed in a cold fluid were numerically investigated. The effects of the Reynolds number (Re) and Darcy number (Da) on the flow and heat-transfer characteristics were investigated in detail. The investigated ranges of the parameters were 10≤Re≤40 and 10−6≤Da≤10−2. It is sophisticated to simulate porous particles with traditional simulation methods because of their complicated structure. Therefore, the lattice Boltzmann method was used to solve the generalized macroscopic governing equations because of its simplicity. The drag coefficient decreased with increasing Re or Da, but the decrease was not prominent in the range 10−6≤Da≤10−4. The heat-transfer efficiency of the front surface was much stronger than that of the rear surface. The heat-transfer efficiency between the particle and the fluid increased with increasing Re or Da. However, for 10−6≤Da≤10−4, the increase was not prominent and the heat-transfer enhancement ratio was slightly larger than one. Furthermore, the effect of Da became more prominent at larger Re. In addition, new correlations for the drag coefficient and surface-averaged Nusselt number were obtained based on the simulated results.

Three-dimensional numerical simulation of ferrofluid magnetoconvection in cubical enclosure heated with an inner spherical hot block

In this study, three-dimensional computational analysis is performed to investigate the magnetoconvection of ferrofluid ([Formula: see text]-water) within a cubical enclosure heated by an inner spherical hot block. The ferrofluid, considered as a working fluid, is modeled as a single-phase fluid. The inner spherical block is put at high temperature while all the remaining walls of the enclosure are exposed to low temperature. Two radii values ([Formula: see text] and [Formula: see text]) of the inner hot sphere are examined. Governing equations with corresponding boundary conditions are solved numerically applying a second-order accurate finite volume method on a staggered grid system, using an accelerated multigrid model. Simulations are carried out based on various flow-governing parameters such as Rayleigh number [Formula: see text], Hartmann number [Formula: see text] and ferrofluid nanoparticle volume fraction [Formula: see text]. The effects of the pertinent parameters in the performance of the system are also studied. The flow and thermal fields, the local and surface-averaged Nusselt numbers on the sphere and the enclosure for both configurations are detailed. The flow remains steady and laminar for all Rayleigh numbers regardless of the sphere radius. Obviously, heat transfer rate improves with [Formula: see text] augmentation and minimizes with Ha decrease. At the highest Ra and lowest Ha, higher inner sphere radius shows significantly better heat transfer rate (more than [Formula: see text]). Useful correlations are presented to quantify the surface-averaged heat transfer rate through the cubical enclosure.

Thermofluid Characteristics on Natural and Mixed Convection Heat Transfer From a Vertical Rotating Hollow Cylinder Immersed in Air: A Numerical Exercise

Abstract Numerical investigations are performed on natural and mixed convection around stationary and rotating vertical heated hollow cylinder with negligible wall thickness suspended in the air. The fluid flow and heat transfer characterization around the hollow cylinder are obtained by varying the following parameters, namely, Rayleigh number (Ra), Reynolds number (ReD), and cylindrical aspect ratio (L/D). The heat transfer quantities are estimated by varying the Rayleigh number (Ra) from 104 to 108 and aspect ratio (L/D) ranging from 1 to 20. Steady mixed convection with active rotation of hollow vertical cylinder is further studied by varying the Reynolds number (ReD) from 0 to 2100. The velocity vectors and temperature contours are shown in order to understand the fluid flow and heat transfer around the vertical hollow cylinder for both rotating and nonrotating cases. The surface average Nusselt number trends are presented for various instances of Ra, ReD, and L/D and found out that the higher rate of heat loss from the cylinder wall occurs at high Ra, low L/D (short cylinder) and high ReD.

Natural convective flow and heat transfer on unconfined isothermal zigzag-shaped ribbed vertical surfaces

Natural convective heat transfer is commonly used as heat transfer mechanism in applications with low heat flux due to its reliability and cost effectiveness. In this study, we introduce zigzag-shaped ribs to vertical, isothermal, heated surfaces with the purpose of increasing natural convective heat transfer. The ribs are characterised by a rib height h, rib length p and vertical pitch distance L. We perform numerical simulations using the Boussinesq approximation by prescribing a linear density-temperature relation and investigate how changes in rib length p/L, rib height h/L affect heat transfer at GrL= 105 and GrL = 106 at Pr = 0.71.The results show how geometric variations affect heat transfer locally. Generally, local heat transfer increases along each outward-facing section and peaks at the tip of each rib. In the limiting case when surface approaches a forward facing step (e.g. p/L = 1), a significant decrease in heat transfer is observed on the horizontal section.A peak in heat transfer is observed for geometries with high rib lengths p/L = 0.9, where the surface-averaged Nusselt number is increased by 4.43% compared to the flat surface. This increases to 11.60% when correcting for the increase in surface area.