Thermal conductivity, radial

Because direct calculation of thermal conductivity is difficulty 1], experimental measurements on composites with nanotubes aligned in the matrix could be a first step for addressing the thermal conductivity of carbon nanotubes. High on-axis thermal conductivities for CCVD high-temperature treated carbon fibers have been obtained, but have not reached the in-plane thermal conductivity of graphite (ref. [3], Fig. 5.11, p. 115). We expect that the radial thermal conductivity in MWNTs will be very low, perhaps even lower than the c-axis thermal conductivity of graphite. 

In the described cylindrical topology (see Figure 11.12), the modeling of the DLC internal construction has given a heat transfer coefficient of 12W/m2/K. The radial thermal conductivity is equal (o 7 = 0.5 W/m/K and is much smaller than the axial thermal conductivity which is equal to... 

V,eff effective radial thermal conductivity of the catal.bed KJ.ir f. K l Yjj = stoichiometric coefficient... 

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). 

Eqns. (3) - (9) enable the effective radial thermal conductivity (kr>eff), the apparent wall heat transfer coefficient (hw>eff) and the overall heat transfer coefficient (U) to be predicted in terms of the physical properties p, kg and Cp of air, together with measurable parameters such as , 6, dt, kp, dp (A), dp>(v), dp>(V/A), e and T, the mean bed temperature. The predicted and observed values of U are compared in Figure 6. The averaged normalised standard error... 

A schematic of an Anter radial thermal conductivity measuring instrument using this principle is shown in Figure 9.2. A specimen in the form of an annular cylinder is placed to surround a central heater. Often the cylinder is made up of a series of stacked rings. Alternatively, a granulated or fibrous form of the specimen may be poured or placed between the central heater ( 1.2 cm OD) and a mullite outer casing ( 10.5 cm ID). 

Figure 9.2 Schematic of radial thermal conductivity apparatus. Specimen dimensions are 2.75 cm in radial thickness and 56 cm in length. Not shown are thermocouples placed axially along the central heater and voltage taps 5 cm apart. Inner and outer thermocouple junctions extend out radially, centered axially between the voltage taps.

Monolith forms can have very high specific surfaces combined with a very low pressure loss. Monoliths with straight, parallel channels, such as used for automobile exhaust control have only very poor radial heat transport properties. Crossed corrugated structures are considerably more favorable for isothermal reaction control. They have a very high radial thermal conductivity which is almost independent of the specific surface area the latter can be varied over a wide range by means of the channel dimensions. 

E modified eddy-diffusion coefficient kp radial thermal conductivity in ex-... 

For non-adiabatic reactors, along with radial dispersion, heat transfer coefficient at the wall between the reaction mixture and the cooling medium needs to be specified. Correlations for these are available (cf. % 10) however, it is possible to modify the effective radial thermal conductivity (k ), by making it a function of radial position, so that heat transfer at the wall is accounted for by a smaller k value near the tube-wall than at the tube center (11). 

Figure 13 A shows some calculated radial temperatures within catalyst beds for some typical cases of hydroprocessing of oils in either a gas phase or a trickle-flow process. In these calculations, effective radial thermal conductivities were used that have been determined from existing correlations involving both static and convective mechanisms of heat transfer. It can be seen that whereas deviations from true iso-thermicity are reasonably small for a bed diameter of 1 cm, isothermal operation is hardly possible at the diameters in the range of pilot-plants, especially if the reactor is relatively short and operated with gas only. Under the latter circumstances, the reactor may even be unstable and temperature may run away.

Bauer R, Schliinder EU (1978) Effective radial thermal conductivity of packings in gas flow. Part 11. Thermal conductivity of the packing fraction without gas flow. Int Chem Eng 18 189-204... 

Effective radial thermal conductivity of the tubular reactor (kJ/h.m.K)... 

A. Matsuura, Y. Hitcka, T. Akehata, and T. Shirai, Effective Radial Thermal Conductivity in Packed Beds with Gas-Liquid Down Flow, Heat Transfer—Japanese Research, (8) 44-52,1979. 

Effective Thermal Conductivity. The effective thermal conductivity signifies the intensity of solids mixing in the interior of the fluidized bed. Muroyama et al. [96] reported that near incipient fluidization the effective thermal conductivity increases sharply with the liquid velocity, passes through a maximum, and then gradually decreases as the liquid velocity is increased. Karpenko et al. [100] reported the effective radial thermal conductivities for liquid fluidized beds of glass and aluminum particles. They obtained correlation Eq. 13.7.5 for predicting the effective thermal conductivity. 

Another equally important property is the heat transfer coefficient for bed-to control fluid heat transfer. If there is no radial heat transfer gradient within the bed (see Figure 12.1), it is assumed that the entire resistance inside the tube is localized within the film adjacent to the wall. This is the value to use in l-D beds. On the other hand, if there is a radial gradient, a 2-D model must be used, and both heat transfer coefficient and effective radial thermal conductivity are involved, as shown in the figure. Several correlations for these have been listed by Kulkarni and Doraiswamy (1980). 

Effective Radial Thermal-Conductivity of Packings in Gas-Flow. 2. Thermal-Conductivity of Packing Fraction without Gas-Flow. International Chemical Engineering, 18(2) 189-204. 

Different expressions are reported in the literature to evaluate the effective radial thermal conductivity [5] and the heat transport coefficient in the first layer near the tube wall fiw [6-9] for the pseudo-homogeneous fluid/solid phase. For ideal models, momentum balance is expressed as ... 

Smirnov El, Kuzmin VA, Zolotarskii lA. Radial thermal conductivity in cylindrical beds packed by shaped particles. Chemical Engineering Research and Design 2004 82 293-296. 

In this heat transport model, the radial heat transfer is represented by a Fourier-like law, using an equivalent radial thermal conductivity of the medium with flowing fluids 4r = A dT/dr. The bed radial effective thermal conductivity was expressed as sum of two terms—conductive contribution and convective contribution A -Aso+ gf In Hashimoto et al. [94] and Mat-suura et al. s [95] approach, the theory of single-phase gas flow... 

Hashimoto K, Muroyama K, Fujiyoshi K, Nagata S. Effective radial thermal conductivity in cocurrent flow of a gas and liquid through a packed bed. Int. Chem. Eng. 1976 16 720. 

Matsuura A, Hitaka Y, Akehata T, Shirai T. Effective radial thermal conductivity in packed beds with downward cocurrent gas-liquid flow. Heat Transf. Jpn Res. 1979 8 44. 

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