The Exact Solutions of Some Stefan Problems With Prescribed Heat Flux

1981 ◽  
Vol 48 (4) ◽  
pp. 732-736 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Heat Flux ◽  
Phase Change ◽  
Exact Solutions ◽  
Stefan Problem ◽  
Supercooled Liquid ◽  
Freezing Temperature ◽  
Similarity Solutions ◽  
Initial Condition ◽  
Stefan Problems ◽  
Before And After

The Stefan problem of a semi-infinite material with arbitrarily prescribed initial and flux conditions is studied. When the surface temperature is initially different from the freezing temperature, there exists a presolidification or a premelting period prior to the occurrence of a phase change. The exact solutions for both periods, before and after the appearance of a phase change, are established. This indicates that the initial condition of the Stefan problem with a prescribed heat flux at the surface cannot be assumed to be constant. Possibilities of similarity solutions of the problem are also examined. A similarity solution exists only when the heat flux is proportional to t−1/2 and the initial and boundary conditions satisfy an inequality. The solidification of a supercooled liquid is also investigated. The exact solution is obtained.

scholarly journals Exact Solutions for Nonclassical Stefan Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Adriana C. Briozzo ◽  
Domingo A. Tarzia
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Exact Solutions ◽  
Convective Boundary Condition ◽  
Convective Boundary ◽  
Flux Boundary Condition ◽  
Similarity Type ◽  
Stefan Problems ◽  
First Case ◽  
Temperature Boundary

We consider one-phase nonclassical unidimensional Stefan problems for a source functionFwhich depends on the heat flux, or the temperature on the fixed facex=0. In the first case, we assume a temperature boundary condition, and in the second case we assume a heat flux boundary condition or a convective boundary condition at the fixed face. Exact solutions of a similarity type are obtained in all cases.

Exact Solution to the One-Dimensional Inverse-Stefan Problem in Nonideal Biological Tissues

1995 ◽  
Vol 117 (2) ◽  
pp. 425-431 ◽  
Author(s):  
Y. Rabin ◽  
A. Shitzer
Keyword(s):  
Steady State ◽  
Phase Change ◽  
Stefan Problem ◽  
Blood Perfusion ◽  
Steady State Solution ◽  
Biological Tissues ◽  
Initial Condition ◽  
Quasi Steady State ◽  
Single Temperature ◽  
The One

A new analytic solution of the inverse-Stefan problem in biological tissues is presented. The solution, which is based on the enthalpy method, assumes that phase change occurs over a temperature range and includes the thermal effects of metabolic heat generation, blood perfusion, and density changes. As a first stage a quasi-steady-state solution is derived, defined by uniform velocities of the freezing fronts and thus by constant cooling rates at those interfaces. Next, the fixed boundary condition leading to the quasi-steady state is calculated. It is shown that the inverse-Stefan problem may not be solved exactly for a uniform initial condition, but rather for a very closely approximating exponential initial condition. Very good agreement is obtained between the new solution and an earlier one assuming biological tissues to behave as pure materials in which phase change occurs at a single temperature. A parametric study of the new solution is presented taking into account property values of biological tissues at low freezing rates typical of cryosurgical treatments.

The Lattice Boltzmann Investigation for the Melting Process of Phase Change Material in an Inclined Cavity

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Zhonghao Rao ◽  
Yutao Huo ◽  
Yimin Li
Keyword(s):  
Heat Flux ◽  
Phase Change ◽  
Lattice Boltzmann ◽  
Melting Process ◽  
Clockwise Rotation ◽  
Linear Distribution ◽  
Left Wall ◽  
Inclined Cavity ◽  
Solid Liquid ◽  
Change Material

The solid–liquid phase change process is of importance in the usage of phase change material (PCM). In this paper, the phase change lattice Boltzmann (LB) model has been used to investigate the solid–liquid phase change in an inclined cavity. Three heat flux distributions applied to the left wall are investigated: uniform distribution, linear distribution, and parabolic symmetry distribution. The results show that for all the heat flux distributions, the slight clockwise rotation of the cavity can accelerate the melting process. Furthermore, when more heat is transferred to the cavity through the middle part (parabolic symmetry distribution) or bottom part (linear distribution) of left wall, clockwise rotation of cavity leads to larger temperature of PCM.

Experimental Study of Heat Transfer in a Reduced Scale Cavity Incorporating Phase Change Material Into Its Vertical Walls

2017 ◽  
Vol 10 (1) ◽  
Author(s):  
Ayoub Gounni ◽  
Mustapha El Alami
Keyword(s):  
Heat Flux ◽  
Heat Source ◽  
Phase Change ◽  
Phase Change Material ◽  
Heat Fluxes ◽  
Surface Temperatures ◽  
Different Depths ◽  
Vertical Walls ◽  
Change Material ◽  
Reduced Scale

In order to really assess the thermal performance of a wall incorporating phase change material (PCM), a reduced scale cavity has been monitored during two heating cycles. For each cycle, the heat source inside the test cell is switched “on” for 5 h and its setpoint is 38 °C and then switched off for 4 h. The outdoor air temperature is kept constant at a low temperature of 20 °C. Two walls are equipped with a PCM layer at different depths in order to study the optimal PCM location. The two other walls are wooden and glass to model a real building. The comparison between the four walls is made based on the absorbed heat fluxes and outside surface temperatures. The results show that the location of the PCM close to the heat source reaches its melting temperature and then reduces the surface temperature. At this location, the PCM layer stores the major part of the inlet heat flux. It takes 10 h to release the absorbed heat flux. However, the PCM layer, practically, does not have an effect on the surface temperatures and absorbed heat fluxes, when it is placed far from the heat source.

Stefan Number-Insensitive Numerical Simulation of the Enthalpy Method for Stefan Problems Using the Space-Time Conservation Element and Solution Element Method

2004 ◽  
Author(s):  
Anahita Ayasoufi ◽  
Theo G. Keith ◽  
Ramin K. Rahmani
Keyword(s):  
Numerical Simulation ◽  
Phase Change ◽  
Numerical Simulations ◽  
Latent Heat ◽  
Space Time ◽  
Enthalpy Method ◽  
Stefan Problems ◽  
Stefan Number ◽  
Original Scheme ◽  
Solution Element

An improvement is introduced to the conservation element and solution element (CE/SE) phase change scheme presented previously. The improvement addresses a well known weakness in numerical simulations of the enthalpy method when the Stefan number, (the ratio of sensible to latent heat) is small (less than 0.1). Behavior of the improved scheme, at the limit of small Stefan numbers, is studied and compared with that of the original scheme. It is shown that high dissipative errors, associated with small Stefan numbers, do not occur using the new scheme.

scholarly journals Application of Phase Change Material and Artificial Neural Networks for Smoothing of Heat Flux Fluctuations

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3531
Author(s):  
Tomasz Tietze ◽  
Piotr Szulc ◽  
Daniel Smykowski ◽  
Andrzej Sitka ◽  
Romuald Redzicki
Keyword(s):  
Neural Networks ◽  
Heat Flux ◽  
Artificial Neural Networks ◽  
Phase Change ◽  
Mass Flow Rate ◽  
Phase Change Material ◽  
Flow Rate ◽  
Smoothing Effect ◽  
Artificial Neural ◽  
Change Material

The paper presents an innovative method for smoothing fluctuations of heat flux, using the thermal energy storage unit (TES Unit) with phase change material and Artificial Neural Networks (ANN) control. The research was carried out on a pilot large-scale installation, of which the main component was the TES Unit with a heat capacity of 500 MJ. The main challenge was to smooth the heat flux fluctuations, resulting from variable heat source operation. For this purpose, a molten salt phase change material was used, for which melting occurs at nearly constant temperature. To enhance the smoothing effect, a classical control system based on PID controllers was supported by ANN. The TES Unit was supplied with steam at a constant temperature and variable mass flow rate, while a discharging side was cooled with water at constant mass flow rate. It was indicated that the operation of the TES Unit in the phase change temperature range allows to smooth the heat flux fluctuations by 56%. The tests have also shown that the application of artificial neural networks increases the smoothing effect by 84%.

Heat Transfer Enhancement in Compact Phase Change Microchannel Heat Exchangers for High Flux Laser Diodes

2018 ◽  
Author(s):  
Jensen Hoke ◽  
Todd Bandhauer ◽  
Jack Kotovsky ◽  
Julie Hamilton ◽  
Paul Fontejon
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Phase Change ◽  
Flow Boiling ◽  
Laser Diodes ◽  
High Heat ◽  
Heat Fluxes ◽  
High Heat Flux ◽  
Maximum Heat ◽  
Average Maximum

Liquid-vapor phase change heat transfer in microchannels offers a number of significant advantages for thermal management of high heat flux laser diodes, including reduced flow rates and near constant temperature heat rejection. Modern laser diode bars can produce waste heat loads >1 kW cm−2, and prior studies show that microchannel flow boiling heat transfer at these heat fluxes is possible in very compact heat exchanger geometries. This paper describes further performance improvements through area enhancement of microchannels using a pyramid etching scheme that increases heat transfer area by ∼40% over straight walled channels, which works to promote heat spreading and suppress dry-out phenomenon when exposed to high heat fluxes. The device is constructed from a reactive ion etched silicon wafer bonded to borosilicate to allow flow visualization. The silicon layer is etched to contain an inlet and outlet manifold and a plurality of 40μm wide, 200μm deep, 2mm long channels separated by 40μm wide fins. 15μm wide 150μm long restrictions are placed at the inlet of each channel to promote uniform flow rate in each channel as well as flow stability in each channel. In the area enhanced parts either a 3μm or 6μm sawtooth pattern was etched vertically into the walls, which were also scalloped along the flow path with the a 3μm periodicity. The experimental results showed that the 6μm area-enhanced device increased the average maximum heat flux at the heater to 1.26 kW cm2 using R134a, which compares favorably to a maximum of 0.95 kw cm2 dissipated by the plain walled test section. The 3μm area enhanced test sections, which dissipated a maximum of 1.02 kW cm2 showed only a modest increase in performance over the plain walled test sections. Both area enhancement schemes delayed the onset of critical heat flux to higher heat inputs.

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