Graetz Number Research Articles

Combined radiation and convection in developing flow in a parallel plate channel with real gas behavior: Contrast between gas cooling and heating

Laminar thermally developing gas flow in a two-dimensional parallel plate channel with real gas radiation has been investigated numerically. Real gas spectral radiation effects are accounted for by using the Spectral Line Weighted-sum-of-gray-gases model. Gas mixtures of H2O and CO2, either alone in mixture with N2 or as a mixture of H2O and CO2 are considered. The phenomena are investigated for a range of differences between wall and inlet gas temperature, mole fraction of the radiating species, and total pressure. Predictions for the cases of gas heating and gas cooling are contrasted with results presented by the classical no-radiation case, commonly known as the Graetz solution. The impact of gas radiation on the local mean temperature, convective and radiative wall heat fluxes, and the convective, radiative, and total local Nusselt numbers are studied.The predictions reveal that the inclusion of participating gas radiation results in a much more rapid change in temperature of the gas in the channel than for the convection-only case, for both the gas heating and gas cooling cases. Further, unique differences in the heat transfer behavior for gas cooling (wall temperature below the inlet gas temperature) and gas heating (wall temperature above the inlet gas temperature) scenarios are shown. However, significant differences between the cases of heating and cooling exist. In the classical Graetz solution, for constant thermophysical properties and using appropriately nondimensionalized heat transfer parameters (e.g., Nusselt number, inverse Graetz number) the predictions for fluid heating and cooling are identical. However, the results of this study including the influence of gas radiation demonstrate that the heat transfer behavior depends heavily on whether the gas is heated or cooled. An apparent thermally fully developed condition may exist for the case of gas cooling, whereas the variation in local Nusselt number with dimensionless axial location exhibits a local minimum for the case of gas heating. Further, the case of gas heating results in a more rapid streamwise change in mean temperature in the channel than for the cooling case. The thermal physics of the developing flow for combined convective-radiative transfer with real gas effects are explored for a range of wall and gas inlet temperatures, radiating species mole fractions, and total pressures.

REVVING UP HEAT-TRANSFER PERFORMANCE OF DOUBLE PIPE HEAT EXCHANGER USING DIVERSE MOLAR Ag-GO HYBRID NANOFLUIDS: AN EMPIRICAL AND NUMERICAL STUDY USING CENTRAL COMPOSITE DESIGN

This study seamlessly integrates empirical and numerical approaches to enhance the efficiency of a double pipe heat exchanger (DPHX) using varied molar concentrations (0.03, 0.06, and 0.09 M) of Ag-doped GO hybrid nanofluids as the working fluids within the heat exchanger's annulus. Remarkable improvements in the heat exchanger's performance were achieved by increasing the molar concentration of Ag-GO hybrid nanoparticles, along with the Reynolds number (ranging from 250 to 1451) and the mass flow rate (ranging from 8 to 47 g/s) of the hybrid nanofluids. The utilization of a 0.09M Ag-GO hybrid nanofluid at a Reynolds number of 1451 and a flow rate of 47 g/s resulted in outstanding enhancements in heat-transfer coefficient (62.9%), and Nusselt number (33.55%) surpassing the base fluid. The empirical results of Nusselt number and heat-transfer coefficient were optimized and analyzed using the central composite design approach (CCD) with response surface method, incorporating the Graetz number, varied molar concentrations of Ag-GO, and thermal conductivity of the hybrid nanofluids as input factors. The optimized second-order polynomial quadratic equation model demonstrated excellent compatibility and optimal performance of the heat exchanger, supported by variance analysis. Additionally, CCD optimization confirmed a notable desirability function (0.99) and emphasized the significant influence of the input factors on the output responses. In conclusion, the Graetz number exhibited prominent influence among the input factors, alongside the molar concentration and thermal conductivity of hybrid nanofluids, effectively enhancing the performance of the DPHX.

Dimensionless numbers on columnar structured electrodes in hydrogen/oxygen polymer electrolyte fuel cells

Platinum columnar growth was found on anode and cathode catalysts after 6 months of operation in hydrogen/oxygen cells (1 A cm−2) from original smooth microspheres layers with loads ca. 1 mg cm−2. The application of symmetric and asymmetric large frequency square wave potential programs in strong sulphuric acid solutions for catalyst preparation showed those effects. For columnar electrodes, current and potential distributions are not those expected for smooth surfaces and constitute the main reason of a mismatching between theory and experimental data. The employ of exact analytical solutions arising from mass/charge balances are a better way to envisage electrochemical profiles instead of numerical solutions. Columnar catalysts can be parameterized by a trochoid curvilinear profile knowing the radius and pass (from ex situ STM images) rendering values ca. ½ for both parameters. Polarization and power density curves were also mimicked from the obtained current and cell potential distributions for the rough columnar surfaces. Finally, Wagner, Damkoehler and Graetz numbers that typify the electrochemical reactor were developed applying these parametric equations using local variations of tangential and normal velocities, from which a general dimensionless equation was obtained.

Two-Dimensional Conjugated Mass Transfer of Carbon Dioxide Absorption in a Hollow-Fiber Gas-Liquid Membrane Contactor.

The absorption efficiencies of CO2 in hollow-fiber membrane contactors using an ethanolamine (MEA) solvent under both concurrent- and countercurrent-flow operations were investigated theoretically and experimentally. Two-dimensional mathematical modeling was developed by Happel’s free surface model, and the resultant partial differential equations were solved analytically using the separated variables method with the use of an orthogonal expansion technique. A simplified expression of Sherwood number variations was reported by employing the relevant operations conditions and expressed in terms of the computed eigenvalues for predicting concentration distribution and absorption efficiency. It is emphasized that, in comparing various fiber packing configurations, both theoretical predictions and experimental results should be compared to find the absorption flux increment accomplished by the CO2/N2 stream passing through the fiber cells under the same mass flow rate. The value of the present mathematical treatment is evident to propose a simplified expression of the averaged Sherwood number variations, and provides the predictions of the absorption flux, absorption efficiency, average Sherwood number with the absorbent Graetz number, inlet CO2 concentration, and absorbent flow rates as parameters. The availability of such concise expressions, as developed directly from the analytical formulations, is the value of the present study. The experiments of the CO2 absorption using MEA with alumina (Al2O3) hollow fiber membranes are also set up to confirm the accuracy of the theoretical predictions. The accuracy derivations between the experimental results and theoretical predictions for concurrent- and countercurrent-flow operations are and , respectively. The operations of the hollow-fiber membrane contactor implementing N = 7 fiber cells and N = 19 fiber cells offer an inexpensive method of improving absorption efficiency by increasing fiber numbers with consideration of device performance.

Numerical investigations on the difference between aiding and opposing flows in the developing regime of laminar mixed convection in vertical tubes

The present article discusses the numerical simulation results of laminar mixed convection flow in vertical tubes. A comparative analysis of the thermal and hydrodynamic features of both buoyancy-assisted and opposed flows was performed for Reynolds number (), Grashof number () and Richardson number () with uniform heat flux boundary condition. 2-D axisymmetric steady state simulations were carried out for a length-to-diameter () ratio of with water as the working fluid. Numerical simulations were performed by employing SIMPLE scheme for pressure–velocity coupling in momentum equations and second order UPWIND scheme for solving energy equations. In case of assisting flow (), at fully developed state the centerline velocity decreases, and velocity is increased near the tube wall due to heat flux induced free convection. With increasing heat flux, the decrement in centerline velocity compared to the no-heat flux condition increases. Further, with increasing heat flux, the increase in friction factor and Nusselt number was observed. Therefore, at the same the variation of heat flux led to unique velocity profiles and temperature gradients. The variation of centerline velocity and temperature in developing region was also studied. While centerline temperature was monotonically increasing with length, the centerline velocity increased up to a maximum in the developing region and then attained the steady state at a lower value in the fully developed state. Subsequently we studied the dependence of on the hydrodynamic and thermal features. For constant friction factor and Nusselt number was observed to increase with increase in from 0.1 to 1.5 range. At fixed variation of heat flux leads to variation of or Hence, the buoyancy effect has a significant role in the entrance length development. The hydrodynamic entry length after an initial increase, attained an almost constant value with increasing On the other hand, the thermal entry length exhibited a decreasing trend with increasing Similarly, at constant Nusselt number increased with increasing for a range of 100–1000. It was evident that for a given and the heat transfer in the developing flow is always higher as that of the fully developed flow. Contrasting observations were observed for buoyancy-opposed flow. Velocity is accelerated at the center as compared to the tube wall in case of opposing flow for same and For a fixed the friction factor and Nusselt number decreased with increasing the Both the hydrodynamic and thermal entry length increases for opposing flows. Finally, we have developed correlations for fully developed friction factor () with as well as and Nusselt number () with for buoyancy-assisting and buoyancy-opposing flows. Two independent correlations are also produced for with Graetz number () for developing and fully developed regimes in both assisting as well as opposing flows.

CFD Analysis of Convective Heat Transfer in a Vertical Square Sub-Channel for Laminar Flow Regime

The development of new practices in nuclear research reactor safety aspects and optimization of recent nuclear reactors needs knowledge on forced convective heat transfer within sub-channels formed between several nuclear fuel rods or heat exchanger tubes, not only in the fully developed regime but also in the developing regime or laminar flow regime. The main objective of this research was to find a new correlation equation for calculating the convective heat transfer coefficient in the vertical square sub-channels. Recently, a simulation study was conducted to find a new heat transfer correlation equation for calculating the convective heat transfer coefficient within a vertical square sub-channel in the developing regime or laminar flow regime for Reynolds number range 400 ≤ Re ≤ 1700. Simulations were carried out using a computational fluid dynamics (CFD) code and modeling already defined in the software. The novelty of the research lies in the analysis of the entrance effect for the sub-channel by proposing a new empirical correlation that can then be inserted into the STAT computer code. The surface temperature distribution around the tangential direction of the active cylinders shows that the implementation of active and dummy cylinders in the current study can simulate sub-channels that exist in a real nuclear reactor core. The current study shows that the flow simulated in this study is in its developing condition (entrance region). A new forced convective heat transfer correlation for the developing region in the form of Nu = 2.094(Gz)0.329 for the Graetz number range 161 ≤ Gz ≤ 2429 was obtained from the current study.

Behavior of weakly adsorbing protein impurities in flow-through ion-exchange chromatography

Flow-through ion-exchange chromatography is frequently used in polishing biotherapeutics, but the factors that contribute to impurity persistence are incompletely understood. A large number of dilute impurities may be encountered that exhibit physicochemical diversity, making the flow-through separation performance highly sensitive to process conditions. The analysis presented in this work develops two novel correlations that offer transferable insights into the chromatographic behavior of weakly adsorbing impurities. The first, based on column simulations and validated experimentally, delineates the relative contributions of thermodynamic, transport, and geometric properties in dictating the initial breakthrough volumes of dilute species. The Graetz number for mass transfer was found to generalize the transport contributions, enabling estimation of a threshold in the equilibrium constant below which impurity persistence is expected. Impurity adsorption equilibria are needed to use this correlation, but such data are not typically available. The second relationship presented in this work may be used to reduce the experimental burden of estimating adsorption equilibria as a function of ionic strength. A correlation between stoichiometric displacement model parameters was found by consolidating isocratic retention data for over 200 protein-pH-resin combinations from the extant literature. Coupled with Yamamoto’s analysis of linear gradient elution data, this correlation may be used to estimate retentivity approximately from a single experimental measurement, which could prove useful in predicting host-cell protein chromatographic behavior.

Conjugated Mass Transfer of CO2 Absorption through Concentric Circular Gas–Liquid Membrane Contactors

A new design of gas absorption that winds the permeable membrane onto an inner concentric tube to conduct a concentric circular gas–liquid membrane module has been studied theoretically in the fully developed region. An analytical formulation, referred to as conjugated Graetz problems, is developed to predict the concentration distribution and Sherwood numbers for the absorbent fluid flowing in the shell side and CO2/N2 gas mixture flowing in the tube side under various designs and operating parameters. The analytical solutions to the CO2 absorption efficiency were developed by using a two-dimensional mathematical modeling, and the resultant conjugated partial differential equations were solved analytically using the method of separation variables and eigen-function expansion in terms of power series. The predictions of CO2 absorption rate by using Monoethanolamide (MEA) solution in concentric circular membrane contactors under both concurrent- and countercurrent-flow operations are developed theoretically and confirmed with the experimental results. Consistency in both a good qualitative and quantitative sense is achieved between the theoretical predictions and experimental results. The advantage of the present mathematical treatment provides a concise expression for the chemical absorption of CO2 by MEA solution to calculate the absorption rate, absorption efficiency, and average Sherwood number. The concentration profiles with the mass-transfer Graetz number, inlet CO2 concentration, and both gas feed and absorbent flow rates are also emphasized. Both theoretical predictions and experimental results show that the device performance of the countercurrent-flow operation is better than that of the concurrent-flow device operation. The availability of such simplified expressions of the absorption rate and averaged Sherwood as developed directly from the analytical solutions is the value of the present study.

Experimental characterization of heat transfer enhancement in a circular tube fitted with Koflo Blade™ inline mixer

This paper addresses the experimental study of heat transfer enhancement of laminar flow in a tube fitted with Koflo Blade™ inline mixer as the swirl-flow generator and under constant surface heat flux. Experiments were carried out with air as heat transfer fluid and two different types of the mixer having blades crossing angles of 90° and 120° The flow swirling in the mixer is mainly due to the centrifugal force induced by the angle between the blades, which is responsible for heat transfer augmentation. The results show that the mixer with 90° blades supplies Nusselt numbers higher than the 120° mixer but, likewise, results in higher pressure drop. The rate of heat transfer in the tube equipped with inserts was augmented up to 78% and 47% for 90°-blades and 120°-blades mixers, respectively. The enhancement in heat transfer was correlated with the Graetz number that supports the experimental data for both mixers by a deviation within ±10%. Furthermore, the tube fitted with a 120°-blades mixer yields higher thermal performance with a maximum value of 1.77. As well, the 120°-blades mixer provides thermal performance factors up to 17% greater than that for 90°-blades mixer for the same pumping power, which implies that heat transfer effects significantly dominate pressure drop effects.

Non-isothermal blade coating analysis of viscous fluid with temperature-dependent viscosity using lubrication approximation theory

Abstract Blade coating process is studied in a nonisothermal analysis of viscous fluid with temperature-dependent viscosity by employing both plane and exponential coaters. The governing expressions are nondimensionalized and simplified under the assumption of lubrication approximation theory. Then, perturbative technique is used to find the solution for velocity, pressure, temperature distribution, and coating thickness. The influence of dimensionless parameter ε, Graetz number Gz, and normalized coating thickness γ on the velocity, maximum pressure, temperature distribution, and pressure gradient is portrayed through graphs, whereas load and coating thickness variations reported in a tabular manner. It is found that maximum pressure, coating thickness, and blade load decreases for temperature variations in viscosity, which leads to improved efficiency of blade coating process and life of the moving substrate.

Analytical and Numerical Study on Thermally Developing Forced Convective Flow in a Channel Filled with a Highly Porous Medium Under Local Thermal Non-Equilibrium

An analytical and numerical study was made on thermally developing forced convective flow in a channel filled with a fluid-saturated porous medium, subject to constant heat flux, under local thermal non-equilibrium. The Brinkman-extended Darcy model was employed to investigate the combined effects of the Darcy number, Biot number, effective thermal conductivity ratio and Graetz number on the longitudinal thermal development in a channel filled with a highly porous medium. 3D convective regime maps were constructed for wide ranges of the parameters, using a 3D coordinate system based on the Biot, Darcy and Graetz number coordinates. The 3D convective regime maps proposed in this study enable one to identify the three distinctive thermal regions possible along the longitudinal coordinate in a channel, namely the entrance developing region, transition region and fully developed region, depending on the relative thicknesses of the fluid and solid thermal boundary layers. It has been found that the Darcy number strongly influences the heat transfer rate in the developing region, yielding a higher Nusselt number for a lower Darcy number, whereas both Biot number and the effective thermal conductivity ratio markedly influence the level of the fully developed Nusselt number. The heat transfer performance evaluation carried out under equal pumping power for the aluminum foam embedded channel saturated with air reveals that a substantial heat transfer argumentation is possible by filling the channel with an aluminum foam.

Correlating laminar convection in slots with developing flow

The problem of combined entry convection between the walls of a duct and a hydrodynamically developing flow is revisited for the geometry of a two-dimensional slot. Accurate calculations of the local and average Nusselt numbers are reported for an extended range of inverse Graetz number (ZH≥10−6), and a wide range of Prandtl number (0.1 ≤ Pr ≤ 500), while considering both constant temperature and constant heat flux wall conditions. Accuracy in this work is achieved by applying a transformation of variables to remove the singularity along the perimeter of the duct inlet before integrating the governing equations. Results of this investigation are correlated for all inverse Graetz numbers and Prandtl numbers Pr ≥ 0.5. Correlations for the local and average Nusselt numbers, and both wall conditions, are accurate to ±2.1% when compared to the benchmark results of this work.

Laminar Convection in Circular Tubes With Developing Flow

Abstract The combined entry problem for the simultaneous development of heat and momentum transfer in a circular tube has been resolved over an extended range of inverse Graetz number ZH≥10−6 and for a wide range of Prandtl numbers 0.1≤Pr≤500. For the historical range of ZH≥5×10−4 and 0.7≤Pr≤50, earlier studies are within 5% of the current benchmark calculations, but for the new extended range of conditions, the best authoritative sources were in error by as much as 33%. Four new correlations are proposed for the local and average Nusselt numbers, and for the constant temperature and constant heat flux wall condition, which are accurate to 2.2% for all values of inverse Graetz number and Pr≥0.5. In contrast, legacy correlations typically had a 10–20% error range when compared to the results of this work, with many exhibiting larger errors and only few achieving errors as low as 5–10%.

Augmentation of Heat Transfer of High Prandtl Number Fe3O4/vacuum pump oil nanofluids flow in a tube with twisted tape inserts in laminar flow

The heat transfer coefficient and friction factor of Fe3O4/vacuum pump oil nanofluids flowing in a tube with twisted tape inserts were analyzed experimentally. The experiments were conducted for different mass flow rates (0.04 kg/s to 0.208 kg/s), volume concentrations (0.05% to 0.5%), Prandtl numbers (440 to 2534), Graetz numbers (500 to 3000) and twisted tape inserts (H/D = 5, 10 and 15). Results reveal that the Nusselt number is enhanced by 8.94% and 13.48% for the 0.5% nanofluid for the mass flow rates of 0.0416 kg/s and 0.208 kg/s, respectively, relative to the base fluid. For the mass flow rate of 0.208 kg/s with 0.5 vol. % nanofluid, the friction factor penalty is 1.21-times compared to the base fluid. By using twisted tape insert of H/D = 5, the Nusselt number is further enhanced by 23.86% and 39.53% for the 0.5% nanofluid for mass flow rates of 0.0416 kg/s and 0.208 kg/s, respectively, relative to the base fluid. For the twisted tape insert of H/D = 5 with mass flow rate of 0.208 kg/s and 0.5 vol. % nanofluid, the friction factor penalty is 1.44-times compared to the base fluid. New Nusselt number and friction factor correlations are proposed.

Laminar convection in rectangular ducts of fully developed flow

Benchmark calculations for the Graetz problem in rectangular ducts are performed for three wall conditions (T – constant temperature, H2 – constant heat flux, and H1 – constant axial heat input with constant peripheral temperature). Results are reported for a wide range of duct aspect ratio (1/1 through 1/10) and over a wide span of inverse Graetz number (Z>10−6). This builds on earlier works that neglect the initial ~20% of thermal boundary layer growth in reporting heat transfer results and that omit the H2 wall condition. Using a new generalized approach, Nusselt number results for the T and H1 wall condition are correlated to better than ±2.5% for the entire range of inverse Graetz number and duct aspect ratios. Limitations on correlating the H2 wall condition by the same method are discussed.

Exact Solutions to the Thermal Entry Problem for Laminar Flow Through Annular Ducts

Abstract The thermal entrance region for laminar-forced convection of a Newtonian fluid in an annular tube is solved by separation of variables using as many eigenvalues and eigenfunctions as needed to report exact results for a specified range of Graetz numbers. Results for the local and average Nusselt numbers are calculated for a wide range of inner to outer wall radius ratios and for convection to either the inner or outer wall, when the opposing wall is adiabatic. The present benchmark results are utilized to critically examine the accuracy of previous extended Lévêque series solutions that are convergent for short axial distances, and Graetz series solutions that are convergent for long axial distances, and to examine the performance of a new correlation for convection in annular tubes.

Fully Developed Transport Constants of Annular Tubes, With Application to the Entrance Region

Abstract Description of the laminar thermal entry problem in annular tubes has historically been limited to a few geometric cases that require piecing together classical Graetz series and Lévêque series solutions to span all values of the Graetz number. The current work uses a recently developed generalized correlation to describe the full range of Graetz numbers for any annular tube geometry. However, the correlation requires fully developed Nusselt number values that have only been accurately reported in tabular and graphical forms. Exact analytic solutions for the constant wall heat flux condition are developed in this work, and simplified correlations are proposed for all wall conditions that reproduce exact Nusselt number solutions to within ± 0.4%. Using these results, a modified version of the generalized Graetz problem correlation is developed to reproduce the most published Nusselt numbers for the thermal entry problem in an annular tube to be within ± 5%.

Development of Heat Transfer Model at Intake System of IC Engine with Consideration of Backflow Gas Effect

Improving thermal efficiency of internal combustion engines has been a priority in the automotive industry. It is necessary to model the heat transfer phenomenon at the intake system and precisely predict intake air’s mass flow rate into the engine cylinder. In the previous studies, the heat transfer at the intake system was modeled as quasi-steady state phenomenon, based on Colburn analogy. Authors developed two empirical equations with the introduction of Graetz and Strouhal numbers. In the present study, further improvements were done by the addition of pressure ratio between the intake manifold and atmospheric pressure, along with Reynolds number in order to characterize the backflow gas effect on intake air temperature. Compared with the experimental results, maximum and average errors of intake air temperature estimations inside the manifold found to be 2.9 % and 0.9 %, respectively.

A local thermal non-equilibrium integral analysis for forced convective thermal boundary development in a channel filled with a fluid-saturated porous medium

An integral solution procedure has been developed to describe the entire evolution of the thermal boundary layer in a channel filled with a fluid-saturated porous medium. The upper and lower walls are heated under a constant heat flux condition, and local thermal nonequilibrium is assumed to apply. The development of the thermal boundary layer in this channel is divided into three distinctive regions, namely, the entrance, transition and the nearly fully-developed regions, in which separate fluid and solid phase thermal boundary layers develop near heated walls with different growth rates. In this integral analysis, each region is considered in terms of the interactions between the fluid and solid thermal boundary layers; this eventually yields a set of algebraic equations for the easy and accurate estimation of the local Nusselt number. The solutions thus obtained for the three regions are combined to reveal the entire development of the local Nusselt number from the entrance to fully-developed stage. This analytic procedure, for the first time, reveals a complete region map showing the locations of the transition from one region to another, and these depend on the Biot number, the thermal conductivity ratio and the Graetz number in a complex manner.

Heat transfer and presser drop of copper oxide–thermal oil in upward single-phase flow in inclined microfin tube under constant wall temperature

The effects of using copper oxide–thermal oil on convective heat transfer and pressure drop in an upward flow in an inclined microfin tube is studied experimentally in this research project. The flow regime and wall temperature are laminar and constant, respectively. The effects of nanofluid, Graetz number, Prandtl number and positive inclination angles on convective heat transfer augment moderately with the boom in nanoparticles mass concentration. The result shows that the forced heat transfer and Darcy friction factor increase with the increment of mass nanoparticle concentration, Reynolds number and inclination angle. The ratio of Darcy friction factor increases up to 64% in microfin tube. The Nusselt number boosts to reach 62 when mass nanoparticle concentration and inclination angle were 1.5% and 30o, respectively. Two correlations are advised to estimate Nusselt number and Darcy friction factor in upward single-phase flow in microfin tube under constant wall temperature and laminar flow in microfin pipe. The maximum aberrations of Nusselt number and Darcy friction factor are 20% and 18%, respectively, which are admissible to predict experimental data. Accompaniment of heat transfer ratio with pumping power ratio is presented in this paper. If the increment of pressure drop is more than heat transfer enhancement, it will not be appropriate to use CuO-thermal oil, inclination angles and microfin tube. The maximum FOM is 59% which is calculated with 1.5% nanoparticle mass concentration and inclination angle of 60o at Prandtl number of 349.