Volume-averaged Temperature Research Articles

Impact of temperature variations on BISON predictions of Ag release in AGR-1 and AGR-2 experiments

Understanding and quantifying the release of fission products like silver (Ag) from TRistructural ISOtropic (TRISO) fuel particles is important to assess the safe operation of advanced high temperature reactors. Although the silicon carbide (SiC) layer of TRISO particles is effective as the main fission product barrier, Ag can be released from intact TRISO particles. A mechanistic model for the effective Ag diffusivity, Deff, was previously developed as a function of temperature and microstructure variables informed by atomistic modeling of Ag diffusivity. In this study, we use this model to explore how experimental temperature uncertainties impact the overall predicted Ag release. This analysis indicates that calculated temperature uncertainties have a significant impact on the overall Ag release predictions. Furthermore, we show that the time average volume average (TAVA) temperature is not an appropriate proxy for the detailed temperature histories to predict fission product release due to the Arrhenius dependence of Ag diffusivity with respect to temperature. The detailed temperature histories, therefore, provide the most accurate results and are of most importance for modeling efforts. Overall, this work shows the importance of considering the experimental uncertainty of the temperature on computational predictions of fission product transport and release and the need for more accurate temperature histories from future experiments.

Design and optimization of an integrated liquid cooling thermal management system with a diamond-type channel

The ability of the battery thermal management system (BTMS) to dissipate heat significantly impacts the operating performance and the service life of lithium-ion batteries (LIBs). Designing an effective and efficient BTMS is difficult due to its multiple parameters. Therefore, an integrated liquid cooling plate with a diamond-type flow channel was designed in this paper. Firstly, two arrangement plans (plan 1 and plan 2) were designed to investigate the effect of coolant flow direction in the channel on the cooling performance of BTMS. The simulation results indicated that plan 2 (the shorter main flow channel serves as the outlet) could lead to better cooling performance. The maximum temperature (Tmax) and the volume-average temperature (VAT) of the battery pack for plan 2 were 313.33 K and 308.79 K, respectively. Then, the surrogate model was developed by the Latin Hypercube Sampling (LHS), and the six parameters (F, Hc, Wb, WL, WS, and WIO) of the BTMS were optimized using the multi-objective genetic algorithm (MOGA). The Tmax of the battery pack decreased from 313.33 K to 308.98 K, and the VAT decreased from 308.79 K to 306.06 K. The average pressure drop (Δp) between the inlets and outlets decreased significantly, from 1708.30 Pa to 520.10 Pa, up to 1188.20 Pa. It demonstrated that the BTMS achieved a balance between thermal performance and power consumption after optimization.

Comprehensive characterization of the adiabatic temperature change and (convection) heat transfer coefficient of a NiTi tube elastocaloric refrigerant

Elastocaloric cooling without global warming substance emission is a promising alternative to the vapor-compression technology. Comprehensive and precise characterization of the adiabatic temperature change of the elastocaloric refrigerant and the heat transfer coefficient between the elastocaloric refrigerant and the surrounding heat transfer fluid/solid is significant for the structural design of elastocaloric coolers. In this article, an analytical solution of the volume-averaged temperature variation in a tubular shape memory alloy elastocaloric refrigerant under cyclic compression was derived using lumped analysis, and a method of comprehensive characterization of the adiabatic temperature change (∆Tad) and (convection) heat transfer coefficient h for the elastocaloric refrigerant based on the analytical solution was proposed. A dimensionless number Ch was defined as the ratio of the latent heat release/absorption rate to the (convection) heat transfer rate, with which the characterization procedure including a series of nonlinear least-square regression tests and data selection criteria were established. The method proposed is applicable to any cross section geometry under both tension and compression, and it was validated using experimental data on a NiTi tube under sinusoidal force-controlled and reverse Brayton cyclic compressions and using existing experimental data of NiTi-based films, strips, and pillars under tension and compression in the literature. The ∆Tad characterized using the proposed method agreed with that using the reverse Brayton cyclic loading method within 5% (absolute value of 1 K). The effect of data selection sequence on the characterization of ∆Tad and h was investigated, and the results showed that proper starting points were significant for the convergence of ∆Tad and h.

Modeling of radio frequency heating of packed fluid foods moving on a conveyor belt: A case study for tomato puree

Computer simulation models were used to study radiofrequency heating of liquid foods moving on a conveyor belt between staggered-through field electrodes of non-uniform electric field distribution. Voltage across the electrodes was estimated using a finite-element model and a one-dimensional model, and these were utilized to simulate heating of tomato puree mixtures of varying dielectric and thermal properties, at various nominal voltages and electrode gaps. The experimental results for volume average temperatures showed good agreement with the outcome of both simulation models. Although the temperature profiles indicate a concentration of heat at the bottom and edges of the bottle, natural convection and the shape of the container helped mitigate heat localization. It was determined that, while simple models could provide accurate volume average temperatures, incorporating convectively enhanced conductivity was necessary to accurately predict temperature distribution. The heating rate was found to decrease with an increase in salt concentration. Industrial relevanceIn-package tunnel pasteurization is preferred over continuous-flow pasteurization for certain commercial products such as carbonated beverages or semi-solid foods of high viscosity. Dielectric heating can be applied in these cases with much lower requirements of floor space, water, and energy, while reducing heat losses and localized overheating. However, the radiofrequency heating of liquid foods has not been extensively studied, especially for in-package thermal treatments. A better understanding of the effect of various processing parameters on temperature distribution and heating rates could support a wider adoption of this technology in industrial processes.

Influence of the Lower Atmosphere on Wave Heating and Evaporation in Solar Coronal Loops

We model a coronal loop as a 3D magnetic cylinder in a realistic solar atmosphere that extends from the chromosphere to the corona. Kink oscillations, believed to be ubiquitous in the solar corona, are launched in the loop. Heating is expected owing to the dissipation of wave energy at small structures that develop from the Kelvin–Helmholtz instability induced by kink oscillations. Increases in temperature and internal energy can be observed in the coronal counterpart of the driven loop. With the presence of thermal conduction, chromospheric evaporation can also be seen. Although the volume-averaged temperature and density changes seem slight (∼4% relative to a nondriven loop), the enthalpy flow from the lower atmosphere redistributes the density and temperature in the vertical direction, thus enhancing the dissipation of wave energy in the corona. The efficient heating in the coronal counterpart of the loop can complement the thermal conductive losses shown in the current model and thus maintain the internal energy in the corona.

Thermal management system for prismatic battery module with biomimetic cephalofoil fin and film heater

For the air-cooling thermal management system of prismatic battery module applied in energy storage system, a form of biomimetic cephalofoil fin coupled with phase change material is presented and integrated with film heaters to conduct the functions of cooling and preheating. The numerical model with experimentally measured heat generation rate and specific heat capacity was established to evaluate performance of the proposed battery module. First, the battery module performance is analyzed in relation to the effects of different filling types and fin structures. As Compared with the case without filling, the phase change material sheet with thickness of 1 mm can reduce the maximum temperature of the battery module by 3.63 °C. Moreover, by combination of cephalofoil fin structure and phase change material sheet, the maximum temperature can be further reduced by 6.65 °C. Then, the investigation of fin structure parameters is carried out and the many-objective optimization and local optimization are conducted subsequently. It shows that the battery module temperature inconsistence can be improved by narrowing the flow path of cooling air or weakening the heat dissipation of the upstream batteries. For the optimal configuration, the maximum temperature and temperature difference can be maintained within 40 °C and 5 °C, respectively, at ambient temperature of 25 °C and C-rate of 2C. Moreover, a minor temperature rises of 5.44 °C is achievable with C-rate of 1.5C at higher ambient temperature of 40 °C. Finally, with the use of film heaters, the volume-average temperature of the battery module can be preheated from −5°C to 20 °C with a good temperature consistency of 1.2 °C within 10 min.

Combustion of Pulverized Coconut Shell in Lab-Scaled Incinerator Rig using CFD

In power generation, particle size distribution (PSD) of pulverized coal used in a power plant was 65%-70% passing 200 mesh or known as 76 microns. Since coal took hundreds of millions of years to form, it is not renewable energy and coal will release harmful gases after-burn. Thus, biomass has become an alternative fuel for reducing the consumption of coal and the emission of harmful gases. Coconut shell has the potential to substitute coal as fuel. In this study, Computational Fluids Dynamics (CFD) is used for simulating the combustion process in the lab-scaled incinerator rig (LSIR). The behaviours of pulverized coconut shells and the effect of excess air during combustion are studied. The results showed that the average carbon monoxide (CO) and carbon dioxide (CO2) mass fraction was calculated at 0.0196 and 0.257 respectively. The combustion efficiency was determined as 92.94%. The percentage of excess air (EA%) was increased by increasing the velocity of inlet air for investigating the responses of combustion efficiency. The EA with 39% was predicted as the most suitable EA for the combustion because the heat generated at 39% EA was the highest and its volume average temperature was recorded as 1198.7K. Besides, the combustion efficiency was increasing when the EA% is increased by calculating from every volume average of mass fraction of CO and CO2.

Evaluation of Steel Slag Optimal Replacement in Asphalt Mixture under Microwave Heating Based on 3D Polyhedral Aggregate Electromagnetic-Thermal Meso-Model

Replacing conventional aggregate with steel slag waste can boost the microwave absorption properties of asphalt mixtures and reduce pollution to protect the environment. In order to achieve the best healing in steel slag asphalt mixture, the optimum particle size and content of steel slag are essential. For this purpose, a high-efficiency algorithm for the random growth and placement of convex polyhedron aggregate is proposed in this paper. The limestone aggregate is replaced with an equal volume of steel slag, and a three-dimensional mesoscale random model of steel slag asphalt mixture is developed. The process of microwave heating is simulated by FEM. The numerical simulations are compared with the reported experimental data, which proves that the model is reliable (R2 = 99.40%). Both the volume average temperature and the uniformity of temperature distribution indicate that the steel slag replacement rate of 60% at 4.75–9.5 mm and 60% at 9.5–13.2 mm is optimal, among which the heat transfer of 4.75–9.5 mm steel slag is more uniform, and the temperature gradient is lower. Steel slag can dramatically increase the heating rate of an asphalt mixture, and the peak of the temperature gradient is around the boundary of steel slag. The reflection properties of steel slag may be related to the dielectric constant, permeability, and particle size. Excess steel slag will cause overheating in most zones of the specimen and will also depress the absorption efficiency of microwaves. The coefficient of variance for spherical (0.36) and polyhedral (0.32) aggregate specimen temperatures indicates that the aggregate’s shape has a negligible effect on the heat transfer of asphalt mixtures.

Characterizing spatial variability in the temperature field to support thermal model validation in a naturally ventilated building

ABSTRACT Night-time passive cooling is an energy-efficient cooling strategy, but the design of passive cooling systems relies on imperfect computational models, which require validation. This paper assesses the importance of spatial variability in the temperature field when performing model validation. Full-scale temperature measurements in a three story atrium building reveal spatial variability of up to on each floor during the natural ventilation process. Validation of a dynamic thermal model with uncertainty quantification reveals accurate volume-averaged air temperature predictions. Discrepancies are on the order of the sensor accuracy (), and are primarily due to slightly under-predicted cooling rates in the model. Importantly, this trend would be identified incorrectly when validating the model against the building's built-in sensors, which consistently record 0.05–1.63 higher temperatures than the volume-averaged air temperature. These findings highlight the importance of spatial variability and careful temperature sensor placement in naturally ventilated buildings.

Optimization of film cooling arrays on a gas turbine vane by using an integrated approach of numerical simulation and parameterized design

• Conjugate heat transfer analysis is performed on an internally and film cooled vane. • A parameterized optimization procedure is developed for film cooling arrays. • All of the film cooling arrays are considered in a multi-objective optimization. • Effects of injection angles and compound angles on film cooling are discussed. To enhance film cooling effectiveness, an optimization of film cooling arrays on a gas turbine vane is carried out by using an integrated approach of numerical simulation and parameterized design. Conjugate heat transfer (CHT) numerical simulation is performed on an internally and film-cooled vane, and the simulation accuracy is validated against the experimental data. The parameterized optimization procedure, which combines CFD simulation with the design of experiment (DOE), back propagation neural network (BPNN), and genetic algorithm (GA), is developed based on the fluid domain of the vane. In the workflow, the data are parameterized and passed across the design, simulation, and optimization. The injection angles and compound angles are optimized. For the adiabatic geometry model used in the optimization process, the downstream average adiabatic film cooling effectiveness achieves a 7.7% increment over the reference design, while the average temperature of the leading edge is decreased by 10.8 K. The obtained optimized design is applied to the CHT computational domain of the vane for further analysis of film cooling characteristics. The area-averaged temperature of the vane surface and the volume-averaged temperature of the solid region are reduced by 7.1 K and 5.8 K, respectively.

Wall-bounded thermal turbulent convection driven by heat-releasing point particles

In this work we investigate the thermal convection driven by heat-releasing point particles. Three-dimensional direct numerical simulations are conducted for$1\times 10^7\le {\textit {Ra}} \le 1\times 10^{10}$and$0.01\le {\textit {St}} \le 10$, where the Rayleigh number${\textit {Ra}}$and Stokes number${\textit {St}}$measure the strengths of the heat releasing rate and the Stokes drag, respectively. A regime at intermediate Stokes numbers is identified with most particles accumulating into the top boundary layer region, while for other cases particles are constantly advected over the entire domain. For the latter state, the flow motions are stronger at the upper part of the domain. The thicknesses of both momentum and thermal boundary layers at the top plate follow the same scaling law with${\textit {Ra}}$and show minor dependences on${\textit {St}}$. The volume-averaged temperature and convective flux exhibit non-monotonic variations as${\textit {St}}$increases and reach their minimums at intermediate${\textit {St}}$. The fraction coefficient of heat flux, i.e. the ratio between the heat flux through the bottom plate and the total flux through both plates, shares the similar dependence on${\textit {St}}$as the convective flux. The relation between these scaling laws can be explained by using the global balance between the dissipation and convective flux. The scaling laws for the transition between different flow regimes are also proposed and agree with the numerical results. The preferential concentration of particles is observed for all cases and is strongest at intermediate Stokes numbers, for which multiscale clustering emerges with small clusters forming larger ones.

CFD study of bio-syngas and coal co-injection in a blast furnace with double lance

To rationally utilize biomass as an auxiliary fuel for pulverized coal injection (PCI) technology, the process of co-injection of bio-syngas/coal into the blast furnace (BF) is designed. In this paper, a 3D CFD model with double lance is established to investigate the co-injection of bio-syngas/coal in the BF. The effects of bio-syngas injection rates and the location of double-lance on the in-furnace thermochemical behavior of pulverized coal (PC) are systematically analyzed. The results show that the combustion of the bio-syngas promotes the devolatilization of the PC, but hinders the volatile matter (VM) combustion. As the bio-syngas injection rate increases, the volume-average temperature of the raceway and the endpoint burnout of PC increases; In addition, a reduction in the distance between the bio-syngas lance and the tuyere, or an increase in the distance between the two lances, can facilitate the PC to obtain more oxygen and have a better endpoint burnout. However, the rise of the coal plume reduces the space for the particles to recirculate, resulting in a reduction in the overall burnout of the PC.

Temperature distribution in a laser-heated diamond anvil cell as described by finite element analysis

Finite element analysis (FEA) is a powerful tool for numerically solving partial differential equations over complex geometries and is thus useful for analyzing heat transport in laser-heated diamond anvil cell (LHDAC) experiments. Our models expand on previously published simulations by calculating the volume-averaged temperatures of both the sample and insulation/pressure media under steady-state heating to determine the thermal pressure of the hot sample. Our goal is to produce an accurate relationship between the measured surface temperature of the absorbing sample and the temperature of the transparent insulating media, which is used to determine thermal pressure but susceptible to steep temperature gradients. We find that in doing so, our FEA models of temperature within the pressure/insulation media can differ from simplified estimates of temperature gradients by more than a factor of 2. We also explore temperature-dependent and temperature-independent thermal conductivity models and find that the volume-averaged temperatures differ by up to a factor of 1.3, forcing the predicted thermal pressures determined to also differ by up to a factor of 1.5 at a temperature of 2000 K at 50 GPa for neon. Higher temperatures exacerbate this difference. We also find that unintentional asymmetric sample insertion and sample heating, which are common in LHDAC experiments, do not have a first-order effect on volume-averaged temperatures. The FEA models, available in both Python and FlexPDE, are versatile across different sample geometries, materials, and heat source laser shapes.

A parametric study of subcooled flow boiling of Al2O3/water nanofluid using numerical simulation and artificial neural networks

ABSTRACT Utilizing an Euler-mixture three-dimensional numerical simulation for Al2O3/water nanofluid subcooled flow boiling in a mini channel, we study the effects of pressure, heat flux, nanoparticle concentration, surface roughness, and subcooled temperature on heat transfer quantities (average and local heat transfer coefficient, average and local vapor volume fraction, and average and local wall temperature) and bubble dynamics quantities (bubble departure diameter, bubble detachment frequency, bubble detachment waiting time, and nucleation site density). The numerical results demonstrate that the nanoparticles particularly impact the bubble dynamics significantly by increasing wettability and decreasing contact angle. In order to reduce the computational burden of such an expensive multiphase flow simulation, we also present a machine learning approach based on artificial neural networks (ANN). The numerical experiments show that using the ANN model, we can achieve highly accurate results with much less computational time and resources.

Full-scale validation of CFD simulations of buoyancy-driven ventilation in a three-story office building

Computational fluid dynamics (CFD) is frequently used to support the design of naturally ventilated buildings; however, the model accuracy should be thoroughly assessed, ideally through validation with full-scale measurements. The present study aims to (1) validate transient CFD simulations with uncertainty quantification (UQ) for buoyancy-driven natural ventilation against full-scale experiments in an operational atrium building, and (2) quantify the sensitivity of the CFD results to the thermal boundary conditions. The UQ and sensitivity analysis consider uncertainties in the initial and boundary conditions for the temperatures. Considering the volume-averaged air temperature on each floor, the predictions and measurements agree well with discrepancies less than 0.3 °C. When considering the temperature averaged over smaller zones on each floor, two trends can be observed. First, in zones not adjacent to windows, the discrepancies between the CFD and measurements can be explained by uncertainty in the boundary conditions and the measurements. Second, in zones adjacent to windows, higher discrepancies are observed due to oscillations in the inflow jets just downstream of the windows, and due to geometrical simplifications in the CFD model. The sensitivity analysis demonstrates that the boundary conditions for the thermal mass surface temperature and the outdoor temperature have a dominant effect on the indoor air temperature predictions, with their relative importance varying as a function of proximity to the windows.

Evaluating the efficiency of fog cooling for climate change adaptation in vulnerable groups: A case study of Daegu Metropolitan City

This study quantitatively analyzed the effect of fog cooling on temperature reduction and conducted an optimization study on the layout for planning a fog cooling system in the Daegu Metropolitan City. First, the design and verification of a single fog cooling unit was identified through an indoor experiment. Second, the temperature reduction effect of the installed fog cooling system was measured based on a sensor and verified through a finite volume method (FVM)-based computational fluid dynamics (CFD) analysis. Third, a numerical analysis of the optimal fog cooling arrangement was designed and performed in the Indongchon area. The temperature reduction design manual and model equations, which can be used by decision-makers, were analyzed through numerical analysis. The measured temperature reduction and simulation values had a high accuracy of R2 ≥ 0.8. Based on the verified model, scenarios A–D were designed and analyzed according to the fog cooling arrangement. Scenario B, in which the fog cooling was arranged vertically to the wind direction, appeared to be the most efficient. In this case, the volume average temperature where fog cooling was installed could be reduced by up to 3.02 °C. It was also found that efficiency is higher when the fog cooling system is positioned at narrow intervals of ≤5 m than at wide intervals of ≥10 m, considering the wind direction. The results of this study can aid decision makers—including urban policymakers and planners—for estimating the potential extent of temperature reduction and increase in humidity by adopting fog cooling.

Understanding, controlling and optimising the cooling of waste thermal treatment beds including STARx Hottpads

STARx (Self-sustaining Treatment for Active Remediation ex situ) is a thermal treatment strategy for contaminated soils and organic wastes. Key to this technology is that organics are embedded in porous matrix beds (e.g. sand). STARx induces a self-sustaining smouldering combustion front that traverses the bed, burning away the embedded contaminants/wastes. The time and cost effectiveness of this technology is largely dictated by the time required for cooling of the hot, clean, porous matrix bed that remains after treatment. This study is the first to explore the cooling of these beds. A suite of novel simulations investigated the influence of key parameters on bed-cooling time. The results reveal that cooling time decreased nearly linearly with decreases of volume-averaged bed temperature and bed bulk density. Increased injection air fluxes led to the non-linear decrease of cooling time. Also, cooling time was negatively impacted by bed temperature inhomogeneity, which influenced preferential air flow through cooler regions of the bed, bypassing hotter regions. From these results, using lower bulk density bed materials, increased air fluxes and enhancing wall insulation to improve bed temperature homogeneity were identified as system optimisations to reduce cooling times. While the aim of this research is to improve the STARx cooling process, the results are also highly applicable to many similar engineering systems that involve hot porous bed cooling.

Numerical modelling of single-bubble acoustic cavitation in water at saturation temperature

Acoustic cavitation plays a major role in the enhancement of many Ultrasound-assisted processes, such as distillation and extraction. This work aims to numerically investigate single-bubble acoustic cavitation in water at saturation temperature. To this end. a new model that describes single-bubble acoustic cavitation was developed. One novel feature of this model is that it resolves the spatial variation of chemical species and energy in the bubble and surrounding liquid with the consideration of shell effect, which allows better prediction of heat and mass transfer during cavitation. This is realized by adopting a unique semi-lagrangian Finite Volume Method with dynamic mesh. Besides, a novel approach, which makes use of interfacial jump conditions that enforce simultaneous energy and mass conservation, is implemented in the model to allow simultaneous coupling of energy and mass transfer across the bubble interface. Dissociation of liquid molecules (chemical reactions) is taken into account. The developed model was used to investigate single-bubble acoustic cavitation in water at room temperature and saturation temperature. The simulation results showed that, despite the much stronger bubble collapse (about twice as strong), the volume-averaged peak temperature reached during cavitation in water at saturation temperature is lower compared to that at room temperature (3877K vs 4600K) due to higher amount of vapor trapped in the bubble during the collapse. Nevertheless, the amount of reactive free radical produced per bubble during the collapse at saturation temperature is much higher (1 – 2 order higher in amount) due to higher vapor molecule concentration in bubble. However, the spatial extent of radical production could be lower due to the much higher bubble expansion. The model and findings could be helpful for the design and optimization of Ultrasound-assisted processes.

Influence of viscosity on the thermal behavior of fluids in a sealed can

Homogeneous fruit juice in a sealed container available in the commercial market exhibits some degree of non-Newtonian behavior with a wide viscosity range. During thermal processing, it is necessary to understand the effect of viscosity on thermal behavior, which may guide insight into the choice of physics involved in the conservation law equations. In this context, the present study considered the Carboxyl Methyl Cellulose (CMC) solutions (a viscosity-enhancing agent) as model foodstuffs to understand the effect of viscosity on the heat transfer mechanism in the thermal treatment of canned liquid products. Computational Fluid Dynamics (CFD) was employed to analyze the thermal behavior of different concentrations of CMC solutions, estimated based on averaged wall heat flux, averaged heat transfer coefficient, volume-averaged temperature, and minimum temperature. The effect of the CMC solutions' rheological properties on the thermal behavior was evaluated by comparing it with the pure conduction simulations. The time–temperature history obtained from the simulations was coupled with the kinetic parameter of microbial destruction to predict the heating time. When the viscosity increases due to increasing the concentration of CMC solution, the natural convection is suppressed, which leads to the heat transfer in almost pure conduction mode.

A two-domain analytical approach for laminar and turbulent forced convection in a transpiration-cooled channel

A two-domain analytical approach has been proposed to attack the conjugate problem associated with forced convective flows in a channel with uniform porous walls subject to transpiration cooling. Both temperature fields in the porous wall and the channel with uniform porous walls are treated simultaneously and matched on the porous wall. General analytical expressions for the temperature fields and wall heat transfer rates, which involve multiple integrals of the velocity profile function, were derived considering the energy balance equation for the two domains in both laminar and turbulent cases. A local thermal non-equilibrium solution was obtained for the volume average temperature of coolant gas phase and that of solid phase within the porous wall, while a local similarity solution was found for the energy equation for the channel flows with transpiration cooling for the first time. For the turbulent flows in a channel, Stevenson's law of the wall was used along with the laminar sublayer velocity profile with transpiration, so as to reveal the effect of the blowing ratio on the temperature fields and heat transfer rates. The present analytical and numerical results agree well with available RANs and DNS results. The present results obtained for both laminar and turbulent flows reveal that Kays and Crawford's correlation for the heat transfer coefficient ratio of with and without blowing holds reasonably well for both laminar and turbulent flow cases. The damping of the heat transfer coefficient ratio, predicted in this analysis, however, is much more substantial than that estimated by Kays and Crawford.