Robert Siegel

An analytical solution for solidification of a moving warm liquid onto an isothermal cold wall.

Abstract Successive solutions for the instantaneous frozen layer thickness and temperature profile in the solidified layer were generated by an analytical iteration technique. Each iteration was more accurate than the preceding one and the succession of solutions converged rapidly. Analytical expressions obtained were of a simple form and agreed with numerical and approximate solutions of other investigators.

Transient heat transfer in a semitransparent radiating layer with boundary convection and surface reflections

Surface convection and refractive index effects are examined during transient radiative heating or cooling of a grey semitransparent layer with internal absorption, emission and conduction. Each side of the layer is exposed to hot or cold radiative surroundings, while each boundary is heated or cooled by convection. Emission within the layer and internal reflections depend on the layer refractive index. The reflected energy and heat conduction distribute energy across the layer and partially equalize the transient temperature distributions. Solutions are given to demonstrate the effect of radiative heating for layers with various optical thicknesses, the behavior of a layer heated by radiation on one side and convectively cooled on the other, and a layer heated by convection while being cooled by radiation. The numerical method is an implicit finite difference procedure with non-uniform space and time increments. The basic method developed in earlier work is expanded to include external convection and incident radiation.

Laminar forced convection in rectangular channels with unequal heat addition on adjacent sides

Abstract Temperature distributions are determined analytically for fully developed laminar heat transfer in channels with aspect ratios from 1 to ∞. The channel walls are uniformly heated, but the heat flux on the short sides is an arbitrary fraction between 0 and 1 of the heat flux on the broad sides. For all cases, the wall temperatures are compared on the basis that the total heat transferred per unit channel length is maintained at a fixed value. The poor convection due to the low velocities in the corners and along the narrow walls always caused the peak temperatures to occur at the corners. The lowest peak temperatures were found when all the heating took place at only the broad walls rather than when heating was partly distributed to the short sides. This results from the fact that, when four sides are heated, more energy is being supplied to the low velocity comer regions. For heating at only the broad walls, the corner temperature decreases rapidly as the aspect ratio is increased to about 10 and insignificantly thereafter. In the limit of infinite aspect ratio, the wall temperature distribution does not approach a constant as is the case for infinite parallel plates.

Forced convection in a channel with wall heat capacity and with wall heating variable with axial position and time

Abstract Heat transfer to a fluid with constant properties is analysed for forced convection between two parallel plates. The plates have a finite heat capacity, and heat is supplied to them in an arbitrary manner with both time and axial position. The wall is assumed sufficiently thin or highly conducting so that the temperature variation through the wall thickness can be neglected. The fluid temperature is variable over the channel cross section, but the fluid velocity is assumed constant throughout the channel (slug-flow condition). The energy equation is laminar. A general method of solution is given and then some illustrative examples are carried out. These include uniform wall heating that varies sinusoidally in time, and heating varying sinusoidally with axial distance and exponentially in time.

Analysis of solidification interface shape during continuous casting of a slab

An analysis was made of the two-dimensional interface shape of a slab ingot being cast continuously by withdrawing it from a mold. The sides of the ingot are being cooled and the upper boundary of the ingot is in contact with a pool of molten metal. The solidification interface shape was determined from the analysis of the heat flow, utilizing the condition that the solidification interface is at constant temperature and must be normal to the lines of heat flow carrying away latent heat of fusion from the interface. The analysis includes the effect of interface curvature which, for a constant rate of withdrawing the cast ingot from the mold, causes the solidification to be nonuniform along the interface. The analysis was carried out by a conformal mapping procedure.

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

Abstract 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.

Two-flux method for transient radiative transfer in a semitransparent layer

The two-flux method was used to obtain transient solutions for a plane layer including internal reflections and scattering. The layer was initially at uniform temperature, and was heated or cooled by external radiation and convection. The two-flux equations were examined as a means for evaluating the radiative flux gradient in the transient energy equation. Comparisons of transient temperature distributions using the two-flux method were made with results where the radiative flux gradient was evaluated from the exact radiative transfer equations. Good agreement was obtained for optical thicknesses from 0.5 to 5 and for refractive indices of 1 and 2. Illustrative results obtained with the two-flux method demonstrate the effect of isotropic scattering coupled with changing the refractive index. For small absorption with large scattering the maximum layer temperature is increased when the refractive index is increased. For larger absorption the effect is opposite, and the maximum temperature decreases with increased refractive index .

Separation of variables solution for non-linear radiative cooling

A separation of variables solution has been obtained for transient radiative cooling of an absorbing-scattering plane layer. The solution applies after an initial transient period required for adjustment of the temperature and scattering source function distributions. The layer emittance, equal to the instantaneous heat loss divided by the fourth power of the instantaneous mean temperature, becomes constant. This emittance is a function of only the optical thickness of the layer and the scattering albedo; its behavior as a function of these quantities is considerably different than for a layer at constant temperature.

Radiative exchange in a parallel-plate enclosure with translucent protective coatings on its walls

Abstract Thermal barrier and protective coatings are used for reducing temperatures in metal walls in high temperature applications, and for protection in corrosive environments. Radiative transfer is important at elevated temperatures, and hot coated surfaces that view each other can exchange radiation. Radiative exchange calculations are usually for surfaces that are opaque, and each surface has a specific radiating temperature. The analysis must be different when some or all of the surfaces have translucent coatings. Some protective coating materials such as zirconia and alumina, are partially transparent to portions of the thermal radiation spectrum. Hence, the radiative exchange process is more complex since radiation from a coated area depends on the temperatures within its coating. These temperatures are unknown, and they depend on the exchange process. The exchange analyzed here is for a parallel-plate enclosure that can have different heating conditions for the internal and external surfaces of each coated wall. Each wall has an opaque outer layer such as a metal plate, and a translucent coating on the inside. The coatings on opposite walls view each other and exchange radiation. An analytical method is developed for obtaining the temperature distribution in each coating and its opaque substrate. Illustrative results are provided for coatings with various optical thicknesses, and are compared with results for opaque coatings.

Some aspects of transient cooling of a radiating rectangular medium

The emission from a gray radiating medium is analyzed for transient cooling in surroundings at a low temperature. The medium is rectangular with no variations in the direction normal to the cross section. The integral equation for the transient temperature distribution is solved numerically using a two-dimensional Gaussian integration subroutine. The emissive ability for a rectangle at uniform temperature is compared with that for transient cooling where the temperature distribution of the region has reached a fully developed shape, as shown by a separation of variables solution. The two solutions provide the upper and lower bounds for the emittance of a rectangle during transient cooling. The emittances for various aspect ratios are presented as a function of the mean length of the rectangle and are compared with results for a plane layer and a cylinder.