Work Done Given Polytropic Index For Compression And Expansion Calculator

Formula Used:

\[ w = \frac{n_c}{n_c-1} \cdot P_1 \cdot V_1 \cdot \left( \left( \frac{P_2}{P_1} \right)^{\frac{n_c-1}{n_c}} - 1 \right) - \frac{n_e}{n_e-1} \cdot P_1 \cdot V_4 \cdot \left( \left( \frac{P_2}{P_1} \right)^{\frac{n_e-1}{n_e}} - 1 \right) \]

Polytropic Index of Compression (n_c):

Polytropic Index of Expansion (n_e):

Suction Pressure (P₁):

Total Volume of Refrigerant in Compressor (V₁):

m³

Discharge Pressure of Refrigerant (P₂):

Expanded Clearance Volume (V₄):

m³

Work Done per Cycle (w):

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1. What is the Work Done Formula?

The work done per cycle formula calculates the total work input required for a refrigeration cycle considering polytropic compression and expansion processes. It accounts for the non-ideal behavior of the refrigerant during these thermodynamic processes.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( w \) — Work done per cycle (J)
\( n_c \) — Polytropic index of compression
\( n_e \) — Polytropic index of expansion
\( P_1 \) — Suction pressure (Pa)
\( V_1 \) — Total volume of refrigerant in compressor (m³)
\( P_2 \) — Discharge pressure of refrigerant (Pa)
\( V_4 \) — Expanded clearance volume (m³)

Explanation: The first term represents work done during compression, while the second term represents work recovered during expansion.

3. Importance of Work Done Calculation

Details: Accurate work calculation is crucial for determining compressor efficiency, system performance, and energy consumption in refrigeration cycles.

4. Using the Calculator

Tips: Enter all values with appropriate units. Polytropic indices must be greater than 1. Pressures and volumes must be positive values.

5. Frequently Asked Questions (FAQ)

Q1: What are typical values for polytropic indices?
A: For refrigeration compressors, n_c typically ranges from 1.1 to 1.4, and n_e from 1.1 to 1.3, depending on the refrigerant and operating conditions.

Q2: Why use polytropic instead of isentropic analysis?
A: Polytropic analysis accounts for heat transfer during compression/expansion, providing more accurate results for real-world compressor performance.

Q3: What is expanded clearance volume?
A: This is the volume occupied by the refrigerant remaining in the compressor after the expansion process, before the next compression cycle begins.

Q4: How does this differ from ideal cycle work calculation?
A: This formula considers the actual polytropic processes rather than assuming ideal isentropic compression and expansion.

Q5: Can this be used for all types of compressors?
A: This formula is particularly useful for reciprocating compressors where clearance volume effects are significant.