The Cyclic Relation

The cyclic relation is a result from calculus that is often used in thermodynamics to generate relations between properties.  Today we’ll talk about the cyclic relation and demonstrate it from tabulated steam table properties.
The cyclic relation can be written for three related properties like this:

For example, we could write a relation between temperature, pressure and enthalpy like this:

Re-arranging, we can get a relationship between temperature and pressure changes along an isenthalpic line:
 

 Using values for enthalpy close to 0.7 bar, 100 °C from the steam tables:

and approximating the partial derivatives with ∆y/∆x, we can find that:

 Graphically, this would be the slope of a line on the p-T coordinates, like this:

We can use the slope to estimate where a line of constant enthalpy, 2680 kJ/kg, in this case, starting from the point T=100°C, p=0.7bar will cross the 1 bar line:

This is very close to the value of 101.88°C that we get from direct interpolation from the table.