Transformed concentration variable

Transformed concentration variables were first introduced by Doherty and co-workers [2, 41] for the steady-state design of reactive distillation processes. In the... 

All methods available for the nonreactive problem (Eq. (4)) can also be applied to the reactive problem in transformed concentration variables (Eq. (5)). 

Finally, it should be noted that the above treatment is only valid for constant flow rates. For processes without solvent (e.g., reactive distillation processes), this assumption is only valid for equimolar reactions. For equimolar reactions the definition of transformed concentration variables introduced by Ung and Doherty [41] reduces to the definition in Eq. (6). For processes with solvent, (e.g., reactive chromatographic processes), the assumption of constant flow rates is also valid in good approximation, if the concentration of the solvent is high compared to the other reacting species. This is also true if one of the reactants is used simultaneously as a solvent, as in many applications of reactive chromatography (see e.g. Refs. [1, 28]). 

If reactant A is taken as the reference component, the following definition of the transformed concentration variables apply... 

Hence, pure product C has a value of zero and pure product B a value of one of the transformed concentration variable. The equilibrium composition of pure reactant A has a value of 0.5. 

The corresponding wave patterns of the transformed concentration variable X for a reactive distillation column are shown in Fig. 5.8. Here, a single feed with pure reactant A is introduced in the middle of the column. As in the nonreactive binary case, the composition profiles consist of a single front in each column section,... 

The profiles of the mole fractions % are easily calculated from the transformed concentration variable by solving Eqs. (10) and (12) for given X. Some characteristic profiles of the mole fractions corresponding to the rectifying section in Fig. 5.8(a) were shown in Fig. 5.3. For the other cases, the reader is referred to Ref. [11]... 

With simultaneous phase and reaction equilibrium the system has only two dynamic degrees of freedom (five solutes - three chemical equilibria) and therefore corresponds again to a nonreactive system with two solutes. If the dimers are taken as reference components the following definition of the transformed concentration variables is found from Eq. (6)... 

Wave models were successfully used for the design of a supervisory control system for automatic start-up of the coupled column system shown in Fig. 5.15 [19] and for model-based measurement and online optimization of distillation columns using nonlinear model predictive control [15], The approach was also extended to reactive distillation processes by using transformed concentration variables [22], However, in reactive - as in nonreactive - distillation, the approach applies only to processes with constant pattern waves, which must be checked first. 

Q transformed concentration variable of the fluid phase [kmol/m3]... 

In the equilibrium regime, the dynamic behavior of an RD column in transformed concentration variables is essentially the same like the dynamic behavior of a non-RD column. Hence it is not surprising that under these conditions we can observe all kinds of multiplicity, oscillations, and wave propagation phenomena like in non-RD. Any novel feature, such as reactive azeotropy, is introduced by the static transformation between physical and transformed concentrations. 

The question of interest in our current context is Which system is more fundamental That is, which variables - Xi or r i - are real Or, which system more naturally describes the real physics In either case, as is also true for any of an infinite number of other possible effective concentration variables yi that we could have chosen, the physical system remains the same, of course. The labels, or variables, with which we choose to describe that system are not fundamental. One is tempted to ask whether substantially greater depths of truth can be mined by considering the set of all possible transformations %j (from one consistent set of variables to another) rather than the set of all possible variables (as is typically done) ... 

For three-dimensional diffusion, if there is spherical symmetry (i.e., concentration depends only on radius), the diffusion equation can be transformed to a one-dimensional type by redefining the concentration variable w = rC. This transformation would work for a solid finite sphere, a spherical shell, an infinite sphere with a spherical hole in the center, or an infinite sphere. 

Experiment Variable i, expressed by the transformed concentration x, Response... 

Note that the bar over the symbol y represents the fact that the dimensionless concentration was transformed. All variables not subjected to direct transformation remain as they were before applying the transform. 

Linear algebra clarifies the use of Legendre transforms in biochemical thermodynamics in the sense that when an independent concentration variable is held constant, its row and column are omitted in the conservation matrix and then redundant columns are eliminated because they indicate pseudoisomer groups. By use of RowReduce many different choices of components can be found. 

The /z-transf orm method was also applied to the calculation of the breakthrough curve for the separation of a binary mixture on a charcoal bed by Tien et al. [36]. By using the li-transform, the course of the separation can be calculated by transforming the concentration variable via the /i-transform into a coordinate system in which algebraic equations describe the process for any number of components. 

The calculation of the bormdary velocities in multicomponent chromatography can be greatly simplified imder certain conditions by an appropriate transform of the variable. The fe-transform was introduced for this very purpose by Helfferich [35]. It is equivalent to the transformation to the "characteristic parameters" that was suggested by Rhee et al. [10] at the same time. The power of the /i-transform is based on the fact that only one of the new dependent variables, h, h2, , hi, , hn for a n-component mixture changes from one side of a boundary to the other side, in contrast with the possible changes of several of the n concentration variables. As a result, the expression giving the velocity of a bormdary depends on only one variable when it is written in terms of the variable hj, whereas an expression for this velocity written in terms of the concentrations will depend in general on all the n concentrations, Q. 

The mean and variance of the LPE were calculated as an estimate of the model bias and the variability of the measured concentrations around the population mean prediction, respectively. In this case the mean and variance of the LPE were 0.0061 and 0.291, respectively, for the IR formulation and 0.0335 and 0.287 for the CR formulation, respectively. In this form, however, interpretation was problematic as it is not clear what 0.0061 means. But, after exponentiation, the model predicted concentrations were on average 0.6% lower and 3.3% higher in the IR and CR formulations, respectively. Two modifications to this approach are often seen. One modification is that instead of using log-transformed concentrations, raw untransformed concentrations are used instead. Bias and variance with untransformed concentrations are directly interpretable and no exponentiation is needed for interpretation. A second modification is to standardize the LPE to the observed concentrations so that a relative LPE is calculated. 

In analytics, nonlinear relationships can be frequently modeled without the application of nonlinear methods. This is feasible by means of transformations of variables, such as signals or concentrations. Remember Beer s law in the form... 

Clearly, we can, in each case, transform to other concentration variables such as mole fraction or molality and obtain the appropriate activity coefficient. We shall not elaborate on this since it requires a relatively simple transformation of variables. We stress, however, that the number density (or the molar concentration) is the more natural choice of a concentration scale, and the corresponding standard chemical potentials enjoy some advantages which are not shared by standard chemical potentials based on either the mole fraction or the molality. More details are given in Section 4.11. 

Since this is still a differential equation, albeit in the variable qy, a simple interpretation is not straightforward. Hence it is necessary to consider the time dependence of Eq. 38, which can be foimd using the method of characteristics. Through a transformation of variables, a first-order partial differential equation is converted into a first-order ordinary differential equation, which can then be solved using standard techniques. Due to the usefulness of this method for solving equations describing coupled shear flow and concentration fluctuation dynamics, it is worth briefly outlining the ideas. If we introduce a variable, t, such that. 

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