Time constant for convective mass transfer

Consider a train of five CSTRs in series that have the same volume and operate at the same temperature. One first-order irreversible chemical reaction occurs in each CSTR where reactant A decomposes to products. Two mass-transfer-rate processes are operative in each reactor. The time constant for convective mass transfer across the inlet and outlet planes of each CSTR is designated by the residence time x = Vjq. The time constant for a first-order irreversible chemical reaction is given >y X = l/k. The ratio of these two time constants,... 

T = 200 (Time constant for convective mass transfer (i.e., average residence time) in seconds)... 

Step 21. Calculate the time constant for convective mass transfer through the packed catalytic tubular reactor in units of minutes, which is equivalent to the residence time ... 

The next objective is to identify a time constant for each important mass transfer rate process and solve the system of equations for the two-phase CSTR in terms of these time constants. This approach allows one to develop generic solutions in dimensionless form. For example, six time constants can be defined for (1) convection in the liquid phase (r), (2) chemical reaction in the liquid phase (X), and (3-6) interphase mass transfer for each component (0y, j = B,C1,M,H). Obviously, these six time constants produce five dimensionless ratios. Remember that time constants represent order-of-magnitude estimates of the time scales of mass transfer rate processes. The time constant for convective mass transfer in the liquid phase is equivalent to the liquid s residence time ... 

Design a two-phase gas-liquid CSTR for the chlorination of benzene at 55°C by calculating the total volume that corresponds to an operating point where r/X = 500 on the horizontal axis of the CSTR performance curve in Figure 24-1. The time constant for convective mass transfer in the liquid phase is r. The time constant for second-order irreversible chemical reaction in the liquid phase is If the liquid benzene feed stream is diluted with an inert, then 7 increases. The liquid-phase volumetric flow rate is 5 gal/min. The inlet molar flow rate ratio of chlorine gas to liquid benzene... 

Put both curves on the same set of axes. The effectiveness factor, interpellet porosity, and time constants for convective mass transfer and chemical reaction are the same in both cases. 

The rate of convective mass transfer relative to the rate of mass transfer via interpellet axial dispersion is eqnivalent to the ratio of the diffusion time constant relative to the residence time for convective mass transfer. The interpellet Damkohler nnmber for reactant A is... 

Time constant ratio (r/A.) = p for convective mass transfer through the reactor (residence time r) relative to the time constant for second-order irreversible chemical reaction in the liquid phase k, where x is incremented as an important design variable. 

Similar time constants can be estimated for momentum and heat transfer also. Table 5.2 shows the time constants for convective transport in the axial direction, the radial transport of momentum, mass, and heat, assuming L = 10 m, n = 0.1 m s and laminar flow. The fluid properties that are typical for a gas are density 1 kg m, diffusivity 10 m s , heat capacity 1000 J kg K . Typical liquid properties are density 1000 kg m , diffusivity 10 m s , heat capacity 1000 J kg K , heat conductivity 0.5 W m K . The main differences between gases and hquids are in the... 

For large rate constants kv of the volume chemical reaction, a thin diffusion boundary layer is produced near the drop surface its thickness is of the order of ky1//2 at low and moderate Peclet numbers, and the solute in this layer has time to react completely. As the Peclet number is increased further, because of the intensive liquid circulation within the drop, there is not enough time to complete the reaction in the boundary layer. The nonreacted solute begins to get out of the boundary layer and penetrate into the depth of the drop along the streamlines near the flow axis. If the circulation within the drop is well developed, a complete diffusion wake is produced with essentially nonuniform concentration distribution that pierces the entire drop and joins the endpoint and the origin of the diffusion boundary layer. In case of a first-order volume chemical reaction, an appropriate analysis of convective mass transfer within the drop for Pe > 1 and kv > 1 was carried out in [150,151]. It should be said that in this case, in view of the estimate (5.4.8), which is uniform with respect to the Peclet number, the mass transfer intensity within the drop is bounded by the rate of volume chemical reaction. 

The solution to this problem requires an analysis of multiple gas-phase reactions in a differential plug-flow tubular reactor. Two different solution strategies are described here. In both cases, it is important to write mass balances in terms of molar flow rates and reactor volume. Molar densities and residence time are not appropriate for the convective mass-transfer-rate process because one cannot assume that the total volumetric flow rate is constant in the gas phase, particularly when the total number of moles is not conserved. In each reaction, 2 mol of reactants generates 1 mol of product. Furthermore, an overall mass balance suggests that the volumetric flow rate is constant only when the overall mass density does not change. This is a reasonable assumption for liquid-phase reactors but not for gas-phase problems when the total volume is not restricted. The exception is a constant-volume batch reactor. 

The sequence of equations presented below is required to solve the isothermal gas-liquid CSTR problem for the chlorination of benzene in the liquid phase at 55°C. After some simplifying assumptions, the problem reduces to the solution of nine equations with nine unknowns. Some of the equations are nonlinear because the chemical kinetics are second-order in the liquid phase and involve the molar densities of the two reactants, benzene and chlorine. The problem is solved in dimensionless form with the aid of five time constant ratios that are generated by six mass transfer rate processes (1) convective mass transfer through the reactor, (2) molecular transport in the liquid phase across the gas-liquid interface for each of the four components, and (3) second-order chemical reaction in the liquid phase. 

Furthermore, Figures 5.12 and 5.13 can also be used to show the dimensionless concentration as a function of dimensionless time and position for the case in which there is resistance to mass transfer at the interface between a solid and a fluid —Bab (9Ca/9z) = kc (Ca, — Caoo), where kc is the convective mass transfer coefficient (Section 4.4), Ca, is the concentration of species A at the interface in the fluid side, and Caoo is the concentration of species A in the fluid far away from the interface. Note that the constant concentration boundary condition referred to in the previous paragraph could be considered as a special case of the convective-type boundary condition for a Sherwood number, Sh (or Nusselt number for diffusion), equal to oo. Also, in Figures 5.12 and 5.13, the Biot (or Nusselt) number for heat transfer should be replaced by kcb/BAB) (l/ ) = (Sh/AT) = Sh, where K is the ratio of the equilibrium concentration in the solid to the... 

Fo being the Fourier number and d the diameter of the disk. The mass transfer coefficient k can be considered as interpolating between the steady-state convective diffusion at large times (t - oo) and unsteady-state diffusion at short times (t — 0 and v = 0). The constants A and B of Eq. (147) follow from the solutions for these two limiting cases. For these two limiting cases... 

FIGURE 18.5 The mechanistic basis of the dispersion element. The three dispersion element parameters are a nondimensional dispersion number that measures the rate of signal diffusion relative to convection t, the apparent mean residence time and a, the signal-to-transcript conversion parameter. For the mRNA and protein compartments in the Hargrove-Schmidt model element, kx, kM, and kp are rate constants for translation, mRNA degradation, and protein degradation, respectively. The gray line indicates that information rather than mass is transferred from the TAT mRNA to the TAT protein compartment (140). 

Statement of the problem. Preliminary remarks. Let us consider the transient convective mass and heat transfer between a spherical drop of radius a and a translational Stokes flow where the resistance to the transfer exists only in the disperse phase. We assume that at the initial time t = 0 the concentration inside the drop is constant and equal to Co, whereas for t > 0 the concentration on the interface is maintained constant and equal to Cs. 

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