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ABSTRACT
To improve the computing precision and efficiency of temperature field analysis in tangential clearance of scroll compressor, the ridgelet finite element method is combined with fuzzy finite element method to construct the fuzzy ridgelet finite element method of analyzing temperature field in tangential clearance. First, the related research progresses on heat transfer of scroll compressor, wavelet finite element method, and fuzzy finite element method are summarized. Second, the leakage flow model of tangential clearance in scroll compressor is studied in depth. Third, the heat transfer model of leakage flow in tangential clearance of scroll compressor is established, and then the fuzzy ridgelet finite element model is constructed, the fuzzy finite element model is transformed to the random model based on information entropy, the corresponding calculating procedure is designed. Finally, the simulation analysis is performed based on fuzzy finite element method, fuzzy Daubechies wavelet finite element method, and the fuzzy ridgelet finite element model, respectively, comparing analysis between simulation and test results shows that the fuzzy ridgelet finite element method has best computing effectiveness on temperature field analysis of tangential clearance of scroll compressor. In addition, the tangential clearance temperature of scroll compressor with and without water cooling has been analyzed based on the fuzzy ridgelet finite element method, and the results show that the water cooling system can reduce the temperature in tangential clearance greatly.
Nomenclature
| = | ridgelet coefficient vector | |
| = | components of | |
| c | = | specific heat of leakage flow medium, J/kg · K |
| C | = | heat capacity matrix |
| Cp | = | constant pressure specific heat capacity, J · kg−1 |
| Cv | = | constant volume specific heat capacity, J · kg−1 |
| = | constant | |
| = | heat capacity matrix of Ridgelet finite element | |
| = | fuzzy heat capacity matrix | |
| e | = | Ridgelet finite element |
| G | = | generation rate of k in a unit volume of leakage flow |
| Gb | = | turbulent kinetic energy generating term caused by buoyancy |
| h | = | convective heat transfer coefficient, |
| = | heat conduction matrix corrected by the heat exchange boundary of Ridgelet finite element | |
| k | = | turbulent kinetic energy |
| kx, ky and kz | = | thermal conductivities in x, y and z directions W/m · K |
| K | = | heat conduction matrix |
| = | heat conduction matrix of Ridgelet finite element | |
| = | fuzzy heat conduction matrix | |
| lp | = | arc length of expansion zone, m |
| ly | = | arc length of compression zone, m |
| m | = | average mass flow rate of compression zone, kg · s−1 |
| mh | = | mass flow rate of inlet for compression zone, kg · s−1 |
| ml | = | the outlet mass flow rate of expansion zone, kg · s−1 |
| mop | = | outlet mass flow rate of nozzle zone, kg · s−1 |
| miy | = | mass flow rate of inlet for nozzle zone, kg · s−1 |
| mip | = | inlet mass flow rate of expansion zone, kg · s−1 |
| moy | = | the mass flow rate of outlet for compression zone, kg · s−1 |
| m′ | = | denotes the average mass flow rate of expansion zone, kg · s−1 |
| nx, ny and nz | = | direction cosines of external normal line |
| pp | = | denotes the inlet pressure of expansion zone, MPa |
| py | = | denotes outlet pressure of compression zone, MPa |
| P | = | temperature load vector |
| = | temperature load caused by element heat source | |
| = | temperature load of Ridgelet finite element on the heat flux boundary | |
| = | is the temperature load of Ridgelet finite element on convective heat transfer boundary | |
| = | given thermal flux on boundary Γ2, W/m2 | |
| Q | = | internal heat density, W/kg |
| = | random vector | |
| = | component of | |
| R1 | = | average curvature radius of compression zone, m |
| Ri | = | average curvature radius of expansion zone, m |
| S | = | linear boundary of Ridgelet finite element |
| t | = | time, s |
| Δt | = | time step |
| T | = | temperature, K |
| Te | = | adiabatic wall temperature on boundary Γ3 |
| Tip | = | denotes the inlet temperature of expansion zone, K |
| Top | = | outlet temperature of nozzle zone, K |
| = | denotes the given temperature on boundary Γ1 | |
| Tl | = | denotes the outlet temperature of expansion zone, K |
| Ty | = | temperature of inlet for nozzle zone, K |
| = | fuzzy temperature variable | |
| V | = | tensor space, |
| Ve | = | volume of Ridgelet finite element |
| x1 and x2 | = | maximum and minimum values of x coordinate for Ridgelet finite element |
| X | = | normal random variable |
| y1 and y2 | = | maximum and minimum values of y coordinate |
| Y | = | fuzzy variable |
| z1 and z2 | = | are the maximum and minimum values of z coordinate. |
| α, β and γ | = | local coordinates in x, y and z directions |
| ε | = | turbulent dissipation rate |
| θ0 | = | denotes the orbiting angle of outlet for expansion zone, ° |
| θy | = | orbiting angle of outlet for compression zone, ° |
| ρ | = | denotes the density of leakage flow medium, kg/m3 |
| λ | = | mean of X |
| μ | = | laminar viscosity coefficient |
| μT | = | turbulent viscosity coefficient |
| σ | = | standard deviation of X |
| σk and | = | turbulent Prandtl number |
| ω | = | rotation speed, rad · s−1 |
| = | weighted functions | |
| ⊗ | = | Kronecker signal |
Nomenclature
| = | ridgelet coefficient vector | |
| = | components of | |
| c | = | specific heat of leakage flow medium, J/kg · K |
| C | = | heat capacity matrix |
| Cp | = | constant pressure specific heat capacity, J · kg−1 |
| Cv | = | constant volume specific heat capacity, J · kg−1 |
| = | constant | |
| = | heat capacity matrix of Ridgelet finite element | |
| = | fuzzy heat capacity matrix | |
| e | = | Ridgelet finite element |
| G | = | generation rate of k in a unit volume of leakage flow |
| Gb | = | turbulent kinetic energy generating term caused by buoyancy |
| h | = | convective heat transfer coefficient, |
| = | heat conduction matrix corrected by the heat exchange boundary of Ridgelet finite element | |
| k | = | turbulent kinetic energy |
| kx, ky and kz | = | thermal conductivities in x, y and z directions W/m · K |
| K | = | heat conduction matrix |
| = | heat conduction matrix of Ridgelet finite element | |
| = | fuzzy heat conduction matrix | |
| lp | = | arc length of expansion zone, m |
| ly | = | arc length of compression zone, m |
| m | = | average mass flow rate of compression zone, kg · s−1 |
| mh | = | mass flow rate of inlet for compression zone, kg · s−1 |
| ml | = | the outlet mass flow rate of expansion zone, kg · s−1 |
| mop | = | outlet mass flow rate of nozzle zone, kg · s−1 |
| miy | = | mass flow rate of inlet for nozzle zone, kg · s−1 |
| mip | = | inlet mass flow rate of expansion zone, kg · s−1 |
| moy | = | the mass flow rate of outlet for compression zone, kg · s−1 |
| m′ | = | denotes the average mass flow rate of expansion zone, kg · s−1 |
| nx, ny and nz | = | direction cosines of external normal line |
| pp | = | denotes the inlet pressure of expansion zone, MPa |
| py | = | denotes outlet pressure of compression zone, MPa |
| P | = | temperature load vector |
| = | temperature load caused by element heat source | |
| = | temperature load of Ridgelet finite element on the heat flux boundary | |
| = | is the temperature load of Ridgelet finite element on convective heat transfer boundary | |
| = | given thermal flux on boundary Γ2, W/m2 | |
| Q | = | internal heat density, W/kg |
| = | random vector | |
| = | component of | |
| R1 | = | average curvature radius of compression zone, m |
| Ri | = | average curvature radius of expansion zone, m |
| S | = | linear boundary of Ridgelet finite element |
| t | = | time, s |
| Δt | = | time step |
| T | = | temperature, K |
| Te | = | adiabatic wall temperature on boundary Γ3 |
| Tip | = | denotes the inlet temperature of expansion zone, K |
| Top | = | outlet temperature of nozzle zone, K |
| = | denotes the given temperature on boundary Γ1 | |
| Tl | = | denotes the outlet temperature of expansion zone, K |
| Ty | = | temperature of inlet for nozzle zone, K |
| = | fuzzy temperature variable | |
| V | = | tensor space, |
| Ve | = | volume of Ridgelet finite element |
| x1 and x2 | = | maximum and minimum values of x coordinate for Ridgelet finite element |
| X | = | normal random variable |
| y1 and y2 | = | maximum and minimum values of y coordinate |
| Y | = | fuzzy variable |
| z1 and z2 | = | are the maximum and minimum values of z coordinate. |
| α, β and γ | = | local coordinates in x, y and z directions |
| ε | = | turbulent dissipation rate |
| θ0 | = | denotes the orbiting angle of outlet for expansion zone, ° |
| θy | = | orbiting angle of outlet for compression zone, ° |
| ρ | = | denotes the density of leakage flow medium, kg/m3 |
| λ | = | mean of X |
| μ | = | laminar viscosity coefficient |
| μT | = | turbulent viscosity coefficient |
| σ | = | standard deviation of X |
| σk and | = | turbulent Prandtl number |
| ω | = | rotation speed, rad · s−1 |
| = | weighted functions | |
| ⊗ | = | Kronecker signal |