Dimensionless model

Plotting the data as In Cai versus t will give - (k x) as the slope and permit calculating k. It should be noted that calculating k from data involves solving the equations, but establishing (k x) as the only important dimensionless group requires only the dimensionless model. 

Consider an nth-order reaction the equivalent dimensionless model for the stirred-tank reactor becomes... 

Program TANK solves the normal dimensional model equation for the problem, whereas TANKD is formulated in terms of the dimensionless model equations. 

The dimensionless model equations are programmed into the ISIM simulation program HOMPOLY, where the variables, M, I, X and TEMP are zero. The values of the dimensionless constant terms in the program are realistic values chosen for this type of polymerisation reaction. The program starts off at steady state, but can then be subjected to fractional changes in the reactor inlet conditions, Mq, Iq, Tq and F of between 2 and 5 per cent, using the ISIM interactive facility. The value of T in the program, of course, refers to dimensionless time. 

Defining Z = z/L, t=T/T0, C=CA/CA0> =Dt/L2, the model equations can be recast into dimensionless terms to give the following dimensionless model equations ... 

The dimensionless model equations are used in the program. Since only two boundary conditions are known, i.e., S at X = l and dS /dX at X = 0, the problem is of a split-boundary type and therefore requires a trial and error method of solution. Since the gradients are symmetrical, as shown in Fig. 1, only one-half of the slab must be considered. Integration begins at the center, where X = 0 and dS /dX = 0, and proceeds to the outside, where X = l and S = 1. This value should be reached at the end of the integration by adjusting the value of Sguess at X=0 with a slider. 

For an anodic current, Eq. (48) should be used, and the dimensionless model will be... 

The dimensionless models keep the same form as with the original although there are differences at. V = 1 on the membrane side only. The two boundary conditions, used to find the roots, Qo or T o, are simplified. 

The dimensionless model equations for an electrochemical reduction reaction in a PBE are in a form of a set of partial ... 

In summary, for three dimensional electrode reactors, where the theoretical dimensionless model developed depends on six dimensionless parameters (ju, s. a, / as well as the aspect ratio C and Peclet number Pe), the nonlinear differential equations) with boundary conditions can be analyzed approximately by ADM. The ADM solution mechanism is easier to use and faster to converge than conventional numerical methods. 

In this section we analyze processes involving the second-order A + B —> 2P reaction. Such processes have been studied, among others, by Luyben andTyreus [10]. It has been noticed [11] that certain control structures lead to state multiplicity and instability. Here, we use dimensionless models to derive general feasibility and stability conditions. The reader is encouraged to check carefully the balance equations, writing them first in the dimensional form, and then deriving the dimensionless versions. To solve these equations, software such as Maple or the symbolic toolbox of Matlab can be used. 

Assuming pure reactant feed za = 1, plug flow, constant physical properties and one-reactant first-order reaction, the following dimensionless model can be derived ... 

To derive a dimensionless model, we introduce the reference concentrations Cjef and Crrf and define the dimensionless concentrations cK by the following relationships ... 

At this point, we anticipate the derivation of the dimensionless model for the gas bulk. This model should be capable of predicting the change of the NO molar flow... 

After inserting dimensionless variables into Eq. (12.7), the following dimensionless model is obtained ... 

The last case we consider is that of a flat velocity profile with linear adsorption and desorption. In this case, the dimensionless model equations are given by... 

These dimensionless groups of fluid properties play important roles in dimensionless modeling equations of transport processes, and for systems where simultaneous transport processes occur. 

The formal solution for this complete dimensionless model can be obtained when the flow equations can be resolved separately. Then we obtain ... 

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