Linear interpolation is a way to fill in the ``holes'' in tables. As an example, if you want to find the saturated pressure of water at a temperature of 40C you can look in Table B.1.1, (p.674), for 40C in the first column. The corresponding desired pressure is then in the next column; in this case, 7.384 kPa. But what if you want to find the saturated pressure at 38C instead of 40C?

A temperature of 38C is not in the table. You could of course just ignore the difference between 38C and 40C, and still take the saturated pressure to be 7.384 kPa. But that is not acceptable in this class; it is too inaccurate. To get an accurate value, you must use linear interpolation. (Though taking the closest value, 40C, is of course better than nothing in case you forgot how to do linear interpolation during an exam.)

Let's introduce a few symbols. Let *g* be your given value,
38C in this example. Let *g _{1}* and

Also, let *d* be our desired value, in our example the saturated
pressure. Let *d _{1}* and

The formula for linear interpolation is:

So, in our example, the desired saturated pressure *d* at 38C
is:

You need two variables to read off the compressed liquid or superheated vapor tables. In the next example, we will find the specific volume of steam at a given temperature of 100C and a given pressure of 20 kPa.

Steam (superheated water vapor) is found in Table B.1.3. We have no difficulty finding the given 100C in that table, but we cannot find the given pressure of 20 kPa. The closest pressures in the table are 10 kPa and 50 Kpa.

So in the linear interpolation formula from the previous section,

we set the given value
The desired quantity *d* is now the specific volume at 100C
and 20 kPa. We set the value *d _{1}* to the specific volume at

Our formula then gives the specific volume at 20 kPa and 100C as:

You might ask what happens to the last example if neither the given pressure nor the given temperature is in the table. For example, to find the specific volume at 20 kPa and 110C, neither 20 kpa nor 110C are in Table B.1.4. I do not think we would do this to you during the exam. But the answer would be to do three linear interpolations: first interpolate a specific volume at 110C and 10 kPa (fill in the 110C ``hole'' in the 10 kPa data), next interpolate a specific volume at 110C and 50 kPa (fill in the 110C ``hole'' in the 50 kPa data), and finally linear interpolate those 110C values in the same way as we did for 100C in the previous section.

Another problem arises if you try to interpolate the specific volume
of steam at 11 kPa and 50C. You can use the B.1.3 entry for
50C and *g _{1}*=10 kPa, giving