Heat transfer modeling

The time constants characterizing heat transfer in convection or radiation dominated rotary kilns are readily developed using less general heat-transfer models than that presented herein. These time constants define simple scaling laws which can be used to estimate the effects of fill fraction, kiln diameter, moisture, and rotation rate on the temperatures of the soHds. Criteria can also be estabHshed for estimating the relative importance of radiation and convection. In the following analysis, the kiln wall temperature, and the kiln gas temperature, T, are considered constant. Separate analyses are conducted for dry and wet conditions. 

Comparisons of the complete heat-transfer model with pilot-scale rotary kiln data are shown iu Figure 5 (21) for moisture levels ranging from 0 to 20 wt %. The tremendous thermal impact of moisture is clearly visible iu the leveling of temperature profiles at 100°C. 

D, W. Blair, CombustFlame 20 (1), 105—9 (1973) CA 78, 113515 (1973) A simple heat-transfer model is coupled with an Arrhenius-type pyrolysis law to study the effect of solid-state heat-transfer losses on burning rates of solid rocket-proplnt strands. Such heat-transfer losses materially affect the burning rates and also cause extinction phenomena similar to some that had been observed exptly. Strand diam and compn, adiabatic burning rate, and the heat-transfer film coeff at the strand surface are important variables. Results of theoretical analysis are applied to AP-based composite solid proplnts... 

In Proceedings of 21st SemiTherm Symposium, San Jose, 15-17 March 2005, pp 1-7 Copeland D, Behnia M, Nakayama W (1997) Manifold micro-channel heat sinks isothermal analysis. IEEE Trans Comp Packag Manuf Technol A 20 96-102 Dupont V, Thome JR, Jacobi AM (2004) Heat transfer model for evaporation in microchannels. 

Thome JR, Dupont V, Jacobi AM (2004) Heat transfer model for evaporation in microchannels. 

Reynolds number. It should be stressed that the heat transfer coefficient depends on the character of the wall temperature and the bulk fluid temperature variation along the heated tube wall. It is well known that under certain conditions the use of mean wall and fluid temperatures to calculate the heat transfer coefficient may lead to peculiar behavior of the Nusselt number (see Eckert and Weise 1941 Petukhov 1967 Kays and Crawford 1993). The experimental results of Hetsroni et al. (2004) showed that the use of the heat transfer model based on the assumption of constant heat flux, and linear variation of the bulk temperature of the fluid at low Reynolds number, yield an apparent growth of the Nusselt number with an increase in the Reynolds number, as well as underestimation of this number. 

Analytical analyses for the growth of a single bubble have been performed for simple geometrical shapes, using a simplified heat transfer model. Plesset and Zwick (1954) solved the problem by considering the heat transfer through the bubble interface in a uniformly superheated fluid. The bubble growth equation was obtained... 

Weislogel MM, Lichter S (1998) Capillary flow in an interior corner. 1 Eluid Mech 373 349-378 Wu PY, Little WA (1984) Measurement of the heat transfer characteristics of gas flow a fine channels heat exchangers used for microminiature refrigerators. Cryogenics 24 415 20 Xu X, Carey VP (1990) Film evaporation from a micro-grooved surface an approximate heat transfer model and its comparison with experimental data. J Thermophys 4(4) 512-520 Yarin LP, Ekelchik LA, Hetsroni G (2002) Two-phase laminar flow in a heated micro-channels. Int J Multiphase Flow 28 1589-1616... 

This paper will discuss the formulation of the simulator for the filament winding process which describes the temperature and extent of cure in a cross-section of a composite part. The model consists of two parts the kinetic model to predict the curing kinetics of the polymeric system and the heat transfer model which incorporates the kinetic model. A Galerkin finite element code was written to solve the specially and time dependent system. The program was implemented on a microcomputer to minimize computer costs. 

The heat transfer model, energy and material balance equations plus boundary condition and initial conditions are shown in Figure 4. The energy balance partial differential equation (PDE) (Equation 10) assumes two dimensional axial conduction. Figure 5 illustrates the rectangular cross-section of the composite part. Convective boundary conditions are implemented at the interface between the walls and the polymer matrix. 

Figure 4. Heat transfer model, energy and material balance equations, boundary and initial conditions plus physical properties.

The use of integrated reactor and heat-transfer models is essential for scale-up. Figure 11.7 shows an early reactor design for the same chemistry that was developed without the use of integrated models. Other unoptimized designs with temperature spikes have also been reported [12,44]. Integrated models were used to... 

No slip Is used as the velocity boundary conditions at all walls. Actually there Is a finite normal velocity at the deposition surface, but It Is Insignificant In the case of dilute reactants. The Inlet flow Is assumed to be Polseullle flow while zero stresses are specified at the reactor exit. The boundary conditions for the temperature play a central role in CVD reactor behavior. Here we employ Idealized boundary conditions In the absence of detailed heat transfer modelling of an actual reactor. Two wall conditions will be considered (1) adiabatic side walls, l.e. dT/dn = 0, and (11) fixed side wall temperatures corresponding to cooled reactor walls. For the reactive species, no net normal flux Is specified on nonreacting surfaces. At substrate surface, the flux of the Tth species equals the rate of reaction of 1 In n surface reactions, l.e. 

A boiling heat transfer model incorporating nucleate boiling, natural convection, and microlayer evaporation was formulated as... 

ORNL small-break LOCA tests Experimental investigation of heat transfer and reflood analysis was made under conditions similar to those expected in a small-break LOCA. These tests were performed in a large, high-pressure, electrically heated test loop of the ORNL Thermal Hydraulic Test Facility. The analysis utilized a heat transfer model that accounts for forced convection and thermal radiation to steam. The results consist of a high-pressure, high-temperature database of experimental heat transfer coefficients and local fluid conditions. 

Mahalingan, M., and Kolar, A. K., Heat Transfer Model for Membrane Wall of a High Temperature Circulating Fluidized Bed, Circ. Fluid. Bed Tech. Ill, 239-246 (1990)... 

Available analytical methods for determining the fire resistance (or fire endurance) of wood members have been reviewed by White (8). A finite-element heat transfer model for wood-frame walls was developed by the University of California-Berkeley (9) with funding from the FPL. 

Ge and Fan (2005) developed a 3-D numerical model based on the level-set method and finite-volume technique to simulate the saturated droplet impact on a superheated flat surface. A 2-D vapor-flow model was coupled with the heat-transfer model to account for the vapor-flow dynamics caused by the Leidenfrost evaporation. The droplet is assumed to be spherical before the collision and the liquid is assumed to be incompressible. 

During the subcooled droplet impact, the droplet temperature will undergo significant changes due to heat transfer from the hot surface. As the liquid properties such as density p (T), viscosity /q(7), and surface tension a(T) vary with the local temperature T, the local liquid properties can be quantified once the local temperature can be accounted for. The droplet temperature is simulated by the following heat-transfer model and vapor-layer model. Since the liquid temperature changes from its initial temperature (usually room temperature) to the saturated temperature of the liquid during the impact, the linear... 

The heat-transfer model described in Sections IV.A.3 and IV.B.2 can be applied to calculate the temperature distribution inside the droplet and energy... 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

At this stage it seemed clear that to improve near-wall heat transfer modeling would require better representation of the near-wall flow field, and how it was connected to bed structure and wall heat transfer rates. Our early models of full beds of spheres at N — 4 were too large for our computational capacity when meshed at the refinement that we anticipated to be necessary for the detailed flow fields that we wanted. We therefore developed the WS approach described above in Section II.B.3. 

For an alloy droplet, the post-recalescence solidification involves segregated solidification and eutectic solidification. 619 Droplet cooling in the region (1),(2) and (6) can be calculated directly with the above-described heat transfer model. The nucleation temperature (the achievable undercooling) and the solid fraction evolution during recalescence and post-recalescence solidification need to be determined additionally on the basis of the rapid solidification kinetics. 154 156 ... 

With the above-described heat transfer model and rapid solidification kinetic model, along with the related process parameters and thermophysical properties of atomization gases (Tables 2.6 and 2.7) and metals/alloys (Tables 2.8,2.9,2.10 and 2.11), the 2-D distributions of transient droplet temperatures, cooling rates, achievable undercoolings, and solid fractions in the spray can be calculated, once the initial droplet sizes, temperatures, and velocities are established by the modeling of the atomization stage, as discussed in the previous subsection. For the implementation of the heat transfer model and the rapid solidification kinetic model, finite difference methods or finite element methods may be used. To characterize the entire size distribution of droplets, some specific droplet sizes (forexample,.D0 16,Z>05, andZ)0 84) are to be considered in the calculations of the 2-D motion, cooling and solidification histories. 

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