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pH Control - Index

System Description

pH control is a common issue in many industrial processes. The basic idea to control the pH variations in some liquid flow, usually making the pH as close to 7 as possible. This influent liquid flows into a tank, where it is mixed with an amount of a concentrated reagent to alter its pH. If the effluent is acidic with a pH of 4, the reagent would be basic with a pH larger than 10. The reagent is more concentrated than the effluent, since it is desirable to as little volume to the effluent as possible.

The pH is controlled in a tank - the effluent flows into the top of the tank, and the reagent is also added to the top of the tank. The tank is well-stirred so that the pH is uniform throughout the tank. The effluent is pumped out the bottom of the tank, as shown in the figure below.

Note that pH is a very difficult control problem due to the non-linearities. This example simplifies the problem somewhat to highlight some points about control system design in general.

Overview of the Example

It is recommended that you follow the example in order, but you may skip ahead if you wish:

Part 1 The basics

Part 2 Flow rate variations

Part 3 Control valve errors

Part 4 A 3 tank solution

Photos of the System

Below is a photo of an industrial pH neutralization system.

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pH Control - Part 1Before continuing, make sure you have read Chapter 19

System Model

From elementary mass balance considerations, it can be shown that an appropriate state space

model for the strong acid-strong base system is

(1)

where

co(t) :    excess hydrogen ion concentration in the effluent stream (mol/L)

cu(t) :    excess hydrogen ion concentration of the reagent (mol/L)

ci(t) :    excess hydrogen ion concentration in the influent stream (mol/L)

u(t) :    flow rate of the reagent (L/s)

q(t) :    flow rate of the influent stream (L/s)

V :    volume of the tank (L)Normally, the reagent concentration is constant, and we control the effluent pH by varying the reagent flow. Note that the concentrations are excess concentrations, meaning that they

measure the concentration of hydrogen ions in excess to that found in water. We can convert an excess concentration C to pH using the following formula.

(2)

From this formula, we can see that an excess concentration of 0 gives a pH of 7 (which is the pH of water, as expected). Also, negative excess concentrations correspond to basic pH, and positive excess concentrations correspond to acidic pH.

NOTE: The reader should already be aware that if a weak acid or weak base is used, the titration curve will be substantially different. The differential equation (1) represents the dynamic equation for a strong acid-strong base titration curve. For any other acid-base combination (e.g. weak acid-strong base), the system model will be substantially different, as will equation (2) for finding concentration from pH. The graph below shows the titration curve for a strong acid-strong base system.

Control Design

We assume that the tank system has a level controller which maintains a constant liquid volume in the tank. The reagent flow is controlled using a simple PI controller as shown in the figure below. Note that the control valve on the reagent pipe has a saturation, since it can be fully open (giving a maximum flow rate) or fully closed (giving a minimum flow rate of zero).

For the purposes of designing the controller, the plant is linearised by assuming that the reagent flow is much smaller than the influent flow (since the reagent is more concentrated). This results in the linear model shown below.

The controller is designed so that the closed-loop bandwidth is twice that of the open-loop bandwidth. This way, the initial peak in reagent flow falls within the limits imposed by the valve. It may seem logical to construct a larger valve so that a faster controller can be implemented, but we will see later that this carries with it serious design tradeoffs which mitigate against going too far in this direction.

For this example, we choose the parameter values as

cu :    -10-2 mol/L

ci :    10-3 mol/L

q :    10 L/min

V :    83.67 L

maximum reagent flow :  0.025 L/sand design a PI controller with kp = 3q / cu and ki = 4q2 / Vcu. Note that the primary concern of the controller is to reject disturbances rather than to follow a set-point, since the set-point will always be a pH of 7. Also, we assume that the control valve allows a maximum reagent flow

rate of 1.5 L / min.

There is an inherent problem with the pH control system: the measured output is pH, but what we want to control is hydrogen ion concentration (using pH as the control variable with a set point of 7 would make the system even more non-linear than it already is). If we did not

know equation (2) for converting between ion concentration and pH, we would have to attempt to estimate the titration curve, possibly linearising it to find ion concentration. The pH curve looks almost vertical in the middle, but upon magnification we find that it is not

linear at all. Thus, linearising the titration curve has serious sensitivity ramifications for the control system.

Fortunately, in this case we know the function for pH in terms of concentration, so we can simply invert this equation to find the ion concentration for the measured pH. This is rarely

possible in practise.

Note that when measuring the effluent pH, there is a delay of the order of 5 seconds. It is safe to ignore this delay considering the time scales we are looking at (i.e. with a plant time

constant of 500 seconds).

Java Applet Simulation

Below is a Java applet which simulates the above pH neutralisation system. The blue plot shows the influent concentration and the green plot shows the effluent concentration. This

process is extremely slow, so the time scale is accelerated somewhat: 1250 simulated seconds pass for each real second of the simulation, giving a horizontal scale of 2500 seconds per

division

Pressing the "Change Parameters" button brings up the parameters window allowing you to change the controller values, the influent pH and the animation speed. The controller has

been implemented in the anti-windup form to avoid the effects of the saturation of the valve. The animation speed can be 25, 12.5 or 6.25 frames per second. This speed is only the rate at which the screen is refreshed - the simulation is unaffected. The speed function is available

for those with slower computers so that the simulations still appear in real-time (if somewhat jerkier). Note that the applet does some auto-detection of the speed of your computer, so if

you select a speed that your computer cannot handle, the applet will reduce the speed accordingly.

 

 

Things to try Things to notice

Look at the initial step response.Observe that the response is different from a normal exponential response.

Change the influent pH to any value you like (the simulation will allow only values between 11 and 7).

Watch the non-linear nature of the system's response.

Design a faster controller.Note the effect of the valve saturation on the response.

The observed non-linear output response is due in part to the non-linear system. However, most of the non-linearity arises from the fact that we are measuring pH which is a

logarithmic function of the excess ion concentration. It is hydrogen ion concentration that we are concerned with controlling, but we measure pH since environmental laws specify the

acceptable effluent pH ranges (the tolerance could be as small as 0.5 of a pH).

In this simulation, it was assumed that both the influent flow and concentration were constant, but this is rarely the case. The next page looks at the effect of varying the influent

flow rate.

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pH Control - Part 2

Flow Variations

If the system's influent flow is from some sort of industrial runoff, then it is highly likely that the flow rate will vary, since the runoff will not be at a constant rate. The variation in influent flow rate presents itself as an input disturbance in the plant model.

where q(t) is the flow rate signal. In this example, we model the fluctuations as a constant flow plus some randomly varying noise. In particular, we set the average flow to the level used in the last example (10 L/min), and the random variations range uniformly between -1 L/min and 1L/min.

We will use the same controller as in the last example, which is based on the nominal flow rate of 10 L/min.  However, we should note that allowing the q(t) input to vary with time makes the system more non-linear than before.

Java Applet Simulation

Below is a Java applet which simulates the pH control system with influent flow rate variations. The blue plot shows the influent concentration and the green plot shows the effluent concentration. This process is extremely slow, so the time scale is accelerated somewhat: 1250 simulated seconds pass for each real second of the simulation, giving a horizontal scale of 2500 seconds per division

Pressing the "Change Parameters" button brings up the parameters window allowing you to change the controller values, the influent pH and the animation speed. The controller has been implemented in the anti-windup form to avoid the effects of the saturation of the valve. The animation speed can be 25, 12.5 or 6.25 frames per second. This speed is only the rate at which the screen is refreshed - the simulation is unaffected. The speed function is available

for those with slower computers so that the simulations still appear in real-time (if somewhat jerkier). Note that the applet does some auto-detection of the speed of your computer, so if you select a speed that your computer cannot handle, the applet will reduce the speed accordingly.

 

 

Things to try Things to notice

Look at the response of the system.Observe the noisy output signal, particularly as the output gets closer to a pH of 7.

Change the influent pH to any value you like (the simulation will allow only values between 11 and 7).

Watch the non-linear nature of the system's response.

Design a faster controller.Note the effect of the valve saturation on the response.

The output variations occur because of the relative size of the tank and the influent valve. If this valve is larger, the influent flow rate increases, and liquid flows through the entire system at a faster rate. This means that the filtering effect of the tank is reduced somewhat. As you might expect, increasing the tank volume would reduce the pH variations in the effluent.  However, building a bigger tank is expensive, and as we will see in the next example, errors in the control valve reduce the effect of a bigger tank anyway.

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pH Control - Part 3

Control Valve Errors

Up until now, we have considered that the control valve was ideal - i.e. the valve provides whatever reagent flow rate we desire (with saturation limits). In reality, things aren't that simple, with most valves displaying backlash characteristics. You should recall the discussion of backlash in the ball and plate control example.

For now, we will assume that the backlash error is 2% of the maximum flow rate allowed by the valve. Often, the valve errors can be as large as 5% or 10%. The block diagram of this

new system is shown below:

With the influent flow variations discussed in the last example, there are now two sources of error in the system. In order to understand them better, we will examine them separately. Since the last example showed the effect of flow variations alone, we will now look at the effect of the valve errors alone.

The nature of the valve hysteresis is such that a simple exponential step response will be largely unaffected. Thus, to highlight the effect of the hysteresis, we will allow extremely small influent flow variations. The influent flow q(t) is then the nominal 10 L/min plus a small sinusoidal variation of amplitude 0.0003 L/min and period 10000 s.

Java Applet Simulation

Below is a Java applet which simulates the pH control system with control valve hysteresis. The blue plot shows the influent concentration and the green plot shows the effluent concentration. This process is extremely slow, so the time scale is accelerated somewhat: 1250 simulated seconds pass for each real second of the simulation, giving a horizontal scale of 2500 seconds per division

Pressing the "Change Parameters" button brings up the parameters window allowing you to change the controller values, the influent pH and the animation speed. The controller has been implemented in the anti-windup form to avoid the effects of the saturation of the valve. The animation speed can be 25, 12.5 or 6.25 frames per second. This speed is only the rate at which the screen is refreshed - the simulation is unaffected. The speed function is available for those with slower computers so that the simulations still appear in real-time (if somewhat jerkier). Note that the applet does some auto-detection of the speed of your computer, so if you select a speed that your computer cannot handle, the applet will reduce the speed accordingly.

The parameters window also allows you to select whether you want flow variations or valve errors. The possibilities are listed in the table below:

flow variations valve error description

off on The default state for the simulation, showing the valve error

characteristics as discussed above.

off offNo valve errors are present, allowing you to see the effect of the small sinusoidal flow variations alone.

on offShows the effect of random flow variations, and is exactly the same as in the previous example.

on onShows the effect of the valve errors in the presence of the random flow variations.

 

 

Things to try Things to notice

Look at the response of the system with valve errors and small sinusoidal flow variations.

Notice that the effluent pH varies between about 8.5 and 5.5

Turn off valve errors (and flow variations)

Observe the response of the small sinusoidal flow variations alone, and compare the output to the system with valve errors.

Turn off the valve errors, and turn on the flow variations

Recall the response from the previous example and note that this response is the same, with the output again varying between 8.5 and 5.5

Turn on both the valve errors and the flow variations

Note the effect of the both the valve hysteresis and the random flow variations.

Try the faster controller that you designed in the previous example

Notice the limited effect of this controller.

The essential point in this example is that both sources of error produce output variations of approximately the same size. If we were to construct a larger tank to reduce the error cause by the influent flow, then the valve error would start to dominate. Since the valve errors are of the same magnitude as the errors caused by the influent flow variations, increasing the tank volume would have virtually no effect on the output variations.

The next idea might be to try to reduce the valve errors. The valve error is a percentage of the valve's size, so to reduce the error, we need a smaller control valve.  However, a smaller control valve will not be able to provide the necessary reagent flow to neutralise the influent.

This highlights the need to consider both sources of error together, rather than trying to eliminating them individually. The next example presents a solution which considers these

factors.

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pH Control - Part 4

Multiple Tanks

To overcome the flow variations and the valve hysteresis problems, we use a 3 tank system where the tanks increase in volume. This setup is shown in the diagram below.

The influent flows into the first tank, which has a large valve to neutralise most of the influent base. The set-point for this control system is set to a pH of 9.6, since the purpose of the 3 tank system is to attack the problem in steps.

The effluent from the first tank flows directly into the second tank, which is 20 times larger. The second tank uses a valve that is 20 times smaller than that used by the first tank. The set-point for this control system is set to pH 8.3.

The effluent from the second  tank flows directly into the third  tank, which is 20 times larger again. The third tank uses a valve that is 20 times smaller than that used by the second tank (and thus 400 times smaller than the first tank). The set-point for this final tank is a pH of 7.

This scheme takes into account the comparable magnitude of the error sources. The first tank is relatively small, since it requires a large valve to reduce the pH to 9.6. As the liquid flows

through the tanks, the variations reduce and the pH gets closer to 7. The final tank performs only minor adjustments, since the second tank has reduced the pH to 8.3. At this final stage, we are able to reduce the valve size and increase the tank volume with no adverse effects.

Having 3 tanks is more expensive than a one tank scheme, but we will see that the 3 tank system is extremely effective. Also, the consideration of the magnitude of the errors ensures that each tank is no larger than it needs to be.

For this simulation, we assume that the first tank has a volume of 83.67 L, and uses a valve which allows a maximum reagent flow rate of 1.5 L / min. Also, the influent flow rate variations are as before with the average flow to the level used in the last example (10 L/min), and the random variations range uniformly between -1 L/min and 1L/min.

Java Applet Simulation

Below is a Java applet which simulates the 3 tank pH control system. The plots shown in the simulation are listed below:

blue influent pH

green effluent concentration for the first (smallest) tank

red effluent concentration for the second tank

purple effluent concentration for the third (largest) tank

This process is extremely slow, so the time scale is accelerated somewhat. There are two speeds at which to view the simulation. The slow speed is the same as that used in the previous simulations, with a horizontal scale of 2500 seconds per division. The fast speed has 10000 simulated seconds passing for each real second of the simulation, giving a horizontal scale of 20000 seconds per division. The plot for the smallest tank loses a lot of information in the fast speed.

Pressing the "Change Parameters" button brings up the parameters window allowing you to change the controller values, the influent pH and the animation speed. The controller has been implemented in the anti-windup form to avoid the effects of the saturation of the valve. The animation speed can be adjusted as before.

Pressing the "Reset" at any time button resets the simulation to its initial state. Also, the "Start/Stop" button allows you to start and stop the simulation at will.

Finally, the pH meter can be attached to any of the tanks in the system by clicking on the appropriate tank in the lower right hand corner of the applet.

 

 

Things to try Things to notice

With the speed set to slow, observe the response of the three tanks

Notice that the effluent pH for the smallest two tanks stays very close to the set point (in steady state)

Set the speed to fastNotice that the effluent pH from the third and largest tank stays very close to 7 (after an initial transient)

Change the set-point for the second tank to something larger than 8.3 (such as 9)

Notice that the controller for the third tank can no longer neutralise the base in the influent

Change the set-point for all tanks to a pH of 7

Note that the controller for the second tank cannot neutralise the effect of the widely varying pH in the first tank's effluent

The three tank system works well when the set-points are uniformly spaced between a pH of 11 and 7. Note that a step of 1.3 in the logarithmic pH scale corresponds approximately to a multiplicative factor of 20 - hence the scaling of the tanks and their valves.

The third tank uses a very small control valve, which is designed to make only the smallest corrections to the pH. If the pH of the liquid coming into the third tank is too high, then the controller will be neutralise the influent effectively, and the pH of the effluent will rise. This comes directly from the saturation limits of the control valve.

Even though this simulation concludes the example, this is not the final solution of the problem. There are many complex issues in industrial pH control, which will not be discussed here.

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