Solutions to the differential equations

The steady-state solution to Pick s first law is a constant concentration or temperature gradient. Only non-steady-state solutions, having simple boundary conditions will be discussed. 

1 Constant surface concentration Cq on a semi-infinite body 

Equation (A.13) is a linear differential equation because it contains only the first power of the differentials. Consequently, any two solutions can be combined to provide a further solution. Let us start with Eq. (A.4) for an initially planar source of impurity atoms. This can be re-expressed in terms of the diffusion coefficient using Eq. (A.8) 

The problem is modelled as an infinite body, in which there is initially a constant concentration 2Cq for x 0. The initial impurity is split up into layers (planar sources) each of strength 2Cq d (Eig. A.3). Impurity reaching 

X on the right-hand side has diffused a distance of at least x from one of the planar sources, and the total concentration is given by summing the individual contributions as 

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A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Another solution to the differential equation for steady state is the follow ing, where the concentration in the supply air, Caj p, is included ... 

The solution to the differential equation at steady state in this case is... 

The restrictions on engineering constants can also be used in the solution of practical engineering analysis problems. For example, consider a differential equation that has several solutions depending on the relative values of the coefficients in the differential equation. Those coefficients in a physical problem of deformation of a body involve the elastic constants. The restrictions on elastic constants can then be used to determine which solution to the differential equation is applicable. 

For the step change condition, shown in Fig. 2.2, the initial conditions are given by y = 0, when t = 0. The solution to the differential equation, with the above boundary conditions, is given by... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

If the incident wave is x-polarized, the -component of A is zero in the xz plane (< > = 0). In this plane, therefore, the field lines are solutions to the differential equation... 

The method of weighted residuals comprises several basic techniques, all of which have proved to be quite powerful and have been shown by Finlayson (1972, 1980) to be accurate numerical techniques frequently superior to finite difference schemes for the solution of complex differential equation systems. In the method of weighted residuals, the unknown exact solutions are expanded in a series of specified trial functions that are chosen to satisfy the boundary conditions, with unknown coefficients that are chosen to give the best solution to the differential equations ... 

Baron, Manning, and Johnstone (B4) have discussed the experimental aspects of the tubular-reactor problem, and an analysis of the results is made by means of the solution to the differential equations for mass... 

The technical conditions on f are quite reasonable if a physical situation has a discontinuity, we might look for solutions with discontinuities in the function f and its derivatives. In this case, we might have to consider, e.g., piecewise-defined combinations of smooth solutions to the differential equation. These solutions might not be linear combinations of spherical harmonics. 

In the limit l->oo a Boltzmann distribution (with a nonequilibrium temperature) is established almost instantaneously and we can use an approach based on two time scales. Assuming that under these circumstances the solution to the differential equation is a(x) exp (- a(r) ) we get, using the known form of... 

Diffusion from spherical silicate samples can be studied readily by observing the loss of volatile components of the silicate as a function of time. Where the sphere is initially uniform in composition and subsequent vaporization allows one to assume a zero surface concentration of the vaporizing component thereafter, the solution to the differential equations and boundary conditions governing concentration-independent diffusion is given by... 

The subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

Find a mathematical function, i <, which is a solution to the differential equation, the Schrodinger equation. 

Let x(t) and V(t) be the actual solutions to these differential equations. In general a given algorithm will replace these differential equations by a particular set of difference equations. These difference equations will then give approximate values of x(t) and V(t) at discrete, equally spaced points in time tu t2,. .., tn where tJ+x = tj + At. The differences between the solutions to the difference equations at tN and the solutions to the differential equations at t N depend critically on the time step At. If At is too large, the system of difference equations may be unstable or be in error due to truncation effects. On the other hand, if At is too small, the solutions to the difference equations may be in error due to the accumulation of machine rounding of intermediate results. 

The parameter 8 is called the form factor or the Frank-Kamenetskii number. When a solution of Equation 13.26 exists, a stationary temperature profile can be established and the situation is stable. When there is no solution, no steady state can be established and the solid enters a runaway situation. The existence, or not, of a solution to the differential Equation 13.26, depends on the value of parameter 8, which therefore is a discriminator. The differential equation can be solved for simple shapes of the solid body, for which the Laplacian can be defined ... 

To solve Eq. (3-1), the separation-of-variables method is used. The essential point of this method is that the solution to the differential equation is assumed to take a product form... 

When we take exponentials of quantities in Equation III.3, we obtain the following solution to the differential equation represented by Equation... 

FIGURE 6 Solution to the differential equation for a CSTR with aggr ation (i.e., equation (6.123). (a) Variation in e for constant j8 and (b) variation in e for constant /3 and (c) variation in e for constant j8 and [Pg.238]

Therefore, while it is relatively easy to calculate the shear stress at the surfece of the rotating cylinder from Equation 3A.2, one can only derive an expression for the difference in shear rates at the surfaces of the inner and outer cylinders from the basic equations of flow. Additional work is required to calculate the corresponding shear rate yi and there have been several approaches to determination it. One approach has been to apply infinite series solution to the differential equation in 3A.12. 

Data obtained under these conditions can be fitted using a least-squares procedure based upon the exact solution to the differential equations describing this mechanism [37, 44]. This yields values for the complex dissociation constant Ky, and the limiting first-order rate constant (a minimum value for the second-order rate constant for complex formation can also be obtained from this analysis). Note that AId refers to the interaction between reduced P and oxidized P, a situation that is observable only by kinetic methodology. 

Solution to the Differential Equation for a First-Order Reaction 746... 

For the case of isothermal operafion with no pressure drop, we were able to obtain an analytical solution, given by equation B, which gives the reactor volume necessary to achieve a conversion X for a gas-phase reaction carried out isothermaliy in a PFR, However, in the majority of situations, analytical solutions to the ordinary differential equations appearing in (he combine step are not possible. Consequently, we include POLYMATH, or some other ODE solver such as MATLAB, in our menu in that it makes obtaining solutions to the differential equations much more palatable, ... 

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