Valve travel

Figure 8-74b is an example of a pneumatic positioner/actuator. The input signal is a pneumatic pressure that (1) moves the summing beam, w ch (2) operates the spool valve amplifier, which (3) provides flow to and from the piston actuator, which (4) causes the ac tuator to move and continue moving until (5) the feedback force returns the beam to its original position and stops valve travel at a new position. Typical positioner operation is thereby achieved. 

When the circulation is started, the poppet valve travels slowly down, generating one pressure pulse when passing each restriction. The measurement range in the standard tool is of 2.5° (also 7° ranges, 1° increments, max. 17°). 

What is the pulse amplitude (pressure surge) for the various flowrates when the puppet valve travels from 0.2 to 0.5 in. At what position will we have a half-height pulse ... 

The proportional band of a particular instrument is expressed as a percent of full range. For example, if full range of an instrument is 200°F and it takes a 50°F change in temperature to cause full valve travel, the percent proportional band is 50°F in 200°F, or 25%. Proportional bands may range from less than 1% to well over 200%. However, proportional bands over 100% cannot cause full valve travel even for full range change of the controlled variable. 

Typically, one specifies the desired response, C(z)/R(z), which yields from equation (12) the required design of the controller, D(z). In practice, however, this design technique results in a controller which requires excessive valve movement, an undesirable situation. Consequently, Kalman (12) developed a Z-transform algorithm which specifies the desired output, C(z), and the desired valve travel, M(z) for a setpoint change. The desired response and valve travel for a unit step change in setpoint is shown in Figure 22. The system response,... 

The transfer functions relating response and valve travel to a unit change in setpoint (R(z) = l/(l-z X)) are then ... 

Figure 22. Desired system response C(z) and valve travel M(z) for a unit step change in setpoint according to Kalman s approach...

Process output, controlled variable, valve travel... 

Each valve will be driven by a valve-positioner, which is a servomechanism designed to drive the valve travel, jc, to its demanded travel, xj. This valve positioner will take a certain time to move the valve, and we will use the simplest possible model of the dynamics of the valve plus positioner, namely a first-order exponential lag ... 

The travel of valve 3 is governed by the action of the level controller. For simplicity, we will suppose that the level controller has a purely proportional action so that the demanded valve travel, xji, is given by... 

Equations (2.21) have been written in the order and manner above to bring out the dynamic interdependence of the states that will normally emerge as a feature of models of typical industrial processes. While the derivative of one state may depend only on the current value of that state, as in the case of the valve travels, x and Xi, others will depend not only on their own state but also on a number of others. This latter situation arises above in the cases of control valve travel, Xi, and the liquid mass in the tank, m. The dependence may be linear in some cases, but in any normal process model, there will be a large number of nonlinear dependencies, as exhibited above by the derivative for tank liquid mass, which is dependent on a term multiplying the square of one state by the square-root of another. This is an important point to grasp for those more accustomed to thinking of linear, multivariable control systems such systems are idealizations only of a nonlinear world. 

Equation (2.21) also shows how state behaviour depends on the forcing variables, in this case the externally determined setpoint for liquid level, / and the demanded valve travels for inlet valve 1, x i, and inlet valve 2, x i-... 

But the relationship between state variables arising from different mathematical descriptions of the same process does not have to be linear. Let us assume that we wish to recast our equations in terms of valve openings rather than valve travels. This is a simple business for the linear valves I and 2, where fractional valve openings are identical with fractional valve travels (equations (2.6) and (2.7)). But the outlet valve has a square-law characteristic ... 

Clearly, our state vector should contain the same information if we substituted valve opening, y, instead of valve travel, X3, but what is the precise effect of the change ... 

To recast our model so that level and valve travels are the new states, we substitute from equations (2.6),... 

To illustrate the difference in stability properties between explicit and implicit integration algorithms, consider again the equation used to describe valve dynamics in Section 2.2. Dropping the subscripts from equation (2.9) for clarity and generality, and setting the demanded valve travel, xj, to zero, indicating a... 

The value of Kc at fully open for a rotary valve, Kc, is considerably lower, typically 0.25. Now, however, the location of the effective throat and the flow field downstream of it are both dependent on valve travel. This explains the dependence of Kc on valve travel, X, for rotary valves, where Kc has been found to obey approximately the following equation set ... 

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