Why is the sign of the formula for calculating shaft work for compressors, pumps and turbines positive? [closed] In this shaft work formula shown above, is the sign of the shaft work positive due to work being done by system on fluid OR work being by fluid on system?

• Jan 26 '21 at 11:25

Turbine work is work done by the system (by the working fluid). Compressor and pump work is done on the system (on the working fluid).

Insofar as which is positive and which is negative is concerned, it depend on what version of the first law is used for an open system. For one version, turbine work is considered positive and compressor and pump work is considered negative. For the other version, it it is the reverse.

But both versions are consistent in that for each version work done by the fluid decreases the energy of the fluid and work done on the fluid increases the energy of the fluid.

Bottom line: IMO. Whether the work is considered positive or negative is unimportant as long as work done by the fluid reduces the energy of the fluid and work done on the fluid increases the energy of the fluid.

Hope this helps.

There is an eternal confusion and related questions on both Physics and Chemistry SE sites.

There are 2 opposite conventions in place:

1. Engineering-like convention, taking as the positive work done by the system.

2. Scientific-like convention, taking as the positive work done on the system.

There is also frequently mentioned distinguishing as

• the former as the convention of physics
• the latter as the convention of chemistry

While it was largely true in past, it is rather obsolete these days, as many physicists use the latter.

As summary, always check the context to realize the explicit or implicit convention, and which of the signs makes sense.

The formula result for the shaft work

$$W_\mathrm{s} = \int_\mathrm{in}^\mathrm{out}{V \cdot \mathrm{d}p}$$

• is positive for compressors or pumps ($$p_\mathrm{in} \lt p_\mathrm{out}$$ ), where the work is system input
• is negative for turbines ($$p_\mathrm{in} \gt p_\mathrm{out}$$ ), where work is the system output.

Therefore the used convention is the latter.

• So is the formula above using the latter form?
– KSP
Jan 26 '21 at 12:17
• @KSP See the A update. Jan 26 '21 at 12:39
• You are confusing shaft work for an open system with total work for a closed system. I suggest you review the open system (control volume) version of the first law of thermodynamics. Moreover, the equation for the shaft work written by the OP applies exclusively to the case where the process is adiabatic and reversible (so that TdS=0 along the device). Jan 26 '21 at 13:16
• @ChetMiller Well, I was not addressing the exact validity of the formula, rather just its sign. Neither do I pretend the shaft work is the total work. The shaft work sign convention is valid regardless of the system being open/closed. But agree with bobd the sign is not important. Jan 26 '21 at 14:55