Calculating Compressor Discharge Temperature

1. The Physics of Gas Compression

When a gas is mechanically compressed by a reciprocating piston or a centrifugal impeller, the mechanical shaft work applied to the gas is converted into internal energy. As a result, the pressure of the gas increases, but inevitably, so does its temperature.

Predicting the exact discharge temperature is critical for process safety and equipment design. If the discharge temperature exceeds typical limits (usually 135 °C to 150 °C / 275 °F to 300 °F), it can degrade lubricating oils, damage internal seals, or trigger high-temperature shutdown interlocks. If temperatures rise too high, engineers must divide the compression into multiple stages and install intercoolers.

2. Isentropic Compression Model

The simplest and most common thermodynamic model used to predict discharge temperature assumes the compression is Isentropic—meaning it is perfectly reversible and adiabatic (no heat is lost to the compressor casing).

For an ideal gas undergoing isentropic compression, the temperatures and pressures are related by the ratio of specific heats ($k = C_p / C_v$): $$\frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\frac{k - 1}{k}}$$

Rearranging to solve for the discharge temperature ($T_2$): $$T_2 = T_1 \cdot \left( \frac{P_2}{P_1} \right)^{\frac{k - 1}{k}}$$ (Note: Temperatures $T_1$ and $T_2$ MUST be in absolute units: Kelvin or Rankine. Pressures must also be absolute: bar(a) or psia).

3. Polytropic Compression (Real-World)

In reality, no compressor is perfectly isentropic. Centrifugal compressors suffer from internal friction, fluid slip, and inefficiencies that generate extra heat, causing the real discharge temperature to be hotter than the isentropic prediction.

To account for this, engineers use the Polytropic model, replacing the isentropic exponent ($k$) with a polytropic exponent ($n$): $$T_2 = T_1 \cdot \left( \frac{P_2}{P_1} \right)^{\frac{n - 1}{n}}$$

The polytropic exponent ($n$) is related to the specific heat ratio ($k$) and the compressor's polytropic efficiency ($\eta_p$): $$\frac{n - 1}{n} = \frac{k - 1}{k \cdot \eta_p}$$

Because efficiency ($\eta_p$) is always less than 1.0 (typically 0.70 to 0.85), the exponent $\frac{n-1}{n}$ becomes strictly larger than $\frac{k-1}{k}$, which mathematically yields a higher predicted discharge temperature.

4. Worked Example: Predicting Discharge Temperature

A centrifugal compressor compresses natural gas ($k = 1.30$) from 2 bar(a) to 8 bar(a). The suction temperature is 30 °C. The vendor states the polytropic efficiency is 75% ($\eta_p = 0.75$).

Calculate the actual discharge temperature.

Step 1: Convert Temperature to Absolute $$T_1 = 30 \text{ °C} + 273.15 = 303.15 \text{ K}$$

Step 2: Calculate the Polytropic Exponent Term $$\frac{k - 1}{k} = \frac{1.30 - 1}{1.30} = 0.2307$$ $$\frac{n - 1}{n} = \frac{0.2307}{0.75} = 0.3076$$

Step 3: Calculate Discharge Temperature $$T_2 = 303.15 \cdot \left( \frac{8}{2} \right)^{0.3076}$$ $$T_2 = 303.15 \cdot (4)^{0.3076}$$ $$T_2 = 303.15 \cdot 1.531 = \mathbf{464.1 \text{ K}}$$

Step 4: Convert back to Celsius $$T_2 = 464.1 - 273.15 = \mathbf{190.9 \text{ °C}}$$

Engineering Verdict: 190.9 °C is much too hot for a single stage! This compressor requires a two-stage design with an intercooler to keep temperatures below the typical 150 °C mechanical limit.