Time dependent analytical solutions

In general, fiiU time-dependent analytical solutions to differential equation-based models of the above mechanisms have not been found for nonhnear isotherms. Only for reaction kinetics with the constant separation faclor isotherm has a full solution been found [Thomas, y. Amei Chem. Soc., 66, 1664 (1944)]. Referred to as the Thomas solution, it has been extensively studied [Amundson, J. Phy.s. Colloid Chem., 54, 812 (1950) Hiester and Vermeiilen, Chem. Eng. Progre.s.s, 48, 505 (1952) Gilliland and Baddonr, Jnd. Eng. Chem., 45, 330 (1953) Vermenlen, Adv. in Chem. Eng., 2, 147 (1958)]. The solution to Eqs. (16-130) and (16-130) for the same boimdaiy condifions as Eq. (16-146) is... 

A closed form similarity solution for the nonlinear time-dependent slow-flow equations has been used as the basis for a simple, time-dependent, analytic model of localized ignition which requires minimal chemical and physical input (8). As a fundamental part of the model, there are two constants which must calibrated the radii, or fraction of the time-dependent simi-... 

For a given extent of reaction, Eq. (3-33) is an equation with the two unknowns r and d. The procedure, in essence, is to measure F at two times and to solve the two simultaneous equations. In practice the problem is more difficult than this because an analytical solution cannot be obtained moreover d is itself dependent upon time. Swain " constructed tables of d (and of log d) as a function of r for three different extents of reaction. Curves of log d vs. log r are plotted. The curve... 

We have given an analytical method of deriving a time-dependent solution to our problem that is complicated but illustrates an important method. Frequently, steady state solutions are all that is needed. 

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 result is valid for variable but not for variable p. It governs a PER with a time-dependent inlet concentration but with other properties constant. The final simplification supposes that is constant so that u is constant. Then Equation (14.13) has a simple analytical solution ... 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

Another commonly used approximation for which there is an analytical solution is to assume that the root acts as a zero sink for uptake. Here the solute concentration at the root surface is taken to be zero and uptake is therefore completely controlled by the diffusive flux to the root (21,40,41). The implicit assumption is that root uptake is very rapid in comparison to resupply by transport and hence the root very rapidly depletes the solute concentration at the root surface to zero and maintains it there. The validity of this assumption depends on the value of X and it is inapplicable unless X is greater than or about 10 (38). For such large X, there is a nondimensional critical time (/,.) after which it is reasonable to assume a zero sink (38,42). Approximate values of /, are... 

ANALYTICAL SOLUTIONS FOR TIME DEPENDENT MELTING MODELS... 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

Numerous systems in science change with time or in space plants and bacterial colonies grow, chemicals react, gases diffuse. The conventional way to model time-dependent processes is through sets of differential equations, but if no analytical solution to the equations is known, so that it is necessary to use numerical integration, these may be computationally expensive to solve. 

The Fokker-Planck equation is a partial differential equation. In most cases, its time-dependent solution is not known analytically. Also, if the Fokker-Planck equation has more than one state variable, exact stationary solutions are... 

Blowdown of Gas Discharge through Orifice An analytic solution is available for blowdown (time-dependent discharge) of an ideal gas from a tank. The time-varying mass of gas in the tank mr is the product of the tank volume VT and the density p ... 

The process sketched out in this section can be used to create an adsorber design. There are often times when one wants to examine the performance of an existing unit. When performance analysis is needed there are several alternatives. Depending on the specifics of the problem there may be an analytical solution for the adsorption problem and that may enable the creation of a satisfactory descrip-hon of the process to use in understanding phenomena that are observed in operation. 

If the Hamiltonian H does not depend on time, this equation has the analytic solution ... 

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