Wing skin temp (was: How much ice...)
Julian Scarfe
possible has occupied a fair amount of bandwith. In my reply to Bill
Levenson I made some comments that I would like to expand on. The following
is long, pretty technical, and probably of interest only to a minority.
Since I spent the time looking at this, I thought I'd share it. If you're
in the 99.8% of readers who don't care for long articles on compressible
aerodynamics, you might want to stop after the executive summary!
Summary:
1) I assert that airframe icing can only occur at points on the airframe
where the static temperature is below freezing. (Mike R has made points
about the rate at which droplets can be cooled, which may be valid, but I'm
assuming that if the airframe is below freezing there is at least the
potential for icing.)
2) Since the pressure over the top surface of a wing is less than the
freestream pressure, the associated reduction in temperature can cause the
airframe to be cooler than the temperature indicated on an OAT probe. (Roy
Smith makes the good point that GA OAT sensors are probably not very
accurate, but I'll assume we have a good one here.)
3) I find that the difference in temperature between indicated OAT and the
coolest part of the wing is proportional to the wing loading of the aircraft
divided by the freestream pressure, and varies a little with AOA.
4) For light aircraft, the difference in temperature between indicated OAT
and the coolest part of the wing is unlikely to exceed about 1 degC. For
big jets, the difference in temperature is unlikely to exceed about 10 degC.
The method:
a) First we ignore the boundary layer and work out the static temperature as
it varies with pressure coefficient over the wing -- this seems consistent
with the methodology outlined by standard texts
b) Then we account for the effect of the boundary layer on the skin
temperature.
c) Then we simplify it by applying some useful relations
d) Then we throw in some numbers
The detail:
a) What is the temperature T at a point where the pressure coefficient is
Cp, when we know T_f (the freestream temperature) and M_f (the freestream
Mach number)?
[Throughout we'll use g is gamma, the ratio of specific heats.]
We need to use three relationships:
1) Adiabatic relation: T is proportional to p^((g-1)/g)
2) The conservation of energy/enthalpy equation:
T_0/T = 1 + (g-1)*M^2/2
so
T_0/T_f = 1 + (g-1)*M_f^2/2
where the subscript _0 refers to the "total" or "stagnation" temperature,
and _f refers to the freestream temperature, in the airflow well away from
the wing.
3) The definition of pressure coefficient
p/p_f = 1 + Cp*g*M_f^2/2
(The incompressible version Cp = (p - p_f)/(0.5*rho*v^2) follows if you
substitute for the speed of sound. Note that Cp is negative for most of the
upper surface where the pressure is below freestream.)
First, ignoring the boundary layer, we find T as a function of Cp using eq 3
in eq 1:
T/T_f = (p/p_f)^((g-1)/g)
= (1 + Cp*g*M_f^2/2)^((g-1)/g)
= 1 + Cp*(g-1)*M_f^2/2 for small Cp*M_f^2 (justified below)
b) Then we consider the effect of the boundary layer:
The solution to the "thermometer problem", where we want to know the
temperature at the adiabatic wall of a laminar compressible boundary layer,
is usually couched as
T_skin = T_e + r*(T_0 - T_e)
where T_e is the temperature at the outer edge of the boundary layer and r
is the "recovery factor".
First let's consider Cp = 0 (i.e. wall parallel to freestream, p = p_f, and
T_e = T_f) This might be a place for the OAT probe, so T_skin_f is what the
OAT probe would read. Applying eq 2:
T_skin_f = T_f + r*(T_0 - T_f)
= T_f*(1 + r*(g-1)*M_f^2/2)
while for general point on the wing where the pressure coefficient is Cp
T_skin = (1-r)*T_e + r*T_0
= T_f*((1-r)*(1 + Cp*(g-1)*M_f^2/2) + r*(1 + (g-1)*M_f^2/2))
= T_f*(1 + (1-r)*Cp*(g-1)*M_f^2/2 + r*(g-1)*M_f^2/2)
= T_f*(1 + (r + (1-r)*Cp)*(g-1)*M_f^2/2)
So subtracting the two:
T_skin - T_skin_f = T_f*(1-r)*Cp*(g-1)*M_f^2/2
[Note that if the OAT probe is placed at a stagnation point rather than
flush to a non-lifting surface, it might show the total temperature T_0. In
that case
T_skin - T_0 = T_f*(1-r)*(Cp-1)*(g-1)*M_f^2/2
which just has Cp-1 instead of Cp and we can continue the analysis as below
remembering to subtract 1 from Cp.]
So there's the answer to the question "how much colder might parts of the
wing skin be than the temperature indicated on the OAT probe?"
Is the difference going to be greater at higher or lower speeds? Cp is
obviously greater at lower speeds, while M_f is proportional to speed. So
let's get slightly smarter...
c) Now we notice that if we average equation 3 (the definition of pressure
coefficient) over the surfaces of the wing we get an expression for the lift
coefficient
wing_loading/p_f = Cl*g*M_f^2/2
So applying that gives us an expression that is relatively independent of
speed
(T_skin - T_skin_f)/T_f = (1-r) * (g-1)/g * (Cp/Cl) * (wing_loading/p_f)
Note that we now have a dependence on Cp/Cl, where Cl is in some sense the
average pressure coefficient, so the biggest effect will be where the -Cp
distribution across the wing is most 'peaky', which will be on the upper
surface of the wing close to the leading edge. I'd expect the ratio to be
greatest at high AOA.
d) OK, let's try some numbers.
For air (1-r) * (g-1)/g ~ 0.04. We'll take T_f ~ 273 K (freezing).
so
(T_skin - T_skin_f)/T_f = 0.04 * (Cp/Cl) * (wing_loading/p_f)
For a light aircraft we expect wing_loading to be less than about
28 psf and p_f at altitude to be about 10 psi or 1400 psf, so the
last term will not exceed about 0.02.
For a big jet, we expect wing_loading to be about
140 psf and p_f at altitude to be about 10 psi or 1400 psf, so the
last term will not exceed about 0.1.
Thus typically for a light aircraft:
T_skin - T_skin_f = (0.02 * 0.04 * 273 K) (Cp/Cl) ~ 0.2 degC * (Cp/Cl)
For a big jet:
T_skin - T_skin_f = (0.1 * 0.04 * 273 K) (Cp/Cl) ~ 1 degC * (Cp/Cl)
So what's the greatest anticipated (negative) value of Cp/Cl? I'd guess a
number between about -1 and -5. I have a Cp distribution from Abbott and
von Doenhoff for a NACA 4412 at 8 degrees (Cl ~ 1.1) which gives a peak
(negative) value of about -2 for Cp/Cl. I also have figures that suggest
that a 'commerical air transport' aerofoil has a peak negative value of
about -6 at 8 degrees, while a 'GA' aerofoil is down at about -2.
So I conclude that the potential for having a part of the airframe more than
about 1 degC cooler than the OAT probe would be pretty extraordinary for a
GA aircraft. By contrast, because of the higher wing loading, a big jet may
have parts of the wing skin below freezing when the OAT probe is reading
several degrees above, perhaps as much as 10 degC in the most unfavourable
circumstances.
Julian Scarfe
jul...@avbrief.com
Ron Parsons
"Julian Scarfe" <jul...@avbrief.com> wrote:
>So what's the greatest anticipated (negative) value of Cp/Cl? I'd guess a
>number between about -1 and -5. I have a Cp distribution from Abbott and
>von Doenhoff for a NACA 4412 at 8 degrees (Cl ~ 1.1) which gives a peak
>(negative) value of about -2 for Cp/Cl. I also have figures that suggest
>that a 'commerical air transport' aerofoil has a peak negative value of
>about -6 at 8 degrees, while a 'GA' aerofoil is down at about -2.
>
>So I conclude that the potential for having a part of the airframe more than
>about 1 degC cooler than the OAT probe would be pretty extraordinary for a
>GA aircraft. By contrast, because of the higher wing loading, a big jet may
>have parts of the wing skin below freezing when the OAT probe is reading
>several degrees above, perhaps as much as 10 degC in the most unfavourable
>circumstances.
Perhaps a bit of observations to go along with your calculations.
Transport aircraft operators pay less attention to OAT than to TAT
(Total Air Temperature) which contains the ram rise. Most GA aircraft
will use a system that has about a .8 factor while TAT is 1.0.
What you conclude is reasonably close for jet engine inlet icing. In
practice, the heat is used any time the TAT is below 10 degC and there
is visible moisture.
As to the wing, the first ice appears in a thin ridge just slightly
above the most forward point leading edge. There is sometimes a second
spot for icing on both the upper and lower surfaces a few inches back on
the wing just past where the wing is cooler than the leading edge which
has been heated by ram rise.
--
Ron
Julian Scarfe
news:jrp59-CF1ADA....@news.bellatlantic.net...
> Perhaps a bit of observations to go along with your calculations.
Thanks Ron
> Transport aircraft operators pay less attention to OAT than to TAT
> (Total Air Temperature) which contains the ram rise. Most GA aircraft
> will use a system that has about a .8 factor while TAT is 1.0.
OK. Substitute Cp-1 for Cp in the result (or about 20% in the temp
differences for typical just-behind-the-leading-edge Cp).
> What you conclude is reasonably close for jet engine inlet icing. In
> practice, the heat is used any time the TAT is below 10 degC and there
> is visible moisture.
I don't know enough about the aerodynmics of intakes to know what sort of
pressures are seen there, but it sounds sensible.
> As to the wing, the first ice appears in a thin ridge just slightly
> above the most forward point leading edge. There is sometimes a second
> spot for icing on both the upper and lower surfaces a few inches back on
> the wing just past where the wing is cooler than the leading edge which
> has been heated by ram rise.
That makes sense, because that is where the greatest (negative) Cp is.
Julian Scarfe