radialkippsegmentlager-Ölzuführungseinfluss ii | tilting

Heft | Issue R594 (2020)

FVV-Informationstagung Turbomaschinen | Frühjahr 2020 – Würzburg FVV Information Sessions Turbomachinery | Spring 2020 – Würzburg

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Radialkippsegmentlager-Ölzuführungseinfluss II | Tilting Pad Journal Bearing Oil Supply Influence II | No 677 II Zwischenbericht | Interim report (ZB)

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Institut für Tribologie und Energiewandlungsmaschinen | Institute of Tribology and Energy Conversion Machines (ITR) Prof. Dr.-Ing. H. Schwarze | Technische Universität Clausthal

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Thema | Full title: Beeinflussung der Strömungen im Schmierstoffzuführbereich von Ra-dialkippsegmentlagern zur Senkung maximaler Lagertemperaturen und Erhöhung der Lagerleistungsdichte | Modification of flow in the space between pads region of tilting pad journal bearings to reduce maximum temperatures and increase bearing power density

Laufzeit | Duration: 01.01.2018 - 31.12.2020

Fördergeber | Funding: Bundesministerium für Wirtschaft und Energie / Arbeitsgemeinschaft in-dustrieller Forschungsvereinigungen e. V. | Federal Ministry for Econo-mic Affairs and Energy / German Federation of Industrial Research Associations eV (BMWi/AiF)

Fördernr. | Funding no: IGF 19926 N

Obmann | Chairperson: Nico Havlik (RENK Aktiengesellschaft Werk Hannover)

Bearbeiter | Coordinators: Sören Wettmarshausen, M.Sc. (ITR)

Vortragende | Lecturers: Sören Wettmarshausen, M.Sc. (ITR)

201

Tilting Pad Journal Bearing Oil Supply Influence II

Danksagung

Dieser Bericht ist das wissenschaftliche Ergebnis einer Forschungsaufgabe, die von der Forschungs-

vereinigung Antriebstechnik (FVA) e. V. gestellt und am Institut für Tribologie und Energiewandlungs-

maschinen (ITR) der Technischen Universität Clausthal unter der Leitung von Prof. Dr.-Ing. H. Schwarze

bearbeitet wurde.

Die Forschungsvereinigung Verbrennungskraftmaschinen (FVV) e. V. dankt Professor Schwarze und

dem wissenschaftlichen Bearbeiter M.Sc. Sören Wettmarshausen (ITR) für die Durchführung des Vor-

habens sowie der Arbeitsgemeinschaft industrieller Forschungsvereinigungen (AiF) e. V. für die finan-

zielle Förderung. Das Vorhaben wurde von einem gemeinschaftlichen Arbeitskreis von FVV und FVA

unter der Leitung von Nico Havlik (RENK Aktiengesellschaft Werk Hannover) begleitet. Diesem projekt-

begleitenden Ausschuss gebührt unser Dank für die große Unterstützung.

Das Forschungsvorhaben wurde im Rahmen des Programms zur Förderung der industriellen Gemein-

schaftsforschung (IGF-Nr. 19926 N) vom Bundesministerium für Wirtschaft und Energie (BMWi) über

die Arbeitsgemeinschaft industrieller Forschungsvereinigungen (AiF) e. V. aufgrund eines Beschlusses

des Deutschen Bundestages gefördert.

Acknowledgement

This report is the scientific result of a research project undertaken by the FVA (The Research Associa-

tion for drive technology eV) and performed by at Institute of Tribology and Energy Conversion Machines

(ITR) of the Technical University of Clausthal Clausthal under the direction of Prof. Dr.-Ing. H. Schwarze.

The FVV (The Research Association for Combustion Engines eV) would like to thank professor

Schwarze and his scientific research assistant – M.Sc. Sören Wettmarshausen (ITR) for the implemen-

tation of the project. Special thanks are due to the AiF (German Federation of Industrial Research As-

sociations eV) for funding the project. The project was conducted by an expert group led by Nico Havlik

(RENK Aktiengesellschaft Werk Hannover). We greatfully acknowledge the support received from the

chairman and from all members of the project user committee.

The research project was carried out in the framework of the industrial collective research programme

(IGF no. 19926 N). It was supported by the Federal Ministry for Economic Affairs and Energy (BMWi)

through the AiF (German Federation of Industrial Research Associations eV) based on a decision taken

by the German Bundestag.

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Kurzfassung

Das Forschungsziel des Projektes “Radialkippsegmentlager Ölzuführungseinfluss II” ist die Unter-

suchung konstruktiver Ölzuführungsvarianten hinsichtlich ihres Einflusses auf den Wärmeübergang im

Lager mit dem Ziel die maximale Temperatur im Lager zu senken. Zu diesem Zweck wurden zwei La-

gerkonfigurationen entwickelt, die einen Einfluss der Segmentströmung im Ölzufuhrbereich auf die Max-

imaltemperatur nachweisen sollen. Zur Erzeugung eines hohen Wärmeübergangskoeffizienten wurde

eine neuartige Ölzuführung entwickelt, welche das thermisch hoch belastete Segmentende direkt an-

strömt und die von der Wellendrehung induzierte Nischenströmung ausnutzt. Ein niedriger Wärmeüber-

gang wird durch die Montage eines Isolators am Segmentende hervorgerufen. Beide Varianten wurden

zunächst rechnerisch untersucht. Eine experimentelle Validierung ist im weiteren Verlauf des Projektes

vorgesehen. In zusätzlichen CFD-Untersuchungen wurden auch die Wärmeübergangskoeffizienten an

den übrigen Segmentflächen quantitativ bestimmt. Für deren Bestimmung in Berechnungen der indus-

triellen Praxis wird ein vereinfachtes Berechnungsverfahren entwickelt, welches in das Gleitlager-

berechnungsprogramm COMBROS R integriert werden soll.

Abstract

The research objective of the project Tilting Pad Journal Bearing Oil Supply Influence II is the investiga-

tion of constructive oil supply variants with regard to their influence on the heat transfer in the bearing

with the aim of reducing the maximum temperature in the bearing. For this purpose, two supply variants

were developed to prove an influence of the oil flow in the space between pads region on the maximum

temperature. To generate a high heat transfer coefficient, a new type of oil feed was developed which

flows directly to the trailing edge of the pad. This makes use of the lid driven cavity induced by the shaft

rotation. Low heat transfer is achieved by mounting an insulator at the trailing edge. Both variants were

initially investigated by simulation. An experimental investigation is planned in the further course of the

project. In additional CFD investigations, the heat transfer coefficients at the remaining pad surfaces

were also determined quantitatively. For their determination in calculations in industrial practice, a sim-

plified calculation method is being developed which is to be integrated into the journal bearing calculation

software COMBROS R.

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1 Introduction

The maximum bearing temperature is an upper limit of the permissible operating range of high-speed

bearings in turbomachinery. For this reason, the reliable prediction of the temperature distribution in

fast-running and highly loaded journal bearings is very important. It is further a basis for reliable predic-

tion of the other parameters of bearing operating behaviour such as the minimum lubrication gap height,

the maximum lubrication film pressure or the stiffness and damping behaviour. On the other hand, a

targeted reduction of the temperature level in the bearing by means of suitable design measures enables

an extension of the bearing operating range and thus an increase in the power density of a tur-

bomachine.

The aim of the project is an experimental proof of a practice-relevant influence of design-conditioned

modifications of the heat transfer coefficients at the trailing edge to influence the maximum temperature

in the bearing as well as an improved theoretical description of the heat transfer between a pad and its

environment.

For this purpose, design variants are developed based on an existing tilting pad journal bearing and the

findings of the fluid mechanics investigations in the previous project, that aim on maximized differences

in the heat transfer coefficients at this interface. Subsequently, the two most permissing variants identi-

fied by CFD analyses that induce highest maximum sensor temperatures, will be manufactured and

experimentally investigated. Within the scope of further theoretical investigations, heat transfer coeffi-

cients at the remaining free pad surfaces are determined.

To transfer the results to design process in industry, a simplified fluid mechanical model will be devel-

oped, to optain the effects identified as essential for characterising heat transfer. This analysis enables

to optain heat transfer coefficients for journal bearing calculation in moderate real time.

Based on measurement data, the bearing size influence on the pad-specific heat transfers will be inves-

tigated and the overall calculation method will be validated.

Finally, the transferability of the research results to axial plain bearings will be assessed. The results will

provide indications of design measures and extended calculation possibilities that can be used cost-

effectively by small- and medium-sized enterprises (SMEs) for new innovative solutions.

2 Designs studies

2.1 High heat transfer

The theoretical investigations of the previous project Tilting Pad Journal Bearing Oil Supply Influence I

[1] show that fluid flow in the space between pads region is dominated by shaft rotation similar to lid

driven cavity flow. While common oil supply variants partly counteract this flow, the new variant should

reinforce it by its inlet to achieve the highest possible flow velocity and thus, directing flow to the entire

trailing edge pad surface.

For the calculation of the heat transfer coefficients, a fully parameterized model of the test bearing

(4 pads, 120 mm shaft diameter) including oil supply is built up. The bearing is resolved completely

(360° circumferential) by utilizing the symmetry to the bearing center plane. The definition of the coordi-

nate system is shown in Fig. 1. In order to enable a local resolution of the thermal boundary layers with

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a moderate number of elements, a block-structured grid is used. It consists of about 36 million hexahe-

dra elements.

In the simulation, the three-dimensional Navier-Stokes equations for an incompressible fluid with tem-

perature-dependent viscosity are solved. The influence of turbulence is considered by a turbulence

model with transition (SST and γ-ϑ model). In addition, the heat conduction equation is solved for the

pads and the bearing liner. As operating boundary condition a specific bearing load of 2 MPa at a speed

of 12000 rpm is given. The fresh oil flow rate is 90 l/min at a supply temperature of 50 °C. Pad tilt angles

and shaft displacement are determined by means of Newton’s method in an outer loop so that the bear-

ing is in mechanical equilibrium for the defined boundary conditions. For this purpose, a new mesh is

generated after each loop pass.

Fig. 1: Definition of the coordinate system

For optimizing the oil supply with regard to the achievable heat transfer coefficient, design parameter

studies were carried out. The position of the inlet device relative to pad 𝑦𝑧, its axial length 𝐵𝑧 over which

the pad is flowed towards, and the width of the inlet ℎ𝑧, which are defined in Fig. 2, as well as the

supplied oil flow rate �̇�𝑠𝑢𝑝 were found to be the most important parameters.

Fig. 2: Design parameters of the developed oil supply device

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The influence of the inlet position on the maximum temperature and the heat transfer coefficients at the

trailing edge (TE) of pads one to four is shown in Fig. 3. For nominal operating conditions the discrete

results of the simulation are approximated by a regression. A higher heat transfer coefficient can be

achieved by shifting the feed position near to the shaft. This can be explained by the fact that the local

heat transfer coefficient decreases with running length and the temperature of the pad increases in the

direction of the shaft. Above a certain level of 𝑦𝑧, there is a strong decrease in heat transfer, as the

remaining inflow area becomes too small. In addition, the flow separates from the end of the pad in the

immediate vicinity of the shaft, which makes it more difficult for the pad to flow in this area, since the

supplied oil does not reach the pad but is deflected in the direction of the shaft. The separation can be

seen in Fig. 6. The optimum inlet position is approximately at 𝑦𝑧 = 8 𝑚𝑚, corresponding to a flow of half

the area of the trailing edge pad surface. However, the temperature reduction compared to a complete

flow towards the trailing edge is small.

Fig. 3: Influence of the position of the inlet on the heat transfer coefficient (n=12000 rpm, pq=2 MPa, Q=90 l/min)

206


A reduction in the width of the inlet causes an increase in the feed pressure and the speed of the oil

supplied, that increases the heat transfer coefficient, as shown in Fig. 4. A suitable compromise must

be found in the design with regards to the purchase and operating costs of the pump.

Fig. 4: Influence of the width of the inlet on the heat transfer coefficient (n=12000 rpm, pq=2 MPa, Q=90 l/min)

Fig. 5 shows that a reduction of the axial length of the inlet leads to a slight increase in the heat transfer

coefficient within the selected parameter range. Simultaneously a strong increase of the oil supply pres-

sure is present. The small increase can be explained by the fact that the flow velocity is increased, but

at the same time the crossflow area is reduced. As a design recommendation, it can be derived that the

fresh oil should flow across almost the entire axial width of the pad.

Fig. 5: Influence of the axial length of the inlet on the heat transfer coefficient (n=12000 rpm, pq=2 MPa, Q=90 l/min)

Based on the results of the parameter study, the inlet was optimised for a oil pressure of 1.4 bar at a oil

flow rate of 90 l/min, which roughly corresponds to the characteristics of the directed lubrication. At this

rate, a heat transfer coefficient of 4.8 kW/(m²K) is achieved at the trailing edge of the thermally highest

207


loaded pad. The temperature distribution in the bearing as well as the flow in the inlet region for the

optimized geometry is shown in Fig. 6. Fig. 7 shows the characteristic of the heat transfer coefficients

and the feed pressure for different oil flow rates. It can be seen that an increase of the oil flow rate leads

to an increase of the heat transfer coefficients at the same time. In the previous project Tilting Pad

Journal Bearing Oil Supply Influence I it could be shown that this behaviour cannot be observed con-

ventionally inlet designs such as directed lubrication by nozzles or spray bars directed to the journal.

Fig. 6: Optimized oil supply, temperature distribution and flow in the gaps

Fig. 7: Influence of the oil flow rate on the heat transfer coefficient

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2.2 Low heat transfer

A low heat transfer is generated by attaching an insulating layer at the trailing edge. An effective heat

transfer coefficient can be estimated in an analytical calculation by adding the area-related thermal re-

sistances. Using a 3 mm thick layer of polycarbonate with 𝑅𝑡ℎ′ = 0.015 𝑚²𝐾/𝑊 and 𝛼 =

1500 𝑊/(𝑚²𝐾), this coefficient is approximately:

𝛼𝑒𝑓𝑓 ≈1

1/𝛼+𝑅𝑡ℎ′ = 63.8

𝑊

𝑚2𝐾 (1)

This value is significantly below the level that could be achieved by a design related reduction of the

inflow velocity at the trailing edge. For better insulation, plastic screws are used instead of metal ones.

Fig. 8 shows a pad with the design implementation of this variant.

Fig. 8: Pad with attached insulating layer at the trailing edge

3 Study on heat transfer of the remaining pad free surfaces

Within the project, the heat transfer coefficients of the remaining pad free surfaces, i.e. the back and the

side faces, are also investigated. In a first step, the influence of the heat transfer coefficients of all four

pad surfaces on the maximum sensor temperature in the pad 𝑇𝑚𝑒𝑠𝑠,𝑚𝑎𝑥 was determined by means of a

parameter study in COMBROS R. For this purpose, one parameter was varied while the others were

kept constant. For 𝛼𝐿𝐸 and 𝛼𝑇𝐸 a value of 1500 W/(m²K) was assumed, for 𝛼𝑆𝐴 and 𝛼𝑆𝑅 a value of 500

W/(m²K) [2]. Fig. 9 shows the predicted maximum sensor temperature as a function of the particular

heat transfer coefficient.

209


Fig. 9: Influence of the heat transfer coefficients on the max. measuring point temperature (n=12000 rpm, pq=2.0 MPa, Q=90 l/min)

The heat transfer coefficient at the trailing edge αTE exhibits the highest influence on the maximum

sensor temperature, while the heat transfer coefficient at the leading edge free pad surface αLE has no

distinct influence. This confirms the results from the previous project. The second largest influence

shows the heat transfer coefficient at the back of the pad αSR. The heat transfer coefficient at the side

surfaces is of minor importance similar to the one at the beginning of the pad. This can be explained on

the one hand by the large area of the pad’s back and on the other hand by its relative proximity to the

measuring point compared to the side surfaces.

In the CFD investigations described above and the ones of the previous project, heat transfers at the

leading edge and the trailing edge pad free surface, as well as their influence on the temperature

distribution in the pad were investigated in detail. For a complete description of the thermal behavior of

a tilting pad journal bearing, additional knowledge of the heat transfers at the remaining pad surfaces,

i.e. the pad side surfaces and the pad backs, is required. Here, CFD calculations of bearings with spray

bar and directed lubrication were carried out. Both variants are shown in Fig. 10. The coordinate system

is defined according to Fig. 1. Calculations for different operating conditions and the results were

evaluated. With regard to these flow forms, the oil supply can be divided into three domains: the domain

of the inlet region between the trailing edge of the upstream pad and the leading edge of the downstream

pad, the domain between the pad side surfaces and the baffle, and the region between pad back and

liner. The oil flow in these domains and the heat transfer at the adjacent pad surfaces will be discussed

in more detail in the following sections.

210


Fig. 10: Geometry of the CFD models (left: directed lubrication, right: spray bar)

Side faces

The pads are radially fixed by axial pins in the baffle. The baffle also seals the inlet region. The test

bearing features a 3mm gap between pads and baffle. The flow in this region is visualized in Fig. 11 and

Fig. 12.

Fig. 11: Flow centred between side faces and pad retaining cover in the r-ϕ-plane at z=37,5mm

(directed lubrication, n=12000 rpm, Q=60 l/min, pq=2 MPa)

Fig. 12: Flow centred between side faces and pad retaining cover in the r-ϕ -plane at z=37,5mm

(directed lubrication at z=37,5mm (spray bar, n=12000 rpm, Q=60 l/min, pq=2 MPa)

Two significant flow components can be identified. On the one hand a flow in circumferential direction

exist that is induced by the rotation of the shaft and decreases in radial direction towards the liner. This

component is more distinctly present for directed lubrication variant than for the spray bar one, since the

spray bar fills almost the entire space between pad region and thus represents a throttling element.

1

2

3

4

1

2

3

4

u

u

211


Fig. 13: Flow between side surfaces and pad retaining cover in the r- z-plane at ϕ=235°

(left: directed lubrication, right spray bar, n=12000 rpm, Q=60 l/min, pq=2 MPa)

The second flow component is directed axially and discharges the hot oil from the bearing particulary.

Both flows have a stochastic character and are overlaid by numerous vortices. In summary, a complex,

three-dimensional flow state exist.

The local heat transfer coefficient on the pad side surfaces is shown in Fig. 14 and Fig. 15 for directed

and spray bar lubrication. There is a strong local dependence of the heat transfer coefficient. It also

tends to be higher near the shaft and decreases in radial direction. However, no direct correlation to the

flow velocity distribution from Fig. 11 and Fig. 12 can be identified. Also a direct relationship between

the heat transfer coefficient and the wall shear stress, as the boundary layer theory suggests, could not

be found here.

Fig. 14: Heat transfer coefficient of the side surfaces (directed lubrication n=12000 rpm, pq=2 MPa, Q=60 l/min)

Fig. 15: Heat transfer coefficient of the side surfaces (spray bar n=12000 rpm, pq=2 MPa, Q=60 l/min)

Rear sides

The test bearing features a line contact between pad back and liner. The pad backs have an outer radius

of 68 mm, the liner an inner radius of 80 mm. If elastic deformations and surface roughness are ne-

glected, both surfaces touch each other in the line contact. However, this idealized contact cannot be

modeled in a flow simulation, as this would lead to invalid element skewness. A detailed resolution of

the contact would be very complex in the modeling procedure and would result in high computational

times. therefore, the contact region was modelled by keeping a minimum distance between pad back

and liner, that is in the order of magnitude of the surface roughness. This enables a complete meshing

of the pad back and describes at least in a first approximation the heat transfer in the contact area.

u

u

212


Fig. 16: Velocity between pad and bearing ring (directed lubrication n=12000 rpm, pq=2 MPa, Q=60 l/min)

Fig. 17: Velocity between pad and bearing ring (spray bar n=12000 rpm, pq=2 MPa, Q=60 l/min)

Fig. 16 and Fig. 17 show the speed field between pad back and bearing ring for the spray bar and

directed lubrication feed variants. Starting from the beginning of the pad, the flow always is directed from

the bearing center to the outside (positive z-direction). In the contact region the speed becomes zero. It

is lower than at the leading edge due to the smaller distance between back of the pad and the liner.

There is no uniform tendency with regards to the flow direction at the trailing edge of the pad.

Fig. 18 shows the axial averaged heat transfer coefficient as a function of the pad angle for spray bar

lubcrication. The reference temperature for the evaluation of heat transfer coefficients was once the

feed temperature (red line) and once the local temperature of the bearing ring (blue line). An analytical

solution of the heat conduction equation is also shown. A better agreement between numerical and

analytical solution is achieved considering the local temperature difference in the contact area. In the

range between 30° and 53° heat transfer is determined by heat conduction.

Fig. 18: Dependence of the heat transfer coefficient at the back side on the angle

Outside this range, convection must be taken into account. The heat transfer coefficient increases in

this area towards both edges of the pad. In the range from 39° to 44° the heat transfer coefficient shows

a maximum value which can be identified with the reciprocal value of the thermal contact resistance.

For an exact determination of the contact resistance, a detailed investigation of the contact, considering

the surface roughness, would be necessary.

u

u

213


4 Development of a simplified model for the computation of HTCs

In the previously presented investigations, heat transfers at the pad surfaces were determined on the

basis of a detailed CFD model. For the calculation of tilting pad journal bearings in industrial practice, a

simplified model has to be developed that predicts heat transfer coefficients a priori.

For this purpose, the three-dimensional Navier-Stokes equations are solved numerically and a velocity

field is determined first. The calculation of the velocity field is carried out isothermally. The viscosity of

the lubricant is assumed to correspond to the fresh oil temperature. In the next step the energy equation

is solved using the previously determined velocity field. The calculation of the temperature is dimension-

less. The temperature on the pads is assumed to be constant. This allows the heat transfer coefficients

to be calculated in advance without knowing the actual temperature distribution. Likewise, the energy

equation does not have to be solved over the entire fluid domain, it is sufficient to solve the solution in

the area of the thermal boundary layer of the pad surfaces.

Momentum equation

The three components of the velocity field shall be determined by solving the momentum equation. The

numerical solution is conductedwith the finite volume method. The formulation in integral form is used:

∰𝜕(𝜌�⃗� )

𝜕𝑡𝑑𝑉 + ∯(𝜌�⃗� �⃗� 𝑇) ⋅ �⃗� 𝑑𝐴 = ∯ 𝝈 ⋅ �⃗� 𝑑𝐴 + ∰ 𝜌�⃗� 𝑑𝑉 (1)

The general momentum equation can be simplified for the application under consideration. The flow can

be assumed as stationary and incompressible, no volume forces occur:

𝜌 ∯(�⃗� �⃗� 𝑇) ⋅ �⃗� 𝑑𝐴 − ∯𝝈 ⋅ �⃗� 𝑑𝐴 = 0 (2)

The non-representational form stress tensor in the incompressible case is given by:

𝜎𝑖𝑗 = 𝜇 (𝜕𝑢𝑖

𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖

) − 𝑝𝛿𝑖𝑗 (3)

For the calculation of journal bearings, the representation in cylindrical coordinates is suitable. In this

case the stress tensor is:

𝝈 =

[ 2𝜇

𝜕𝑢

𝜕𝑟− 𝑝 𝜇 (

𝜕𝑣

𝜕𝑟+

1

𝑟

𝜕𝑢

𝜕𝜑) 𝜇 (

𝜕𝑤

𝜕𝑟+

𝜕𝑢

𝜕𝑧)

𝜇 (𝜕𝑣

𝜕𝑟+

1

𝑟

𝜕𝑢

𝜕𝜑)

2𝜇

𝑟

𝜕𝑣

𝜕𝜑− 𝑝 𝜇 (

1

𝑟

𝜕𝑤

𝜕𝜑+

𝜕𝑣

𝜕𝑧)

𝜇 (𝜕𝑤

𝜕𝑟+

𝜕𝑢

𝜕𝑧) 𝜇 (

1

𝑟

𝜕𝑤

𝜕𝜑+

𝜕𝑣

𝜕𝑧) 2𝜇

𝜕𝑤

𝜕𝑧− 𝑝

]

(4)

214


Pressure equation

Another equation is necessary to determine the pressure. By introducing the continuity equation into the

momentum equation, the Poisson equation for the pressure can be obtained [4]. Here Δ denotes the

Laplace operator.

Δ𝑝 = −𝜌∇ ⋅ [∇(�⃗� �⃗� 𝑇 − 𝝈) − �⃗� +𝜕𝜌�⃗�

𝜕𝑡] (5)

By assuming a stationary, incompressible flow, this equation is simplified as follows:

Δ𝑝 = −𝜌Δ(�⃗� �⃗� 𝑇) (6)

Energy equation

The energy equation in its general form is:

𝜕𝜌𝐸

𝜕𝑡+ ∇ ⋅ (𝜌�⃗� 𝐻) = 𝜌 �⃗� ⋅ �⃗� + ∇ ⋅ (𝜎 ⋅ �⃗� − 𝜆∇𝑇) (7)

Considering the previous assumptions (stationary, incompressible, no volume forces), the energy equa-

tion can be simplified. Furthermore, the dissipation in the inlet region can be neglected. Thus the energy

equation represents a linear partial differential equation of 2nd order.

𝜌𝑐𝑝∇ ⋅ (�⃗� 𝑇) + 𝜆Δ𝑇 = 0 (8)

In cylindrical coordinates the equation is:

𝜌𝑐𝑝 [𝑢𝑟

𝜕𝑇

𝜕𝑟+ 𝑇

𝜕𝑢𝑟

𝜕𝑟+

𝑢𝑟𝑇

𝑟+

𝑢𝜑

𝑟

𝜕𝑇

𝜕𝜑+

𝑇

𝑟

𝜕𝑢𝜑

𝜕𝜑+ 𝑢𝑧

𝜕𝑇

𝜕𝑧+ 𝑇

𝜕𝑢𝑧

𝜕𝑧] + 𝜆 (

1

𝑟

𝜕𝑇

𝜕𝑟+

𝜕2𝑇

𝜕𝑟2+

1

𝑟2

𝜕2𝑇

𝜕𝜑2+

𝜕2𝑇

𝜕𝑧2) = 0

(9)

The numerical solution can be carried out using the finite difference method.

215


Bibliography

[1] Radialkippsegmentlager-Ölzuführungseinfluss, Abschlussbericht, IGF Nr. 17373

[2] Dokumentation Radialgleitlagerberechnungsprogramm COMBROS R,

Dr.-Ing. Thomas Hagemann

[3] ANSYS CFX-Solver Theory Guide, ANSYS, Inc.

[4] Numerische Strömungsmechanik, J.H. Ferziger, M. Peric

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