Gross Weight

When we left off, we had almost determined the empty weight fraction, and we’re finally ready to put all of our efforts together to estimate gross weight. Before we do that, though, let’s review the ground we’ve covered so far. Here is what we know (or have assumed) about the airplane:

performance

cruise speed (target)

$V_{\tiny cruise}=200\text{ mph}$

stall speed (target)

$V_{S0}=58 \text{ mph}$

range (target, including 45-min reserve)

$R=800 \text{ mi}$

lift-to-drag ratio

$L/D = 7.93$

weight

crew weight

$W_{crew}=460 \text{ lb}$

payload weight

$W_{payload}=100 \text{ lb}$

fuel fraction

$W_f/W_0=0.152$

empty weight fraction

$W_e/W_0=1.1 \cdot W_{0}^ {-0.09}$

loading

wing loading

$W/S = 17.2 \text{ lb/ft}^{2}$

power loading*

$W/P = 8.72 \text{ lb/hp}$

*I’m a little suspicious of this because planes of similar weight and speed seem to get by with higher power loading figures. We’ll wait and see once we can calculate performance later.

propulsion

power

$P=200 \text{ hp}$

specific fuel consumption

$c_{bhp}=0.425 \text{ lb/hp/hr}$

fuel burn rate (cruise)

$10.6 \text{ gal/hr}$

propeller efficiency

$\eta_p=0.85$

Recall that in the section on weight fractions, we found the gross weight of the airplane is:

$$\scriptsize W_0=\frac{W_{crew}+W_{payload}}{1-(W_{fuel}/W_0)-(W_{empty}/W_0)}$$

Substituting in the known values from the weight table above, we find:

$$\scriptsize W_0=\frac{460 +100}{1-0.152-1.1 \cdot W_{0}^ {-0.09}}$$

$$\scriptsize W_0=\frac{560 }{.848-1.1 \cdot W_{0}^ {-0.09}}$$

Because $W_0$ appears on both sides of the equation in a way that’s inconvenient to solve for $W_0$ , we can either:

Iterate by plugging in some trial values for $W_0$ until the calculated value is the same as the value we plug in, or
Solve graphically.

Using the first method of iteration, we find that the value is somewhere between 1,900 lb and 1,950 lb. We continue guessing until it converges on 1,924 lb.

$$\scriptsize W_0 \text{ (trial value)}$$	$$\scriptsize W_0 \text{ (calculated)}$$
$\scriptsize 1,900 \text{ lb}$	$\scriptsize 1,928 \text{ lb}$
$\scriptsize 1,924 \text{ lb}$	$\scriptsize 1,924 \text{ lb}$
$\scriptsize 1,950 \text{ lb}$	$\scriptsize 1,920 \text{ lb}$

However, it takes time to zero in on the solution (even if you’re using a spreadsheet to do the calculating for you). A quicker method is to solve graphically. This is done by graphing two functions:

$$\scriptsize y=\frac{560 }{.848-1.1 x^ {-0.09}}$$

$$\scriptsize y=x$$