Peclet number for mass transfer

Dimensionless group Dimensionless group Dimensionless concentration Peclet number for mass transfer Peclet number for heat transfer Dimensionless temperature Dimensionless length Activation energy group Dimensionless time... 

Answer The product of Re and Sc is the mass transfer Peclet number, Pcmt, where the important mass transfer rate processes are convection and diffusion. Since the dimensional scaling factors for both of these rate processes do not contain information about the constitutive relation between viscous stress and velocity gradients, one concludes that PeMT is the same for Newtonian and non-Newtonian fluids. Hence, the mass transfer Peclet number for species in a multicomponent mixture is... 

TABLE 22-3 Effect of the Interpellet Damkohler Number on the Critical Value of the Mass Transfer Peclet Number for Second-Order Irreversible Chemical Kinetics in Packed Catalytic Thbular Reactors"... 

Based on this requirement, one obtains the correlation shown in Table 22-4 between the interpellet Damkohler number and the critical value of the mass transfer Peclet number for ideal response. For interpellet Damkohler numbers between 100 and 500 [i.e., 100 < (1 interpellet) ( A, intrapeIlet) A, interpellet — ... 

As the first step in calculating kg, estimate an average value for the mass-transfer Peclet number Pe,. for the bed, and then evaluate the effective diffusivity JD.. 

When multiple reactions occur in the gas phase, the mass balance for component i is written for an ideal tubular reactor at high mass transfer Peclet numbers in the following form, and each term has units of moles per volume per time ... 

This analysis begins with the unsteady-state mass balance for component i in the well-mixed reactor. At high-mass-transfer Peclet numbers, which are primarily a function of volumetric flow rate q, the rate processes of interest are accumulation, convective mass transfer, and multiple chemical reactions. Generic subscripts are... 

Now, the coupled mass and thermal energy balances can be combined and integrated analytically to obtain a linear relation between temperature and conversion under nonequilibrium (i.e., kinetic) conditions because it is not necessary to consider the temperature and conversion dependence of (Cp mixture)- At high-mass-transfer Peclet numbers, axial diffusion can be neglected relative to convective mass transfer, and the mass balance is expressed in terms of molar flow rate F, and differential volume dV for a gas-phase tubular reactor with one chemical reaction ... 

The mass balance for a differential plug-flow reactor (equation 3-17) that operates at high-mass-transfer Peclet numbers allows one to replace dtw, in (3-32) ... 

At high-mass-transfer Peclet numbers, the steady-state mass balance for component i, with units of moles per time, is expressed in terms of its molar flow rate Fi and differential volume dV = ttR dz for a tubular reactor. If species i... 

Coupled mass and thermal energy balances are required to analyze the nonisother-mal response of a well-mixed continuous-stirred tank reactor. These balances can be obtained by employing a macroscopic control volume that includes the entire contents of the CSTR, or by integrating plug-flow balances for a differential reactor under the assumption that temperature and concentrations are not a function of spatial coordinates in the macroscopic CSTR. The macroscopic approach is used for the mass balance, and the differential approach is employed for the thermal energy balance. At high-mass-transfer Peclet numbers, the steady-state macroscopic mass balance on reactant A with axial convection and one chemical reaction, and units of moles per time, is... 

For unsteady-state diffusion into a quiescent medium with no chemical reaction, the mass transfer Peclet number does not appear in the dimensionless mass transfer equation for species i because it is not appropriate to make variable time t dimensionless via division by L/ v) if there is no bulk fluid flow (i.e., (d) = 0). In this case, the first term on each side of equation (10-11) survives, which corresponds to the unsteady-state diffusion equation. However, the characteristic time for diffusion of species i over a length scale L, given by L /50i,mix. replaces L/ v) to make variable time t dimensionless. Now, the accumulation and diffusional rate processes scale as CAo i.mix/A, with dimensions of moles per volume per time. Since both surviving mass transfer rate processes exhibit the same dimensional scaling factor, there are no dimensionless numbers in the mass transfer equation which describes unsteady-state diffusion for species i in nonreactive systems. 

Consider a non-Newtonian fluid with power-law index n and consistency index m. Construct appropriate dimensionless representations for the Reynolds, Schmidt, and mass transfer Peclet numbers. 

The only assumption is that the physical properties of the fluid (i.e p and A.mix) are constant. The left-hand side of equation (11-1) represents convective mass transfer in three coordinate directions, and diffusion is accounted for via three terms on the right side. If the mass balance is written in dimensionless form, then the mass transfer Peclet number appears as a coefficient on the left-hand side. Basic information for dimensional molar density Ca will be developed before dimensionless quantities are introduced. In spherical coordinates, the concentration profile CA(r,6,4>) must satisfy the following partial differential equation (PDE) ... 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

If the Reynolds number is based on the sphere diameter, as defined earlier, then the group of terms prior to the integral in (11-93) is proportional to the inverse of the mass transfer Peclet number. The general expression for the mass transfer boundary layer thickness is... 

If there is only one chemical reaction on the internal catalytic surface, then vai = — 1 and subscript j is not required for all quantities that are specific to the yth chemical reaction. When the mass transfer Peclet number which accounts for interpellet axial dispersion in packed beds is large, residence-time distribution effects are insignificant and axial diffusion can be neglected in the plug-flow mass balance given by equation (22-11). Under these conditions, reactor performance can be predicted from a simplified one-dimensional model. The differential design equation is... 

All dimensionless numbers that are required to analyze mass transfer in packed catalytic tubular reactors can be expressed in terms of three time constants t, CO, and d. For example, the mass transfer Peclet number is... 

When the intrapellet Damkohler number for reactant A is large enough and the catalyst operates in the diffusion-limited regime, the effectiveness factor is inversely proportional to the Damkohler number (i.e., Aa, intrapeiiet)- Under these conditions, together with a large mass transfer Peclet number which minimizes effects due to interpellet axial dispersion, the following scaling law is valid ... 

Convergence is obtained when the appropriate guess for d p./di at the reactor inlet predicts the correct Danckwerts condition in the exit stream, within acceptable tolerance. To determine the range of mass transfer Peclet numbers where residence-time distribution effects via interpellet axial dispersion are important, it is necessary to compare plug-flow tubular reactor simulations with and without axial dispersion. The solution to the non-ideal problem, described by equation (22-61) and the definition of Axial Grad, at the reactor outlet is I/a( = 1, RTD). The performance of the ideal plug-flow tubular reactor without interpellet axial dispersion is described by... 

The numerical examples in Sections 22-4.1 and 22-4.2 compare I a( = 1, RTD) and 4 a( = 1, ideal) for several mass transfer Peclet numbers when the kinetics are either first-order or second-order and irreversible. 

Non-ideal simulations satisfy the Danckweits boundary condition for the outlet concentration gradient. Real and ideal tubular reactor performance at various mass transfer Peclet numbers is compared when the product of the effectiveness factor, the interpeilet Damkohler number, and the catalyst filling factor is 5. 

At seven different values of the interpellet Damkohler number for which real and ideal packed catalytic tubular reactor performance is summarized in Table 22-2, it is possible to identify a critical value of the mass transfer Peclet number (Re Sc)cnticai, above which the effects of interpellet axial dispersion are insignificant for second-order irreversible chemical kinetics. For example, if ideal performance is justified when the outlet conversion of reactants under real and ideal conditions differs by less than 0.5%,... 

If one operates a packed catalytic tubular reactor below the critical value of the mass transfer Peclet number where ideal performance is not achieved, then the following empirical linear correlation allows one to predict the dimensionless molar density of reactant A in the exit stream, 4first-order irreversible chemical kinetics ... 

Problem. Think about the overall strategy that must be implemented to account for the effect of interpellet axial dispersion on ihe outlet concentration of reactant A when Langmuir-Hinshelwood kinetics and Hougen-Watson models are operative in a packed catalytic tubular reactor. Residence-time distribution effects are important at small mass transfer Peclet numbers. 

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