You say THP is thrust x velocity. I agree. But I say that only works for a specific set of assumptions. Static thrust is not the only scenario that doesn’t conform to the assumptions. Here is another:
Aircraft flying straight and level at speed Va, pilot sets new power to accelerate to Vb. Assume constant propeller efficiency between Va and Vb due variable prop. Final THP = thrust x Vb, we agree that much. Using the hypothesis you outlined above we can draw some conclusions:
The new power is set, the BHP has gone to the prop but THP doesn’t catch up until the plane reaches Vb?!
Thrust horsepower during the acceleration is less than final THP when you get to Vb?!
Thrust must have increased, but this didn’t reflect an increase in THP, this was because power was wasted in the propwash?!
OK lets’ look at your scenario and see if it really proves your point.
To simplify the situation and to avoid any ambiguities we will assume the following:
1. The aircraft is initially in steady non-accelerated flight. This means that the power required is equal to the THP being generated.
2. The initial speed Va is greater than Vmd so any increase in speed will produce an increase in drag and an increase in power required.
3. There is no reduction gearbox between the engine and the propeller, so the full engine torque is applied to the propeller.
4. The propeller is of the constant speed –variable pitch type.
5. To increase the power, the pilot simply pushes the throttle forward (selecting a higher RPM would just complicate the argument, but not really change the outcome).
6. The pilot has decided to increases airspeed simply by increasing the engine power output (you have already intimated this, but it is best to avoid unnecessary arguments about how best to increase airspeed).
With the aircraft in steady flight at constant RPM the engine torque (causing the propeller to rotate) will be equal and opposite of the propeller torque (the component of propeller total reaction that acts in the plane of rotation to oppose rotation in powered flight). This balance of two opposing torques will maintain a constant propeller RPM.
When the pilot opens the throttle the power output (BHP) of the engine will increase. The increased power will initially take the form of an increase in engine torque. This will cause engine torque to become greater than propeller torque, so the propeller RPM will start to increase.
When the propeller constant speed unit senses this increase in RPM, it will command the pitch change unit to increase the pitch of the propeller blades. This will increase the angle of attack of the blades, thereby increasing propeller total reaction. This in turn will increase both the propeller torque and the thrust. The increased propeller torque will cause the RPM to return to its initial (selected) value. So the overall effects of the pilot opening the throttle will be an increase in thrust, which will cause an immediate increase in THP.
Prior to the pilot opening the throttle, the aircraft was in steady balanced flight, so the thrust was equal to the drag and the power required (drag x TAS) was equal to the THP (thrust x TAS).
The increased thrust caused by the pilot opening the throttle would have the following effects:
1. The thrust would be greater than the drag.
2. The THP being generated would be greater than the power required.
The combination of thrust greater than drag and THP greater than power required would cause the airspeed to increase. As the airspeed increases, it would cause the drag and hence the power required to increase. The acceleration would continue until the increasing drag was once again equal to the thrust and the increasing power required was equal to the THP being generated. The aircraft would then settle at the new into balanced flight at the new higher speed.
Now let’s look again at your comments:
The new power is set, the BHP has gone to the prop but THP doesn’t catch up until the plane reaches Vb?!
Thrust horsepower during the acceleration is less than final THP when you get to Vb?!
The situation is actually one in which the THP increase occurred before the airspeed increase and contributed to it. The only thing that you have proved by proposing this scenario, is that you have given insufficient thought to the subject.
Static thrust is not the only scenario that doesn’t conform to the assumptions. Here is another:
I presume that this is a reference to your earlier suggestion that an aircraft attempting to accelerate from brake release was another “impossible chicken and egg” situation. I am quite happy to explain to you why this is not the impossibility that you suggest. But you will probably gain more by working it out for yourself.
work done on the propwash isn’t wasted if it’s giving thrust.
As an aircraft flies through the air it is continuously losing energy due to the drag force. If the airspeed and altitude are to be maintained, this lost energy must be replenished at a rate that is equal to the loss rate. Looked at in this context the purpose of the propulsion system is to take energy from the fuel, convert it into kinetic energy and transfer it to the aircraft. Any energy that is lost to the propwash (or jet efflux) represents a waste of energy.
The enormous improvements in fuel efficiency that have been achieved in turbojets / turbofans over the past few decades have been brought about by reducing the amount of kinetic energy that is wasted in the jet efflux. The old turbojets were quite able to produce thrust, but their very high exhaust velocities meant that they were throwing away enormous amounts of kinetic energy. The fact that modern engines are able produce the same (or even more) thrust, while throwing away much less energy, shows that the energy loss is wasteful.
I am quite happy to respond to the remainder of your post when you have stated the equation that you consider to be correct for propeller efficiency.