Shape effects thermal conductivity

Thiele(I4>, who predicted how in-pore diffusion would influence chemical reaction rates, employed a geometric model with isotropic properties. Both the effective diffusivity and the effective thermal conductivity are independent of position for such a model. Although idealised geometric shapes are used to depict the situation within a particle such models, as we shall see later, are quite good approximations to practical catalyst pellets. 

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. 

Besides these models there are many others models, but no single model explains the effective thermal conductivity in all cases. Besides the thermal conductivities of the base fluid and nanoparticles and the volume fraction of the particles, there are many other factors influencing the effective thermal conductivity of the nanofluids. Some of these factors are the size and shape of nanoparticles, the agglomeration of particle, the mode of preparation of nanofluids, the degree of purity of the particles, surface resistance between the particles and the fluid. Some of these factors may not be predicted adequately and may be changing with time. This situation emphasizes the importance of having experimental results for each special nanofluid produced. 

Figure 23.3. Gas pressure dependence of the effective thermal conductivity of the pore gas nitrogen in relation to the thermal conductivity of the free gas according to (23.10) for different pore diameters D and II = 1. Typical s-shaped curves can be observed in the linear-log representation.

Radiation in the transmitting and emitting media of the molten glass is a significant contributor to the overall heat transfer and cannot be accurately simulated by an effective thermal conductivity at this small size scale. The addition of a full discrete ordinates (DO) treatment of the radiative heat transfer should also improve the accuracy of the free surface shape development, but at a very high cost for simulation speed and resources required. 

Other models take into consideration the effects of shape, size, and interfacial resistance on thermal conductivity. However, these models are unable to predict the effective thermal conductivity accurately if contact among filler particles exists. The Cheng and Vachon (Tavman 2003) model assumes a parabolic distribution of disperse phase (spheres or fibers) in a solid matrix. When k, > kp, thermal conductivity of the polymer composite is given by equation (11.7) ... 

The Lewis-Nielsen model considers the effect of the shape of the filler and the orientation or type of packing for a two-phase system or single-phase reinforcement composite, resulting in equation (11.8) for effective thermal conductivity (Tavman... 

Tekce H. Serkan, Kumlutas Dilek, and Tavman Ismail Hakki. Effect of particle shape on thermal conductivity of copper reinforced polymer composites. J. Reinforced Plast. Compos. 26 no. 1 (2007) 113—121. 

Curves of Figure 19 compare the data published for (a) boron nitride [37,40] (b) aluminium (c) diamond-[37-39] (d) aluminium nitride [37-42] (e) crystalline silica. It can be seen that, at 45 vol.%, the maximum thermal conductivity achieved with diamond powder is 1.5 W m K, while crystalline boron nitride at 35 vol.% affords 2.0Wm K. The thermal conductivity of silver-filled adhesives was studied by using silicon test chips attached to copper and molybdenum substrates [43]. The authors outline the importance of the shape factor A, related to the aspect ratio of the particles, to achieve the highest level of thermal conductivity. Another study reports the variation of the effective thermal resistance, between a test chip and the chip carrier, in relation to the volume fraction of silver and the thickness of the bond layer [44]. The ultimate value of bulk thermal conductivity is 2 W m at 25 vol.% silver. However, the effective thermal conductivity, calculated from the thermal resistance measurements, is only one-fifth of the bulk value when the silicon chip is bonded to a copper substrate. 

The representation of our experimental data in terms of an effective thermal conductivity rather than an effective emissivity is justified by the results. The temperature distribution through a sample has the shape that would be predicted... 

Powder Insulation A method of reahzing some of the benefits of multiple floating shields without incurring the difficulties of awkward structural complexities is to use evacuated powder insulation. The penalty incurred in the use of this type of insulation, however, is a tenfold reduction in the overall thermal effectiveness of the insulation system over that obtained for multilayer insulation. In applications where this is not a serious factor, such as LNG storage facihties, and investment cost is of major concern, even unevacuated powder-insulation systems have found useful apphcations. The variation in apparent mean thermal conductivity of several powders as a function of interstitial gas pressure is shown in the familiar S-shaped curves of Fig. 11-121. ... 

Effectively, Eqs. (86) and (87) describe two interpenetrating continua which are thermally coupled. The value of the heat transfer coefficient a depends on the specific shape of the channels considered suitable correlations have been determined for circular or for rectangular channels [100]. In general, the temperature fields obtained from Eqs. (86) and (87) for the solid and the fluid phases are different, in contrast to the assumptions made in most other models for heat transfer in porous media [117]. Kim et al. [118] have used a model similar to that described here to compute the temperature distribution in a micro channel heat sink. They considered various values of the channel width (expressed in dimensionless form as the Darcy number) and various ratios of the solid and fluid thermal conductivity and determined the regimes where major deviations of the fluid temperature from the solid temperature are found. 

Several other more complex dependences accounting for the shape of grains and pores have been proposed. However, it should be pointed out that in most of these models the disperse phase consists of solid particles, while the dispersion phase is gas in the form of pores. Such models do not correspond to the process for they do not account for the effect of the liquid phase on heat transfer. Considering a cubic foam model Manegold [5] has suggested a relation between the thermal conductivity of a foam and the liquid content in it... 

However, for wet wood samples (H37), we observe a significant effect of the shape and of the species but without significant interaction between both factors. Energy flows are delayed for blocks 16 compared with cubes 4 and for poplar compared with beech, The effect of the lengthening of the wood blocks in the direction of the fibres as well as the effect of the species, act on the physical properties of wood samples in carbonization (decrease of the thermal conductance and of the permeability to gases). These effects could be enhanced by the moisture content of wood. 

Combination of several properties is becoming increasingly important in modem industry. One example may be taken from electronics, where in addition to mechanical properties and electric resistance, themial stability and conductivity are important requirements. It was estimated that the increase of temperature by 10°C reduces time to failure by the factor of two." A finite analysis model was developed which accounts for the following properties of filled composites microstructure, effect of particle shape, formation of conductive chains, effect of filler aspect ratio, and interfacial thermal resistance. The predictions of the model indicate the most... 

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