Appendix B:  How The Temperature Difference Reference Was Computed

 

This is a bit esoteric, so if you're not a math person, you can probably safely ignore this.  But if you want to validate the numbers in the table for your own satisfaction and confidence, or compute your own reference table for different pressure units, or compute your own reference table specifically for your circumstances, here is how they were computed.

 

We start off with 2 equations, each of which asserts a truth by stating 2 values are equal.

 

Eq 1:  P₂ = T₂ * (P₁/T₁)

 

This is from above, with temperatures in °K.

 

Eq 2:  P₂ + 1 = (T₂ + Δ°K) * (P₁/T₁)

 

This asserts that some change in T₂ (Δ°K) will result in P₂ rising by 1 (in whatever units P₁ is in).

 

Now we use the "substitution method" for solving simultaneous equations:  we substitute "T₂ * (P₁/T₁)" from the first equation for the value of P₂ in second equation.

 

(T₂ * (P₁/T₁)) + 1 = (T₂ + Δ°K) * (P₁/T₁)

 

Next we carry out the multiplication of "(T₂ + Δ°K)" by "(P₁/T₁)" on the right side by distribution:

 

(T₂ * (P₁/T₁)) + 1 = (T₂ * (P₁/T₁)) + (Δ°K * (P₁/T₁))

 

Note now that we have the value (T₂ * (P₁/T₁)) on both sides of the equation.  So we can eliminate it by subtracting (T₂ * (P₁/T₁)) from both sides, leaving only:

 

1 = Δ°K * (P₁/T₁)

 

Now we solve for Δ°K to change PSI by 1 by dividing both sides by (P₁/T₁).

 

Δ°K = 1 / (P₁/T₁)

 

Note that 1/(P₁/T₁) is the reciprocal of (P₁/T₁). We can simplify this expression by simply reversing the order of the division: 1/(P₁/T₁) = T₁/P₁. Thus:

 

Δ°K = T₁ / P₁

 

(As an aside, you can compute the change in PSI expected when °K is changed by 1 by the computing reciprocal:  P₁ / T₁.)

 

Finally, Δ is in °K, and to be useful to us on the road we want °F.  Since Δ is a "span" or "difference", not a temperature, we only need to compute the equivalent span in °F.  The span of 1 °K is equivalent to the span of 1.8°F.  Thus:

 

Δ°F = 1.8 * T₁ / P₁

 

If you are computing a table to read results in °C, then you don't need to multiply by 1.8 because a "span" or "difference" of 1°K is equivalent to the span of 1°C.  Thus:

 

Δ°C = T₁ / P₁