Perturbation variables

We will find it very useful in practically all the linear dynamics and control studies in the rest of the book to look at the changes of variables away from steadystate values instead of the absolute variables themselves. Why this is useful will become apparent in the discussion below. 

Since the total variables are functions of time, x, , their departures from the steadystate values x will also be functions of time, as sketched in Fig. 6.3. These departures from steadystate are called perturbations or perturbation variables. We will use, for the present, the symbol xf,. Thus the perturbation in x is defined  

The equations describing the linear system can now be expressed in terms of these perturbation variables. When this is done, two very useful results occur  

The terms in the ordinary differential equation with just constants in them drpp out. 

1 The initial conditions for the perturbation variables are all equal to zero if the starting point is the steadystate operating condition around which the equations have been linearized. 

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System stability ean also be analysed in terms of the linearised differential model equations. In this, new perturbation variables for concentration C and temperature T are defined. These are defined in terms of small deviations in... 

Bar above symbol refers to dimensionless variable Refers to perturbation variable, superficial velocity or stripping section... 

Deviation variables are analogous to perturbation variables used in chemical kinetics or in fluid mechanics (linear hydrodynamic stability). We can consider deviation variable as a measure of how far it is from steady state. 

Therefore the last term in Eq. (6.37) is equal to zero. We end up with a linear ordinary differential equation with constant coefficients in terms of perturbation variables. 

Since we will be using perturbation variables most of the time, we will often not bother to use the superscript p. It will be understood that whenever we write the linearized equations for the system all variables will be perturbation variables. Thus Eqs. (6.39) and (6.41) can be written... 

Note that the initial conditions of all these perturbation variables are zero since all variables start at the initial steadystate values. This will prove to simplify things significantly when we start using Laplace transforms in Part IV. 

In this example we have used total variables. If we convert Eq. (6.46) into perturbation variables, we get... 

The last term in the equation above is lero. Therefore Eqs. (6.54) and (6,46) are identical, except one is in terms of total variables and the other is in terms of perturbations, Whenever the original ODE is already Hnear, either total or perturbation variables can be used. Initial conditions will, of course, differ by the steady stale values of all variables. 

L2. Linearize the ODE describing the conical Utnk modeled in Prob. 3.1 and convert to perturbation variables. 

Derive the second-order equation describing the ctosedloop process in terms of perturbation variables. Show that the damping coefllcient is... 

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. 

Example 9Ji, Consider the isothermal CSTR of Example 6.6. The equation describing the system in terms of perturbation variables is... 

The variables can be either total or perturbation variables since the equations are linear (all Jc s and r s arc constant). Let us use perturbation variables, and therefore the initial conditions for all variables are zero. 

Example 9Ji. The nonisothermal CSTR modeled in Sec. 3.6 can be linearized (sec Prob. 6.12) to give two linear ODEs in teinis of perturbation variables. 

Equation (9.107) is written in terms of total variables. If we are dealing with perturbation variables, we simply drop the bias term. Laplace transforming gives... 

Equation (9,110) is in terms of total variables. Converting to perturbation variables and Laplaoe-transforming give... 

Note the above equations arc in terms of perturbation variables. 

The Oi/s are all constants made up of the steadystate holdups, flow rates and compositions. Table 12.2 gives their values. The variables in Eqs. (12.72) to (12.79) are all perturbation variables. is the hydraulic constant, the linearized relationship between a perturbation in liquid holdup on a tray,, and the perturbation in the liquid flow rate L leaving the tray. 

K is the linearized relationship between the perturbations in vapor composition y and liquid composition x,. Note that this is not the same K value" used in VLE calculations which relates total x and y variables. The K s in Table 12.2 are the slopes of the equilibrium line and relate perturbation variables. 

The input disturbance for a pulse test begins and ends at the same value. In terms of perturbation variables, the input m ) is initially zero and is returned to zero after some time, ... 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

The front is inherently unstable, however, and this is often studied by a linear stability analysis. Infinitesimal perturbations are applied to all of the variables to simulate reservoir heterogeneities, density fluctuations, and other effects. Just as in the Buckley-Leverett solution, the perturbed variables are governed by force and mass balance equations, and they can be solved for a perturbation of any given wave number. These solutions show whether the perturbation dies out or if it grows with time. Any parameter for which the perturbation grows indicates an instability. For water flooding, the rate of growth, B, obeys the proportionality... 

It is assumed that the averaging holds for the perturbation variables as well, and that the hold-up and pressure are independent of Y. 

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