Partial differential equations temperature

Packed Red Reactors The commonest vessels are cylindrical. They will have gradients of composition and temperature in the radial and axial directions. The partial differential equations of the material and energy balances are summarized in Table 7-10. Example 4 of Modeling of Chemical Reactions in Sec. 23 is an apphcation of such equations. 

If the reaetion rate is a funetion of pressure, then the momentum balanee is eonsidered along with the mass and energy balanee equations. Both Equations 6-105 and 6-106 are eoupled and highly nonlinear beeause of the effeet of temperature on the reaetion rate. Numerieal methods of solution involving the use of finite differenee are generally adopted. A review of the partial differential equation employing the finite differenee method is illustrated in Appendix D. Eigures 6-16 and 6-17, respeetively, show typieal profiles of an exo-thermie eatalytie reaetion. 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

In the present case the state variables are most conveniently chosen as crystal size, moments of the distribution and solution concentration (which, as shown above, give rise to ordinary rather than partial differential equations, equations 7.14-7.18) while the control is solution temperature. The performance measure adopted is to maximize the terminal size of the S-crystals (i.e. those originating as added seeds those from nuclei being A -crystals )... 

It is impossible to solve these partial differential equations with the varied input conditions resulting from the California cycle. A computer was used in the study, where distance is divided into increments of 0.08 to 0.30 in. in thickness, and time is divided into increments of 0.4 sec. They obtained the temperature profile in the bed as a function of time from a cold start. This procedure could result in a very long computation, as the California cycle has a duration of 16 min, requiring 2400 time intervals and computations. 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

This partial differential equation is most conveniently solved by the use of the Laplace transform of temperature with respect to time. As an illustration of the method of solution, the problem of the unidirectional flow of heat in a continuous medium will be considered. The basic differential equation for the X-direction is ... 

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) ... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

These coupled partial differential equations can be solved for the temperature and composition at any point in the catalyst bed by using numerical procedures to solve the correspon-... 

Let us consider the semi-infinite (thermally thick) conduction problem for a constant temperature at the surface. The governing partial differential equation comes from the conservation of energy, and is described in standard heat transfer texts (e.g. Reference [13]) ... 

At the macroscopic level, matter is described in terms of fields such as the velocity, the mass density, the temperature, and the chemical concentrations of the different molecular species composing the system. These fields evolve in time according to partial differential equations of hydrodynamics and chemical kinetics. 

Similar considerations concern the irreversible processes of diffusion and reaction in mixtures [5]. A system of M different molecular species is described by the three components of velocity, the mass density, the temperature, and (M — 1) chemical concentrations and is ruled by M + 4 partial differential equations. The M — 1 extra equations govern the mutual diffusions and the possible chemical reactions... 

The mathematical model developed in the preceding section consists of six coupled, three-dimensional, nonlinear partial differential equations along with nonlinear algebraic boundary conditions, which must be solved to obtain the temperature profiles in the gas, catalyst, and thermal well the concentration profiles and the velocity profile. Numerical solution of these equations is required. 

In our original system of partial differential equations, to obtain a pseudohomogeneous model the two energy balances can be combined by eliminating the term (Usg/Vb)(Ts — Tg), that describes the heat transfer between the solid and the gas. If the gas and solid temperatures are assumed to be equal (Ts = Tg)19 and the homogeneous gas/solid temperature is defined as T, the combined energy balance for the gas and solid becomes... 

A system of conservation equations, whose solution describes the velocity, temperature, and composition fields. These equations usually take the form of partial differential equations that are derived from physical laws governing the conservation of mass, momentum, and energy. 

It is readily apparent that the system of equations is a coupled system of nonlinear partial differential equations. The independent variables are time t and the spatial coordinates (e.g., z, r, 0). For the fluid mechanics alone, the dependent variables are mass density, p, pressure p, and V. In addition the energy equation adds either enthalpy h or temperature T. Finally the mass fractions of chemical species are also dependent variables. 

The boundary-layer equations represent a coupled, nonlinear system of parabolic partial-differential equations. Boundary conditions are required at the channel inlet and at the extremeties of the y domain. (The inlet boundary conditions mathematically play the role of initial conditions, since in these parabolic equations x plays the role of the time-like independent variable.) At the inlet, profiles of the dependent variables (w(y), T(y), and Tt(y)) must be specified. The v(y) profile must also be specified, but as discussed in Section 7.6.1, v(y) cannot be specified independently. When heterogeneous chemistry occurs on a wall the initial species profile Yk (y) must be specified such that the gas-phase composition at the wall is consistent with the surface composition and temperature and the heterogeneous reaction mechanism. The inlet pressure must also be specified. 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

Therefore, any result that follows from considerations of the form of Fick s second law applies to evolution of heat as well as concentration. However, the thermal and mass diffusion equations differ physically. The mass diffusion equation, dc/dt = V DVc, is a partial-differential equation for the density of an extensive quantity, and in the thermal case, dT/dt = V kVT is a partial-differential equation for an intensive quantity. The difference arises because for mass diffusion, the driving force is converted from a gradient in a potential V/u to a gradient in concentration Vc, which is easier to measure. For thermal diffusion, the time-dependent temperature arises because the enthalpy density is inferred from a temperature measurement. 

Continuum Dynamics. In this appruach, fluid properties, such as velocity, density, pressure, temperature, viscosity, and conductivity, among others, arc assumed to be physically meaningful functions of three spatial variables t. . 1 . and. n. and lime i. Nonlinear partial differential equations are set up to relate these variables. Such equations have nil general solutions even for the most restrictive boundary conditions. Bui solutions are carried out for very idealized flows. Couetle flow is one of these. See Fig. I. 

We will consider a dispersed plug-flow reactor in which a homogeneous irreversible first order reaction takes place, the rate equation being 2ft = k, C. The reaction is assumed to be confined to the reaction vessel itself, i.e. it does not occur in the feed and outlet pipes. The temperature, pressure and density of the reaction mixture will be considered uniform throughout. We will also assume that the flow is steady and that sufficient time has elapsed for conditions in the reactor to have reached a steady state. This means that in the general equation for the dispersed plug-flow model (equation 2.13) there is no change in concentration with time i.e. dC/dt = 0. The equation then becomes an ordinary rather than a partial differential equation and, for a reaction of the first order ... 

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