This section will present specific examples of lags to help transcend the gap from the general concept to the actual system. The organisation of material is mixed between type of lag and type of equipment or process.

Dead Time

Most likely, all time delays without any dynamics are related to distance and velocity. Usually, both the distance and velocity are readily available, making it possible to determine the time delay. Although the velocity may vary with process conditions, a new time delay can be computed. A couple of instances would be determining the coating weight on a moving film and calculating the actual weight after casting or forming the sheet, such as in polyester sheet material or paper manufacturing. These time delays usually last for several minutes due to the vast distances and velocities involved.

Another instance where dead time is present is in fluid flow, specifically in a brine recirculating system around a jacketed reactor. A colder brine is introduced into the loop, and the temperature of the circulating brine is frequently measured. The distance from the fresh brine entrance to the temperature measurement point creates a dead time, which is typically only a few seconds with a standard design velocity. However, the lag from brine entrance to temperature measurement is more complicated than a pure dead time, as the pipe temperature needs to be changed, causing the pipe to absorb some of the energy that would otherwise be transported. It’s challenging to estimate how much this adds to the lag time since it depends on the system’s parameters. In general, it is smaller than typical temperature measurement lags.

Sample lines used for analyzers, such as for IR or chromatography, introduce a dead time as well. If the flow is turbulent, the dead time of composition to the analyzer is almost pure since there is little longitudinal mixing. On the other hand, if the flow is laminar, such as in a sample line taken off a viscometer, there is considerable longitudinal mixing, and the dead time is not pure. It has a pure dead time element that is shorter than that computed from the average velocity because the velocity in the center of the pipe is higher than the average velocity. Then, the response has a lag beyond that as the material along the edge of the pipe comes along, but this is not pure dead time. However, in the viscosity case, it is generally acceptable to assume pure dead time calculated from distance and average velocity. The consequences of this simplification depend on the other lags in the loop.

When it comes to using a chromatograph in a control loop, it introduces an effective dead time. For example, assume the unit has a cycle time of 10 minutes, which introduces a minimum effective dead time of half that or 5 minutes. However, there is also the time between when the sample valve opens and when the peak being controlled gets measured and held, which may take 3 minutes. Therefore, the chromatograph adds an effective dead time of 8 minutes to the loop dynamics, excluding any lag in the sampling line.

It’s worth mentioning that a lot of people outside the control field assume that a process with “a lot of dead time” is uncontrollable. This is not true. Such processes are just as controllable as any other process. In fact, an argument can be made that they are more controllable since the natural period may be closer to than . The view that processes with pure dead time are hard to control stems from the fact that they frequently have long pure dead times that last several minutes. However, the dead time itself is not the issue. The speed and severity of the upsets relative to the dead time are what’s crucial.

Controllers

Most controllers are typically fast enough that their lags are negligible, even in the case of pneumatic controllers. However, in systems operating with natural periods of one second or less, this may not always be true. In such cases, additional knowledge beyond what is covered in these notes may be required. Nowadays, most instrument manufacturers can provide the dynamic characteristics of their equipment.

In situations where a pneumatic controller is used to operate a valve without a positioner, or has to fill a large volume, there may be a small lag. However, in cases where dynamic performance is a concern, a positioner is typically used with the valve, which reduces the volume that the controller and transmission line need to fill, making the controller’s lag negligible.

It’s important to note that when discussing lags in a controller, we are typically referring to the proportional-action component. The integral action introduces its own lag, but we have control over it. On the other hand, the derivative action provides a lead, which is also within our control. When analyzing the lags in a complete control system, it’s preferable to consider the controller as having no lags (or leads), since their contribution is taken into account during the tuning process.

Pneumatic Transmission Lags

The calculation of lags in transmission lines is a highly intricate mathematical process that takes into account a multitude of factors, such as length, inside diameter, absolute pressure, temperature, and termination volume. When the termination volume increases with pressure, as is the case with an un-positioned valve operator, the lag time becomes even longer. This is because it takes a certain number of molecules to increase the pressure of the existing volume and fill the incrementally new volume. It’s essential to recognize that the new volume needs molecules to fill it from zero absolute pressure, not just to change an incremental amount, which can constitute a significant part of the total effective volume. However, except for un-positioned valves, termination volumes of pneumatic equipment are typically small enough to be disregarded.

In order to simplify the discussion of transmission lags, we often approximate them with a first-order lag time, assuming negligible termination volume, even though the calculations are much more complex. Figure 5.1 illustrates this approximation of the transmission lag.

 

Figure 5.1 [The lag of transmission tubing may be approximated by a first order lag.]

Using a larger tubing may require a booster to generate sufficient air flow to fill it, depending on the air-flow capacity of the pneumatic instrument. However, this topic falls outside the scope of this discussion.

In cases where the pneumatic transmission lag constitutes a substantial portion of the overall lags in the system, and it’s necessary to reduce these lags, it may be advantageous to switch to electronic transmission instead.

Valves

This discussion will not cover un-positioned valves. If a control loop has poor performance and includes an un-positioned valve, the first step is to install a positioner to determine if the performance issue is resolved. This advice is for those who may not have easy access to control loop performance experts. It is often more cost-effective to install a positioner than to hire a specialist to diagnose the problem. In fact, a specialist may recommend installing a positioner first before further investigating the issue.

When it comes to transmission lags, positioned valves can have complex lags. For significant changes, the system may experience “velocity limiting,” while for small changes, it may have a dead band. For moderate changes, typically between 1% to 10%, the valve lag may range from 0.2 to 1 or 2 seconds. The larger valves usually have longer lags. The valve lag can be approximated as a first order lag, or as two first order lags whose sum is the chosen value. Positioner manufacturers should provide dynamic response information for their equipment on specific valves or classes of valves.

Positioning systems can be made faster by using a booster between the positioner output and the valve operator. While newer positioners are better than older ones, even the latest positioners can be made faster with boosters. This technique is seldom necessary, so the specifics required for optimal operation are beyond the scope of this discussion. Using a booster can reduce the time lag to 0.2 seconds even on relatively large valves, and to 0.1 seconds on small valves. Additionally, using a booster can mitigate the negative effects of velocity limiting and dead band, which is sometimes the main reason for using a booster.

Transmitters

Like controllers, transmitters are typically fast enough that their lags can be disregarded. However, in the case of temperature measurement, the measurement process may be slow, but the transmitter’s response is usually fast enough. Some transmitters have dampening or adjustable dampening, which is sometimes referred to as a filter, and is essentially a first-order lag. It is recommended to consult the equipment’s instruction manual, and if needed, reach out to the vendor for assistance. Smart transmitters usually have a filtering function, but setting the time constant too high can negatively impact control performance.

Temperature Measurement

The time delay in measuring temperature can vary widely, ranging from less than a second to several minutes. When a thermocouple is placed in a well, the system typically behaves as two first-order lags rather than one, and the lags increase with the diameter of the sensor and/or well. In addition, the presence of more thermal mass and lower fluid velocities further slows down the measurement. The use of wells can also result in longer lags in gas service, but adding a heat transfer oil or paste can improve the speed and accuracy of the measurement.

In liquid service, it’s often possible to set up an experiment in the instrument shop to determine the lag time. By suddenly immersing the thermal system in a bucket of water at a different temperature and observing the time it takes to respond 63% of the way, you can estimate the lag time, which can be treated as a first order lag or two equal first order lags.

It’s important to note that a measurement lag, such as that in temperature measurement, can impact the dynamic performance of a loop, making it appear better than it actually is. The actual variations are always larger than the observed, which can be significant in some cases. By improving the speed of measurement, significant improvements in control performance can sometimes be achieved.

Tanks, Liquid Flow Lag

Consider the systems in Figure 5.2:

Figure 5.2 [The lag of a tank is not always it’s volume divided by it’s throughput.]

Let’s assume that each tank contains 200 litres and that the outflow rate depends linearly on the head and is not pumped out. It may seem reasonable to estimate the lag as 200 litres divided by 100 litres per minute, which gives two minutes. However, this approach is incorrect when considering volumetric flow instead of composition. If the flow rate in is increased to, for example, 120 litres per minute, the level will increase until the outflow rate matches the inflow rate. While it is possible to derive the lag from the differential equations, it may not be the preferred method for most people. Fortunately, there is a simpler way to determine the lag.

To find the lag, calculate the new level based on the new flow rate and then calculate the change in inventory. Let’s say this calculation results in a change of 10 litres. The lag is often 10 litres divided by 20 litres per minute, which gives 0.5 minutes. This lag can be treated as first order if the flow through the exit restriction is laminar, which means that the flow is linear with the head.

This calculation reveals that the actual volume in the tank is not relevant. What matters is the change in volume for a change in flow rate. This approach applies even if the level is on proportional-only control, and it is much simpler than the more fundamental derivation.

Tanks, Compositional Lag

The diagram in Figure 5.3 illustrates a system in which the composition in the tank will track the feed composition with a first order lag, which can be calculated based on the hold-up volume and flow rate. Temperature can be treated as a special case of composition. If the tank is perfectly agitated and the level control is accurate, then the lag will be a pure first order lag. However, since the first two assumptions are only approximations, the lag is also an approximation.

In a vessel with a jacket, the temperature lag in the jacket can be estimated as the volume divided by the flow rate, assuming good mixing. However, if mixing is poor, the total lag must be divided between an apparent dead time and a first order lag.

When a chemical reaction is occurring, there is no simple answer, and the differential equations provide the only means to solve the problem.

Figure 5.3 [The lag of a stirred tank to composition changes is its volume divided by its throughput.]

Thermal Process Lags

The term “process lags” is a broad category that encompasses all lags in a process except for measurement lags. This can include heat exchangers and other types of equipment, which can be complex and varied. Due to the diverse range of equipment and the complexity of the processes involved, it is difficult to provide typical lag times for this category. Generally, lags become longer as the surface-to-volume ratio decreases, and can range from a few seconds to several hours.

Typical Natural Periods

Control loops of the same variable type generally have similar natural periods, although there can be exceptions. Flow loops usually have a natural period of 1 to 2 seconds when transmission lags are short, but this can increase to 5 to 10 seconds for long pneumatic lines. These loops should return to normal quickly after an upset, typically within 5 to 10 seconds for the faster loops. Pressure loops also tend to be fast, with higher gain than flow loops. Temperature loops, on the other hand, tend to be much slower, with natural periods ranging from half a minute to several minutes, consistent with the discussion on lags. However, there are many exceptions to this general trend. Composition control loops are typically even slower, with natural periods ranging from several minutes to several hours.