What is the compressor impeller head coefficient (pressure coefficient)?

The head coefficient, often called the pressure coefficient, is a dimensionless number that compares the polytropic work achieved during compression to the kinetic energy associated with the impeller's tip speed. It's a key indicator of how effectively an impeller converts rotational energy into pressure rise. Higher values generally indicate more effective pressure generation. Typical values for centrifugal impellers range from 0.5 to 0.6.

How is the flow coefficient defined for a compressor impeller?

The flow coefficient is a dimensionless parameter that characterizes the volumetric flow rate through the impeller relative to its dimensions and tip speed. It's calculated using the compressor's volumetric flow rate, impeller tip speed, and impeller radius. It indicates the capacity or throughput capability of the impeller. Centrifugal impellers typically have flow coefficients between 0.04 and 0.2, with higher values for mixed-flow designs.

What factors influence the compression ratio of a single compressor impeller?

The compression ratio of an impeller is primarily influenced by the head coefficient, the impeller tip speed, the polytropic efficiency, the gas's isentropic coefficient (k) and molar mass (M), and the suction temperature. Importantly, the compression ratio is theoretically independent of the suction pressure, meaning an impeller will produce the same ratio regardless of the absolute inlet pressure, assuming other parameters remain constant. Higher tip speeds and head coefficients generally lead to higher compression ratios.

Centrifugal compressor : head coefficient, flow coefficient and compression ratio

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Section summary
1. Performance of an impeller
2. Head coefficient of pressure coefficient
3. Flow coefficient
4. Compression ratio calculation

1. Performance of an impeller

It is possible to estimate the performance of a centrifugal compressor, in terms of compression ratio or flowrate if the manufacturer has given some information about the impeller, in addition to basic dimensional data : the head coefficient, the flow coefficient and the polytropic efficiency.

This page aims at answering to the following questions : what is the head coefficient ? What is the flow coefficient ? What is the relation between the head and flow coefficient ? How to calculate the compression ratio given by an impeller ?

⚠️ ENGINEERING NOTICE & EDUCATIONAL DISCLAIMER: This interactive calculator is provided exclusively for preliminary estimation and educational purposes. It is not intended for detailed design or equipment procurement without certified vendor rating. No warranty, expressed or implied, is provided, and no liability is assumed.

Polytropic Work (W_p)

J/kg

Impeller Tip Speed (u₂)

m/s

Impeller Outside Radius (R₂)

Compressor Volumetric Flowrate (Q_v)

m³/s

Isentropic Coefficient (k)

Molar Mass (M)

g/mol

Polytropic Efficiency (η_p)

Suction Temperature (T_suction)

°C

Calculation Results

Head Coefficient (μ) 0.500

Flow Coefficient (δ) 0.100

Compression Ratio (τ) 1.821

2. Head coefficient or pressure coefficient

The head coefficient is an adimensional number that is comparing the polytropic work that one gets during compression to the impeller tip speed

\[ \mu = \frac{W_p}{u_2^2} \]

Calculation of compressor impeller head coefficient

Equation 1 : calculation of compressor head coefficient

With

μ = pressure coefficient
W_p = polytropic work in J/kg
u₂ = tip speed of the impeller in m/s

⚙️ Practical Plant Engineering Rules of Thumb & Safety Limits

Typical Head Coefficients: For centrifugal flow impellers, the head coefficient (μ) generally ranges from 0.5 to 0.6. Higher values might indicate more aggressive designs or specific applications.
Tip Speed Limitations: Impeller tip speed (u₂) is often limited by material strength (mechanical integrity) and the Mach number at the blade tip (aerodynamic performance, preventing choking and excessive noise). Typical limits can be around 300-450 m/s for high-speed industrial compressors, but can vary widely. Exceeding sonic velocity at the blade tip can lead to significant performance degradation and vibration.
Polytropic Work (Wp): This value is dependent on the gas properties and the achieved pressure ratio. For common gases like air, Wp will typically be in the range of 30,000 to 100,000 J/kg for a single stage.
Input Validation: Ensure all inputs are positive. Zero or negative tip speed would result in division by zero and is physically meaningless.

Order of magnitude of the pressure coefficient :

Centrifugal flow impeller : 0.5 to 0.6
Mixed flow impeller : 0.35
Axial flow impeller : 0.05 to 0.25
Peripheral flow impeller : 1.8

3. Flow coefficient

The flow coefficient is calculated the following way :

\[ \delta = \frac{Q_v}{u_2 \cdot R_2^2} \]

Equation 2 : flow coefficient calculation

With

δ = flow coefficient
u₂ = tip speed of the impeller in m/s
R₂ = impeller outside radius in m
Qv = compressor flowrate in m3/s

⚙️ Practical Plant Engineering Rules of Thumb & Safety Limits

Typical Flow Coefficients: For centrifugal impellers, the flow coefficient (δ) typically ranges from 0.04 to 0.2. Mixed-flow impellers can go up to 0.6, while axial flow designs can be much higher (0.8 to 1.2).
Impeller Radius (R₂): The impeller radius significantly impacts both tip speed and the area available for flow. Larger radii at a given RPM mean higher tip speeds.
Volumetric Flowrate (Q_v): This value is highly dependent on the compressor's operating point and the gas density at suction conditions. Ensure consistency with the actual inlet conditions.
Relation between μ and δ: There's an inverse relationship between the head coefficient (μ) and flow coefficient (δ). Generally, designs optimized for high pressure rise (high μ) tend to have lower flow capacities (lower δ), and vice-versa. This characteristic is often depicted in compressor performance maps.

Order of magnitude of the flow coefficient :

Centrifugal flow impeller : 0.04 to 0.2 with until 0.6 for mixed flow
Axial flow impeller : 0.8 to 1.2
Peripheral flow impeller : 0.04

Note that there is relation in between μ and δ : the lower the flow coefficient, the higher will be the pressure coefficient. The manufacturer can deliver the curve μ=f(δ) to help size the compressor but one must be careful that the relation will change if the gas handled changed or if the speed of the compressor varies beyond a certain limit

4. Compression ratio calculation

Thanks to the head coefficient it is possible to calculate which compression ratio will be given by an impeller. This calculation is based on the definition of the pressure coefficient and the definition of the polytropic work.

\[ \tau = \left[ \frac{(k-1) \cdot \mu \cdot u_2^2 \cdot M}{k \cdot \eta_p \cdot R \cdot T_{suction}} + 1 \right]^{\frac{k \cdot \eta_p}{k-1}} \]

Compressor Impeller compression ratio calculation

Equation 3 : compression ratio calculation

With

τ = compression ratio
k = isentropic coefficient
μ = pressure coefficient
u₂ = tip speed of the impeller in m/s
M = molar mass of the gas
η_p = polytropic coefficient
R = ideal gas constant
T_suction = suction temperature in K

⚙️ Practical Plant Engineering Rules of Thumb & Safety Limits

Isentropic Coefficient (k): Also known as the heat capacity ratio (C_p/C_v). This value is gas-specific; e.g., for air it's around 1.4, for methane 1.3, for CO₂ 1.3. Accurate 'k' values are crucial for correct compression ratio calculation.
Polytropic Efficiency (η_p): Represents how effectively the compressor converts input power into useful compression work. Typical values for industrial centrifugal compressors range from 0.70 to 0.85. Higher efficiencies are desirable for reduced energy consumption.
Molar Mass (M): This is gas-specific. For air, it's approximately 28.97 g/mol. Heavier gases generally require more work for a given compression ratio.
Suction Temperature (T_suction): Must be in absolute units (Kelvin or Rankine). Colder suction temperatures generally result in higher compression ratios for the same polytropic work input.
Compression Ratio per Stage: For a single centrifugal impeller, the compression ratio (τ) is typically limited to 3-5. Achieving very high ratios in a single stage can lead to very high discharge temperatures and mechanical stresses. Multi-stage compressors are used for higher overall pressure ratios.

Note that the compression ratio is actually independent from the suction pressure.