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Thermodynamics: Moving Boundary Work

In this section I will show how to calculate the amount of energy going to into a system, such as a cylinder, that is required to make that system expand or contract resulting in mechanical work. An example of this in real life would be the cylinders inside a car's engine expanding due to the explosions caused by the spark plugs.

Recall the fundamental method of calculating work is by multiplying the distance an object traveled due to a force that was applied by to the object. Refer to the equation below.

General Work Equation (1)

Now when dealing with something like a cylinder the work is done by pressure change that would cause the volume inside the cylinder to change moving a piston. To calculate this equation 1 would have to be modified. Refer to equation 2.

Boundary Work of a closed system equation (2)

Notice that for both equations 1 and 2 I showed the calculation of work as an integral due to a distance change or volume changed multiplied by a force or pressure. By using an integral, the area under the process curve is found, and this area represents the boundary work. Refer to the figure below.

Work under the process curve

As I've stated above, the above image represents the work for a process. However, in most cases you want this process to repeat which means another process has to occur to result in a complete cycle. To return a piston inside a cylinder back to its original position a certain amount of work is required. The difference between the work for the two processes will result in a net work. Refer to the equation and figure below.

Net Work Equation (3)

Net work of a cycle

Finally, the above equation 2 assumes that the system is in a state of quasi-equilibrium throughout the process. For this to happen the process has to occur at a slow pace, which means for most real life situation equation 2 will only provide rough estimates of the true work required to move the piston inside of a cylinder. To calculate the true boundary work tests would have to be conducted on the system so that measurements can be obtained.

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