Hyperbolic equation continuity

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

The transient continuity equations and the combined momentum equation constitute a set of hyperbolic equations. The formulation is well-posed provided the equations possess real characteristics. The conditions of well-posedness of averaged two-fluid models were extensively discussed in the literature (e.g., Lyczkowski et al. [106], Ramshaw and Trapp [107], Banerjee and Chan [56], Drew [108], Jones and Prosperetti [109], Prosperetti and Jones [110], Moe [111]). The condition under which the characteristic roots of Equations 1, 2, 7 are real reads, (derived in 43 for C, = 0) ... 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elhptic ) or diffusion and heat transfer ( para-bohc ). Prototypes are ... 

This equation gives the differential yield of V for a porous catalyst at a point in a reactor. For equal combined diffusivities and the case where hT approaches zero (no diffusional limitations on the reaction rate), this equation reduces to equation 9.3.8, since the ratio of the hyperbolic tangent terms becomes y/k2 A/ki v As hT increases from about 0.3 to about 2.0, the selectivity of the catalyst falls off continuously. The selectivity remains essentially constant when both hyperbolic tangent terms approach unity. This situation corresponds td low effectiveness factors and, in tliis case, equation 12.3.149 becomes... 

The system of hyperbolic and parabolic partial differential equations representing the ID or 2D model of monolith channel is solved by the finite differences method with adaptive time-step control. An effective numerical solution is based on (i) discretization of continuous coordinates z, r and t, (ii) application of difference approximations of the derivatives, (iii) decomposition of the set of equations for Ts, T, c and cs, (iv) quasi-linearization of... 

For steady-state analysis (i.e., no time variation) the coupled system is essentially elliptic, with some hyperbolic characteristics. The continuity equation alone is clearly hyperbolic, having only first-order derivatives. That is, it carries information about velocity from an inlet boundary, across a domain, to an outlet boundary. By itself, the continuity equation has no way to communicate information at the at the outlet boundary back into the domain. Based on the second-derivative viscous terms, the momentum equation is elliptic in velocity. However, because it is first order in pressure, there is also a hyperbolic character to the momentum equation. Moreover the convective terms have a hyperbolic character. There are situations, for example in high-speed flow, where the viscous terms diminish or even vanish in importance. As this happens, and the second-derivative terms become insignificant relative to the first-derivative terms, the systems changes characteristics to hyperbolic. 

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

Continuity and momentum remain coupled and together form a third-order system. The system appears to have both parabolic and hyperbolic character, owing to the first derivatives in pressure and velocity. However, this appearance may be misleading. Taking the divergence of the momentum equation, we have... 

We can extend the hyperbolic model to cases in which the solute diffuses in more than one phase. A common case is that of a monolith channel in which the flow is laminar and the walls are coated with a washcoat layer into which the solute can diffuse (Fig. 4). The complete model for a non-reacting solute here is described by the convection-diffusion equation for the fluid phase coupled with the unsteady-state diffusion equation in the solid phase with continuity of concentration and flux at the fluid-solid interface. Transverse averaging of such a model gives the following hyperbolic model for the cup-mixing concentration in the fluid phase ... 

The continuity equations for mass and energy will be used to derive the hyperbolic partial differential equation model for the simulation of moving bed coal gasifier dynamics. Plug flow (no axial dispersion) and adiabatic (no radial gradients) operation will be assumed. 

The continuity equations for mass and energy were used to derive an adiabatic dynamic plug flow simulation model for a moving bed coal gasifier. The resulting set of hyperbolic partial differential equations represented a split boundary-value problem. The inherent numerical stiffness of the coupled gas-solids equations was handled by removing the time derivative from the gas stream equations. This converted the dynamic model to a set of partial differential equations for the solids stream coupled to a set of ordinary differential equations for the gas stream. 

An element for the stress components composed of 16 sub-elements (4x4) on which bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28]. This leads to a continuous C° approximation of the three variables. The velocity is approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([17]) made a mathematical analysis of the Stokes problem for this three-field formulation. They conclude that the polynomial approximations of the different variables should satisfy the generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they proved it was the case for the Marchal and Crochet element. In order to take into account the hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG). Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in [13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this problem, wiggles in the stress and the velocity field were obtained. In the SU method, the modified weighting function only applies to the convective terms in the constitutive equations. 

It is noted that the given pressure-correction method can be applied for arbitrary Mach number flows. At low Mach numbers (almost incompressible flow), the Laplacian term dominates and we recover the Poisson equation. On the other hand, at high Mach number (highly compressible flow), the convective term dominates, reflecting the hyperbolic nature of the flow. Solving the pressure-correction equation is then equivalent to solving the continuity equation for density. Thus, the pressure correction method automatically adjusts... 

The governing equations (1) and (2) are of a mixed parabolic-elliptic nature. A key feature of incompressible flow is that that the time derivative of pressure vanishes from the equations. Hence the equations do not transmit any pressure history directly, and it is as if a new pressure field is established at each step. This situation does not arise for compressible flow where, owing to the presence of the time derivative of the pressure term in the continuity equation, one can solve the coupled hyperbolic system by advancing in time. In the absence of such a term, the algebraic system of equations becomes singular. This is also why attempts to solve the incompressible flow problem as a low Mach-number, compressible-flow problem lead to ill-conditioned algebraic systems with poor algorithmic efficiency and accuracy. For a detailed discussion of these issues, see Ref. 74, p. 642. 

The first term In equation (14), representing the gross rate of biomass production, Is Identical with the function Monod (25) originally adopted "to express conveniently the relation between exponential growth rate and concentration of an essential nutrient." Such a rectangular hyperbolic function has been derived many times from various reaction mechanisms (26-30). but none has addressed the present case of continuous culture systems where y j and K have been observed to vary with temperature and dilution rate. 

We note that eq. (90) gives the (differential) yield of B for a porous catalyst at a point in a reactor, just as eq. (87) gives this for a nonporous catalyst. Thus the two equations should agree exactly for the case of small h, since this corresponds to a pore surface completely available to reaction. Making h small in (90) converts it exactly into 7) since for greater than about 3.0 the selectivity stays at a constant low level, since the hyperbolic tangent terms in (90) assume the constant values of 1.0. For this case of an active catalyst with small and long pores, eq. (90) becomes ... 

In order to describe adequately the hydrodynamics of the experimental fixed bed reactor, it is necessary to take into account the axial dispersion in the mathematical model. The time dependent continuity equation including axial dispersion for a fixed bed reactor is given by a partial differential equation (pde) of the parabolic/hyperbolic class. These types of pde s are difficult to solve numerically, resulting in long cpu times. A way to overcome these difficulties is by describing the fixed bed reactor as a cascade of perfectly stirred tank reactors. The axial dispersion is then accounted for by the number of tanks in series. For a low degree of dispersion (Bo < 50) the number of stirred tanks, N, and the Bodenstein number. Bo, are related as N Bo/2 [8].The fixed bed reactor is now described by a system of ordinary differential equations (ode s). No radial gradients are taken into account and a onedimensional model is applied. Mass balances are developed for both the gas phase and the adsorbed phase. The reactor is considered to be isothermal. 

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