Water: Thermal Expansion vs Temperature and Salinity

As discussed in Water: Density vs Temperature and Salinity, the density of water changes as a function of thermodynamic properties like temperature and pressure.  Since density and volume have a simple inverse relationship, the volume of water also changes as those properties vary.  One commonly studied instance of this is the Coefficient of Thermal Expansion: the rate at which a substance's volume changes as the temperature changes and pressure is held constant.  For Water this Coefficient is particularly interesting because it starts off negative (volume decreases as you increase temperature) and then transitions to positive (volume increases as you increase temperature).

Unfortunately, finding high-accuracy values for this Coefficient as a function of Temperature and Salinity on the internet proved difficult.  However, since we already did research and found high-accuracy values for Density we can use that data to numerically determine the Coefficient of Thermal Expansion.  Here we provide the results of that analysis, as well as compare the calculated values to the sparse data available on the internet.  These values are intended for Armchair Scientists: those of us without access to professional journals or specialized python libraries, but who still want to do calculations relating to volume changes of water.

Thermal Expansion of Water vs Temperature and Salinity

Conclusions

We have the Suggested Sources for Water Density from our previous post on Water Density:

Pure Water Density

• Suggested Source: https://webbook.nist.gov/chemistry/fluid/
  • Accurate for Temperatures (Melting Point, 1273 K), Pressures up to 1000 MPa

Sea Water Density

• Suggested Source: https://os.copernicus.org/articles/5/91/2009/os-5-91-2009.pdf
  • Accurate for Temperatures (T) (0 - 90) ℃ and Salinity (S) (0 - 70) g / kg
  • Provides a modification to Pure Water Density, that we provide here so readers don't have to dive into the article:
    • 𝞺(T,S) = 𝞺_o(T) + A(T) * S + B(T) * S^1.5 + C(T) * S²
    • 𝞺_o(T) = Pure Water Density at the desired Temperature
    • A(T) = 8.197247e-01 - 3.779454e-03 * T + 6.821795e-05 * T² - 8.009571e-07 * T³ + 6.158885e-09 * T⁴ - 2.001919e-11 * T⁵
    • B(T) = -5.808305e-03 + 5.354872e-05 * T - 4.714602e-07 * T²
    • C(T) = 5.249266e-04
  • These coefficients are only valid when units are specified as:
    • 𝞺: kg / m³
    • T: ℃
    • S: g / kg
  • Sea Water is typically specified with S = 35.16504

We inverted these Density values to get Volume, and then used Finite Difference Methods to calculate the Coefficient of Thermal Expansion for Water for the ranges of Temperature and Salinity specified above: Temperatures (T) (0 - 90) ℃ and Salinity (S) (0 - 70) g / kg.

Coefficient of Thermal Expansion

• The Suggested Sources have been assembled into an Excel 2019 Spreadsheet with outright data for Pure Water, but with the above formula for Sea Water coded in.  Users only need to specify the Salinity they're interested in and Density and Thermal Expansion Coefficients will be populated for temperatures at 0.01 ℃, 0.1 ℃, and 1.0 ℃ increments.

Water Property Calculator - Excel 2019 spreadsheet to calculate Water Density and Coefficient of Thermal Expansion vs Temperature for a user-specified Salinity.  ~1.4 Mb in size.

We combined all of this data into a Surface Plot of Coefficient of Thermal Expansion versus Temperature and Salinity:

Water Thermal Expansion vs Temperature and Salinity

Methodology

We started by googling things like "thermal expansion of water vs temperature" and got a much smaller list of sources than we got for density.  We got even fewer sources when we tried to get Coefficient values that took Salinity into account.  Without any complete online sources, the next best method is to calculate the Coefficients ourselves from the Density values that we determined in our Density of Water research.  That research provided Density values as a function of Temperature and Salinity with 0.01 ℃ resolution, which should be sufficient for our purposes.

The formula for the Coefficient of Thermal Expansion is:

Starting with Density values, we can invert them to get Volume values.  From there we can use Finite Difference Methods to estimate the derivative of Volume with respect to Temperature.  We used the fourth-order central finite difference methods described in Wikipedia.  The formulas for these methods are encoded in the Water Property Calculator Excel 2019 spreadsheet linked to above.

We can compare the calculated values to the few online sources we found for the Coefficient of Thermal Expansion as a function of Temperature and Salinity, although those sources tended to be sparsely populated and inaccurate.

Pure Water

For Pure Water we compare our calculated values to the following sources:

1. MIT.PURE - Massachusetts Institute of Technology
   • http://web.mit.edu/seawater/2017_MIT_Seawater_Property_Tables_r2b.pdf
2. TOOLBOX1.PURE - The Engineering ToolBox, first page
   • https://www.engineeringtoolbox.com/volumetric-temperature-expansion-d_315.html
3. TOOLBOX2.PURE - The Engineering ToolBox, second page
   • https://www.engineeringtoolbox.com/water-density-specific-weight-d_595.html
4. UCPRESS.PURE - A book about the Physics and Chemistry of the Oceans from 1942
   • https://publishing.cdlib.org/ucpressebooks/view?docId=kt167nb66r&chunk.id=d3_4_ch03&toc.depth=1&toc.id=ch03&brand=eschol
5. CALC.PURE - Our Finite Difference Method

Pure Water Thermal Expansion vs Temperature (0-90) ℃

TOOLBOX2.PURE clearly diverges from the other sources and is simply a bad source at high temperatures.  This is not uncommon when sources use an Equation of State outside of its accurate range.  The divergence between CALC.PURE and MIT.PURE is due to the MIT Equation of State not being particularly good: as commented in the Density post, it isn't accurate enough to give the right temperature of maximum density.

Overall, the calculated values for Coefficient of Thermal Expansion of Pure Water look good, and given the accuracy of the Density values we can have faith in these calculated values.

Sea Water

For Sea Water we compare our calculated values to the following sources:

1. MIT.SEA - Massachusetts Institute of Technology
   • http://web.mit.edu/seawater/2017_MIT_Seawater_Property_Tables_r2b.pdf
2. UCPRESS.SEA - A book about the Physics and Chemistry of the Oceans from 1942
   • https://publishing.cdlib.org/ucpressebooks/view?docId=kt167nb66r&chunk.id=d3_4_ch03&toc.depth=1&toc.id=ch03&brand=eschol
3. FEISTEL.SEA - a paper by R. Feistel from 2011
   • https://www.researchgate.net/publication/275701512_Feistel_2011_EOLSS_appendix
4. CALC.SEA - Our Finite Difference Method

Sea Water Thermal Expansion vs Temperature (0-90) ℃

Once again, the MIT.SEA data diverges from our calculated values as for the aforementioned reasons.  Otherwise, the calculated values for Sea Water look good.

Overall, using Fourth Order Central Difference Finite Methods to calculate the Coefficients of Thermal Expansion versus Temperature and Salinity from our Density data seems to have worked.

Related Pages

• Water Main Page
  • Description of Full Research Project
• Water: Density vs Temperature and Salinity
  • How does Water's density vary as a function of Temperature and Salinity?
• Water Property Calculator
  • Excel 2019 spreadsheet to calculate Water Density and Coefficient of Thermal Expansion vs Temperature for a user-specified Salinity.  ~1.4 Mb in size.