Step Change In Wall Temperature Research Articles

The Graetz problem extended to Jeffery-Hamel flow through a convergent plate channel with step-change in wall temperature and streamwise conduction

In this article, we analyze analytically the issue of thermally developing steady laminar of Jeffery-Hamel flow through a convergent-plate channel, including streamwise-conduction with step change of uniform wall temperature. The physical characteristics have been calculated under the assumption that the flow is symmetric and purely radial. The answer is largely based on the strong approach of self-adjoint formalism developed by Papoutsakis and Ramkrishna for the Graetz problem extended to Jeffery-Hamel flow. This mathematical method is the result of decomposing the energy equation into a first-order system of partial differential equations. An extension of previous research is represented by the analytical results, considering the streamwise-conduction impact in the radial direction. The analysis reveals that the streamwise-conduction in the flow and the aperture angle 2ψ between the two plane walls of the channel have a substantial impact on the physical parameters within the warming section. Furthermore, it can be shown that for usual fluids, in the laminar subsonic flow regime, the effect of viscous dissipation is negligible along the channel due to small values of Brinkman number Br except in a very small region at the outlet of order r/r1<<ψBr/Pe; where r1 denotes the heating section's length and Pe the Péclet number.

Simulation to study the effect of natural convection in the voids of a 2D granular packed bed of cylindrical particles on effective thermal conductivity

Natural convection can occur in the voids of a granular material due to temperature difference between neighboring particles. Such an effect of natural convection in voids of a granular material can have a significant impact on effective thermal conductivity of that granular material. In the present study, effective thermal conductivity of a packed bed of solid cylindrical steel rods filled with air and water in the voids is analyzed by simulation. Effective thermal conductivity is obtained for the two systems from a response to step change in wall temperature. In order to quantify the effect of natural convection, a two dimensional array of squarely packed cylinders is simulated with and without the natural convection effect. The simulation is carried out using COMSOL Multiphysics. The temperature versus time data for various particle sizes of 4mm, 8mm and 12mm made of steel are obtained. The maximum velocity magnitude generated in the voids is recorded as a function of time as a measure of natural convection. The effective thermal conductivity is determined for all the conditions using curve fitting between simulation data and analytical function. The effective thermal conductivity calculated from simulation data for two conditions namely with and without natural convection are compared with the literature correlations for both the systems. Role of Grashof number on the intensity of natural convection in voids is discussed. Aspect ratio effect of the packed bed and mesh size effect are also studied.

Experimental determination of effective thermal conductivity of granular material by using a cylindrical heat exchanger

Granular material in general has lower thermal conductivity than solid material. This is due to the limited contact between particles and the presence of air gaps. In the present study, a cylindrical heat exchanger is utilized to obtain temperature versus time response at the central location for a step change in wall temperature. Steel balls in spherical form are studied for estimation of effective thermal conductivity. Particle sizes studied are 12mm, 8mm, 4mm, 3mm, 2mm and 1mm. It is anticipated that a considerable variation in thermal conductivity would be obtained over this size range of particles. The governing equation for unsteady heat conduction in cylindrical co-ordinates incorporates the thermal diffusivity as a parameter. Hence, an analytical solution to the temperature dynamics is obtained by guessing the value of thermal diffusivity and it is used as predicted profile. The guess value of thermal diffusivity is varied and the standard deviation of error between experimental and predicted temperature profiles is minimized to find the optimum thermal diffusivity value. Later, the thermal conductivity of granular material is calculated using the definition of thermal diffusivity which involves density and specific heat capacity also. The overall temperature–time profile in dimensionless form is again compared to evaluate the deviations if any. The present results of effective thermal conductivity are also compared with prediction by Bruggeman’s equation for granular material.

Numerical analysis of transient laminar forced convection of nanofluids in circular ducts

In this study, forced convection heat transfer characteristics of nanofluids are investigated by numerical analysis of incompressible transient laminar flow in a circular duct under step change in wall temperature and wall heat flux. The thermal responses of the system are obtained by solving energy equation under both transient and steady-state conditions for hydro-dynamically fully-developed flow. In the analyses, temperature dependent thermo-physical properties are also considered. In the numerical analysis, Al2O3/water nanofluid is assumed as a homogenous single-phase fluid. For the effective thermal conductivity of nanofluids, Hamilton–Crosser model is used together with a model for Brownian motion in the analysis which takes the effects of temperature and the particle diameter into account. Temperature distributions across the tube for a step jump of wall temperature and also wall heat flux are obtained for various times during the transient calculations at a given location for a constant value of Peclet number and a particle diameter. Variations of thermal conductivity in turn, heat transfer enhancement is obtained at various times as a function of nanoparticle volume fractions, at a given nanoparticle diameter and Peclet number. The results are given under transient and steady-state conditions; steady-state conditions are obtained at larger times and enhancements are found by comparison to the base fluid heat transfer coefficient under the same conditions.

Variational formulation on Joule heating in combined electroosmotic and pressure driven microflows

The present study attempts to analyze the extended Graetz problem in combined electroosmotic and pressure driven flows in rectangular microchannels, by employing a variational formulation. Both the Joule heating and axial conduction effects are taken into consideration. Since assuming a uniform inlet temperature profile is not consistent with the existence of these effects, a step change in wall temperature is considered to represent physically conceivable thermal entrance conditions. The method of analysis considered here is primarily analytical, in which series solutions are presented for the electrical potential, velocity, and temperature. For general treatment of the eigenvalue problem associated with the solution of the thermal field, an approximate solution methodology based on the variational calculus is employed. An analytical solution is also presented by considering thin electrical double layer limits. The results reveal non-monotonic behaviors of the Nusselt number such as the occurrence of singularities in the local Nusselt number values when the fluid is being heated from the wall. Moreover, the effect of increasing the channel aspect ratio is found to be increasing both the temperature difference between the wall and the bulk flow and the Nusselt number. In addition, higher wall heat fluxes are obtained in the entrance region by increasing the Peclet number.

Electroosmotic Flow of Viscoelastic Fluids Through a Slit Microchannel With a Step Change in Wall Temperature

Thermally developing electroosmotically generated flow of two viscoelastic fluids, namely the PTT and FENE-P models, through a slit microchannel is considered. Both the viscous dissipation and Joule heating effects are taken into account and a step change in wall temperature is considered to represent physically conceivable thermal entrance conditions. Expressions for the dimensionless temperature and Nusselt number in the form of infinite series are presented. In general, the resultant eigenvalue problem is solved numerically; nevertheless, an analytical solution is presented for the regions close to the entrance. A parametric study reveals that increasing amounts of the Peclet number result in higher wall heat fluxes. The results also indicate higher wall heat fluxes for non-Newtonian fluids in comparison with Newtonian fluids and the difference is increased with increasing the level of elasticity. Furthermore, based on the value of the dimensionless Joule heating parameter, the Nusselt number may be either an increasing or a decreasing function of the axial coordinate or even both of them in the presence of a singularity point. The viscous heating effects are also found to be negligible.

Thermally developing electroosmotic flow of power-law fluids in a parallel plate microchannel

The present investigation considers the thermally developing electroosmotic flow of power-law fluids through a parallel plate microchannel. Both the viscous dissipation and Joule heating effects are taken into account and a step change in wall temperature is considered to represent physically conceivable thermal entrance conditions. Expressions for the dimensionless temperature and Nusselt number in the form of infinite series are presented. In general, the resultant eigenvalue problem is solved numerically; nevertheless, an analytical solution is presented for the regions close to the entrance. A parametric study reveals that increasing amounts of the Peclet number result in higher wall heat fluxes, whereas the opposite is true for the flow behavior index. Furthermore, based on the value of the dimensionless Joule heating parameter, the Nusselt number may be either an increasing or a decreasing function of the axial coordinate or even both of them in the presence of a singularity point. The viscous heating effects are also found to be negligible.

Graetz Problem Extended to Mixed Electroosmotically and Pressure Driven Flow

Thermally developing mixed electroosmotically and pressure-driven flow in a parallel plate microchannel with a step change in wall temperature is considered in the framework of an extended Graetz problem. Both Joule heating and viscous dissipation effects are taken into consideration. Expressions for the dimensionless temperature and Nusselt number in the form of infinite series are presented. In general, the associated eigenvalue problem is solved numerically. Nevertheless, an analytical solution is also presented for axial locations close to the entrance. Comparisons aremade between the present results and those obtained by approximating the electroosmotic velocity with the Helmholtz–Smoluchowski velocity. It is revealed that, for a pressure-assisted flow with a sufficiently high magnitude of pressure gradient, the Nusselt number is weakly dependent on the dimensionless Debye–Huckel parameter. Accordingly, the approximate solution is valid for such cases.However, unless a sufficiently large value of the dimensionless Debye–Huckel parameter is considered, the incurred error rapidly increases when the pressure gradient is absent or acts adversely. The approximate solution also fails to predict accurate results near the singularity points that occur in theNusselt numberswhen thewall temperature in the downstream is higher than that of upstream.

Semi-analytical solution of the extended Graetz problem for combined electroosmotically and pressure-driven microchannel flows with step-change in wall temperature

Analytical solutions are obtained for the temperature and Nusselt number distribution in the thermal entrance region of a parallel plate microchannel under the combined action of pressure-driven and electroosmotic transport mechanisms, by taking into account the effects of viscous dissipation, Joule heating and axial conduction simultaneously, in the framework of an extended Graetz problem. Step changes in wall temperature are considered to represent physically conceivable thermal entrance conditions. The solution of the temperature distributions at the various channel sections essentially involves the determination of a set of eigenvalues and eigenfunctions corresponding to a Sturm Liouiville problem with non self-adjoint operators. The resultant eigenfunctions are non orthogonal in nature, and are obtained in the forms of hypergeometric functions. Parametric variations on the effects of the relative strengths of the pressure gradients and the electric field, ratio of the rate of heat generation to the rate of wall heat transfer, and the Peclet number are analyzed in details, in terms of their influences on the temperature field as well as the Nusselt number distribution.

Effects of viscous dissipation and fluid axial heat conduction on heat transfer for non-Newtonian fluids in ducts with uniform wall temperature: Part II. Annular ducts

The present paper, which is an extension of a previous study [O. Jambal, T. Shigechi, D. Ganbat, S. Momoki, Int. Commun. Heat Mass Transf. 32(9) (2005) 1165] on laminar heat transfer to non-Newtonian fluids in parallel plates and circular ducts, deals with concentric annular ducts subjected to a step change in wall temperature. A finite-difference scheme is applied to determine the velocity and temperature fields. Developing Nusselt numbers are graphically shown for various Brinkman numbers and Peclet numbers and the effects of viscous dissipation, fluid axial heat conduction, non-Newtonian behavior together with the radius ratio effect on heat transfer are discussed.

Supersonic Turbulent Boundary Layer Subjected to Step Changes in Wall Temperature

An investigation has been made into the effect of a step change in wall temperature on a turbulent boundary layerina supersonicowata Mach numberof2.3.Measurements of themean and turbulentow® elds were made. Theresultsshowthattherelaxation behaviorin general isslowexceptnearthewallwherethelogarithmicbehavior of the thermal and dynamical ® eld is unaffected. Perhaps more interestingly, it appears that the effect of strong heating on the velocity ® eld can be explained largely in terms of changes induced in the localuid properties, and speci® c compressibility effects appear to be small. A mixing length argument is suggested for scaling the Reynolds stresses and the comparison with measurements of the longitudinal Reynolds stress is encouraging.

Supersonic turbulent boundary layer subjected to step changes in wall temperature

Supersonic turbulent boundary layer subjected to step changes in wall temperature

TRANSIENT FREE CONVECTION WITH MASS TRANSFER ON A VERTICAL PLATE EMBEDDED IN A HIGH-POROSITY MEDIUM

Abstract The problem of transient free convection is investigated in a high-porosity medium adjacent to a vertical semi-infinite flat plate with a simultaneous step change in wall temperature and wall concentration. The non-Darcian effects of convection, boundary, and inertia are all considered. The coupled nonlinear partial differential equations are solved by using a cubic spline collocation method. The numerical results show that the Darcy model overestimates both the transient heat and the mass transfer rate for a high-porosity medium. When the inertia effect is neglected, there is a minimum in the temporal transient Nusselt and Sherwood numbers before steady state is achieved. The present analysis also investigates the effects of the following parameters on the time required to reach steady state: buoyancy force ratio N, Darcy number Da, inertia coefficient T, and Lewis number Le. The time required to reach steady state decreases as|N| or Da increases and increases as T increases. When Le < 1, the ti...

Laminar Natural Convection From a Vertical Plate With a Step Change in Wall Temperature

The study of natural convection heat transfer from a vertical flat plate in a quiescent medium has attracted a great deal of interest from many investigators in the past few decades. The plate with various thermal conditions that allow similarity transformations as well as those that are continuous and well defined have been examined. However, practical problems often involve wall conditions that are arbitrary and unknown a priori. To understand and solve problems involving general nonsimilar conditions at the wall, it is useful to investigate problems subjected to a step change in wall temperature. The problems impose a mathematical singularity and severe nonsimilar conditions at the wall. In this paper, a new analytical model that can deal with a discontinuous wall temperature variation is presented. The method results in a set of approximate solutions for temperature and velocity distributions. The validity and accuracy of the model is demonstrated by comparisons with the results of the aforementioned investigators. The agreement is excellent and the results obtained with the solution of this work are remarkably close to existing numerical data of Hayday et al. and the perturbation series solution of Kao.

Effects of heat generation and axial heat conduction in laminar flow inside a circular pipe with a step change in wall temperature

This study analyzes the effect of internal heat generation and axial heat conduction on the convective heat transfer characteristics of laminar flow in a circular pipe subjected to a step-change in wall temperature. Additionally, the flow is considered to be hydrodynamically fully developed and contains an internal heat generation density. The temperature field in the circular pipe is obtained analytically/ numerically from the energy equation by employing a straightforward approach using the Fourier transform technique. The effects of axial heat conduction and internal heat generation are included in this expression in both upstream and downstream directions for Peclet numbers ranging from 0 to ∞. The representative curves illustrating the variations of the bulk temperature and Nusselt numbers with pertinent parameters are plotted. In addition, the ratio of the Nusselt number with heat generation to that without heat generation is plotted against the dimensionless longitudinal coordinate. Furthermore, the Nusselt number is plotted against the dimensionless heat generation number for various Peclet numbers.

Heat transfer between a circular free impinging jet and a solid surface with non-uniform wall temperature or wall heat flux—2. Solution for the boundary layer region

The heat transfer in the boundary layer region of a circular free jet impinging on a flat solid surface with non-uniform wall temperature or wall heat flux is investigated analytically. The flow is laminar, incompressible and steady. The analysis begins with obtaining the solution to the problem with a step change in wall temperature or wall heat flux. The solution to the corresponding problem with arbitrary wall temperature or wall heat flux is obtained by superposition. This solution is then matched with that for the stagnation region obtained in the first part of this study so that the Nusselt number throughout the stagnation region and the boundary layer region is obtained. The results indicate that the Nusselt number for increasing wall temperature or wall heat flux can be considerably higher than that for constant wall temperature or wall heat flux outside the stagnation region. For the special case of constant wall temperature, the result is in good agreement with the integral solution ofChaudhury.

Transient free convection with mass transfer from an isothermal vertical flat plate embededded in a porous medium

The problem of transient free convection in a porous medium adjacent to a vertical semi-infinite flat plate with a simultaneous step change in wall temperature and wall concentration is investigated. Nondimensionalization of the transient boundary-layer equations results in three governing parameters: (1) Le, the Lewis number, (2) N, the bouyancy ratio, and (3) ε, the value of the porosity of the porous medium divided by the ratio of heat capacity of the saturated porous medium to that of the fluid. The resulting nonlinear partial differential equations are solved by an explicit finite-difference method. The numerical results are presented for 0.3≤Le≤100, 0≤ N≤10 and for ε = 0.5, 1, and 2. It is shown that for a given Le and ε the time required to reach the steady state decreases as N increases; for a given N and ε, when Le<1, the time decreases as Le increases, while for Le ≥ 1, the reverse trend is true; and for a given N and Le, the time increases as ε increases. The final steady-state profiles are in good agreement with similarity solutions. Moreover, a simple relation of predicting the length of time for which a one-dimensional heat/mass transport will exist is obtained.

Unsteady thermal entrance heat transfer of laminar duct flows with wall suction and injection

Abstract The unsteady thermal entrance heat transfer problem is analyzed by the instant‐local similarity method for the system of fully‐developed laminar incompressible flow into the region of porous wall which permits uniform suction or injection and with a step change in wall temperature. Both the circular tube and flat duct between parallel plates are considered in this work. Numerical solutions of the transient heat transfer rates and the development of temperature profiles are presented for some selective values of suction Reynolds number, Prandtl number, and Graetz number. The numerical solutions at the asymptotic small and large time are in agreement excellently with the transient condution solution and steady‐state solution, respectively. The corresponding steady‐state heat transfer problems are also analyzed by the local similarity and local nonsimilarity methods. The numerical solutions of these methods are in agreement very closely.

Transient forced convection in laminar channel flow with stepwise variations of wall temperature

AbstractTransient laminar forced convection of Newtonian fluids inside circular ducts and parallel‐plate channels subjected to stepwise variations of wall temperature is considered. Approximate analytical solutions are developed by making use of the generalized integral transform and the classical Laplace transform techniques. Higher order solutions are given for the case of a step change in wall temperature. A second‐order accurate finite‐difference solution is developed to assess the accuracy of the available approximate solutions. The Sign‐Count method is used to solve the resulting eigenvalue problems and determine as many eigenvalues and eigenfunctions as needed. Results are presented for the local Nusselt number, average fluid temperature and the wall heat flux over a wide range of the dimensionless axial coordinate.

Unsteady Stagnation Point Heat Transfer with Blowing or Suction

The effects of blowing and suction on unsteady heat transfer at a stagnation point due to a step change in wall temperature are examined. Two asymptotic solutions for the temperature field at large and small Prandtl numbers are presented. It is shown that the asymptotic solution for large Prandtl number gives sufficiently accurate results for the surface heat transfer even for the moderate values of Prandtl number if Euler transformation is applied to the series.