Dimensionless concentrations first-order kinetics

Little is known about the kinetics of the bioprocesses. A reasonable assumption is that the reaction rates are proportional to the amount of micro-organisms catalyzing the reactions. The influence of the other reactants is more complex, for example a nutrient in high concentration often has an inhibiting effect. Moreover, factors such as pH, salt concentrations, temperature, can have effects that are difficult to quantify. For this reason, we assume first-order kinetics and include all the other factors influencing the process rate in two Damkohler numbers. The following dimensionless reactor model is obtained ... 

Figure 16-6 Plot of In [A] versus time for a reaction A —> products that follows first-order kinetics. The observation that such a plot gives a straight line would confirm that the reaction is first order in [A] and first order overall, that is, rate = [A], The slope is equal to —ak. Because a and k are positive numbers, the slope of the line is always negative. Logarithms are dimensionless, so the slope has the units (time) L The logarithm of a quantity less than 1 is negative, so data points for concentrations less than 1 molar would appear below the time axis.

Note the similarities and differences between this steady-state equation and the transient one, Eq. (6-23). The boundary conditions for the two expressions are the same, that is, Eqs. (6-25) and (6-26). For first-order kinetics the solution is straightforward. Introducing r = k C and using a dimensionless concentration C — C/Cq and reactor length z — z/L, we obtain for the differential equation and boundary conditions... 

In these equations, v and a denote, respectively, the constant rate of synthesis of ATP and the maximum velocity of adenylate cyclase, both divided by the Michaelis constant of the substrate for the active form of the enzyme a represents the concentration of ATP divided by )8 and y are the concentrations of intracellular and extracellular cAMP, divided by the dissociation constant Kf of extracellular cAMP for the receptor (a, jS and y are thus dimensionless) q = KJK -, L is the allosteric constant of the receptor-enzyme complex /c, and k denote, respectively, the apparent first-order kinetic constants for the transport of cAMP into the extracellular medium (Dinauer, MacKay ... 

Under the hypothesis of first-order kinetics, the dimensionless concentration profile is calculated with the assumption that j3 1 ... 

Fig. 3. Dimensionless concentration as a function of the dimensionless position in a catalyst pellet during simultaneous reaction and diffusion for first-order kinetics and slab...

Suppose one has performed experiments with the mixture under consideration in a batch reactor, and one has obtained experimentally the overall kinetics—the R ) function such that dCldt = —R(C). For instance, one could obtain R C) = if the intrinsic kinetics are in fact first order and the initial concentration distribution is (l,x) = exp(—x). If one were to regard R(C) as a true (rather than an apparent) kinetic law, one would eonclude that in a CSTR with dimensionless residence time T the exit overall concentration is delivered by the (positive) solution of TC +C = 1. The correct value is in fact C = , and the difference is not a minor one. (To see that easily, consider the long time asymp-... 

Now suppose the tip-generated species is not stable and decomposes to an elec-troinactive species, such as in the case (Chapter 12). If R reacts appreciably before it diffuses across the tip/substrate gap, the collection efficiency will be smaller than unity, approaching zero for a very rapidly decomposing R. Thus a determination of /x//s as a function of d and concentration of O can be used to study the kinetics of decomposition of R. In a similar way, this decomposition decreases the amount of positive feedback of O to the tip, so that ij is smaller than in the absence of any kinetic complication. Accordingly, a plot of ij vs. d can also be used to determine the rate constant for R decomposition, k. For both the collection and feedback experiments, k is determined from working curves in the form of dimensionless current distance (e.g., did) for different values of the dimensionless kinetic parameter, K = kcP ID (first-order reaction) or = k a CQlD (second-order reaction). 

If we were to change the kinetics so that the first reaction was second order in A and the second reaction was first order in B, then we would see largely the same picture emerging in the graphs of dimensionless concentration versus time. There would of course be differences, but not large departures in the trends from what we have observed for this all first-order case. But what if the reactions have rate expressions that are not so readily integrable What if we have widely differing, mixed-order concentration dependencies In some cases one can develop fully analytical (closed-form) solutions like the ones we have derived for the first-order case, but in other cases this is not possible. We must instead turn to numerical methods for efficient solution. 

Mass flux of reactant A into the catalyst across its external surface is employed to develop analytical expressions for the effectiveness factor in terms of the intrapellet Damkohler nnmber. Reactant molar density profiles for diffusion and first-order irreversible reaction have been developed in three coordinate systems, and these profiles in Chapter 17 represent the starting point to calculate the dimensionless concentration gradient on the external surface of the catalyst. In each case, the reader should verify these effectiveness factor results by volumetri-cally averaging the dimensionless molar density profile throughout the pellet via equations (20-47) with n = 1, realizing that it is not necessary to introduce a critical dimensionless spatial coordinate when the kinetics are first-order. 

Figure 27-1 Effect of the thermal energy generation paramete on dimensionless reactant concentration profiles as one travels inward toward the center of a porous catalyst with rectangular symmetry. The chemical kinetics are first-order and irreversible, and the reaction is exothermic. All parameters are defined in Table 27-4. The specific entries for P = 0.6 and = 1.0 are provided in Table 27-6.

Estimate the dimensionless concentration gradient on the external surface of the catalyst at = 1 or f = 0 which yields a zero gradient at the center of the catalyst for the following set of important dimensionless parameters when the chemical kinetics are first-order and irreversible in porous catalysts with rectangular symmetry Aa, intranet = 2, = 0.65, y = 8-6. 

As expected, < =

first order. Note that R (l) is the derivative of the dimensionless reaction rate with respect to the dimensionless concentration, evaluated when the latter variable is 1 (at the surface). For power-law kinetics. 

Figure 5.75. Dimensionless plot of reaction rate k = v /Uq versus substrate concentration in the bulk liquid A (cf. Equ. 5.254a). Zero-, first-, and half-order reactions (wg = 0,1,1/2) are demonstrated in their range of applicability. Note how Us = 1/2 may erroneously be interpreted as the saturation effect of Monod kinetics. B and E are explained in Equs. 5.254b and c (Harremoes, 1977).

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