(a) A 1500 litre fuel storage tank initially contains 750 litres of liquid fuel with 20g...

 

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(a) A 1500 litre fuel storage tank initially contains 750 litres of liquid fuel with 20g of chemical (C) additive in it. Fuel flows into the tank at a rate of 30 litres/hour. The fuel flowing into the tank has a time dependent concentration of the above mention chemical of ¿) g/litre, where t is time in seconds. Assuming a well-mixed solution leaves the tank at a rate of 15 litres/hour outline an ordinary differential equation (ODE) that relates how the amount the Chemical (C) varies with time (t). Solve this ODE analytically to find a relationship between C and t. Calculate C when t-30seconds. Plot graphs of your result. Show your full working in full. [9 marks] (b) You are test driving a new electric car on a straight and level race-track at a steady speed of 5 m/s. As you cross the starting line you put your foot down fully on the accelerator. A constant force F is applied to the car (by the wheels) and you accelerate to a constant terminal velocity, VER- During this motion, the car is subjected only to the engine force and the drag force from the air flowing past the car. The drag force on the car can be approximated as F₁=C4pV², where C is the drag coefficient of the car, V is the car velocity, A is the car frontal area, and p is the air density. The motion of the car is governed by Newton's second law ΣF=ma, where m is the mass of the car and a is its acceleration in the x-direction. You can assume that C is constant. Applying Newton's second law with the two applied forces (the constant wheel force F and the opposing drag forceFD) results in the equation F-F₂=ma which governs the car's motion. As the velocity increases, the drag force on the car increases, and the velocity eventually approaches a maximum value at the terminal velocity, V TERM- The following are known: F-4000 N; m=1300 kg; A=3 m²; C = 0.6; p = 1.2 kgm³; (1) Calculate what the terminal velocity, V TERM will be [4 marks] Hint: the car's acceleration is zero when it reaches the terminal velocity. (ii) Calculate the time, at which the terminal velocity is reached, this is defined as the time at which the velocity exceeds 99% of V TERM Hint: this will involve solving an ODE to find V for different values of t. You can choose to solve analytically or numerically and can use any method you choose, but must show/explain your approach and working. You can use computational tools such as Matlab or Excel if you wish [7 marks] (a) A 1500 litre fuel storage tank initially contains 750 litres of liquid fuel with 20g of chemical (C) additive in it. Fuel flows into the tank at a rate of 30 litres/hour. The fuel flowing into the tank has a time dependent concentration of the above mention chemical of ¿) g/litre, where t is time in seconds. Assuming a well-mixed solution leaves the tank at a rate of 15 litres/hour outline an ordinary differential equation (ODE) that relates how the amount the Chemical (C) varies with time (t). Solve this ODE analytically to find a relationship between C and t. Calculate C when t-30seconds. Plot graphs of your result. Show your full working in full. [9 marks] (b) You are test driving a new electric car on a straight and level race-track at a steady speed of 5 m/s. As you cross the starting line you put your foot down fully on the accelerator. A constant force F is applied to the car (by the wheels) and you accelerate to a constant terminal velocity, VER- During this motion, the car is subjected only to the engine force and the drag force from the air flowing past the car. The drag force on the car can be approximated as F₁=C4pV², where C is the drag coefficient of the car, V is the car velocity, A is the car frontal area, and p is the air density. The motion of the car is governed by Newton's second law ΣF=ma, where m is the mass of the car and a is its acceleration in the x-direction. You can assume that C is constant. Applying Newton's second law with the two applied forces (the constant wheel force F and the opposing drag forceFD) results in the equation F-F₂=ma which governs the car's motion. As the velocity increases, the drag force on the car increases, and the velocity eventually approaches a maximum value at the terminal velocity, V TERM- The following are known: F-4000 N; m=1300 kg; A=3 m²; C = 0.6; p = 1.2 kgm³; (1) Calculate what the terminal velocity, V TERM will be [4 marks] Hint: the car's acceleration is zero when it reaches the terminal velocity. (ii) Calculate the time, at which the terminal velocity is reached, this is defined as the time at which the velocity exceeds 99% of V TERM Hint: this will involve solving an ODE to find V for different values of t. You can choose to solve analytically or numerically and can use any method you choose, but must show/explain your approach and working. You can use computational tools such as Matlab or Excel if you wish [7 marks]