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ME 2124 Work - = = = 1 2 1 1 2 1 1 1 2 1 12 ln V V V...

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ME 2124 Fall 2011 © Al Kornhauser Pressure-Volume Work for Closed Systems Pressure-volume work done on the environment by a system going from state 1 to state 2 is defined by = 2 1 12 PdV W . This definition gives work out of the system as positive. It is only valid when the process is quasistatic, so that the system is at uniform pressure during each step of the process. In many real processes this equation cannot be integrated in closed form. For some machines – steam engines, internal combustion engines, and reciprocating compressors – pressure and volume data from tests is often numerically integrated to determine work. For a few simple but important cases the integral can be calculated analytically. Constant volume: 0 2 1 12 = = PdV W Constant pressure: ( ) 1 2 2 1 12 V V P PdV W - = = Constant temperature ideal gas:
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Unformatted text preview: = = = ∫ ∫ 1 2 1 1 2 1 1 1 2 1 12 ln V V V P dV V V P PdV W = = 1 2 1 2 2 2 ln ln V V mRT V V V P Polytropic ( PV n = const.) ideal gas: ( ) n V V V P dV V V P PdV W n n n n n--= = =--∫ ∫ 1 1 1 1 2 1 1 2 1 1 1 2 1 12 ( ) ( ) n V P V P n V V P V V P n n n n--=--=--1 1 1 1 2 2 1 1 1 1 1 2 2 2 ( ) n T T mR--= 1 1 2 For any quasistatic process in a closed system, the work is proportional to the area under the curve of a pressure-volume diagram. If volume is increasing during the process (1 to 2 on the figure at right) work will be positive, while if it is decreasing (2 to 1 on the figure) it will be negative. P V W 1 2...
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