By Jim Fuller

Summary: Residential water treatment dealers may take it for granted when installing a tank for a reverse osmosis (RO) unit that there’s more than water to consider when determining the proper pressure for sizing purposes.

Air sizing. Air sizing? What would any self-respecting water quality professional need to know about air sizing? Answer: everything. You need it every time you troubleshoot a problem job. It explains the energy absorption of water hammer arrestors, and you use it every time you size a water storage tank. Be it non-pressurized or pre-pressurized, sizing the proper tank involves the knowledge of sizing air.

I’ve been teaching class after class of water storage tank sizing and application. The question that all students are asked is: “What’s the reason a tank is installed in the system?” The answers range from the tongue-in-cheek “to make more money…” to the practical “to store water.” While most answers are valid, the best answer any student put forth was by V. Hernandez from New York who said, “…to add a cushion of air to the system.” I thought about this answer for weeks, months and ultimately years, and Mr. Hernandez may have given close to the perfect answer.

Basic terms
Let’s start our discussion with some simple terms and definitions. The terms most used in our industry involve certain factors and pressures. Simplified definitions follow:

• Gauge pressure: Pressure read from a gauge (such as a tire-type gauge) referred to in
as psig, or pounds per square inch gauge.
• Absolute pressure: The total pressure of the gas referred to as psia, or pounds
per square inch absolute. It’s equal to the gauge pressure plus atmospheric pressure.
• Atmospheric pressure: The pressure exerted by the atmosphere (14.7 psia at sea level).
• Acceptance: The amount of water that enters between two pressure conditions in a
storage vessel.
• Acceptance factor (AF): The percentage (expressed as a decimal) of the total tank
volume that stored water will occupy between two pressure conditions in a storage vessel.
• Max. Acceptance factor (MAF): The maximum percentage (expressed as a decimal) of
the total tank volume that stored water will occupy before the inner membrane is possibly stretched.

Boyle’s Law
About 350 years ago, scientist Robert Boyle discovered and presented a study on compressible gases. These findings were published in a book entitled The Spring of Air. By careful experimentation, he established Boyle’s Law, which states that the volume of a given amount of gas varies inversely with its pressure, if temperature remains constant. Boyle’s Law is most often expressed through the following:

Equation 1: P1·V1 = P2·V2

Where:
P1 : Pressure at initial state (condition 1)
V1 : Volume at initial state (condition 1)
P2 : Pressure at final state (condition 2)
V2 : Volume at final state (condition 2)

This means that as the pressure of gas is increased, the volume will decrease by the exact proportion. If you double the pressure, you’ll end up with half the volume. If you triple the pressure, you’ll have one-third of the original volume, and so on.

Consider a cylinder filled with air to one-gallon in size, and 0 psig of air pressure, as shown in Figure 1. Even though the gauge reads 0 psig, the air in the cylinder truly has some pressure relative to a pure vacuum. This pressure is atmospheric pressure, which is 14.7 psi at sea level. Boyle’s law is based on absolute pressure. Absolute pressure (psia) is equal to gauge pressure (psig) plus atmospheric pressure (0 psig). Figure 1 shows the cylinder fitted with a gauge that shows both gauge pressure and absolute pressure. As the volume of air is compressed, the pressure rises proportionally. Figure 2 reveals that as the air volume is “squeezed” and reduced to 1/4 gallon, the absolute pressure increased to 60 psia, which is equal to 45 psig.

The identical relationship occurs in a water storage tank. Instead of using a piston to compress the air, the incoming water is used. Figure 3 shows a 10-gallon tank of air at atmospheric pressure. As water enters the tank, the air volume is “squeezed” raising the pressure proportionally. To achieve a pressure of 15 psig (29.7 psia), the entering water must squeeze the cushion of air until it reaches 5 gallons. To further increase the air pressure to 30 psig (44.7 psia) the air cushion must be compressed to 3.3 gallons. This means 6.7 gallons of water had to enter the tank to compress the air to this volume and pressure. Another way to look at this example is that there is 5.0 gallons of water stored between 15 and 0 psig, and 2.7 gallons of water stored between 30 and 15 psig. As water exits the tank, the air cushion expands, and the pressure drops—exactly as expressed by Boyle’s Law!

Pre-pressurized vessels
Most tanks used in today’s home and commercial systems are pre-pressurized vessels using a flexible membrane to keep the air cushion permanently captured. This allows for more usable pressurized water storage per given gallon of tank. Consider the same tank of 10 gallons (see Figure 4). In this example, the tank will be pre-charged to 15 psig (29.7 psia). To compress the air to 30 psig (44.7 psia), the cushion of air must be compressed to 5 gallons. This means that 5 gallons of water entered the tank to squeeze the air cushion to 30 psig. The non pre-charged tank stored 2.7 gallons of water between 15 and 30 psig. The pre-charged tank nearly doubled the amount of water stored between these two pressures.

Because we’re ultimately searching for the amount of water stored in a storage tank, Boyle’s Law can be rewritten to reflect the water’s effect on the compressed air cushion:

Equation 2: Vw = V1· (1-P1/P2)

Where:
Vw is the volume of water needed to compress the air cushion V1 from absolute pressure P1 to P2.

Equation 2 is a common relationship used to size and apply a variety of vessels for water hammer energy absorption and water storage applications. Water hammer occurs when water flow is suddenly and abruptly stopped. This occurs in plumbing systems that use quick closing valves, such as flush valves and solenoid valves. The resulting rapid stoppage of flow creates a spike in line pressure that can severely damage piping and fixtures.

By definition, the term acceptance factor (AF) is introduced to replace (1-P1/P2). Thus, Equation 2 becomes:

Equation 3: Vw = V1·AF

Or, if you need to find the total tank size (V1) needed to store water Vw, then you may use:

Equation 4: V1 = Vw/AF

The term acceptance factor has long been used in our industry to help size a vessel for water storage. The acceptance factor has been simplified into tabular format, such as that shown in Table 1. The tables have made the selection quite simple, yet knowing the straightforward equation that’s the origin of the table will give you confidence of a correct answer every time.

Critical questions
Sizing a proper air cushion to support the water needed for the job can now be simplified by answering the following three critical sizing questions:

1. How much water is needed in storage (Vw)?
2. What is the low pressure required at the tank (P1)?
3. What is the maximum high pressure at the tank (P2)?

Using Table 1 and Equation 4, we can complete the critical sizing for any application.

Example A: 12 gallons of RO water for daily usage must be stored in a vessel for use in a home. The minimum pressure required at the storage tank is 20 psig. A pump is used to transfer the treated water to the tank until a pressure switch shuts the pump off at 35 psig. What’s the minimum vessel size required to store the 12 gallons?

Solution: Answering the three questions for critical sizing shows:

1. How much water is needed in storage (Vw)? Vw = 12 gallons
2. What is the low pressure required at the tank (P1)? P1 = 20 psig
3. What is the maximum high pressure at the tank (P2)? P2 = 35 psig

Using Equation 4 and Table 1, we can now size the correct initial volume of air needed (which, of course, is equal to the tank size needed to store our 12 gallons of water):

V1 = 12 gallons/AF

From Table 1, the acceptance factor AF = 0.30. The equation further simplifies to:

V1 = 12 gallons/0.30
V1 = 40 gallons

This means a 40-gallon tank (or larger) is needed to allow us to store 12 gallons of water between 20 psig and 35 psig. In terms of Boyle’s Law, a 40-gallon air cushion is needed, pre-charged to 20 psig. As water enters the vessel, the air is squeezed until the air cushion reaches 35 psig. To make this happen, 12 gallons of water enter the vessel to squeeze the air cushion to 28 gallons. The resulting air cushion is now at 35 psig and at rest. As water is used from the vessel, the air cushion expands and the pressure begins to drop. This will continue until the air cushion reaches its original state of 40 gallons, at which point the pressure will reach the initial state of 20 psig.

Conclusion
Boyle’s Law of compressible gases has long been utilized to size hydro-pneumatic water storage vessels for many different applications. Although many companies have simplified the physical law to make it easier for the layperson to use easy-tank sizing, the true professional wants the knowledge of what’s behind the charts and tables. With greater knowledge comes greater confidence. And, who knows, you may not have these simplified charts and tables, or the understanding and confidence to explain them, when you most need them.