Problem Statement:

The energy contained in fuel is often expressed as an energy density or heating value of the fuel. The energy density expresses how much energy is released per unit mass of fuel when the fuel is burned. The energy usually appears as thermal energy when the burning fuel heats its surroundings. The energy density of aviation fuels is about 43.5 MJ/kg. That is, 43.5 MJ of energy is released for each kilogram of fuel burned.

At takeoff, a Boeing 747-400 contains 216,840 L (216.8 m³) of fuel with a density of about 820 kg/m³. Based on the amount of energy stored in the fuel, what is the maximum speed the aircraft could obtain? Use the fact that the total mass of the aircraft (not including the fuel) is m_aircraft = 190,000 kg.

Solution:

The First Law of Thermodynamics involves changes in energy, so let's start there. The First Law tells us

Since we are interested in the best possible scenario (i.e., maximum possible kinetic energy of the aircraft), let's ignore any work done by friction or drag on the aircraft (W = 0). Let's also ignore energy loss due to heat transfer (Q = 0). So the best we can do is

That is, the change in total energy of the aircraft during the burning of fuel has to be zero. Total energy is the sum of all types of energy, namely, potential, kinetic, thermal, and chemical. We will assume the aircraft is flying horizontally so there is no change in potential energy. We will also assume no change in thermal energy, which is the same as saying that any thermal energy obtained from combustion has been perfectly converted into kinetic energy of the aircraft and exhaust. As a result, the first law reduces to

The change in chemical energy is the initial energy stored in the fuel minus the energy left after it is burned (i.e., zero). Thus,

The change in kinetic energy is the initial aircraft kinetic energy minus the final kinetic energy. Since the aircraft starts from rest

We only include the mass of the aircraft and not the mass of the fuel in the calculation of the final kinetic energy because we assume that the burnt fuel (which is ejected from the engines) has no net speed with respect to the ground. Putting it all together we have

or

Of course, this is a ridiculously high number because we have ignored key limiting factors like friction, drag, and heat loss. We also assumed that all thermal energy is converted into kinetic energy of the aircraft and that the fuel leaves the engine at zero speed relative to the ground. These last two assumptions correspond to assuming 100% thermal and propulsive efficiency, respectively. We will discuss thermal and propulsive efficiency on the next page, but the maximum values they can have are 1.0 (or 100%).