Specific Heat Ratio

The specific heat ratio, k, is defined as the ratio of the constant pressure specific heat to the constant volume specific heat.  It depends weakly (over small temperature ranges) on temperature.  In this post, we’ll look at the effect of that temperature dependence.  This figure shows the value of the specific heat ratio of air as a function of temperature.

For calculations near room temperature, or for rough estimates, it is usually accurate enough to employ the so-called “cold gas assumption” and just use the value of k at room temperature for all calculations.  However, for the most accurate calculations, the variation with temperature must be used.
As an example, this figure shows the compression heating curve for air from the last post (where the cold gas assumption was employed) along with the curve that would result from taking into account the variation of specific heat with temperature.  The percentage error due to the cold gas assumption is also shown on the plot in green.  Over this modest temperature range, the cold gas assumption is accurate to within about 0.5%.

As another example, this figure shows the speed of sound in air [speed of sound=sqrt(k*R*T)] as a function of temperature for the cold gas assumption, and for variable specific heat.  Note that the temperature range in this case is much larger, and the cold gas assumption results in error of about 2.5% at the high end of the range.  The percentage error is again shown in green.

This fourth order polynomial in temperature (T) provides an estimate for the specific heat ratio of air over a range of 200K to 5000K.  The error of the estimate is less than 1% over that range.

k=1.43857 +
(-1.43062E-04)*(1/K) *T +
(5.13047E-08)*(1/K^2)*T² +
(-8.60273E-12)*(1/K^3)*T³ +
(5.43445E-16)*(1/K^4)*T⁴