When failsafe isn't enough
An orderly shutdown is imperative: These equations give a quick way to check the recommended-volume tank size or to do the sizing oneself
By Bryce Elliott
Many times, processes will require reserve volumes of air for valve actuation on failure of the air header.
Typical reasons for needing this additional volume are when the failure position of the valve is not the “failsafe” position or when operating requirements dictate a more orderly shutdown than having the valve immediately going to its failure position.
A volume tank needs to be in place to supply a reserve volume for actuating the valve. The question then comes of how to size this tank.
The volume of the tank, Vt, has to be large enough and under sufficient pressure, Pi, to fill the volume of the actuator, Va, at the minimum pressure required by the actuator, Pf, for the number of strokes required, s.
The values of Va and Pf need to come from the actuator manufacturer. Pf will change depending on the torque required to stroke the valve, so input from the valve vendor may also be necessary.
Typically, the valve vendor will calculate the required torque, which will vary depending on the individual valve type, packing design, shutoff differential pressure, and required leakage, and choose the actuator accordingly.
The actuator manufacturer should supply a table that relates torque to supply pressure, and the engineer can select the appropriate pressure based on the required torque the valve vendor has given. Pi will be the normal operating pressure of the air header.
s is the number of times the valve will need to stroke before the pressure reduces below the point at which the valve can no longer actuate. This will depend on the operating philosophy for this valve. The process, operations, and safety groups may have input into determining an adequate value for s.
To begin developing an equation for sizing the tank, start with the simplest case, a single stroke.
The gas in the volume tank will expand to fill the volume Vt + Va. There are two known pressures, Pi and Pf. Since the gas expansion will be fairly quick, little heat will come from or go into the environment. For this reason, we can take the gas expansion to be adiabatic in our model.
An adiabatic process is one in which no heat is exchanged with the surroundings.
The other extreme case is isothermal, where the expansion takes place slowly enough that the gas stays at constant temperature; this results in smaller calculated values than the adiabatic assumption. Reality lies somewhere in between. Calculating the process as adiabatic will provide some of the “margin for error.”
Thermodynamics tells us, the PVk is a constant for an adiabatic process. k is the ratio of the specific heats, CP/CV, which in the case of air at pressures and temperatures of interest, is approximately 1.4.
The equation is:
Pi, Pf, Va, and k are knowns, and we can readily solve for Vt :
Keep in mind the pressure units must be absolute (e.g. psia). Aside from that, the units—volume and pressure—need only be consistent. If the actuator volume is given in cubic inches, the tank volume solved for will be in cubic inches (actuator volume in gallons will yield tank volume in gallons, etc.).
If multiple strokes are required, the procedure is similar, but a bit more complicated.
The air in the volume tank expands to fill the volume in the volume tank and the actuator. The volume in the actuator is then exhausted, and the remaining air in the tank expands again to fill the volume in the volume tank and the actuator. For two strokes, the two equations are:
P2 is the intermediate pressure in the volume tank after the first stroke. Solve each equation for P2.
Set (5) and (6) equal to one another, eliminating P2.
Solving (7) for Vt yields:
For more than two strokes, a similar system of equations can be set up, with the intermediate pressures eliminated algebraically. The general formula is:
For multiple strokes, the strokes probably will not be in quick succession, which would allow the tank air to warm to ambient temperature between strokes (it cools slightly when it expands to fill the actuator). This will slightly reduce the amount of necessary air because the pressure in the volume tank will increase with the temperature increase. Because we cannot know ambient temperature in advance, it is impossible to calculate this effect precisely. Since it is not significant, we can neglect it.
Another margin comes by the fact that tanks are available in discrete sizes. If one calculates a volume of 8 gallons, 10 gallons is the best tank size, so 25% extra is automatically built-in.
Some of the smaller standard sizes offered are 10, 15, 20, 30, 60, 80, and 120 gallons. Note: Tubing volume is typically a negligible consideration. However, for long tubing runs (greater than 75 feet), we may need to factor the volume in.
It is possible to reduce the air necessary by putting a downstream pressure regulator between the volume tank and the actuator. In this case, the set pressure of the regulator is set equal to, or very slightly higher than, the minimum pressure of the actuator.
Doing so gives a similar adiabatic expansion, but since the actuator is being filled at the same pressure each time, the end result is as though the air in the volume tank at the starting pressure takes up Vt + sVa at the ending pressure, or:
Solving for Vt gives:
This is the result of equation (2) multiplied by the number of strokes. Also, in the examples below, reducing the required air by adding a pressure regulator usually did not reduce the selected tank size.
A volume tank for a throttling control valve requires a more complex analysis than what we are looking at here. A throttling valve will have partial strokes. It may also have a positioner, which is a constant bleed device, meaning the volume in the tank will leak out over a fairly short period of time.
This analysis requires knowing the bleed rate (which varies depending on input pressure), the amount of time the valve is expected to be available (multiplying these two will yield a mass of air, though the variable bleed rate may require some integration, either piecewise or continuous), and some estimate of the number of strokes required.
This will not necessarily be a whole number; round up. One can then apply the same sort of analysis given here to come up with air necessary to stroke the valve. Add the air required by the bleed rate to the air required to stroke the valve, taking care to keep consistent units.
For sizing volume tanks for on/off valves, use equation (9) or (11), as appropriate. A nice result of these equations is it is not necessary to include a “safety factor,” as the safety factor is a part of the simplifying assumptions. These will give the engineer a quick way either to check the volume tank size recommended by the valve vendor or to do the sizing oneself.
ABOUT THE AUTHOR
Bryce H. Elliott, P.E. (email@example.com) has a B.S. in engineering and applied sciences and a master’s degree in chemical engineering. He is a project engineer for Shaw Energy & Chemicals Group in Houston.
Adiabatic process occurs with no exchange of heat between the system and its environment.
Fail-safe or fail-secure describes a device or feature, which in the event of failure, responds in a way that will cause no harm or at least a minimum of harm to other devices or danger to personnel.
Actuator is a device to convert an electrical control signal to a physical action. Actuators may be for flow-control valves, pumps, positioning drives, motors, switches, relays, and meters.
Pi = initial pressure (equal to the air supply header pressure)
Pf = required actuator supply pressure (provided by the valve vendor)
k = ratio of specific heats CP/CV
s = number of strokes
Va = actuator volume
Vt = tank volume