One-Dimensional Heat Conduction Equation

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

ONE-DIMENSIONAL HEAT CONDUCTION EQUATION WITH CONSTANT COEFFICIENTS... 

In this section we consider the one-dimensional heat conduction equation with constant coefficients and difference schemes in order to develop various methods for designing the appropriate difference schemes in the case of time-dependent problems. 

Recall that in the case of the one-dimensional heat conduction equation a similar implicit scheme is associated on every layer with the difference... 

Here Go(a ,t) is a function of the heat source of the Cauchy problem associated with the one-dimensional heat conduction equation... 

The general methodology provides proper guidelines for the selection rules in studying one-dimensional heat conduction equations... 

This is the one-dimensional heat-conduction equation. To treat more than onedimensional heat flow, we need consider only the heat conducted in and out of a unit volume in all three coordinate directions, as shown in Fig. l-3a. The energy balance yields... 

Consider an insulated composite rod, which is formed of two parts of equal length. The thermal conductivities of parts a and b are a and kh, for 0 < x < 1/2. The nondimensional transient, one-dimensional heat conduction equations over the length x of the rod are expressed by the following equations... 

In the first approach, the temperature-time profiles of the combustion wave are measured using thin thermocouples (Zenin et al, 1980 Dunmead et al, 1992a,b,c). In the work of Zenin et al (1980), 7-/im-diameter microthermocouples protected by a layer of boron nitride were used. The temperature-time profiles were analyzed using the one-dimensional heat conduction equation with heat generation (see Section IV,A, 1) taking numerical derivatives of the spatial temperature distribution. An implicit assumption in the analysis is that the combustion wave is stable and planar at both the macro- and microscopic scales. 

FIGURE 2-51 Basic steps involved in the soluiion of the steady one-dimensional heat conduction equation in cylindrical coordinates. 

Transient temperature distribution during quenching of cylindrical steel (about 0.45 percent carbon) test specimens have been predicted [204] by solving numerically the transient, one-dimensional heat conduction equation... 

One-dimensional heat conduction equation with constant coefficients... 

Here we assume photothermal mechanism is responsible even for 248 nm excitation, and estimate numerically temperature rise and decay dynamics. As large excitation energy is confined at toe thin surface layers of 27 nm and 830 nm thickness for 248 nm and 351 nm excitation, respectively, assuming Lambert-Beer law hold, temperature elevation is extremely large. Here the surface of the 20 nm thickness is numerically examined for both excitation wavelengtos. Surface temperature rise as a function of delay time was calculated by solving one-dimensional heat conduction equation. In calculation it was... 

Examples of One-Dimensional Heat Conduction Equation 22.3.3.1 Large Slab with Internal Heat Generation... 

The initial stage of vaporization is usually controlled by the heat transfer from the groimd. This is especially true for a spill of liquid with a normal boiling point below ambient temperature or ground temperature (i.e., boiling liquid). The heat transfer from the ground is modeled with a simple one-dimensional heat conduction equation given by... 

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