When water is heated, it expands. If this expansion occurs in a closed system, dangerous water pressures can be created. A domestic hot water system can be a closed system when the hot water fixtures are closed and the cold water supply piping has backflow preventers or any other device that can isolate the domestic hot water system from the rest of the domestic water supply.

These pressures can quickly rise to a point at which the relief valve on the water heater unseats, thus relieving the pressure, but at the same time compromising the integrity of the relief valve.

A relief valve installed on a water heater is not a control valve, but a safety valve. It is not designed or intended for continuous usage. Repeated excessive pressures can lead to equipment and pipe failure and personal injury.

When properly sized, an expansion tank connected to the closed system provides additional system volume for water expansion while ensuring a maximum desired pressure in a domestic hot water system. It does this by utilizing a pressurized cushion of air (figure below). The following discussion explains how to size an expansion tank for a domestic hot water system and the theory behind the design and calculations. It is based on the use of a diaphragm or bladder-type expansion tank, which is the type most commonly used in the plumbing industry. This type of expansion tank does not allow the water and air to be in contact with each other.

## Expansion of Water

A pound of water at 140°F has a larger volume than the same pound of water at 40°F. To put it another way, the specific volume of water increases with an increase in temperature.

If the volume of water at a specific temperature condition is known, the expansion of water can be calculated as follows:

Where:

- Vew = Expansion of water, gallons
- Vs
_{1}= System volume of water at temperature 1, gallons - Vs
_{2}= System volume of water at temperature 2, gallons

Temp., ∘ F | Specific Volume, ft^{3}/lb |

40 | 0.01602 |

50 | 0.01602 |

60 | 0.01604 |

70 | 0.01605 |

80 | 0.01607 |

90 | 0.01610 |

100 | 0.01613 |

110 | 0.01617 |

120 | 0.01620 |

130 | 0.01625 |

140 | 0.01629 |

150 | 0.01634 |

160 | 0.01639 |

Vs_{1} is the initial system volume and can be determined by calculating the volume of the domestic hot water system. This entails adding the volume of the water-heating equipment to the volume of the piping and any other part of the hot water system.

Vs_{2} is the expanded system volume of water at the design hot water temperature. Vs_{2} can be expressed in terms of Vs_{1}. To do that, look at the weight of the water at both conditions.

The weight (**W**) of water at temperature 1 (T_{1}) equals the weight of water at T_{2}, or W_{1} = W_{2}. At T_{1}, W_{1} = Vs_{1}/vsp_{1}, and similarly at T_{2}, W_{2} = Vs_{2}/vsp_{2}, where vsp equals the specific volume of water at the two temperature conditions. (See Table above for specific volume data.)

Since W_{1} = W_{2}, then:

Solving for Vs_{2}:

**Example 1:**

A domestic hot water system has 1,000 gallons of water. How much will the 1,000 gallons expand from a temperature of 40°F to a temperature of 140°F?

From Table above, vsp_{1} = 0.01602 (at 40°F) and vsp_{2} = 0.01629 (at 140°F). Utilizing Equation:

**Note that this is the amount of water expansion and should not be confused with the size of the expansion tank needed.**

## Expansion of Material

**Will the expansion tank receive all of the water expansion?**

The answer is **no**, because not just the water is expanding. The piping and water-heating equipment expand with an increase in temperature as well. Any expansion of these materials results in less of the water expansion being received by the expansion tank. Another way of looking at it is as follows:

Venet = Vew – Vemat

**Venet**= Net expansion of water received by the expansion tank, gallons**Vew**= Expansion of water, gallons**Vemat**= Expansion of material, gallons

To determine the amount of expansion each material experiences per a certain change in temperature, look at the **coefficient of linear expansion** for that material.

For copper, the coefficient of linear expansion is 9.5 × 10^{–6} inch/inch/°F, and for steel it is 6.5 × 10^{–6} inch/inch/°F.

From the coefficient of linear expansion, a material’s coefficient of volumetric expansion can be determined. **The coefficient of volumetric expansion is three times the coefficient of linear expansion**:

**ß=3α**

ß = Volumetric coefficient of expansion

α = Linear coefficient of expansion

Thus, the volumetric coefficient for copper is 28.5 × 10^{–6} gallon/gallon/°F, and for steel it is 19.5 × 10^{–6} gallon/gallon/°F. The material will expand proportionally with an increase in temperature.

**Vemat = Vmat × ß (T2 – T1)**

**Venet = Vew – [Vmat _{1}×ß_{1} (T_{2} – T_{1})+Vmat_{2}×ß_{2} (T_{2} – T_{1})]**

Nominal Volume of Piping

Volume of Pipe, gal/linear ft of pipe | |

Pipe Size, in. | 0.02 0.02 0.02 |

1 / 2 1 / 2 1//2 | 0.03 0.03 0.03 |

3 / 4 3 / 4 3//4 | 0.04 0.04 0.04 |

1 | 0.07 0.07 0.07 |

11 / 4 11 / 4 11//4 | 0.10 0.10 0.10 |

11 / 2 11 / 2 11//2 | 0.17 0.17 0.17 |

2 | 0.25 0.25 0.25 |

21 / 2 21 / 2 21//2 | 0.38 0.38 0.38 |

3 | 0.67 0.67 0.67 |

4 | 1.50 1.50 1.50 |

6 | 2.70 2.70 2.70 |

8 |

**Example 2:**

A domestic hot water system has a water heater made of steel with a volume of 900 gallons. It has 100 feet of 4-inch piping, 100 feet of 2-inch piping, 100 feet of 1½-inch piping, and 300 feet of ½-inch piping. All of the piping is copper. Assuming that the initial temperature of the water is 40°F and the final temperature of the water is 140°F, (1) **how much will each material expand**, and (2) **what is the net expansion of water that an expansion tank will see**?

Utilizing Equation **Vemat ** for the steel (material no. 1), Vmat_{1} = 900 gallons and Vemat_{1} = 900 (19.5 × 10^{–6})(140 – 40) = 1.8 gallons. For the copper (material no. 2), first look at Table above to determine the volume of each size of pipe:

- 4 inches = 100 x 0.67 = 67 gallons
- 2 inches = 100 x 0.17 = 17 gallons
- 1½ inches = 100 x 0.10 = 10 gallons
- ½ inch = 300 x 0.02 = 6 gallons

Total volume of copper piping = 100 gallons Utilizing Equation **Vemat ** for copper, Vmat_{2} = 100 gallons and Vemat_{2} = 100 (28.5 × 10–6)(140 – 40) = 0.3 gallon.

The initial system volume of water (Vs_{1}) equals Vmat_{1} + Vmat_{2}, or 900 gallons + 100 gallons. From Last example, 1,000 gallons of water going from 40°F to 140°F expands 16.9 gallons. Thus, utilizing Equation Venet, Venet = 16.9 – (1.8 + 0.03) = 15 gallons. This is the net amount of water expansion that the expansion tank will see. **Once again, note that this is not the size of the expansion tank needed**.

## Boyle’s Law

After determining how much water expansion the expansion tank will see, it is time to look at how the cushion of air in an expansion tank allows the designer to limit the system pressure.

Boyle’s law states that at a constant temperature, the volume occupied by a given weight of perfect gas (including for practical purposes atmospheric air) varies inversely as the absolute pressure (gauge pressure + atmospheric pressure). It is expressed by the following:

**P _{1}V_{1} = P_{2}V_{2} **

where

- P
_{1}=Initial air pressure, pounds per square inch absolute (psia) - V
_{1}=Initial volume of air, gallons - P
_{2}=Final air pressure, psia - V
_{2}=Final volume of air, gallons

**How does this law relate to sizing expansion tanks in domestic hot water systems? **

The air cushion in the expansion tank provides a space into which the expanded water can go. The volume of air in the tank decreases as the water expands and enters the tank. As the air volume decreases, the air pressure increases.

Utilizing Boyle’s law, the initial volume of air (i.e., the size of the expansion tank) must be based on:

- Initial water pressure,
- Desired maximum water pressure, and
- Change in the initial volume of the air.

To utilize the above equation, realize that the pressure of the air equals the pressure of the water at each condition, and make the assumption that the temperature of the air remains constant at condition 1 and condition 2 in Figure above.

This assumption is reasonably accurate if the expansion tank is installed on the cold water side of the water heater. **Remember, in sizing an expansion tank, the designer is sizing a tank of air, not a tank of water.**

Referring to Figure above, at condition 1 the tank’s initial air pressure charge, P_{1}, equals the incoming water pressure on the other side of the diaphragm. The initial volume of air in the tank, V_{1}, is also the size of the expansion tank. The final volume of air in the tank, V_{2}, also can be expressed as V_{1} less the net expansion of water (Venet).

The pressure of the air at condition 2, P_{2}, is the same pressure as the maximum desired pressure of the domestic hot water system at the final temperature, T_{2}. **P _{2} should always be less than the relief valve setting on the water heater**.

Utilizing Boyle’s law:

- V
_{1}= Size of expansion tank required to maintain the desired system pressure, P2, gallons - Venet= Net expansion of water, gallons P1 = Incoming water pressure, psia (Note: Absolute pressure is gauge pressure plus atmospheric pressure, or 50 psig = 64.7 psia.)
- P2 = Maximum desired pressure of water, psia

**Example 3:**

Looking again at the domestic hot water system described in Example 2, if the cold water supply pressure is 50 psig and the maximum desired water pressure is 110 psig, what size expansion tank is required?

Example 2 determined that Venet equals 15 gallons. Converting the given pressures to absolute and utilizing Equations described above, the size of the expansion tank needed can be determined as:

Note: When selecting the expansion tank, make sure the tank’s diaphragm or bladder can accept 15 gallons of water (Venet).