Numerical and experimental studies on feasibility of a cryogenic turboexpander rotor supported on gas foil bearings

Abstract

Gas foil bearings are gaining popularity for their compliance properties in various high-speed turbomachinery applications such as air cycle machine, turbocompressor, turbocharger, turboexpander etc. A modest attempt is made in the current research to study the feasibility of gas foil bearing for a turboexpander rotating at 1,75,000 rpm. The turboexpander rotor with 16 mm diameter and 91 mm length used for experimentation is supported by a pair of gas foil journal bearings and mounted with turbine and compressor wheels at both ends of the rotor. The feasibility study was performed based on comparison of rotodynamic analysis and experimental data for the critical speed of the rotor and unbalance response at bearing locations. The critical speeds and the unbalance response are predicted using the finite element analysis, which takes into account the gyroscopic effect, shear deformation, internal damping, inertia of the rotor and the dynamic coefficients of the gas foil bearing. The predicted and experimental variation of critical speed is found to be within a relative error of 3–6%; similarly, the variation of unbalance response was found with a relative error of 2–9%. The low relative errors suggest that the experiment and prediction methodology are credible. The author believes that the rotodynamic analysis methodology will be quite valuable for researchers working in the area of high-speed rotors supported with gas foil bearings.

Appendix

1.1 A.1. Shaft element matrices

$$ [M_{e}^{T} ]_{s} = \frac{{\uprho AL_{e} }}{{840(1 +\Phi )^{2} }}\left[ {\begin{array}{*{20}c} {m_{1} } & 0 & 0 & {m_{2} } & {m_{3} } & 0 & 0 & {m_{4} } \\ 0 & {m_{1} } & { - m_{2} } & 0 & 0 & {m_{3} } & { - m_{4} } & 0 \\ 0 & { - m_{2} } & {m_{5} } & 0 & 0 & {m_{4} } & {m_{6} } & 0 \\ {m_{2} } & 0 & 0 & {m_{5} } & { - m_{4} } & 0 & 0 & {m_{6} } \\ {m_{3} } & 0 & 0 & { - m_{4} } & {m_{1} } & 0 & 0 & { - m_{2} } \\ 0 & {m_{3} } & {m_{4} } & 0 & 0 & {m_{1} } & {m_{2} } & 0 \\ 0 & { - m_{4} } & {m_{6} } & 0 & 0 & {m_{2} } & {m_{5} } & 0 \\ {m_{4} } & 0 & 0 & {m_{6} } & { - m_{2} } & 0 & 0 & {m_{5} } \\ \end{array} } \right] $$

(A.1)

$$ [M_{e}^{R} ]_{s} = \frac{{{\uprho {\rm I}}}}{{30L_{e} (1 +\Phi )^{2} }}\left[ {\begin{array}{*{20}c} {m_{7} } & 0 & 0 & {m_{8} } & { - m_{7} } & 0 & 0 & {m_{8} } \\ 0 & {m_{7} } & { - m_{8} } & 0 & 0 & {m_{7} } & { - m_{8} } & 0 \\ 0 & { - m_{8} } & {m_{9} } & 0 & 0 & {m_{8} } & {m_{10} } & 0 \\ {m_{8} } & 0 & 0 & {m_{9} } & { - m_{8} } & 0 & 0 & {m_{10} } \\ { - m_{7} } & 0 & 0 & { - m_{8} } & {m_{7} } & 0 & 0 & { - m_{8} } \\ 0 & { - m_{7} } & {m_{8} } & 0 & 0 & {m_{7} } & {m_{8} } & 0 \\ 0 & { - m_{8} } & {m_{10} } & 0 & 0 & {m_{8} } & {m_{9} } & 0 \\ {m_{8} } & 0 & 0 & {m_{10} } & { - m_{8} } & 0 & 0 & {m_{9} } \\ \end{array} } \right] $$

(A.2)

$$ {[}G_{e} {]}_{s} = \frac{{{\uprho {\rm I}}}}{{15L_{e} (1 +\Phi )^{2} }}\left[ {\begin{array}{*{20}c} 0 & { - g_{1} } & {g_{2} } & 0 & 0 & {g_{1} } & {g_{2} } & 0 \\ {g_{1} } & 0 & 0 & {g_{2} } & { - g_{1} } & 0 & 0 & {g_{2} } \\ { - g_{2} } & 0 & 0 & { - g_{3} } & {g_{2} } & 0 & 0 & { - g_{4} } \\ 0 & { - g_{2} } & {g_{3} } & 0 & 0 & {g_{2} } & {g_{4} } & 0 \\ 0 & {g1} & { - g_{2} } & 0 & 0 & { - g_{1} } & { - g_{2} } & 0 \\ { - g_{1} } & 0 & 0 & { - g_{2} } & {g_{1} } & 0 & 0 & { - g_{2} } \\ { - g_{2} } & 0 & 0 & { - g_{4} } & {g_{2} } & 0 & 0 & { - g_{3} } \\ 0 & { - g_{2} } & {g_{4} } & 0 & 0 & {g_{2} } & {g_{3} } & 0 \\ \end{array} } \right] $$

(A.3)

$$ [K_{e} ]_{s} = \frac{{{{\rm E}{\rm I}}}}{{L_{e}^{3} (1 +\Phi )}}\left[ {\begin{array}{*{20}c} {12} & 0 & 0 & {6L_{e} } & { - 12} & 0 & 0 & {6L_{e} } \\ 0 & {12} & { - 6L_{e} } & 0 & 0 & { - 12} & { - 6L_{e} } & 0 \\ 0 & { - 6L_{e} } & {(4 +\Phi )L_{e}^{2} } & 0 & 0 & {6L_{e} } & {(2 -\Phi )L_{e}^{2} } & 0 \\ {6L_{e} } & 0 & 0 & {(4 +\Phi )L_{e}^{2} } & { - 6L_{e} } & 0 & 0 & {(2 -\Phi )L_{e}^{2} } \\ { - 12} & 0 & 0 & { - 6L_{e} } & {12} & 0 & 0 & { - 6L_{e} } \\ 0 & { - 12} & {6L_{e} } & 0 & 0 & {12} & {6L_{e} } & 0 \\ 0 & { - 6L_{e} } & {(2 -\Phi )L_{e}^{2} } & 0 & 0 & {6L_{e} } & {(4 +\Phi )L_{e}^{2} } & 0 \\ {6L_{e} } & 0 & 0 & {(2 -\Phi )L_{e}^{2} } & { - 6L_{e} } & 0 & 0 & {(4 +\Phi )L_{e}^{2} } \\ \end{array} } \right] $$

(A.4)

Where,

$$\begin{aligned} m_{1} & = 312 + 588\Phi + 280\Phi ^{2} , \\ m_{2} & = (44 + 77\Phi + 35\Phi ^{2} )L_{e} , \\ m_{3} & = 108 + 252\Phi + 140\Phi ^{2} , \\ m_{4} & = - (26 + 63\Phi + 35\Phi ^{2} )L_{e} , \\ m_{5} & = (8 + 14\Phi + 7\Phi ^{2} )L_{e}^{2} , \\ m_{6} & = - (6 + 14\Phi + 7\Phi ^{2} )L_{e}^{2} , \\ m_{7} & = 36, \\ m_{8} & = (3 - 15\Phi )L_{e} , \\ m_{9} & = (4 + 5\Phi + 10\Phi ^{2} )L_{e}^{2} , \\ m_{{10}} & = ( - 1 - 5\Phi + 5\Phi ^{2} )L_{e}^{2} , \\ g_{1} & = 36, \\ g_{2} & = (3 - 15\Phi )L_{e} , \\ g_{3} & = (4 + 5\Phi + 10\Phi ^{2} )L_{e}^{2} , \\ g_{4} & = ( - 1 - 5\Phi + 5\Phi ^{2} )L_{e}^{2} . \\ \end{aligned}$$

1.2 A.2. Bearing matrices

$$ [C]_{B} = \left[ {\begin{array}{*{20}c} {C_{xx} } & {C_{xy} } \\ {C_{yx} } & {C_{YY} } \\ \end{array} } \right]\;\;\;\;[K]_{B} = \left[ {\begin{array}{*{20}c} {K_{xx} } & {K_{xy} } \\ {K_{yx} } & {K_{YY} } \\ \end{array} } \right] $$

(A.5-6)

1.3 A.3. Disc matrices

$$ [M_{e} ]_{D} = \left[ {\begin{array}{*{20}c} {m_{d} } & 0 & 0 & 0 \\ 0 & {m_{d} } & 0 & 0 \\ 0 & 0 & {I_{d} } & 0 \\ 0 & 0 & 0 & {I_{d} } \\ \end{array} } \right]\;\;\;\;[G_{e} ]_{D} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - I_{p} } \\ 0 & 0 & {I_{p} } & 0 \\ \end{array} } \right] $$

(A.7-8)