Question

Steam at a given state enters a turbine operating at steady state and expands adiabatically to a specified lower pressure. Would you expect the power output to be greater in an internally reversible expansion or an actual expansion?

Short answer

The power output is greater in an internally reversible expansion than in an actual expansion.

Step-by-step solution

Understand the Problem

Identify that the problem involves a turbine where steam expands adiabatically from a high pressure to a low pressure state. The goal is to compare power output between an internally reversible expansion and an actual expansion.

Internally Reversible Expansion

In an internally reversible process, the expansion occurs in a manner where no entropy is generated, and the process is isentropic (entropy remains constant). The efficiency of this expansion is maximized because there are no dissipative effects.

Actual Expansion

An actual expansion in a turbine is not perfectly efficient and involves irreversible processes. These include friction, turbulence, and heat losses, leading to an increase in entropy. Thus, the work output is less than that of the internally reversible process.

Compare Work Outputs

Use the first law of thermodynamics for both cases. For the internally reversible process: \[ W_{\text{rev}} = \text{h1} - \text{h2s} \] where \( h1 \) is the initial specific enthalpy and \( h2s \) is the specific enthalpy in the isentropic state after expansion. For the actual process: \[ W_{\text{act}} = \text{h1} - \text{h2a} \] where \( h2a \) is the specific enthalpy in the actual state after expansion. Generally, \( h2a > h2s \), so \( W_{\text{act}} < W_{\text{rev}} \).

Conclusion

Since the internally reversible process maximizes efficiency and minimizes entropy generation, the work output for an internally reversible expansion is greater than that of an actual expansion.

Key concepts

Isentropic Process

In thermodynamics, an isentropic process is a particular type of process where entropy remains constant. This means there is no entropy generation within the system. An isentropic process is ideal because it represents a fully reversible process, free from any dissipative effects like friction or turbulence.
For a turbine, an isentropic expansion means that the steam expands from a higher pressure to a lower pressure without any loss of energy due to internal friction or heat transfer.
This implies that the efficiency of an isentropic expansion is maximized. Mathematically, we describe an isentropic expansion by ensuring the entropy at the start and end of the process remains constant:
\( s1 = s2 \)
where \( s \) is the entropy at those states.
Utilizing the First Law of Thermodynamics for an isentropic process, the work output, \( W_{\text{rev}} \), is determined as follows:
\[ W_{\text{rev}} = h1 - h2s \]
Here, \( h1 \) is the initial specific enthalpy and \( h2s \) is the specific enthalpy after isentropic expansion.

Entropy Generation

Entropy generation occurs in any real process where irreversibilities like friction, turbulence, or heat loss are present. These factors cause the process to deviate from the ideal behavior observed in an isentropic process. In practical turbine operation, actual expansion is not entirely efficient due to these effects.
As a result, the actual expansion process generates entropy, denoted by \( \Delta S > 0 \). A higher entropy state means that some potential work is lost, and the process is less efficient.
In the formula for work output for an actual process, the specific enthalpy in the post-expansion state, \( h2a \), is higher due to entropy generation:
\[ W_{\text{act}} = h1 - h2a \]
Generally, \( h2a \) is greater than \( h2s \), indicating that \( W_{\text{act}} \) is less than \( W_{\text{rev}} \). The increase in enthalpy post-expansion reflects energy losses due to irreversibilities.

First Law of Thermodynamics

The First Law of Thermodynamics is a fundamental principle that describes the conservation of energy. It states that the energy within a closed system remains constant. For any process happening in a system, the change in internal energy is equal to the heat added to the system minus the work done by the system.
Mathematically, the First Law is expressed as:
\[ \Delta U = Q - W \]
For an adiabatic turbine where heat transfer, \( Q \), is zero, the First Law simplifies to:
\[ \Delta U = -W \]
With the turbine operating under steady-state conditions, the work done becomes directly related to change in specific enthalpy, \( h \):
\[ W = h1 - h2 \]
For an internally reversible (isentropic) process, this modifies to:
\[ W_{\text{rev}} = h1 - h2s \]
And for an actual process plagued by irreversibilities:
\[ W_{\text{act}} = h1 - h2a \]
By comparing the two, it is evident that \( W_{\text{rev}} \) exceeds \( W_{\text{act}} \) because \( h2a > h2s \. \). This underlines the importance of minimizing entropy generation to maximize the power output of turbines.