Continuum heat transfer models

Previous one-phase continuum heat transfer models (1), (5), (10), (11), which are all based upon "large diameter tube" heat transfer data, fail to extrapolate to narrow diameter tubes. These equations systematically underpredict the overall heat transfer coefficient by 40 - 50%, on average. When allowance is made in the one-phase model for the effect of tube diameter on the apparent solid conductivity (kr>s), Eqn. (7), the mean error is reduced to 18%. However, the best predictions by far (to within 6.8% mean error) are obtained from the heterogeneous model equations. 

Homogeneous Continuum Heat Transfer Models We shall confine our study to homogeneous continuum models... 

The combined CFD-DEM approach, incorporated with heat transfer models of convection, conduction, and radiation, has been developed and can be used in the study of heat transfer in packed and fluidized beds. It has various advantages over the conventional experimental techniques and continuum simulation approaches. For example, the detailed conductive heat transfer between particles can be examined and the factors such as local particle-fluid structure and materials properties in determining heat transfer can be investigated. 

Contact Drying. Contact drying occurs when wet material contacts a warm surface in an indirect-heat dryer (15—18). A sphere resting on a flat heated surface is a simple model. The heat-transfer mechanisms across the gap between the surface and the sphere are conduction and radiation. Conduction heat transfer is calculated, approximately, by recognizing that the effective conductivity of a gas approaches 0, as the gap width approaches 0. The gas is no longer a continuum and the rarified gas effect is accounted for in a formula that also defines the conduction heat-transfer coefficient ... 

Steady state models of the automobile catalytic converter have been reported in the literature 138), but only a dynamic model can do justice to the demands of an urban car. The central importance of the transient thermal behavior of the reactor was pointed out by Vardi and Biller, who made a model of the pellet bed without chemical reactions as a onedimensional continuum 139). The gas and the solid are assumed to have different temperatures, with heat transfer between the phases. The equations of heat balance are ... 

Pressure drop and heat transfer in a single-phase incompressible flow. According to conventional theory, continuum-based models for channels should apply as long as the Knudsen number is lower than 0.01. For air at atmospheric pressure, Kn is typically lower than 0.01 for channels with hydraulic diameters greater than 7 pm. From descriptions of much research, it is clear that there is a great amount of variation in the results that have been obtained. It was not clear whether the differences between measured and predicted values were due to determined phenomenon or due to errors and uncertainties in the reported data. The reasons why some experimental investigations of micro-channel flow and heat transfer have discrepancies between standard models and measurements will be discussed in the next chapters. 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

The application of CFD to packed bed reactor modeling has usually involved the replacement of the actual packing structure with an effective continuum (Kvamsdal et al., 1999 Pedernera et al., 2003). Transport processes are then represented by lumped parameters for dispersion and heat transfer (Jakobsen... 

The inclusion of radiative heat transfer effects can be accommodated by the stagnant layer model. However, this can only be done if a priori we can prescribe or calculate these effects. The complications of radiative heat transfer in flames is illustrated in Figure 9.12. This illustration is only schematic and does not represent the spectral and continuum effects fully. A more complete overview on radiative heat transfer in flame can be found in Tien, Lee and Stretton [12]. In Figure 9.12, the heat fluxes are presented as incident (to a sensor at T,, ) and absorbed (at TV) at the surface. Any attempt to discriminate further for the radiant heating would prove tedious and pedantic. It should be clear from heat transfer principles that we have effects of surface and gas phase radiative emittance, reflectance, absorptance and transmittance. These are complicated by the spectral character of the radiation, the soot and combustion product temperature and concentration distributions, and the decomposition of the surface. Reasonable approximations that serve to simplify are ... 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

Therefore, an attempt was made to determine the kinetic reaction scheme and effective heat transfer as well as kinetic parameters from a limited number of experimental results in a single-tube reactor of industrial dimensions with side-stream analysis. The data evaluation was performed with a pseudohomo-geneous two-dimensional continuum model without axial dispersion. The model was tested for its suitability for prediction. 

Extensive experimental determinations of overall heat transfer coefficients over packed reactor tubes suitable for selective oxidation are presented. The scope of the experiments covers the effects of tube diameter, coolant temperature, air mass velocity, packing size, shape and thermal conductivity. Various predictive models of heat transfer in packed beds are tested with the data. The best results (to within 10%) are obtained from a recently developed two-phase continuum model, incorporating combined conduction, convection and radiation, the latter being found to be significant under commercial operating conditions. 

Mechanistic equations describing the apparent radial thermal conductivity (kr>eff) and the wall heat transfer coefficient (hw.eff) of packed beds under non-reactive conditions are presented in Table IV. Given the two separate radial heat transfer resistances -that of the "central core" and of the "wall-region"- the overall radial resistance can be obtained for use in one-dimensional continuum reactor models. The equations are based on the two-phase continuum model of heat transfer (3). 

The overall heat transfer coefficient U in Eqn. (3) is based on the measured temperature difference between the central axis of the bed and the coolant. It is derived by asymptotic matching of thermal fluxes between the one-dimensional (U) and two-dimensional (kr,eff kw,eff) continuum models of heat transfer. Existing correlations are employed to describe the underlying heat transfer processes with the exception of Eqn. (7), which is a new result for the apparent solid phase conductivity (k g), including the effect of the tube wall. Its derivation is based on an analysis of stagnant bed conductivity data (8, 9), accounting for "central-core" and wall thermal resistances. 

In this paper the coupled elliptic partial differential equations arising from a two-phase homogeneous continuum model of heat transfer in a packed bed are solved, and some attempt is made to discriminate between rival correlations for those parameters not yet well-established, by means of a comparison with experimental results from a previous study (, 4). 

Bennon WD, Incropera FP (1987) A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems-II. Application to solidification in a regular cavity. Int J Heat Transfer 30(10) 2171-2187... 

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

Both radial and axial diffusion can be taken into account and the final equations to be solved are relatively simpler than those of the continuum model. Although, the equations of the model at steady state are algebraic equations, the dimensionality of the system increases considerably. McGuire and Lapidus (1965) used this model for the study of the stability of a packed bed reactor which included both interphase and intraparticle mass and heat transfer resistances. 

Specifically, theoretical equations and correlations of data are presented for evaluating the local rate of heat transfer between the surface of a body and an encompassing fluid at different temperatures and in relative motion. Forced convection requires either that the fluid be pumped past the body, as for a model in a wind tunnel, or the body be propelled through the fluid, as an aircraft in the atmosphere. The methods presented apply equally to either situation when velocities are expressed relative to the body. Gravity forces are usually negligible under these conditions. Further, the contents of this chapter are confined to those conditions where the fluid behaves as a continuum. 

Summary. In conclusion, some suggestions are made on how to model the problem of radiative heat transfer in porous media. First, we must choose between a direct simulation and a continuum treatment. Wherever possible, continuum treatment should be used because of the lower cost of computation. However, the volume-averaged radiative properties may not be available in which case continuum treatment cannot be used. Except for the Monte Carlo techniques for large particles, direct simulation techniques have not been developed to solve but the simplest of problems. However, direct simulation techniques should be used in case the number of particles is too small to justify the use of a continuum treatment and as a tool to verify dependent scattering models. 

Rarefied analyses of thermal coupling between the micro-nozzle with the surrounding substrate have also been reported [10]. In this work, steady low Reynolds number gas flows were again modeled by the DSMC approach, and the substrate transient thermal response was governed by the heat conduction equation. It was shown that propulsive efficiencies of the micronozzle decreased with higher nozzle wall temperatures and vice versa. These results are in agreement with the continuum-based heat transfer results previously discussed. 

In this chapter we start with fundamental aspects of local blood tissue thermal interaction. Discussions on how the blood effect is modeled then follow. Different approaches to theoretically modeling the blood flow in the tissue are shown. In particular the assumptions and validity of several widely used continuum bioheat transfer equations are evaluated. Different techniques to measure temperature, thermophysical properties, and blood flow are then described. The final part of the chapter focuses on one of the medical applications of heat transfer, hyperthermia treatment for tumors. 

In continuum models, blood vessels are not modeled individually. Instead, the traditional heat conduction equation for the tissue region is modified by either adding an additional term or altering some of the key parameters. The modification is relatively simple and is closely related to the local vasculature and blood perfusion. Even if the continuum models cannot describe the point-by-point temperature variation in the vicinity of larger blood vessels, they are easy to use and allow the manipulation of one or several free parameters. Thus, they have much wider applications than the vascular models. In the following sections, some of the widely used continuum models are introduced and their validity is evaluated on the basis of the fundamental heat transfer aspects. 

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