Non-dimensional Temperature Profiles Research Articles

Reynolds stresses and turbulent heat fluxes in fan-shaped and cylindrical film cooling holes

Large eddy simulation (LES) is carried out to predict the thermal performance, unsteady flow patterns, and turbulence characteristics of the fan-shaped hole and the cylindrical hole cooling films. Velocity fields, Reynolds stresses, and turbulent heat flux distributions are the factors that ultimately determine the film cooling effectiveness by deciding the concentration of coolant and heat near the surface. Accurate prediction of these processes can provide a reliable data set for the design of film cooling systems. It can also help solve difficulties of improving RANS-based simulations in terms of their inaccuracies in matching measured flow rates for a given pressure drop across the film and the mixing process once the coolant is ejected from the film. Credible estimates and modest costs of the current LES originate from the multiblock hexahedra meshing, the turbulence inflow generator, and the numerical schemes implemented. Multiblock hexahedra meshes are adopted and generated by a quasi-automatic mesh generation algorithm utilizing template-based multi-block construction followed by elliptic smoothing. Computational resources are utilized in a way that mesh refinement exists only near cooling decorations of small size and wall boundaries, without the need for associated clustering in the far field which is commonly used in commercial packages. The high orthogonality and continuous smoothness throughout the computational domain are maintained to facilitate turbulence capturing. The synthetic inflow generator produces a random field matching a realistic set of two-point statistics based on eigen-reconstruction in a computationally inexpensive processing step. To prove that the current LES simulations of modest cost can reproduce with high accuracy, the discharge characteristics, velocity, and turbulence intensity profiles of cylindrical and fan-shaped cooling films are compared with published measured data on the same flow configurations. Results reveal that the set of modeling methodologies is valid for calculating film-cooling performance within an acceptable range of accuracy. The energy-carrying turbulence structures in and near the cooling holes are shown and their behaviors are linked to turbulent Reynolds stresses. Reynolds stresses and turbulent heat fluxes are compared for different blowing ratios and two types of cooling holes in the parametric study. Similarity patterns of velocity profiles, non-dimensional temperature profiles, Reynolds stress and turbulent heat flux profiles are discussed.

Iterative Solutions for the Nonlinear Heat Transfer Equation of a Convective-Radiative Annular Fin with Power Law Temperature-Dependent Thermal Properties

The temperature distribution in a conductive-radiative rectangular profiled annular fin with internal heat generation is scrutinized in the present investigation. The nonlinear variation of thermal conductivity and heat transfer coefficient governed by the power law is considered. The analytical approximation for the non-dimensional temperature profile is obtained using the differential transform method (DTM)-Pade approximant. The nondimensionalization of the governing energy equation using dimensionless terms yields a nonlinear ordinary differential equation (ODE) with corresponding boundary conditions. The resulting ODE is analytically solved with the assistance of the DTM-Pade approximant procedure. Furthermore, the impact of thermal parameters on the temperature field and thermal stress is elaborated with graphs. The important results of the report divulge that temperature distribution greatly enhances with an augmentation of the heat generation parameter, but it gradually reduces with an increment in the magnitude of the thermogeometric and radiative-conductive parameter.

Convective-radiative thermal investigation of a porous dovetail fin using spectral collocation method

An inverted or reversed trapezoidal profiled fin (dovetail fin) is a fin with an irregular cross-sectional area that is used for improved performance in double pipe heat exchanger applications. The current investigation focuses on the temperature distribution in an internal heat-generating longitudinal dovetail porous fin. A second-order ordinary differential equation (ODE) governs the heat transfer problem of a dovetail fin, and dimensionless terms are used to reduce it to a non-dimensional form. The resulting ODE is solved numerically using the spectral collocation method (SCM) via a local linearization strategy, and the temperature distribution in the dovetail profiled fin is discretized using Chebyshev-Gauss-Lobatto (CGL) collocation points and Chebyshev polynomials. The effect of various thermal parameters on the non-dimensional temperature profile and fin efficiency is graphically depicted for a fin with both convective and insulated tip conditions. As a result, for higher values of the convective-conductive and porosity parameter, the temperature in the dovetail fin distributes in a declining manner.

NUMERICAL STUDY OF THE INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON THE UNSTEADY LAMINAR FLOW AND HEAT TRANSFER OF A VISCOUS INCOMPRESSIBLE FLUID DUE TO A ROTATING DISC

In this article, the effect of temperature-dependent viscosity (TVD) on the unsteady laminar flow and heat transfer (HT) of a viscous incompressible fluid due to a rotating disc (RD) has been investigated numerically by exploiting an in-house numerical code. A set of time-dependent, axisymmetric, and non-linear partial differential equations which govern the fluid flows and heat transfer are reduced to non-linear local non-similarity ordinary differential equations by introducing a newly developed group of transformations for different time regimes. Three different solution methods, such as, (i) perturbation solution method for small t, (ii) asymptotic solution method for large t, and (iii) implicit finite difference method for the entire t regime, have been applied to solve the resulting equations treating t as the time-dependent rotating parameter. The local radial skin friction, tangential skin friction and the heat transfer are computed at the surface of the disc for different numerical parameters, such as, Prandtl number, Pr and the viscosity-variation parameter, e. Besides, the key dimensionless quantities such as velocity and temperature profiles, which are inherently linked with the boundary layer thickness, are presented graphically for different values of e while Pr = 0.72. It is found that the dimensionless radial, tangential and axial velocity profiles decrease as e increases, and consequently, the momentum boundary layer thickness is decreased. On the other hand, the non-dimensional temperature profiles are increased owing to the increasing values of e, and this effect eventually leads to a small increment in the thermal boundary layer thickness.

Mathematical Simulation of Casson MHD Flow through a Permeable Moving Wedge with Nonlinear Chemical Reaction and Nonlinear Thermal Radiation.

The influence of the chemical interaction and dynamic micropolar convective heat transfer flow of Casson fluid caused by a moving wedge immersed in a porous material was explored. The Joule heating owing to magnetized porous matrix heating was also deliberated. The mathematical formulation for mass conservation, momentum, energy, and concentration profiles was expressed in the form of partial differential equations. The dimensionless set of ordinary equations was reduced from modeled equations via a transformation framework and then solved by the RK4 built-in function in MATLAB SOFTWARE by taking a step size of . The existing work was compared with the published work. The iteration procedure was stopped until all of the nodes in the η-direction met the convergence condition 10−5. The physical appearance of material parameters on the flow field, temperature, concentration, drag force, and Nusselt number was discussed through plots. The numerical results were obtained for limiting circumstances. The unsteadiness factor thinned the velocity boundary layer but decreased the thermal and concentration boundary layers. By increasing the Eckert number, the nondimensional temperature profile was enhanced. The novelty of the present study is that no one has numerically investigated the magnetized Casson fluid over a moving wedge in the presence of a chemical reaction and thermal radiation.

Boundary-Layer Flow and Heat Transfer over a Rotating Porous Disk in a Non-Newtonian Williamson Nanofluid

Objectives: The present study analyses the effects of nanoparticles over steady, incompressible boundary-layer flow of non-Newtonian Williamson fluid flowing over a rotating porous disk. Methods/statistical analysis: The analysis used nonlinear partial differential equations associated with the non-Newtonian Williamson fluid flow and used an abridged, simplified version of ordinary differential equations under the Boussines q and boundary-layer approximations. The resulting system of nonlinear ordinary differential equations is then solved analytically using the homotopy analysis. Findings: The behavior of three non-dimensional velocity profiles and the non-dimensional temperature profile for important physical parameters like Williamson number, Prandtl number and surface suction/ injection parameters is tabulated and the results are graphed. Application/improvements: Nanoparticles like Copper, Alumina and titania are commonly used with liquid coolants to increase the heat-dissipating capability. Keywords: Boundary-Layer Flow; Heat Transfer; Porous Disk; non-Newtonian; Williamson Nanofluid

The effect of typical buoyant flow elements and heat load combinations on room air temperature profile with displacement ventilation

Typically vertical temperature gradient is modelled to be linear over the room height. More advanced models consist of several nodes that allow different slopes for the temperature profile between the nodes. Validation and development of all those models have been based mainly on measurement using low ceiling height (below 3 m). Also, the previous studies have not covered typical flow elements that exist in office buildings. In this study, the performance of a displacement ventilation system is studied using 3.3 m and 5.1 m ceiling heights in a variety of load conditions. Typical buoyant flow elements and heat load combinations were measured in a simulated office room. The experimental study included room air temperature measurements at different heights and locations over the occupied zone in addition to surface measurement and supply and exhaust air temperature measurements. The measurement data was compared with current models. The results show that the major part of the vertical temperature gradient occurs already at low level. With some typical buoyant flow elements there is no benefit if the ceiling is lower or higher level. Also, measurements depict that modelled non-dimensional temperature profile using low ceiling height (about 3 m) is not valid for high ceiling applications (more than 4 m). Multi-node models works quite well with several buoyant flow elements. Still, the proposed multi-node models did not give good estimation of the vertical temperature gradient when warm window surface or heat gains at ceiling level were introduced in the room space.

Effect of heat transfer on rotating electroosmotic flow through a micro-vessel: haemodynamical applications

This paper theoretically analyzes the heat transfer characteristics associated with electroosmotic flow of blood through a micro-vessel having permeable walls. The analysis is based on the Debye–Huckel approximation for charge distributions and the Navier–Stokes equations are assumed to represent the flow field in a rotating system. The velocity slip condition at the vessel walls is taken into account. The essential features of the rotating electroosmotic flow of blood and associated heat transfer characteristics through a micro-vessel are clearly highlighted by the variation in the non-dimensional flow velocity, volumetric flow rate and non-dimensional temperature profiles. Moreover, the effect of Joule heating parameter and Prandtl number on the thermal transport characteristics are discussed thoroughly. The study reveals that the flow of blood is appreciably influenced by the elctroosmotic parameter as well as rotating Reynolds number.

Effect of Entry Temperature on Forced Convection Heat Transfer With Viscous Dissipation in Thermally Developing Region of Concentric Annuli

Steady laminar forced convection heat transfer in the thermal entrance region of concentric annuli has been studied considering viscous dissipation characterized by the Brinkman number. The inner and outer pipes have been kept at constant and equal temperature. Two cases of entry temperatures have been considered, case 1: an entry temperature that varies with the radial coordinate, obtained by an adiabatically prepared fluid, i.e., attained by the fluid due to viscous dissipation in an adiabatic concentric annular duct and case 2: the conventional uniform entry temperature. The numerical results presented include the nondimensional temperature profiles, Nusselt numbers, and heat transferred from (or to) the inner and outer pipes. It has been shown from the numerical solutions that it is necessary to employ the dissipative entry temperature in place of conventional uniform entry temperature for higher Brinkman numbers. The results for circular pipes follow when the radius ratio takes the limiting value of zero.

Analytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces

AbstractIn this study, heat transfer and temperature distribution equations for logarithmic surface are investigated analytically and numerically. Employing the similarity variables, the governing differential equations have been reduced to ordinary ones and solved via Homotopy perturbation method (HPM), Variational iteration method (VIM), Adomian decomposition method (ADM). The influence of the some physical parameters such as rate of effectiveness of temperature on non-dimensional temperature profiles is considered. Also the fourth-order Runge-Kutta numerical method (NUM) is used for the validity of these analytical methods and excellent agreement are observed between the solutions obtained from HPM, VIM, ADM and numerical results.

Thermal characteristics of electromagnetohydrodynamic flows in narrow channels with viscous dissipation and Joule heating under constant wall heat flux

The present paper critically analyzes, for the first time, the heat transfer characteristics associated with thermally fully developed combined electromagnetohydrodynamic flows through narrow flow conduits, considering electrokinetics effects, for the constant wall heat flux condition. In order to maintain the generality of the problem under consideration, the liquid flow is actuated by a combination of imposed pressure-gradient, electrokinetic effects and additional electromagnetic interactions. The alterations in the thermal transport phenomenon, induced by the variations in the imposed electromagnetic effects are thoroughly explained, in a pin-pointed manner, through an elegant analytical formalism. The essential features of the electromagnetohydrodynamic flow, and associated heat transfer characteristics, through the narrow confinement, are clearly highlighted by the variations in the non-dimensional flow velocity, non-dimensional temperature profile and the Nusselt number. Moreover, the influence of the imposed electromagnetic effects on micro/nano-scale thermal transport characteristics is clearly differentiated from other inherent effects like Joule heating and viscous dissipation. The implications of the constant wall temperature condition are also briefly discussed. The present endeavor has far-reaching implications towards the development of state-of-the-art electro-magneto-mechanical devices, with improved efficiency and functionality, for applications in the field of micro-scale thermal management.

Diffusive-ballistic heat transport in thin films using energy conserving dissipative particle dynamics

Diffusive-ballistic heat transport in thin films was simulated using energy conserving dissipative particle dynamics (DPDe). The solution domain was considered to be two-dimensional with DPD particles distributed uniformly under constant temperature boundary conditions at the top and bottom walls and periodic boundaries at the side walls. The effects of phonon mean free path were incorporated by its relation to the cutoff radius of energy interaction. This cutoff radius was based on the Knudsen number using the existing phonon-boundary scattering models. The simulations for 0.1<Kn<10 were obtained with the different modifications of the cutoff radius. The results were presented in form of a nondimensional temperature profile across the thin film and were compared with the semi-analytical solution of the equation of phonon radiative transport (EPRT). When the phonon-boundary scattering is not considered, the DPDe simulation results have more discrepancies compared with the EPRT solution as Kn increases, indicating that the phonon-boundary scattering plays an important role when the heat transport across the film becomes more ballistic. The results demonstrate that the DPDe can simulate the diffusive-ballistic heat transport for a broad range of Kn, but phonon-boundary scattering should be considered for the accurate simulation of the ballistic heat transport.

Computational study of film cooling from single and two staggered rows of novel semi-circular cooling holes including coolant plenum

Computational analysis of film cooling effectiveness using novel semi-circular hole shapes with streamwise inclination of 30° has been carried out. Velocity profiles from the separate coolant plenum geometry are used for inlet condition to cooling holes. Realizable k−ε turbulence model with enhanced wall treatment is used for turbulence modeling in simulations. It is observed that coolant jet heights for one row and two staggered rows of semi-circular holes are lower than that for one row of circular holes. The centerline and laterally averaged effectiveness values from a row of semi-circular holes are found almost same as that from a row of circular holes due to the less jet lift-off in semi-circular case. Semi-circular hole utilizes half of the coolant mass required for full circular hole for producing the same blowing ratio, so same level of effectiveness from semi-circular holes as that from full circular holes is great advantage. The centerline and laterally effectiveness values from two staggered rows of semi-circular holes are found much higher than a single row of full circular hole case at all streamwise regions at all blowing ratios tested. Along with centerline effectiveness and laterally averaged effectiveness values, the non-dimensional temperature profiles at ( x/ D, z/ D) = (5.0, 0.0), non-dimensional temperature contours and velocity vectors at plane x/ D = 5.0 are also presented. Counter rotating vortex pairs from row of semi-circular holes are found to be weak and of lower size as compare to that from a row of full circular holes.

Mean temperature profiles in turbulent Rayleigh–Bénard convection of water

We report an investigation of temperature profiles in turbulent Rayleigh–Bénard convection of water based on direct numerical simulations (DNS) for a cylindrical cell with unit aspect ratio for the same Prandtl number Pr and similar Rayleigh numbers Ra as used in recent high-precision measurements by Funfschilling et al. (J. Fluid Mech., vol. 536, 2005, p. 145). The Nusselt numbers Nu computed for Pr = 4.38 and Ra = 108, 3 × 108, 5 × 108, 8 × 108 and 109 are found to be in excellent agreement with the experimental data corrected for finite thermal conductivity of the walls. Based on this successful validation of the numerical approach, the DNS data are used to extract vertical profiles of the mean temperature. We find that near the heating and cooling plates the non-dimensional temperature profiles Θ(y) (where y is the non-dimensional vertical coordinate), obey neither a logarithmic nor a power law. Moreover, we demonstrate that the Prandtl–Blasius boundary layer theory cannot predict the shape of the temperature profile with an error less than 7.9% within the thermal boundary layers (TBLs). We further show that the profiles can be approximated by a universal stretched exponential of the form Θ(y) ≈ 1 − exp(−y − 0.5y2) with an absolute error less than 1.1% within the TBLs and 5.5% in the whole Rayleigh cell. Finally, we provide more accurate analytical approximations of the profiles involving higher order polynomials in the approximation.

HEAT TRANSFER ANALYSIS FOR A VAPOR BUBBLE

In this work, we review the basic heat-transfer mechanisms of a vapor bubble immersed in an unbounded and subcooled liquid. The governing equations for the vapor and liquid phases, together with appropriate initial and boundary conditions, are presented in dimensionless form. In particular, if the characteristic inertial times are smaller than the characteristic process times, the evolution for the radius of the bubble depends basically on the Jakob number, Ja, and the vapor densities ratio, ρ. Recognizing that the Jakob number is very large compared with unity, the resulting governing equations are modeled by a thermal boundary layer, adjacent to the radius of the bubble. We show that higher order corrections due to curvature effects, associated with conductive and convective heat terms of the energy equation for the liquid, characterize the collapse of the bubble. For different values of Ja and ρ, the numerical results for the evolution of the nondimensional radius of the bubble,a, and for the corresponding nondimensional temperature profiles, θ, show that the ending collapse state has a singular behavior, which we have denoted as a “thermal runaway.”

Cooling of a Heat-Generating Strip Immersed in a Laminar Channel Flow

An asymptotic and numerical analysis was conducted for the cooling of a heated strip immersed in a laminar channel flow. The strip material, which can represent an electronic chip, generates heat internally at a uniform rate. Taking into account the finite thermal conductivity of the strip, the nondimensional temperature profile in the strip, the maximum longitudinal temperature differences, and the corresponding Nusselt number have been obtained as functions of the nondimensional parameters a and Pe. The heat conduction parameter a represents the competition between the longitudinal heat conduction in the strip and the heat convection in the laminar cooling flow. The parameter Pe is the well-known Peclet number of the problem. The numerical and asymptotic results for these show a strong dependence of the parameters a and Pe. Therefore, the proposed cooling mechanism serves to identify the role of longitudinal heat conduction effects in the strip. This mechanism can seriously affect the thermal performance of this simplified model of an electronic circuit board.

Numerical analysis of an asymptotic model for the collapse of a vapor bubble

In this work, we analyze the thermal collapse of a vapor bubble immersed in a unbounded and subcooled liquid. In this thermal regime, controlled basically by Jakob number (Ja), we present an asymptotic limit of the governing equations by identifying the appropriate temporary and spatial scales to solve numerically the mathematical model. In the limit of Ja ≫ 1, the governing equations describe the spatial and temporal evolution of the adjacent thermal boundary layer to the radius of the bubble. In particular, we prove that the influence of curvature effects due to conductive and convective heat terms of the energy equation for the liquid are responsible to characterize the thermal collapse regime. The numerical results for the evolution of the nondimensional radius of the bubble, a, and the corresponding nondimensional temperature profiles, θ, for different values of the Ja, show that the ending collapse state has a singular behavior, which we have denoted as a “thermal runaway”.

Model predictive control of diffusion-reaction processes

Parabolic partial differential equations naturally arise as an adequate representation of a large class of spatially distributed systems, such as diffusion-reaction processes, where the interplay between diffusive and reaction forces introduces complexity in the characterization of the system, for the purpose of process parameter identification and subsequent control. In this work we introduce a model predictive control (MPC) framework for the control of input and state constrained parabolic partial differential equation (PDEs) systems. Model predictive control (MPC) is one of the most popular control formulations among chemical engineers, manly due to its ability to account for the actuator (input) constraints that inevitably exist due to finite actuator power and its ability to handle state constraints within an optimal control setting. In controller synthesis, the initially parabolic partial differential equation of the diffusion reaction type is transformed by the Galerkin method into a system of ordinary differential equations (ODEs) that capture the dominant dynamics of the PDE system. Systems obtained in such a way (ODEs) are used as the basis for the synthesis of the MPC controller that explicitly accounts for the input and state constraints. Namely, the modified MPC formulation includes a penalty term that is directly added to the objective function and through the appropriate structure of the controller state constraints accounts for the infinite dimensional nature of the state of the PDE system. The MPC controller design method is successively applied to control of the diffusion-reaction process described by linear parabolic PDE, by demonstrating stabilization of the non-dimensional temperature profile around a spatially uniform unstable steady-state under satisfaction of the input (actuator) constraints and allowable non-dimensional temperature (state) constraints.

Laser schlieren measurement of vertical flow past a heated horizontal circular cylinder

Flow past a heated horizontal circular cylinder in the vertically upward direction has been experimentally studied using a monochrome schlieren technique. Both free convection (( Gr) 1/3 Re)=0 and mixed convection (( Gr) 1/3 Re)=1011, 1055, 1095 and 1133 cases have been studied. The Reynolds number based on the cylinder diameter is set at 102 for the mixed convection, and four heating levels have been utilized with Grashof numbers of Gr=975, 1105, 1240 and 1370. The temperature distribution of the plume, the Strouhal number and the schlieren images have been reported. The vortex shedding frequency decreases with increasing Grashof number and a complete suppression of vortex shedding takes place at Grashof number of 1370. The wake is seen to become visibly narrow during the suppression of vortex shedding. The nondimensional temperature profile inside the plume is a strong function of Grashof number for free convection in comparison to that of mixed convection.

Thermal Stresses in a Circular Cylinder with Temperature Dependent Properties

The thermal stresses in a long circular cylinder are analyzed by two approximate methods. An interpolation formula is then used to develop a semi-empirical solution for thermal stresses which is then validated by a test case of a nondimensional temperature profile with temperature dependent modulus of elasticity, coefficient of thermal expansion and Poisson’s ratio. The interpolation formula is compared with the numerical solution by a finite element code (ABAQUS). It is shown that the interpolated approximation compares well with the finite element solution for a range of parameters that are typical for processing of billets.