Univocity condition

Particularized mathematical model. The univocity conditions given by the system geometry, the material conditions and the initial and frontiers state of the process variables have to be related with the models shown in Table 3.1 ... 

The univocity conditions that complete this general mathematical model can be written as follows ... 

For a full definition of the model of transport through the particle, it is necessary to set up the univocity conditions for the above equations ... 

The coupling of the univocity conditions to the problem established at the end of the probability balance. 

To solve the model obtained, it is necessary to link it with the univocity conditions. They are obtained from the physical meanings of the problem ... 

The SDE and transport equation can be used with the same univocity conditions. For simple univocity conditions and functions such as Di-a(Fa), the transport equations have analytical solutions. Comparison with the numerical solutions of stochastic models allows one to verify whether the stochastic model works properly. The numerical solution of SDE is carried out by space and time discretization into space subdivisions called bins. In the bins j of the space division i, the dimensionless concentration of the property (F = Fa/Faq) takes the Fj value. Taking into consideration these previous statements allows one to write the numerical version of relation (4.118) ... 

Indeed, we can easily produce the evolution of the outputs of the process when the univocity conditions and parameters of the process are correctly chosen. 

After the establishment of the univocity conditions, we can begin the numerical treatment of the model. For this purpose, we can use the simplified model form (see relations (4.145)) or its original form (4.141)). 

The change of a continuous poly stochastic model into its numerical form is carried out using the model described by Eq. (4.71) rewritten in Eq. (4.146). The solution of this model must cover the variable domain 0 z z, 0 t T. In accordance with the previous discussion, the following univocity conditions must be attached to this stochastic model. Here, fk(z),g (T) and h(t) are functions that must be specified. 

It is important to notice that the univocity conditions must adequately correspond to the process reality. Concerning the numerical discretisation of each variable space, model (4.146) gives the following assembly of numerical relations ... 

This mathematical model has to be completed with realistic univocity conditions. In the literature, a large group of stochastic models derived from the model described above (4.150), have already been solved analytically. So, when we have a new model, we must first compare it to a known model with an analytical solution... 

The analysis of the univocity conditions attached to the model shows that, here, we have an unsteady model where nonsymmetrical conditions are dominant. 

So as to show how we use the integral transformation in an actual case, we simplify the general model relation (4.150) and its attached univocity conditions to the following particular expressions ... 

Figure 4.12 Univocity conditions of the model of diffusive and unidirectional displacement (4.153).

This last result can be written as Eq. (4.164) and completed with the univocity conditions (4.165) resulting from the Laplace transformation of the original conditions written with relation (4.147) ... 

The univocity conditions, necessary to solve the model, are established by the following considerations ... 

In order to solve the model equation, we must complete it with the univocity conditions. In some cases, relations (3.100)-(3.107) can be used as solutions for the model particularized for the process. The equivalence between both expressions is that c(x,t)/C(j appears here as P(x,t). Extending the equivalence, we can establish that P(x, t) is in fact the density of probability associated with the repartition function of the residence time of the liquid element that evolves inside a uniform porous structure. 

This model is of interest because it can be easily reduced to a hyperbolic form of the transport model of one property. With some particular univocity conditions, this hyperbolic model accepts analytical solutions, which are similar to those of an equivalent parabolic model. The hyperbolic model for the transport of a property is obtained by coupling the equation P(x, t) = Pj(x, t) + P2(x, t) to relations (4.267) and then eliminating the terms Pj(x, t) and Pj(x, t). The result can be written as ... 

The hyperbolic model shows a fast evolution of the probability P(x.t) at the spatial distance x = yr with respect to x = 0 or more precisely at x/[v T/a]° = l/2exp(—ar). At moderate or large time, we cannot observe a difference between the predicted values of P(x.t) from the models. This is due to the rapid decrease with time of the magnitude of the rapid evolution of the predicted probability P(x.t) in the hyperbolic model. It is important to specify that the hyperbolic model keeps a fast evolving probability P(x.r) for all possible univocity conditions at small time. It is difficult to demonstrate experimentally the prediction of the stochastic hyperbolic model for the liquid dispersion inside a porous solid because the predicted skip is very fast P(x.r) and not easily measurable. 

Relation (4.283), obtained by coupling Eq. (4.282) and (4.280), presents the time derivation results in (4.284). Replacing the term dCgg/dt in (4.284) by (4.280) results in the famous Mint model equation (4.285). Relations (4.286) and (4.287) are the most commonly used univocity conditions of this model (i) before starting fdtration, the bed does not contain any retained solid (ii) during filtration, the bed is fed with a constant flow rate of suspension, which has a constant concentration of solid. ... 

By considering the combined variable z = x — xj2, we remove the mixed partial differential term from Eq. (4.293). The transformation obtained is the hyperbolic partial differential equation (4.294). This equation represents a new form of the stochastic model of the deep bed filtration and has the characteristic univocity conditions given by relations (4.295) and (4.296). The univocity conditions show that the suspension is only fed at times higher than zero. Indeed, here, we have a constant probability for the input of the microparticles ... 

In this stochastic model, the values of the frequencies skipping from one state to another characterize the common deep bed filtration. This observation allows the transformation of the above-presented hyperbolic model into the parabolic model, given by the partial differential equation (4.297). With the univocity conditions (4.295) and (4.296) this model [4.5] agrees with the analytical solution described by relations (4.298) and (4.299) ... 

The resulting asymptotic model is described by the following equation and the univocity conditions given by relations (4.295) and (4.296) ... 

With the univocity conditions given in relations (4.312) and (4.313), the stochastic model becomes ready to be used in simulation. 

The univocity conditions can be obtained from Eq.(4.311) which, at x = 0, results in the problem described by Eq. (4.317), which presents solution (4.318). This last relation represents the initial condition from the univocity problem of model (4.316) ... 

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