# How do I compute the saturation temperature of liquid water?

I'm trying to create a code to calculate the saturation temperature of liquid water, when I'm only given the pressure. There are calculators online, but there's no further details about what formula they used. If anyone could give me references on how to find the saturation temperature, that would be great.

Thanks!

• When I needed to do that, I typed the properties into an excel table from the steam tables and then wrote an interpolating lookup function - linear of course. A good exercise to improve your skills. Apr 22, 2022 at 5:55
• That method seems a little bit tedious. How long did it take you to do that? Edit: But then again, I don't think I need to write that many values since I only need the saturation temperatures over a range of about 100 values.. Apr 22, 2022 at 6:01
• Tedious? Doing all the calculations for 1 run on the engine project took 8 hours. Once I had the spreadsheet done it was 10 mins of typing the primary values - then even the charts were complete. Had over 80 runs. Apr 22, 2022 at 6:07
• One second thought, I might go with your plan. There's a nifty website that allows you to get the thermophysical properties of different substances, including water. I've compared the values to my steam table and they seem to just be very similar: webbook.nist.gov/chemistry/fluid Apr 22, 2022 at 7:11

Any equation of liquid/vapor water properties is an emperical estimation of the observed properties. This is why steam tables exist, and not steam equations.

W C Reynolds published an equation for Psat as function on temperature in his book Thermodynamic Properties in SI (1979, Dept of Mechanical Engineering, Stanford University). It is a bit complicated, like 8 constants of 8 significant figures.

A quick search "steam table equations" located an article available at sciencedirectassets.com called "Simplified Equations for Saturated Steam Properties for Simulation Purpose" (authors from Malaysia). Their equation is simpler.

$$ln P = a+bln(T_r)+c[ln T_r]^2+d[lnT_r]^4+eT_r^4$$

$$T_r$$ is $$T/647.096$$, the number is water critical temperature in Kelvin.

A search engine will let you find the original paper.

T = ( [ (P - a)(1/e) - d ] / b)(1/c)

From inverting this equation, which computes the vapor pressure of water in Torr over the range 40°F to 120°F:

P = a + (b $$\cdot$$Tc + d)e

a = 4.623310605E+00
b = 4.501250993E-03
c = 1.541209347E+00
d = 1.397979393E-01
e = 2.211835619E+00

P is saturation vapor pressure in Torr
T is water temperature in °F


The calculated P is within +/- 0.7 Torr of the value from NIST.

The coefficients are from a least-squares fit of NIST data at 1°F increments using Excel's Solver.

For greater accuracy (within less +/- 1$$\cdot$$10-5 Torr of the NIST value), apply the correction below to the P value above (from a BASIC program):

    LET P = P + poly_5(T)

FUNCTION poly_5(t)
LET a = 1.27036280E-09
LET b = -1.47540377E-07
LET c = -1.11584346E-05
LET d = 1.15323358E-03
LET e = 4.83696142E-02
LET f = -3.47478756E+00

LET poly_5 = a*t^5 + b*t^4 + c*t^3 + d*t^2 + e*t + f
END FUNCTION


Inverting "P = P + poly_5(T)" to obtain T as a function of P, which was requested at the top of this thread, is left to others.