Heat flux estimation in journal bearings using inverse heat transfer method
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ORIGINAL
Heat flux estimation in journal bearings using inverse heat transfer method Davoud Alibeyki 1 & Reza Mehryar 1 Received: 17 December 2019 / Accepted: 8 September 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Journal bearings are used in industrial equipment at high rotational speed. Normal and also fault operations of bearing increase the temperature of the bearing. Estimation of heat flux on the inner surface of the bearing and prediction of temperature of the bearing can prevent the damage to the bearing. In this research, these quantities are estimated in normal operation by an inverse method. The sequential function specification method is applied and the best number of temperature sensors and their positions are obtained. The temperature sensors are embedded in the bearing. The results show that the accuracy of heat flux estimation has a district relation with the number and the position of the sensors and also with the measurement frequency. Keywords Inverse method . Journal bearing . Heat flux
Nomenclature L Length of bearing Cr Radial clearance C Specific heat e Eccentricity ebias Bias error erms Root mean square error evar Variance error H1 Ambient heat transfer coefficient H2 Oil heat transfer coefficient k Thermal conductivity m Time index Nm Number of time steps Ns Number of sensors Np Number of segments of unknown heat flux OB Center of the journal bearing OS Center of the shaft P Pressure q Heat flux vector qin Inlet heat flux
* Reza Mehryar [email protected] Davoud Alibeyki [email protected] 1
Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran
R RBi RBO RS r Sh T b T Tair Ts T0 t t+ ur uθ X X+ Y Z+
Radial coordinates Inner radius of the journal bearing Outer radius of the journal bearing Shaft radius Number of future time steps Heat source Vector of calculated temperatures Calculated temperature Ambient temperature Supply oil temperature Initial temperature Time Dimensionless time The velocity component in r The velocity component in θ Sensitivity coefficient matrix Dimensionless sensitivity coefficient matrix Measured temperature Kind of dimensionless sensitivity coefficient matrix
Greek symbols Δt Time step θ Tangential coordinate μ Dynamic viscosity as a function of temperature μ0 Dynamic viscosity at 303 K ρ Density σ Standard deviation of fluctuation ψ The dissipation function φ Attitude angle
Heat Mass Transfer
ω α
Rotational speed Coefficient of thermal expansion
Subscripts i Time index j Sensor index mean Measured opt Optimum S Shaft B Bearing f Fluid
1 Introduction Journal bearings are widely used in industrial equipment, such as turbines, generators, and compressors under high loads and high rotating speeds. A thin lubricant film between the shaft and the bearing prevents overheating of bearing surface. The lubricant is heated by viscous heating due to high velocity gradient between shaft and bearing. The lubricant is circulated continuously and cooled in a heat exchanger [1–7]. Seve
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