Steady-State Problems

Note that the above approximation is a first order approximation. If we were to use a central difference, we would increase the order, but contrary to what is expected, this choice will adversely affect the accuracy and stability of the solution due to the fact the information is forced to travel in a direction that is not supported by the physics of the problem. How convective problems are dealt will be discussed in more detail later in this chapter. The following sections will present steady state, transient and moving boundary problems with examples and applications. 

This section will illustrate the tools taught in the above sections in the form of examples applied to steady state problems. Example 8.3 applies the finite difference method to a simple one-dimensional fin cooling problem and illustrates the nature of the system of equations that is normally achieved. Example 8.4 present a 2D compression molding problem where an iterative solution method is introduced. 

Steady-state temperature profile of a fin of uniform cross-sectional area. In a fin 

For a material with constant properties and heat transfer coefficient, the energy equation is reduced to 

Let s pretend that this solution is not known and proceed to arrive at a solution using finite differences. Here, the first step is to create the grid, which is illustrated in Fig. 8.5. For simplicity, we are going to use only n = 5 points (or nodes). The grid size is given by Ax = L/(n — 1). The FD expression of eqn. (8.35) is written as 

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Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

In steady-state problems 6/S.l = 1 and the time-dependent term in the residual is eliminated. The steady-state scheme will hence be equivalent to the combination of Galerkin and least-squares methods. 

Law Simplified flux equations that arise from Eqs. (5-181) and (5-182) can be used for nnidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions and I Aft at the left-hand and right-hand sides of a hypothetical layer having thickness Az. The principal restric tion of the following equations is that the concentration and diffnsivity are assumed to be constant. As written, the flux is positive from left to right, as depic ted in Fig. 5-25. 

Equations (3.77) and (3.78) give the solutions to the steady-state problem, given... 

The method of false transients converts a steady-state problem into a time-dependent problem. Equations (4.1) govern the steady-state performance of a CSTR. How does a reactor reach the steady state There must be a startup transient that eventually evolves into the steady state, and a simulation of... 

Find the analytical solution to the steady-state problem in Example 4.2. 

The accumulation term is zero for steady-state processes. The accumulation term is needed for batch reactors and to solve steady-state problems by the method of false transients. 

The program can solve both steady-state problems as well as time-dependent problems, and has provisions for both linear and nonlinear problems. The boundary conditions and material properties can vary with time, temperature, and position. The property variation with position can be a straight line function or or a series of connected straight line functions. User-written Fortran subroutines can be used to implement more exotic changes of boundary conditions, material properties, or to model control systems. The program has been implemented on MS DOS microcomputers, VAX computers, and CRAY supercomputers. The present work used the MS DOS microcomputer implementation. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

Of course, depending on the system, the optimum state identified by the second entropy may be the state with zero net transitions, which is just the equilibrium state. So in this sense the nonequilibrium Second Law encompasses Clausius Second Law. The real novelty of the nonequilibrium Second Law is not so much that it deals with the steady state but rather that it invokes the speed of time quantitatively. In this sense it is not restricted to steady-state problems, but can in principle be formulated to include transient and harmonic effects, where the thermodynamic or mechanical driving forces change with time. The concept of transitions in the present law is readily generalized to, for example, transitions between velocity macrostates, which would be called an acceleration, and spontaneous changes in such accelerations would be accompanied by an increase in the corresponding entropy. Even more generally it can be applied to a path of macrostates in time. 

Most plant simulations have been steady-state simulations. This is to be expected, since just as a baby must learn to crawl before he can walk, so the simpler steady-state problems must be solved before the unsteady-state ones can be tackled. However, unsteady- state plant simulations are being attempted, and undoubtedly sometime in the future this will be a common tool for plant designers. 

A radically different approach to the steady-state problem was investigated by Hsing (H6). In this approach the steady-state flow problem was formulated as the following constrained minimization problem ... 

Steady state problems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name steady state. Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 

The steady-state problem yields a system of simultaneous linear algebraic equations that can be solved by Gaussian elimination and back substitution. I shall turn now to calculating the time evolution of this system, starting from a phosphate distribution that is not in steady state. In this calculation, assume that the phosphate concentration is initially the same in all reservoirs and equal to the value in river water, 10 I 3 mole P/m3. How do the concentrations evolve from this starting value to the steady-state values just calculated ... 

This dynamic approach to equilibrium method is used in later examples to illustrate its further application to the solution of complex steady state problems. 

We have chosen the initial conditions as the reference vector. For steady-state problems, o = 0 and any non-zero inlet vector can be chosen. 

Solution of the entire pseudo-steady state problem (commonly referred to as the shrinking core model) is achieved by analytical integration of eqn. (53) and substitution of the result into eqn. (55), subsequently eliminating the unknown Ca by the use of eqn. (54). Substitution into eqn. (56) then gives the overall reaction rate in terms of CAg, and r. This result is not particuleirly useful, however, until the shrinking core radius, r, is related to time. Recalling the chemical stoichiometric relationship [eqn. (50)] the rate of consumption of A in terms of the core radius is... 

Thus far diffusion in nonflow systems has been discussed. We now turn our attention to forced-convection problems. Only steady state problems are considered here, and it is assumed that they can all be described by the differential equation [see Eq. (50)]... 

In addition to the vorticity transport equation, a relationship between vorticity and stream function can be developed for two-dimensional steady-state problems. Continuing to use the r-6 plane as an example, the stream function is defined to satisfy the continuity equation exactly (Section 3.1.3),... 

For this steady-state problem, there can be no net mass entering or exiting any specified volume. Does the result from the previous question obey overall mass conservation ... 

Here, for a steady-state problem and parallel flow, there is no acceleration. In general, the substantial derivative for the cylindrical system is... 

Solve the nondimensional steady-state problem and plot the radial nondimensional velocity and temperature profiles versus the nondimensional radius. 

If available, use Twopnt or other user-oriented boundary-value software to solve the steady-state problem. 

The plug-flow problem is formulated as a steady-state problem—that is, nothing varies as a function of time. Consequently the surface composition at any point on the channel... 

In the steady stagnation-flow formulation the thermodymanic pressure may be assumed to be constant and treated as a specified parameter. The small pressure variations in the axial direction, which may be determined from the axial momentum equaiton, become decoupled from the system of governing equations (Section 6.2). The small radial pressure variations associated with the pressure-curvature eigenvalue A are also presumed to be negligible. While this formulation works very well for the steady-state problem, it can lead to significant numerical difficulties in the transient case. A compressible formulation that retains the pressure as a dependent variable (not a fixed parameter) relieves the problem [323],... 

I. The steady-state problem. Diffusion and reaction, Chem. Eng. Sci. 1957, 6, 262-268. 

For steady state problems the following two types of spatial boundary conditions are identified ... 

Direct use of Fick s first law is limited to steady-state (or nearly steady-state) problems in which the variation of Ac /Ax over the concentration range of concern can be neglected. 

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