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I know shaft work in is expressed as $$ \int v dp $$

In a turbine with fluid in at P1 and V1, and fluid out at P2 and V2, the fluid 1 exerts pressure on the turbine blades equal to P1. The blades must then exert pressure back such that P_blades+P2 = P1. As the fluid undergoes expansion over the blades, it does boundary work $$\int p dv $$

since p is always equal to P_blades+P2, clearly $$\int p_{blades} dv = \int p dv -\int p_2 dv$$

And the integral of p_blades is considered the shaft work.

Thus it seems the shaft work the boundary work - atmospheric work. However when I do problems with this method I do not get the right answer (the answer with the vdp integral).

Why is this model not correct?

user2958456
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  • The open system version of the 1st law of thermodynamics, which is involved in modeling a continuous flow turbine, includes both work to push gas into and out of the turbine at the entrance and exit plus the shaft work. In your argument, you failed to consider the former. – Chet Miller May 14 '18 at 04:56
  • You need to get $\Delta H$ across the turbine and multiply by turbine efficiency to calculate shaft work. – David White Feb 21 '21 at 00:35

2 Answers2

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In applying the open system version of the first law to a continuous flow turbine, we are not explicitly calculating the shaft work done from turbine forces and entry/exit forces. We are treating the turbine as a black box, and backing out what the work done is from the inlet and exit conditions. Thus, the steady state energy balance equation reduces to: $$W_S=-\dot{m}\Delta h$$ where $\dot{m}$ is the mass flow rate through the turbine and $\Delta h$ is the change in enthalpy per unit mass of the fluid. The change in enthalpy is calculated by integrating the fundamental property equation $dh=Tds+vdP$ from inlet to outlet of the turbine, where s is the entropy per unit mass of the gas and v is the specific volume. If the gas parcels passing through the turbine experience a basically adiabatic reversible expansion, then Tds = 0, and dh reduces to $dh=vdP$. Thus, the shaft work is given by $$W_S=-\dot{m}\int{vdP}$$If the initial pressure and volume are known, then, since the gas parcels that have passed through the turbine are each assumed to have had an adiabatic reversible expansion (by whatever interaction they have had with the turbine blades), we can write (for an ideal gas approximation) that $$Pv^{\gamma}=P_0v_0^{\gamma}$$ on average throughout the turbine. This gives us what we need to integrate to get the shaft work, for a given exit pressure.

Chet Miller
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In steam turbine plant like Rankine cycle(for example):

I think this question has similarities What is vdp work and when do I use it?

I.Omar
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