Question:
Liquid fuel is to be stored in a closed pressure vessel during the shipping process. It is assumed that the pressure vessel can be modelled as a cylinder. It is initially stored at a temperature of 30°C and must never rise above 60°C or fall below 5°C. The shipping route will take 12 days.
a) By analysing data from previous journeys, the vertical displacement of the ship can be modelled as:
By determining the rate of change, find an equation for the vertical velocity of the ship, so that engineers can assess the stability of the vessel on the ship.
b) Newton’s Law of cooling states that the rate of change of cooling is proportional to the temperature difference between the object and its surroundings. If 𝑡 is time in hours and the surroundings are kept at a constant temperature of 0°C, calculate the temperature of the fuel on arrival by formulating a prediction of exponential decay using integration methods. Initial tests show that the temperature cools by 1°C in the first 21.45 hours.
c) The pressure vessel is held closed during shipping using a metal clip. To open the vessel, the metal clip must be heated to allow it to expand and be removed. The rate of change of length of the clip with respect to temperature is directly proportional to the length. At 0°C, the length of the clip is 70 mm and at 20°C, it is 70.73 mm. The clip will open when it is anything longer than 72 mm. Formulate a prediction of exponential growth using integration methods to find the temperature at which the clip will allow the vessel to be opened. Use your result to check if this will risk over-heating the oil inside the vessel.
Answer:
1a)
1b)
When T = 29°C, t = 21.45h and A = 30°C...
When t = 288h, A = 30°C and k = 0.00158...
1c)
When T = 20°C, A = 70mm and L = 70.73mm...
When A = 70mm, L = 72mm and k = 0.000519...
As the temperature does not rise over 60°C the oil is not at risk of overheating.