Boundary temperatures, fictitious

A closely related method is that of Boley (B8), who was concerned with aerodynamic ablation of a one-dimensional solid slab. The domain is extended to some fixed boundary, such as X(0), to which an unknown temperature is applied such that the conditions at the moving boundary are satisfied. This leads to two functional equations for the unknown boundary position and the fictitious boundary temperature, and would, therefore, appear to be more complicated for iterative solution than the Kolodner method. Boley considers two problems, the first of which is the ablation of a slab of finite thickness subjected on both faces to mixed boundary conditions (Newton s law of cooling). The one-dimensional heat equation is once again... 

T is a fictitious boundary to the assumed gas temperature profile. Since one of the profile boundary temperatures and the mean temperature are defined, the remaining profile boundary temperature, TT, is fixed but lot necessarily equal to the gas inlet temperature or saturation temperature. 

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Theory of the fictitious temperature field allows us to analyze the problems of residual stresses in glass using the mathematical apparatus of thermoelasticity. In this part we formulate the boundary-value problem for determining the internal stresses. We will Lheretore start from the Duhamel-Neuinan relations... 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

The second example is the solidification of liquid enclosed between parallel boundaries. Here it is necessary to deal with two fictitious bodies of constant dimensions, one representing the liquid and the other the solid. For the solid phase one deals not with an initially finite body whose dimensions are shrinking, but with a body whose dimensions increase from zero. For this reason an imaginary initial temperature turns out to be more appropriate than an imaginary heat input. 

We have thus far obtained the distribution functions for the incoming and reflected particles of reduced mass in different regions in terms of the unknown constants A and B and the unknown dimensionless surface temperature 8. The surface temperature was then related to the internal energy and the number density of the fictitious particle at the reflecting surface. Now, in order to determine the constants A and B, wc must specify the boundary conditions for the mass and the energy flux at the sphere of influence. 

Using the zero flux condition [13b] and the boundary condition [13a], the parameter r can be expressed in terms of the flux of the fictitious particles incident upon the reflecting surface A, and the surface temperature Tr as,... 

Heat convection for gaseous flow in a circular tube in the slip flow regime with uniform temperature boundary condition was solved in [23]. The effects of the rarefaction and surface accommodation coefficients were considered. They defined a fictitious extrapolated boundary where the fluid velocity does not slip by scaling the velocity profile with a new variable, the shp radius, pj = l/(l + 4p.,Kn), where is a function of the momentum accommodation coefficient, and defined as p, =(2-F,j,)/F,j,. Therefore, the velocity profile is converted to the one used for the... 

Tbe preload in the hold-down bolts is introduced by means of thermal prestrain. This involves imposing artificial temperature in the bolts sufficiently lower than the assembly temperature to achieve the desired preload. By this method the bolt preload can be monitored very effectively. The temperatures in the reactor vessel support system are obtained using a thermal analysis that employs a model identical to Figure 12.8. The ledge temperature of 66° C is used as a boundary condition in the thermal analysis. The bolt temperature distribution is also evaluated in the thermal analysis. The computed bolt temperature is superimposed onto the fictitious bolt preload temperature. The stresses are evaluated in the stress analysis model using the computed temperatures. 

Although the Navier-Stokes equations are not valid in the Knudsen layer, due to a nonlinear stress/strain-rate behavior in this small layer [4], their use with appropriate boundary velocity slip and temperature jump conditions proved to be accurate for predicting mass flow rates [5] ( methods for flow rate measurements) and velocity profiles out of the Knudsen layer. Classically, the real flow is not simulated within the JCnudsen layer, but the influence of the Knudsen layer on the flow outside this non-equilibrium layer is taken into account, replacing the no-slip condition at the wall with a slip-flow condition. For that purpose, a fictitious slip velocity MsUp is introduced (Fig. 2). Real slip at the wall, gas — wall. is due to the fact that gas molecules very close to the wall have actually a mean... 

The second but fictitious test case is related to a frozen particle being placed in an upward adiabatic low temperature flow as shown in Fig. 10.27. Particles of different size (i.e. between 5 and 500 pm) are considered and their heat-up time is determined. The inflow velocity was adjusted to the terminal velocity of the considered particle size so that the particles are levitated. At the side faces of the computational domain, slip boundary conditions are applied (Fig. 10.27). The particle diameter was always resolved by ten grid nodes and also the heat... 

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