Dimensional scaling factor

Equation 4.4.12 is the dimensionless, reaction-based design equation of a CSTR, written for the mth-independent reaction. The factor (ter/Q) in Eqs. 4.4.11 and 4.4.12 is a dimensional scaling factor that converts the design equations to dimensionless forms. 

The equation of motion for a generalized incompressible fluid (8-39) is written in terms of dimensionless variables and dimensional scaling factors ... 

Notice that the accumulation rate process and convective forces scale as pV /L, whereas viscous, pressure, and gravity forces scale as pV/L . If one takes the ratio of these two dimensional scaling factors, an important dimensionless number is obtained ... 

DIMENSIONAL SCALING FACTORS FOR THE MASS TRANSFER RATE PROCESSES... 

Introduce dimensionless variables (i.e., C, = Cps fi, etc.) and write each mass transfer rate process in terms of these dimensionless variables and the corresponding dimensional scaling factor. This scaling factor contains all the dimensions of, as well as an order-of-magnitude estimate for, the particular mass transfer rate process. For example, the left-hand side of equation (10-4) is written as follows, where all the variables are dimensionless ... 

Divide the entire mass transfer equation by the scahng factor for diffusion (i.e ,mixCAo/i )- This is an arbitrary but convenient choice. Any of the r + 2 dimensional scaling factors can be chosen for this purpose. When the scaling factor for the diffusion term in the dimensional mass transfer equation is divided by ,mixCAo/T, the Laplacian of the molar density contains a coefficient of unity. When the remaining r - -1 scaling factors in the dimensional mass transfer equation are divided by i,mixCAo/T, the dimensionless mass transfer equation is obtained. Most important, r -h 1 dimensionless transport numbers appear in this equation as coefficients of each of the dimensionless mass transfer rate processes, except diffusion. Remember that the same dimensionless number appears as a coefficient for the accumulation and convective mass transfer rate processes on the left-hand side of the equation. 

As mentioned above, the dimensionless transport numbers in the mass transfer equation are generated from ratios of dimensional scaling factors. If one divides the scaling factor for convective mass transfer by the scaling factor for diffusion, the result 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. 

Notice that the molar density profiles for these problems are not affected by any dimensionless numbers because either there is only one mass transfer rate process for steady-state analysis, or both rate processes are described by the same dimensional scaling factor. These qualitative trends should be considered before one seeks quantitative information about a particular mass transfer problem. 

If N components (1 < / < N) participate in r independent chemical reactions (1 < 7 < r), then the previous discussion illustrates the methodology to generate r -h 1 dimensionless numbers from r + 2 dimensional scaling factors in the mass balance for component i. This process is repeated by analyzing the mass balance for each component in the mixture. The characteristic molar density of key-limiting reactant A, Cao, is employed to make all molar densities in the reactive mixture dimensionless, as follows ... 

Hence, the dimensional scaling factor for convective mass transfer is the same in each mass balance. Similarly, dimensional scaling factors for all of the independent chemical reactions do not change from one mass balance to the next. However, when the r - - 2 dimensional scaling factors in the mass balance for component i are divided by the dimensional scaling factor for component i s rate of diffusion (i.e., i.mixCAo/L ), one obtains r - -1 dimensionless numbers... 

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... 

Division of the dimensional scaling factors for these two momentum transfer rate processes, or the corresponding scaling factors in the equation of motion, yields an expression for the Reynolds number. Hence,... 

This is a mathematical expression for the steady-state mass balance of component i at the boundary of the control volume (i.e., the catalytic surface) which states that the net rate of mass transfer away from the catalytic surface via diffusion (i.e., in the direction of n) is balanced by the net rate of production of component i due to multiple heterogeneous surface-catalyzed chemical reactions. The kinetic rate laws are typically written in terms of Hougen-Watson models based on Langmuir-Hinshelwood mechanisms. Hence, iR ,Hw is the Hougen-Watson rate law for the jth chemical reaction on the catalytic surface. Examples of Hougen-Watson models are discussed in Chapter 14. Both rate processes in the boundary conditions represent surface-related phenomena with units of moles per area per time. The dimensional scaling factor for diffusion in the boundary conditions is... 

Notice that the rate law becomes dimensionless via division by 2 (Ca, surface), which is exactly the same as the dimensional scaling factor for irreversible second-order chemical reaction in the numerator of the intrapellet Damkohler... 

Since 1 a is only a function of spatial coordinate r, the partial derivative in (19-38) is replaced by a total derivative, and the dimensionless concentration gradient evaluated at the external surface (i.e., ] = 1) is a constant that can be removed from the surface integral in the numerator of the effectiveness factor. In terms of the Hougen-Watson kinetic model and the dimensional scaling factor for chemical reaction that agree with the Langmuir-Hinshelwood mechanism described at the beginning of this chapter ... 

---