Phy 201
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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My question is on the following question I was trying during a practice test:
Analyze the pressure vs. volume of a 'bottle engine' consisting of 8 liters of an ideal gas as it operates between minimum temperature 200 Celsius and maximum temperature 360 Celsius, pumping water to half the maximum possible height. Sketch a pressure vs. volume graph from the original state to the maximum-temperature state and use the graph to determine the useful work done by the expansion. Then, assuming a diatomic gas, determine the thermal energy required to perform the work and the resulting practical efficiency of the process.
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I understand the pressure vs volume graph. Does this question basically mean that I should find d'Q(v) and d'Q(p) and if so, what temperatures do I use for the temperature change. Then once i find d'Q, how do i find work. It was stated that it is the area under the curve, but is this the same as the equation d'Q(p)-d'Q(v). Also efficiency is found by taking Max temp-min temp/max temp, so I know how to do that, but why would this change with the amount of energy needed to perform work. I am very confused on this problem.
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Very good questions.
Note also that the Bottle Engine is addressed fairly extensively in Class Notes between #08 and #12, and is the subject of the two video experiments to be viewed as part of Assignment 11.
For the situation in question the maximum pressure possible, operating the system between 200 C and 360 C, is about
T_max / T_min * P_min = 623 K / (473 K) * 100 kPa = 132 kPa.
This would allow us to support a column of water which exerts a pressure of 32 kPa. This column would be about 3.2 meters high (easily found using Bernoulli's equation).
To raise water to half this height would require a temperature of about 280 C, after which the gas would expand by factor
623 K / (553 K) = 1.14.
The volume would change by this factor by displacing an equal volume of water, which would be raised to the 1.6 meter height.
The temperature of the gas would be raised from 200 C to 280 C at constant volume, then from 280 C to 360 C at constant pressure.
The efficiency we calculate here is the 'practical efficiency', which is the ratio of the mechanical work done to the thermal energy added to the system. The mechanical work is the raising of the water, so is equal to the PE change of the water which is raised to the 1.6 m height. The thermal energy added is the energy required to heat the gas, first at constant volume then at constant pressure.