Transformed concentrations

PCB Polychlorinated biphenyls. Aka polychloro-biphenyls. Difficult to remediate chemical used in old-style transformers. Concentrated PCBs used to be referred to as "1268". 

The age determination allowed Fourier transforms to be made, transforming concentration versus depth into signal power and amplitude versus the period expressed in years. The periods found are listed in Table 1. The sea core spans 700 years, with each sample containing seven years of sediment, so that evidence for periods between 40 years and 200 years should be meaningful [48]. 

Consider, for example, a model that is being used to predict the log-concentration of a chemical in some environmental medium (e.g., the average log-transformed concentration in the muscle tissue of fish exposed to the chemical at a particular site). 

In this example, the likelihood function is the distribution on the average of a random sample of log-transformed tissue residue concentrations. One could assume that this likelihood function is normal, with standard deviation equal to the standard deviation of the log-transformed concentrations divided by the square root of the sample size. The likelihood function assumes that a given average log-tissue residue prediction is the true site-specific mean. The mathematical form of this likelihood function is... 

Notice that the right-hand side of Eq. (34) is equal to the ratio of the transformed concentration at the second measurement point to the transformed concentration at the first measurement point. In the terminology of control engineering, this quantity is the transfer function of the system between Xo and Xm- The Laplace-transform method is possible because the diffusion equation is a linear differential equation. Thus, the right-hand side of Eq. (34) could in principle be used in a control-system analysis of an axial-dispersion process. 

Substituting the reaction rate, Q, into Eq. 2, the general solution of the Laplace transformed concentration can be given for each differential equation as follows ... 

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

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]... 

Logarithmically transformed, concentration-dependent pharmacokinetic parameters should be analysed using analysis of variance (ANOVA). Usually the ANOVA model includes the formulation, period, sequence or carry-over and subject factors. 

Eor a type 111 phase enviromnent, there possibly exist the left lobe [type ll(-i-)] and the right lobe [type 11(-)]. The plait point must vary between 0 and CfpL, the left plait point for type n(-p), or between 1 and CfpR, the right plait point for 11(-). The idea to calculate the phase compositions in the lobes is to follow the approach for type ll(-) and type II(-i-) phase environments with the transformed concentrations. Before that, however, we need to define how the plait points and invariant point move. 

As shown in Figure 7.18, the transformed concentrations (denoted by superscript prime) are (UTCHEM-9.0, 2000)... 

For the left and right lobes in a type III phase environment, calculation of the phase compositions follows the approach for type II(-i-) and type II(-) phase environments with the transformed concentrations that are described earlier in the previous section. 

If we Fourier transform the one-dimensional version of eqn (7.18), the resulting differential equation in the Fourier transformed concentration is... 

This represents an initial concentration field in which all of the diffusing species is concentrated at the origin. To find Co of eqn (7.21), we require c k, 0). If we Fourier tranform the initial concentration profile given in eqn (7.22), the result is c(k,0) = 1. We are now prepared to invert the transformed concentration profile to find its real space representation given by... 

U = average convective travel time, cm s 1 u = reduced average convective travel time, cm s l x = distance from the inlet, cm Y = transformed concentration function, g m-3 A = frequency constant, h l 0 = porosity, also angle in radians oo = loading frequency, h-1... 

Table 5.5 Simple linear regression summary of Ln-transformed AUC against Ln-transformed concentration using data in Table 5.3.

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. 

Because the transformed concentration profiles, (5.4.45) and (5.4.46), contain y, which is a function of they are not readily inverted to produce a equations covering all time regimes. However, we can obtain limiting cases for the early transient and steady-state regimes simply by recognizing limiting forms, just as we did earlier. 

One normally analyzes the effluent stream in terms of mole fractions of components present at the reactor outlet. This allows us to perform mass and atomic balances on the effluent and thereby ensure the consistency of the data. On the other hand, mechanistic rate expressions are normally developed in terms of concentrations. What is required is therefore a method of transforming concentration-based rate expressions to a fractional conversion form. This is particularly important in treating data from a temperature scanning reactor (TSR) (see Chapter 5) but is important in dealing with the analytical data from most reactors before the data can be fitted to mechanistic rate expressions. The procedure for doing this is well defined. In liquid and solid phase reactions, where there is no change in volume with reaction, the problem can be ignored in the gas phase one proceeds as follows. 

Example 5.19 Transformation concentration to absorbance for a consecutive photoreaction For the reaction... 

In the more general case with Eq. 12.5.b-2, it is readily seen that the (Laplace transformed) concentration in region 2 is given by ... 

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