Calculate the net work output of a heat engine following path ABCDA in Figure 14.30, where V1 = 5.0 10-3 m3 and V2 = 20.0 10-3 m3.
The net work done is the area inside the closed loop in the PV diagram.
W = (1.4 * 3)*(106 N/m2)(10-3 m3) = 4.2*103 J, using area = average height * width.
or
area inside the loop = area of center rectangle + area of to triangle - area of bottom triangle
= (1.4*15 + 0.6*15/2 - 0.4*15/2)*(106 N/m2)(10-3 m3) = 22.5*103 J
To calculate the net work output of the heat engine, we need to find the area enclosed by the path ABCDA on the P-V diagram (Figure 14.30). The net work output is equal to the work done by the engine on its surroundings during one complete cycle.
To find the work done during each segment of the cycle, we can use the equation for work done by a gas:
Work = P * ΔV
where P is the pressure and ΔV is the change in volume.
Let's calculate the work done during each segment:
Segment AB:
During this segment, the volume changes from V1 to V2. The pressure remains constant, so the work done is:
Work_AB = P * ΔV_AB
Segment BC:
During this segment, the volume remains constant at V2, so the work done is zero:
Work_BC = 0
Segment CD:
During this segment, the volume changes from V2 to V1. The pressure remains constant, so the work done is:
Work_CD = P * ΔV_CD
Segment DA:
During this segment, the volume remains constant at V1, so the work done is zero:
Work_DA = 0
Now, let's calculate the values of ΔV_AB and ΔV_CD:
ΔV_AB = V2 - V1 = 20.0 * 10^-3 m^3 - 5.0 * 10^-3 m^3 = 15.0 * 10^-3 m^3
ΔV_CD = V1 - V2 = 5.0 * 10^-3 m^3 - 20.0 * 10^-3 m^3 = -15.0 * 10^-3 m^3
Since we know that the pressure remains constant throughout the process, we can say that P * ΔV_AB = P * ΔV_CD.
Therefore, the net work done during one complete cycle is:
Net Work = Work_AB + Work_BC + Work_CD + Work_DA
= P * ΔV_AB + 0 + P * ΔV_CD + 0
= P * (ΔV_AB + ΔV_CD)
= P * (15.0 * 10^-3 m^3 - 15.0 * 10^-3 m^3)
= 0
Thus, the net work output of the heat engine following the path ABCDA is zero.
To calculate the net work output of a heat engine following path ABCDA, we need to determine the area enclosed by the path on a P-V (pressure-volume) diagram. The net work output is equal to the area enclosed.
The P-V diagram is not provided in the question, so we'll need to refer to Figure 14.30 in the mentioned source. However, since I am an AI text-based bot, I cannot see any figure. Therefore, I can only provide you with a general approach to solve this type of problem.
Here's a general step-by-step guide to calculate the net work output of a heat engine:
1. Identify the path ABCDA on the P-V diagram.
2. Break down the area enclosed by the path into simpler shapes (rectangles, triangles, etc.).
3. Calculate the area of each individual shape.
4. Sum up the areas of all the shapes to find the total area enclosed.
5. The net work output is equal to the total area enclosed.
Make sure to check the provided source for Figure 14.30 to get a clear visual representation of the path and its respective area.