Most every pilot will tell you that a good total energy (TE) compensated variometer is essential for soaring flight. But how and why does it work? Total energy compensation, in the form commonly used in sailplanes today (capacity bottle and bent probe), was developed and patented by Oran Nicks (Patent Link) in 1977. Prior to that, G.E. Moore had done extensive research (Link 1, Link 2) investigating a way to combine both pitot and static pressure signals to compensate changes in static pressure (height gain/loss) with changes in airspeed (pitot pressure) to give the pilot an indication of the energy state, or total energy gain/loss, of the sailplane. The problem was that the device was complicated, expensive, and needed to be tested and adjusted to achieve accurate readings. It was Oran Nicks that saved us from this situation by discovering that the total energy of a sailplane is related to a unique pressure that can easily be measured independent of the pitot and static pressure input. The following proof will derive that pressure. For further reading see page 149 in Helmut Reichmann's Cross-Country Soaring.
Problem: We want to find a pressure (P*) that directly correlates with the total Energy (TE) of the sailplane.
Find: The change in Total Energy of the sailplane (delta_TE) as a function of an measurable change in pressure delta_P*.
Assume: We are flying in still air in a glider with an infinite glide ratio (no energy is lost to drag). In a glider with an infinite glide ratio, all kinetic energy (KE) can be transferred to potential energy (PE) and back to KE with out any losses (KE<==>PE, delta_TE=0).
1) TE=KE+PE (total energy is the sum of the kinetic and potential energy)
2) delta_TE=delta_KE+delta_PE (the change in total energy is the sum of the change in both kinetic and potential energy)
3) 0 = delta_KE + delta_PE (since we assumed delta_TE = 0)
4) 0 = 1/2*m*delta_v^2 + m*g*delta_h (delta_KE = change in dynamic pressure [airspeed], delta_PE = change in static pressure [altitude], m = mass of air, g = gravitational constant)
5) 0 = 1/2*rho*delta_v^2 + rho*g*delta_h (4 divided by volume of air, rho = air density [assumed constant])
6) 0 = delta_P_dynamic - delta_P_stat (We know dynamic pressure is 1/2*rho*v^2, we also know static pressure is -rho*g*h, negative because pressure decreases with increasing h)
7) 0 = -delta_P_dynamic + delta_P_stat (multiply both sides by -1)
Read this to understand the source of equation 8.
8) Cp* = (P* - P_stat)/(P_dynamic) (the coefficient of pressure Cp* is the difference between the target pressure (P*) and static pressure divided by dynamic pressure.
9) Cp* = (delta_P* - delta_P_stat)/(delta_P_dynamic) (take the change in pressures)
10) delta_P* = (Cp*) * delta_P_dynamic + delta_P_stat (solve 9 for delta_P*)
11) if Cp* = -1, delta_p* = -delta_P_dynamic + delta_P_stat
This is where we see that if Cp* = -1, equation 11 and equation 7 are the same. This is what we want because delta_P* should represent the change in total energy, which we have already said is zero.
You can re-derive these equation without the assumption that delta_TE = 0 and you will find that delta_TE = -delta_P* (the negative originates for step 7). This was the amazing discovery that allowed for simple total energy compensation. delta_P*, the change in pressure associated with a coefficient of pressure of -1 represents the change in total energy of the sailplane. The result that a Cp* of -1 is required for total energy compensation is the reason for the shape of the total energy probe. One final point is required, rho (air density) was assumed constant in step 5, which is not exactly the case, however additional steps can be made in the design of a TE system to help achieve a relatively constant air density. Those techniques will be address along with how a Cp of -1 relates to the design of the TE probe in a future blog post.
This proof is a bit lengthy, however it is was an important discovery to soaring, please post any questions or comments and I'll do my best to answer them.