Initial Weight Estimation

Initial Weight Estimation#

This section describes initial weight estimation method and demonstrates it on the example aircraft. The maximum takeoff weight (\(W_{TO}\)) can be divied into four components for this initial estimation: crew weight (\(W_c\)), payload weight (\(W_p\)), empty weight (\(W_e\)), and fuel weight (\(W_f\)). Mathematically, it can be written as

\[ W_{TO} = W_c + W_p + W_e + W_f. \]

Note that \(W_p\) includes both passenger and baggage weight. The \(W_c\) and \(W_p\) are usually provided in the RFP. Following subsections outline methods for computing \(W_e\), \(W_f\) and \(W_{TO}\).

Below code snippet imports required python packages:

import numpy as np

Empty Weight Fraction#

During the intial phase of design, the empty weight is usually computed based on the historical data for similar type of aircraft. Most aircraft design books will provide some sort of empirical formula to compute \(W_e\) or empty weight fraction (\(W_e/W_{TO}\)). For the example airplane, empty weight trend provided in Nicolai and Carichner’s book for light propeller airplane (Pg. 763) is used. The empty weight is computed using

\[ W_e = 0.911 \times W_{TO}^{0.947}. \]

NOTE: In the above formula, \(W_{TO}\) should be in pounds (lbs)

Following code defines a simple python function for computing \(W_e\) based on a given \(W_{TO}\).

def compute_empty_weight(wto):
    """
        Function to compute empty weight

        Note: wto should be in pounds (lbs)
    """
    
    # empty weight fraction (from Nicolai and Carichner, Pg. 763)
    we = 0.911 * wto ** 0.947
    
    return we

Fuel Fraction#

The fuel fraction is computed using the mission segment weight fraction method. In this method, weight fractions are computed for each segment of the mission profile and are multiplied to obtain the ratio of final weight (\(W_{final}\)) to initial weight i.e. maximum takeoff weight (\(W_{TO}\)). Mathematically, it can be written as

\[ \frac{W_{final}}{W_{TO}} = \prod_{i=1}^{n} \frac{W_{i}}{W_{i-1}}, \]

where \(W_{i-1}\) and \(W_{i}\) are the weights at the start and end of mission segment \(i\), respectively, and \(n\) is the total number of mission segments. Note that \(W_0\) denotes maximum takeoff weight. The weight fractions can be computed using either historical data or simple analytical formulas, most aircraft design books will provide these details. Once final weight fraction is obtained, one can compute fuel fraction using

\[ \frac{W_f}{W_{TO}} = 1.06 * \Big ( 1 - \frac{W_{final}}{W_{TO}} \Big) \]

Note that a \(6\%\) factor is multiplied to account for trapped and unusable fuel based on the recommendations provided in Raymer’s book.

This method is now used for computing fuel weight fraction for the example airplane. Based on the mission profile, there are 8 mission segments in total. The value of weight fractions for most mission segments is based on historical data, except for cruise and loiter mission. The cruise and loiter weight fractions are computed using following equations, refer to lecture notes or any of the aircraft design books for more details.

\[\begin{split} \begin{aligned} \text{cruise weight fraction:} \quad & \frac{W_3}{W_2} = \exp \Bigg[ - \frac{ RC_{{bhp}_{cr}} }{ 375\eta_{p_{cr}} (L/D)_{cr} } \Bigg] \\ \text{loiter weight fraction:} \quad & \frac{W_6}{W_5} = \exp \Bigg[ - \frac{ E V_{lo} C_{{bhp}_{lo}} }{ 375 \eta_{p_{lo}} (L/D)_{lo} } \Bigg] \end{aligned} \end{split}\]

NOTE: The above cruise and loiter equations are valid for propeller-driven airplanes only

The R and E are the range and endurance time, respectively. Note that range should be in miles and endurance should be in hour. The C_bhp denotes brake specific fuel consumption for the piston-engine which is the pounds of fuel per hour to produce one horsepower at propeller shaft. The factor of 375 is used to ensure consistency of units across all the terms. The \(\eta\) denotes propeller efficiency while \(V\) is the speed of the airplane in mph. The subscript \(cr\) and \(lo\) denotes cruise and loiter, respectively. For a propeller-driven airplane, the cruise and loiter \(L/D\) can be approximated as \(L/D_{max}\) and \(0.866 \times L/D_{max}\), respectively. The maximum \(L/D\) for a subsonic airplane can be computed following formula provided Nicolai and Carichner.

\[ \bigg(\frac{L}{D}\bigg)_{max} = \frac{1}{2\sqrt{C_{D_0}K}} \]

Here, \(C_{D_0}\) is the zero-lift drag coefficient and \(K\) is induced drag factor given by

\[ K = \frac{1}{\pi Ae}. \]

The \(A\) is the aspect ratio and \(e\) is oswald efficiency factor which is computed based on a statistical estimation provided in Raymer. The value of \(A\) and \(C_{D_0}\) is assumed based on similar type of aircraft at this stage of design. Below table summarizes various fixed values used in fuel fraction computation.

Parameter	Value	Source
Aspect ratio, \(A\)	8	Assumed using similar aircraft
Zero-lift drag coeff, \(C_{D_0}\)	0.028	Nicolai and Carichner, Table 5.2
Cruise propeller efficiency, \(\eta_{p_{cr}}\)	0.82	Roskam Part 1, Table 2.2
Cruise brake specific fuel consumption, \(C_{{bhp}_{cr}}\)	0.4 lb/hr/bhp	Raymer, Table 3.4
Loiter propeller efficiency, \(\eta_{p_{lo}}\)	0.72	Roskam Part 1, Table 2.2
Loiter brake specific fuel consumption, \(C_{{bhp}_{lo}}\)	0.4 lb/hr/bhp	Raymer, Table 3.4
Loiter velocity, \(V_{lo}\)	120 knots	Assumed

Using the above equation and data, fuel fraction is computed for each segment of the mission profile for the example airplane and outlined in the below table. The historical data for segments other than cruise and loiter is obtained from Roskam Part 1 Table 2.2 (Pg 12).

Mission segment	Weight fraction	Value
Warmup, taxi, takeoff	\(W_1/W_0\)	0.984
Climb	\(W_2/W_1\)	0.990
Cruise	\(W_3/W_2\)	0.880
Descent	\(W_4/W_3\)	0.992
Climb	\(W_5/W_4\)	0.990
Loiter	\(W_6/W_5\)	0.988
Descent	\(W_7/W_6\)	0.992
Land, taxi, shutdown	\(W_8/W_7\)	0.992

Below code block defines function for computing fuel weight based on a given \(W_{TO}\).

def compute_fuel_weight(wto):
    """
        Function to compute fuel weight
    """

    ######### define and compute variables

    # Range
    range = 1200 # nmi
    range = range * 1.15077945 # miles

    # Endurance
    E = 45/60 # in hour

    # Loiter speed
    V_lo = 120 # knots, assumed
    V_lo = V_lo * 1.15077945 # in mph

    aspect_ratio = 8 # assumed

    # Propeller efficieny
    np_cr = 0.82
    np_lo = 0.72

    # C_bhp
    C_bhp_cr = 0.4
    C_bhp_lo = 0.4

    e = 1.78*(1 - 0.045*aspect_ratio**0.68) - 0.64 # oswald efficiency, Raymer equation 12.48

    K = 1 / np.pi / aspect_ratio / e # induced drag factor

    zero_lift_drag_coeff = 0.028

    L_by_D_max = 1 / 2 / (zero_lift_drag_coeff*K)**0.5 # maximum L_by_D

    L_by_D_cr = L_by_D_max

    L_by_D_lo = 0.866*L_by_D_max

    ######### Mission segment weight fractions

    w1_by_wto = 0.984 # warmup, taxi, takeoff

    w2_by_w1 = 0.99 # climb

    w3_by_w2 = np.exp(-range*C_bhp_cr/375/np_cr/L_by_D_cr) # cruise

    w4_by_w3 = 0.992 # descent

    w5_by_w4 = 0.99 # climb

    w6_by_w5 = np.exp(-E*V_lo*C_bhp_lo/375/np_lo/L_by_D_lo) # loiter

    w7_by_w6 = 0.992 # descent

    w8_by_w7 = 0.992 # land, taxi, shutdown

    w8_by_wto = w1_by_wto * w2_by_w1 * w3_by_w2 * w4_by_w3 \
                * w5_by_w4 * w6_by_w5 * w7_by_w6 * w8_by_w7

    wf_by_wto = 1.06 * (1 - w8_by_wto) # 6% is for trapped unusable fuel

    return wf_by_wto * wto

MTOW Estimation#

Once empty weight fraction and fuel weight fraction are computed, one can follow a simple iterative procedure to compute \(W_{TO}\). The procedure starts by guessing a value of \(W_{TO}\) and computing the empty weight and fuel weight. The \(W_e\) and \(W_p\), along with \(W_c\) and \(W_p\), can be used to compute \(W_{TO}\). Then, the error between the guessed and estimated \(W_{TO}\) can be computed. If this error is more than a desired tolerance, then this process is repeated again. The computed \(W_{TO}\) can be used as the guess value for \(W_{TO}\) in next iteration.

Below code block performs this iterative procedure for the example aircraft and prints the value of \(W_{TO}\), \(W_e\), and \(W_f\).

# Variables
wto_guess = 5000
error = np.inf
tolerance = 0.01

# Crew weight and payload weight - from req.
wc = 200 # lbs
wp = 1000 # lbs

while error > tolerance:

    wto = wc + wp + compute_empty_weight(wto_guess) + compute_fuel_weight(wto_guess)

    error = np.abs(wto - wto_guess)

    wto_guess = wto

print(f"Maximum takeoff weight: {wto} lbs")
print(f"Empty weight: {compute_empty_weight(wto)} lbs")
print(f"Fuel weight: {compute_fuel_weight(wto)} lbs")

Maximum takeoff weight: 5353.848970299187 lbs
Empty weight: 3094.3197451478054 lbs
Fuel weight: 1059.5366475387418 lbs

Following table summarizes the results for the initial weight estimation.

Parameter	Value
Maximum takeoff weight	5354 lbs
Empty weight	3094 lbs
Fuel weight	1060 lbs
Payload weight	1000 lbs
Crew weight	200 lbs
Empty weight fraction	0.578
Fuel weight fraction	0.198

NOTE: The estimated weight values should be rounded to the nearest integer.

Initial Weight Estimation

Contents

Initial Weight Estimation#

Empty Weight Fraction#

Fuel Fraction#

MTOW Estimation#