Nusselt Number Correlations Research Articles

Fast thermodynamic simulation of electrical control cabinets via thermal resistor networks

A fine-grid Computational Fluid Dynamics (CFD) simulation without thermo-electric coupling provides an accurate prediction of temperatures inside low-voltage control cabinets. But due to its high computational demand and complexity, the approach is not eligible for a widespread use in pre-production design. This study presents a lightweight and robust model based on the thermal-electrical analogy and Kirchhoff’s current law. The thermal resistor network is created for each control cabinet individually. An isothermal coarse-grid CFD simulation is used to approximate flow resistances. Convective resistances are calculated with Nusselt number correlations. The approach was tested on 120 randomly generated 2D control cabinet configurations. Component temperature errors were evaluated based on a normal distribution approximation. For moderate temperature rises up to 20 K the model had a mean error of 0.38 K and a standard error of 1.51 K compared to the reference fine-grid CFD model and was up to 12.8 times faster.

Development of correlations of Nusselt number and friction factor of solar thermal collector equipped with hybrid rib roughness

In this advanced and comprehensive research, a pioneering hybrid rib configuration, amalgamating arc and V-shaped ribs, is precisely engineered to delve into the complex synergistic effects between this cutting-edge rib design and the heat transfer and fluid flow behavior within a solar thermal collector (STC) specifically utilized for direct air heating. The focal point of this investigation is the impact of the rib pattern on pivotal performance indicators, notably the Nusselt number (Nu) and the friction factor (f). This specific rib roughness design emanates from an exhaustive analysis of a spectrum of non-dimensional roughness parameters: the arc angle (α), the attack angle (β), the relative positioning of the V ribs (D/w), the relative spacing of the ribs (p/e), and the rib height (e/D). These parameters are studied within defined ranges: 30̊‐60̊, 30̊‐60̊, 0.2–1, 8–12, and 0.023–0.043, respectively. The results of highlight a significant improvement in heat transport, as indicated by a significant increase in the Nusselt number (Nu), which has risen impressively by a factor of 3.08. Furthermore, this paper presents development of correlations for Nu and f, taking into account the roughness parameters of the hybrid ribs and the flow Reynolds number, in order to precisely forecast these key performance measurements.

Thermal and hydraulic performance of volumetrically heated triply periodic minimal surface heaters

Advanced heat transfer surfaces, mainly Triply Periodic Minimal Surfaces (TPMSs) have great potential to increase heat transfer performance over traditional heat transfer surfaces due to their tortuous flow path and higher surface area-volume ratio. The emergence of additive manufacturing of nuclear fuel motivates experimental investigation of such advanced geometries in the context of a nuclear reactor core. This study employed a novel method to explore the performance of gyroid and diamond TPMS geometries. Volumetrically heated heaters were additively manufactured from conductive polymer filament, and airflow was used to simulate the convective heat transfer from the core within a nuclear reactor. The mass flow and volumetric generation rates were varied, local Nusselt number correlations were obtained using embedded thermocouples to measure the solid and fluid temperature, and friction factor correlations were developed using differential pressure measurements. The gyroid TPMS presented a friction factor 1.5 times higher than the diamond TPMS and 90 times that of the rod bundle. TPMS geometries showed Nusselt numbers 8–10 times higher than that of the rod bundle, also maintaining colder internal temperatures, which suggested that they could sustain 250–275 % higher power density than the rod bundle for a given centerline temperature. Hence, having the potential to greatly increase the safety margins and power output of a nuclear reactor core with similar volume.

Nusselt number analysis of printed circuit heat exchangers with straight and zigzag channels

Printed Circuit Heat Exchangers (PCHEs) are applied in processes involving high temperatures and working pressures, such as the supercritical CO2 Brayton cycle. These devices are typically designed with semi-circular channels and zigzag paths to enhance thermal performance. In recent years, several Nusselt number correlations for zigzag channels have been proposed to estimate the thermal performance. However, under similar operating conditions and zigzag angles, these correlations yield divergent results for supercritical and non-supercritical condition. No study has compared these correlations to identify the most accurate one. Therefore, this research performed a thermal analysis of PCHEs operating with straight and zigzag channels, using fluids in normal and supercritical conditions. Additionally, it was evaluated existing Nusselt number correlations to determine which ones presents the best result. This study examined 37 Nusselt number correlations, analyzed four experimental studies available in the literature and evaluated a total of 374 experimental data. For straight channels, the error ranged from 3% to 27%, whereas for zigzag channels, the error varied between 3% and 84%. This demonstrates how correlations developed for the same application can lead to discrepant results.

Machine learning-based Nusselt number prediction for falling-film evaporators in absorption refrigeration systems

The evaporator serves as a pivotal component of an absorption refrigeration system (ARS), bolstered by improved wettability and mitigation of the irregular arrangement of the liquid film. The complexities of the evaporation heat transfer characteristics of falling films have led to numerous empirical correlations for falling-film Nusselt number (Nu) estimation, demanding substantial computational resources and experimental expenses and limiting their universal applicability. This study used machine learning to predict Nu of a falling-film evaporator. Artificial neural network (ANN) and random forest regression (RFR) models were used to predict heat transfer characteristics and establish a novel Nu correlation using an experimentally derived dataset. Correlations of falling-film Nu were established for six tube types using empirical data, ANN, and RFR predictions and then compared based on experimental falling-film Nu. The ANN and RFR provided high prediction performance within 10 % error and an R2 of 0.985 and 0.999, respectively. The Nu correlations derived from ANN and RFR demonstrated satisfactory agreement, typically within ±15 % error of experimental results lower than that of the empirical correlations. The optimized ANN and RFR demonstrates their promising ability to predict heat transfer characteristics and establish Nu correlations, thereby reducing the need for immense experimental efforts.

Natural Convection Immersion Cooling of the Cylinders in Nanofluids: Developing a New Nusselt Number Correlation

Natural Convection Immersion Cooling of the Cylinders in Nanofluids: Developing a New Nusselt Number Correlation

Tilt Angle Effect on Natural Convection Heat Transfer from an Inclined Array of Square Cylinders

Natural convection heat transfer from an inclined array of square cross-section cylinders is experimentally studied. Three cylinders of side length D = 0.02 m and 1 m length are used to form the inclined array. Three-cylinder axis inclination angles to the horizontal of 30°, 45°, and 60° are considered. The cylinders are heated using an internal heating element of constant heat flux. Surface temperatures are measured along the three surfaces of each cylinder; upper, lower, and the front side at nine points spaced evenly by 10 cm. The local circumference average temperature is obtained at each point. Four center-to-center distances to side length S/D = 1.25, 1.75, 2.25, and 2.75 are used. Local Nusselt numbers and the modified Rayleigh numbers are obtained for each cylinder at each circumference averaged temperature point. Results show that the lower cylinder heat transfer is least affected by the array compared to that of its single one followed by the middle and the upper cylinder for all S/D and the inclination angles used. The Nusselt number is degraded from that of the single cylinder at a small S/D and that degradation decreases as the S/D increases for all angles. It is observed that the Nusselt number enhances at the small cylinder axis tilt angle of 30°, followed by 45°, and 60°, especially at high modified Rayleigh numbers. New novel general empirical correlations for Nusselt numbers are obtained for each S/D using the cylinder axis tilt angle, the modified Rayleigh numbers, and the cylinder number in the array as parameters. A new general correlation is obtained for the array using the modified Rayleigh number, S/D, the cylinder axis tilt angle, and the cylinder sequential order number as parameters. These new correlations will help any engineering applications using such a configuration of the array.

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.

Effect of turbulent fluctuation on the ignition of millimeter particle: Experimental studies and numerical modelling with a new correlation of nusselt number

Understanding the influencing mechanism of turbulent fluctuation on the ignition characteristics of millimeter coal particles is essential. In this work, to study the effect of turbulent fluctuation on ignition time, millimeter coal particles are subjected to a specific flow field, generated in a furnace with symmetric fans. A one-dimensional model with the new proposed correlation and the Ranz-Marshall (R-M) correlation for Nu (Nusselt number) is established to simulate the coal ignition process. In addition, the effects of fan speed, temperature, particle diameter, particle distance and coal type on the ignition time are investigated. It is found that an increase in fan speed from 0 to 3000 rpm leads to a particle Reynolds number Rep increase from 0 to 22.5, and a turbulent particle Reynolds number Ret∗ increase from 0 to 71.5. With a consideration of the fluctuation effect, the new correlation of Nu gives a better prediction of ignition time compared to the R-M correlation. Moreover, the ignition time is revealed to decrease with an increasing fan speed and an elevating temperature. While the ignition time shows merely an initial boost with enlarging particle distance, it exhibits a linearity with the term of particle diameter dp1.3–1.7 and Reynolds numbers (Nu∗/Nu)–0.6 (Nu∗ is turbulent Nusselt number). Based on this relationship, the difference of predicted ignition time is calculated at different Rep and Ret∗. It is shown that at low Rep or high Ret∗ values, the new correlation should substitute for the R-M correlation.

Computation of natural convection heat transfer from an infrared suppression device (IRS) with multiple louvered cylindrical funnels

The current computational work investigates the buoyancy-driven flow and heat transfer in and around an Infrared suppression device (IRS) with perforated cylindrical funnels. It reports the effects of three critical parameters: the louver size ( 0.025 ≤ d / D ≤ 0.2 ) , diameter ratio of the funnel ( 1.02 ≤ DR ≤ 1.3 ) , and Rayleigh number ( 10 10 ≤ Ra ≤ 10 12 ) on the heat transfer rate, mass suction rate, thermal plume, and velocity vectors. It also proposes suitable correlations for the average Nusselt number and mass suction rate, having an accuracy of 5%, and 6%, respectively. The study’s outcomes show that the heat transfer rate increases with the louver size, only up to a particular value of diameter ratio, beyond which it reduces. However, the heat transfer rate with a louvered funnel is more than the plane funnels, except when DR =1.15 and 1.2 and d/D = 0.2 (approximately). The work also reports the temperature versus time curve for the IRS.

Impacts of hydrocarbons impurities on heat transfer deterioration

Hydrocarbon impurities in Liquified Natural Gas can lead to unwanted occurrences like heat transfer deterioration and pseudo-boiling phenomena. It can pose a challenge for cooling channels in rocket engines. This study examines the impacts of hydrocarbon impurities on heat transfer deterioration in cooling channels. The Reynolds averaged Navier Stokes equation is solved numerically for different commercial LNG flows in circular pipes under supercritical conditions. The thermophysical and transport properties are calculated using the 2008-GERG Equation of states and extended corresponding states models. The simulations reveal that hydrocarbon impurities significantly impact heat transfer deterioration. These impacts vary depending on the concentrations of hydrocarbons. A Nusselt number correlation equation is derived numerically, which describes the heat transfer deterioration of multiple LNGs. In addition, the effects of roughness heights to minimize heat transfer deterioration are studied in detail.

A universal ANN-based approach predicting PCHEs’ off-design performance across various operating conditions of sCO2 RCBCs

This paper proposes a universal approach that employs artificial neural networks (ANN) to predict the off-design performance of printed circuit heat exchangers (PCHEs) using supercritical carbon dioxide (sCO2) fluid. PCHEs are widely studied precoolers and recuperators of sCO2 power cycles, for which predicting their off-design heat transfer performance is challenging due to the fluid’s dramatic non-linear property variations. Due to their narrow valid range and constrained expressing ability, the existing empirical correlations might introduce significant out-of-scope errors that result in diverged simulation results, slow convergence and abnormal quitting. This work addresses the issue by proposing an ANN-based Nusselt number correlation, ANNu, to predict the off-design performance of PCHEs quickly and universally over the vast operating range of sCO2 cycles. Multiple techniques were adopted to promote ANNu’s prediction accuracy and generalization, such as consolidating 494 training data, data-driven input selection, and global optimization for the networks’ architecture and hyperparameters. This paper comprehensively evaluates ANNu’s performance by comparing its predictions with ten empirical correlations over four typical real-world scenarios, where ANNu features an overall MSE of 83.4222 and overall MAPE of 16.285 % that rank the second best and third best among eleven candidates (even overpassing two references), respectively. Moreover, ANNu exhibits good generalization and reliability, featuring MAPEs ranging from 9.89 % to 27.73 % when processing unmet experiment data from sCO2 PCHE applications.

An open-source porous media modelling approach to investigate thermohydraulic features of compact printed circuit heat exchangers

An experimental air-foil printed circuit heat exchanger (PCHE) with CO2 as a working fluid is numerically modelled and developed within the OpenFOAM environment as a freely available package for more general PCHE designs. The conjugate heat transfer solver (chtMultiRegionFoam) is adapted to include both the hot and cold fluid streams of the PCHE, along with the solid recuperator body, within three uniquely specified overlapping mesh regions. The fluid stream momentum equation is adapted to incorporate porosity, with the additional streamwise air-foil drag (friction factor) accounted for by the Darcy–Forcheimer porous media model. A simple linear ad-hoc model for the transverse friction factor is evaluated to determine the dispersion of the momentum across the flow. Heat transfer between the fluid streams and the solid body is driven by a volumetric thermal resistance with a cell volume-weighted interpolation method (volume-to-volume coupling), with experimentally determined Nusselt number correlations applied. Temperature-dependent parameters based on isobaric NIST data for CO2 are tabulated as a user library and integrated within the coding package. The model predictions of the solid body temperature distributions are assessed and validated against experimental data with eight equidistant fibre optical measurements across the PCHE core and compared against finite-element-based MATLAB numerical results. Thermal stresses are evaluated and qualitatively evaluated against experimental data, demonstrating the capability of the model to highlight potential design features for improvement.

Numerical investigation of unsteady heat transfer from a vertical helical coil

A comprehensive investigation was conducted utilizing three-dimensional transient Computational Fluid Dynamics (CFD) to analyze conjugate heat transfer, with a specific focus on determining the cooling time of a hot vertical helical coil. The heat transfer characteristics of the helical coil are assessed numerically with respect to parameters within certain ranges such as Rayleigh number 104≤Ra≤108, surface emissivity 0≤ε≤1, and geometrical parameters of the coil, such as coil-to-rod diameter ratio 8≤D/d≤24, and pitch-to-rod diameter ratio 3≤p/d≤7.5. Furthermore, a lumped heat capacitance model is also proposed, which significantly reduces computational time while maintaining accuracy by utilizing the steady-state correlation of Nusselt number vs. Rayleigh number (Nu vs. Ra) and other pertinent geometrical parameters of the coil. Comparative analysis between the lumped heat capacitance model and transient simulations reveals a remarkable correspondence, with deviations not exceeding 2%. Validation of the computational model is conducted against experimental data, showing good agreement. Results indicate that coils with higher pitch-to-rod diameter ratio (p/d) and coil-to-rod diameter ratio (D/d) dissipate heat more quickly, while higher initial Rayleigh numbers lead to faster cooling rates. Temperature distribution and Nusselt number variations during the cooling process are analyzed and compared between steady and unsteady behavior. The study provides valuable insights into the cooling behavior of helical coils, crucial for industrial applications requiring efficient heat transfer.

Thermo-hydraulic performance of solar air heater with built-in one-eighth sphere vortex generators

The primary objective of optimizing solar air heaters is to enhance their thermal efficiency, and the incorporation of perturbing elements into the absorber plate has proven to be an effective approach. The present study employed an electric heating plate to simulate solar heating and proposed a novel one-eighth sphere vortex generator to create strong longitudinal vortices and weak transverse vortices, thereby achieving enhanced overall performance. The experimental investigation focused on examining the impact of parameters, including deflection angle (α), relative transverse pitch (S/H), relative longitudinal pitch (L/H), relative height (e/H), and Reynolds number (Re), on both heat transfer and flow characteristics. The results show that new vortex generator performs best at deflection angles of 180°. The maximum thermo-hydraulic performance parameter of 2.03 was observed in this study. Additionally, it was found that the maximum Nusselt number (Nu) and friction factor (f) were 2.45 and 1.96 times higher than those obtained for a smooth duct, respectively. New correlations of Nusselt number (Nu) and friction factor (f) were developed to predict the heat transfer and flow resistance characteristics of solar air heater with built-in one-eighth sphere vortex generators. This study proposes a new idea to enhance the performance of solar air heaters.

Experimentally determining the thermophysical properties, heat transfer and friction factor Fe3O4-TiO2 magnetic hybrid nanofluids in a mini-heat sink under magnetic field: Proposing new correlations

The heat transfer and friction factor for the Fe3O4-TiO2/water nanofluids circulating in a mini-heat sink of comparable magnitude to personal computer or electronic gadgets with and without a magnetic field have been estimated experimentally. The Fe3O4-TiO2 nanomaterial was prepared using chemical coprecipitation and sol–gel technique. Synthesized Fe3O4-TiO2 nanoparticles were characterized by X-ray diffraction, and vibrating sample magnetometer. The water-based hybrid nanofluids were then used in a series of experiments for assessing heat transfer, and friction factor of a mini-heat sink assembly of five circular channels with 4 mm in diameter and 50 mm in length. The experiments were performed in the operating range: 238.66 < Re < 1873.36; 0 <ϕ < 2 %; and 0 < Gauss < 1000, respectively. Constant heat flux boundary condition was maintained at the bottom surface of the heat sink and an external magnetic field was provided with an electromagnet. Results indicate that, by using the Fe3O4-TiO2 hybrid nanofluid of 2 % vol. the Nusselt number, with no magnetic field, is enhanced over that of water by 38.16 %, and by 88.93 % when the magnetic field of 1000 gauss is applied. However, the hybrid nanofluid of 2 vol% imposed a penalty in increase in friction factor of 25.87 % without magnetic field, and 67.64 % when magnetic field of 1000 G is applied at a Reynolds number of 1873.33 over water data. The experimental data were used to develop generalized Nusselt number and friction factor correlations for a mini-heat sink subjected to an external magnetic field.

Numerical investigation of natural convection flow inside a square enclosure filled with different nanofluids by using two-phase Eulerian–Eulerian model: A new correlation for Nusselt number

Numerical investigation of natural convection flow inside a square enclosure filled with different nanofluids by using two-phase Eulerian–Eulerian model: A new correlation for Nusselt number

Study on the thermal performance of the cross-flow plate heat exchangers

The exhaust gas emitted by a stenter machine will lead to energy waste and environmental heat pollution. Therefore, designing an efficient waste heat recovery apparatus for exhaust gas not only protects the environment but also saves production costs. In the article, the heat transfer performance and pressure drop of three types of heat exchangers: flat-plate heat exchanger (PHE), wavy-PHE, and zigzag-PHE were numerically studied in the context of waste heat recovery from the exhaust gas of stenter machines. The heat transfer rate (Q), Nusselt number (Nu), Colburn factor (j), pressure drop, Fanning friction factor (f), and comprehensive performance factor (JF) were investigated while Reynolds numbers ranged from 1000 to 9000. An experimental measurement was devised to validate the accuracy of the numerical simulation. The results of the research indicate that the wavy-PHE exhibits improved heat transfer efficiency and reduced pressure drop, primarily attributed to the substantial generation of secondary flow. Three new correlations for Nusselt number in different PHEs were developed based on commonly existing empirical formulas. The prediction accuracy for the Nusselt number based on the new correlations has been significantly improved in the present study.

Laminar Flow Heat Transfer Studies in a Twisted Square Duct for Constant Wall Temperature

Abstract: Three dimensional analysis of steady fully developed laminar flow inside twisted duct of square cross section flow area is studied for Reynolds number range of 100 – 2000 using a commercially available software. Twist ratio used are 2.5, 5, 10 and 20. Heat transfer results are generated for a uniform wall temperature case for Prandtl number range of 0.7 to 20. Maximum values for the product of friction factor and Reynolds number is observed for a twist ratio of 2.5 and Reynold number of 2000. Maximum Nusselt number is observed for the same values along with Prandtl number of 20. Correlations for friction factor and Nusselt number are developed involving swirl parameter. Local distribution of friction factor ratio and Nusselt number across a cross-section is presented. Heat transfer enhancement for Reynold number of 2000 and Prandtl number of 0.7 for twist ratio of 2.5, 5, 10, and 20 are 33 %, 18 %, 10 % and 7 % respectively. The results are significant because it will contribute to development of compact heat exchanger.

Thermohydraulic performance parameter based experimental investigation of frustum shaped roughened solar air heater

An outdoor experimental investigation was conducted to explore the thermohydraulic characteristic (Nusselt no. and friction factor) of artificial roughened solar air heater (ARSAH) having frustrum shaped roughened element on absorber plate. Data were recorded and further processed to generate Nusselt number and friction factor correlation to predict the effect of flow and roughness parameter on performance of ARSAH. The investigation was conducted by varying Reynolds number in range of 2500–12,500. Dimensionless parameter for Frustrum shaped roughness was relative frustum height (e/Dh) from 0.013 to 0.054, relative frustum pitch (p/e) from 8 to 14, Relative frustum height to base diameter (e/d1) from 0.37 to 0.75, relative frustum diametral ratio (d1/d2) from 1 to 3. Maximum performance for ARSAH was obtained corresponding to the optimum value of flow and geometric parameter. The geometric orientation of the proposed roughness geometry produced tremendous rise in Nu with an acceptable rise in friction (f). Maximum enhancement in Nu for varying p/e, e/Dh, e/d1, and d1/d2 was, 4.58, 5.21, 6.62 and 6.09 times and that of friction was 5.59, 4.69, 4.92 and 4.66 times respectively over conventional SAH. Maximum Thermohydraulic performance was 3.7, 3.6, 3.9 and 3.8 at p/e = 12, e/Dh = 0.054, e/d1 = 0.75 and d1/d2 = 3.