1015SCG: Quantitative Reasoning Project 2: Developing a Mathematical Model Due Date: Friday 26 May (Week 12) Total Marks: 30% Fuel Usage Model From your report to the aviation industry, it is evident that there are many factors that influence fuel consumption on a plane, for example, travel distance, payload weight, drag and so on. We have been tasked by the industry to develop a model that simulates fuel consumption for a typical passenger aircraft on a 2 hour flight. It has been estimated that the aircraft has an average fuel consumption of 4 litres per second. Given that weight directly affects fuel consumption, the solution variable for the model will be fuel (F) measured in kilograms (kg). Note that aviation fuel weighs 0.721 kg per litre. Task 1: Developing the Fuel Model (Total Marks: 8%) When developing any mathematical model, we start with the simplest version of the model. We then verify that the model works before adding more complexity. As in project 1, we will be using the input/output framework. The solution variable will be the weight of fuel (F) measured in kilograms (kg) and we will only be considering one input and output for the model. The output for the model will be fuel usage (u) due to running the aircraft as well as physical effects like drag. The input to the model will be fuel savings (s) which is caused by environmental effects such as tail winds. For this task, you will need to: 1) derive the recursive equation for the fuel usage model assuming that fuel usage and fuel savings is directly proportional to the weight of the fuel. Note that this will result in a fuel usage parameter (u) and a fuel savings (s) parameter that will initially be constant; 2) Identify the part of the recursive equation that determines how the fuel changes over time i.e. the “change process” part of the equation; 3) Using the change process, derive the conditions where the system is increasing, decreasing and does not change; 4) To verify your model, run your model under the conditions in part 3) to ensure it exhibits the right behaviour; and 5) Discuss which conditions are relevant for this model. Note that we will be assuming a time scale of minutes i.e. in our recursive equation our time step will be ∆𝐠= 1 minute. Therefore, fuel usage will be the amount of fuel consumed per minute and fuel savings will be the amount of fuel saved per minute.Task 2: Calibrate Model using a standard flight (Total Marks: 10%) We will now use the model above to estimate the fuel usage parameter (u) for a standard flight. As mentioned above, the flight will be 2 hours in duration and the aircraft has an average fuel consumption of 4 litres per second. We will also assume that there will be no physical effects that will give a fuel saving during the trip. For this task you will need to do the following: 1) For the initial condition (at t = 0), calculate the fuel required for the flight given the duration of travel and the average fuel consumption above. Remember to convert from litres to kilograms. 2) Using the information above, assign the initial fuel weight from your calculation in 1) and assign the fuel savings parameter to 0; 3) To run the model, we will arbitrarily assign our fuel usage parameter to 0.03 and run the model to determine how much fuel is left at the end of the trip. Since we want to be as efficient as possible, we will aim to have 100kg of fuel left at the end of the trip; 4) Using trial and error, run the model several times for different values of the fuel usage parameter to determine the amount of fuel left for each value. Record the different fuel usage parameters and its corresponding fuel amount in a table. You will need to do at least five different simulations to create the table; 5) From the table in 3), plot fuel usage parameter verses fuel and explore ways to refine your estimate for the fuel usage parameter i.e. to get as close as possible to 100kg of fuel; 6) Report on your findings and discuss the validity of the result given the assumptions of the model. Task 3: The effect of tail winds on fuel consumption (Total Marks: 12%) The effect of tail winds will change over time during the flight of the aircraft and, therefore, the fuel savings parameter (s) will also change with time. The industry provided the following equation that describes how the fuel savings parameter will change during the flight: 𝐠= 1.2 × 10−5(120𝐠− 𝐲) For this task you will need to: 1) Change your model to incorporate the equation above for the fuel savings parameter. Note that the initial condition and all other parameters will not change from Task 1; 2) Run your model to see how this affects the fuel consumption of the aircraft; 3) Using the new change process in the recursive equation, derive the conditions where the system is increasing, decreasing and does not change; 4) Discuss your results from 2) and 3) detailing the effect the above equation has on fuel consumption.