# Math Help - Double interpolation.

1. ## Double interpolation.

I am trying to find enthalpy values from a steam table that gives enthalpy as a function of temperature and pressure. I need to find the enthalpy at 689 kPa and 171.6 Celsius, neither of which are listed thus making a double interpolation necessary. I cannot understand how to do this. A walkthrough would be greatly appreciated.

2. It's hard to say anything without knowing the precise data or what kind of interpolation you want. If you want to do a linear interpolation, the simplest kind, then you would use a formula like $K= a(t- t_0)+ b(p- p_0)+ K_0$ where $t_0$ is the temperature, listed in the table, just less than the temperature you are given, $p_0$ is the pressure, listed in the table, just higher than the pressure you are given, and $K_0$ is the enthalpy given by the table at $t= t_0$ and $p= p_0$. You can then determine the two coefficients, a and b, by using the temperature and pressure just above the temperature and pressure you are given to get two linear equations to solve for a and b. Once you have found a and b, put the temperature and pressure you are given into $K= a(t- t_0)+ b(p- p_0)+ K_0$ to find the (interpolated) value of K.

(Actually, you could use any two of $K(t_0, p_1)$, $K(t_1, p_0)$ or $K(t_1, p_1)$ where $t_1$ and $p_1$ are the temperature and pressure given in the table just above the temperature and pressure you are given. If the formula were really linear, those would all give the same result but are more likely to get 3 slightly different results. There is no "correct" answer here- interpolation is an approximation.