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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Characterization and Scaling of Forced Convective Swirl in Sinusoidal Wavy-Plate-Fin Cores of Compact Heat Exchangers

The characterization and scaling of the thermal-hydraulic performance in wavy plate-fin compact heat exchanger cores, based on the understanding of the physical phenomena and heat transfer enhancement mechanism is delineated. Experimental data are presented for forced convection in air (Pr = 0.71) with flow rates in the range 50 ≤ Re ≤ 4000. A variety of wavy-fin cores that span viable applications, with geometrical attributes described by the cross section aspect ratio α (ratio of fin spacing to height), fin corrugation aspect ratio γ (ratio of 2× corrugation amplitude to wave pitch), and fin spacing ratio ζ (ratio of fin spacing to wave pitch), are considered. To characterize and correlate the vortex-flow mixing in interfin spaces, a Swirl number Sw is introduced from the balance of viscous, inertial and centrifugal forces. It is shown that the laminar, transitional and turbulent flow regimes can be explicitly identified by this Swirl number. Based on the experimental results and extended analysis, new correlations for Fanning friction factor f and Colburn factor j are developed with Sw, α, γ, and ζ as scaling parameters. Requisite expressions are devised by a superposition of both enhancement components due to the corrugated surface area enlargement and induced swirl flow field, and they are combined to cover the laminar, transitional and turbulent regimes by the method of asymptotic matching. The resulting generalized correlations are further shown to predict all available experimental data for f and j factors to within ±20% and ±15%, respectively.

42 ENGINEERING↗
DOE OSTI2021

Experimental Study on Flow Behavior and Heat Transfer Enhancement With Distinct Dimpled Gas Turbine Blade Internal Cooling Channel

Abstract The study presents the investigation on heat transfer distribution along a gas turbine blade internal cooling channel. Six different cases were considered in this study, using the smooth surface channel as a baseline. Three different dimples depth-to-diameter ratios with 0.1, 0.25, and 0.50 were considered. Different combinations of partial spherical and leaf dimples were also studied with the Reynolds numbers of 6,000, 20,000, 30,000, 40,000, and 50,000. In addition to the experimental investigation, the numerical study was conducted using a large eddy simulation (LES) to validate the data. It was found that the highest depth-to-diameter ratio showed the highest heat transfer rate. However, the deepest dimpled channel shows the highest pressure drop, which affects the overall thermal performance of the cooling channel. The results showed that the leaf dimpled surface is the best cooling channel based on the highest Reynolds number's heat transfer enhancement and friction factor. However, at the lowest Reynolds number, partial spherical dimples with a 0.25 depth-to-diameter ratio showed the highest thermal performance.

Energy & Fuels↗
DOE OSTI2022

Investigation of Gas Turbine Internal Cooling Using Supercritical CO 2 —Effect of Surface Roughness and Channel Aspect Ratio

In this paper, an experimental and numerical investigation of internal cooling channels with rib turbulators is presented with sCO 2 as the working fluid at process conditions (pressure-20.7 MPa and temperature up to 150 °C). The effect of channel aspect ratio up to 2:1 on thermal-hydraulic performance is explored in additively manufactured rectangular channels and square channels, both with and without 60 deg ribs on the top and bottom sides. The Wilson-plot method is employed to experimentally measure channel-averaged Nusselt number over a Reynolds number range up to 370,000. The friction factor is calculated from pressure drop and mass flow rate and additionally, the overall thermal performance factor (TPF) is reported. A companion computational fluid dynamics (CFD) simulation is performed for the rib turbulated cooling configurations reported in the experiments using the Reynolds average Navier–Stokes-based turbulence model. The objective of the numerical study is to gain insight into the local heat transfer augmentation in the ribbed channels as a result of varying the aspect ratio, channel configuration (square versus rectangular), operating conditions (Reynolds number) and the surface roughness, an inherent outcome of the additive manufacturing process. Surface roughness is simulated using sand grain roughness height (K S ) calculated from the experimental data, and a comparison is presented with the corresponding channel configuration with varying surface roughness heights starting from smooth surfaces (K S = 0). Experimental results indicate that the heat transfer augmentation is negligible in the rectangular channels with ribs on the long side compared to the square channel. However, it is enhanced by 60% in comparison to placing ribs on the shorter side. The TPF remains constant at around 1 for the entire range of Reynolds numbers consistent with prior work at the National Energy Technology Laboratory (NETL). Here, the simulation results highlight that increased surface roughness can have a favorable considerable influence on Nusselt number and overall thermal performance enhancement.

42 ENGINEERING↗
DOE OSTI2022

Design of an Additively Manufactured Recuperator With 800 °C Inlet Temperature for sCO2 Power Cycle Application

Abstract Recuperators with design temperatures at and above 800 °C can further increase the thermal efficiency of supercritical CO2 power cycles by enabling higher turbine exhaust temperatures. Mar-M247 is a well-suited nickel-based superalloy for high temperature service due to its high creep strength that prevents excessive material thickness being required for pressure containment. Additive manufacturing using a high-speed laser-directed energy deposition (L-DED) process presents a promising solution, with build trials demonstrating the ability to produce nonconventional flow channels for enhanced heat transfer. A design process is presented that includes aerothermal and mechanical evaluation to maximize performance within the constraints of the manufacturing process. A 2-D heat transfer network and pressure drop code allows prediction of flow distribution and its effect on overall thermal performance. Established literature correlations, along with CFD simulation, inform the prediction of heat transfer coefficients and friction factors for the flowpaths and enhancement features in the heat exchanger core. Mechanical evaluation using finite element analysis (FEA) modeling with the intent of the ASME Boiler & Pressure Vessel Code (BPVC) Section VIII, Div. 2 assesses the operational safety of the design. The detailed design features annular finned passages that take advantage of helical flow paths to distribute the flow from separated headers to shared heat transfer surfaces. Performance predictions for the recuperator at a 50 kW scale provide insights into the feasibility of the additively manufactured (AM) process to produce recuperators on a commercial scale that extend existing operating envelopes.

Engineering↗
DOE OSTI2023

Helical Fins for Concentrated Solar Receivers: Design Optimization and Entropy Analysis

Concentrated solar power (CSP) with thermal energy storage (TES) has the potential to achieve grid parity. This can be realized by operating CSP systems at temperatures above 700 °C with high-efficiency sCO 2 power cycles. However, operating CSP systems at such temperatures poses several challenges, among which the design of solar receivers to accommodate increased thermal loads is critical. To this end, this work explores and optimizes various swirl-inducing internal fin designs for solar receiver tubes. These fin designs not only improve the thermal performance of receiver tubes but also levelize temperature unevenness caused by non-uniform thermal loading. In this work, the geometric parameters of the fin designs are optimized to maximize the Nusselt number with a constraint on the friction factor. This optimization, however, is computationally intensive, requiring hundreds of simulation calls to computational fluid dynamics (CFD) models. Here, to circumvent this problem, surrogate models are used to approximate the simulation outputs needed during the optimization. In addition, this study also examines the fin designs from an entropy generation perspective. To this end, the entropy contributions from thermal and viscous effects are quantitatively compared while varying the operational Reynolds number.

14 SOLAR ENERGY↗
DOE OSTI2022

Phase 1 Specifications and preliminary sensitivity analyses of the OECD-NRC liquid metal fast reactor core thermal-hydraulics benchmark

The developments in wire-wrapped fuel bundle thermal-hydraulics research have become increasingly diverse and this makes comparisons across studies difficult. This paper describes an ongoing benchmark that will address this issue by directly comparing computational results with experimental data from two unique wire-wrapped thermal-hydraulic facilities. Preliminary subchannel sensitivity / uncertainty analyses show largely expected behavior for the presented cases, where rod diameter and friction factor show the greatest sensitivities to geometrical perturbations. Guidelines for computational fluid dynamics evaluations have been prepared and highlight the need for consolidated modeling strategies. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2019

A User Guide to PARET/ANL

PARET was originally created in 1969 at what is now Idaho National Laboratory (INL), to analyze reactivity insertion events in research and test reactor cores cooled by light or heavy water, with fuel composed of either plates or pins. The use of PARET is also appropriate for fuel assemblies with curved fuel plates when their radii of curvatures are large with respect to the fuel plate thickness. The PARET/ANL version of the code has been developed at Argonne National Laboratory (ANL) under the sponsorship of the U.S. Department of Energy/NNSA since the inception of the Reactor Conversion Program. Since Reduced Enrichment for Research and Test Reactors (RERTR) began in 1978, PARET/ANL has been benchmarked to experimental data including SPERT testing, and used to determine the expected transient behavior of a large number of reactors regardless of enrichment both inside and outside the Reactor Conversion Program. This document provides the pertinent information for the use of PARET/ANL Version 7.6. PARET/ANL models the various fueled regions of a reactor core as channels. Each of these channels consists of a single flat fuel plate/pin (including cladding and, optionally, a gap) with water coolant on each side. In slab geometry the coolant channels for a given fuel plate are of identical dimensions (mirror symmetry), but they can be of different thickness in each channel. There can be many channels, but each channel is independent and coupled only through reactivity feedback effects to the whole core. The time-dependent differential equations that represent the system are replaced by an equivalent set of finite-difference equations in space and time, which are integrated numerically. PARET/ANL uses fundamentally the same numerical scheme as RELAP5 for the time-integration of the point-kinetics equations. The one-dimensional thermal-hydraulic model includes temperature-dependent thermal properties of the solid materials, such as heat capacity and thermal conductivity, as well as the transient heat production and heat transfer from the fuel meat to the coolant. Temperature- and pressure-dependent thermal properties of the coolant such as enthalpy, density, thermal conductivity, and viscosity are also used in determining parameters such as friction factors and heat transfer coefficients. The code first determines the steady-state solution for the initial state. Then the solution of the transient is obtained by integration in time and space. Multiple heat transfer, DNB and flow instability correlations are available. The code was originally developed to model reactors cooled by an open loop, which was adequate for rapid transients in pool-type cores. An external loop model appropriate for Miniature Neutron Source Reactors (MNSR’s) was also added to PARET/ANL to model natural circulation within the vessel, heat transfer from the vessel to pool and heat loss by evaporation from the pool. PARET/ANL also contains models for decay heat after shutdown, control rod reactivity versus time or position, time-dependent pump flow, and loss-of-flow event with flow reversal as well as logic for trips on period, power, and flow. Feedback reactivity effects from coolant density changes and temperature changes are represented by tables. Feedback reactivity from fuel heat-up (Doppler Effect) is represented by a four-term polynomial in powers of fuel temperature. Photo-neutrons produced in beryllium or in heavy water may be included in the point-kinetics equations by using additional delayed neutron groups.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2019

RCCS Testing with the Water-based NSTF: Year-1 Single-Phase Results

Under support from the Department of Energy (DOE) and the Office of Advanced Reactor Technologies (ART), a large scale test facility has been constructed to generate validation data for passive decay heat removal systems of advanced reactors. Located at Argonne National Laboratory, the Natural convection Shutdown heat removal Test Facility (NSTF) reflects key features of a ½ scale, water-based, Reactor Cavity Cooling System (RCCS). This report serves as a summary of the single-phase testing results obtained during the programs first year of matrix testing. An introduction to the high-level program objectives is first presented, followed by a brief summary of facility design, instrumentation, and control systems. Details of the NQA-1 driven test method are included, which govern the programs quality assurance requirements and provide a background for the testing strategy and procedures. Checkout activities, including verification of gross system mass and energy balances, heat loss quantification, and isothermal facility characterization, were performed early in the first year and results presented in this report. These tests have identified relevant facility characteristics such as K-factors, frictional pressure losses, parasitic heat loss rates, and transient heating response. Results are then presented from four accepted matrix tests where the facility established a steady-state, single-phase mode of flow operation. All matrix tests were performed with an identical top-level procedure: first initiating a heater power ramp to target levels, and then holding power levels constant until the facilities structural members and network piping reached thermal equilibrium. Upon obtaining stable thermal and hydraulic conditions, active cooling was initiated off of the bulk storage tank to maintain constant riser inlet temperatures of 30°C. This steady-state period was then held for a minimum required duration, at which point the heaters were ramped down and test concluded. The two baseline tests completed reflect conditions prototypic to the full scale plant at both normal operation, and design basis Depressurized Loss of Forced Cooling (DLOFC) conditions. These parameters are defined by full RCCS decay heat loads of 1.4 MWt and 2.1 MWt, which translate to NSTF scale thermal power levels of 34.4 kWt and 51.6 kWt, respectively. Two additional tests were also conducted, which repeated baseline cases but included modification by single parameter variations in decay heat load, or heater power, at 700 kWt, and 2.8 MWt. Finally, a scoping test was performed to demonstrate the facilities ability to operate in a two-phase mode of operation. The facility successfully operated for a 3-hour period at this mode, reaching peak inventory temperatures of 105°C and condensing approximately 50 gallons of boil-off inventory. The testing parameters for this demonstration were based off the design basis DLOFC condition, and will be the reference for new baseline cases as the program moves forward with two-phase matrix testing.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

A User Guide to PARET/ANL

PARET was originally created in 1969 at what is now Idaho National Laboratory (INL), to analyze reactivity insertion events in research and test reactor cores cooled by light or heavy water, with fuel composed of either plates or pins. The use of PARET is also appropriate for fuel assemblies with curved fuel plates when their radii of curvatures are large with respect to the fuel plate thickness. The PARET/ANL version of the code has been developed at Argonne National Laboratory (ANL) under the sponsorship of the U.S. Department of Energy/NNSA since the inception of the Reactor Conversion Program. Since Reduced Enrichment for Research and Test Reactors (RERTR) began in 1978, PARET/ANL has been benchmarked to experimental data including SPERT testing, and used to determine the expected transient behavior of a large number of reactors regardless of enrichment both inside and outside the Reactor Conversion Program. This document provides the pertinent information for the use of PARET/ANL Version 7.6. PARET/ANL models the various fueled regions of a reactor core as channels. Each of these channels consists of a single flat fuel plate/pin (including cladding and, optionally, a gap) with water coolant on each side. In slab geometry the coolant channels for a given fuel plate are of identical dimensions (mirror symmetry), but they can be of different thickness in each channel. There can be many channels, but each channel is independent and coupled only through reactivity feedback effects to the whole core. The time-dependent differential equations that represent the system are replaced by an equivalent set of finite-difference equations in space and time, which are integrated numerically. PARET/ANL uses fundamentally the same numerical scheme as RELAP5 for the time-integration of the point-kinetics equations. The one-dimensional thermal-hydraulic model includes temperature-dependent thermal properties of the solid materials, such as heat capacity and thermal conductivity, as well as the transient heat production and heat transfer from the fuel meat to the coolant. Temperature- and pressure-dependent thermal properties of the coolant such as enthalpy, density, thermal conductivity, and viscosity are also used in determining parameters such as friction factors and heat transfer coefficients. The code first determines the steady-state solution for the initial state. Then the solution of the transient is obtained by integration in time and space. Multiple heat transfer, DNB and flow instability correlations are available. The code was originally developed to model reactors cooled by an open loop, which was adequate for rapid transients in pool-type cores. An external loop model appropriate for Miniature Neutron Source Reactors (MNSR’s) was also added to PARET/ANL to model natural circulation within the vessel, heat transfer from the vessel to pool and heat loss by evaporation from the pool. PARET/ANL also contains models for decay heat after shutdown, control rod reactivity versus time or position, time-dependent pump flow, and loss-of-flow event with flow reversal as well as logic for trips on period, power, and flow. Feedback reactivity effects from coolant density changes and temperature changes are represented by tables. Feedback reactivity from fuel heat-up (Doppler Effect) is represented by a four-term polynomial in powers of fuel temperature. Photo-neutrons produced in beryllium or in heavy water may be included in the point-kinetics equations by using additional delayed neutron groups.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

Full Length Assembly Testing in PELICAN (Final Report)

In support of the development of the U.S. Department of Energy (DOE) Versatile Test Reactor (VTR), a thermal hydraulics test facility was constructed to generate experimental measurement of the pressure drop across a single full-scale assembly containing prototypic axial reflectors, fuel, and plena components. Constructed and operated at Argonne National Laboratory, the Pressure drop Experimental Loop for Investigations of Core Assemblies in Nuclear reactors (PELICAN) facility was designed to achieve hydraulic conditions identical to those anticipated for a full-scale fuel assembly located in the VTR core in the region with the highest flow rate. Using water as surrogate for liquid sodium, the flow loop was operated at elevated temperatures and pressures to match the thermophysical properties of liquid sodium and ensure matching Reynolds and Euler numbers. The measurement objectives for data generated from this test facility was driven primarily by the validation needs for code calculations and simulations of the reference VTR core. These objectives focused on the need to validate pressure drop results across the various segments of the fuel assembly as they relate directly to the pumping power and safety of the reactor. Presented in this report are experimental results and analytical comparisons based on testing of a full-length assembly in PELICAN. Housed within a hexagonal test section extending 3.4 m in length, the tested assembly features a prototypic lower reflector, grid plates, wire-wrapped rod bundle, upper reflector, and exit region. The rod bundle extends over 1.5 m in length and contains 217 individual wire-wrapped rods with dimensions that best reflect the reference VTR design. The as-tested bundle assembly was fabricated using 316 stainless steel 0.25-inch (6.35-mm) diameter rods wrapped with 0.04-inch (1.016-mm) diameter wire at a helical pitch of 10.51 inch (26.6 cm). Details of the method for in-house wire-wrapping, assembly, and installation are provided later in this report. Experimental measurements of pressure drop at 19 positions along the test assembly were recorded for a range of flow conditions, with special attention paid to key locations within the assembly, including component inlet and outlet, transition, and wire-wrapped rod bundle regions. Testing conditions were based on 110°C water with flow rates ranging from 50 to 450 GPM (3 to 27 kg/s) at the inlet of the test assembly generating Reynolds numbers and velocities up to ~8.0×10 4 and ~7.8 m/s, respectively, within the rod bundle region. Non-dimensional values for the friction factor were then calculated based on these experimental measurements and compared against those predicted by various analytical correlations available from open literature. Predictions by the upgraded Cheng and Todreas, Rehme, and Novendstern correlations fell within 4% to those values measured experimentally.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

Heat Transfer of Supercritical Carbon Dioxide in 3D-Printed Tubes with Internal Ribs and Yttria Stabilized Zirconia Coating

This technical report describes measurements performed in NETL’s Heat Exchanger Experimental Test (HEET) facility using supercritical carbon dioxide as the working fluid. 3D printed tubes with internal ribs and yttria stabilized zirconia (YSZ) exterior coatings were employed in the heat exchanger. Heat transfer and friction factor enhancements from the internal ribs were characterized over a range of flow conditions. Additionally, the thermal resistance from the YSZ coating was characterized for both a freshly applied coating and a coating subjected to thermal aging.

36 MATERIALS SCIENCE↗
DOE OSTI2025

BlueCRAB Domain Overlapping Coupling: Theory & Verification

Coupling low- and high-fidelity codes is a useful way to model complex engineering systems. Thanks to the high-fidelity code, complex phenomena can be resolved in areas of the system where this is required, yet the efficiency of the lower-fidelity code is still retained in modeling the rest of the system. This document details the theory and implementation of coupling two different thermal-hydraulic codes: the system thermal-hydraulics (STH) code System Analysis Module (SAM) and the coarse-mesh computational fluid dynamics (CFD) code Pronghorn. Both applications are included in the Comprehensive Reactor Analysis Bundle (BlueCRAB) code suite and are based on the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework. The domain overlapping (DO) coupling approach was adopted, as it offers proven advantages over more conventional domain decomposition methods. In the DO coupling, SAM provides Pronghorn with boundary conditions that depend on the system-level simulation of the entire plant. In return, the overlapping coupled SAM components–termed “surrogate components”–receive friction factors and source terms computed online based on the Pronghorn simulation. The framework is designed to be generic and enable coupling regardless of geometry and the number of inlet/outlet boundaries in the DO coupled domain. The developed method leads to consistent pressure drops, enthalpies, and scalar concentrations when comparing the coupled SAM and Pronghorn simulations. This document presents the DO coupling approach, along with two verification and two demonstration cases. The proposed problems explore different physical aspects relevant to nuclear reactor analysis, including buoyancy-driven flows, complex flow patterns, and multiple inlets and outlets. Periodically, new versions of this “BlueCRAB Domain Overlapping Coupling: Theory & Verification” report will be issued to reflect future developments and verification tests.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2024

LES simulations of a vacuum membrane distillation channel with geometric alterations

3D LES simulations were carried out to study the performance of a vacuum membrane distillation module. A wiggly wall profile with/without embedded stiffeners was considered to alleviate axial and radial temperature polarization, the cause of performance decrease in membrane distillation. Results of the flow field show the wiggles create unsteady vortex shedding inducing intense mixing in the channel. Vortex shedding intensity increases as the Reynolds number increases or stiffeners are added. 3D results show a 56% improvement in flux when moving from a flat sheet membrane to a wiggly membrane with stiffeners at a constant Reynolds number, corresponding to an improvement from 11.7 to 41.7 in the Nusselt number, showing that the alterations improved the flux performance by enhancing the heat transfer along the membrane surface and therefore alleviating temperature polarization, a critical bottleneck in membrane distillation systems. Further, a merit criterion was defined based on the Nusselt number and friction factor, and a 40% increase in merit was shown switching from a flat to a wiggly channel, while a 97% merit increase was seen going from flat to wiggly with stiffeners. A 42% enhancement in the flux was also seen moving from a straight channel to a wiggly channel at a higher Reynolds number which highlights the importance of strategically choosing the mass flow rate, as well as inducing flow separation and vortex shedding in the channel to promote mixing and dissipate the thermal boundary layer. The variation in results between the wiggly module with stiffeners and the wiggly module at high Reynolds numbers suggests enhancing mixing structures can be more impactful on flux for this geometry than increasing the flow rate to a turbulent/transitional regime. However, both are preferable for peak system flux performance. Furthermore, a 2D approximation was used to perform simulations on more extended channels to examine the length degradation. The modules with wiggly channels performed at the same flux level with a doubling of the length. On the other hand, the flat sheet modules experienced length degradation by temperature polarization and dropped in flux yield by around 13% of the short-channel value. This work illustrates that modeling the system and understanding how the performance decreases as the membrane surface area increases are critical for a larger module (scaling up from a lab to a prototype module) to maintain high flux performance.

42 ENGINEERING↗
DOE OSTI2022

Laminar Flow Heat Transfer In Helical Oval-Twisted Tube for Heat Exchanger Applications

The heat transfer performance of a novel tube configuration that combines the swirling velocity induced by oval-twisting and secondary flow generated by a helical geometrical flow path is presented. The Nusselt number (Nu) and friction factor (f) are compared for the laminar flow regime (Re = 250- 2000) under isothermal wall conditions. Under the same flow and boundary conditions, the oval-twisted helical tube increased the Nu and slightly increased the f over the circular helical tube. The best performance with the highest Nuand lowest f occurs at the coil curvature ratio (d h /D) of 0.17. The quantified enhancement performance factor (h) shows a 46 - 56% increase for the oval-twisted helical tube over the circular helical tube with d h /D = 0.17. Correlations for the Nu and f are developed and agreed with the numerical results: ± 6% and ± 5% for the Nu and f, respectively.

42 ENGINEERING↗
NASA NTRS1968

Internal low Reynolds number turbulent heat transfer

The results of a semi-analytical and experimental study of adiabatic tube flow and an analytical and experimental investigation of the thermal entrance region for gas flowing through electrically heated circular tubes are presented. Emphasis is placed on the low Reynolds number turbulent flow regime--defined as fully turbulent flow at bulk Reynolds numbers from 3,000 to about 15,000. Adiabatic air velocity and friction data and localized heat transfer measurements for air and helium, at low heating rates, are presented for this range. The adiabatic data were obtained in a 1.61 inch 1D tube for flow at bulk Reynolds numbers from 3,000 to 15,000. A continuous, Reynolds number-dependent, profile is developed from the data by using a modification of Reichardts' wall and middle law eddy diffusivity expressions. The velocity profile satisfies continuity. It is valid for all Reynolds numbers in excess of 3,000 for which the flow is fully turbulent and the Blasius friction factor expression is valid. The thermal entrance problem for a fully developed velocity profile is solved analytically by the method of Sparrow, Hallman, and Siegel. The solution is based on the profile developed from the velocity study. Tabular values of the eigenvalues and normalized Nusselt numbers for gases are presented for it range of Reynolds numbers from 3,000 to 50,000. The axial variation of Nusselt number is found to be correlated by [Nu/Nu_(∞)] = 1 + 0.8 (1+ 70,000 Re^(-3/2)) ((x/D)^(-1)) to within ± 5 per cent for x/D ≥2. The fully developed value agrees with the Dittus - Boelter correlation, Nu_(∞) = 0.021 Re^(0.8) Pr^(0.4) For the eigenvalues, λ^(2)_(n), and the associated constants, A_(n), correlations of the form λ^(2)_(n) =A_(1,n) Re^(-b_(1,n) + C_(1,n) Re^(-d_(1,n)) A_(n) = -A_(2,n) Re^(-b_(2,n) + C_(2,n) Re^(-d_(2,n)) are obtained. The coefficients and powers are presented in tabular form. Heat transfer data are presented, primarily for helium, for the low Reynolds number turbulent range. A one-quarter inch, resistively heated, vertical, circular tube was used for the study. The data cover an axial range from 1.2 to 96 diameters; wall-to-bulk temperature ratios vary from 1 to 1.4. In the low Reynolds number turbulent regime, these data clearly support the present analytical solution rather than the prediction obtained by applying the eddy diffusivity distribution used by Sparrow, Hallman, and Siegel.

Harold C. Reynolds, Jr.↗