Input steady

Frequeney domain analysis is eoneerned with the ealeulation or measurement of the steady-state system output when responding to a eonstant amplitude, variable frequeney sinusoidal input. Steady-state errors, in terms of amplitude and phase relate direetly to the dynamie eharaeteristies, i.e. the transfer funetion, of the system. 

The two most common temporal input profiles for dmg delivery are zero order (constant release), and half order, ie, release that decreases with the square root of time. These two profiles correspond to diffusion through a membrane and desorption from a matrix, respectively (1,2). In practice, membrane systems have a period of constant release, ie, steady-state permeation, preceded by a period of either an increasing (time lag) or decreasing (burst) flux. This initial period may affect the time of appearance of a dmg in plasma on the first dose, but may become insignificant upon multiple dosing. 

AUC is the area under the curve or the integral of the plasma levels from zero to infinite time. Conversely, equation 1 may be used to calculate input rates of dmg that would produce steady-state plasma levels that correspond to the occurrence of minor or major side effects of the dmg. 

Material and energy balances are based on the conservation law, Eq. (7-69). In the operation of liquid phase reactions at steady state, the input and output flow rates are constant so the holdup is fixed. The usual control of the discharge is on the liquid level in the tank. When the mixing is adequate, concentration and temperature are uniform, and the effluent has these same properties. The steady state material balance on a reacdant A is... 

RGA Method for 2X2 Control Problems To illustrate the use of the RGA method, consider a control problem with two inputs and two outputs. The more general case of N X N control problems is considered elsewhere (McAvoy, Interaction Analysis, ISA, Research Triangle Park, North Carohna, 1983). As a starting point, it is assumed that a linear, steady-state process model is available. 

Since this is a steady condition, the rate of reac tion in the film equals the rate of input to the film,... 

Steady eonditions will exist when the aetual and desired temperatures are the same, and the heat input exaetly balanees the heat loss through the walls of the building. 

The total response of the system is always the sum of the transient and steady-state eomponents. Figure 3.1 shows the transient and steady-state periods of time response. Differenees between the input funetion X[ t) (in this ease a ramp funetion) and system response Xo t) are ealled transient errors during the transient period, and steady-state errors during the steady-state period. One of the major objeetives of eontrol system design is to minimize these errors. 

The first term in equation (3.39) represents the input quantity, the seeond is the steady-state error and the third is the transient eomponent. When time t is expressed as a ratio of time eonstant T, then Table 3.3 and Figure 3.15 ean be eonstrueted. In Figure 3.15 the distanee along the time axis between the input and output, in the steady-state, is the time eonstant. 

Consider a second-order system whose steady-state gain is K, undamped natural frequency is Wn and whose damping ratio is (, where C < 1 For a unit step input, the block diagram is as shown in Figure 3.18. From Figure 3.18... 

For a first-order plant, PI eontrol will produee a seeond-order response. There will be zero steady-state errors if the referenee and disturbanee inputs r (t) and f2(f) are either unehanging or have step ehanges. The proeess of ineluding an integrator within the eontrol loop to reduee or eliminate steady-state errors is diseussed in more detail in Chapter 6 under system type elassifieation . 

Fig. 6.1 Steady-state input and output sinusoidal response.

Flence, for a sinusoidal input, the steady-state system response may be calculated by substituting. v = )lu into the transfer function and using the laws of complex algebra to calculate the modulus and phase angle. 

These are first-order systems where the phase of the output (in steady-state) leads the phase of the input. The transfer funetion of a first-order lead system is... 

Table 6.4 Relationship between input funetion, system type and steady-state error...

Table 6.4 shows the relationship between input funetion, system type and steady-state error. From Table 6.4 it might appear that it would be desirable to make most systems type two. It should be noted from Figure 6.22(e) that type two systems are unstable for all values of K, and will require some form of eompensation (see Example 6.6). 

The compensated and uncompensated open-loop frequency response is shown in Figure 6.41. From this Figure the compensated gain margin is 12.5 dB, and the phase margin is 48°. In equation (6.117), K does not need to be adjusted, and can be set to unity. When responding to a step input, the steady-state error is now 4.6%. 

Sinee there is a steady state operation, INPUT RATE = OUTPUT RATE... 

Steady-state operation (i.e., accumulation in the reactor is zero) Constant fluid mixture density Stirrer input energy is neglected Wj,... 

Chemostat A bioreaetor in whieh steady-state growth of miero-organisms is maintained over prolonged periods of time under sterile eonditions by providing the eells with eonstant input of nutrients and eontinuously removing effluent with eells as output. 

Quality-assurance procedures have to be established for the checking of both input and results checks of energy balances, plausibility tests, and comparison with steady-state calculations and with results from similar cases. These checks are demanding and time consuming and thus prone to be omitted but are mandatory for reliable simulations. 

These show a wide variation. Since meat wiii keep for severai days at 2°C, rework case i on the basis of a steady input of 50 t/day aii coming from the abattoir. 

Despite the advantages of continuous cultures, the technique has found little application in the fermentation industry. A multi-stage system is the most common continuous fermentation and has been used in the fermentation of glutamic add. The start-up of a multi-stage continuous system proceeds as follows. Initially, batch fermentation is commenced in each vessel. Fresh medium is introduced in the first vessel, and the outflow from this proceeds into the next vessel. The overall flow rate is then adjusted so that the substrate is completely consumed in the last vessel, and the intended product accumulated. The concentration of cells, products and substrate will then reach a steady state. The optimum number of vessels and rate of medium input can be calculated from simple batch experiments. 

Equations (299) and (300) depict the input-output relationships for the concentrations and the temperature in each phase for a given continuous steady-flow dispersed system. Therefore, (299) and (300) can be used in predicting the input-output relationships for a multistage multicomponent gas-liquid system with several continuous stirred vessels in series. 

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