3.3.3. Fighter Plane Refueling Problem

A fighter plane being refuelled in flight at the rate of 568 liters per minute of fuel having a specific gravity of 0.68. What additional thrust does the plane need to develop to maintain the constant velocity it had just before hookup? The inside diameter of the flexible pipe is 0.127 meters. The fluid pressure in the pipe at the entrance to the plane is 27 kilopascal.

This section introduces a problem where a fighter plane is being refueled while in motion. We know the rate at which the fuel is supplied, its specific gravity, and the dimensions of the equipment involved (like the diameter of the pipe). The challenge here is to determine how much additional thrust the plane needs to stay at the same speed while taking on fuel. The specific gravity tells us how dense the fuel is compared to water, and the pressure gives insights into how the fuel flows into the aircraft.

Imagine driving a car while filling up its gas tank using a nozzle. If the gas starts pouring into the tank, you need to make sure you press the accelerator more to compensate for the extra weight of gas; similarly, the fighter plane needs more thrust to maintain its speed while refueling.

Neglect the mechanical forces on the plane directly from the flexible pipe itself and consider the unit weight of water will be 1000 kg per meter cube. The problem can be considered as one-dimensional, unsteady flow, turbulent, and incompressible.

In this chunk, we outline our assumptions for simplifying the calculations. By neglecting the mechanical forces from the flexible pipe, we focus solely on pressure and velocity changes caused by the refueling process. The flow is thought to be unsteady, which means that it can change with time, but we model it in a simpler way by treating it as steady due to the rapid nature of inflight refueling. The flow is also turbulent (which is common in aviation), and we keep in mind that the fuel is incompressible, meaning its density doesn’t change significantly under the conditions provided.

Think of how water flows from a hose: if the hose is moving around, the water doesn’t change its nature but flows in a turbulent pattern. Similarly, the fuel entering the plane is like this turbulent flow that doesn’t compress. We assume it behaves consistently over time.

The pressure distributions enter outside the control surface can be considered atmospheric pressure. Velocity distributions can be taken as average values with beta equal to 1.

Here, we delve into the pressure at the entry point of the fuel into the plane. We treat the outer environment as having atmospheric pressure, which simplifies our calculations because the fuel is being injected at a specific pressure that is above this atmospheric value (27 kilopascals). When we say 'beta equal to 1', we imply that we assume the flow velocity is uniform across the pipe’s diameter, allowing us to simplify our calculations significantly.

It is like when you drink from a straw; as long as you suck hard enough, you can assume the liquid flows at an average rate through the whole straw. Similarly, if we consider beta to be one, it means we’re assuming fuel flows evenly from the pipe into the plane; it simplifies our work, just like drinking from the straw simplifies drinking a quick beverage.

The additional thrust required can be calculated based on the flow rate of fuel and the exit conditions.

Finally, we calculate how much additional thrust the plane requires. This calculation involves using the flow rate of fuel (568 liters per minute) to determine how much mass of fuel is being added to the plane every second and how this mass affects the plane’s velocity. We then factor in the pressure of the fuel as it enters the plane to establish the new thrust required to maintain a constant velocity. This thrust will compensate for the additional weight of fuel being loaded.

Imagine you’re riding a bicycle while carrying a backpack full of books—if a friend starts adding more books to your backpack, you need to pedal harder to maintain your speed. The calculation of thrust needed is similar: you determine how 'heavy' the bike is getting due to the added weight of the fuel and adjust your efforts (or thrust) accordingly.

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