Thermal Peclet number

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

Fig.4. Effect of thermal Peclet number on maximum reactor temperature and methanol conversion.

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

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 ... 

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

Ferry (1992) and Ague (1994b) used the dimensionless thermal Peclet number (B) of Brady (1988) to assess whether or not the large fluxes needed to make regional quartz vein sets elsewhere in the Acadian orogen of New England may have also transported heat ... 

Basic dimensionless parameters. The diffusion and thermal Peclet numbers in the convective mass and heat transfer equations (3.1.8) and (3.1.33) are related to the Reynolds number Re = at/ jv (where v is the kinematic viscosity of the fluid) on the right-hand side in the Navier-Stokes equations (1.1.12) by the formulas... 

Numerous experiments and numerical calculations show that the laminar hydrodynamic boundary layer occurs for 5 x 102 < Rex 5 x 105 to 106 [427]. In this region the thermal Peclet number Pe = Pr Rex is large for gases and common liquids. For liquid metals, there is a range of Reynolds numbers, 104 < Rex 106, where the Peclet numbers are also large. 

The thermal Peclet number for a noncirculatory flow past bodies of various shape... 

Table 4.6 gives expressions for the thermal Peclet number occurring in... 

We note that the coupling between the hydrodynamic, diffusion, and thermal problems is provided by the convective terms in the equations of diffusion and heat conduction together with the two boundary conditions (5.10.5) and (5.11.2). To obtain the leading terms of the expansion with respect to low diffusion and thermal Peclet numbers, one can neglect the convective terms, so that the coupling between these problems will be provided only by the boundary conditions. 

C. J. Hsu, An Exact Analysis of Low Peclet Number Thermal Entry Region Heat Transfer in Transversely Nonuniform Velocity Fields, AIChEJ., (17) 732-740,1971. 

Both UL/a and UL/D are termed the Peclet number and usually given the same symbol. When there is no reason for confusion, we shall do the same otherwise we distinguish between the two by reference to the thermal Peclet number or the diffusion Peclet number. The Peclet number plays a similar role in heat and mass transport as the Reynolds number in momentum transport. The thermal and diffusion Peclet numbers may be written somewhat differently to bring out their relation to the Reynolds number in particular. 

This parameter is termed the Marangoni number. As discussed beJow, if Ma exceeds a critical value, an unstable convective flow will develop. The Marangoni number can also be interpreted as a thermal Peclet number (Eq. 3.5.16) if the characteristic velocity for the surface tension driven viscous flow is taken to be that of Eq. (10.5.5). We emphasize that this velocity is not a given parameter but rather a derived quantity. Expressing this velocity in terms of the imposed uniform temperature gradient p, with the aid of continuity, we arrive at Eq. (10.6.10). Interpreted as a Peclet number, the Marangoni number is a measure of the heat transport by convection due to surface tension gradients to the bulk heat transport by conduction. 

Show that the Rayleigh number (Eq. 10.6.1) can, like the Marangoni number, be interpreted as a thermal Peclet number. 

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... 

Thermal axial dispersion must be treated with care. Even if axial dispersion of mass is negligible, the same may not be true for heat transport. The dispersion coefficient that appears in the thermal Peclet number is very different from the dispersion coefficient of the mass Peclet number. The combination of a plug-flow model for the mass balance and a dispersion... 

The analogous heat transfer solution is valid at large Prandtl and thermal Peclet numbers. Independent variables r and 9 are replaced by y and x, respectively, such that... 

The Peclet s number is defined as thermal Peclet s number (5.113) and the diffusion one (5.114). At Per 1 the basic contribution to the heat transfer comes from convection, and at Per 1 - from the thermal conductivity. Similar conclusions refer to Pep. 

For a microsystem, it is tempting to assume the Peclet number to be small due to the small value of T indicating the negligible role of convection in microsystems. Let us consider the typical size of a microsystem to be 100 pm with velocity of the order of meters per second. For kinematic viscosity of water at 30 °C (v = 0.801 x 10 m /s), Re = 124. For water with thermal diffiisivity, a = 0.143 x 10 m /s, Pe,i, 700. Thus, the thermal Peclet number and Reynolds number cannot generally be considered to be small in microsystem convective heat transfer. 

A thermal Peclet number Pe can be defined by using the flow velocity V and the interface bending Ax ... 

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 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. 

Forced-Convection Flow. Heat transfer in pol3rmer processing is often dominated by the uVT flow advectlon terms the "Peclet Number" Pe - pcUL/k can be on the order of 10 -10 due to the polymer s low thermal conductivity. However, the inclusion of the first-order advective term tends to cause instabilities in numerical simulations, and the reader is directed to Reference (7) for a valuable treatment of this subject. Our flow code uses a method known as "streamline upwindlng" to avoid these Instabilities, and this example is intended to illustrate the performance of this feature. 

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. 

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