Multidimensional systems, conduction

We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution. 

The basic solutions for the infinite plates and infinitely long cylinders can be used to obtain solutions for multidimensional systems such as long rectangular plates, cuboids, and finite circular cylinders with end cooling. The texts on conduction heat transfer [4,11, 23, 29, 38, 49,56, 87] should be consulted for the proofs of the method and other examples. 

Bubble dynamics in microchannels is of great interest in a number of applications, including microchannel evaporators, high-flux heat removal systems for chip cooling applications, ink-jet printers, atomization nozzles and bubble pumps. The high heat transfer coupled with multidimensional heat conduction in the channel wall, and the rapid growth of the bubble at the microscale make it difficult to verify the analytical models. Future research is needed in understanding the heat transfer mechanism... 

These limitations have recently been eliminated using solid-state sources of femtosecond pulses. Most of the femtosecond dye laser teclmology that was in wide use in the late 1980s [11] has been rendered obsolete by tliree teclmical developments the self-mode-locked Ti-sapphire oscillator [23, 24, 25, 26 and 27], the chirped-pulse, solid-state amplifier (CPA) [28, 29, 30 and 31], and the non-collinearly pumped optical parametric amplifier (OPA) [32, 33 and 34]- Moreover, although a number of investigators still construct home-built systems with narrowly chosen capabilities, it is now possible to obtain versatile, nearly state-of-the-art apparatus of the type described below Ifom commercial sources. Just as home-built NMR spectrometers capable of multidimensional or solid-state spectroscopies were still being home built in the late 1970s and now are almost exclusively based on commercially prepared apparatus, it is reasonable to expect that ultrafast spectroscopy in the next decade will be conducted almost exclusively with apparatus ifom conmiercial sources based around entirely solid-state systems. 

The theoretical work that exploited the advantages of the multidimensional separation format appears to have been developed much later than the original experimental work. One of the earliest studies was conducted by Connors (1974), who assumed that the distribution of spots on a two-dimensional thin-layer chromatography (2DTLC) plate could be modeled using a Poisson distribution of data on each retention axis. He then constructed equations that related the number of chromatographic systems needed to resolve a specific number of compounds. One... 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

In the last section we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Most heat transfer problems encountered iu practice can be approximated as being onedimensional, and we mostly deal with such problems in tliis text. However, this is not always the case, and sometimes we need to consider heat transfer in other directions as well. In such cases heal conduction is said to be multidimensional, and in this section we develop the governing differential equation in such systems in rectangular, cylindrical, and spherical coordinate systems. 

The GAMMA-F code (Lim, 2014) has been developed by KAERI for system and safety analysis of VHTR. The code has the capabilities for multidimensional analyses of the fluid flow and heat conduction as well as the chemical reactions related to the air or steam ingress event in a multicomponent mixture system. As a system thermo-fluid and network simulation code, GAMMA-F includes a nonequilibrium porous media model for pebble-bed and prismatic reactor core, thermal radiation model, point reactor kinetics, and special component models such as pump, circulator, gas turbine, valves, and more. 

Simulation of multidimensional RHT in a participating medium remains so far difficult. Detailed comparison with experimental data [24] shows that the diffusion approximation of RHT via an additional radiative thermal conductivity (for example in Ref [25] to study Cap2 melt growth), does not describe the correct temperature distribution in the growth system. Thus, advanced models, such as the discreet exchange factor method [26] or the characteristics method [27, 28] are needed. 

It is almost impossible to cover the entire range of models in Figure 25.1, and in this chapter we will limit ourselves to the different modeling approaches at the continuum level (micro-macroscopic and system-level simulations). In summary, there are computational models that are developed primarily for the lower-length scales (atomistic and mesoscopic) which do not scale to the system-level. The existing models at the macroscopic or system-level are primarily based on electrical circuit models or simple lD/pseudo-2D models [17-24]. The ID models are limited in their ability to capture spatial variations in permeability or conductivity or to handle the multidimensional structure of recent electrode and solid electrolyte materials. There have been some recent extensions to 2D [29-31], and this is still an active area of development As mentioned in a recent Materials Research Society (MRS) bulletin [6], errors arising from over-simplified macroscopic models are corrected for when the parameters in the model are fitted to real experimental data, and these models have to be improved if they are to be integrated with atomistic... 

SCDAP/RELAP5/M0D3 uses a one-dimensional, two-fluid, nonequilibrium, six-equation hydrodynamic model with a simplified capability to treat multidimensional flows. This model provides continuity, momentum, and energy equations for both the liquid and vapor phases within a control volume. The energy equation contains source terms that couple the hydrodynamic model to the heat structure conduction model by a convective heat transfer formulation. The code contains special process models for critical flow, abrupt area changes, branching, crossflow junctions, pumps, accumulators, valves, core neutronics, and control systems. A flooding model can be applied at vertical junctions. 

---