LaPlace transformation equation

Eq. 4 is amenable to solution techniques based on the numerical inversion of Laplace-transformed equations these calculations can be performed rapidly and are therefore suitable for calibration. In Figure 1, typical soil/bentonite column predictions are shown to highlight the effect of the influent mixing zone on the spatial contaminant distributions for low-flow systems. The simulation results, which were generated for column conditions described by Khandelwal et al. (1998), indicate that the mixing zone has a significant influence on the shape of the spatial contaminant distribution and, therefore should be considered explicitly in estimating sorption parameters from spatial column data. 

Another approach presented by Yamaoka, Yano, and Tanaka, uses a fast inverse Laplace transform to generate the integrated equation data. Thus the model is described in terms of the Laplace transform equations and solved numerically. 

The flux boundary condition has to be considered while taking the Laplace transform. Equation (4.3) is solved in Maple below ... 

This result obtained from the incremental inverse bond graph can be verified by derivation of the transfer matrix H from the direct bond graph in Fig. 4.11 and by differentiating its inverse H with respect to the parameter Rm- Derivation of Laplace transformed equations from the direct bond graph yields... 

Laplace transforming Equation (A.45) and the boundary conditions (A.47), see above, we obtain... 

As we did in the membrane, again, we linearize and Laplace transform equation (8.61) to get the electrode impedance. 

We can now introduce deviation variables Sc and take the Laplace transform. Equation (13.6) can then be written as ... 

To solve this equation using Laplace transforms we use the following definition of the Laplace transform (Equation 3.79) and standard transforms of functions, e.g. Coughanowr and Koppel [9] ... 

These equations are perturbed, linearized, and Laplace-transformed from the time domain to the frequency domain to evaluate the transfer functions between various thermal-hydraulic parameters. The Laplace-transformed equations are solved simultaneously by means of a matrix equation. 

The linearized and Laplace-transformed equations of the models described above are used to evaluate the various system transfer functions as functions of the Laplace variables s = cr + jco, where a is the real part and co is the imaginary part of the complex variable s. a refers to the damping constant (or damped exponential frequency) and co refers to the resonant oscillation frequency of the system. 

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see Integral Transforms (Operational Methods) ]. The one-sided Laplace transform indicated by L[f t)] is defined by the equation L[f t)] = /(O dt. It has... 

Example The equation 3c/3f = D(3 c/3a. ) represents the diffusion in a semi-infinite medium, a. > 0. Under the boundary conditions c(0, t) = Cq, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t,... 

Equations of Convolution Type The equation u x) = f x) + X K(x — t)u(t) dt is a special case of the linear integral equation of the second land of Volterra type. The integral part is the convolution integral discussed under Integral Transforms (Operational Methods) so the solution can be accomplished by Laplace transforms L[u x)] = E[f x)] + XL[u x)]LIK x)] or... 

Other applications of Laplace transforms are given under Differential Equations. ... 

The z -transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. 

Equations readily solvable by Laplace transforms. For example ... 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of second-order reactions of Figs. 7-5n and 7-5c required numerical integrations. 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

The term in parentheses in Eq. (8-17) is zero at steady state and thus it can be dropped. Next the Laplace transform is taken, and the resulting algebraic equation solved. Denoting X s) as the Laplace transform of and X,(.s) as the transform of 4, the final transfer Function can be written as ... 

Elimination of Ci and C3 from these equations will result in the desired relation between inlet Cj and outlet Co concentrations, although not in an exphcit form except for zero or first-order reactions. Alternatively, the Laplace transform could be found, inverted and used to evaluate segregated or max mixed conversions that are defined later. Inversion of a transform hke that of Fig. 23-8 is facilitated after replacing the exponential by some ratio of polynomials, a Pade approximation, as explained in books on hnear control theory. Numerical inversion is always possible. 

For operation with an inert tracer, the material balances are conveniently handled as Laplace transforms. For a stirred tank, the differential equation... 

Differential equations and their solutions will be stated for the elementary models with the main lands of inputs. Since the ODEs are linear, solutions by Laplace transforms are feasible. 

The Laplace transform technique also allows the reduction of the partial differential equation in two variables to one of a single variable In the present case. 

Tables of Laplace ti ansforms for a large number of functions have been calculated, and can be obtained from published data. In the present example, the transformed equation is...

A transfer function is the Laplace transform of a differential equation with zero initial conditions. It is a very easy way to transform from the time to the. v domain, and a powerful tool for the control engineer. 

Find the Laplace transform of the following differential equation given ... 

G(.v) is the transfer function, i.e. the Laplace transform of the differential equation for zero initial conditions. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

The Laplace transformation converts a function of t, F(t), into a function of s, f s), where s is the transform variable. The quantity/(s) is called the Laplace transform of F(t). Equation (3-66) shows several equivalent symbolic representations of the Laplace transform of the function y = F(t). 

The Laplace transform of a derivative dy/dt is found by application of Equation (3-65) and integration by parts ... 

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

This is the procedure From the postulated kinetic scheme we write the differential rate equations. Take the Laplace transforms of the differential equations. Solve the resulting set of algebraie equations for the transforms of the concentrations. Then take the inverse transforms to obtain the coneentrations as funetions of time. 

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