Step change input

In this case, three time constants in series, X, %2 and X3, determine the form of the final outlet response C3. As the number of tanks is increased, the response curve increasingly approximates the original, step-change, input signal, as shown in Fig. 2.12. The response curves for three stirred tanks in series, combined with chemical reaction are shown in the simulation example CSTR. 

F Residence time distribution function for step change input... 

The static mixer produces an outlet concentration profile nearly as sharp as a step-change input in a plug-flow device. 

Dynamic Response of RCL System to a Step-Change Input... 

W th little or no damping a step-change input will cause an oscillatory RCL system to respond at its natural frequency f . The oscillations decrease with time, and this decay may be defined in terms of the logarithmic decrement or exponential decay ratio. At critical damping the response is similar to that of a linear system subjected to the same step input. With large amounts of damping, the response is non-oscillatory... 

Fig. 7. Residence time distributions where U = velocity, V = reactor volume, t = time, = UtjV, Cj = tracer concentration to initial concentration and Q = reactor volume (a) output responses to step changes (b) output responses to pulse inputs.

In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients are simply the values of the output variable at the samphng instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of bump tests are required to compensate for unanticipated disturbances, process nonhnearities, and noisy measurements. 

FIG. 8-43 Step response for u, a unit step change in the input. 

Step An input in which the concentration of tracer is changed to some constant value Cy at time zero and maintained at this level indefinitely. The symbol Cju(t — a) represents a step of magnitude Cy beginning 3.i t = a. The resulting effluent concentration is designated C. ... 

Lag time The time interval between a step change in input concentration and the first observable corresponding change in response. 

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. 

The Piston Flow Reactor. Any input signal of an inert tracer is transmitted through a PFR without distortion but with a time delay of F seconds. When the input is a negative step change, the output will be a delayed negative step change. Thus, for a PFR,... 

In testing process systems, standard input disturbances such as the unit-step change, unit pulse, unit impulse, unit ramp, sinusoidal, and various randomised changes can be employed. 

First-Order Response to an Input Step-Change Disturbance... 

Figure 2.14. Schematic drawing of a process with immediate (dashed line) and time-delayed responses to a step change of an input signal.

If for some reason the outlet pump slows down, the liquid level in the tank will back up until it overflows. Similarly, if the outlet pump speeds up, the tank will be drained. The tank level will not reach a new steady state with respect to a step change in the input. 

The time variations of the effluent tracer concentration in response to step and pulse inputs and the frequency-response diagram all contain essentially the same information. In principle, any one can be mathematically transformed into the other two. However, since it is easier experimentally to effect a change in input tracer concentration that approximates a step change or an impulse function, and since the measurements associated with sinusoidal variations are much more time consuming and require special equipment, the latter are used much less often in simple reactor studies. Even in the first two cases, one can obtain good experimental results only if the average residence time in the system is relatively long. 

Generalized response of an arbitrary reactor to a step change in input tracer concentration. 

Derive the F(t) curve for a CSTR by considering its response to a step change in the input tracer concentration. Let Wq and represent the weight fraction tracer in the feed before and... 

For the example shown in Figure 19, the step change has an amplitude of 10%, and the constant of the integrator causes the output to change 0.2% per second for each 1% of the input. 

The integrator acts to transform the step change into a gradually changing signal. As you can see, the input amplitude is repeated in the output every 5 seconds. As long as the input remains constant at 10%, the output will continue to ramp up every 5 seconds until the integrator saturates. 

For a step change, a material-balance criterion, analogous to equation 19.3-2 for a pulse input, is that the steady-state inlet and outlet tracer concentrations must be equal, both before and after the step change. Then, it may be concluded that the response of the system is linear with respect to the tracer, and that there is no loss of tracer because of reaction or adsorption. 

Unlike the response to a pulse input, which is related to E(t), the response to a step increase is related to F(t). The normalized response, which is equal to F(t), is obtained as follows. Consider, for simplicity, a step-change in tracer A from c, = 0 to cA in =... 

The value of cAo depends upon the input function (whether step or pulse), and the initial condition (cAj(0)) for each reactor must be specified. For a pulse input or step increase from zero concentration, cA,-(0) is zero for each reactor. For a washout study, Ai(O) is nonzero (Figure 19.5b), and cAo must equal zero. For integer values of N, a general recursion formula may be used to develop an analytical expression which describes the concentration transient following a step change. The following expressions are developed based upon a step increase from a zero inlet concentration, but the resulting equations are applicable to all types of step inputs. 

Determine the open-loop response of the output of the measuring element in Problem 7.17 to a unit step change in input to the process. Hence determine the controller settings for the control loop by the Cohen-Coon and ITAE methods for P, PI and PID control actions. Compare the settings obtained with those in Problem 7.17. 

A theoretical situation in which the system oscillates in response to a step change in the input value and the amplitude of the oscillations does not diminish with time the damping coefficient is 0. 

The step change in input value from positive down to baseline initiates a change in the output reading. The system is un-damped because the output value continues to oscillate around the baseline after the input value has changed. The amplitude of these oscillations would remain constant, as shown, if no energy was lost to the surroundings. This situation is, therefore, theoretical as energy is inevitably lost, even in optimal conditions such as a vacuum. 

The ratio of the change in the steadystate value of the output divided by the magnitude of the step change made in the input is called the steadystate gain of the process K,. 

The dynamic response of most transmitters is usually much faster than the process and the control valves. Consequently we can normally consider the transmitter as a simple gain (a step change in the input to the transmitter gives an instantaneous step change in the output). The gain of the pressure transmitter considered above would be... 

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