Analytical solution steady-state

Exact Analytical Solution (Steady-State Approximation)... 

The above ordinary differential equations (ODEs), Eqs. (19-11) and (19-12), can be solved with an initial condition. For an isothermal first-order reaction and an initial condition, C(0) = 0, the linear ODE may be solved analytically. At steady state, the accumulation term is zero, and the solution for the effluent concentration becomes... 

In the active parts of a gas diffusion electrode, the pore electrolyte is contained in the small pores between the catalyst particles and the carbon support particles. The larger pores are then filled with gas. Gas has to diffuse through a thin film of electrolyte and in the small pores that contain the pore electrolyte. This introduces an additional mass transfer resistance that can be described with an agglomerate model. Such a model is described with a diffusion-reaction equation for the gaseous species dissolved in the pore electrolyte with the charge transfer reactions as source or sink. The solution to this reaction diffusion model is used to calculate new reaction terms that replace the reaction terms in the balance of charge, material, mass, and energy. These models can often be solved analytically for steady-state if the reactions are of... 

In many books, radial flow theory is studied superficially and dismissed after cursory derivation of the log r pressure solution. Here we will consider single-phase radial flow in detail. We will examine analytical formulations that are possible in various physical limits, for different types of liquids and gases, and develop efficient models for time and cost-effective solutions. Steady-state flows of constant density liquids and compressible gases can be solved analytically, and these are considered first. In Examples 6-1 to 6-3, different formulations are presented, solved, and discussed the results are useful in formation evaluation and drilling applications. Then, we introduce finite difference methods for steady and transient flows in a natural, informal, hands-on way, and combine the resulting algorithms with analytical results to provide the foundation for a powerful write it yourself radial flow simulator. Concepts such as explicit versus implicit schemes, von Neumann stability, and truncation error are discussed in a self-contained exposition. 

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

In principle, given expressions for the crystallization kinetics and solubility of the system, equation 9.1 can be solved (along with its auxiliary equations -Chapter 3) to predict the performance of continuous crystallizers, at either steady- or unsteady-state (Chapter 7). As is evident, however, the general population balance equations are complex and thus numerical methods are required for their general solution. Nevertheless, some useful analytic solutions for design purposes are available for particular cases. 

This simple steady-state form is amenable to analytical solution and the application of an integrating factor followed by subsequent integration yields... 

Continuous flow devices have undergone careful development, and mixing chambers are very efficient. Mixing is essentially complete in about 1 ms, and half-lives as short as 1 ms may be measured. An interesting advantage of the continuous flow method, less important now than earlier, is that the analytical method need not have a fast response, since the concentrations are at steady state. Of course, the slower the detection method, the greater the volumes of reactant solutions that will be consumed. In 1923 several liters of solution were required, but now reactions can be studied with 10-100 mL. 

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. 

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

Set the time derivatives in Example 12.6 to zero to find the steady-state design equations for a CSTR with a Michaelis-Menten reaction. An analytical solution is possible. Find the solution and compare it with the solution in Example 12.3. Under what conditions does the quasisteady solution in Example 12.3 become identical to the general solution in Example 12.6 ... 

Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

Assuming constant coefficients, both the dynamic and steady-state equations describing this system can be solved analytically, but the case of varying coefficients requires solution by digital simulation. 

The analytical solution shows that the approach to steady state is very rapid when Vq is small and that the concentration in the tank is always constant, when starting with a relatively empty tank. 

For completeness it should be mentioned that some of the theoretical conclusions for SECMIT are analogous to earlier treatments for the transient and steady-state response for a membrane-covered inlaid disk UME, which was investigated for the development of microscale Clark oxygen sensors [62-65]. An analytical solution for the steady-state diffusion-limited problem has also been proposed [66,67]. 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

Partial) dialysis in flow analysis. The sample solution flows along one side of the membrane, while the analyser solution passing (often in counter-current) on the other side takes up the diffused components from the sample. A dynamic equilibrium is reached (under steady-state conditions) in the leaving analyser solution, which is then analysed and from the result of which the analyte content can be derived via calibration with standard solutions treated in exactly the same way. This is a common procedure, e.g., in Technicon AutoAnalyzers, and has also been applied in haemoanalysis by Ammann et al.154 as described above. 

Under steady-state conditions, equation 14.10 has a simple analytical solution, which allows the calculation of the pressure Pr at several radial distances from the well ... 

The initial condition is C(0) = Cjs, where Cjs is the value of the steady state solution. The inlet concentration is a function of time, Cin = Cm(t), and will become our input. We present the analytical results here and will do the simulations with MATLAB in the Review Problems. 

PESTAN (12) is a dynamic TDE soil solute (only) model, requiring the steady-state moisture behavior components as user input. The model is based on the analytic solution of equation (3), and is very easy to use, but has also a limited applicability, unless model coefficients (e.g., adsorption rate) can be well estimated from monitoring studies. Moisture module requirements can be obtained by any model of the literature. 

If steady state is assumed the two equations become algebraic and direct analytical solution is possible. An intermediate situation can exist if the amount in the water is small compared to the amount in the sediment, a steady state water situation can be assumed. The water differential equation then becomes algebraic and can be substituted into the sediment equation. Details of these equations and their solutions are given by Mackay et al (15)... 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

The first equation could be integrated analytically and the result substituted into the second equation which may not be integrable. Therefore, the plotted solution was obtained by numerical integration of the pair. Both the cells and the penicillin producing entity reach steady state in a few hundred hours... 

Analytical solution is possible only for the first stage. The plotted and tabulated results for four stages are obtained with ODE, with the constants C0 = 1, k = 0.5 and x = 5, and with solvent only in the tank at the start. The later stages approach steady state more slowly. ... 

A large number of analytical solutions of these equations appear in the literature. Mostly, however, they deal only with first order reactions. All others require solution by numerical or other approximate means. In this book, solutions of two examples are carried along analytically part way in P7.02.06 and P7.02.07. Section 7.4 considers flow through an external film, while Section 7.5 deals with diffusion and reaction in catalyst pores under steady state conditions. 

Steady-state mathematical models of single- and multiple-effect evaporators involving material and energy balances can be found in McCabe et al. (1993), Yannio-tis and Pilavachi (1996), and Esplugas and Mata (1983). The classical simplified optimization problem for evaporators (Schweyer, 1955) is to determine the most suitable number of effects given (1) an analytical expression for the fixed costs in terms of the number of effects n, and (2) the steam (variable) costs also in terms of n. Analytic differentiation yields an analytical solution for the optimal n, as shown here. 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

Other geometries can be substantially more difficult to solve [45], mostly when the one-dimensional approach is not appropriate. The steady-state analytical solution through a multilayer medium towards a disc surface is available [47], but for most problems (especially transient ones) only numerical simulation is feasible. 

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