scholarly journals The solution of Problems Involving the Melting and Freezing of Finite Slabs by a Method due to Portnov

1964 ◽  
Vol 14 (2) ◽  
pp. 109-128 ◽  
Author(s):  
F. Jackson
Keyword(s):  
Exact Solution ◽  
Steady State ◽  
Temperature Distribution ◽  
Heat Fluxes ◽  
Poisson Integral ◽  
Time Dependent ◽  
Arbitrary Time ◽  
Infinite Slab ◽  
The Face ◽  
Steady State Solutions

In a recent paper (1) Portnov used a form of Poisson integral to find the exact solution for the temperature distribution in a freezing semi-infinite slab occupying the region x > 0, and having an arbitrary time dependent temperature applied at the face x = 0. Previously, Boley (2) had used a method based on Duhamel's theorem to find solutions for problems involving melting, in both finite and semi-finite regions, caused by time dependent heat fluxes. Steady-state solutions have been investigated by Landau (3), Masters (4) and others (5).

Numerical Experiments of Coolant Mixing in a Lower Plenum of PWR Under Asymmetric Thermal- Hydraulics Conditions

2006 ◽  
Author(s):  
Masanori Ohtani ◽  
Akito Kozuru ◽  
Yasuyuki Kashimoto ◽  
Mitsuto Montani ◽  
Koutaro Takeda ◽  
...  
Keyword(s):  
Steady State ◽  
Temperature Distribution ◽  
Turbulent Mixing ◽  
Numerical Experiments ◽  
Time Dependent ◽  
Inlet Temperature ◽  
Velocity Fluctuations ◽  
The Core ◽  
State Calculation ◽  
Lower Temperature

Asymmetric thermal-hydraulic conditions among primary loops during a postulated steam line break (SLB) induce a non-uniform temperature distribution at a core inlet. When coolant of lower temperature intrudes into a part of core, it leads to a reactivity insertion and a local power increase. Therefore, an appropriate model for the core inlet temperature distribution is required for a realistic SLB analysis. In this study, numerical experiments were conducted to examine the core inlet temperature distribution under the asymmetric thermal-hydraulic coolant conditions among primary loops. 3D steady-state calculations were carried out for Japanese standard Pressurized Water Reactor (PWR) such as 2, 3, 4 loop types and an advanced PWR. Since the flow in a reactor vessel involves time-dependent velocity fluctuations due to a high Reynolds number condition and a complicated geometry of flow path, the turbulent mixing might be enhanced. Hence, the turbulent thermal diffusivity for the steady-state calculation was examined based on experimental results and another transient calculation. As a result, it was confirmed that (1) the turbulent mixing in a downcomer and a lower plenum were enhanced due to time-dependent velocity fluctuations and therefore the turbulent thermal diffusivity for steady-state calculation was specified to be greater, (2) the core inlet temperature distribution predicted by a steady-state calculation reasonably agreed with a experimental data, (3) the patterns of core inlet temperature distribution were comprehended to be dependent on the plant type, i.e. the number of primary loop and (4) under a low flow rate condition, the coolant of lower temperature appeared on the opposite side of the affected loop due to the effect of a natural convection.

scholarly journals Thermal ignition in a reactive slab with unsymmetric boundary temperatures

1983 ◽  
Vol 24 (3) ◽  
pp. 279-288 ◽  
Author(s):  
K. K. Tam ◽  
P. B. Chapman
Keyword(s):  
Steady State ◽  
Critical Role ◽  
Upper And Lower Solutions ◽  
Lower Solution ◽  
Time Dependent ◽  
Thermal Ignition ◽  
Time Dependent Problem ◽  
Steady State Solutions

AbstractThe problem of thermal ignition in a reactive slab with unsymmetric temperatures equal to 0 and T is considered. Steady state upper and lower solutions are constructed. It is found that T plays a critical role. Results similar to the case with symmetric boundary temperatures are expected when T is small. When T is sufficiently large, there is only one steady state upper or lower solution. The time dependent problem is then considered. Phenomena suggested by studying the upper and lower steady state solutions are confirmed.

Laser Beam-Hygroscopic Aerosol Interactions

1977 ◽  
Vol 99 (2) ◽  
pp. 281-286 ◽  
Author(s):  
G. E. Caledonia ◽  
J. D. Teare
Keyword(s):  
Steady State ◽  
Laser Beam ◽  
Water Droplet ◽  
Atmospheric Aerosol ◽  
Phase Shifts ◽  
Time Dependent ◽  
Small Water ◽  
Steady State Solutions

A model for the prediction of the temperature and vapor fields created about a small water droplet undergoing irradiation by a laser beam has been developed. Time-dependent and steady-state solutions of the model are discussed. Estimates of characteristic phase shifts to be expected in propagating through standard atmospheric aerosol distributions are also presented. While the model is quite general, the calculations are limited to DF laser wavelengths.

Steady-State Temperature Distribution in a Rotating Roll Subject to Surface Heat Fluxes and Convective Cooling

1981 ◽  
Vol 103 (1) ◽  
pp. 36-41 ◽  
Author(s):  
E. J. Patula
Keyword(s):  
Steady State ◽  
Temperature Distribution ◽  
Cold Rolling ◽  
Heat Input ◽  
Surface Heat ◽  
Heat Fluxes ◽  
Solution Technique ◽  
Convective Cooling ◽  
Roll Temperature ◽  
Steady State Temperature

With the higher rolling speeds used in modern cold-rolling mills, proper roll cooling has become a critical factor in avoiding problems of excessive roll spalling and poor thermal crowning. Poor thermal crowning of rolls can severely affect the shape and profile of sheet and strip products. To determine the influence of cooling practices on roll temperature, a mathematical model was developed that determines the two-dimensional (radial and circumferential) steady-state temperature distribution in a rotating roll subject to constant surface heat input over one portion of the circumference and convective cooling over another portion of the circumference. The model is analytical in nature, as opposed to a direct numerical simulation, which enables extensive parametric studies to be performed conveniently. The solution technique can be used to solve numerous problems involving any combination of surface boundary conditions that have, at most, a linear dependence with respect to the surface temperature. With the use of the principle of superposition, the present solution can be utilized to solve problems where various regions of the surface have constant heat fluxes. Results of the present analysis indicate that for normal cold-rolling situations during steady operation, the penetration of the effects of the surface heating and cooling that occur during every roll revolution is usually less than 4 percent of the radius. Furthermore, the bulk of the roll is at a uniform temperature that can be calculated quite accurately by neglecting all internal temperature gradients. The location of the cooling regions relative to the heat-input regions has little effect on the bulk roll temperature in this situation. This approximation would be useful for computing bulk roll temperature, which could be utilized in future models for determining thermal crowns, but would not be suited for determining accurate temperatures at the roll surface.

scholarly journals The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number

1993 ◽  
Vol 256 ◽  
pp. 561-605 ◽  
Author(s):  
Phillip M. Lovalenti ◽  
John F. Brady
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Stokes Flow ◽  
Slip Velocity ◽  
Hydrodynamic Force ◽  
Time Dependent ◽  
Rigid Particle ◽  
Flowing Fluid ◽  
Arbitrary Time ◽  
Unsteady Stokes Flow

The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t-2.

Time-Dependent Simulations of Shear-Thinning Elastic Flows Through Contractions

2001 ◽  
Author(s):  
Paulo J. Oliveira
Keyword(s):  
Steady State ◽  
Unsteady Flow ◽  
Experimental Work ◽  
Shear Thinning ◽  
Time Dependent ◽  
Numerical Work ◽  
Viscoelastic Models ◽  
Vortex Merging ◽  
Constant Viscosity ◽  
Steady State Solutions

Abstract Past numerical work with constant-viscosity viscoelastic fluids through planar contractions has enabled steady-state solutions to be obtained at relatively high Weissenberg numbers (We up to 10). Those solutions for the upper-convected Maxwell and Oldroyd-B models showed steady flow patterns for increasing We with the appearance of lip-vortices, vortex growth, and comer-lip vortex merging, in agreement with recent simulations and some experimental work. When similar simulations were performed with two shear-thinning viscoelastic models, namely the Phan-Thien – Tanner and the Giesekus models, steady state solutions could not be obtained beyond a certain value of We, but the solution was seen to follow a periodic unsteady flow pattern. The present paper reports preliminary numerical results which highlight the formation of the lip vortices through a sequence of time dependent events.

Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks

1965 ◽  
Vol 21 (4) ◽  
pp. 623-633 ◽  
Author(s):  
Carl E. Pearson
Keyword(s):  
Steady State ◽  
Viscous Flow ◽  
Numerical Solutions ◽  
Preceding Paper ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Main Body ◽  
Rotating Disks ◽  
High Reynolds ◽  
Steady State Solutions

The nature of the steady-state viscous flow between two large rotating disks has often been discussed, usually qualitatively, in the literature. Using a version of the numerical method described in the preceding paper (Pearson 1965), digital computer solutions for the time-dependent case are obtained (steady-state solutions are then obtainable as limiting cases for large times). Solutions are given for impulsively started disks, and for counter-rotating disks. Of interest is the fact that, at high Reynolds numbers, the solution for the latter problem is unsymmetrical; moreover, the main body of the fluid rotates at a higher angular velocity than that of either disk.

Heat Transfer in Mini∕Microchannels With Combustion: A Simple Analysis for Application in Nonintrusive IR Diagnostics

2008 ◽  
Vol 130 (12) ◽  
Author(s):  
Ananthanarayanan Veeraragavan ◽  
Christopher P. Cadou
Keyword(s):  
Exact Solution ◽  
Temperature Distribution ◽  
Fourth Order ◽  
Heat Fluxes ◽  
Reacting Flow ◽  
Parallel Plates ◽  
Gas Temperature ◽  
Order Polynomial ◽  
Measured Absorption ◽  
Fourth Order Polynomial

An analytical solution for the temperature distribution in 2D laminar reacting flow between closely spaced parallel plates is derived as part of a larger effort to develop a nonintrusive technique for measuring gas temperature distributions in millimeter and submillimeter scale channel flows. The results show that the exact solution, a Fourier series, which is a function of the Peclet number, is approximated by second and fourth order polynomial fits to an R2 value of almost unity for both fits. The slopes of the temperature near the wall (heat fluxes) are captured to within 20% of the exact solution using a second order polynomial and to within 2% of the exact solution using a fourth order polynomial. The fits are used in a nonintrusive Fourier transform infrared spectroscopy technique and enable one to infer the temperature distribution along an absorbing gas column from the measured absorption spectrum. The technique is demonstrated in a silicon-walled microcombustor.

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