Laplace transform, process transfer

In Chapters 3-6, we will consider analytical solutions of linear dynamic models using Laplace transforms and transfer functions. These useful techniques allow dynamic response characteristics to be analyzed on a more systematic basis. They also provide considerable insight into common characteristics shared by complex processes. 

Chapters 2 and 3 have considered dynamic models in the form of ordinary differential equations (ODE). In this chapter, we introduce an alternative model form based on Laplace transforms the transfer function model. Both types of models can be used to determine the dynamic behavior of a process after changes in input variables. The transfer function also plays a key role in the design and analysis of control systems, as will be considered in later chapters. 

Part II (Chapters 3 through 7) is concerned with the analysis of the dynamic (unsteady-state) behavior of processes. We still rely on the use of Laplace transforms and transfer functions, to characterize the dynamic behavior of linear systems. However, we have kept analytical methods involving transforms at a minimum and... 

Instead of spacing out in the Laplace-domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. But wouldn t it be nice if we could actually trace the time profile without having to do the reverse Laplace transform ourselves Especially the response with respect to step and impulse inputs Plots of time domain dynamic calculations are extremely instructive and a useful learning tool.1... 

A particular vessel behavior sometimes can be modelled as a series or parallel arrangement of simpler elements, for example, some combination of a PFR and a CSTR. Such elements can be combined mathematically through their transfer functions which relate the Laplace transforms of input and output signals. In the simplest case the transfer function is obtained by transforming the linear differential equation of the process. The transfer function relation is... 

Most of you have probably been exposed to Laplace transforms in a mathematics course, but we will lead off this chapter with a brief review of some of the most important relationships. Then we will derive the Laplace transformations of commonly encountered functions. Next we will develop the idea of transfer functions by observing what happens to the differential equations describing a process when they are Laplace-transformed. Finally, we will apply these techniques to some chemical engineering systems. 

Our primary use of Laplace transformations in process control involves representing the dynamics of the process in terms of "transfer functions." These are output-input relationships and are obtained by Laplace-transfonning algebraic... 

J0i Use Laplace transforms to prove mathematically that a P controller produces steadystate ofiMt and that a PI controller does not. The disturbance is a step change in the load variable. The process openloop transfer functions, Gm and G[, are both liist-order lags with dUTerent gains but identical time constants. 

Notice that the right-hand side of Eq. (34) is equal to the ratio of the transformed concentration at the second measurement point to the transformed concentration at the first measurement point. In the terminology of control engineering, this quantity is the transfer function of the system between Xo and Xm- The Laplace-transform method is possible because the diffusion equation is a linear differential equation. Thus, the right-hand side of Eq. (34) could in principle be used in a control-system analysis of an axial-dispersion process. 

In general, it is not possible to characterize all processes exactly, so several approximations have to be made. Normally one assumes that the dynamic characteristic can be reproduced by a process of the first order plus dead time. The Laplace transformation for this assumption (transfer to the s plane) is approximated ... 

Appendix B. Laplace Transform Method Solution for the Application of a Constant Potential to a Simple Charge Transfer Process at Spherical Electrodes When the Diffusion Coefficients of Both Species are Equal... 

Block Diagram Analysis One shortcoming of this feedforward design procedure is that it is based on the steady-state characteristics of the process and, as such, neglects process (Ramies (i.e., how fast the controlled variable responds to changes in the load and manipulated variables). Thus, it is often necessary to include dynamic compensation in the feedforward controller. The most direct method of designing the FF dynamic compensator is to use a block dir rram of a general process, as shown in Fig. 8-34, where G, represents the disturbance transmitter, (iis the feedforward controller, Cj relates the disturbance to the controlled variable, G is the valve, Gp is the process, G is the output transmitter, and G is the feedback controller. All blocks correspond to transfer fimetions (via Laplace transforms). 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

We had no difficulty in formulating the problem. However, the solution process is considerably involved and remains beyond the scope of this text. For example, the use of Laplace transforms (see Chapter 7 of Conduction Heat Transfer by Arpaci) conveniently leads to... 

Consider the block diagram of a direct digital feedback control loop shown in Figure 29.9. Such loops contain both continuous- and discrete-time signals and dynamic elements. Three samplers are present to indicate the discrete-time nature of the set point j/Sp( ), control command c(z), and sampled process output y(z). The continuous signals are denoted by their Laplace transforms [i.e., y(s), Jn(s), and d(s)]. Furthermore, the continuous dynamic elements (e.g., hold, process, disturbance element) are denoted by their continuous transfer functions, H(s), Gp(s), and GAs), respectively. For the control algorithm, which is the only discrete element, we have used its discrete transfer function, D(z). 

In the next three chapters we develop methods of analysis of dynamic systems, both openloop and closedloop, using Laplace transformation. This form of system representation is much more compact and convenient than the time-domain representation. The Laplace-domain description of a process is a transfer function. This is a relationship between the input to a system and the output of the system. Transfer functions contain all the steady-state and dynamic information about a process in a very compact form. 

In this chapter we demonstrate the significant computational and notational advantages of Laplace transforms. The techniques involve finding the transfer function of the openloop process, specifying the desired performance of the closedloop system (process plus controller), and finding the feedback controller transfer function that is required to do the job. 

Openloop process transfer function. These three ODEs are linear, so we do not have to linearize. Converting to perturbation variables, Laplace transforming, and solving for the transfer function between the controlled variable T, and the manipulated variable Q give... 

The mass transfer process is again governed by equation 10.66, but the third boundary condition is applied at y = L, the film thickness, and not at y = oo. As before, the Laplace transform is then ... 

This equation describes many transient heat and mass transfer processes, such as the diffusion of a solute through a slab membrane with constant physical properties. The exact solution, obtained by either the Laplace transform, separation of variables (Chapter 10) or the finite integral transform (Chapter 11), is given as... 

Experience in the process industries indicates that there are a limited number of expected dynamic behaviors that actually influence the controller design step. These behaviors can be categorized using the step response and are based on a transfer function representation of the process model, which is assumed to be linear or a linear approximation of a nonlinear model. A transfer function is found by taking the Laplace transform of the ordinary differential equation that describes the system the mathematical definition of the Laplace transform is... 

Laplace transform used in the development of transfer functions, which are the most widely used model form in process control studies. The Laplace transform converts an ordinary dilferential equation (ODE) to an algebraic equation and, likewise, converts a partial dilferential equation (PDE) into an ordinary dilferential equation (ODE). 

The stable transfer matrices Tu s), T is) and T2i s) are determined by the plant, the Youla parameter Q s) is a free stable transfer matrix. In the optimization, the Youla parameter 5) is expressed as a finite series expansion in terms of suitable fixed transfer matrices qi s) and variable coefficients x [4], In the evaluation of the cost function, the time-domain equivalent of (16) is needed which for a better numerical efficiency can be reformulated via Laplace-transforms to avoid time intensive computations during the optimizations (see e.g. [44, 45]). The advantage of this formulation is that it is linear in the unknown x, and all other quantities can be computed before the iterations performed in the optimization process. This leads to... 

It follows from the last example that when controlling the potential, its value is to be related to i and surface concentration of EAC. The latter term appears in the kinetic equation. For this purpose, it is necessary to know mechanism and kinetic parameters of the charge transfer process. However, this complicated way may be avoided, if the experimentally obtained i(t)-function is utilized. Then, to find concentration profiles, the Laplace transform of this function is to be obtained ... 

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