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Laplace domain

Concentration Profiles In the general case but with a hnear isotherm, the concentration profile can be found by numerical inversion of the Laplace-domain solution of Haynes and Sarma [see... [Pg.1535]

The above example shows why it is mathematically more convenient to apply step changes rather than delta functions to a system model. This remark applies when working with dynamic models in their normal form i.e., in the time domain. Transformation to the Laplace domain allows easy use of delta functions as system inputs. [Pg.546]

Example 15.8 used a delta function input in the Laplace domain to find Tfi result was Equation (15.38). Comparison with... [Pg.563]

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. [Pg.478]

The inverse transform Xp(t) in the time domain can be obtained by means of the method of indeterminate coefficients, which was presented above in Section 39.1.6. In this case the solution is the same as the one which was derived by conventional methods in Section 39.1.2 (eq. (39.16)). The solution of the two-compartment model in the Laplace domain (eq. (39.77)) can now be used in the analysis of more complex systems, as will be shown below. [Pg.488]

Do not worry if you have forgotten the significance of the characteristic equation. We will come back to this issue again and again. We are just using this example as a prologue. Typically in a class on differential equations, we learn to transform a linear ordinary equation into an algebraic equation in the Laplace-domain, solve for the transformed dependent variable, and finally get back the time-domain solution with an inverse transformation. [Pg.10]

Figure 2.1. Relationship between time domain and Laplace domain. Figure 2.1. Relationship between time domain and Laplace domain.
Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or linearized differential equation, dy/dt = f(y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y(s). The time domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. [Pg.11]

What we argue (of course it is tme) is that the Laplace-domain function Y(s) must contain the same information as y(t). Likewise, the function G(s) contains the same dynamic information as the original differential equation. We will see that the function G(s) can be "clean" looking if the differential equation has zero initial conditions. That is one of the reasons why we always pitch a control problem in terms of deviation variables.1 We can now introduce the definition. [Pg.11]

Even as we speak of time-domain analysis, we invariably still work with Laplace transform. Time-domain and Laplace-domain are inseparable in classical control. [Pg.45]

In establishing the relationship between time-domain and Laplace-domain, we use only first and second order differential equations. That s because we are working strictly with linearized systems. As we have seen in partial fraction expansion, any function can be "broken up" into first order terms. Terms of complex roots can be combined together to form a second order term. [Pg.45]

Be careful with the notation. Upper case C is for concentration in the Laplace domain. The boldface upper case C is the output matrix. [Pg.72]

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... [Pg.228]

Note In the text, we emphasize the importance of relating pole positions of a transfer function to the actual time-domain response. We should get into the habit of finding what the poles are. The time response plots are teaching tools that reaffirm our confidence in doing analysis in the Laplace-domain. So, we should find the roots of the denominator. We can also use the damp () function to find the damping ratio and natural frequency. [Pg.229]

It has been shown that linear mammillary compartment models can readily be represented by products of input and disposition functions in the Laplace domain [20], Solutions for the drug concentration or amount in any compartment are obtained by taking the inverse of the Laplace function. This approach avoids the use of differential equations and their potentially tedious solution. The La-... [Pg.77]

Concentration Profiles In the general case but with a linear isotherm, the concentration profile can be found by numerical inversion of the Laplace-domain solution of Haynes and Sarma [see Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no axial dispersion, an explicit time-domain solution is also available in the cyclic steady state for repeated injections of arbitrary duration tE followed by an elution period tE with cycle time tc = tE + tE [Carta, Chem. Eng. Sci, 43, 2877 (1988)]. For the linear driving force mechanism, the solution is... [Pg.44]

This involves obtaining the mean-residence time, 0, and the variance, (t, of the distribution represented by equation 19.4-14. Since, in general, these are related to the first and second moments, respectively, of the distribution, it is convenient to connect the determination of moments in the time domain to that in the Laplace domain. By definition of a Laplace transform,... [Pg.475]

Note that the determination of moments in the Laplace domain in this case is much easier than in the time domain, since tedious integration of a complicated function is replaced by the easier operation of differentiation. The result for 0 we could have anticipated, but not the very simple result for tr2 in equation 19.4-26. [Pg.476]

As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give functions of time is a time domain teehnique. Use of Laplace transforms to characterize the dynamics of systems is a Laplace domain technique. Frequency response methods provide another approaeh to the problem. [Pg.13]

I a this section we will study the time-dependent behavior of some chemical. engineering systems, both openloop (without control) and closedloop (with controllers included). Systems will be described by diflerential equations, and solutions will be in terms of time-dependent functions. Thus, our language for this part of the book will be English. In the next part we will learn a little Russian in order to work in the Laplace domain where the notation is more simple than in English. Then in Part V we will study some Chinese because of its ability to easily handle much more complex systems. [Pg.165]

Thus Laplace transformation converts functions from the time domain (where t is the independent variable) into the Laplace domain (where s is the independent variable). The advantages of using this transformation will become clear later in this chapter. [Pg.304]

After transforming equations into the Laplace domain and solving for output variables as functions of s, we sometimes want to transform back into the time domain. This operation is called mission or inverse Laplace tran ormation. We are translating from Russian into English. We will use the notation. [Pg.308]

The result is the most useful of all the Laplace transformations. It says that the operation of differentiation in the time domain is replaced by multiplication by s in the Laplace domain, minus an initial condition. This is where perturbation variables become so useful. If the initial condition is the steadystate operating level, all the initial conditions like are equal to zero. Then simple multiplication by s is equivalent to differentiation. An ideal derivative unit or a perfect differentiator can be represented in block-diagram form as shown in Fig. 9.3. [Pg.313]

The above can be generalized to an Nth-order derivative with respect to time. In going ffom the time domain into the Laplace domain, d xjd is replaced by 5". Therefore an Nth-order differential equation becomes an Nth-order algebraic equation. [Pg.313]

The operation of integration is equivalent to division by s in the Laplace domain, using zero initial conditions. Thus, integration is the inverse of differentiation. Figure 9.4 gives a block-diagram representation. [Pg.315]

Thus time delay or deadtime in the time domain is equivalent to multiplication by in the Laplace domain. [Pg.316]

By definition, steadystate corresponds to the condition that all time derivatives are equal to zero. Since the variable s replaces d/dt in the Laplace domain, letting s go to zero is equivalent to the steadystate gain. [Pg.328]


See other pages where Laplace domain is mentioned: [Pg.488]    [Pg.81]    [Pg.84]    [Pg.79]    [Pg.62]    [Pg.13]    [Pg.302]    [Pg.302]    [Pg.308]    [Pg.309]    [Pg.312]    [Pg.314]    [Pg.314]    [Pg.316]    [Pg.316]    [Pg.317]    [Pg.327]    [Pg.328]    [Pg.329]    [Pg.335]    [Pg.340]    [Pg.340]    [Pg.344]   
See also in sourсe #XX -- [ Pg.488 ]

See also in sourсe #XX -- [ Pg.263 ]




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