Model rates

With a larger N, the width of the elution curve becomes narrower at a given V r, and a better resolution will be attained. The dependence of the value of N on the characteristics of packing material and operating conditions is not clear with the stage model, because the value of N is obtained empirically. The dependence of Hs on these parameters can be obtained by the following rate model. 

This treatment is based on the two differential material balance equations on a fixed bed and packed particles. Although these equations, together with an 

The first term of the right-hand side of Equation 11.20 shows the contribu- 

This treatment is based on the two differential material balance equations on a fixed-bed and packed particles. Although these equations, together with an adsorption isotherm and suitable boundary and initial conditions, describe precisely the performance of chromatography, the complexity of their mathematical treatment sometimes makes them insolvable. If the distribution coefficient K given by Equation 11.1 is constant, then Hs can be correlated to several parameters by use of the first absolute moment and the second central moment, as given by Equation 11.20 [3]. 

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The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

Ryan et al. [Chem. Eng. Progr, 90(8X 83 (1994)] showthat separate mass and heat transfer-rate modeling of an HCl absorber predicts 2 percent fog in the vapor. The impact is equivalent to lowering the stage efficiency to 20 percent. 

Full rate modeling Accurate description of transitions Appropriate for shallow beds, with incomplete wave development General numerical solutions by finite difference or collocation methods Various to few... 

FIG. 16-35 EluHon curves under trace conditions with a constant separation factor isotherm for different feed loadings and N = 80. Solid lines, rate model dashed line, local equilihrium theory for Xp= 0.4. 

Important exceptions to McCabes gro th-rate model have been noted by Bramson, by Randolph, and by Abegg. These are discussed by Canning and Randolph, Am. Inst. Chem. Ting. J., 13, 5 (1967). 

No industrial process enjoys a knowledge of mechanism and kinetics so complete that models can be compared to it. Aris (1975) and Cropley (1978) simulated experimental results using a rate model. From the data a new model was derived and compared with the original. 

Although experimental results could be fitted well with irreversible rate models, ignoring thermodynamic facts could be disastrous. Although reversibility moderated the maximum temperature at runaway, it was not the most important qualitative result. In fact, the one dimensional (directional, or irreversible, correctly) model was not realistic at these conditions. For the prediction of incipient runaway and the AT ax permissible before runaway, the reversibility was obviously important. 

Several methods are used to fit rate models, the two most common of which often give erroneous results. The first is the transformation of a proposed rate model to achieve a model form that is linear in the parameters. An example is the nonlinear model ... 

For two-phase flow through pipes, an overall dimensionless dis-eharge eoeffieient, /, is applied. Equation 12-11 is referred to as the equilibrium rate model (ERM) for low-quality ehoked flow. Leung [28] indieated that Equation 12-11 be multiplied by a faetor of 0.9 to bring the value in line with the elassie homogeneous equilibrium model (HEM). Equation 12-11 then beeomes... 

The method includes the mass unit vent flow capacity per unit area. G. This allows using any applicable vent capacity calculation method. The method incorporates the equilibrium rate model (ERM) for vent flow capacity when friction is negligible. Additionally, a coiTection factor is used for longer vent lines of constant diameter and with negligible static head change. ... 

Confidence Estimation for the Constant Failure Rate Model... 

BFR - Binomial Failure Rate (model of common cause system interactions). 

The part stress analysis prediction section contains failure rate models for a broad variety of parts used in electronic equipment. This method includes the effects of part quality factors and environmental factors. The tabulated values of the base failure rate are "cut off" at the design temperature and stress of the part. 

The rate model for a biological process is given by a Monod rate model... 

CASE STUDY OXYGEN TRANSFER RATE MODEL IN AN AERATED TANK FOR PHARMACEUTICAL WASTEWATER... 

Figure 3.7 shows the growth of R. rubrum in a batch fermentation process using a gaseous carbon source (CO). The data shown follow the logistic model as fitted by (3.14.2.11) with the solid lines, which also represent an unstructured rate model without any lag phase. The software Sigma Plot was used to fit model (3.14.2.11) to the experimental data. An increase in concentration of acetate in the prepared culture media did not improve the cell dry weight at values of 2.5 and 3 gT-1 acetate, as shown in Figure 3.7. However, the exponential growth rates were clearly observed with acetate concentrations of 0.5-2 g-F1 hi the culture media.

Table 3.1 shows the kinetic parameters for cell growth, rate models with or without inhibition and mass transfer coefficient calculation at various acetate concentrations in the culture media. The Monod constant value, KM, in the liquid phase depends on some parameters such as temperature, initial concentration of the carbon source, presence of trace metals, vitamin B solution, light intensity and agitation speeds. The initial acetate concentrations in the liquid phase reflected the value of the Monod constants, Kp and Kp. The average value for maximum specific growth rate (/xm) was 0.01 h. The value... 

Table 3.1. Kinetic parameters and rate models with and without inhibition mass...

The reaction rate model is based on total enzyme, substrate and inhibitor concentrations. 

Table E.2.3. Data collected and calculated for rate model...

For Sa << Km, the rate model is reduced to a first-order rate equation The Michaelis-Menten rate equation is ... 

Pesticide inhibition on an active enzyme has been reported, which caused enzyme activities to reduce. The collected data with and without inhibition are presented hi Table E.5.1. Determine the rate model with and without inhibitor (see Table E.5.1). Also define the type of inhibition. 

Rearrangement of the rate model and the equation is solved for substrate S ... 

A Monod rate model is used to demonstrate the rate of biomass generation. We neglect the cell death rate. Let us denote the ratio of biomass rate of generation to biomass concentration, rJX, that is the specific growth rate /a also denotes the dilution rate D is defined as number of tank volumes passed through per unit time, FIV. After substitution of D and /a into (6.8.1), the following equation is obtained ... 

The Monod rate model is valid for a CSTR bioreactor with maximum specific growth rate of 0.5 li 1 and K, 2 g-1. What would be a suitable dilution rate at steady-state condition, where there is no cell death if initial substrate concentration is 50g-l-1 and yield of biomass on substrate is 100%. 

The performance of a biochemical reactor is designed and evaluated based the reaction rate equation. The rate of biomass generation is based on the Monod rate model ... 

We wish to compare the performance of two reactor types plug flow versus CSTR with a substrate concentration of Csf = 60g-m 3 and a biomass yield of Y = 0.1. In a plug flow bioreactor with volume of 1 m3 and volumetric flow rate of 2.5 m -li what would be the recycle ratio for maximum qx compared with corresponding results and rate models proposed for the chemostat ... 

In plug flow reactor the value for Cs is reduced from 12 to 3 g-m 3, then the retention time and rate model with recycle ratio in a plug flow reactor can be written as ... 

Tlie biochemical reaction rate followed the Monod rate model with a Monod rate constant of ks = 6.2 X 10 6g-cm 3 and a specific growth rate of vmlx 6.67 X 10 7g-cm 3-s. Design the bioreactor with a suitable heat transfer area. 

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