Peclet number temperature

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

The bulk temperature Tb onb is close to saturation temperature Ts, when the values calculated using Eqs. (6.32), (6.33) and (6.34) do not differ significantly from unity. In Fig. 6.11 the experimental results reported by Kennedy et al. (2000) are presented as the dependence of the value /c,onb (Eq. 6.32) on the Peclet number. The data may be described by the single line of /c,onb = 0.96. In this case the bulk temperature 2b,onb, at ONE point should not differ significantly from 7s. Experimental results given in Table 6.4 support this statement. 

Water at room temperature is flowing through a 1.0-in i.d. tubular reactor at Re= 1000. What is the minimum tube length needed for the axial dispersion model to provide a reasonable estimate of reactor performance What is the Peclet number at this minimum tube length Why would anyone build such a reactor ... 

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

Study the effect of varying mass transfer and heat transfer diffusivities (D and X, respectively) and hence Peclet numbers (Pi and P2) on the resulting dimensionless concentration and temperature reactor profiles. 

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration c,. Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion-advection mass transport [48,49], The relative contribution of advection to total transport is characterised by the nondimensional Peclet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh - 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. 

Simulations show that the radial and axial temperature and bulk concentration profiles are effectively not influenced by these modeling differences. Figure 9 shows the radial concentration profiles at = 0.38 and at the reactor outlet. Even with very high Peclet numbers, the differences between the radial concentration profile across the relatively small bed and the assumed uniform profile are minimal. Under typical operating conditions with small Peclet numbers, there is no benefit to increasing the number of radial collocation points, especially in light of the increased dimensionality of the resulting system. 

The model discretization or the number of collocation points necessary for accurate representation of the profiles within the reactor bed has a major effect on the dimensionality and thus the solution time of the resulting model. As previously discussed, radial collocation with one interior collocation point generally adequately accounts for radial thermal gradients without increasing the dimensionality of the system. However, multipoint radial collocation may be necessary to describe radial concentration profiles. The analysis of Section VI,E shows that, even with very high radial mass Peclet numbers, the radial concentration is nearly uniform and that the axial bulk concentration and radial and axial temperatures are nearly unaffected by assuming uniform radial concentration. Thus model dimensionality can be kept to a minimum by also performing the radial concentration collocation with one interior collocation point. 

Figure 17 shows that a more uniform mass distribution can be achieved at higher filtration velocities, both for the low and the high porous material. Finally, Fig. 18 shows the utilized capacity of the filter wall for all the cases in study, computed with a gas temperature of 280°C and a primary and aggregate particle size of 20 and 90 nm, respectively, as a function of the Peclet number. It is seen (Fig. 18) that the more porous materials with a smaller wall thickness can attain a better usage of the capacity of the filter wall, when the Peclet number increases. 

Figure 8.16 presents the temperature profile for different values of Pe as predicted by eqn. (8.64). The figure demonstrates how an increase in Peclet number leads to deviations in the temperature profile of the pure conductive problem. As a matter of fact, for Pe > 1 we will have an additional length scale in the problem, 5, the... 

The analytical solution for convective heat transfer from an isolated particle in a Stokes flow can be obtained by using some unique perturbation methods, noting that the standard perturbation technique of expanding the temperature field into a power series of the Peclet number (Pe = RepPr) fails to solve the problem [Kronig and Bruijsten, 1951 Brenner, 1963]. The Nup for the thermal convection of a sphere in a uniform Stokes flow is given by... 

Here we have denoted y conversion, 0 Frank-Kameneckii dimensionless temperature, Da Damkohler number, Pe Peclet number for axial mass dispersion, Pe Peclet number for Sxial heat dispersion, Y dimensionless activation energy, B dimensionless adiabatic temperature rise, 3 dimensionless cooling parameter, 6 temperature of the cooling medium, A mass capacity, AT heat capacity. 

Using values as low as R x 1/tm, Ap = 10-2g/cm3, a = g (gravitational acceleration), and L = 0.1 cm at room temperature (T = 300K) one obtains a Peclet number of 102. This means that for most practical situations the diffusional motion in the bulk can be neglected compared with the motion induced by the external field. 

Consider convection with incompressible, laminar flow of a constant-temperature fluid over a flat plate maintained at a constant temperature. With the velocity distributions found in either Prob. 10.1 or Prob. 10.2, compute the dimensionless temperature distribution within the thermal boundary layer for the Peclet number equal to 0.1,1.0,10.0,100.0. Use the ADI method. 

The notation is the same as before, with i representing the variable dimensionless temperature within the disturbance, x representing composition variable. The reaction is first order and the Peclet numbers for heat and mass transfer are assumed to be equal. xQ represents the composition in the normal region, equal to 1 - (t/S) with t and S defined as before. 

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