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Screw Compressors- Mathematical Modelling And Performance Calculation Direct

REPORT: Mathematical Modelling and Performance Calculation of Screw Compressors

Date: October 26, 2023 Subject: Technical Review of Thermodynamic and Geometric Modelling Techniques

3.2 Flow Through Clearances

For small gaps, flow is modelled as compressible, isentropic flow through a nozzle or viscous slit flow. The mass flow rate ( \dotmleak ) for a given clearance area ( Aleak ) and upstream/downstream pressures ( p_u, p_d ):

If ( \fracp_dp_u > \left(\frac2\kappa+1\right)^\frac\kappa\kappa-1 ) (subsonic):

[ \dotm = A_leak \cdot p_u \sqrt\frac2\kappa(\kappa-1)RT_u \left[ \left( \fracp_dp_u \right)^\frac2\kappa - \left( \fracp_dp_u \right)^\frac\kappa+1\kappa \right] ]

If chocked (sonic flow at throat), the pressure ratio is replaced by the critical pressure ratio. Lobe count (z₁, z₂): Typically 4/6 or 5/6 (male/female)

For very narrow slits (height < 50 µm), viscous laminar flow models are more accurate:

[ \dotm = \frac\rho_m \cdot (p_u - p_d) \cdot h^3 \cdot w12 \mu L_path ]

Where ( h ) = clearance height, ( w ) = width, ( \mu ) = viscosity, ( L_path ) = leakage path length.

1.1 Key Geometric Parameters

Why it’s solid:

7.1 Oil-Injected vs. Dry Compressors

Oil-injected models require two-phase flow (gas + oil droplets). The oil absorbs compression heat, reducing discharge temperature. Additional equations for oil mass fraction, droplet size, and heat transfer between phases are needed:

[ \dotQoil = hoil-gas \cdot a_oil \cdot (T_gas - T_oil) \cdot V_chamber ] Volumetric efficiency η_v: dry 0.6–0.85

5.2 Key Performance Metrics

A. Volumetric Efficiency ($\eta_v$): This measures the effectiveness of the compressor in moving gas. It is reduced by leakage and heating of the intake gas. $$ \eta_v = \frac\dotmactual\dotmtheoretical = \frac\dotmactualVdisp \cdot N \cdot \rho_suc $$ Where $V_disp$ is the displaced volume per revolution and $N$ is the rotational speed.

B. Indicated Power ($P_i$): The power developed inside the compression chamber, derived from the area enclosed by the P-V diagram. $$ P_i = \fracn60 \oint P , dV $$ Where $n$ is the rotational speed in RPM.

C. Isentropic Efficiency ($\eta_is$): Compares the actual work to the ideal isentropic compression work. $$ \eta_is = \fracW_idealW_actual = \frac\dotm(h_dis,isentropic - h_suc)P_shaft $$ (Note: Shaft power $P_shaft$ includes mechanical losses due to bearings and timing gears, whereas Indicated Power does not).

14. Typical Parameter Ranges (practical reference)

2.2 Real Gas Equation of State

For many gases (especially refrigerants like R134a or hydrocarbons), ideal gas law fails. A real gas equation like Peng-Robinson or NIST REFPROP correlations is used:

[ p = \fracRTv - b - \fraca(T)v(v+b) + b(v-b) ] use vendor maps.

Where ( v ) is specific volume, ( a(T) ) and ( b ) are fluid-specific parameters.

5. Leakage Models

Leakage paths significantly affect volumetric efficiency:

| Path | Description | Significance | |------|-------------|--------------| | Blow-hole | Triangular gap at rotor end | High | | Seal line | Between rotor lobes | Medium | | Radial gap | Between rotor tip and casing | Medium | | End face gaps | Between rotor face and housing | Low |

Leakage flow equation (compressible flow, orifice model): $$ \dotmleak = C_d \cdot Agap \cdot \sqrt \frac2R T_up \cdot \frac\kappa\kappa-1 \left[ \left( \fracP_downP_up \right)^\frac2\kappa - \left( \fracP_downP_up \right)^\frac\kappa+1\kappa \right] $$

If $P_down/P_up \le P_critical$, use choked flow: $$ \dotmchoked = C_d \cdot Agap \cdot P_up \sqrt \frac\kappaR T_up \left( \frac2\kappa+1 \right)^\frac\kappa+1\kappa-1 $$

Typical discharge coefficient $C_d = 0.6 - 0.8$.