Temperature field

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. 

Many authors have shown that residual stresses in glass articles can be formally considered as the thermal stresses due to a certain fictitious temperature field. 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]. 

Step 2 an initial configuration representing the partially filled discretized domain is considered and an array consisting of the appropriate values of F - 1, 0.5 and 0 for nodes containing fluid, free surface boundary and air, respectively, is prepared. The sets of initial values for the nodal velocity, pressure and temperature fields in the solution domain are assumed and stored as input arrays. An array containing the boundary conditions along the external boundaries of the solution domain is prepared and stored. 

As comparison of the simulated temperature fields shows, fluid slippage results in temperature peak shifting from a location furthest away from the... 

OUTPUT Prints out the computed velocity and temperature fields. For postprocessing the user should store the output in suitable files... 

Figures 7.3 and 7.4 show, respectively, the computed velocity and temperature fields, generated by PPVN.f for this example.

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. 

With contact temperature measurement, placing the measurement probe in contact with the object of measurement (duct, surface, etc.) produces an additional route for heat conduction to or from the object. This perturbation error changes the initial temperature field in the vicinity of the contact point and creates measurement errors. 

Here U = T — T )Cp/L is the appropriately rescaled temperature field T measured from the imposed temperature of the undercooled melt far away from the interface. The indices L and 5 refer to the liquid and solid, respectively, and the specific heat Cp and the thermal diffusion constant D are considered to be the same in both phases. L is the latent heat, and n is the normal to the interface. In terms of these parameters,... 

Reliable micro-scale measurement and control of the temperature are required in developing thermal micro-devices. Available measurement techniques can be largely classified into contact and non-contact groups. While the resistance thermometer, thermocouples, thermodiodes, and thermotransistors measure temperature at specific points in contact with them, infrared thermography, thermochromic liquid crystals (TLC), and temperature-sensitive fluorescent dyes cover the whole temperature field (Yoo 2006). 

In the thermochromic liquid crystal (TLC) the dominant reflected wavelength is temperature-dependent and it has been employed for full-field mapping of temperature fields for over three decades. Although it is non-intrusive and cost effective, there are some problems in applying it to micro-scale measurements, because of size (typically tens of micrometers) and time response (from a few milliseconds to several hundred milliseconds depending on the material and the form). Examples of application are micro-fabricated systems (Chaudhari et al. 1998 Liu et al. 2002) and electronic components (Azar et al. 1991). 

Fig. 2.64a,b Temperature field on heater. Reprinted from Mishan et al. (2007) with permission... 

Fig. 2.65 Comparison between numerical simulation and experimental results for the temperature field on the heater surface. The solid line represents simulation, and triangles (A) experimental results (line 1 in Fig. 2.64) dotted line represents simulation, and squares experimental results (line 2 in Fig. 2.64). Reprinted from Mishan et al. (2007) with permission...

Qu et al. (2000) carried out experiments on heat transfer for water flow at 100 < Re < 1,450 in trapezoidal silicon micro-channels, with the hydraulic diameter ranging from 62.3 to 168.9pm. The dimensions are presented in Table 4.5. A numerical analysis was also carried out by solving a conjugate heat transfer problem involving simultaneous determination of the temperature field in both the solid and fluid regions. It was found that the experimentally determined Nusselt number in micro-channels is lower than that predicted by numerical analysis. A roughness-viscosity model was applied to interpret the experimental results. 

A detailed study of the influence of viscous heating on the temperature field in micro-channels of different geometries (rectangular, trapezoidal, double-trapezoidal) has been performed by Morini (2005). The momentum and energy conservation equations for flow of an incompressible Newtonian fluid were used to estimate... 

Heat transfer in micro-channels occurs under superposition of hydrodynamic and thermal effects, determining the main characteristics of this process. Experimental study of the heat transfer in micro-channels is problematic because of their small size, which makes a direct diagnostics of temperature field in the fluid and the wall difficult. Certain information on mechanisms of this phenomenon can be obtained by analysis of the experimental data, in particular, by comparison of measurements with predictions that are based on several models of heat transfer in circular, rectangular and trapezoidal micro-channels. This approach makes it possible to estimate the applicability of the conventional theory, and the correctness of several hypotheses related to the mechanism of heat transfer. It is possible to reveal the effects of the Reynolds number, axial conduction, energy dissipation, heat losses to the environment, etc., on the heat transfer. 

Parameters 7c,onb, s.onb, and 74,onb change in the range of 0 < 7 < 1. They account for a specific temperature field in heated micro-channels and are criteria for the relative micro-channel length. Note, if 7 < 1 the value of parameter 7 is significantly less than unity. The paper by Celata et al. (1997) reports the results of experimental research of the onset of subcooled water boiling in the circular... 

The bubble dynamics in a confined space, in particular in micro-channels, is quite different from that in infinity still fluid. In micro-channels the bubble evolution depends on a number of different factors such as existence of solid walls restricting bubble expansion in the transversal direction, a large gradient of the velocity and temperature field, etc. Some of these problems were discussed by Kandlikar (2002), Dhir (1998), and Peng et al. (1997). A detailed experimental study of bubble dynamics in a single and two parallel micro-channels was performed by Lee et al. (2004) and Li et al. (2004). 

ReveUin R, Thome J. (2008) A theoretical model for the prediction of the critical hat flux in heated micro-channel. Int. J. Heat and Mass Transfer 51 1216-1225 Roach GM, Abdel-Khahk SI, Ghiaasiaan SM, Dowling MF, Jeter SM (1999) Low-flow critical heat flux in heated microchannels. Nucl Sd Eng 131 411 25 Robinson AJ, Judd RL (2001) Bubble growth in a uniform and spatially distributed temperature field. Int J Heat Mass Transfer 44 2699-2710... 

Zuber N (1961) The dynamics of vapor bubbles in non-uniform temperature fields. Int J Heat Mass Transfer 2 83-98... 

The flow in a heated capillary depends on a number of parameters including the channel geometry, physical properties of the liquid and the heat flux. An immediate consequence of the liquid heating and evaporation is convective motion of both phases. The latter leads to a velocity and temperature field fransformation and a change in fhe meniscus shape. 

The temperature distribution in the capillary slot is presented in Fig. 10.18. These data show the wall superheat influence on temperature fields in liquid and vapor domains. In these cases, significant heterogeneity of temperature fields is observed. 

Numerical Solution. The momentum Equation 5 is solved simultaneously along with the energy Equation 6 to obtain axial velocity, v, and temperature fields. The continuity equation with the known axial velocity is used to... 

The temperature field for the polymer melt can be determined using an energy balance which includes terms for convection, conduction and viscous dissipation. 

Figure 3 Temperature field triangulation at the inlet using 64 triangles. The dotted line indicates the polymer-metal interface, and the dimensions are in centimeters.

PIV velocity measurements made it possible to evaluate the flame temperature field [23], following the method demonstrated in Ref. [25]. The calculated thermal structure of lean limit methane flame is shown in Figure 3.1.7. The differences between the structures of lean limit methane and propane flames are fundamental. The most striking phenomenon seen from Figure 3.1.7 is the low temperature in the stagnation zone (the calculated temperatures near the tube axis seem unrealistically low, probably due to very low gas velocities in the stagnation core). 

The calculated flow and temperature fields for a flame propagating in a channel with the walls separated by Dq = 4 mm is shown in Figures 6.1.9 and 6.1.10. 

Numerical simulation of the temperature field during flame propagation in a channel with Dq = 4mm = 0.73). 

From the 2D instantaneous Rayleigh temperature fields, such as shown in Figure 7.1.8, isotemperature contours can be obtained and they clearly show that the distance between the isothermal contours strongly varies at different locations, being deeply perturbed by turbulence, especially on the fresh reactants side. 

Perturbed flamelet structure as obtained from 2D instantaneous Rayleigh temperature fields for the case FI (jet exit velocity is 65m/s). (Reproduced from Dumont, J.P., Durox, D., and Borghi, R., Combust. Sci. Tech., 89,219,1993. With permission. Figure 3.1, p. 233, copyright Gordon Breach Science Publishers (Taylor and Francis editions).)... 

A bluff-body stabilized flame of CH4/H2 in air (designated HMl by Dally et al. [22]) (a) time-averaged photograph of flame luminosity, (b) time-averaged streamlines from LES, (c) instantaneous visualization of OH "luminosity" from LES, and (d) instantaneous temperature field from LES. (b and d are adapted from Raman, V. and Pitch, H., Combust. Flame, 142,329,2005. With permission.)... 

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