Effective axial heat conductivity

The effective axial heat conductivity of monolith substrates kg,a is readily estimated as... 

Effect of axial heat conduction in the channel wall... 

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 dependence of the local Nusselt number on non-dimensional axial distance is shown in Fig. 4.3a. The dependence of the average Nusselt number on the Reynolds number is presented in Fig. 4.3b. The Nusselt number increased drastically with increasing Re at very low Reynolds numbers, 10 < Re < 100, but this increase became smaller for 100 < Re < 450. Such a behavior was attributed to the effect of axial heat conduction along the tube wall. Figure 4.3c shows the dependence of the relation N /N on the Peclet number Pe, where N- is the power conducted axially in the tube wall, and N is total electrical power supplied to the tube. Comparison between the results presented in Fig. 4.3b and those presented in Fig. 4.3c allows one to conclude that the effect of thermal conduction in the solid wall leads to a decrease in the Nusselt number. This effect decreases with an increase in the... 

The numerical and experimental study of Tiselj et al. (2004) (see Fig. 4.17) was focused on the effect of axial heat conduction through silicon wafers on heat transfer in the range of Re = 3.2—84. Figure4.17 shows their calculation model of a triangular micro-channels heat sink. The results of calculations are presented in Fig. 4.18. 

The numerical experiment started at a steady-state value of 200 C for both temperature nodes with an output of 16.89% for both heaters output number 1 was then stepped to 19.00%. If both outputs had been stepped to 19%, then both nodes would have gone to 220 C. The temperature of node 5 does not go as high, and the temperature of node 55 goes too high. In the reduced order model, the time constant x represents the effect of radial heat conduction, while the time constant X2 represents the effect of axial heat conduction. SimuSolv estimates these two parameters of the dynamic model as ... 

State with no entrance effects or radial velocity components body forces are neglected axial heat conduction is small compared to radial conduction Region I of Smith-Ewart kinetics (i.e., when micelles are first forming) is neglected and the initiator concentration is constant. The model may be summarized as ... 

When the effect of the fluid axial heat conduction is considered, the thermal entrance length increases for decreasing Peclet number (Pe = Re Pr). 

At low Re and when conjugate effects have to be considered, the temperature distribution along the microchannel is not linear. Under constant heat flux boundary conditions, Nu decreases with decreasing ratio of outer to inner channel diameter, approaching the constant temperature solution. A decrease in Nu is also seen with increasing wall conductivity. For constant temperature boundary conditions, Nu will increase approaching the constant heat flux solution with axial heat conduction in the wall. The values for local Nusselt number for the conjugated problem lie between the values for the two boundary conditions constant heat flux and constant temperature. 

Question by J. A. Clark, University of Michigan What would be the influence of the transient term in the differential equation on the results In a paper by P. J. Schneider [Trans, ASME, Vol. 79, p. 765 (1957)] the effect of axial heat conduction on heat transfer in entrance regions in laminar flow was investigated. 

The length and diameter of the tube and the particle size (hydratdic diameter) also affect flow distrihution within the packed tube. If the ratio of the tube diameter to that of the particle diameter is above 30, radial variations in velocity can be n lected, and plug (piston) flow behavior can be assumed. The ratio of the tube length to particle diameter is also important if this ratio exceeds 50, axial dispersion and axial heat conduction effects can be ignored. These efiects bring notable simplifications into the modeling of PBRs, which are discussed in Chapter 3. 

A parametric study on the effects of axial heat conduction in the solid matrix has shown that i) such effects are negligible in ceramic monoliths (cordierite, kj = 1.4 w/m/K) but expectedly significant in metallic monoliths (Fecralloy, k i = 35 W/m/K) when a constant heat flux is imposed at the external matrix wall ii) however, the influence of axial conduction in metallic monoliths is much less apparent if a constant wall temperature condition is applied, since the monolith tends to an isothermal behavior. Metallic matrices exhibit very flat axial and radial temperature profiles, which seems promising for their use as catalyst supports in non-adiabatic chemical reactors. 

The calculations revealed that full conversion of the fuel was no longer jjossible with more than 10% load of the reactor due to the increasing effect of heat losses and an increasing contribution of axial heat conduction along the axis of the reactor length, which flattened its temperature profile. Increasing the O/C ratio of the feed from 0.75 to 1 compensated for these effects, as shown in Figure 5.18. 

Axial dispersion of heat In the case of strong exothermic or endothermic reactions, axial temperature profiles will occur even in intensively cooled reactors, and axial (longitudinal) dispersion smoothens these profiles. This dispersion can be described by an effective axial thermal conductivity Xax that combines heat conduction via the gas and solid phase. 

The problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

The following, well-acceptable assumptions are applied in the presented models of automobile exhaust gas converters Ideal gas behavior and constant pressure are considered (system open to ambient atmosphere, very low pressure drop). Relatively low concentration of key reactants enables to approximate diffusion processes by the Fick s law and to assume negligible change in the number of moles caused by the reactions. Axial dispersion and heat conduction effects in the flowing gas can be neglected due to short residence times ( 0.1 s). The description of heat and mass transfer between bulk of flowing gas and catalytic washcoat is approximated by distributed transfer coefficients, calculated from suitable correlations (cf. Section III.C). All physical properties of gas (cp, p, p, X, Z>k) and solid phase heat capacity are evaluated in dependence on temperature. Effective heat conductivity, density and heat capacity are used for the entire solid phase, which consists of catalytic washcoat layer and monolith substrate (wall). 

When the internal diffusion effects are considered explicitly, concentration variations in the catalytic washcoat layer are modeled both in the axial (z) and the transverse (radial, r) directions. Simple slab geometry is chosen for the washcoat layer, since the ratio of the washcoat thickness to the channel diameter is low. The layer is characterized by its external surface density a and the mean thickness <5. It can be assumed that there are no temperature gradients in the transverse direction within the washcoat layer and in the wall of the channel because of the sufficiently high heat conductivity, cf., e.g. Wanker et al. 

The gas velocity affects heat and mass transfer between the particles and flowing gas as well as the axial dispersion and heat conduction phenomena. For a reactor operating near at the extinction boundary, an increase of inlet velocity results in a sudden decrease of exit conversion. Sometimes this effect is called the blow-out phenomenon (Fig. 17). On the other hand, for a very active catalyst a decrease of inlet velocity leads to an ignited upper steady state. 

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