COMSOL 6.4 - Thermal Stress in a Rotor due to Bearing Heat Loss

Thermal Stress in a Rotor due to Bearing Heat Loss

The lubricant in hydrodynamic bearings gets sheared due to the relative velocity between the surfaces of the journal and the bushing, which causes viscous heat dissipation. This heat is conducted to the neighboring journal and bearing housing. Surfaces of the rotor and the housing finally convects this heat to the atmosphere. In a steady state situations, the temperature profile in the system is such that the heat generated in the bearings is equal to the heat convected to the atmosphere. A varying temperature profile in the system causes a thermal deformation as well as thermal stresses.

In this model, a rotor-bearing system is considered to study the thermal equilibrium due to heat dissipation in bearings by performing a stationary analysis. The resulting stresses and deformations are also studied.

Model Definition

The model consists of a rotor supported by two hydrodynamic bearings. The housing of each bearing is also modeled. The rotor is supported by thrust bearings at its ends, and another thrust bearing at the collar on the rotor between both the two journal bearings. The geometry of the system is shown in Figure 1.

Figure 1: Rotor geometry.

Modeling of the problem requires coupling between the different physical phenomena and physics interfaces; these are listed in the following:

•

Hydrodynamic Bearing: Models the pressure distribution in the lubrication film corresponding to the static weight on each bearing. Viscous heat dissipation due to shearing of the lubricant is also computed.

•

Heat Transfer in Films: Models the heat transfer in the lubricant film. Heat is convected due to churning of the lubricant in the bearing. The convective velocity and heat dissipation are obtained from the Hydrodynamic Bearing interface. The top and bottom surfaces in the film are in contact with the journal and bushing, respectively. A continuity condition is used to connect the heat flow from film to the rotor and the housing.

•

Heat Transfer in Solids: Models the heat transfer in the rotor and the bearing housings. Heat flowing in from the lubricant film is convected to the atmosphere from the surfaces of the rotor and the housings. The temperature profile in the steady state is obtained from the balance of heat flowing in and out of the solids.

•

Solid Mechanics: Models the thermal expansion and stress in the rotor and the housings.

•

Thermal Expansion Multiphysics Feature: Transfers the temperature from the Heat Transfer in Solids interface to Solid Mechanics to compute the thermal strain.

Note that the multiphysics interface automatically adds the and interfaces together with the multiphysics coupling. Therefore, the multiphysics interface can be used instead of adding them individually.

Heat Convection

The rotor and bearing housings both convect heat to the atmosphere. The housings are stationary and only natural convection takes place. A nominal value of 5 W/(m2·K) is used as a heat transfer coefficient. Heat convection from the rotor is more complicated due to the rotation of the rotor. The heat transfer coefficient for the external surfaces of the rotor is given by

where kair is the thermal conductivity of the air, R is the radius of the rotor, Re,air is the Reynolds number of the rotor given by

where Ω is the angular speed of the rotor, D is the diameter of the rotor and νair is the kinematic viscosity of the air. The Prandtl number, Pr,air, is given by

The properties of the bearing are given in Table 1, and the properties of air in Table 2.

Table 1: Bearing properties.
Parameter	value
Journal diameter, dJ	0.05 m
Bushing length, Lb	0.04 m
Bearing clearance, C	0.001dJ
Journal speed, N	3000 rpm
Static load, W	2000 N
Oil viscosity, μO	0.0028 Pa.s
Oil density, ρO	866 kg/m3
Oil heat conductivity, kO	0.13 W/m.K
Oil heat capacity, CpO	2000 J/kg.K
Oil heat capacity ratio, γO	1.4

Table 2: Air properties.
Parameter	value
Air density, ρair	1.225 kg/m3
Air kinematic viscosity, νair	17·10-6 m2/s
Air thermal conductivity, kair	0.028 W/m.K
Air heat capacity, Cp, air	1.006 kJ/kg.K
Gas constant for air, Rd	287 J/kg.K

This model only considers a unidirectional coupling in which the viscous heat dissipation changes the temperature of the system, causing thermal deformation and stress. In general, viscous heat dissipation can depend on the temperature through the viscosity and density of the lubricant, and on thermal deformation changing the clearance of the bearing. These effects are ignored in this model. They can, however, easily be included by making the lubricant properties functions of temperature and adjusting the bearing clearance to incorporate the change of the gap due to thermal expansion of the materials.

Results and Discussion

The pressure in the bearings is shown in Figure 2. The maximum pressure is located at the bottom of the bearing to support the static load.

Figure 2: Pressure in the bearings.

The viscous heat dissipation in the bearing is shown in Figure 3. The maximum dissipation occurs where the pressure gradient is the largest.

Figure 3: Viscous heat dissipation in the bearings.

The temperature profile in the bearing is shown in Figure 4. Significant heat dissipation occurs only at the location of the minimum film thickness. The temperature profile in the bearing is uniform due to the heat convection through the flow of the lubricant. Note that the temperature in both bearings is slightly different due to asymmetry in the heat convection from the rotor. The temperature in the rotor and the bearing housings is shown in Figure 5.

Figure 4: Temperature in the film.

Figure 5: Temperature in the rotor and the bearing housings.

The stress in the rotor and the housings is shown in Figure 6. In the housings, the maximum stress occurs in the connection holes. As the connection holes are fixed, thermal expansion is resisted at these locations. In the rotor, the largest stresses appear near the thrust bearings. The rotor will expand radially as well as axially due the temperature increase, and the axial expansion is restricted by the thrust bearings giving rise to high stresses in these locations.

Figure 6: Stress in the rotor and the bearing housings.

Rotordynamics_Module/Tutorials/rotor_thermal_stress

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

The problem involves the bearing modeling together with heat transfer in the film and heat transfer in the housing and shaft. Add corresponding interfaces to model all the physical phenomena.

In the tree, select > > , > > , and > > .

Click .

Click

In the tree, select > .

Click

Geometry 1

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file rotor_thermal_stress.mphbin.

Click

Form Union (fin)

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Create an imprint on the shaft to use it as journal surface.

Select the checkbox.

In the toolbar, click

Disable the analysis of the geometry as the remaining small geometric details can be kept.

In the window, click .

In the window for , locate the section.

Clear the checkbox.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file rotor_thermal_stress.txt.

Create some selections for later use.

Definitions

Bearing Housing

In the toolbar, click

Select Domains 28 and 29 only.

In the window for , type Bearing Housing in the text field.

Rotor

In the toolbar, click

In the window for , type Rotor in the text field.

Locate the section. Under , click

In the dialog, select in the list.

Click .

Identity Boundary Pair 1 (ap1)

In the window, under > click .

In the window for , locate the section.

Click

In the dialog, type Journal 1 in the text field.

Click .

In the window for , locate the section.

Click

In the dialog, type Bearing 1 in the text field.

Click .

Identity Boundary Pair 2 (ap2)

In the window, click .

In the window for , locate the section.

Click

In the dialog, type Journal 2 in the text field.

Click .

In the window for , locate the section.

Click

In the dialog, type Bearing 2 in the text field.

Click .

Housing foundation

In the toolbar, click

In the window for , type Housing foundation in the text field.

Locate the section. From the list, choose .

Select Boundaries 367, 409, 430, and 472 only.

Select the checkbox.

Journals

In the toolbar, click

In the window for , type Journals in the text field.

Locate the section. From the list, choose .

Locate the section. Under , click

In the dialog, in the list, choose and .

Click .

Bearings

In the toolbar, click

In the window for , type Bearings in the text field.

Locate the section. From the list, choose .

Locate the section. Under , click

In the dialog, in the list, choose and .

Click .

Shaft Exterior

In the toolbar, click

In the window for , locate the section.

Under , click

In the dialog, select in the list.

Click .

In the window for , type Shaft Exterior in the text field.

Housing Exterior

Right-click and choose .

In the window for , locate the section.

In the list box, select .

Under , click

In the dialog, select in the list.

Click .

In the window for , type Housing Exterior in the text field.

Convective boundaries (Shaft)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. Under , click

In the dialog, select in the list.

Click .

In the window for , locate the section.

Under , click

In the dialog, select in the list.

Click .

In the window for , type Convective boundaries (Shaft) in the text field.

Convective boundaries (Housing)

Right-click and choose .

In the window for , type Convective boundaries (Housing) in the text field.

Locate the section. In the list box, select .

Under , click

In the dialog, select in the list.

Click .

In the window for , locate the section.

In the list box, select .

Under , click

In the dialog, select in the list.

Click .

Exterior Edges (Journal)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Under , click

In the dialog, select in the list.

Click .

In the window for , locate the section.

From the list, choose .

In the text field, type Exterior Edges (Journal).

Interior Edges (Journal)

Right-click and choose .

In the window for , type Interior Edges (Journal) in the text field.

Locate the section. From the list, choose .

Select the checkbox.

Create variable for convection coefficient on the shaft surface.

Variables 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
d_rot	2*sqrt(X^2+Z^2)	m	Diameter of the shaft
Re_air	2pifr*d_rot^2/nu_air		Reynolds number for the flow
Pr_air	Cp_airrho_airnu_air/k_air		Prandtl number for air
h_shaft	k_airRe_air^(2/3)Pr_air^(1/3)/(15*d_rot/2)	W/(m²·K)	Heat transfer coefficient of the shaft

Add a operator to pick the temperature of the housing as a boundary temperature in lubricant film.

General Extrusion 1 (genext1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type X.

In the text field, type Y.

In the text field, type Z.

Locate the section. From the list, choose .

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Material 2 (mat2)

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Heat Transfer in Films (htlsh)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Hydrodynamic Bearing (hdb)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Materials

In the window, under > click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Heat capacity at constant pressure	Cp	CpO	J/(kg·K)	Basic
Density	rho	rhoO	kg/m³	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	kO	W/(m·K)	Basic
Thickness	lth	C	m	Shell
Rotation	lrot	0.0	deg	Shell
Mesh elements	lne	2	1	Shell
Dynamic viscosity	mu	muO	Pa·s	Basic

In the text field, type Material: Oil.

Heat Transfer in Films (htlsh)

Fluid 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Use the mean flow velocity of the film as a convective velocity for the heat transfer.

Locate the section. From the u list, choose .

Heat Source 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Use the viscous heat dissipation from bearing as a heat source in the film.

Locate the section. From the Q0 list, choose .

Temperature is continuous between the lubricant film and the surrounding boundaries in contact with the film. Assign the temperature from the shaft as temperature at the top interface of the film. Similarly, assign the temperature of the housing as the temperature at the bottom interface of the film.

Temperature, Interface 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the T0 text field, type T2.

Locate the section. From the list, choose .

Temperature, Interface 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the T0 text field, type genext1(T2).

Since, the film is modeled on the journal boundary, a operator is used to pick the temperature from the housing boundary.

Hydrodynamic Bearing (hdb)

Click the

button in the toolbar.

In the dialog, in the tree, select the checkbox for the node > .

Click .

In the window, under click .

In the window for , locate the section.

From the list, choose .

Hydrodynamic Journal Bearing 1

In the window, under > click .

In the window for , locate the section.

In the C text field, type C.

From the Xc list, choose .

Bearing pressure supports the static load on the shaft. Assign half of the static load in each bearing.

Locate the section. From the list, choose .

Specify the Wj vector as


0	x
0	y
-W/2	z

In the Ω text field, type 2*pi[rad]*fr.

Duplicate the current feature and change the selection to model the second bearing.

Hydrodynamic Journal Bearing 2

Right-click > > and choose .

In the window for , locate the section.

From the list, choose .

Bearing Orientation 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Specify the V vector as


1	x
0	y
0	z

Solid Mechanics (solid)

In the window, under click .

Fixed Constraint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Add a to model the thrust bearings. Two of the thrust bearings are at rotor ends and one is located in between the journal bearings at collars.

Prescribed Displacement 1

In the toolbar, click

and choose .

Select Boundaries 9, 10, 41, 42, 89, 90, 165, 168, 226, 231, 354, and 355 only.

In the window for , locate the section.

From the list, choose .

Add another node at the axial edges of the rotor in the journal bearings. This is to restrict the motion of the rotor in the lateral directions at the bearing locations.

Prescribed Displacement 2

In the toolbar, click