|
Just so that we are all on the same page
here.
First the disclaimer:
The following assumes that you are flying a
subsonic airplane where compressibility effects can be neglected, and that you
are not putting out 1000+hp. I think these are fairly safe assumptions. The
other thing is it must be steady state with stabilized readings relative to
time. The math to look at this case is really very simple. If you can pass
algebra, you can do the math for this problem. Just keep the units consistent
and use absolute temps for calculating DT (deg Kelvin or Rankine depending on
metric or english units)
The job of the cooling system is to remove enough
waste heat to keep the engine below its maximum temp in all operating ranges.
This must be done while keeping the weight and drag to a minimum.
The conflict is always between hot day climb and
cruise efficiency.
The airplane spends most of its time in
cruise, therefore cruise efficiency is the most important design point.
Hot day climb only happens once in a while. It is
the secondary concern. Adding a spray bar or some other band aid to get past
this one design point is a perfectly logical and acceptable solution. It is
not "cheating".
In steady state cruise there is some heat flux Q
from the engine that must be disposed of to keep the engine at its desired
operating temp. There is no "optimum" coolant delta T (DT)across the
radiator for all conditions to accomplish this. There is only ONE
coolant DT for any given steady state operating point that will achieve this and
it is found by the equation
Q = Mdot x Cp x DT
Q = The heat flux that must be removed
Mdot = mass flow rate of the water per
unit time
Cp = The specific heat of the coolant (how much
heat each unit of mass of the coolant absorbs per unit of temp increase)
DT = T1-T2 The change in temp of the water
across the radiator (must be equal to the water DT across
the engine block for steady state operation)
Now for the optimum part:
The absolute best possible performance would be a
system that used just enough air Mdot to heat the cooling
air to the same temperature as the water radiator exit temp ( water
going back to the engine from the radiator). This condition is impossible to
achieve in practice. So you try to get as close as you can. How close you get is
called the heat exchanger effectiveness. The closer the air and water exit
temps are, the better the effectiveness.
The air side is also governed by the same
equation:
Q = Mdot x Cp x DT
This is because Qwater = Qair for steady state
operation
The Cp for air is much less than for water.
The density of water is also much greater than
air. Because of this the water DT will always be much less
than the Air DT
The optimum condition is to minimize Mdot air (thus
drag) for a given heat exchanger effectiveness to get the required Q.
This results in the smallest cooling system and the smallest drag/weight in
cruise.
The catch is you must oversize the system just
enough so that a cowl flap can increase Mdot air enough in hot day climb to
adequately cool the engine. In this regime there will be more drag and less Air
DT. Who cares! Cruise is where its at.
Transient operation can only be analyzed using
differential equations, or piecewise analysis and a computer code. Climb is
transient in nature, that is the problem with trying to analyze it. The
transient nature of climb is why the thermal mass of the engine and coolant
help us. It absorbs heat as it comes up to temp, This is all Q that the rad
does not have to reject to the air. Taking off with the engine already at
red line temp is a different matter entirely. In this case you have no big heat
sink to help you. So use a lower rate of climb, a higher speed, open cowl flaps
and a spray bar. Or better yet, let the engine cool before you take off then do
all those things.
For my design I prefer a light low drag
installation that is optimized for cruise with some Band-Aids to get me past the
once in a while condition that I rarely see.
If you are designing a glider tug or a STOL plane
for use in North Africa, you will have to adjust your design
accordingly.
Monty
|