91Ó°ÊÓ

In thermodynamics, the steady flow energy equation (SFEE) is a cornerstone for analyzing systems where mass flows steadily through a control volume, such as compressors, turbines, or heat exchangers.
The SFEE is an adaptation of the first law of thermodynamics for open systems and is expressed as:
\[ \text{ \ \frac{\text{d}}{\text{d}t} \bigg[ m \bigg( h + \frac{V^2}{2} + gz \bigg) \bigg] = \frac{\text{d}Q}{\text{d}t} - \frac{\text{d}W}{\text{d}t} \ } \]
This equation essentially states that the rate of change of the total energy of a system is equal to the heat added to the system minus the work done by the system.

### Applying SFEE for compressors:
In the context of compressors, several simplifying assumptions can be made:

• Negligible changes in kinetic and potential energy
• Steady-state process

Thus, the SFEE can be simplified to:
\[ \text{ \ \frac{\text{d}Q}{\text{dt}} - \frac{\text{d}W}{\text{dt}} = \text{m} \bigg( h_2 - h_1 \bigg) \ } \] where:

• \text{ \ \text{dQ/dt} = Heat transfer rate \ }
• \text{ \ \text{dW/dt} = Work done by the compressor \ }
• \text{ \ \text{m} = Mass flow rate \ }
• \text{ \ h_2, h_1 = Specific enthalpy at the outlet and inlet \ }

Understanding this equation helps predict outlet conditions when you know the power input and heat interactions in a compressor.

The ideal gas law is crucial for understanding and solving problems related to gases in various thermodynamic processes. The law is a simplified equation of state for an ideal gas, expressed as:
\[ \text{ \ PV = nRT \ } \] where:

• \text{ \ P = Pressure \ }
• \text{ \ V = Volume \ }
• \text{ \ n = Number of moles \ }
• \text{ \ R = Universal gas constant \ }
• \text{ \ T = Temperature \ }

When dealing with a specific gas like air, the equation can be modified to:
\[ \text{ \ P = \frac{\text{mRT}}{\text{V}} \ } \] Applying to the problem, this law helps calculate mass flow rate through the compressor:
\[ \text{ \ \text{m} = \frac{\text{PV}}{\text{RT}} \ } \]
This allows us to express mass flow rate (\text{ \dot{m}\ }) in terms of volumetric flow rate (\text{ \dot{V} \ }), pressure (\text{ \ P \ }), and temperature (\text{ \ T \}). Which simplifies our calculations and steps for further analysis.

Specific enthalpy (\text{ \(h\) }) is a measure of the total energy of a fluid per unit mass and is a combination of internal energy and the product of pressure and volume. It is defined as:
\[ \text{ \ h = u + Pv \ } \]
where:

• \text{ \ u = Specific internal energy \ }
• \text{ \ P = Pressure \ }
• \text{ \ v = Specific volume \ }

For an ideal gas, specific enthalpy can be simplified using its specific heat at constant pressure (\text{ \(c_p\)}):
\[ \text{ \ h = c_pT \ } \]

### Calculating Specific Enthalpy in Compressors:
In the problem, we first find the specific enthalpy at the inlet as:
\[ \text{ \ h_1 = c_pT_1 \ } \] Then, using the steady flow energy equation:
\[ \text{ \ h_2 = h_1 + \frac{\text{dW/dt}}{\text{m}} + \frac{\text{dQ/dt}}{\text {m}} \ } \] where:

• \text{ \ \text {dW/dt} = Compressor power input \ }
• \text{ \ \text {dQ/dt} = Heat transfer rate \ }
• \text{ \ \text {m} = Mass flow rate \ }

This step is crucial for calculating the exit temperature of air, as specific enthalpy directly ties the changes in heat and work to temperature variations in the compressor. Finally, we convert enthalpy back to temperature using:
\[ \text{ \ T_2 = \frac{h_2}{c_p} \ } \]