Range Of Prandtl Numbers Research Articles

Scale law analysis of the curved boundary layer evolving around a horizontal cylinder at Pr > 1

The convective boundary layer flow on the external surface of an isothermally heated horizontal cylinder is investigated in this study. Numerical simulations are first carried out for a wide range of flow parameters, i.e., Rayleigh and Prandtl numbers, and scale relations quantifying the boundary layer flow are then determined from the simulation data. The numerical results suggest that the curved boundary layer experiences an initial growth state, a transitional state, and a developed state, which are essentially identical to the extensively studied flat boundary layers. Scale relations quantifying the local flow variables are obtained, and the proposed scale laws indicate that during the initial growth, the present curved boundary layer flow follows a two-dimensional growth rather than the well-known one-dimensional startup of flat boundary layers. It is further demonstrated that the characteristic velocity of the boundary layer flow maximizes at π/2, but its thickness is circumferential location independent. In the steady state, however, the maximum streamwise velocity of the boundary layer shifts to approximately 7π/9 and the thickness consistently increases with the circumferential location. It is also shown that the thickness of the inner viscous boundary layer could be obtained by properly considering the Prandtl number effect, i.e., by the term (1 + Pr−1/2)−1. The proposed scale relations could reasonably describe the curved boundary layer flow, and all regression constants are above 0.99.

CFD MODELING OF SLURRY PIPELINE AT DIFFERENT PRANDTL NUMBERS

The present work shows the slurry flow characteristics of glass beads having density 2470 kg/m3 at different Prandtl number through a horizontal pipeline. The simulation is conducted by Eulerian two-phase model using RNG k-ε turbulence closure in available commercial software ANSYS FLUENT. The transportation of solid particulates has the settling behaviour in the slurry pipeline and that leads to the sedimentation and blockage of the pipeline resulting more power and pressure drop in the pipeline. Therefore, it is important to know the transport capability of the solid particulates at different Prandtl fluids to minimise the pressure loss. The fluid properties at four Prandtl numbers i.e. 1.34, 2.14, 3.42 and 5.83 is used to carry the solid concentration ranges from 30-50 % (by volume) at mean flow-velocity ranging from 3 to 5 ms-1 . The obtained computational results are validated with the published data in the literature. The results show that the pressure-drop rises with escalation in flow velocity and solid concentration at all Prandtl number. It is found that the suspension stability enhancement is considerable for lower range of Prandtl number and decreases for higher range of Prandtl number. Finally, glass beads concentration contours, velocity contours, concentration profile, velocity profiles and pressure drop are predicted to understand the slurry flow for chosen Prandtl numbers.

Conjugate heat transfer in the unbounded flow of a viscoelastic fluid past a sphere

This work addresses the conjugate heat transfer of a simplified PTT fluid flowing past an unbounded sphere in the Stokes regime (Re = 0.01). The problem is numerically solved with the finite-volume method assuming axisymmetry, absence of natural convection and constant physical properties. The sphere generates heat at a constant and uniform rate, and the analysis is conducted for a range of Deborah (0 ≤ De ≤ 100), Prandtl (100 ≤ Pr ≤ 105) and Brinkman (0 ≤ Br ≤ 100) numbers, in the presence or absence of thermal contact resistance at the solid–fluid interface and for different conductivity ratios (0.1 ≤ κ ≤ 10). The drag coefficient shows a monotonic decrease with De, whereas the normalized stresses on the sphere surface and in the wake first increase and then decrease with De. A negative wake was observed for the two solvent viscosity ratios tested (β = 0.1 and 0.5), being more intense for the more elastic fluid. In the absence of viscous dissipation, the average Nusselt number starts to decrease with De after an initial increase. Heat transfer enhancement relative to an equivalent Newtonian fluid was observed for the whole range of conditions tested. The dimensionless temperature of the sphere decreases and becomes more homogeneous when its thermal conductivity increases in relation to the conductivity of the fluid, although small changes are observed in the Nusselt number. The thermal contact resistance at the interface increases the average temperature of the sphere, without affecting significantly the shape of the temperature profiles inside the sphere. When viscous dissipation is considered, significant changes are observed in the heat transfer process as Br increases. Overall, a simplified PTT fluid can moderately enhance heat transfer compared to a Newtonian fluid, but increasing De does not necessarily improve heat exchange.

A Multiple-Grid Lattice Boltzmann Method for Natural Convection under Low and High Prandtl Numbers

A multi-distribution lattice Boltzmann Bhatnagar–Gross–Krook (BGK) model with a multiple-grid lattice Boltzmann (MGLB) model is proposed to efficiently simulate natural convection over a wide range of Prandtl numbers. In this method, different grid sizes and time steps for heat transfer and fluid flow equations are chosen. The model is validated against natural convection in a square cavity, since extensive benchmark solutions are available for that problem. The proposed method can resolve the computational difficulty in simulating problems with very different time scales, in particular, when using extremely low or high Prandtl numbers. The technique can also enhance computational speed and stability while keeping the simplicity of the BGK method. Compared with the conventional lattice Boltzmann method, the simulation time can be reduced up to one-tenth of the time while maintaining the accuracy in an acceptable range. The proposed model can be extended to other lattice Boltzmann collision models and three-dimensional cases, making it a great candidate for large-scale simulations.

Effects of Prandtl number in quasi-two-dimensional Rayleigh–Bénard convection

Abstract

The Generalized Lévêque Equation and its application to flat tubes without and with passive inserts

Searching for an analogy concept between heat transfer and pressure drop is anything but new, nevertheless it is still the subject of researches as it enables the prediction of heat transfer from pressure drop data, which helps reducing experimental effort for heat transfer measurements. The Generalized Lévêque Equation (GLE) is a widely-used analogy concept, which allows the prediction of heat transfer from pressure drop data, if hydrodynamic boundary layer is developed or still developing. This analogy concept has been successfully applied to flow situations with practical relevance like chevron-type plate heat exchangers, packed beds and in tube bundles, where the boundary layers are forced to reform continuously. In this contribution, the GLE is applied to flat tube flow without and with passive inserts, investigating the impact of passive inserts, which force the boundary layers to reform continuously, on the application of the GLE.This paper presents results of an experimental investigation on pressure drop and heat transfer in flat tubes without and with passive inserts, covering a wide range of Prandtl number (8 < Pr < 60) and Reynolds number values (40 < Re < 3500). These parameter ranges and the passive inserts represent the typical conditions and design used for the liquid side of single-phase flat tube heat exchangers used in the automotive industry. 238 experimental data points for heat transfer coefficients are compared to heat transfer coefficients calculated from pressure drop data with the GLE.The results show that the GLE is not applicable to flat tubes without passive inserts, yet becomes applicable with passive inserts, which are forcing a re-formation of boundary layers in the whole range of investigated Reynolds and Prandtl numbers.

Forced Convective Flow of Bingham Plastic Fluids in a Branching Channel With the Effect of T-Channel Branching Angle

Abstract This work aims to explore the T-channel momentum and heat transfer characteristics with the combined effect of Bingham plastic fluids (0.01 ≤ Bn ≤ 20) behavior and geometrical variation in terms of branching angle (30 deg ≤ α ≤ 90 deg). The problem has been solved over a wide range of Reynolds number (50 ≤ Re ≤ 300) and Prandtl number (10 ≤ Pr ≤ 50). For the momentum flow, qualitative and quantitative features are analyzed in terms of streamlines, structure of yielded/unyielded regions, shear rate contours, plug width and length variation, and local pressure coefficient. These features have been represented in terms of isotherm patterns, temperature profile, Nusselt number, and its asymptotic value for heat transfer characteristics. The recirculating flows have been presented here in the vicinity of T-junction, which promote mixing and heat transfer. Broadly, the size of this zone bears a positive dependence on Re and α. However, fluid yield stress tends to suppress it. The critical Reynolds and Bingham numbers were found to be strong functions of the pertinent parameters like α. The inclination angle exerts only a weak effect on the yielded/unyielded regions and on the recirculation length of main branch. Results show a strong relationship of the plug width and length with key parameters and branches. The Nusselt number exhibits a positive relationship with α, Bn, and Re but for lower Pr in the T-junction vicinity for both branches. Such length indicates the required optimum channel length for thermal mixing.

Non-equilibrium wall functions for large Eddy simulations of complex turbulent flows and heat transfer

In this paper novel unified wall function formulations for large eddy simulation of complex turbulent flows and heat transfer are presented. In contrast to existing wall functions, the proposed analytical expressions for velocity and temperature are: (1) valid over the whole range of dimensionless wall distance y+, (2) applicable to complex flow situations that include local non-equilibrium effects, (3) suitable for a wide range of molecular Prandtl numbers, (4) thermodynamically consistent with the second law of thermodynamics, and (5) easily extendable to account for contributions of additional source term e.g. gravity, radiation, chemical reactions. Such an universal and consistent wall function formulation is particularly useful for computational fluid dynamics applications in providing appropriate wall boundary conditions for complex high Reynolds number flows. The accuracy and thermodynamic consistency of the proposed wall treatment is first testified by comparison with experimental and direct numerical simulation data. Second, in order to highlight the applicability and performance of the proposed wall functions for computational fluid dynamics, results of large eddy simulations of various complex turbulent flows including heat transfer relevant to internal combustion engine applications are presented and evaluated. It turns out that the suggested wall function approach features a superior physics modeling accuracy in terms of well-known key parameters in near-wall bounded flows in comparison to state-of-the-art wall functions.

Heat Transfer Performance of Different Fluids During Natural Convection in Enclosures with Varying Aspect Ratios

The heat transfer process takes place in numerous applications through the natural convection of fluids. Investigations of the natural convection heat transfer in enclosures have gained vital importance in the last decade for the improvement in thermal performance and design of the heating/cooling systems. Aspect ratios (AR=height/length) of the enclosures are one of the crucial factors during the natural convection heat transfer process. The investigated fluids consisting of air, water, engine oil, mercury, and glycerine have numerous engineering applications. Heat transfer and fluid flow characteristics are studied in 3-dimensional rectangular enclosures with varying aspect ratios (0.125 to 150) using computational fluid dynamics (CFD) simulations. Studies are carried out using the five different fluids having Prandtl number range 0.01 to 4500 in rectangular enclosures with the hot and cold surface with varying temperature difference 20K to 100K. The Nusselt number and heat transfer coefficients are estimated at all conditions to understand the dependency of ARs on the heat transfer performance of selected fluids. Temperature and velocity profiles are compared to study the flow pattern of different fluids during natural convection. The Nusselt number correlations are developed in terms of aspect ratio and Rayleigh number to signify the natural convection heat transfer performance.

Revisiting Reynolds and Nusselt numbers in turbulent thermal convection

In this paper, we extend Grossmann and Lohse’s (GL) model [S. Grossmann and D. Lohse, “Thermal convection for large Prandtl numbers,” Phys. Rev. Lett. 86, 3316 (2001)] for the predictions of Reynolds number (Re) and Nusselt number (Nu) in turbulent Rayleigh–Bénard convection. Toward this objective, we use functional forms for the prefactors of the dissipation rates in the bulk and boundary layers. The functional forms arise due to inhibition of nonlinear interactions in the presence of walls and buoyancy compared to free turbulence, along with a deviation of the viscous boundary layer profile from Prandtl–Blasius theory. We perform 60 numerical runs on a three-dimensional unit box for a range of Rayleigh numbers (Ra) and Prandtl numbers (Pr) and determine the aforementioned functional forms using machine learning. The revised predictions are in better agreement with the past numerical and experimental results than those of the GL model, especially for extreme Prandtl numbers.

Hot-wire experimental investigation on turbulent Prandtl number in a rotating non-isothermal turbulent boundary layer

This experiment used a parallel array of hot wire probes to simultaneously measure the temperature and velocity fields in the non-isothermal turbulent boundary layer of a rotating straight channel. The Reynolds numbers are 15,000 and 25,000, respectively. The rotation numbers are 0, 0.07, 0.14, 0.21 and 0.28, respectively. The purpose of this study is to calculate the turbulent Prandtl number in a rotating non-isothermal turbulent boundary layer. Due to the difficulty in measuring local turbulent Prandtl numbers, this study focuses on the average turbulent Prandtl numbers in the logarithmic region instead. Under static conditions, this value is taken as 0.9 normally. This research finds that rotation conditions can affect the turbulent Prandtl number by affecting the properties of velocity and temperature boundary layers. The change range of the turbulent Prandtl number is roughly 0.6–1.1. The influence of the leading side is greater than that of the trailing side, especially at high rotation numbers. This can provide validation and guidance for numerical simulation. Other information within the turbulent boundary layer is also discussed. It is hoped that this study would enhance our understanding of the mechanism of turbulent flow in the turbulent layer at rotating conditions.

Polarization force and geothermal viscosity driven unsteady Bödewadt transport phenomenon over a ferrofluid saturated disk

This work aims to investigate mass transport phenomena on time dependent Bödewadt flow of magnetic Nanoliquid embedded in the porous medium with vertical polarization force and geothermal viscosity variations. The resultant nonlinear coupled system of partial differential equations is solved numerically. For thorough understanding of flow and transport phenomenon, a wider range of Prandtl number is taken to analyze the effects of various physical entities including polarization force, geothermal viscosity, permeability and rotation. Eventually, some new marvels are found. The diffusion rate enhances significantly for higher values of Prandtl, Schmidt numbers and the boundary layer thickness decimates with all parameters except depth dependent viscosity. Besides, frictions on the surface of the plate have also been computed and found them very high for all above mentioned entities. In nutshell, to have the realistic view, this investigation reveals that the polarization force and viscosity variation due to the temperature and depth have imperative role on the unsteady transport phenomenon in ferrofluid flow.

Numerical Simulation of MHD Natural Convection Flow Within Porous Trapezoidal Cavity With Heated Triangular Obstacle

A steady laminar two-dimensional finite element based Galerkin weighted residual numerical approach (GWRNA) has been adopted to solve magnetohydrodynamic (MHD) natural convective flow filled with porous medium in a trapezoidal cavity with various aspect ratio of heated triangular obstacle $$ (1/3 \le AR \le 1) $$ . The boundary location of the cavity consists of non-uniform heating of bottom wall whilst the two vertical walls are maintained at a cold temperature and the top wall is thermal insulated. For the simulation purpose, wide range of parameters, such as, Hartmann number (Ha = 50), Prandtl number (0.026 ≤ Pr ≤ 1000), Rayleigh number (103 ≤ Ra ≤ 106) and Darcy number (10−5 ≤ Da ≤ 10−1) are used in the cavity with various AR. The study demonstrated that the aforementioned parameters have strong influence on the stream functions and isotherms for different AR. A stronger flow circulation is observed for higher Ra and Da. Besides these, a symmetry for fluid flow simulations and isotherm contours are viewed. The result too unveils that the heat transfer rates as well as local and average Nusselt numbers strongly depend on various AR and abovementioned parameters.

Numerical investigation of thermal and hydraulic characteristics of sCO2-water printed circuit heat exchangers with zigzag channels

Since the precooler and the recuperator are the largest components of a supercritical carbon dioxide Brayton cycle, their design can substantially affect the performance and size of the whole system. Although the design of a precooler with zigzag channel geometry as an alternative to straight channels can reduce its size significantly, the applicability of available correlations (e.g. 0.7 < Pr<2.2) for the zigzag channel geometries is limited to the operational range of recuperators only. The current study, therefore, aims to develop correlations and understating of the complex flow and heat transfer characteristics in the zigzag channel printed circuit heat exchangers (PCHEs) operating under precooler conditions (2.2<Pr<13) of supercritical carbon dioxide Brayton cycle(sCO2-BC). Thermal and hydraulic characteristics of the PCHEs are computed numerically for a wide range of Reynolds numbers (5000<Re<7000) and Prandtl number (2.2<Pr<13). Also, a new data reduction method based on segmental averaged values has been proposed to handle adverse variations in the thermophysical properties of CO2 under precooler conditions. To ensure accurate evaluations, steep variations in the thermophysical properties of CO2 are implemented by supplying high-resolution real gas (RGP) property tables. Results suggest that thermal and hydraulic characteristics associated with zigzag channel vary substantially along the length of heat exchanger thus the conventional data reduction methods based on the channel average values cannot be used for true evaluations. Instead, segmental average values are used to develop pressure drop and heat transfer correlations for a broader range of Reynolds number and Prandtl number. The proposed correlations should be useful in the design of compact heat exchanger systems using zigzag channels for a wider range of cooling loads.

Numerical study on the effect of corrugation angle on thermal performance of cross corrugated plate heat exchangers

In the present work, the single-phase thermohydraulic performance of a cross-corrugated plate heat exchanger (PHE) with different corrugation angles 30°≤β≤80° is studied numerically. A three-dimensional numerical model using the finite volume method is developed and validated assuming laminar and turbulent flows in the Reynolds number range of 400–10000. The two-equation k-ω SST model is employed to analyse the turbulent flows in the PHE. It is found that for all the configurations of the PHEs, the pressure drop per unit length increases with the Reynolds number and the corrugation angle. The friction factor of the PHE with β = 80°–80° is higher by a maximum of 2.8 times than that of the flat PHE. The Nusselt number is found to be higher in the corrugated PHE as compared to that of the flat plate PHE. The Nusselt number is increased by a maximum of 3 times for the PHE with β = 80°–80° than that of the flat PHE. Further, single phase generalized correlations of Nusselt number with Reynolds number, geometrical parameters of the cross-corrugated plate heat exchanger and fluid properties are developed for high cross-corrugated angles and a wide range of Prandtl numbers within a maximum accuracy of 40%.

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%.

Minimization of entropy generation due to MHD double diffusive mixed convection in a lid driven trapezoidal cavity with various aspect ratios

Entropy generation minimization has significant importance in fluid flow, heat and mass transfer in an enclosure to get the maximum efficiency of a system and to reduce the loss of energy. In the present study, the analysis of mixed convection fluid flow, heat and mass transfer with heat line and mass line concept and entropy generation due to the effects of fluid flow, heat flow, mass flow and magnetic field in a trapezoidal enclosure with linearly heated and diffusive left wall, uniformly heated and diffusive lower wall, cold and nondiffusive right wall, adiabatic and zero diffusion gradient top wall have been reported. Parametric studies for the wide range of Prandtl number (Pr = 0.7 for air cooling system and Pr = 1000 for the engines filled with olive or engine oils), Rayleigh number (Ra = 103–105), aspect ratio (A = 0.5–1.5) and inclination angle of the cavity (ϕ = 45°–90°) have been performed, which help to construct the perfect shape of cavity in many engineering and physical applications so that the entropy is minimum to get the maximum efficiency of any system. The finite-difference approximation has been used to find out the numerical solutions. Biconjugate Gradient Stabilized (BiCGStab) method is used to solve the discretized nonhomogeneous system of linear equations.

Finite element method visualization about heat transfer analysis of Newtonian material in triangular cavity with square cylinder

Current effort is devoted to investigate the thermophysical aspects of viscous material in a triangular enclosure interacting with a heated square cylinder placed at central position. To view thermal aspects in explicit manner laminar, steady, incompressible viscous material for two different thermal situations are considered bottom wall is (a) uniformly heated (b) non-uniformly heated. Mathematical modelling of phenomenon is attained in the form of dimensional partial differential expressions. To handle manipulated partial differential system an efficient finite element method is implemented. Discretization of domain at multiple levels is generated in the form hybrid grid containing triangular and quadrilateral elements. Velocity, pressure and temperature are approximated by LBB-stable element. Impact of flow controlling parameters on velocity and thermal profiles for spacious range of Rayleigh number (Ra) and Prandtl number (Pr) are considered. Impact of the sundry parameters on coefficient of convective heat transfer, vortex centres and kinetic energy is also computed and shown in tabular format. It is deduced from heat is more efficiently and effectively supplied for non-uniform heating base wall than the uniformly heated wall. It is worthwhile to mention that discontinuity appears in case of uniform heat supply whereas this singularity is removed in the situation where non-uniform heat is provided. Positive trends in Nusselt number and kinetic energy are disclosed for uplifting magnitude of Rayleigh number. It is also found that velocity magnitude at all locations in cavity increments for intensifying values of Rayleigh number.

Mixed convection in the heated semi-circular lid-driven cavity for non-Newtonian power-law fluids: Effect of presence and shape of the block

The present work delineates the hydrodynamics and thermal characteristics due to mixed convection in the lid-driven semi-circular cavity affected by the presence of the adiabatic block at its geometric center for two-dimensional, steady-state, laminar and for non-Newtonian power-law fluids. The semi-circular cavity has a diameter of D . The horizontal wall/lid is sliding with a uniform horizontal velocity ( u = U ) and is subjugated to the ambient thermal condition; while the curved surface is subjugated to a higher isothermal temperature. The convective characteristics inside the system is explored for the broad range of Richardson number (0.1 ≤ Ri ≤ 10), Prandtl number (1 ≤ Pr ≤ 100) and non-Newtonian power-law index (0.5 ≤ n ≤ 1.5) at a constant Grashof number of 10 4 . Apart from this, the effect of shape (cross-section) of the inserted block, i.e. , circular, square and triangular on heat transfer characteristics has also been explored. It is observed that the shear thickening fluids display better cooling characteristics. Besides, the cavity with immersed triangular block shows better heat transfer results than the circular and square blocks. The deviations observed in the flow and heat transfer characteristics in the cavity by inserting an adiabatic block as compared with cavity without block have been ascertained by calculating normalized Nusselt number ( Nu N ). The presence of the block was found to have a diminishing effect on the heat transfer due to convection in the cavity. In the end, the results of the study are summarized in the form of a predictive correlation exhibiting the functional dependence of average Nusselt number with Prandtl number, power-law index, and Richardson number.