Sorry for asking this kind of question. Do you recongnize this formula? $$V(T) = 0.0000679T^3+0.0085043T^2- 0.0624T+999.87$$ when $V$ is volume of water in $\mathrm{ml}$, $T$ is temperature in celsius. This is an approximation for volume of 1kg of water with temperatures between 0~30 celsius.

This formula is simple exercise from calculus, finding out extrema. However, I've seen this approximation so many times, even in many different books. So I googled it, and found those approximations were quite accurate with some conditions. (Elements of Physical Hydrology 46p table 3.1)

Is this an appropriate approximation? How can I get this approximation? Any references will be a great help. Thank you for reading

  • $\begingroup$ Isn't it just like a Taylor expansion (about $T=0$) for the expression $\rho'=\rho/(1+\beta(T'-T))$? engineeringtoolbox.com/… $\endgroup$ – typesanitizer Jul 18 '16 at 15:16
  • $\begingroup$ It is a fit from experimental data, ant it is probably good at atmospheric pressure. $\endgroup$ – valerio Jul 18 '16 at 16:25

The equation is a cubic polynomial curve fit. There are many mathematical curve fitting programs that can give this formula, and they merely require that you list four or more points of volume vs. temperature throughout the range of interest. The exact constants that result from the data depend on the units on volume and temperature, so the equation that is listed in the posting can have different coefficients if the units are not deg C and ml.

Whether or not the approximation is appropriate depends on the degree of precision needed. The curve fit that is obtained can minimize total error vs. the measured data, or it can minimize total percent error vs. the measured data. In any event, the general trend for such data indicate that the error in the equation vs. the measured data gets better as the data range gets smaller. And, of course, since the equation is derived from measured data, there is always some small amount of error in the starting data, and that information would be important to know when you are deciding if the equation is valid for the data range of interest.


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