Three-dimensional heat transfer

Chapter 4 is devoted to single-phase heat transfer. Data on heat transfer in circular micro-tubes and in rectangular, trapezoidal and triangular ducts are presented. Attention is drawn to the effect of energy dissipation, axial conduction and wall roughness on the thermal characteristics of flow. Specific problems connected with electro-osmotic heat transfer in micro-channels, three-dimensional heat transfer in micro-channel heat sinks and optimization of micro-heat exchangers are also discussed. 

Three-Dimensional Heat Transfer in Micro-Channel Heat Sinks... 

Qu W, Mudawar I (2002) Analysis of three-dimensional heat transfer in micro-channel heat sinks. Int J Heat Mass Transfer 45 3973-3985... 

When tackling a two- or three-dimensional heat-transfer problem first try to reduce it to a one-dimensional problem. An example is a cylinder with length much larger than its diameter. 

The heat produced in this manner is transferred to the surrounding explosive material. The heat transfer rate is dependent upon temperature as well as thermal conductivity, heat capacity, and density. One of the classical three-dimensional heat transfer equations that relates the rate of heat production to the rate of temperature rise of the reacting material and to its surroundings is the Frank-Kamenetskii (FAT) equation (Ref 1). 

Qu, W. and Mudawar, 1., Analysis of Three-Dimensional Heat Transfer in Micro-Channel Heat Sinks, Int. J. Heat Mass Transfer, 2002, 45, 3973-3985. 

X. G Xu, Mathematical Modeling of Three-Dimensional Heat Transfer from the Flame in Combustion Chambers, 18th Symposium (International) on Combustion, the Combustion Institute, pp. 1919-1925,1981. 

The three-dimensional heat transfer within the concrete is governed by the differential equation ... 

The thermal growth of fast reactions in exothermic systems is described quantitatively by a three-dimensional heat-transfer equation which includes a factor for the rate of evolution of heat (taken for simplicity to have an Arrhenius dependence) ... 

The sausages were considered as infinite cylinders and thermal diffusivity was calculated equal to 3.846 x 10" m s". Constant thermal diffusivity and homogeneity of the food were also the simplifying assumptions necessary to develop a model to predict internal temperature profile in chicken pieces fried under pressure [60]. For the chicken pieces a three-dimensional heat transfer was considered and the predicted results were closer to experimental ones obtained at the center of the pieces, than those obtained near the boundaries. Better simulation was also obtained for more regularly shaped pieces. 

Three-dimensional heat transfer analysis can be performed using a computational program such as Calore. It is built upon the SIERRA finite element framework to run on a computer in a manner that relates mesh management, external field... 

Equation (6-45) may be solved numerically for a variety of boundary conditions, such as constant T at the walls, or constant or prescribed beat flux. Because of complicated three-dimensional heat transfer pathways (shown in Figure 6-23), calculation of heat fluxes and temperature profiles in a fuel cell stack requires 3-D numerical simulation. Figure 6-25 shows temperature distribution in a representative cross-section of a fuel cell obtained by 3-D numerical simulation [28]. From Figure 6-25, it is obvious that there are significant temperature variations inside a fuel cell stack. Because most heat in a fuel cell stack is produced in the cathode catalyst layer, that layer expectedly has the highest temperature. 

Figure 127. Example of a real, three-dimensional heat storage. 4 flat plate heat exchangers connected exchange heat between the storage medium and the heat transfer fluid (picture ZAE Bayern)...

The ihennal resistance concept or the electrical analogy can also be used to solve steady heat transfer problems that involve parallel layers or combined series-parallel arrangements. Although such problems are often two- or even three-dimensional, approximate solutions can be obtained by assuming one-dimensional heat transfer and using the Ihennal resistance network. 

The stability criterion that requires the coefficient of 7), in the Tjf expression to be greater than or equal to zero for all nodes is equally valid for two-or three-dimensional cases and severely limits the size of the lime step Ar that can be used with the explicit method. In the case of transient two-dimensional heat transfer in rectangular coordinates, the coefficient of 7)j, in the I if expression is 1 4t, and thus the stability criterion for all interior nodes in this... 

Mat M. D, Kaplan Y, Aldas K. Investigation of Three-Dimensional Heat and Mass Transfer in a Metal Hydride Reactor. Int. J. Energy Research, 2002 26 973-986. 

In the experimental study by Zhu et al. (1998), the heating pattern induced by a microwave antenna was quantified by solving the inverse problem of heat conduction in a tissue equivalent gel. In this approach, detailed temperature distribution in the gel is required and predicted by solving a two- or three-dimensional heat conduction equation in the gel. In the experimental study, all the temperature probes were not required to be placed in the near field of the catheter. Experiments were first performed in the gel to measure the temperature elevation induced by the applicator. An expression with several unknown parameters was proposed for the SAR distribution. Then, a theoretical heat transfer model was developed with appropriate boundary conditions and initial condition of the experiment to study the temperature distribution in the gel. The values of those unknown parameters in the proposed SAR expression were initially assumed and the temperatiue field in the gel was calculated by the model. The parameters were then adjusted to minimize the square error of the deviations theoretically predict from the experimentally measured temperatures at all temperature sensor locations. 

In microchannels, conjugate heat transfer leads to a complex three-dimensional heat flow pattern and Poiseuille flow may no longer be accurate [43]. Numerical simulations show that axial conduction in the channel wall does lower the Nusselt number but it is still in the range of conventional values [38]. The work of Gamrat et al. [44], in contrast, could not explain the lower Nusselt number by the axial conduction in the channel walls by numerical simulations. 

Fig. 1. The postulated flame stmcture for an AP composite propellant, showing A, the primary flame, where gases are from AP decomposition and fuel pyrolysis, the temperature is presumably the propellant flame temperature, and heat transfer is three-dimensional followed by B, the final diffusion flame, where gases are O2 from the AP flame reacting with products from fuel pyrolysis, the temperature is the propellant flame temperature, and heat transfer is three-dimensional and C, the AP monopropellant flame where gases are products from the AP surface decomposition, the temperature is the adiabatic flame temperature for pure AP, and heat transfer is approximately one-dimensional. AP = ammonium perchlorate.

This implies that the LMTD or M I D as computed in equations 20 through 26 may not be a representative temperature difference between the two heat-transferring fluids for aU tubes. The effective LMTD or M ID would be smaller than the value calculated, and consequentiy would require additional heat-transfer area. The tme value of the effective M I D may be determined by two- or three-dimensional thermal—hydrauUc analysis of the tube bundle. Baffle—Tube Support PlateXirea. The portion of a heat-transfer tube that passes through the flow baffle—tube support plates is usuaUy considered inactive from a heat-transfer standpoint. However, this inactive area must be included in the determination of the total length of the heat-transfer tube. 

Patankar, S. V., and D. B. Spalding. 1972. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat and Mass Transfer. 15 1787-1806. 

Advanced two- and three-dimensional computer analysis methods are used today in the analyses of all critical components to verify aerodynamic, heat transfer, and mechanical performance. Additionally, the reduction of leakage paths in the compressor, as well as in the gas turbine expander, results in further plant efficiency improvements. At the compressor inlet, an advanced inlet flow design improves efficiency by reducing pressure loss. Rotor air cooler heat utilization and adt anccd blade and vane cooling arc also used. 

Heat transfer in the furnace is mainly by radiation, from the incandescent particles in the flame and from hot radiating gases such as carbon dioxide and water vapor. The detailed theoretical prediction of overall radiation exchange is complicated by a number of factors such as carbon particle and dust distributions, and temperature variations in three-dimensional mixing. This is overcome by the use of simplified mathematical models or empirical relationships in various fields of application. 

One particular characteristic of conduction heat transfer in micro-channel heat sinks is the strong three-dimensional character of the phenomenon. The smaller the hydraulic diameter, the more important the coupling between wall and bulk fluid temperatures, because the heat transfer coefficient becomes high. Even though the thermal wall boundary conditions at the inlet and outlet of the solid wall are adiabatic, for small Reynolds numbers the heat flux can become strongly non-uniform most of the flux is transferred to the fluid at the entrance of the micro-channel. Maranzana et al. (2004) analyzed this type of problem and proposed the model of channel flow heat transfer between parallel plates. The geometry shown in Fig. 4.15 corresponds to a flow between parallel plates, the uniform heat flux is imposed on the upper face of block 1 the lower face of block 0 and the side faces of both blocks... 

Kroeker CJ, Soliman HM, Ormiston SJ (2004) Three-dimensional thermal analysis of heat sinks with circular cooling micro-channels. Int J Heat Mass Transfer 47 4733 744 Lee PS, Garimella SV, Liu D (2005) Investigation of heat transfer in rectangular micro-channels. Int J Heat Mass Transfer 48 1688-1704... 

Lelea D, Nishio S, Takano K (2004) The experimental research on micro-tube heat transfer and fluid flow of distilled water. Int J Heat Mass Transfer 47 2817-2830 Li J, Peterson GP, Cheng P (2004) Three-dimensional analysis of heat transfer in a micro-heat sink with single phase flow. Int J Heat Mass Transfer 47 4215-4231 Lin TY, Yang CY (2007) An experimental investigation by method of fluid crystal thermography. Int. J. Heat Mass Transfer 50(23-24) 4736-4742... 

The combined fiuld fiow, heat transfer, mass transfer and reaction problem, described by Equations 2-7, lead to three-dimensional, nonlinear, time dependent partial differential equations. The general numerical solution of these goes beyond the memory and speed capabilities of the current generation of supercomputers. Therefore, we consider appropriate physical assumptions to reduce the computations. 

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