Peclet number, wall heat transfer

Hot gas at the center will also move radially carrying heat to the walls. The Peclet number for heat transfer includes a turbulent diffusion conductivity ... 

Equations (8) are based on the assumption of plug flow in each phase but one may take account of any axial mixing in each liquid phase by replacing the molecular thermal conductivities fc, and ku with the effective thermal conductivities /c, eff and kn eff in the definition of the Peclet numbers. The evaluation of these conductivity terms is discussed in Section II,B,1. The wall heat-transfer terms may be defined as... 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

The interfacial heat transfer coefficient can be evaluated by using the correlations described by Sideman (S5), and then the dimensionless parameter Ni can be calculated. If the Peclet numbers are assumed to be infinite, Eqs. (30) can be applied to the design of adiabatic cocurrent systems. For nonadiabatic systems, the wall heat flux must also be evaluated. The wall heat flux is described by Eqs. (32) and the wall heat-transfer coefficient can be estimated by Eq. (33) with... 

From this discussion of parameter evaluation, it can be seen that more research must be done on the prediction of the flow patterns in liquid-liquid systems and on the development of methods for calculating the resulting holdups, pressure drop, interfacial area, and drop size. Future heat-transfer studies must be based on an understanding of the fluid mechanics so that more accurate correlations can be formulated for evaluating the interfacial and wall heat-transfer coefficients and the Peclet numbers. Equations (30) should provide a basis for analyzing the heat-transfer processes in Regime IV. 

If one were to attempt to determine any communality in the discussion of models given in this chapter, about the best would be to say that the parameters invoked are derivatives of the model, as would be inferred from the titles of the previous sections. For example, there is the overall heat-transfer coefficient, h, that appears in the nonisothermal, one-dimensional axial dispersion model, which is not to be confused with the wall heat transfer coefficient, a y, that belongs to the radial dispersion model. Similarly, would the bed thermal conductivity be the same in an axial dispersion model as in a radial dispersion model What is the difference between a mass Peclet number and a thermal Peclet number and so on. In fact, let us take a moment... 

Dimensionless effective radial thermal conductivity X and effective wall heat transfer coefficient h as function of Peclet-number (a measured under reacting conditions, b calculated according to /42/for nonreacting conditions)... 

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. 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

A basic element of the thermal dynamics of the DPF is the heat transfer between the gas in the channel and the porous wall. In case of a porous wall having small wall thermal Peclet number PeT (as is always the case for a DPF as shown by Bissett and Shadman (1985)) the problem degenerates to the following modified Graetz problem ... 

Yaglom A. M. and Kader B. A. (1974) Heat and mass transfer between a rough wall and turbulent flow at high Reynolds and Peclet numbers. J. Fluid Mech. 62, 601 -623. 

In this lecture, the effects of the abovementioned dimensionless parameters, namely, Knudsen, Peclet, and Brinkman numbers representing rarefaction, axial conduction, and viscous dissipation, respectively, will be analyzed on forced convection heat transfer in microchannel gaseous slip flow under constant wall temperature and constant wall heat flux boundary conditions. Nusselt number will be used as the dimensionless convection heat transfer coefficient. A majority of the results will be presented as the variation of Nusselt number along the channel for various Kn, Pe, and Br values. The lecture is divided into three major sections for convective heat transfer in microscale slip flow. First, the principal results for microtubes will be presented. Then, the effect of roughness on the microchannel wall on heat transfer will be explained. Finally, the variation of the thermophysical properties of the fluid will be considered. 

Radial dispersion of mass and heat in fixed bed gas-solid catalytic reactors is usually expressed by radial Peclet number for mass and heat transport. In many cases radial dispersion is negligible if the reactor is adiabatic because there is then no driving force for long range gradients to exist in the radial direction. For non-adiabatic reactors, the heat transfer coeflScient at the wall between the reaction mixture and the cooling medium needs also to be specified. 

Convective Heat Transfer in Microchannels, Table 3 Nusselt numbers Nut3 for laminar fully developed flow as a fimction of the dimensionless wall thermal resistance and of the Peclet number for the T3 honndary condition... 

Three mechanisms influence the heat balance in an extruder heat transferred through the wall, heat transported by convection, and heat generated by viscous dissipation. The relative importance of these three mechanisms is generally expressed by the Brinkmann and the Peclet number. 

The Brinkmann number gives the ratio between heat generated by viscous dissipation and the heat conduction to the wall. The Peclet number is the ratio between convective heat transfer and conductive heat transfer. For extrusion, however, the use of these two numbers would lead to... 

Other relationships based on conventional penetration theory for packed bed have deduced that Nu = 2 i -JPe and for low Peclet numbers to the order of 10, Tscheng and Watkinson (1979) deduced an empirical correlation, where Nu = 11.6 x7e°. Any of these would suffice in estimating the wall-to-bed heat transfer coefficient as functions of the kiln s rotational speed, w, and the dynamic angle of repose, The calculated values of Nusselt numbers using Perron and Singh s... 

Yaglom, A.M. and Kader, B.A. (1974). Heat and Mass Transfer Between a Rough Wall and Turbulent Fluid Flow at High Rejmolds and Peclet Numbers. J. Fluid Mech., Vol. 62, pp. 601-623. 

Heat transfer coefficient for a one-dimensional model (ht) Wall coefficient for a two-dimensional model (/ ,) Radial Peclet number for mass dispersion ((Pe)r)... 

Here x is the extent of the reaction (or scaled concentration of the reagent B), X2 is the normalized temperature of the complex liquid-solid medium, Pei and Pc2 are the Peclet numbers for mass and heat transport, Le is the Lewis number, is the longitudinal spatial coordinate. Da is the Damkohler number, 7 is the normalized activation energy of the reaction, /) is the transverse residence time of fluid in the reactor determined by the rate of cross-flow, b is the adiabatic temperature rise for the empty reactor (without packing), / iv is the surface heat transfer coefficient, and X2w the temperature of the reactor walls [22],... 

---