The Laplace Transformation

The Laplace transformation has proved an effective tool for the solution of the linear heat conduction equation (2.110) with linear boundary conditions. It follows a prescribed solution path and makes it possible to obtain special solutions, for example for small times or at a certain position in the thermally conductive body, without having to determine the complete time and spatial dependence of its temperature field. An introductory illustration of the Laplace transformation and its application to heat conduction problems has been given by H.D. Baehr [2.25]. An extensive representation is offered in the book by H. Tautz [2.26]. The Laplace transformation has a special importance for one-dimensional heat flow, as in this case the solution of the partial differential equation leads back to the solution of a linear ordinary differential equation. In the following introduction we will limit ourselves to this case. 

This is known as the Laplace transform2 of the temperature and depends on s and x. We use the symbol ) when we are stating theorems, while u is an abbreviation for in the solution of concrete problems. Often () will be called 

To apply the Laplace transformation several theorems are required, which have been put together, without proofs, in Table 2.2. In addition a table of functions of t with their Laplace transforms a is also needed. This correspondence table is generated by the evaluation of the defining equation (2.111). 

Further correspondences of this type, which are important for the solution of the heat conduction equation are contained in Table 2.3. More extensive tables of correspondences can be found in the literature, e.g. [2.1], [2.26] to [2.28]. 

In order to explain the solution process we will limit ourselves to linear heat flow in the x-direction and write the heat conduction equation as 

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). 

Some authors use the letter p instead of s for the transform variable. 

We will obtain some Laplace transforms by applying the definition, Eq. (3-65). Suppose F(t) = a, where a is a constant. Then 

Next consider the exponential function, which is important in Idnetics. Let F(t) = 

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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]. 

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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