# Expansion Tank Sizing Formulas

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Expansion tanks are a necessary part of all closed hydronic systems to control both minimum and maximum pressure throughout the system. Expansion tanks are provided in closed hydronic systems to (1) accept changes in system water volume as water density changes with temperature to keep system pressures below equipment and piping system component pressure rating limits. Also, (2) maintain a positive gauge pressure in all parts of the system to prevent air from leaking into the system. (3) Maintain sufficient pressures in all parts of the system to prevent boiling, including cavitation at control valves and similar constrictions. (4) Maintain net positive suction head required (NPSHR) at the suction of pumps.

The latter two points generally apply only to high temperature (greater than approximately 210°F [99°C]) hot water systems. For most HVAC applications, only the first two points need to be considered.

## Tank Styles

There are four basic styles of expansion tanks:

### Vented or open steel tanks

Since they are vented, open tanks must be located at the highest point of the system. Water temperature cannot be above 212°F (100°C), and the open air/water contact results in a constant migration of air into the system, causing corrosion. Accordingly, this design is almost never used anymore.

### Closed steel tanks

Also called plain steel tanks or compression tanks by some manufacturers.

This is the same tank style as the vented tank, but with the vent capped. This allows the tank to be located anywhere in the system and work with higher temperatures. But they still have the air/water contact that allows for corrosion, and sometimes a gradual loss of air from the tank as it is absorbed into the water.

Unless precharged to the minimum operating pressure prior to connection to the system, this style of tank also must be larger than precharged tanks. Accordingly, this design is also almost never used anymore.

### Diaphragm tanks

This was the first design of a compression tank that included an air/water barrier (a flexible membrane, to eliminate air migration) and that was designed to be precharged (to reduce tank size). The flexible diaphragm typically is attached to the side of the tank near the middle and is not field replaceable; if the diaphragm ruptures, the tank must be replaced.

Bladder tanks use a balloon-like bladder to accept the expanded water. Bladders are often sized for the entire tank volume, called a “full acceptance” bladder, to avoid damage to the bladder in case they become waterlogged. Bladders are gener ally field replaceable. This is now the most common type of large commercial expansion tank.

## Sizing Formulas

The general formula for tank sizing, Equation 1 (with variable names adjusted to match those used in this article), from basic principles assuming perfect gas laws:

V_(t)=(V_(s)(E_(w)-E_(p)))/((P_(s)T_(c))/(P_(i)T_(s))-(P_(s)T_(h))/(P_(max)T_(s))-E_(wt)[1-(P_(s)T_(c))/(P_(max)T_(s))]+E_(t))-0.02V_(s)

Where

Vt = tank total volume

Vs = system volume

Ps = starting pressure when water first starts to enter the tank, absolute

Pi = initial (precharge) pressure, absolute

Pmax = maximum pressure, absolute

Ew = unit expansion ratio of the water in the system due to temperature rise = (νhc-1)

vh = the specific volume of water at the maximum temperature, Th.

vc = the specific volume of water at the minimum temperature, Tc .

Ep = unit expansion ratio of the piping and other system components in the system due to temperature rise = 3α(Th-Tc )

α = coefficient of expansion of piping and other system components, per degree

Th = maximum average water temperature in the system, degrees absolute

Tc = minimum average water temperature in the system, degrees absolute

Ts = starting air temperature in the tank prior to fill, degrees absolute

Ewt = unit expansion ratio of water in the tank due to temperature rise

Et = unit expansion ratio of the expansion tank due to temperature rise

The last term (0.02 Vs ) accounts for additional air from desorption from dissolved air in the water. This equation can be simplified to Equation below by ignoring small terms and assuming tank temperature stays close to the initial fill temperature (typically a good assumption, assuming no insulation on the tank or piping to it, which is a common, and recommended, practice):

V_(t)=(V_(s)[((v_(h))/(v_(c))-1)-3alpha(T_(h)-T_(c))])/((P_(s))/(P_(i))-(P_(s))/(P_(max)))

This equation includes the credit for the expansion of the piping system. This term is also relatively small and the expansion coefficients are hard to determine given the various materials in the system, but it is included in Equation above since it is included in the ASHRAE Handbook sizing equations. This term is also included in some, but not most, expansion tank manufacturers’ selection software. Most manufacturers conservatively ignore this term since it is small and no larger than the terms already ignored in the above Equation. Ignoring this term results in Equation below:

V_(t)=(((v_(h))/(v_(c))-1)V_(s))/((P_(s))/(P_(i))-(P_(s))/(P_(max)))

The numerator is the volume of the expanded water, Ve , as it warms from minimum to maximum temperatures, so the equation can be written:

V_(t)=(V_(e))/((P_(s))/(P_(i))-(P_(s))/(P_(max)))

Where:

V_(e)=(v_(h)//v_(c)-1)V_(s)

The equation can be further simplified based on the style of tank used.

### Vented tank

For vented tanks, the pressures are all the same and the dominator limits to 1, so the tank size is simply the volume of expanded water:

V_(t)=V_(e)

### Closed Tank (no precharge)

For unvented plain steel tanks, the starting pressure is typically atmospheric pressure with the tank empty (no precharge). The tank is then connected to the makeup water, which pressurizes the tank to the fill pressure by displacing air in the system, essentially wasting part of the tank volume. So the sizing equation is:

V_(l)=(V_(e))/((P_(a))/(P_(i))-(P_(a))/(P_(max)))

Where, Pa = atmospheric pressure

### Precharged Tank

For any tank that is precharged to the required initial pressure, including properly charged diaphragm and bladder tanks, but also including closed plain steel tanks if precharged, Ps is equal to Pi so the sizing equation reduces to:

V_(t)=(V_(e))/(1-(P_(i))/(P_(max)))

Note that this equation only applies when the tank is precharged to the required Pi . Tanks are factory charged to a standard precharge of 12 psig (83 kPag).

### Closed Tank

For higher desired precharge pressures, either a special order can be made from the factory or the contractor must increase the pressure with compressed air or a hand pump. But it is not uncommon for this to be overlooked. This oversight can be compensated for by sizing the tank using Equation below (assuming atmospheric pressure at sea level):

V_(t)=(V_(e))/((26.7)/(P_(i))-(26.7)/(P_(max)))

(12 psig/26.7 psia [83 kPag/184 kPaa] precharge). This will increase the tank size vs. a properly precharged tank.

### ASME Boiler and Pressure Vessel Code-2015, Section VI

ASME Boiler and Pressure Vessel Code-2015, Section VI, includes sizing equations (as do the UMC and IMC, which extract the equations verbatim), as shown in Equation below, with variables revised to match those used in this article:

V_(t)=(V_(s)(0.00041T_(h)-0.0466))/((P_(a))/(P_(i))-(P_(a))/(P_(max)))

Comparing the denominator of this Equation to Equation for Closed Tank (no precharge), this formula is clearly for sizing a nonprecharged tank; it will overestimate the size of a precharged tank. The numerator is a curve fit of Ve ; it assumes a minimum temperature of 65°F (18°C) and is only accurate in the range of about 170°F to 230°F (77°C to 110°C) average operating temperature. Therefore, this equation cannot be used for very high temperature hot water (e.g. 350°F [177°C]), closed-circuit condenser water, or chilled water systems.

Author: Steven T. Taylor, PE

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