Solid conductivity, dimensionless

The dimensionless ratio Fd/k, where d is the particle diameter, is known to be correlated with particle Reynolds number, solid conductivity, void fraction, and particle shape (2). It is a complicated but weak function of these quantities, which approaches a constant value of about 10 for large Reynolds numbers and low solid conductivity. This limiting value can be used for our purpose, leading to, P =10 L/d. ... 

FIGURE 9.15 Effect of dimensionless solid conductivity on the dimensionless radiant conductivity for (a) diffuse particle surface and (b) specular particle surface [9],... 

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

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. 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

Figure 8. The dimensionless thermal conductivity, b1tljcjljX,(t), at p = 0.8 and T0 = 2. The symbols are the simulation data, with the triangles using the instantaneous velocity at the end of the interval, X/(t), Eq. (277), and the circles using the coarse velocity over the interval, Eq. (278). The solid line is the second entropy asymptote, essentially Eq. (229), and the dotted curve is the Onsager-Machlup expression om(t)> Eq. (280). (Data from Ref. 6.)...

Convection Heat Transfer. Convective heat transfer occurs when heat is transferred from a solid surface to a moving fluid owing to the temperature difference between the solid and fluid. Convective heat transfer depends on several factors, such as temperature difference between solid and fluid, fluid velocity, fluid thermal conductivity, turbulence level of the moving fluid, surface roughness of the solid surface, etc. Owing to the complex nature of convective heat transfer, experimental tests are often needed to determine the convective heat-transfer performance of a given system. Such experimental data are often presented in the form of dimensionless correlations. 

Kp/a thermal conductance of solid across depth a Equations (4.1) and (4.2) can be reduced to dimensionless forms as 1... 

ZT Y r A A A A A AC dimensionless thermoelectric figure of merit electronic coefficient of heat capacity (1+ZT)F2 crystal field singlet non-Kramers doublet (crystal field state) crystal field triplet crystal field triplet hybridization gap jump in heat capacity at Tc K KL -min P 6>d X JCO total thermal conductivity of solid thermal conductivity of electrons or holes thermal conductivity of lattice minimum lattice thermal conductivity electrical resistivity Debye temperature magnetic susceptibility magnetic susceptibility at T = 0... 

Let us now consider a catalytic packed bed reactor , i.e. a tubular reactor filled with a grained catalyst through which the gas mixture flows. With the particle diameter of the catalyst, dp, an additional dimensionless number dp/d is added to the pi-space the Reynolds number is now expediently formed with dp. The reaction rate is related to the unit of the bulk volume and characterized by an effective reaction rate constant ko,eff = k . The thermal conductivity (k) also has to be valid for the gas/bulk solids system and diffusion can be considered as being negligible (Sc is irrelevant). The complete pi-space is therefore ... 

Figure 3 presents the variation of the electrophoretic mobility with the dimensionless distance X when a sphere moves perpendicularly towards a conducting plane. The solid line represents the results from the bipolar coordinate method and the dash curve is the approximate results from the reflection method. A good agreement between the results from the both methods is attained. The electrophoretic velocity of the sphere decreases monotonically with increasing X and is expected to vanish as the particle... 

FIG. 3. Dimensionless electrophoretic velocity U/Uq of a sphere vs. distance parameter A for particle s motion normal to a conducting plane. Solid line is the exact numerical results and dashed line represents the approximate results from the method of reflections. 

Two-phase continuum models, in which the solid particles and their associated stagnant fillets of fluid are regarded as a continuous pseudo-solid phase, are to be preferred to the more traditional cell model of a particle bathed in fluid, which does not allow conduction from particle to particle. In previous studies, such models have been simplified by considering a one-dimensional model only (25). This study considers the full equations, which are, in dimensionless form ... 

FIGURE 4-27 Dimensionless temperature distribution for transient conduction in a semi-infinite solid whose surface is maintained at a constant temperature T,. 

The temperature held is dependent on this number when heat transfer takes place into a fluid. The Biot number has the same form as the Nusselt number defined by (1.36). There is however one very significant difference, A in the Biot number is the thermal conductivity of the solid whilst in the Nusselt number A is the thermal conductivity of the fluid. The Nusselt number serves as a dimensionless representation of the heat transfer coefficient a useful for its evaluation, whereas the Biot number describes the boundary condition for thermal conduction in a solid body. It is the ratio of L0 to the subtangent to the temperature curve within the solid body, cf. Fig. 2.4, whilst the Nusselt number is the ratio of a (possibly different choice of) characteristic length L0 to the subtangent to the temperature profile in the boundary layer of the fluid. 

The equations given here for the temperature distribution and the average temperature are especially easy to evaluate if the dimensionless time t+ is so large, that the solution can be restricted to the first term in the infinite series, which represent the temperature profile in the plate and t Qy in the long cylinder. The equations introduced in section 2.3.4.5 and Table 2.6 can then be used. The heating or cooling times required to reach a preset temperature in the centre of the thermally conductive solids handled here can be explicitly calculated when the series are restricted to their first terms. Further information is available in [2.37],... 

In Table 1.10 those dimensionless groups that appear frequently in the heat and mass transfer literature have been listed. The list includes groups already mentioned above as well as those found in special fields of heat transfer. Note that, although similar in form, the Nus-selt and Biot numbers differ in both definition and interpretation. The Nusselt number is defined in terms of thermal conductivity of the fluid the Biot number is based on the solid thermal conductivity. 

Weisz modulus = m potential in conductive solid phase (V) potential in solution phase (V) dimensionless potential variable = dimensionless potential atX= 1 dimensionless limiting current density = ///o overpotential (V) applied overpotential (V)... 

The model Eqs. 11.78 describe the concentration distribution in a particle where adsorption is taking place with a linear adsorption isotherm. The external film mass transfer is reflected in the dimensionless Biot (Bi) number. The model equations can also describe heat conduction in a solid object and the heat transfer coefficient is reflected in the dimensionless Bi parameter. 

The dimensionless conductivity 0/0 and coupling coefficients depend linearly upon C The conductivity and electro-osmotic coefficients 6- = (cr/a - l)/c ) and are plotted in Figs 4a and 4b. respectively, versus the solid volume concentrations ( ) for the three reduced surface potentials = -1,0, +1. For = 0, one can see that 6- tends to -3/2 as (j) 0. Moreover, for uncharged particles, the... 

FIG. 5 (a) Derivative of the reduced conductivity dGla° )/d(, and (b) coupling coefficient 377 in a cubic array of contacting spheres (solid lines) or oblate ellipsoids (dashed lines) along the horizontal (O) and vertical (+) directions as functions of the inverse dimensionless double layer thickness kR. The dash-dotted line in (a) is the result for spheres of a matched asymptotic expansions technique [13,27],... 

FIGURE 1.2 Dependence of the conductivity ofF) on the magnitude of tension external electric field E for ideal CNT (10,0) - solid line and the CNT (10,0) with hydrogen adatom -dashed line. A -axis is a dimensionless quantity of the external electric fieldii (unit corresponds to 4.7x10 V/m), they-axis is dimensionless conductivity a E) (unit corresponds to 1.9x10 S/m). 

Equation 3.43g compares the timescale for radial heat dispersion in the solid phase with the one for internal heat conduction. For catalysts with good heat conduction properties and low particle-to-bed diameter ratios, A l. In this case, the surface boimdary condition is homogeneous and of Robin type, as given by the first terms on each side of (3.42b). A similar dimensionless number related with dispersion in the axial direction also appears, but its magnitude is considered much smaller than that of the other parameters in Equation 3.43, due to the geometrical reasons explained earlier. Note that Equations 3.32 and 3.34 are obtained by integrating Equation 3.41 with respect to over the pellet domain and using Equation 3.42 as boundary conditions. 

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