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Journal articles on the topic 'Nusselt number correlation'

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Published: 4 June 2021
Last updated: 7 February 2022

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1

Lehnen, M. V., C. Y. Y. Lee, and F. L. D. Alves. "Nusselt number correlation for synthetic jets." Journal of the Brazilian Society of Mechanical Sciences and Engineering 38, no. 7 (March 25, 2015): 2161–71. http://dx.doi.org/10.1007/s40430-015-0337-1.

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2

Sato, A. I., C. A. C. Altemani, and V. L. Scalon. "MEAN NUSSELT NUMBER CORRELATION FOR TISE HEATSINK THERMAL DESIGN." Revista de Engenharia Térmica 19, no. 1 (September 9, 2020): 24. http://dx.doi.org/10.5380/reterm.v19i1.76427.

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This work was developed from a review of the technical literature for the thermal design of parallel plates heat sinks with uniform cross section cooled by airflow with the TISE (Top Inlet Side Exit) configuration. Due to an observed lack of agreement of the literature correlations among themselves and also with the available experimental results, numerical simulations were then performed to evaluate the forced convective heat transfer in the channels of these heat sinks with the TISE configuration. The simulations encompassed a range of heatsink airflow rates, considering distinct channel sizes and also the effect of a partial opening for the airflow entrance at the heat sink top. The obtained numerical results were employed to evaluate the average convective heat transfer coefficient inside the heatsink’s channels, based on the flow mixed mean temperature. A new empirical correlation was then proposed for the average Nusselt number as a function of the airflow Reynolds number and three dimensionless channel geometric parameters. The new correlation was compared with available experimental data.
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3

Meola, Carosena. "A New Correlation of Nusselt Number for Impinging Jets." Heat Transfer Engineering 30, no. 3 (February 2009): 221–28. http://dx.doi.org/10.1080/01457630802304311.

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4

Mortean, M. V. V., and M. B. H. Mantelli. "Nusselt number correlation for compact heat exchangers in transition regimes." Applied Thermal Engineering 151 (March 2019): 514–22. http://dx.doi.org/10.1016/j.applthermaleng.2019.02.017.

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5

Basha, Mehaboob, and C. S. Nor Azwadi. "Regularized Lattice Boltzmann Simulation of Laminar Natural Convection in Entrance Region of 2D Channels." Applied Mechanics and Materials 307 (February 2013): 267–70. http://dx.doi.org/10.4028/www.scientific.net/amm.307.267.

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This paper presents a numerical study of incompressible laminar natural convection in entrance region of two dimensional vertical and inclined channels using regularized lattice Boltzmann Bhatnaghar-Gross-Krook method. Individual distribution functions with lattice types D2Q9 and D2Q5 are considered to solve fluid flow and thermal fields, respectively. Rayleigh number and inclination angle are varied from 10e2 to 10e6 and 0 to 60°, respectively. Distribution functions are introduced to mimic Bernoulli’s equation for calculating pressure at the inlet. Predicted Nusselt numbers are compared with Nusselt numbers correlation. Averaged Nusselt numbers compare well with Nusselt number correlation of Bar-Cohen & Rohsenow.
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6

da Silva, Raquel da Cunha Ribeiro, Carlos Salinas Sedano, Kamal A. R. Ismail, and Paúl Adrian Delgado Maldonado. "Empirical Heat Transfer Correlations during Frost Deposition on a Triangular Tube Banks Arrangement." Defect and Diffusion Forum 366 (April 2016): 88–96. http://dx.doi.org/10.4028/www.scientific.net/ddf.366.88.

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An experimental study was reported earlier on the development of frost formation by humid flow passing over the cylinder. In this study, dimensionless correlations used in previous experimental data, and reported empirical correlations of the Nusselt number, were used. This paper reports results of an experimental and numerical investigation where the emphasis was placed on obtaining empirical correlation for the Nusselt number. In this work some experimental results of the frost thickness around every cylinder in a triangular arrangement are presented, an estimated experimental correlation to find Nusselt number. This correlation is based on the experimental measurements in a wind tunnel situated in the Laboratory of Thermal storage and Fluids in the Mechanical Engineering Faculty at Unicamp. A numerical study is performed to study the frost formation in the cylindrical system.
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7

Lin, Ta-Wei, Ming-Chang Wu, Li-Kang Liu, Chun-Jen Fang, and Ying-Huei Hung. "Cooling Performance of Using a Confined Slot Jet Impinging onto Heated Heat Sinks." Journal of Electronic Packaging 128, no. 1 (June 4, 2005): 82–91. http://dx.doi.org/10.1115/1.2161426.

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A series of experimental investigations on the studies related to transient- and steady-state cooling performance from the horizontally heated heat sinks with a confined slot jet impingement have been conducted. The relevant parameters influencing the transient convective cooling performance include the steady-state Grashof number, ratio of jet separation distance to nozzle width, ratio of heat sink height to nozzle width, and jet Reynolds number. The transient heat transfer behaviors such as the temperature distribution, local and average Nusselt numbers on the heated heat sinks have been systematically explored. Two empirical correlations of steady-state local and average Nusselt numbers are presented. Furthermore, a complete composite correlation of steady-state average Nusselt number for mixed convection due to jet impingement and buoyancy is proposed. This empirical correlation obtained by data regression is in good agreement with the experimental data. The maximum and average regression errors are 7.46% and 2.87%, respectively.
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8

TSUZUKI, Nobuyoshi, Yasuyoshi KATO, Yasushi MUTO, Takao ISHIZUKA, Motoaki UTAMURA, and Masanori ARITOMI. "Nusselt Number Correlation of Microchannel Heat Exchanger with S-Shaped Fins." Transactions of the Atomic Energy Society of Japan 9, no. 1 (2010): 13–20. http://dx.doi.org/10.3327/taesj.j09.021.

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9

Taler, Dawid, and Anna Korzen. "Numerical modeling transient response of tubular cross flow heat exchanger." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 1 (January 2, 2018): 81–91. http://dx.doi.org/10.1108/hff-10-2016-0406.

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Purpose The paper aims to present the mathematical modeling of plate fin and tube heat exchanger at small Reynolds numbers on the water side. The Reynolds number of the water flowing inside the tubes was varied in the range from 4,000 to 12,000. Design/methodology/approach A detailed analysis of transient response was modeled for the following changes in the operating parameters of the heat exchanger: a reduction in the water volume flow, an increase in the water volume flow and an increase in the water volume flow with a simultaneous reduction in the air flow velocity. Findings The results of the numerical simulation of a heat exchanger by using experimentally determined water-side heat transfer correlation and theoretical correlation derived for the transition tube flow agree very well. The relationship to calculate the air-side Nusselt number was determined experimentally. The correlation for the air-side Nusselt number was the same for the theoretical and experimental water side correlation. Research limitations/implications The correlation for the air-side Nusselt number as a function of the Reynolds and Prandtl numbers is based on the experimental data and was determined using the least squares method. Originality/value The form of the relationship that was used to approximate experimentally determined water-side Nusselt numbers is identical to the theoretically derived formula for the transition range. The experiments show that the relationship for the water-side Nusselt number in transition and turbulent flow regime that was obtained using theoretical analysis gives quite satisfactory results.
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10

Basit, Romana, Xinyang Li, Zheqing Huang, and Qiang Zhou. "Heat Transfer Studies of Arrays of Prolate Particles in Gas-Solid Flows." Mathematical Problems in Engineering 2020 (November 11, 2020): 1–12. http://dx.doi.org/10.1155/2020/6639172.

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Numerical study of forced convection heat transfer from arrays of prolate particles is performed using the second-order Immersed Boundary-Lattice Boltzmann Method (IB-LBM). Prolate particle is studied with aspect ratio of 2.5 with solid volume fraction variation from 0.1 to 0.3. For each solid volume fraction, arrays of prolate particles are generated and simulations have been performed to calculate Nusselt number for four different Hermans orientation factors and various Reynolds numbers. From the simulation results, it has been observed that, for any specific value of Hermans orientation factor, Nusselt number increases with the increase of the Reynolds number and solid volume fraction. More importantly, it is found that the effect of orientations on Nusselt number is significant. Nusselt number correlation is developed for ellipsoidal particles as function of Reynolds number, Prandtl number, solid volume fraction, and orientation factors. This correlation is valid for 0.1 ≤ c ≤ 0.3 and 0 < Re ≤ 100 .
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11

Daza, Silvia Araujo, Ronald Jaimes Prada, José R. Nunhez, and Guilherme J. Castilho. "Nusselt number correlation for a jacketed stirred tank using computational fluid dynamics." Canadian Journal of Chemical Engineering 97, no. 2 (December 21, 2018): 586–93. http://dx.doi.org/10.1002/cjce.23385.

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12

Lee, Myoungwoo, Hyun Jung Kim, and Dong-Kwon Kim. "Nusselt number correlation for natural convection from vertical cylinders with triangular fins." Applied Thermal Engineering 93 (January 2016): 1238–47. http://dx.doi.org/10.1016/j.applthermaleng.2015.10.105.

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13

Park, Kuen Tae, Jin Woo Lee, Moon Gu Lee, Hyun Jung Kim, and Dong-Kwon Kim. "Nusselt number correlation for vibration-assisted convection from vertically oriented plate fins." International Journal of Heat and Mass Transfer 78 (November 2014): 522–26. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.015.

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14

Hamdan, Mohammad. "An empirical correlation for isothermal parallel plate channel completely filled with porous media." Thermal Science 17, no. 4 (2013): 1061–70. http://dx.doi.org/10.2298/tsci120419015h.

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This study reports a simple empirical correlation for friction factor and Nusselt number for laminar, steady state, hydraulically and thermally fully developed flow in isothermal parallel plate channel completely filled with porous media. The study is carried out using a finite difference numerical analysis. The Darcy-Brinkman-Forchheimer model is used to model the flow inside the porous media. The empirical correlations are developed to relate friction factor and Nusselt number to Darcy and Forchheimer coefficient.
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15

Keyhani, M., F. A. Kulacki, and R. N. Christensen. "Experimental Investigation of Free Convection in a Vertical Rod Bundle—A General Correlation for Nusselt Numbers." Journal of Heat Transfer 107, no. 3 (August 1, 1985): 611–23. http://dx.doi.org/10.1115/1.3247468.

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Free convection in two vertical, enclosed rod bundles has been experimentally investigated for a wide range of Rayleigh numbers. A uniform power dissipation per unit length is supplied to each rod, and the enclosing outer cylinder is maintained at constant temperature. Nusselt numbers for each rod, as well as an overall value for each bundle, have been obtained as a function of Rayleigh number. Comparison of the results for air and water as the working fluid indicate that, for a fixed Rayleigh number, an increase in the Prandtl number produces a reduction in the Nusselt number. This is contrary to what has been reported for vertical cavities and is attributed to curvature effects. Furthermore, the data reveal the interesting fact that it is quite possible for the individual rods in the bundle to exchange energy with the working fluid via different but coexisting regimes at a given power dissipation. Also, as the Rayleigh number is increased, the rods each tend to assume nearly the same heat transfer coefficient. Finally, a correlation for the overall convective Nusselt number is developed in terms of Rayleigh number and geometric parameters.
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16

da Cunha Ribeiro da Silva, Raquel, C. T. Salinas, P. A. Delgado, and K. A. R. Ismail. "Estimation of a Nusselt Correlation for the Numerical Prediction of Frost Thickness on a Tube Banks of Triangular Arrangement." Defect and Diffusion Forum 353 (May 2014): 121–25. http://dx.doi.org/10.4028/www.scientific.net/ddf.353.121.

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This paper presents a frost numerical model for predicting the frosting behavior in terms of the correlation of the heat transfer coefficient, Nusselt number. In this work the correlations of the Nusselt number from various references is examined and compared with experimental data. In this study, a mathematical model that can predict the behavior of the frost growth on a bank of triangularly arranged tubes is developed. Because of the complex heat transfer during frost formation on a tube bank, this subject is receiving much attention. Heat transfer in flow across a bank of tubes has particular importance in the design of heat exchangers. For this study various empirical relations are analyzed to determine the Nusselt number and compared with experimental data.
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17

Bazydola, Sarah M., and Mohammad E. Taslim. "An Experimental Investigation of a Staggered Array of Heatsinks in the Hydrodynamic and Thermal Entrance Regions of a Duct." Journal of Electronic Packaging 115, no. 1 (March 1, 1993): 106–11. http://dx.doi.org/10.1115/1.2909288.

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The need for methods to predict temperatures of electronic packages is becoming more important as time to market decreases and the need to reduce engineering costs increases. Geometric parameters that are commonly found in a card cage computer design are duct height to package height ratio (H/B) and longitudinal to latitudinal spacing (L/D). The velocity of the air flowing through the card cage is also an important parameter. The purpose of this experimental research is to develop correlations that can be used to predict heat transfer coefficients and thus temperatures of electronic packages in a staggered array in the hydrodynamic and thermal entrance regions of a duct. The ten components in the array are mounted on one wall of the duct to simulate a printed circuit board in a card cage. Results are presented for three values of Reynolds number (20250, 50000 and 65000) for a range of H/B from 1 to 5 by increments of 2 while L/D is varied from 1 to 3 by increments of 0.5. Comparisons are made between upstream and downstream packages. It was determined that one correlation could be used to predict the Nusselt number for upstream heatsinks. This correlation is a function of Reynolds number and H/B. The results also indicate that three correlations are necessary to predict the Nusselt number of downstream heatsinks. When H/B = 1, the Nusselt number is only a function of Reynolds number; when H/B is greater than 1 and L/D is less than 2, the Nusselt number is dependent on Reynolds number, H/B and L/D; when H/B is greater than 1 and L/D is greater than or equal to 2, the Nusselt number is a function of Reynolds number and H/B.
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18

Kang, Byeong Dong, Hyun Jung Kim, and Dong-Kwon Kim. "Nusselt Number Correlation for Vertical Tubes with Inverted Triangular Fins under Natural Convection." Energies 10, no. 8 (August 10, 2017): 1183. http://dx.doi.org/10.3390/en10081183.

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19

Habib, M. A., A. M. Attya, S. A. M. Said, A. I. Eid, and A. Z. Aly. "Heat transfer characteristics and Nusselt number correlation of turbulent pulsating pipe air flows." Heat and Mass Transfer 40, no. 3-4 (February 1, 2004): 307–18. http://dx.doi.org/10.1007/s00231-003-0456-6.

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20

Pallares, J., and F. X. Grau. "A modification of a Nusselt number correlation for forced convection in porous media." International Communications in Heat and Mass Transfer 37, no. 9 (November 2010): 1187–90. http://dx.doi.org/10.1016/j.icheatmasstransfer.2010.07.014.

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21

Ahmed, G. Refai, and M. M. Yovanovich. "Experimental Study of Forced Convection From Isothermal Circular and Square Cylinders and Toroids." Journal of Heat Transfer 119, no. 1 (February 1, 1997): 70–79. http://dx.doi.org/10.1115/1.2824102.

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Experimental studies of forced convection heat transfer from different body shapes were conducted to determine the effects of Reynolds number and different characteristic body lengths on the area-averaged Nusselt number. Although the bodies differed significantly in their shapes, they had approximately the same total surface area, A = 11,304 mm2 ± 5%. This ensured that for a given free stream velocity and total heat transfer rate all bodies had similar trends for the relationship of Nusselt and Reynolds numbers. The experimental program range was conducted in the Reynolds number range 104≤ReA≤105 and Prandtl number 0.71. Finally, the empirical models for forced convection heat transfer were developed. These empirical models were valid for a wide range of Reynolds numbers 0≤ReA≤105. The present experimental correlations were compared with available correlation equations and experimental data. These comparisons show very good agreement.
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22

SAID, Abdessadek AIT HAJ, Mahfoud ELFAGRICH, and Omar ABOUNACHIT. "Numerical investigation of free convection through a horizontal open-ended axisymmetric cavity." Indian Journal of Science and Technology 14, no. 13 (April 9, 2021): 1081–96. http://dx.doi.org/10.17485/ijst/v14i13.2259.

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Objectives: The purposes of this work are to investigate the free convective heat transfer in an axis-symmetric open-ended cavity heated from below and to propose useful correlations of Nusselt number. Methods: The governing equations that model the fluid flow and the temperature field are solved using a control volume-based finite differences method. Under steady state condition, the natural convective flow is considered to be laminar, incompressible and axisymmetric. The Boussinesq approximation with constant thermophysical properties is adopted. Numerical experimentations are performed to deduce the optimum sizes of the calculation domain and the mesh grid. Findings: the obtained results indicate that when Rayleigh number (Ra) and aspect ratio (A) are low the heat transfer is weak and mainly conductive. The increase of Ra and A enhances the convective heat transfer mode thereby the heat transfer is ameliorated. Unlike the Rayleigh Bénard convection, the transition from conduction to convection produces at critical value of Rayleigh number (Rac) that is variable dependent on A. Novelty: To the best of authors knowledge, the formula of (Rac) elaborated in this work for the studied cavity is the first attempt. As well, correlation of Nusselt numbers (Nu) for the cold upper plate in terms of Ra and A is performed. Comparisons between Nu at the lower plate given in previous work and Nusselt number at the upper plate is conducted. Keywords: free convection; circular plates; Nusselt number correlations; open ended cavity; critical Rayleigh number
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23

An, Byoung Hoon, Hyun Jung Kim, and Dong-Kwon Kim. "Nusselt number correlation for natural convection from vertical cylinders with vertically oriented plate fins." Experimental Thermal and Fluid Science 41 (September 2012): 59–66. http://dx.doi.org/10.1016/j.expthermflusci.2012.03.010.

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24

Lacroix, M. "ANALYSIS OF NATURAL CONVECTION HEAT TRANSFER BETWEEN TWO HORIZONTAL CYLINDERS AND THEIR ENCLOSURE." Transactions of the Canadian Society for Mechanical Engineering 16, no. 1 (March 1992): 17–32. http://dx.doi.org/10.1139/tcsme-1992-0002.

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A numerical study has been conducted for natural convection heat transfer for air around two horizontal heated cylinders placed inside a rectangular enclosure cooled from the side. Three cylinder spacings were investigated. The local and overall Nusselt numbers were determined over the range of Rayleigh numbers from 104 to 106. It is found that the thermal performance of the unit is strongly influenced by the Rayleigh number and, to a lesser extent, by the cylinder spacing. A correlation is suggested for the overall Nusselt number.
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25

Zeitoun, O., M. Shoukri, and V. Chatoorgoon. "Interfacial Heat Transfer Between Steam Bubbles and Subcooled Water in Vertical Upward Flow." Journal of Heat Transfer 117, no. 2 (May 1, 1995): 402–7. http://dx.doi.org/10.1115/1.2822536.

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In two-fluid modeling, accurate prediction of the interfacial transport of mass, momentum, and energy is required. Experiments were carried out to obtain a data base for the development of interfacial transport models, or correlations, for subcooled water–steam bubbly flow in vertical conduits. The experimental data of interest included the interfacial area concentration, interfacial condensation heat transfer, and bubble relative velocity. In the present investigation, bubble condensation in subcooled water–steam flow in a vertical annulus at low flow rate and low pressure is investigated experimentally. A high-speed video system (up to 1000 frame/s) was used to visualize two orthogonal views of the flow simultaneously. A digital image processing technique was used to track and measure the velocity and size of the collapsing bubbles. The axial void fraction distribution was also measured by a single beam gamma densitometer. The results were compared with existing correlations and a new correlation for bubble condensation Nusselt number was obtained based on the present data. In the proposed correlation, the bubble condensation Nusselt number is expressed in terms of the bubble Reynolds number, Jakob number, and void fraction. The latter term accounts for the multi-bubble effect. This correlation can predict the present data within ±20 percent with a correlation coefficient of 96 percent.
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26

Babaee, H., P. Perdikaris, C. Chryssostomidis, and G. E. Karniadakis. "Multi-fidelity modelling of mixed convection based on experimental correlations and numerical simulations." Journal of Fluid Mechanics 809 (November 21, 2016): 895–917. http://dx.doi.org/10.1017/jfm.2016.718.

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For thermal mixed-convection flows, the Nusselt number is a function of Reynolds number, Grashof number and the angle between the forced- and natural-convection directions. We consider flow over a heated cylinder for which there is no universal correlation that accurately predicts Nusselt number as a function of these parameters, especially in opposing-convection flows, where the natural convection is against the forced convection. Here, we revisit this classical problem by employing modern tools from machine learning to develop a general multi-fidelity framework for constructing a stochastic response surface for the Nusselt number. In particular, we combine previously developed experimental correlations (low-fidelity model) with direct numerical simulations (high-fidelity model) using Gaussian process regression and autoregressive stochastic schemes. In this framework the high-fidelity model is sampled only a few times, while the inexpensive empirical correlation is sampled at a very high rate. We obtain the mean Nusselt number directly from the stochastic multi-fidelity response surface, and we also propose an improved correlation. This new correlation seems to be consistent with the physics of this problem as we correct the vectorial addition of forced and natural convection with a pre-factor that weighs differently the forced convection. This, in turn, results in a new definition of the effective Reynolds number, hence accounting for the ‘incomplete similarity’ between mixed convection and forced convection. In addition, due to the probabilistic construction, we can quantify the uncertainty associated with the predictions. This information-fusion framework is useful for elucidating the physics of the flow, especially in cases where anomalous transport or interesting dynamics may be revealed by contrasting the variable fidelity across the models. While in this paper we focus on the thermal mixed convection, the multi-fidelity framework provides a new paradigm that could be used in many different contexts in fluid mechanics including heat and mass transport, but also in combining various levels of fidelity of models of turbulent flows.
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27

Wei, Hui Ming, Xuan Zhang, and Tai Li Liu. "Heat Transfer Correlation in Small Channels." Advanced Materials Research 516-517 (May 2012): 308–11. http://dx.doi.org/10.4028/www.scientific.net/amr.516-517.308.

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In this study, a study on heat transfer is performed for water upward flow in small channels to provide a systematic database for low pressure and mass flow velocity conditions. An empirical correlation on heat transfer in small channels for predicting the average Nusselt number has been derived by using Genetic Algorithm (GA).
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28

Zolghadri, Amir, Heydar Maddah, Mohammad Hossein Ahmadi, and Mohsen Sharifpur. "Predicting Parameters of Heat Transfer in a Shell and Tube Heat Exchanger Using Aluminum Oxide Nanofluid with Artificial Neural Network (ANN) and Self-Organizing Map (SOM)." Sustainability 13, no. 16 (August 6, 2021): 8824. http://dx.doi.org/10.3390/su13168824.

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This study is a model of artificial perceptron neural network including three inputs to predict the Nusselt number and energy consumption in the processing of tomato paste in a shell-and-tube heat exchanger with aluminum oxide nanofluid. The Reynolds number in the range of 150–350, temperature in the range of 70–90 K, and nanoparticle concentration in the range of 2–4% were selected as network input variables, while the corresponding Nusselt number and energy consumption were considered as the network target. The network has 3 inputs, 1 hidden layer with 22 neurons and an output layer. The SOM neural network was also used to determine the number of winner neurons. The advanced optimal artificial neural network model shows a reasonable agreement in predicting experimental data with mean square errors of 0.0023357 and 0.00011465 and correlation coefficients of 0.9994 and 0.9993 for the Nusselt number and energy consumption data set. The obtained values of eMAX for the Nusselt number and energy consumption are 0.1114, and 0.02, respectively. Desirable results obtained for the two factors of correlation coefficient and mean square error indicate the successful prediction by artificial neural network with a topology of 3-22-2.
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29

Rangasamy, Rajavel. "Experimental and numerical studies of a spiral plate heat exchanger." Thermal Science 18, no. 4 (2014): 1355–60. http://dx.doi.org/10.2298/tsci130317131r.

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An experimental and numerical study of heat transfer and flow characteristics of spiral plate heat exchanger was carried out. The effects of geometrical aspects of the spiral plate heat exchanger and fluid properties on the heat transfer characteristics were also studied. Three spiral plate heat exchangers with different plate spacing (4mm, 5mm and 6 mm) were designed, fabricated and tested. Physical models have been experimented for different process fluids and flow conditions. Water is taken as test fluid. The effect of mass flow rate and Reynolds number on heat transfer coefficient has been studied. Correlation has been developed to predict Nusselt numbers. Numerical models have been simulated using CFD software package FLUENT 6.3.26. The numerical Nusselt number have been calculated and compared with that of experimental Nusselt number.
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30

Sk, Ansar Ali, Pardeep Kumar, and Sandeep Kumar. "Effect of impeller diameter on Nusselt number in mechanically agitated vessel." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 4 (August 19, 2019): 2225–35. http://dx.doi.org/10.1108/hff-05-2019-0435.

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Purpose The purpose of the study is to developed the effect of Nusselt number on impeller diameter in agitated vessel, which is beneficial to find out the heat transfer coefficient in the process industry. A comparison has been done between the experimental and calculated Nusselt numbers with standard deviation found to be 8.03 per cent. Design/methodology/approach For studying the effect of impeller diameter on Nusselt Number, the heat transfer measurements were made with three different impellers of diameter. Although the diameter of impeller, Da shows its effect in Reynolds number, an attempt has been made to find the relationship between the impeller diameter and Nusselt number. A correlation between (NNuj/N″Pra1/3 N″Rea2/3) vs Da/DT and (NNuoc/N″Pra1/3 N″Rea2/3) vs Da/Dc in which data of three fluids [1, 2 and 4 per cent carboxy methyl cellulose solution of A type (CMC-A) solutions] have been plotted. Findings The heat transfer data for agitated Newtonian and non-Newtonian fluids have been successfully correlated by using the viscosity of the fluid evaluated at the impeller tip assuming a cylinder of diameter equal to that of impeller rotating in an infinite fluid. Data of 1, 2 and 4 per cent CMC-A, for three impeller diameters, have been correlated by equations. Using the above concepts of Reynolds and Prandtl numbers, Nusselt Numbers and Da/DT, it is also possible to correlate the available published data for other non-Newtonian fluids obtained with different impeller geometries. Originality/value A set up was made for studying the effect of impeller diameter, the heat transfer measurements were made with three impellers of diameter 7.5, 12.7 and 18.35 cm respectively. Although the diameter of impeller, Da shows its effect is Reynolds number, an attempt has been made to find the effect of Da/DT ratio on Nusselt number.
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31

Naylor, D., and J. D. Tarasuk. "Natural Convective Heat Transfer in a Divided Vertical Channel: Part II—Experimental Study." Journal of Heat Transfer 115, no. 2 (May 1, 1993): 388–94. http://dx.doi.org/10.1115/1.2910690.

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An interferometric study has been conducted on two-dimensional laminar natural convection heat transfer in an isothermal vertical divided channel. Interferograms were obtained for air and a plate-to-channel length ratio of Lp/Lc= 1/3. Data are presented for the dividing plate located at the bottom (Li/Lc = 0) and top of the channel (Li/Lc=2/3). Comparisons of local and average Nusselt numbers are made with the numerical predictions from Part I. Although the experimental average Nusselt numbers are typically about 10 percent lower than the numerical results, the general trends of the data are in good agreement. Average Nusselt number correlation equations are presented.
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32

Fang, C. J., M. C. Wu, Y. M. Kuo, C. Y. Lee, C. H. Peng, and Y. H. Hung. "Heat Transfer Behavior for a Stationary or Rotating MCM Disk With an Unconfined Round Jet Impingement." Journal of Electronic Packaging 129, no. 4 (December 28, 2006): 400–410. http://dx.doi.org/10.1115/1.2804087.

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A series of experimental investigations on the studies related to fluid flow and transient mixed convection from a horizontally unconfined stationary or rotating ceramic-based multichip module (MCM) disk with unconfined jet impingement have been successfully conducted. The fluid flow and heat transfer behavior explored includes the streamwise velocity and turbulence intensity distributions, transient dimensionless temperature distribution on the MCM disk surface, transient heat flux distribution of input power, and transient chip and average heat transfer characteristics on the MCM disk surface. Besides, two new correlations of transient stagnation and average Nusselt numbers in terms of jet Reynolds number, ratio of jet separation distance to nozzle diameter and time elapsed during the transient period, are presented for the cases of stationary MCM disk. For the cases of rotating MCM disk, a complete composite correlation of steady-state average Nusselt number for mixed convection due to jet impingement, disk rotation and buoyancy is proposed. As compared with the steady-state results, if the transient chip and average heat transfer behaviors may be considered as a superposition of a series of quasisteady states, the transient chip and average Nusselt numbers in all the present transient experiments can be properly predicted by the existing steady-state correlations when t>6min in the power-on transient period.
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33

Balaji, C., M. Hölling, and H. Herwig. "Nusselt Number Correlations for Turbulent Natural Convection Flows Using Asymptotic Analysis of the Near-Wall Region." Journal of Heat Transfer 129, no. 8 (December 1, 2006): 1100–1105. http://dx.doi.org/10.1115/1.2737485.

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In this paper, we propose a general methodology by which a universal temperature profile, derived by matching temperature gradients in the overlap region of the wall layer and the core layer, that is valid for high Rayleigh number flows, can be recast into a correlation for the Nusselt number. We then evaluate its adequacy for three types of flows: (a) fully developed flows (e.g., the infinite channel), (b) developing flows (e.g., the vertical flat plate boundary layer), and (c) complex flows (e.g., Rayleigh-Bénard convection and the differentially heated square cavity). The correlation for the Nusselt number, in general, contains a logarithmic term, usually missing in earlier studies, with which we are able to match existing direct numerical simulations or experimental data very well for both fully developed and complex flows.
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34

Chen, T. S., B. F. Armaly, and N. Ramachandran. "Correlations for Laminar Mixed Convection Flows on Vertical, Inclined, and Horizontal Flat Plates." Journal of Heat Transfer 108, no. 4 (November 1, 1986): 835–40. http://dx.doi.org/10.1115/1.3247020.

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Local Nusselt numbers for laminar mixed convection flows along isothermal vertical, inclined, and horizontal flat plates are presented for the entire mixed convection regime for a wide range of Prandtl numbers, 0.1 ≤ Pr ≤ 100. Simple correlation equations for the local and average mixed convection Nusselt numbers are developed, which are found to agree well with the numerically predicted values and available experimental data for both buoyancy assisting and opposing flow conditions. The threshold values of significant buoyancy effects on forced convection and forced flow effects on free convection, as well as the maximum increase in the local mixed convection Nusselt number from the respective pure convection limits, are also presented for all flow configurations. It is found that the buoyancy or forced flow effect can increase the surface heat transfer rate from pure forced or pure free convection by about 20 percent.
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35

da Cunha Ribeiro da Silva, Raquel, Carlos T. Salinas, and Kamal A. R. Ismail. "Estimation of a Nusselt Correlations for Numerical Prediction of Frost Thickness Growth over a Cold Cylinder." Advanced Materials Research 875-877 (February 2014): 771–75. http://dx.doi.org/10.4028/www.scientific.net/amr.875-877.771.

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This paper evaluates numerically some of the parameters involved in modeling the process of frost formation over a cold cylinder surface subject to the flow of humid air. Was utilized for numerical predictions the empirical Nusselt correlation from the literature, obtained in experimental studies on frost formation phenomena of frost grow over a cold cylinder surface. To predict frosting process a numerical solution was utilized, and a new correlation for Nusselt number based on the experimental correlation of Kim was estimated. For the new Nusselt correlation an optimization method that adjusts the numerical solution of modeling the frost formation process with experimental results of the frost layer thickness was used. The calculation procedure allows the estimation of the parameter K of equation. The modeling process was validated by comparison with available experimental data.
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36

Elshamy, M. M., and M. N. Ozisik. "Numerical Study of Laminar Natural Convection From a Plate to its Cylindrical Enclosure." Journal of Solar Energy Engineering 113, no. 3 (August 1, 1991): 194–99. http://dx.doi.org/10.1115/1.2930492.

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The steady-state laminar natural convection for air bounded by a hot plate and a cold cylindrical enclosure has been studied numerically for the case of cold isothermal cylinder and hot isothermal plate. A correlation is presented for the average Nusselt number over the range of Rayleigh number from 105 to 106 for different values of the width-aspect ratio Sw and thickness aspect-ratio St of the plate. It is found that the average Nusselt number increases with increasing Sw and Rayleigh number. A two-cell pattern is observed for Sw=1.5 and less. The effect of Sw on the average Nusselt number is found to be stronger than that of St.
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37

Persoons, Tim, Alan McGuinn, and Darina B. Murray. "A general correlation for the stagnation point Nusselt number of an axisymmetric impinging synthetic jet." International Journal of Heat and Mass Transfer 54, no. 17-18 (August 2011): 3900–3908. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.04.037.

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38

Bahmani, Alireza, and Hadi Kargarsharifabad. "Magnetohydrodynamic free convection of non-Newtonian power-law fluids over a uniformly heated horizontal plate." Thermal Science 24, no. 2 Part B (2020): 1323–34. http://dx.doi.org/10.2298/tsci190102110b.

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The MHD free convection flow of non-Newtonian power-law fluids over a horizontal plate subjected to a constant heat flux is studied. The results are presented for various values of the three influential parameters, i. e. the generalized Hart?mann number, the generalized Prandtl number, and the non-Newtonian power-law viscosity index. Increasing the Hartmann number increases the thermal boundary-layer thickness and the surface temperature and consequently decreases the wall skin friction and Nusselt number. A lower generalized Prandtl number results in a larger skin friction coefficient and higher wall temperature as well as thicker thermal boundary-layer. The viscosity index is predicted to influence the flow conditions depending on the value of generalized Hartmann number. At high generalized Prandtl number numbers, by decreasing non-Newtonian power-law index, the wall skin friction, temperature scale, and thermal boundary-layer thickness are increased and the Nusselt number is decreased, while the opposite trend is observed for low generalized Prandtl number. A general correlation for the Nusselt number is derived using the numerical results
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39

Toghraie, Davood, and Ehsan Shirani. "Numerical simulation of water/alumina nanofluid mixed convection in square lid-driven cavity." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 5 (July 8, 2019): 2781–807. http://dx.doi.org/10.1108/hff-02-2019-0114.

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Purpose The purpose of this paper is to investigate the mixed convection of a two-phase water–aluminum oxide nanofluid in a cavity under a uniform magnetic field. Design/methodology/approach The upper wall of the cavity is cold and the lower wall is warm. The effects of different values of Richardson number, Hartmann number, cavitation length and solid nanoparticles concentration on the flow and temperature field and heat transfer rate were evaluated. In this paper, the heat flux was assumed to be constant of 10 (W/m2) and the Reynolds number was assumed to be constant of 300 and the Hartmann number and the volume fraction of solid nanoparticles varied from 0 to 60 and 0 to 0.06, respectively. The Richardson number was considered to be 0.1, 1 and 5. Aspect ratios were 1, 1.5 and 2. Findings Comparison of the results of this paper with the results of the numerical and experimental studies of other researchers showed a good correlation. The results were presented in the form of velocity and temperature profiles, stream and isotherm lines and Nusselt numbers. The results showed that by increasing the Hartmann number, the heat transfer rate decreases. An increase from 0 to 20 in Hartmann number results in a 20 per cent decrease in Nusselt numbers, and by increasing the Hartmann number from 20 to 40, a 16 per cent decrease is observed in Nusselt number. Accordingly, it is inferred that by increasing the Hartmann number, the reduction in the Nusselt number is decreased. As the Richardson number increased, the heat transfer rate and, consequently, the Nusselt number increased. Therefore, an increase in the Richardson number results in an increase of the Nusselt number, that is, an increase in Richardson number from 0.1 to 1 and from 1 to 5 results in 37 and 47 per cent increase in Nusselt number, respectively. Originality/value Even though there have been numerous investigations conducted on convection in cavities under various configurations and boundary conditions, relatively few studies are conducted for the case of nanofluid mixed convection in square lid-driven cavity under the effect of magnetic field using two-phase model.
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40

Rustum, I. M., and H. M. Soliman. "Experimental Investigation of Laminar Mixed Convection in Tubes With Longitudinal Internal Fins." Journal of Heat Transfer 110, no. 2 (May 1, 1988): 366–72. http://dx.doi.org/10.1115/1.3250493.

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Experiments were performed to study the pressure drop and heat transfer characteristics for laminar flow in a smooth tube and four tubes with internal longitudinal fins, with emphasis on showing how the experimental results relate to previous analytical predictions. Measured quantities include the fully developed friction factor, local and fully developed Nusselt numbers. Good agreements were obtained between the friction factor results and previous analytical predictions, and between Nusselt number results for the smooth tube and previous experiments. Free convection is shown to have a strong influence on heat transfer in finned tubes and the results approach the forced-convection predictions as Rayleigh number decreases. Internal fins appear to retard the onset of significant free convective currents; however, once initiated, a faster rate of heat transfer enhancement occurs compared to the smooth tube. An empirical correlation of fully developed Nusselt number is presented taking into account the influences of tube geometry.
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41

NAZARI, M., M. H. KAYHANI, and R. MOHEBBI. "HEAT TRANSFER ENHANCEMENT IN A CHANNEL PARTIALLY FILLED WITH A POROUS BLOCK: LATTICE BOLTZMANN METHOD." International Journal of Modern Physics C 24, no. 09 (August 18, 2013): 1350060. http://dx.doi.org/10.1142/s0129183113500605.

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The main goal of the present study is to investigate the heat transfer enhancement in a channel partially filled with an anisotropic porous block (Porous Foam) using the lattice Boltzmann method (LBM). Combined pore level simulation of flow and heat transfer is performed for a 2D channel which is partially filled with square obstacles in both ordered and random arrangements by LBM which is not studied completely in the literature. The effect of the Reynolds number, different arrangements of obstacles, blockage ratio and porosity on the velocity and temperature profiles inside the porous region are studied. The local and averaged Nusselt numbers on the channel walls along with the respective confidence interval and comparison between results of regular and random arrangements are presented for the first time. For constant porosity and block size, the maximum value of averaged Nusselt number in the porous block is obtained in the case of random arrangement of obstacles. Also, by decreasing the porosity, the value of averaged Nusselt number is increased. Heat transfer to the working fluids increases significantly by increasing the blockage ratio. Several blockage ratios with different arrangements are checked to obtain a correlation for the Nusselt number.
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42

Touahri, Sofiane, and Toufik Boufendi. "Numerical study of the conjugate heat transfer in a horizontal pipe heated by Joulean effect." Thermal Science 16, no. 1 (2012): 53–67. http://dx.doi.org/10.2298/tsci101120080t.

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The three dimensional mixed convection heat transfer in a electrically heated horizontal pipe conjugated to a thermal conduction through the entire solid thickness is investigated by taking into account the thermal dependence of the physical properties of the fluid and the outer heat losses. The model equations of continuity, momentum and energy are numerically solved by the finite volume method. The pipe thickness, the Prandtl and the Reynolds numbers are fixed while the Grashof number is varied from 104to107. The results obtained show that the dynamic and thermal fields for mixed convection are qualitatively and quantitatively different from those of forced convection, and the local Nusselt number at the interface solid-fluid is not uniform: it has considerable axial and azimuthally variations. The effect of physical variables of the fluid depending on temperature is significant, which justifies its inclusion. The heat transfer is quantified by the local and average Nusselt numbers. We found that the average Nusselt number of solid-fluid interface of the duct increases with the increase of Grashof number. We have equally found out that the heat transfer is improved thanks to the consideration of the thermo dependence of the physical properties. We have tried modelling the average Nusselt number as a function of Richardson number. With the parameters used, the heat transfer is quantified by the correlation: NuA=12.0753 Ri0.156
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43

Friedman, J., P. Koundakjian, D. Naylor, and D. Rosero. "Heat Transfer to Small Horizontal Cylinders Immersed in a Fluidized Bed." Journal of Heat Transfer 128, no. 10 (March 22, 2006): 984–89. http://dx.doi.org/10.1115/1.2345425.

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Heat transfer to horizontal cylinders immersed in fluidized beds has been extensively studied, but mainly in the context of heat transfer to boiler tubes in coal-fired beds. As a result, most correlations in the literature have been derived for cylinders of 25-50mm diameter in vigorously fluidizing beds. In recent years, fluidized bed heat treating furnaces fired by natural gas have become increasingly popular, particularly in the steel wire manufacturing industry. These fluidized beds typically operate at relatively low fluidizing rates (G∕Gmf<5) and with small diameter wires (1-6mm). Nusselt number correlations developed based on boiler tube studies do not extrapolate down to these small size ranges and low fluidizing rates. In order to obtain reliable Nusselt number data for these size ranges, an experimental investigation has been undertaken using two heat treating fluidized beds; one a pilot-scale industrial unit and the other a lab-scale (300mm diameter) unit. Heat transfer measurements were obtained using resistively heated cylindrical samples ranging from 1.3 to 9.5mm in diameter at fluidizing rates ranging from approximately 0.5×Gmf (packed bed condition) to over 10×Gmf using aluminum oxide sand particles ranging from dp=145-330μm (50–90 grit). It has been found that for all cylinder sizes tested, the Nusselt number reaches a maximum near 2×Gmf, then remains relatively steady (±5-10%) to the maximum fluidizing rate tested, typically 8-12×Gmf. A correlation for maximum Nusselt number is developed.
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44

Hader, M. A., and M. A. Jog. "Effect of Drop Deformation on Heat Transfer to a Drop Suspended in an Electrical Field." Journal of Heat Transfer 120, no. 3 (August 1, 1998): 682–89. http://dx.doi.org/10.1115/1.2824337.

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Heat transfer to a drop of a dielectric fluid suspended in another dielectric fluid in the presence of an electric field is investigated. We have analyzed the effect of drop deformation on the heat transport to the drop. The deformed drop shape is assumed to be a spheroid and is prescribed in terms of the ratio of drop major and minor diameter. Results are obtained for both prolate and oblate shapes with a range of diameter ratio b/a from 2.0 to 0.5. The internal problem where the bulk of the resistance to the heat transport is in the drop, as well as the external problem where the bulk of the resistance is in the continuous phase, are considered. The electrical field and the induced stresses are obtained analytically. The resulting flow field and the temperature distribution are determined numerically. Results indicate that the drop shape significantly affects the flow field and the heat transport to the drop. For the external problem, the steady-state Nusselt number increases with Peclet number for all drop deformations. For a fixed Peclet number, the Nusselt number increases with decreasing b/a. A simple correlation is proposed to evaluate the effect of drop deformation on the steady-state Nusselt number. For the internal problem, for all drop deformations, the maximum steady-state Nusselt number becomes independent of the Peclet number at high Peclet number. The maximum steady-state Nusselt numbers for an oblate drop are significantly higher than that for a prolate drop.
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45

Sobota, Tomasz. "Experimental and numerical analysis of heat transfer in the helically coiled heat exchanger." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 6 (May 29, 2019): 2935–51. http://dx.doi.org/10.1108/hff-12-2018-0793.

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Purpose The knowledge of the heat transfer coefficient is important for the proper design of heat exchangers as well as for the determination of the working medium outlet temperatures. This paper aims to present a method of simultaneous determination of coefficients in correlation formulas for the Nusselt number on both sides of the heat transfer surface. Design/methodology/approach The idea of the developed method is based on determining such a values of the coefficients in Nusselt number correlations that fulfill the condition of equality between the measured and calculated temperature at the outlet of heat exchanger in terms of least squares method. To test the proposed method, a special experimental installation was built. The heat transfer in helically coiled tube-in-tube heat exchanger was examined for the wide range of temperature changes and volumetric flow rates of working fluid. Findings The simulation results were validated with an experimental data. The results show that the heat transfer coefficient of the counter-current is higher than the co-current flow in helically coiled heat exchanger. This phenomenon can be beneficial particularly in the laminar flow regime. Research limitations/implications The correlation for the Nusselt number as a function of the Reynolds and Prandtl numbers for hot and cold liquid was obtained with the least squares method for the experimental data. Practical implications The presented method allows for the simultaneous determination of heat transfer coefficient on both sides of the wall without the necessity of indirect calculation of the overall heat transfer coefficient. The presented method can be used in the thermal design of various type heat exchangers. Originality/value This work presents the new methodology of determination correlations for the helically coiled tube-in-tube heat exchanger for co-current and counter-current arrangement, which can be used in thermal design.
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46

Kim, Tae Hoon, Dong-Kwon Kim, and Kyu Hyung Do. "Correlation for the fin Nusselt number of natural convective heat sinks with vertically oriented plate-fins." Heat and Mass Transfer 49, no. 3 (December 8, 2012): 413–25. http://dx.doi.org/10.1007/s00231-012-1100-0.

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47

Gbadebo, S. A., S. A. M. Said, and M. A. Habib. "Average Nusselt number correlation in the thermal entrance region of steady and pulsating turbulent pipe flows." Heat and Mass Transfer 35, no. 5 (October 21, 1999): 377–81. http://dx.doi.org/10.1007/s002310050339.

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48

Saha, S. K., and D. N. Mallick. "Heat Transfer and Pressure Drop Characteristics of Laminar Flow in Rectangular and Square Plain Ducts and Ducts With Twisted-Tape Inserts." Journal of Heat Transfer 127, no. 9 (November 21, 2004): 966–77. http://dx.doi.org/10.1115/1.2010493.

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Abstract The present paper reports the results of an experimental investigation of the heat transfer and pressure drop characteristics of laminar flow of viscous oil through horizontal rectangular and square plain ducts and ducts inserted with full-length twisted tapes, short-length twisted tapes, and regularly spaced twisted-tape elements. Isothermal pressure drop measurements were taken in acrylic ducts. Heat transfer measurements were taken in electrically heated stainless-steel ducts imposing uniform wall heat flux boundary conditions. The duct aspect ratios AR were 1, 0.5, and 0.333. The twist ratios of the twisted tapes were y=2.692, 5.385, 2.597, 5.193, 2.308, and 4.615. Short-length tapes were 0.9, 0.7, and 0.5 times the duct length. The space ratios were s=2.692, 5.385, 2.597, 5.193, 2.308, and 4.615. Both friction factor and Nusselt number increase with decreasing y and AR for AR⩽1 and increasing Re, Sw, and Pr. As the tape-length decreases, both friction factor and Nusselt number decrease. Friction factor increases as s decreases, and Nusselt number increases as s increases. Isothermal friction factor correlation and comprehensive Nusselt number correlation have been developed to predict data reasonably well in the entire range of parameters. Performance evaluation says that short-length twisted tapes are worse and regularly spaced twisted-tape elements are better than the full-length twisted tapes.
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49

Chen, K., and G. A. Karim. "Evaluation of the instantaneous unsteady heat transfer in a rapid compression-expansion machine." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 212, no. 5 (August 1, 1998): 351–62. http://dx.doi.org/10.1243/0957650981536925.

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Investigation is made of the instantaneous unsteady heat transfer within a pneumatically driven rapid compression-expansion machine that offers simple, well-controlled and known boundary conditions. Values of the instantaneous apparent overall heat flux from the cylinder gas to the wall surfaces were calculated using a thermodynamic analysis of the experimentally measured pressure and volume temporal development. Corresponding heat flux values were also estimated through the application of a three-dimensional computational fluid dynamics code, KIVA3. Correlation of the derived data using the mean piston speed and cylinder bore diameter as the characteristic parameters for the Reynolds and the Nusselt numbers resulted in Nu = 0.010 × Re1.188 for all compression ratios (8.4–24.3). Correlation of the derived data when using the instantaneous height between the piston top and the cylinder head, the maximum gas velocity and the kinetic mean gas velocity obtained when using the KIVA3 code as the characteristic parameters for the Reynolds and the Nusselt numbers was found to produce values of the Nusselt number that were almost independent of the corresponding calculated values of the Reynolds number during the major part of the compression and expansion strokes. A power relationship between the cylinder pressure and gas temperature and the apparent heat transfer coefficient was demonstrated.
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50

Yigit, Sahin, Timothy Graham, Robert J. Poole, and Nilanjan Chakraborty. "Numerical investigation of steady-state laminar natural convection of power-law fluids in square cross-sectioned cylindrical annular cavity with differentially-heated vertical walls." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 1 (January 4, 2016): 85–107. http://dx.doi.org/10.1108/hff-01-2015-0030.

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Purpose – Numerical simulations have been used to analyse steady-state natural convection of non-Newtonian power-law fluids in a square cross-sectioned cylindrical annular cavity for differentially heated vertical walls for a range of different values of nominal Rayleigh number, nominal Prandtl number and power-law exponent (i.e. 103 < Ra < 106, 102 < Pr < 104 and 0.6 < n < 1.8). The paper aims to discuss these issues. Design/methodology/approach – Analysis is carried out using finite-volume based numerical simulations. Findings – Under the assumption of axisymmetry, it has been shown that the mean Nusselt number on the inner periphery Nu i increases with decreasing (increasing) power-law exponent (nominal Rayleigh number) due to strengthening of thermal advection. However, Nu i is observed to be essentially independent of nominal Prandtl number. It has been demonstrated that Nu i decreases with increasing internal cylinder radius normalised by its height r i /L before asymptotically approaching the mean Nusselt number for a two-dimensional square enclosure in the limit r i /L→infinity. By contrast, the mean Nusselt number normalised by the corresponding Nusselt number for pure conductive transport (i.e. Nu i /Nu cond ) increases with increasing r i /L. Originality/value – A correlation for Nu i has been proposed based on scaling arguments, which satisfactorily captures the mean Nusselt number obtained from the steady-state axisymmetric simulations.
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