Step Change In Wall Temperature Research Articles

Unsteady heat transfer from a circular cylinder immersed in a Darcy flow

Analyses are made for unsteady heat transfer from a circular cylinder immersed in a porous medium through which a liquid is flowing according to Darcy's law. Asymptotic solutions for large and small Peclet numbers (Pe) are obtained for the case where the unsteady temperature field is produced by a step change in wall temperature. The former is valid for Pe≳200 and the latter for Pe≲0.1. The series solution for small time, which is valid for all values of Pe, is also obtained. By applying the Euler transformation to the series, its convergence is greatly improved, and it appears that the Eulerized series determines the mean Nusselt number adequately for most values of time.

Molecular Gas Radiation in the Thermal Entrance Region of a Duct

The effect of molecular gas radiation upon the thermal development downstream from a step change in wall temperature is examined for both laminar and turbulent flow in a black-walled flat-plate duct. The exponential-tailed band model is used to represent spectral variations in gas absorption and emission. Values of total and radiative Nusselt numbers, cold-wall-layer transmission factors, and dimensionless bulk temperatures are reported for several dimensionless axial locations and for various sets of the dimension-less controlling parameters. Even in the entrance region, self absorption by wall layer gas blocks significantly the radiation exchange between the gas core and wall. An approximate correlation is proposed for both plane-duct and pipe turbulent entrance flows.

The development of turbulent thermal layers on flat plates

Measurements of mean velocity and temperature are presented for a turbulent boundary layer developing on a uniform temperature flat smooth plate. An analysis for determining the virtual origins of the aerodynamic and thermal layers is presented. Using this analysis, the virtual origins of the aerodynamic and thermal layers were made coincident. A thermal defect law and thermal law of the wall were found to be applicable. Mean temperature measurements were made for the case of an internal thermal layer originating at a step change in wall temperature downstream of the origin of the aerodynamic layer. It was found that the thermal law of the wall became applicable, for the case studied, a small streamwise distance after the step.

Stability of heated laminar boundary layers in water with nonuniform surface temperature

As part of a continuing study of the stability of heated laminar boundary layers in water, an experiment has been performed for the case of nonuniform surface temperature where disturbances are introduced by a vibrating ribbon and the response measured using temperature compensated hot-film anemometry. Two types of nonuniform wall temperature distributions have been studied. These are: (a) power law distributions of the form ΔT= (Tw−T∞) =Axn, and (b) step changes in wall temperature. Measured mean velocity and temperature profiles agree with solutions of the mean boundary layer equations. Measured stability characteristics for power law temperature distributions indicate that, at a fixed Reynolds number and local ΔT, both the disturbance growth rate and the band of frequencies receiving amplification are reduced relative to the uniform wall temperature case when n<0 and increase relative to the uniform wall temperature case when n≳0. The maximum growth rate for ΔT =2.78°C (5°F) and n=−0.5 is negative indicating damped disturbances at all frequencies. Disturbance growth rates for step changes in wall temperature at first decrease and then increase relative to those for uniform wall temperature as the step change location is moved downstream.

Convective Instability in a Melt Layer Heated From Below

Consideration is given to the onset of convective motions in a horizontal melt layer created by solid-to-liquid phase change. The melt layer is heated at its lower bounding surface either due to convective transfer from an adjacent fluid medium or to a step change in wall temperature. The analysis is carried out for liquid melts whose densities decrease with increasing temperature. Linear stability theory is employed to determine the conditions marking the onset of motion. The results of the analysis are expressed in terms of two Rayleigh numbers. One of these, the internal Rayleigh number, is based on the instantaneous thickness and instantaneous temperature difference across the layer. The other, the external Rayleigh number, is more convenient to use in applications problems since it contains quantities which are constant and a priori prescribable. For a melting problem where the external Rayleigh number is large, instability occurs soon after the start of heating. At smaller external Rayleigh numbers, the duration time of the regime of no motion increases markedly. At large times, the stability results for convective heating coincide with those for stepped wall temperature. In addition to the results for the stability problem, results for conduction phase change (in the absence of motion) are also presented for the surface convection boundary condition.

The effect of wall cooling on a compressible turbulent boundary layer

Experimental results are presented for two turbulent boundary-layer experiments conducted at a free-stream Mach number of 4 with wall cooling. The first experiment examines a constant-temperature cold-wall boundary layer subjected to adverse and favourable pressure gradients. It is shown that the boundary-layer data display good agreement with Coles’ general composite boundary-layer profile using Van Driest's transformation. Further, the pressuregradient parameter βK found in previous studies to correlate adiabatic highspeed data with low-speed data also correlates the present cooled-wall high-speed data. The second experiment treats the response of a constant-pressure highspeed boundary layer to a near step change in wall temperature. It is found that the growth rate of the thermal boundary layer within the existing turbulent boundary layer varies considerably depending upon the direction of the wall temperature change. For the case of an initially cooled boundary layer flowing onto a wall near the recovery temperature, it is found that δT ∼ x whereas the case of an adiabatic boundary layer flowing onto a cooled wall gives δT ∼ x½. The apparent origin of the thermal boundary layer also changes considerably, which is accounted for by the variation in sublayer thicknesses and growth rates within the sublayer.

Thermal response of an unsteady laminar boundary layer on a flat plate due to step changes in wall temperature and in wall heat flux

The prediction of the entire time-history of the heat transfer process due either to a step change in plate surface temperature or to heat flux is studied for two-dimensional, incompressible laminar flows. The study utilizes a very efficient numerical method for solving the nonsteady energy equation. Numerical results for several Prandtl numbers are compared with analytical results obtained with the restriction of small and large time values, and the range of applicability of these analytical solutions is established.

An Analysis of Laminar Free and Forced Convection between Finite Vertical Parallel Plates

Approximate analytical solutions are presented for constant-property laminar free- and forced-convection flows between finite vertical parallel plates. For free convection, the thermal boundary conditions considered include the thermally symmetric channel with uniform wall temperature or step change in wall temperature and the unsymmetric channel with uniform but unequal wall temperatures. For forced convection and combined free and forced convection, the thermally symmetric and uniform thermal boundary condition is considered. Where possible, results are compared with available numerical and experimental results. Particular attention is given to heat-transfer results, which cover a wide range of Rayleigh and Prandtl numbers. For combined convection the heat-transfer results are related to the impressed pressure difference and flow rate.

A solution to the turbulent Graetz problem by matched asymptotic expansions - I The case of uniform wall temperature

Consideration of heat transfer to a constant property fluid in fully developed turbulent flow in a pipe with a step change in wall temperature leads to a Sturm-Liouville problem. A first order asymptotic technique applied to this problem leads to estimates of heat transfer rates that are in poor agreement with numerical calculations [1]. Presented here is an extension of the first order analysis which yields higher approximations for the lower eigenfunctions. The extension is an unusual application of the method of matched asymptotic expansions in which the expansion parameter is the eigenvalue and matching of both amplitude and phase of functions are required. Analytical expressions are derived but no numerical calculations are given.

Two-dimensional unsteady incompressible laminar duct flow with a step change in wall temperature

Analytical solutions are obtained of transient heat transfer for unsteady incompressible laminar flow between parallel plates. The transient is caused by simultaneously changing with time the driving pressure of the fluid and the wall temperature. The solution is first obtained for the case where the inside surfaces of the channel walls undergo a specified step in temperature, that is, the heattransfer resistance of the wall is neglected. Then some results are given where the temperature is specified at the outside surfaces of the walls and the transient heat conduction through the walls is taken into account. A few numerical examples are carried out to illustrate the method.

Heat Transfer for Laminar Flow in Ducts With Arbitrary Time Variations in Wall Temperature

An analysis is made for laminar forced-convection heat transfer in a circular tube or a parallel plate channel whose walls may undergo arbitrary time variations in temperature. The time-varying process can begin from an already established steady-state situation with heat transfer taking place, or the fluid and walls can be initially at the same uniform temperature. The fluid velocity distribution is fully developed and unchanging with time. At any instant during the transient the wall temperature is spatially uniform, that is, all portions of the wall simultaneously undergo the same temperature-time variation. The greater part of the analysis is concerned with the response to a step change in wall temperature, and the time required to reach steady state is given for this type of transient. Then the results are generalized to apply for arbitrary variations with time.

Transient Heat Transfer for Laminar Forced Convection in the Thermal Entrance Region of Flat Ducts

An analysis is made for transient laminar heat transfer in the thermal entrance region of a flat duct (parallel plate channel) whose bounding surfaces are subjected to an arbitrary time variation of temperature or of heat flux. Initially, the system may be either in an already established steady-state heat-transfer situation, or else, the fluid and duct walls may be at the same uniform temperature. The velocity distribution in the flow is taken to be fully developed and unchanging with time. The solution for arbitrary time-dependent conditions is obtained by generalizing the thermal response to a unit step change in wall temperature or in wall heat flux. This step-function response is found by using the method of characteristics. Heat-transfer results are presented as simple analytical expressions. The time required to achieve steady state after a unit step is also given. Working formulas are summarized at the end of the paper.