Quasilinear partial differential equation

In the nonreactive case, r is equal to zero and Eq. (3) reduces to a homogeneous system of first-order quasilinear partial differential equations... 

In the reactive case, r is not equal to zero. Then, Eq. (3) represents a nonhmoge-neous system of first-order quasilinear partial differential equations and the theory is becoming more involved. However, the chemical reactions are often rather fast, so that chemical equilibrium in addition to phase equilibrium can be assumed. The chemical equilibrium conditions represent Nr algebraic constraints which reduce the dynamic degrees of freedom of the system in Eq. (3) to N - Nr. In the limit of reaction equilibrium the kinetic rate expressions for the reaction rates become indeterminate and must be eliminated from the balance equations (Eq. (3)). Since the model Eqs. (3) are linear in the reaction rates, this is always possible. Following the ideas in Ref. [41], this is achieved by choosing the first Nr equations of Eq. (3) as reference. The reference equations are solved for the unknown reaction rates and afterwards substituted into the remaining N - Nr equations. 

Characteristic lines The existence of these lines is an essential feature of the theory of quasilinear partial differential equations of the type of the mass balance equation in the ideal model dC/dt - - u/(l -f Fdq/dC)dC/dz = 0, Eq. 7.2), and a key element of their solutions. The lines (dx/dt) = u/(l + Fdq /dC) in the (x, f) plane are called characteristic lines or characteristics. C is constant along each of these lines. See Chapter 7, Section 7.2.2. 

The behavior of a reactive wave depen ds on the flow of its reacting and product-gases. The conservation laws lead to systems of partial differential equations of the first order which are quasilinear, ie, equations in which partial derivatives appear linearly. In practical cases special symmetry of boundary and initial conditions is often invoked to reduce the number of independent variables. 

In analogy with quasilinear as used for partial differential equations. 

M. First-Order Partial Differential Equations (with H.-K. Rhee and N.R. Amundson) (2 vols.). VoL 1 Theory and Application of Single Equations. Vol. 2 Theory and Application of Hyperbolic Systems of Quasilinear Equations. Englewood Cliffs, NJ Prentice-Hall, 1986 1988. 

The outline of this chapter is as follows First, some basic wave phenomena for separation, as well as integrated reaction separation processes, are illustrated. Afterwards, a simple mathematical model is introduced, which applies to a large class of separation as well as integrated reaction separation processes. In the limit of reaction equilibrium the model represents a system of quasilinear first-order partial differential equations. For the prediction of wave solutions of such systems an almost complete theory exists [33, 34, 38], which is summarized in a second step. Subsequently, application of this theory to different integrated reaction separation processes is illustrated. The emphasis is placed on reactive distillation and reactive chromatography, but applications to other reaction separation processes are also... 

We consider a first-order partial differential equation in a function z of two variables x and y. The quasilinear partial differential eqnation is given by... 

Rhee, H.K., Aris, R., and Amundson, N.R. (1986) First-order partial differential equations, in Theory and Application of Hyperbolic Systems of Quasilinear Equations, vol. I, Prentice-Hall, New Jersey. 

Simulation of the adiabatic tubular reactor consists of the solution of the partial differential equations which describe the system. The nonlinear nature of these equations makes solution difficult. A powerful technique developed in recent years for the solution of nonlinear partial differential equations is that of quasilinearization. 

Partial differential equations are categorized into linear, quasilinear, and nonlinear equations. Consider, for example, the following second-order equation ... 

---