Correlation For Average Nusselt Number Research Articles

Numerical simulation of radial duel-stage nozzle vortex cooling at the leading edge of a gas turbine blade

Thermal failure is one of the major challenges faced by the leading edge of gas turbines, which significantly affects the improvement of their efficiency. The vortex cooling technique has become a promising solution to address this issue. Radial duel-stage nozzle vortex cooling (RDNVC) model, featured with a vortex tube and two sets of nozzles, is proposed based on the radial single-stage nozzle vortex cooling (RSNVC) configuration. This novel structure aims to intensify turbulence within the cooling fluid at the nozzle inlets, thereby improving heat transfer at the leading edge. The model improves turbulence in the cooling fluid, promoting better heat transfer at the leading edge. Numerical simulations with the SST k-ω turbulence model were performed to analyze heat transfer and flow characteristics at varying Reynolds numbers (Re). The results show that the non-uniform inlet velocity in RDNVC increases turbulence and vorticity inside the vortex tube, leading to improved heat transfer. At the same cooling airflow rate, RDNVC increased the global average Nusselt number by 23.56% compared to RSNVC. Nu increases with Re. Among the three models, RDNVC(6–2) (nozzle heights of 6 mm and 2 mm) showed the best thermal performance, followed by RDNVC(2–6), while RSNVC had the lowest thermal performance. For the same pressure drop, RDNVC(2–6) achieved an 8.42% improvement in heat transfer efficiency over RSNVC. Based on Re and geometry parameters, the empirical correlation for the global average Nusselt number of RDNVC is also proposed.

Jet array impingement heat transfer in a rectangular cavity with effusion holes

Various researchers have studied jet array impingement heat transfer in impingement/effusion cooling systems. However, there is a lack of research on impingement/effusion cooling systems installed within rectangular cavities that focus on the impact of the proximity of the jet hole to the cavity sidewalls on cooling performance. The main objective of this study is to investigate the flow and heat transfer characteristics of jet array impingement with effusion holes in a rectangular cavity, considering various spacings between the cavity sidewalls and the outermost jet hole. The design parameters in this study include the ratio of the jet hole pitch to jet hole diameter of 7.1, 10.0, and 16.7, and the ratio of the distance between the jet and impingement plates to jet hole diameter of 2, 6, and 10, with the Reynolds number based on the jet hole diameter ranging from 2500 to 15,000. Heat transfer characteristics in the stagnation region and wall jet region were examined using local Nusselt number distributions on the impingement surface, measured by liquid crystal thermography. The local Nusselt number was high in the stagnation region and decreased radially from the stagnation region as the wall jet region formed. The closer the outermost jet hole is to the sidewall, the higher the Nusselt number on the impingement surface near the sidewall. Moreover, the flow structure in the rectangular cavity was numerically investigated, and the velocity vectors and streamlines showed that primary and secondary vortices were generated in the middle of two neighboring jets and near the sidewall, respectively. This study also assessed previous average Nusselt number correlations. Based on experimentally determined average Nusselt number data with 54 center unit cells and 1296 side unit cells, new correlations to predict the average Nusselt number on the impingement surface in a rectangular cavity with effusion holes were developed.

Local Heat Transfer Distribution On Thin Metal Foil Impinged by Array of Free Surface Jets

Abstract The present study focuses on the measurement of the local heat transfer distribution of a smooth flat plate impinged by an array of free surface jets on a thin metal foil. Local Nusselt number distributions are measured for a fixed jet diameter (d = 3 mm). The jet arrays consist of perfectly round apertures arranged in a square pattern, with a uniform spacing of 4d between adjacent jets in both the streamwise and spanwise direction. A wide range of Reynolds numbers varying from 1000 to 12500 are covered in this study. The nozzle to plate spacing (z/d) is varied between 1 to 10. The effect of the Reynolds number and nozzle to plate spacing on the local and spanwise average Nusselt number distributions are studied. The local Nusselt number exhibits an increase with nozzle-to-plate spacing within the low Reynolds number range (Re = 1000 to 2500). However, for Reynolds numbers exceeding 2500, the influence of nozzle-to-plate spacing on Nusselt number distributions remains negligible up to a nozzle-to-plate distance (z/d) of 5. Beyond this point, there is a gradual decrease in the Nusselt number value. The Nusselt number value gradually decreases beyond z/d of 5. At a Reynolds number of 1500, the Nusselt number increases by 71% for z/d = 10 in comparison to z/d = 1. Empirical correlations for local and spanwise average Nusselt number are proposed which satisfactory predict the local as well as spanwise average Nusselt number distributions.

Local and area average Nusselt number correlation for a circular impinging jet over a flat plate

Local and area average Nusselt number correlation for a circular impinging jet over a flat plate

Heat transfer characteristics for multi-silicon ingots irradiation in a typical research reactor

A computational fluid dynamics (CFD) study is conducted for silicon ingot irradiation facility in a typical material testing reactor (MTR). The ingots are placed inside the irradiation device in staggered and in-line bundles. The configurations are varied according to the variations of longitudinal and transverse pitches Sl and St, respectively. For in-line bundles, Sl and St are changed in the range of 1.1, 1.2, 1.3, 1.4, and 1.5 of the ingot diameter. For staggered bundles, Sl is changed in the range of 1.1, 1.2, 1.3, 1.4, and 1.5 of the ingot diameter while St is changed in the range of 0.95, 1.0, 1.05, 1.1, and 1.15 of the ingot diameter. The silicon ingot volumetric heat generation Qv is allowed to change around its nominal value in the range of 0.25, 0.5, 0.75, 1.0, and 1.25 W/g. The effects of rotation of ingots during irradiation at velocities of 15, 30, and 60 rpm are investigated. The computational fluid dynamics code ANSYS FLUENT is employed to carry out the calculations. The temperature profile inside the silicon ingot during irradiation is of significant importance for uniform doping. Therefore, the impact of heat generation and bundle pitches on the temperature profile through the ingot is calculated and reported for all cases of this study. The results show that the maximum ingot temperature does not exceed the limits for each bundle at maximum heat generation. Two correlations for average Nusselt number for both aligned and staggered configurations are generated to describe heat transfer rates during natural convection regime. The rotation of ingots during irradiation showed considerable enhancement of heat transfer rates.

Secondary Flow and Heat Transfer Enhancement Characteristics of a Channel Formed by Circular Tube Bank Fins Stamped with Convex/Concave Pyramids

Thermal and flow characteristics of a channel formed by circular tube bank fins stamped with convex/concave pyramids were numerically investigated by ANSYS Fluent software, and the parametric studies on the thermal performance of the pyramid fin are examined for varying pyramid height (H p), pyramid length (L p), and pyramid width (W p) as Reynolds number (Re) ranged from 1297 to 4541. The pyramid fin provides superior thermal performance to the counterpart plain fin, which is ascribed to the punched pyramids effectively expanding heat transfer area and inducing the secondary flow to better fluid mixing, and the action of the convex/concave pyramids lead to the area of undesirable heat transfer dead zone behind the circular tube being significantly reduced. Under identical pumping power constraint, the thermal performance factor of 1.41 is achieved for H p = 1 mm, L p = 5.74 mm, and W p = 3 mm at Re = 3892, and the optimal pyramid height and width fin respectively are H p = 1 mm and W p = 3 mm for studied cases, while the optimal pyramid length is associated closely with Re. The correlations of average Nusselt number and friction factor are proposed to estimate thermal performance of the pyramid fin used in a finned circular tube heat exchanger.

Numerical studies of natural convection phenomena for a vertical cylinder with multiple lateral baffles in triangular and hexagonal enclosures

Numerical case studies for natural convection were conducted for triangular and hexagonal enclosures as a follow-up to a previous study of natural convection for a baffled cylinder in a square enclosure. For a Prandtl number of 0.71, a range of the length ratio corresponding to the spatial ratio was varied from 0.4 to 0.9, and Rayleigh numbers from 2.5 × 103 to 5.0 × 107 was considered. To model the buoyancy force, the Boussinesq approximation was adopted. The boundary condition of constant wall temperature was applied to the cylinder and enclosure surfaces. Overall trends of the heat transfer performance were separated into conduction- and convection-dominated regimes, similar to the previous study. Because of the wider space of the triangular enclosure, however, an advantage of the convection effect was observed in the triangular enclosure. Due to the short distance between the cylinder and enclosure, the hexagonal enclosure had a more conduction-dominated characteristic than that of the triangular enclosure. A strong correlation was identified between the location of the high-energy vortex and the Nusselt number distribution. Using the calculation results, an improved average Nusselt number correlation is proposed to include triangular, square and hexagonal enclosures.

Mixed Convection Heat Transfer From Swirling Open Spherical Cavity

Abstract This work reports a numerical study on mixed convection flows around a swirling spherical shaped open vessel in air within the laminar regime. This investigation is quite important and relevant in various industrial operations like centrifugal casting, formation of shield surfaces, thermal processing of different food stuffs, etc. This study aims to characterize the fluid flow and heat transfer behavior from both inner and outer surfaces of the open cavity. Governing differential equations, such as continuity, momentum, and energy are solved by using finite volume technique to describe the effect of relevant pertinent parameters over wide range: Rayleigh number (103≤Ra≤107), height to diameter ratio (0.15≤h/D≤0.95), and Reynolds number (0≤ReD≤300). It is observed that the plume is deformed greatly by swirling effect at higher ReD and lower Ra for a fixed h/D. The percentage of increase of heat transfer rate from ReD=0 to ReD≠ 0 is significantly higher at lower Ra for all cases of h/D. Lastly, a suitable correlation for average Nusselt number is proposed as a function of Ra, h/D, and ReD, which shows a satisfactory agreement with numerical data. This correlation is expected to be helpful for academic and industrial purposes.

Study on Flow and Heat Transfer Performance of a Rectangular Channel Filled with X-Shaped Truss Array under Operating Conditions of Gas Turbine Blades

In this investigation, the heat transfer and flow capabilities of an X-shaped truss array cooling channel under various operating conditions of gas turbine blades were thoroughly studied. The influence laws of the inlet Reynolds number, inlet turbulence intensity, wall heat flux and cooling medium (air, steam) on the heat transfer and flow performance of the X-shaped truss array channel were analyzed and summarized. The empirical correlations of friction coefficients and average Nusselt numbers with maximum deviations less than ± 14% were fitted. The results show that the inlet Reynolds number has the most significant effect on the flow and heat transfer performance of the X-shaped truss array channel. When the inlet Reynolds number increases from 20,000 to 200,000, the average Nusselt number of the X-shaped truss array channel is increased by 3.92 times, the friction coefficient is decreased by 12.88%, and the comprehensive thermal coefficient is decreased by 31.19%. Compared with the medium turbulence intensity of Tu = 5%, the average Nusselt number, friction coefficient and comprehensive thermal coefficient of the X-shaped truss array channel at Tu = 20% are increased by 3.70%, 2.51% and 2.79%, respectively. With the increase in the wall heat flux, the friction coefficient of the X-shaped truss array channel roughly shows a trend of first decreasing and then increasing, while the average Nusselt number and the comprehensive thermal coefficient show a trend of first rapidly increasing and then slightly decreasing or remaining unchanged. Compared with air cooling, the average Nusselt numbers of the X-shaped truss array channel of steam cooling are increased by 6.30% to 9.54%, and the corresponding friction coefficients and comprehensive thermal coefficients are decreased by 0.11% to 0.55% and 2.63% to 5.59%, respectively.

The Forced Convection Analysis of Water Alumina Nanofluid Flow through a 3D Annulus with Rotating Cylinders via κ−ε Turbulence Model

We investigated the dynamics of nanofluid and heat transfer in a three-dimensional circular annular using the κ−ε turbulence model and energy equations. The pipe contained two concentric and rotating cylinders with a constant speed in the tangential direction. A heat flux boundary condition was executed at the inner cylinder of the annular. The pipe was settled vertically, and water alumina nanofluid was allowed to enter, with the initial velocity depending on the Reynolds number, ranging from 30,000 to 60,000. The volume fraction of the solid particles was tested from 0.001 to 0.1. The speed of the rotation of the cylinders was tested in the range from 0.5 to 3.5. The simulations were developed using COMSOL Multiphysics 5.6, adopting the finite element procedure for governing equations. The results were validated using the mesh independent study and the average Nusselt number correlations. We found that the average Nusselt number in the middle of the channel decreases linearly with the increase in the volume fraction of the water alumina nanofluid. The novelty of the present work is that various correlations between the average Nusselt number and volume fraction were determined by fixing the Reynolds number and the rotation of the inner cylinder. We also found that fixing the Reynolds number and the volume fraction improves the average Nusselt number at the outlet linearly. In addition, it was stated that the increase in the total mass of the nanofluid would decrease the average temperature at the outer cylinder of the annular. Moreover, the maximum average improvement percentage in the average Nusselt number, which is about 21%, was observed when the inner cylinder rotation was changed from 1.5 to 2.5 m/s.

Free Convection Heat Transfer From a Spherical Shaped Open Cavity

Abstract A computational work is performed on laminar free convection from an isothermally heated spherical shaped open cavity with negligible wall thickness suspended in the air. Fluid flow and heat transfer are analyzed in detail by solving governing differential equations (continuity, momentum, and energy) numerically over wide ranges of the relevant dimensionless parameters, namely, Rayleigh number, 104 ≤ Ra ≤ 108; and height to diameter ratio, 0.15 ≤ h/D ≤ 0.95. The detailed behavior of thermal and flow fields is delineated by suitable visualization techniques for different Ra and h/D. The influence of Ra and h/D on the local and average Nusselt number is also predicted and it is observed that the average Nusselt number on both outer and inner surfaces decreases with the increase of h/D for a constant value of Ra. A suitable correlation for the net average Nusselt number is obtained for the spherical-shaped open vessel surface as a function of Ra, and h/D based on the computed data points, which is expected to be relevant for various academic and industrial operations. This study can be helpful in various industrial operations, such as heat treatment of foodstuffs, shield surfaces, thermal insulations, melting of polymer pellets, and fluidized reactors.

Natural-Convection Heat Transfer from Upward/Downward/Sideward Converged or Parallel Finned Hemispheres

Natural-convection heat transfer from a finned hemisphere is solved in the laminar range of Rayleigh number ( to ) to predict the flowfield and the temperature distribution around it. Two different types of fins, namely, converged and parallel fins, have been employed with fin height varied from to and fin number from 8 to 40. The hemisphere is placed either upward, downward, or sideward to observe the orientation effect on the heat transfer. It is found that Nusselt number for the downward and sideward finned hemisphere is less than that of the upward finned hemisphere for greater than 20. The hemisphere with parallel fins results in greater Nusselt number than converged fins at any Rayleigh number for upward and downward hemispheres. The situation reverses for the downward hemisphere at a high of . For a sideward hemisphere, the Nusselt number for converged fins is more compared to parallel fins. The hemisphere with converged fins is found to be more effective in heat transfer because such fins generate more surface area for heat transfer. Eventually, the correlations for average surface Nusselt number against various input pertinent parameters have been proposed for different orientations of the hemisphere.

Rarefaction Effect, Heat Transfer, and Drag Coefficient for Gas Flow Around Square Cylinder in Transition Flow Regime

Abstract In this work, rarefaction effect, heat transfer, and drag coefficient for gas flow around a square cylinder in transition flow regime are numerically studied using unified gas kinetic scheme (UGKS). To reduce computational cost, a mirror symmetry boundary treatment for UGKS is proposed and applied in this study. It is found that: (1) velocity slip is not obvious on upwind and downwind surfaces of square cylinder due to the tangential gas flow being weak in these surfaces; (2) with the increase of local Knudsen number, velocity slip on upper surface always increases, but temperature jump can decrease, which indicates that Knudsen number is not the decisive parameter to characterize temperature jump; (3) the heat transfer between gas and square cylinder enhances with the increase of inflow Knudsen number and cylinder temperature due to the increase of temperature jump and deteriorates with the increase of inflow Mach number on account of gas stagnation; and (4) the drag coefficient increases with the increase of inflow Knudsen number, the decrease of inflow Mach number, and the increase of cylinder temperature. To further predict the variation of average Nusselt number and drag coefficient, correlations for average Nusselt number and drag coefficient with inflow Mach number ranging from 0.05 to 0.3, inflow Knudsen number ranging from 0.1 to 10, and cylinder temperature ranging from 320 K to 380 K are proposed. This research can improve the understanding for mechanisms of gas flow and heat transfer in micro-electromechanical system (MEMS) devices.

Magneto-turbulent natural convection and entropy generation analyses in liquid sodium-filled cavity partially heated and cooled from sidewalls with circular blocks

Liquid sodium is a promising candidate working fluid in the new solar system and high-density heat sources in industrial applications. In this study free convective flow, heat transfer, and entropy generation of liquid sodium in a hot square enclosure with 16 and 64 cylindrical solid blocks are studied. The cavity is partially heated and cooled from sidewalls with three different locations of heater and cooler named CONF1–3. The effects of the magnetic field in the form of Hartmann number (Ha = 0 and 100), and flow condition in the form of the Rayleigh number (Ra = 106, and 107) are investigated. For tuning the CFD code primitive parameter, the dimensionless forms of the governing equations are used. The ANSYS Fluent equations solver conjugated with a nondimesionlization scheme are used in the turbulent region due to the behavior of liquid metal. The grid verification and validation tests are carried out to ensure the accuracy of the data. Two correlations for average Nusselt number and entropy generation are proposed to simplify calculations. The results discovered that the average Nusselt number and entropy generation increase with increasing the Rayleigh number for each value of the Hartmann number.

Numerical study of natural convection heat transfer of vertical cylinder with multiple lateral baffles in square enclosure for sodium beta-alumina batteries

A computational study was carried out to assess the natural convection around a cylinder having multiple lateral baffles in a square enclosure for length ratios from 0.4 to 0.9, Rayleigh numbers from 2.5 × 103 to 5.0 × 107, and Prandtl number 0.71. Using a three-dimensional calculation with the Boussinesq approximation, the average Nusselt numbers on the cylinder and the enclosure surfaces were summarized. For low Rayleigh numbers, a conduction-dominated regime was found, showing that the average Nusselt number on the cylinder surface is mainly dependent on the distance between the hot and cold surfaces. For high Rayleigh numbers, the thermal field became convection-dominated, and the case with large cavity spacing showed large heat transfer. A new average Nusselt number correlation was proposed with and without lateral baffles on the cylinder surface. The flow pattern in the cavity was identified for various conditions, and the stratified temperature distribution and corresponding local Nusselt number distributions were summarized. Finally, the contribution of the vortices in the enclosure to the Nusselt number was assessed by adopting Q-criterion analysis. Presents results will provide useful information for thermal design of hot box modules that contain stacked planar battery cells.

Performance enhancement of latent energy storage system using effective designs of tubes and shell

This study aims to numerically investigate the effects of geometric designs of tubes and shell on thermal performance enhancement of latent thermal energy storage system (LTESS). Stearic acid is used as a phase change material (PCM) while water acts as heat transfer fluid (HTF). Starting with a base case consisting of three circular HTF tubes within a circular shell, the tube and shell geometries are modified systematically. First, the effect of tube shapes and their orientations are investigated in detail. The circular exteriors of HTF tubes are modified with hexagonal, pentagonal, square and triangular shapes. The performance of triangular tubes with the vertex pointing downward exceeds all the other tube configurations. It augments the melting rate of the PCM by 27.2% and the energy storage capacity of the LTESS by 3.72%, as compared to the base case. The bottom vertex angle of the best HTF tube design is then varied yielding 45∘ as the optimum triangular tube configuration. It improves the energy storage capability of the LTESS by 7.61% and the melting rate of the PCM by 41.4%. Following the optimum HTF tube design, the triangulated shell designs with various bottom vertex angles are explored. The 75∘ bottom vertex angle of the shell offers maximum improvement. It accelerates the melting rate of the PCM by 66.9% while enhancing the energy storage capacity of the LTESS by 23.7% in comparison to the base case. Lastly, two new correlations of melting Fourier number and average Nusselt number are proposed for the optimum LTESS design configuration.

An experimental investigation of the flow-field and thermal characteristics of synthetic jet impingement with different waveforms

Present experimental investigation reports the flow and thermal behavior of electromagnetically actuated circular orifice synthetic jet involving sinusoidal and square waveforms. The effect of Reynolds number (1656–2218), dimensionless axial distance (1–25), and the excitation frequency (100–175 Hz) on the thermal performance is studied here. The flow field is investigated using the hot wire anemometry and particle image velocimetry technique to couple the thermal behavior with fluid dynamics. The surface temperature is measured using an infrared thermal imaging camera. The average Nusselt number for synthetic jet with sinusoidal waveform is found to be 20.86% higher compared to the synthetic jet with square waveform. However, in the stagnation region, the square waveform exhibits 18.94% higher heat transfer compared to the sinusoidal waveform. The flow dynamics reveal the underline flow physics for the variation in the heat transfer rate. The velocity contours show a smooth and wide top hat with the sinusoidal waveform. However, in the case of square waveform, the velocity profile is narrow, which results in a decrease in entrainment of fresh ambient fluid into the jet. The sine wave exhibits 73.68% higher mass flow rate compared to square wave pattern at its diaphragm resonance frequency (150 Hz). The power spectral density of the velocity signal, obtained from the hot wire anemometry, is analyzed to understand the formation and propagation of vortex rings. A unified correlation for average Nusselt number is developed for both sinusoidal and square waveforms. The results obtained from the present experimental investigation provide significant guidance for the efficient design of synthetic jets for space-constrained electronics cooling.

Performance enhancement of double-wall-heated rectangular latent thermal energy storage unit through effective design of fins

This study aims to numerically investigate the influence of different designs of fins on the performance enhancement of a double-wall-heated rectangular thermal energy storage unit (TESU). Stearic acid is used as the phase change material (PCM) which is enclosed in a rectangular enclosure with two vertical heated isothermal walls and two adiabatic horizontal walls. The rectangular horizontal fins assist the heated walls during the melting process of the PCM. The design parameter variations of fins include angles (A), shapes (S), lengths (L) while maintaining constant PCM volume. The base case is modeled by embedding heated six horizontal fins, three on each of the heated walls. configuration-A, which incorporates three variations of fin angles, significantly increases the melting time of PCM by 16% and reduces the averaged energy storage rate by 14.2%. Whereas, the combination of straight and angled type hybrid fins of configuration-S show appreciable improvement in thermal performance of TESU by enhancing the melting rate by 18% and average energy storage rate by 19.8%. The configuration-L, encompassing combinations of straight fins with different lengths and thicknesses is found to be most effective of all. The optimum case of configuration-L augments the melting performance of the TESU by 39.5% and the average energy storage rate by 65.07%. Moreover, for optimum configuration, two new correlations of melting Fourier number and average Nusselt number as a function of Stefan number and Rayleigh number, respectively, are proposed for a wide range of wall temperatures.

Investigations of Flow and Heat Transfer Characteristics in a Channel Impingement Cooling Configuration with a Single Row of Water Jets

The present study experimentally and numerically investigates the effect of channel height on the flow and heat transfer characteristics of a channel impingement cooling configuration for various jet Reynolds numbers in the range of 2000–8600. A single array consisting of eleven jets with 0.8 mm diameter injects water into the channel with 2 mm width at four different channel heights (3, 4, 5, and 6 mm). The average heat transfer coefficients at the target surface are measured by maintaining a temperature difference between the jet exit and the target surface in the range of 15–17 °C for each channel height. The experimental results show the average heat transfer coefficient at the target surface increases with the jet Reynolds number and decreases with the channel height. An average Nusselt number correlation is developed based on 85 experimentally measured data points with a mean absolute error of less than 4.31%. The numerical simulation accurately predicts the overall heat transfer rate within 10% error. The numerical results are analyzed to investigate the flow structure and its effect on the local heat transfer characteristics. The present study advances the primary understanding of the flow and heat transfer characteristics of the channel impingement cooling configuration with liquid jets.

A novel strategy to enhance the cooling effectiveness in a confined porous jet impingement using non-Newtonian fluids

Jet impingement configuration is commonly employed in the thermal management of the gas turbine engines, rocket launchers, and high-density electrical equipment cooling to remove a large amount of heat. Owing to such overwhelming applications, significant research efforts have been devoted to enhance the cooling performance of jet impingement configuration. In this work, the influence of shear-thinning behaviour in overall heat transfer characteristics have been investigated as a function of pertinent dimensionless numbers as 10−6 ≤ Da ≤ 10−2 (Darcy number), 1 ≤ Pe ≤ 102 (Péclet number), 0 ≤ Ri ≤ 4 (Richardson number), 0.4 ≤ n ≤ 1 (shear-thinning index). The numerical results on local thermal non-equilibrium (LTNE) parameter, streamlines and isotherms, local and average Nusselt number are reported for a range of power-law index, Péclet number and Richardson number. All in all, shear-thinning fluids offer 50–60% enhancement in Nusselt number with reference to the Newtonian fluid at a higher Darcy number. A correlation for average Nusselt number is put forward to enable its use for process design calculations.