I am having a little difficulty understanding what you are proposing. Here is why.

Looking at the units, this is ((number) x Force) / (number x number x Area), i.e force per unit area. Let M be mass units, L be length units, T be time units. Force is then M x L / T^2; Area is L^2, so this comes down to M x L / T^2 x L^2, which is M / T^2 x L.

Whereas in the usual definition of c_L, which you quote as c_L has no units: (M x L / T^2) / ( (M / L^3) x (L/T)^2 x L^2 ) all cancels out.

PBL

PBL;

Yes, you're quite right. c_L is dimensionless. The formula interprets W as mass.

regards,
HN39

PM
The constant just after the equal sign is equal to 2*g/(rho*c^2), where:
g = acceleration of gravity at SL = 9.80665 m/s^2
rho = air density at SL std temp = 1.225 kg/m^3
c = speed of sound at SL std temp = 340.3 m/s

Sorry, numbers have no dimensions to me.

Terminology difference. Yes, it is now clear to me that your numbers are constants. However, I still have difficulty with equating weight with mass. Weight is to me still a force (opposed in the case of level flight to lift, and one which I feel clearly every time I pick up my suitcase), so still M x L / T^2.

PBL

PBL: Sorry I missed your post while editing mine. I trust the re-edit clears things up.

An alternative (and perhaps simpler?) way into this one is to note that

q = 1/2*gamma*M^2*P

where P is staitc pressure, Hence:

c_L = weight/(1/2*gamma*M^2*P*S)

If you substitute P= delta*101325 Pa, weight = mass*9.80665 m/s^2 and gamma=1.4 you get HN39's formula.

In any event c_L (and hence alpha) increase as you cimb at constant Mach. Would Mcrit for a wing increase with alpha? I don't know but I would think it might.

AoA and altitude
 

This graph: Variation of AoA with altitude shows angle of attack at constant weight and Mach (0.65) for a typical new technology widebody airplane.

regards,
HN39

The Prandtl relation and the more commonly used Karman Tsien have two main applications...
1. determine the lift slope curve dCl/dalpha
2. to find the highest incompressible Lift coefficient where the free stream
MN =1.00,...or Mcrit---usually in a graphical tabulation of some kind

:)