Finding an expression for the minimum drag coefficient

[ryan88]

Homework Statement

I need to find an expression for the minimum drag coefficient, $$C_D$$, of an aircraft during straight and level flight.

Homework Equations

$$C_D=C_{D_0}+kC_L^2$$

Where:
$$C_{D_0}$$ is the profile drag coefficient
$$C_L$$ is the lift coefficient
$$k$$ is just a constant

The Attempt at a Solution

I started off by plotting the equation for $$C_D$$ shown above and got the following:
http://g.imagehost.org/view/0218/drag

So I thought that the minimum drag would be when:
$$\frac{dC_D}{dC_L}=0$$

However, I remember my lecturer saying something about I would have to do:
$$\frac{d}{dC_L}\left(\frac{C_D}{C_L}\right)=0$$

But I am unsure why, I was wondering if someone could shed some light on this please?

Thanks,

Ryan

 

[minger]

What do your variables stand for?

 

[ryan88]

Sorry, I have updated my first post.

Ryan

 

[minger]

Maybe I'm still not following you. It seems as if you have two independent variables, which may or may not be related. Are the two drag coefficients (lift and profile) functions of anything or what?

Otherwise we simply have two positive numbers adding each other.

 

[ryan88]

The profile drag coefficient is constant and the lift coefficient is dependent upon the angle of attack of the aircraft. Therefore as the angle of attack changes, so does the lift coefficient and also the drag coefficient.

Ryan

 

[minger]

...OK now we're getting somewhere. How do the coefficients change as a function of angle of attack?

 

[ryan88]

Ok, I have just seen my lecturer and cleared this up.

Differentiating $$C_D$$ with respect to $$C_L$$ does find the minimum drag, however this happens when $$C_L = 0$$. Since the question states that the aircraft is in straight and level flight, it is obvious that $$C_L \neq 0$$. The minimum drag during flight would be when the performance of the aircraft (i.e. the lift-to-drag ratio) is at its maximum, i.e:

$$\frac{d^2C_L}{dC_D^2} < 0$$

Drag is defined as:

$$C_D=C_{D_0}+kC_L^2$$

Dividing through by $$C_L$$ gives:

$$\frac{C_D}{C_L}=\frac{C_{D_0}}{C_L}+kC_L$$

Since this is the reciprocal of the performance (lift-to-drag ratio), finding the minimum of this function will be the same as finding the maximum of the performance, which will give the minimum drag of the aircraft.

Hence to answer this question you need to check for when:

$$\frac{d\frac{C_D}{C_L}}{dC_L}=0$$

Not:

$$\frac{dC_D}{dC_L}=0$$

Thanks minger for your help on this.

Ryan

 

[minger]

Oh, I thought we were acutally finding the expression, which is why I wanted to know the equations.

Either way, glad you got it figured out.