The Laplace transform

The Laplace transform is essential in order to transform a partial differential equation into a total differential equation. After solving the equation the transform is inverted in order to obtain the solution to the mathematical problem in real time and space. 

The occurrence of partial differential equations in electrochemistry is due to the variation of concentration with distance and with time, which are two independent variables, and are expressed in Fick s second law or in the convective-diffusion equation, possibly with the addition of kinetic terms. As in the resolution of any differential equation, it is necessary to specify the conditions for its solution, otherwise there are many possible solutions. Examples of these boundary conditions and the utilization of the Laplace transform in resolving mass transport problems may be found in Chapter 5. 

Normally, instead of calculating the transform or its inverse, a table is consulted which gives the result directly. Table Al.l shows useful examples, some of them easy to verify. 

The Laplace transform has some properties which are extremely useful in aiding in the resolution of equations in electrochemistry and other branches of science. 

If F(t) is redefined such that F(0) = 0, then F(0) disappears. An example would be a dimensionless concentration variable 

Appendix 2 is an example of the way in which the transfer function of a flow-mixing system may be used to predict the conversion of reactant which will be achieved in that system when a reaction with first-order kinetics is occurring. This short-cut route is used extensively later in this chapter. 

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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,... 

Sufficient Conditions for the Existence of Laplace Transform Suppose/ is a function which is (1) piecewise continuous on eveiy finite intei val 0 < t exponential growth at infinity, and (3) Jo l/t)l dt exist (finite) for every finite 6 > 0. Then the Laplace transform of/exists for all complex numbers. s with sufficiently large real part. 

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

To illustrate how Laplace transforms work, consider the problem of solving Eq. (8-2), subjec t to the initial condition that = 0 at t = 0, and Cj is constant. If were not initially zero, one would define a deviation variable between and its initial value (c — Cq). Then the transfer function would be developed using this deviation variable. Taking the Laplace transform of both sides of Eq. (8-2) gives ... 

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 ... 

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

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. 

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. 

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 ... 

JLo(.v) is the Laplace transform of the output function, or system response. 

Tustin s Rule Tustin s rule, also called the bilinear transformation, gives a better approximation to integration since it is based on a trapizoidal rather than a rectangular area. Tustin s rule approximates the Laplace transform to... 

Experimentally, the absorbance A(5) of a band is measured as a function of the angle of incidence B and thus of S. Two techniques can be used to determine a(z). A functional form can be assumed for a(z) and Eqs. 2 and 3 used to calculate the Laplace transform A(5) as a function of 8 [4]. Variable parameters in the assumed form of a(z) are adjusted to obtain the best fit of A(5) to the experimental data. Another approach is to directly compute the inverse Laplace transform of A(5) [3,5]. Programs to compute inverse Laplace transforms are available [6]. 

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). 

Application of the definition shows that the Laplace transform is a linear oper-ator " this property is represented in Eqs. (3-67) and (3-68). 

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

Applying Eq. (3-68) to the right side and Eq. (3-69) to the left side provides the Laplace transform ... 

Thus the Laplace transformation constitutes a method of integration, and a table of Laplace transforms plays a role in this process that is analogous to a table of... 

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. 

Then Eqs. (3-130) are substituted into Eqs. (3-128), giving Ca. and cc as functions of time. The final expressions are not written here because we have already derived them by the Laplace transform method they are Eqs. (3-99), (3-101), and (3-103), with X2 and X3 replacing a and p. 

In the context of chemical kinetics, the eigenvalue technique and the method of Laplace transforms have similar capabilities, and a choice between them is largely dependent upon the amount of algebraic labor required to reach the final result. Carpenter discusses matrix operations that can reduce the manipulations required to proceed from the eigenvalues to the concentration-time functions. When dealing with complex reactions that include irreversible steps by the eigenvalue method, the system should be treated as an equilibrium system, and then the desired special case derived from the general result. For such problems the Laplace transform method is more efficient. 

What are the units of the Laplace transform variable s when applied to a first-order reaction ... 

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