Nusselt Correlation Research Articles

Improved gas heaters for supercritical CO2 Rankine cycles: Considerations on forced and mixed convection heat transfer enhancement

To improve heat transfer performance of the gas heater in the supercritical CO2 Rankine cycle, heat transfer characteristics of supercritical CO2 heated in Internally Ribbed Tubes (IRT) was experimentally and numerically investigated over a wide range of buoyancy effect. Experiments reveal that for IRT with a geometric parameter of β=109, serious local heat transfer deterioration did not occur and buoyancy consistently improved heat transfer. IRT showed far superior performance over smooth tubes. Empirical Nusselt correlations were recommended for both upward and downward flows. A numerical model was then developed and verified against experimental data from IRT with different geometries. Further numerical studies on 7 different IRT demonstrate that buoyancy effect relates closely to rib geometries. For IRT with β=0.48–4, heat transfer to supercritical CO2 in upward flow was impaired when Bo is around 10−4, though the impairment became weaker as β increased. When β reached 16, buoyancy-induced impairment disappeared and Nusselt number increased gradually as buoyancy became stronger. Further increase in β up to 109 didn’t lead to obvious heat transfer enhancement of supercritical CO2, indicating that rib geometries play a smaller role in heat transfer than buoyancy for IRT with β>16. Finally, design strategy for IRT was provided for the optimization of gas heaters.

Sub-grid models for heat transfer in gas-particle flows with immersed horizontal cylinders

Simulating full-scale heated fluidized bed reactors can provide invaluable insight to the design process. Such simulations are typically computationally intractable due to their complex multi-physics and various length-scales. While it may be possible to simulate some large-scale systems, they require significant computing resources and do not lend themselves well to design optimization methods. To overcome these problems coarse-grid simulations can be used with supplementary constitutive sub-grid models to approximate the unresolved physics. This study details the development, implementation, and verification of a sub-grid model for heat transfer in gas-particle flows with immersed horizontal cylinders. Using the two-fluid model for multiphase flow, small periodic unit-cell domains were simulated over a wide range of flow and geometry conditions. The results were filtered and fit using nonlinear regression to build a Nusselt correlation based on the solids fraction, solids velocity, cylinder geometry (diameter and spacing), and the Peclet number. The proposed model is highly nonlinear and includes power-law contributions from each parameter. The model was verified using a nearly orthogonal experiment design where the input parameters were varied randomly to generate combinations not previously considered. The predicted filtered Nusselt numbers agreed well with the observed (simulated) values. Work is on-going to further expand the capabilities of the model, including 3D simulations, vertical cylinders, and uncertainty quantification.

Experimental and Computational Analysis of Single Phase Flow Coiled Flow Inverter Focusing on Number of Transfer Units and Effectiveness

Abstract The coiled flow inverter (CFI) is an enhanced heat exchanger. This device uses the principle of flow inversion to increase heat transfer and has potential for industrial applications. A mathematical model based on experimental and numerical data was developed for the case of single phase flow with water as working fluids. This model includes (ε-NTU) Effectiveness Number of Transfer Units curves, Nusselt correlations, for this specific neither parallel nor counter flow setup. A range of Reynolds numbers from 2,000 to 18,000 in the tube side, and 500 to 2,000 in the shell side was considered. The coiled flow inverter is made of coils and 90◦ bends, inserted in a closed shell. The shell side is cylindrical. The average temperatures at input and output of the heat exchanger were reported for different tube and shell side flow rates. Overall heat-transfer coefficients (Uf) were calculated as well as the Number of Transfer Units (NTU) and Effectiveness (ε) at various process conditions. A Nusselt correlation was proposed for the shell side of this configuration. The ε-NTU curves of the selected heat exchanger have a high resemblance to the parallel flow tube heat exchanger, with a maximum of 2.1 % error. The coiled flow inverter has increases of 200 %, 300 % and 500 %, respectively for effectiveness, number of transfer units and overall heat transfer coefficient compared to a regular parallel flow heat exchanger at the same conditions. Correlations for the CFI in its shell were proposed. This heat exchanger provides higher Uf, reduces cumbersomeness and length of piping.

Heat exchanger selection and design analyses for metal hydride heat pump systems

This study presents a design analysis for the development of highly efficient heat exchangers within stationary metal hydride heat pumps. The design constraints and selected performance criteria are applied to three representative heat exchangers. The proposed thermal model can be applied to select the most efficient heat exchanger design and provides outcomes generally valid in a pre-design stage. Heat transfer effectiveness is the principal performance parameter guiding the selection analysis, the results of which appear to be mildly (up to 13%) affected by the specific Nusselt correlation used. The thermo-physical properties of the heat transfer medium and geometrical parameters are varied in the sensitivity analysis, suggesting that the length of independent tubes is the physical parameter that influences the performance of the heat exchangers the most. The practical operative regions for each heat exchanger are identified by finding the conditions over which the heat removal from the solid bed enables a complete and continuous hydriding reaction. The most efficient solution is a design example that achieves the target effectiveness of 95%.

Heat transfer to supercritical water in a vertical tube with concentrated incident solar heat flux on one side

A solar tower power plant with supercritical water as a heat-transfer medium in the central receiver is potentially one of the most promising solar thermal power technologies due to its high solar-to-electric efficiency. In this paper, the heat transfer of supercritical water in a vertical tube of a solar tower receiver has been investigated. A 3D mathematical model has been developed to investigate the distribution of heat flux in the circumferential direction of a circular tube heated by concentrated incident solar flux on one side. The RNG k–ε model with the standard wall function is employed to describe the turbulent flow of water from a liquid-like state to a gas-like state, and the results are validated with experimental data. For supercritical water in a tube heated on one side by concentrated incident solar heat flux, the maximum wall temperature is located on the sunward outside wall where the incident heat flux is at a maximum. In the pseudo-critical flow region, due to the drastic turbulent diffusion of supercritical water, the temperature distribution of the supercritical water in the tube is evenly distributed at the same flow cross section. To avoid deterioration in the heat transfer of supercritical water, the relation between the threshold incident solar heat flux and mass flux for the supercritical water has been provided. Furthermore, the Nusselt correlation which can be used to predict heat transfer coefficient of the vertical tube heated non-uniformly on one side by concentrated solar flux has been confirmed.

Modeling of high pressure steam condensation in inclined horizontal tubes of PAFS in APR+

ABSTRACTA Single-tube COndensation exPeriment (SCOP) was performed at Korea Advanced Institute of Science and Technology (KAIST) to simulate the condensation phenomena of Passive Auxiliary Feedwater System (PAFS), which is introduced in Advanced Power Reactor Plus. SCOP tests were conducted for two pipes of different diameters [22.6 and 44.8 mm inside diameter] with 8.2 m length and 2.5–4 degrees of the inclined angles of the upper and lower section of each tube. The experiment were performed at 1.5–6.5 MPa of steam pressure and 0.12–0.45 kg/s of steam flow. Based on the Kim and No correlation (KAIST, 2000), we developed a new turbulent–condensation correlation for the turbulent region of the liquid film in the inclined tube. We introduced the factors to account for the different inclined angle effect, different diameter effect. The Nusselt correlation is adopted for condensation near the tube inlet where the laminar region of the liquid film is dominant due to the small amount of condensed water. The present correlation was compared with the Shah and RELAP5 condensation model using the SCOP data. The root mean square errors of the present correlation, Shah model, and RELAP5 model are 17.41%, 57.88%, and 43.07%, respectively: the present correlation can predict well the condensation characteristics in comparison with other correlations in PAFS.

On heat and moisture transfer characteristics of a desiccant dehumidification unit using fin tube heat exchanger with silica gel coating

In this paper, the heat and moisture transfer characteristics of a desiccant dehumidification unit using fin-tube heat exchanger with silica gel coating have been experimentally investigated. The present study focuses on the analysis of air side heat and moisture transfer performance in dehumidification mode. The empirical formula on Nusselt and Sherwood numbers were deducted and curve fitted from the test results under various operation conditions, using the non-linear least square method. The typical operation condition is that the air side Reynolds number ranges between 220 and 550 and tube side cooling water flow rate is about 7.7 L/min. Further tests indicate that the correlations of Nusselt and Sherwood number on air side of the unit agree well with the test results, and the mean errors are about ±10% and ±12%, respectively. Finally, the improvements of the operation and configuration parameters of this novel heat and moisture transfer components are discussed with those formulas. It is found that the correlated formulas provide the theoretical basis and practical experience for design and optimization of the desiccant dehumidification unit using fin tube heat exchanger with silica gel coating.

Natural convection on inclined QFN32 electronic package generating constant volumetric heat flux

Qualification and quantification of the natural convective phenomena are examined in the case of a Quad Flat Non-lead (QFN32). This active electronic package is inclined with respect to the horizontal plane by an angle varying between 0° and 90° corresponding to the horizontal and vertical position respectively. It generates during its operation a constant volumetric heat flux leading to Rayleigh numbers varying in the range 1.31x107‐1.01x108. The walls of the large air-filled cubic cavity containing this device are maintained isothermal. The temperature and velocity fields are presented for different combinations of the Rayleigh number and inclination angle. The convective heat transfer concerning the whole component exchange surface is determined for all the treated configurations. Correlations of Nusselt–Rayleigh type are proposed. They allow optimizing the thermal design of electronic assemblies used in various engineering domains.

Use of a fictitious Marangoni number for natural convection simulation

In this paper, a method based on the use of a fictitious Marangoni number is proposed for the simulation of natural thermocapillary convection as an alternative to the traditional effective diffusivity approach. The fundamental difference between these two methods is that the new method adopts convective mass flows in simulating natural convection. Heat transfer in the natural convection simulation is calculated through the mass transport. Therefore, empirical Nusselt numbers correlations required in the effective diffusivity method are eliminated. This represents a clear advantage in the context of design studies where flexibility in varying the geometry unconstrained by the availability of appropriate correlations is highly desirable. The new method is demonstrated using a simple geometrical model. An analytical expression of the fictitious Marangoni number associated with convection between vertical plates is derived and a computational fluid dynamics (CFD) simulation is performed to study the efficacy of the proposed method. The results show that the new method can approximate real natural convection quite accurately and can be used to simulate the convective flow in complex, obstructed or finned structures where the specific Nusselt correlation is not known.

Comparison between heat transfer to supercritical water in a smooth tube and in an internally ribbed tube

Numerical studies on heat transfer to supercritical water in an internally ribbed tube were performed and compared with those in a smooth tube. The Shear–Stress Transport k–ω model was adopted and validated against experimental data from tubes with different geometries. Comparisons between upward and downward flows show that the Jackson buoyancy criterion, which was proposed based on analysis and experiments in smooth tubes, can be used to evaluate effect of buoyancy in internally ribbed tubes. Radial profiles of turbulence and property reveal that in both smooth tube and internally ribbed tube, forced convection heat transfer is mainly influenced by specific heat, and effects of thermal conductivity and viscosity cancel each other out. As heat flux and bulk temperature increased, changes in the integral effect of specific heat resulted in variation in the heat transfer coefficient. The Jackson Nusselt correlation for smooth tubes can also accurately predict heat transfer coefficients of forced convection in an internally ribbed tube. Under conditions of mixed convection, buoyancy had a weaker impact on heat transfer in the internally ribbed tube. Detailed velocities and turbulence explained this weaker effect: the sharp drops in axial velocity gradient and turbulent kinetic energy, which occurred in a heated smooth tube at a radial position similar to the law of the wall region for isothermal flow, did not occur in a heated internally ribbed tube. Moreover, friction factors for internally ribbed tubes were always higher than that for smooth tubes in both forced convection and mixed convection.

A synthesis of correlations on quantification of free convective heat transfer in inclined air-filled hemispherical enclosures

Recent works have examined the thermal and dynamical aspects of natural convection occurring in tilted hemispherical cavities whose disk constitutes the hot active wall. The influence of some important aspects on the natural flows have been considered such as the Rayleigh Number, the disk inclination angle with respect to the horizontal plane and the thermal boundary condition on the disk (Dirichlet, Neumann). Combinations of these parameters have been treated in steady state as well as is transient regime. Correlations of Nusselt–Rayleigh type (steady state) and Nusselt–Rayleigh–Fourier type (transient regime) that allow quantifying the heat transfer are proposed in these surveys. The objective of this paper is to synthesize and summarize these works in order to facilitate their implementation in engineering applications.

Natural convection in inclined hemispherical cavities with isothermal disk and dome faced downwards. Experimental and numerical study

The objective of this work is to propose correlations of Nusselt–Rayleigh type for controlling the thermal state of electronic devices used in various engineering sectors such as thermoregulation of electronic devices, solar energy, aeronautics or safety and security. The assemblies are contained in hemispherical air-filled cavities whose disk is maintained isothermal and inclined at an angle varying between 90° (vertical position) and 180° (disk horizontal with isothermal dome oriented downwards). The numerical approach performed by means of the finite volume method for Rayleigh numbers varying between 104 and 5 × 108 is validated by measurements for various combinations of inclination angle and Rayleigh number. The results of this work show that the natural convective heat transfer are between 13% and 21% higher when the disk is maintained isothermal as compared to the case corresponding to imposed heat flux on the disk, for the same inclination angle and Rayleigh ranges.

Empirical Nu–Ra–Fo relationships for natural convection in air-filled hemispherical enclosures. Isothermal and inclined disk with dome oriented downwards

Correlations of Nusselt–Rayleigh–Fourier type proposed in this work allow quantifying of the transient convective heat transfer occurring in air-filled hemispherical cavities. The disk, initially at ambient temperature, is suddenly heated and kept isothermal. Throughout the heating process, the dome is maintained at ambient temperature. The radius of the cavity, associated with the temperature difference imposed between the disk and the dome, involves a large Rayleigh number range, varying between 104 and 5×108. The disk can be inclined with respect to the horizontal plane at an angle varying between 90° (vertical disk) and 180° (horizontal disk with dome downwards) by steps of 10°. The numerical approach is based on the finite volume method. The proposed empirical relationships in transient regime are linked to the steady state Nusselt–Rayleigh ones. They are obtained via the analysis of results corresponding to many combinations of the Rayleigh number and the disk inclination angle. The relationships are new since the considered geometry associated with the inclinations of the isothermal disk has not been treated previously. They constitute an important tool for the thermal design of engineering systems involved as they allow determining the convective heat transfer during the transient regime. They can be applied in several fields such as nuclear technology, solar energy, security and safety electronics, building, domotics or aeronautics.

Mechanistic modeling of modular co-rotating twin-screw extruders

In this study, we present a one-dimensional (1D) model of the metering zone of a modular, co-rotating twin-screw extruder for pharmaceutical hot melt extrusion (HME). The model accounts for filling ratio, pressure, melt temperature in screw channels and gaps, driving power, torque and the residence time distribution (RTD). It requires two empirical parameters for each screw element to be determined experimentally or numerically using computational fluid dynamics (CFD). The required Nusselt correlation for the heat transfer to the barrel was determined from experimental data. We present results for a fluid with a constant viscosity in comparison to literature data obtained from CFD simulations. Moreover, we show how to incorporate the rheology of a typical, non-Newtonian polymer melt, and present results in comparison to measurements. For both cases, we achieved excellent agreement. Furthermore, we present results for the RTD, based on experimental data from the literature, and found good agreement with simulations, in which the entire HME process was approximated with the metering model, assuming a constant viscosity for the polymer melt.

Heat transfer to supercritical water in circular tubes with circumferentially non-uniform heating

Heat transfer was numerically investigated for flow of supercritical water in a vertical circular tube which was heated on one side. The Shear-Stress Transport k-ω (SST) model was employed and preliminarily validated against experimental results under both circumferentially uniform and non-uniform heating. Numerical studies were performed for a pressure of 24.82 MPa, mass velocity of 404 kg/(m2s) and heat flux between 158 and 370 kW/m2. Within the range studied, empirical Nusselt correlations for uniform heating can adequately predict the circumferentially maximum wall temperature of a one-side heated tube. The circumferential difference of inner wall temperature was large (>60 K at high heat flux) due to poor mixing of fluid between the hot and cold side. At low heat flux, maximum wall temperature barely differed from that under uniform heating. While at high heat flux, maximum wall temperature dropped significantly compared with the uniformly heated case. With further heat input, buoyancy-induced local deterioration of heat transfer occurred, though not as severe as uniformly heated cases. Detailed axial velocity and turbulence show that turbulent quantities in the region, which is similar to the law of the wall region for isothermal flow, play an important role in heat transfer. Local deterioration did not happen when turbulent diffusion of this region was intense. Results also indicate that in a one-side heated tube, buoyancy effect is much stronger in upward flow than in downward flow.

Estimation of a Nusselt Correlation for the Numerical Prediction of Frost Thickness on a Tube Banks of Triangular Arrangement

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.

Cooling wheel features and amorphous ribbon formation during planar flow melt spinning process

A numerical model incorporating actual production conditions of planar flow melt spinning process is presented. Knowledge of the process conditions at which an amorphous ribbon with uniform thickness can be obtained is important to reduce the manufacturing costs. Cooling wheel conditions play a significant role during rapid solidification of the melt. Hence a heat transfer analysis is performed to investigate the influence of cooling wheel temperature on ribbon formation. Nusselt's correlation is used for the first time for selecting a convective heat transfer coefficient between rotating wheel and surrounding air. This approach assists in predicting the conditions at which a continuous/broken/no ribbon formation is obtained during the process. The model could predict whether the ribbon to be obtained is amorphous or non-amorphous before the experiment is actually performed at a set of process conditions. Various internal and external conditions of the cooling wheel are tested, and they show little influence on the ribbon thickness up to 0.5s (10 rotations). Broken and nonamorphous ribbons are obtained for poor cooling conditions of the wheel with increase in time of cast from 0.5s to 1s (20 rotations). Amorphous ribbon with uniform thickness can be obtained for a wheel of 20mm wall thickness, when the inner and outer surfaces of the cooling wheel are maintained at a temperature of at least 300K.

The Development of Turbine Volute Surface Temperature Models for 3D CFD Vehicle Thermal Management Simulations: Part 3: Exhaust Radial Turbine Volute Systems

Modern exhaust systems contain not only a piping network to transport hot gas from the engine to the atmosphere, but also functional components such as the catalytic converter and turbocharger. The turbocharger is common place in the automotive industry due to their capability to increase the specific power output of reciprocating engines. As the exhaust system is a main heat source for the under body of the vehicle and the turbocharger is located within the engine bay, it is imperative that accurate surface temperatures are achieved. A study by K. Haehndel [1] implemented a 1D fluid stream as a replacement to solving 3D fluid dynamics of the internal exhaust flow. To incorporate the 3D effects of internal fluid flow, augmented Nusselt correlations were used to produce heat transfer coefficients. It was found that the developed correlations for the exhaust system did not adequately represent the heat transfer of the turbocharger. This paper addresses the fluid flow phenomena present in the turbine volute and applies augmented Nusselt correlations to accurately represent the heat transfer coefficients of the internal volute surface. Due to the broad range of operating conditions that are applicable to the turbocharger and the varied states of fluid flow that occur, algorithms are used to apply the appropriate Nusslet correlations and augmentations. Furthermore, the turbocharger extracts enthalpy from the working fluid; therefore to accurately calculate surface temperatures of downstream components and that of the turbocharger itself, an energy extraction model is used. Validation was conducted with four vehicle configurations. The hot-end of each configuration was aimed to be distinctly different to test the robustness of the prediction model. A tolerance range of +50/-20K was used for the study, however temperature differences were generally well within the tolerance range.

Estimation of a Nusselt Correlations for Numerical Prediction of Frost Thickness Growth over a Cold Cylinder

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.

Nusselt Correlations in a Trailing Edge Cooling System with Long Pedestals and Ribs

Detailed TLC measurements have been carried out to estimate the heat transfer coefficient in stationary conditions under high Reynolds numbers in a trailing edge cooling system of a high pressure gas turbine blade. The investigated geometry consists of a 30:1 PMMA scaled model reproducing the typical wedge-shaped discharge duct of a trailing edge cooling system with one row of 7 enlarged pedestals. The section of the channel upstream the pedestal region is used to guide the airflow from the radial hub inlet to the tangential trailing edge outlet; in this section three different surfaces have been studied: one is smooth and the other two are ribbed with +60° and -60° angled ribs respect to the radial direction. This work focuses on the pedestal outlet section, by giving a correlation of the variation of the Nusselt number as a function of the Reynolds number, from 10000 to 40000, in the different inter-pedestal regions along the radial direction. These correlations give the regionally averaged heat transfer coefficients, from the hub to the tip of the model for both smooth and ribbed cases. The interest of these results is the use for the design of the trailing edge blade cooling systems, under the investigated Reynolds number range, that it is the typical case of the industrial application.