Understanding Expansion Factor
Expansion factor is deﬁned as the fractional change of the volume of a substance from a change in its temperature. Simply said, as a substance temperature increases, its volume increases. This is important when determining the amount of ﬂuid that will expand in a hydronic system and subsequently the expansion tank needed to accommodate this increase.
We will look at ﬂuid expansion with an end towards what is considered the accurate estimation as documented in the ASHRAE Fundamentals Handbook. The equation is fundamentally:
Net expansion = gross expansion of ﬂuid minus expansion of piping and components
This is typically expressed as a fraction that is then multiplied by the total system volume of the hydronic system to calculate the expanded ﬂuid. To that end, the gross expansion is the difference in speciﬁc volumes at two temperatures divided by the starting (lower temperature) speciﬁc volume.
Gross Expansion Factor of Fluid = ((v2v1))/v1 = v2/v11
Where:
v1: specific volume of the fluid at temperature t1 (higher temperature)
v2: specific volume of the fluid at temperature t2 (lower temperature)
The expansion of the piping and components can be cumbersome to calculate because different component materials (boilers/chillers, air separators, heat exchangers, etc.) may be at potentially different temperatures, so an estimation is needed to be made. The traditional approach is to make all components materials homogeneous with the system piping and all at the same stop/start temperatures. The formula for calculating the change in volume of a solid or liquid due to thermal expansion is:
Expansion Factor of Piping and Components = β(t2t1)
Where:
β: coefficient of the ratio of volumetric expansion of the piping material ((in^{3}/in^{3})^{o}F) The coefficient can be estimated:
β≅3α
Making the equation:
Expansion Factor of Piping and Components = 3α(t2t1)
Where:
α: Coefficient of the ratio of linear expansion of the piping material (in/in^{o}F)
For the exactingminded, the actual equation for the growth factor of the expansion of piping and components is:
Expansion Factor of Piping and Components = (α(t2t1)+1)^31
This is the integration of the circumferential and linear growth of a hollow cylinder (pipe). The approximation of 3α(t2t1) follows the equation with less than 1 percent difference over the temperature range from 100^{o}F to 350^{o}F, which makes the 3α(t2t1) a more convenient equation estimation.
Linear expansion coefficients (α) for common materials are shown in the following table:
Material 
α (in./in./^{o}F) 
Ductile Iron 
6.20E06 
PVC 
3.00E05 
Cast Iron 
6.11E06 
Steel 
6.67E06 
HDPE 
8.00E05 
Concrete 
5.50E06 
Copper 
9.44E06 
And yes, this paper uses English units because who could argue with the exact science behind the inch, foot, yard, and pound and Fahrenheit? USA, Liberia, and Myanmar can’t all be wrong! Even the United Kingdom abandoned and switched to Celsius decades ago. Oh well, go ‘Merica!
It is important to review the speciﬁc weight of water and discuss the differences of heating and chilled water systems. The chart shows the proﬁle of speciﬁc weight and the typical chilled water and typical heating system zones. The fact that water density in a heating water system decreases at a considerably greater rate than in a chilled water system, shows that a single linear approximation equation is not a n accurate option. In fact, as will be shown, the piping can expand greater than the water at lower temperatures.
The net expansion of water starting at 70^{o}F in a hydronic heating system as inﬂuenced by the piping material is shown in the following chart for steel piping:
Charts can be deceiving in assessing the inﬂuence of the component expansion. This impact of steel pipe expansion on the net expansion of the water in the piping system is shown in the following table.
Heating System 

mp (^{o}F) 
Gross Exp. 
Net Exp. 
% Reduction 
70 
0.00% 
0.00% 
– 
80 
0.13% 
0.11% 
15.55% 
90 
0.31% 
0.27% 
13.08% 
100 
0.48% 
0.42% 
12.40% 
110 
0.71% 
0.63% 
11.25% 
120 
0.96% 
0.86% 
10.46% 
130 
1.22% 
1.10% 
9.85% 
140 
1.50% 
1.36% 
9.34% 
150 
1.81% 
1.65% 
8.82% 
160 
2.13% 
1.95% 
8.45% 
170 
2.48% 
2.28% 
8.05% 
180 
2.84% 
2.62% 
7.75% 
190 
3.23% 
2.99% 
7.43% 
200 
3.63% 
3.37% 
7.17% 
This reveals a dramatic impact on the expansion of the water in the heating system. Employing the same methodology to the chilled water system starting at 40^{o}F, the impact of the steel piping has an even greater inﬂuence on the net expansion.
Chilled Water System 

Temp (^{o}F) 
Gross Exp. 
Net Exp. 
% Reduction 
40 
0.00% 
0.00% 
– 
50 
0.02% 
0.00% 
24.82% 
60 
0.10% 
0.06% 
58.43% 
70 
0.19% 
0.13% 
68.85% 
80 
0.32% 
0.24% 
75.11% 
90 
0.50% 
0.40% 
79.96% 
100 
0.68% 
0.56% 
82.29% 
110 
0.91% 
0.77% 
84.54% 
This shows that at times the lower temperature piping expands faster that the water.
Propylene Glycol
The mixture of propylene glycol has the effect of increasing speciﬁc weight and increasing the slope of the rate of change.
The resulting net expansion of various propylene concentrations are shown for typical heating and chilled hydronic loops. The chart for heating shows the expansion factor with 70oF as the starting temperature.
Net Expansion Heating System 

Temp (^{o}F) 
Water 
10%PG 
20%PG 
30%PG 
40%PG 
50%PG 
60%PG 
70 
0.00% 
0.00% 
0.00% 
0.00% 
0.00% 
0.00% 
0.00% 
80 
0.11% 
0.22% 
0.27% 
0.33% 
0.38% 
0.40% 
0.45% 
90 
0.27% 
0.44% 
0.55% 
0.66% 
0.76% 
0.81% 
0.91% 
100 
0.42% 
0.67% 
0.83% 
0.99% 
1.15% 
1.22% 
1.37% 
110 
0.63% 
0.93% 
1.14% 
1.34% 
1.53% 
1.67% 
1.86% 
120 
0.86% 
1.21% 
1.48% 
1.70% 
1.93% 
2.14% 
2.36% 
130 
1.10% 
1.49% 
1.82% 
2.07% 
2.32% 
2.62% 
2.86% 
140 
1.36% 
1.78% 
2.16% 
2.44% 
2.72% 
3.10% 
3.37% 
150 
1.65% 
2.12% 
2.51% 
2.84% 
3.15% 
3.56% 
3.85% 
160 
1.95% 
2.47% 
2.85% 
3.25% 
3.58% 
4.02% 
4.34% 
170 
2.28% 
2.82% 
3.20% 
3.65% 
4.01% 
4.48% 
4.84% 
180 
2.62% 
3.16% 
3.57% 
4.06% 
4.47% 
4.95% 
5.35% 
190 
2.99% 
3.48% 
3.95% 
4.48% 
4.94% 
5.43% 
5.88% 
200 
3.37% 
3.81% 
4.34% 
4.90% 
5.42% 
5.91% 
6.42% 
The chart for chilled hydronic systems shows the net expansion with the lowest temperature that concentration of propylene glycol is anticipated to start expanding. For example, 40% propylene glycol starts expanding from 10oF (its lowest anticipated temperature).
Net Expansion Chilled System 

Temp (^{o}F) 
Water 
10%PG 
20%PG 
30%PG 
40%PG 
50%PG 
60%PG 
50 






0.00% 
40 






0.08% 
30 





0.00% 
0.16% 
20 





0.11% 
0.32% 
10 




0.00% 
0.26% 
0.51% 
0 



0.00% 
0.09% 
0.42% 
0.78% 
10 


0.00% 
0.01% 
0.23% 
0.60% 
1.08% 
20 


0.01% 
0.11% 
0.49% 
0.87% 
1.42% 
30 

0.00% 
0.10% 
0.35% 
0.70% 
1.17% 
1.72% 
40 
0.00% 
0.13% 
0.27% 
0.58% 
1.01% 
1.52% 
2.11% 
50 
0.02% 
0.30% 
0.49% 
0.83% 
1.30% 
1.87% 
2.52% 
60 
0.10% 
0.46% 
0.71% 
1.07% 
1.60% 
2.22% 
2.92% 
70 
0.19% 
0.64% 
0.94% 
1.34% 
1.92% 
2.59% 
3.34% 
80 
0.32% 
0.86% 
1.22% 
1.67% 
2.31% 
3.00% 
3.81% 
90 
0.50% 
1.09% 
1.50% 
2.00% 
2.70% 
3.42% 
4.29% 
100 
0.68% 
1.31% 
1.78% 
2.34% 
3.09% 
3.84% 
4.76% 
110 
0.91% 
1.57% 
2.10% 
2.70% 
3.49% 
4.30% 
5.26% 
There are many aspects of expansion that can be further explored as to their signiﬁcance:
 Is the component and piping expansion coefficient truly linear?
 Should the ﬁnal temperature of a heating loop and the starting temperatures of a chilled hydronic loop be better represented as (ts+tr)/2? Where ts = supply temperature and tr = return temperature?
The entire speciﬁc volumes are included at the ﬁnal pages of this paper and can be used to develop a simple spreadsheet to use lookup functions and piping material inputs to accurately calculate the expansion factor for any system input (between 50^{o}F and 240^{o}F). For other heat transfer and antifreeze ﬂuids, simply add these to your spreadsheet for future reference.
Sam Fuller
Technical Engineer
Wessels Company