Direct and inverse heat transfer in non-contacting face seals

International Journal of Heat and Mass Transfer 90 (2015) 710–718

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Direct and inverse heat transfer in non-contacting face seals Slawomir Blasiak a,⇑, Anna Pawinska b a Kielce University of Technology, Faculty of Mechatronics and Machine Building, Division of Mechanical Engineering and Metrology, Aleja Tysiaclecia Panstwa Polskiego 7, 25-314 Kielce, Poland b Kielce University of Technology, Faculty of Management and Computer Modelling, Chair of Applied Informatics and Mathematics, Aleja Tysiaclecia Panstwa Polskiego 7, 25-314 Kielce, Poland

a r t i c l e

i n f o

Article history: Received 30 April 2015 Received in revised form 1 July 2015 Accepted 1 July 2015

Keywords: Mechanical seal Non-contacting face seal Direct heat transfer Inverse heat transfer Trefftz functions Bessel function Fourier–Bessel series

a b s t r a c t The subject of this paper was to develop a mathematical model of a non-contacting face seal describing the phenomenon of the heat transfer in the system: sealing rings – fluid film. The function of non-contacting face seals used in rotor machines is to separate the working agent from the external environment. The nature of operation of non-contacting face seals allows the fluid to leak through the clearance not bigger than a few micrometres. During the operation of the rotor machine between the co-operating rings, an intense conversion of mechanical energy into heat occurs. At first, the heat flux generated in the fluid film is channelled to the sealing rings and then to the surrounding fluid. The solution of the presented model was conducted with the use of analytical methods for the direct and the inverse heat transfer problem. The distribution of the temperature fields in the sealing rings for the direct heat transfer problem was determined with the use of Fourier–Bessel series as the surface function of two variables ðr; hÞ for the cross-section of a ring. The inverse heat transfer problem was solved with the use of Trefftz functions. The presented computational methods allow a more detailed identification of the phenomenon of the heat transfer in non-contacting face seals and indicate a direction of further research and preparation of new computational methods. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Determining the temperature distribution in the lubricating fluid and in the structural elements (seal rings) is of great practical importance due to the limitation of failures caused by significant increase of temperature and evaporation of agent as well as the thermo-elastic deformities which are the consequence of the occurrence of large temperature gradients. Formulating models which allow possibly the most precise reproduction of the occurring physical phenomena, i.e. heat transfer, thermal deformations and changes of physiochemical features of the fluid passing through the radial clearance is extremely significant already at the stage of designing the devices in which non-contacting face seals are applied. The issue of the heat transfer in non-contacting mechanical seals was the subject of a number of academic papers whose results have been published in recent years. For instance, Dumbrava and Morariu [1] presented the thermohydrodynamic ⇑ Corresponding author. E-mail addresses: [email protected] (S. Blasiak), [email protected] (A. Pawinska). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.004 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

(THD) analysis for the mechanical face seal in which they considered the changes of the physical features of the agent and the heat conductivity to the rings limiting the clearance, as well as the heat transfer as convection with the fluid surrounding the rings. Lebeck [2] in his book summarized various heat transfer mechanisms in mechanical seals. Pascovici and Etsion [3,4] presented the method of determining the heat flow in the area of sealing rings and the THD analysis for a double seal in the ‘‘face to face’’ configuration. As a result of an analytical solution, they obtained a radial temperature distribution in the fluid film and in the rotating ring (rotor) assuming that the surface of the second ring (stator) is insulated and all the heat generated in the film is channelled through the rotor to the surrounding fluid. Subsequent thermohydrodynamic (THD)} [5–7] and thermoelastohydrodynamic (TEHD) [8] models of a non-contacting face seal can be found for instance in the paper of Tournerie et al. [9] where basic equations of THD lubrication and the equations of heat conductivity were presented. In this paper the problem of the heat transfer in a non-contacting face seal was solved with the use of two methods. The first analytical method consisted in separating the variables in a partial differential equation which described the heat transfer in a cylindrical system. As a consequence, ordinary differential

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711

Nomenclature C 1 ; C 2 ; C 3 ; C 4 constants csn unknown coefficient of the linear combination h clearance geometry in the direction of the radial coordinate r ho nominal clearance height hðrÞ function of clearance height J 0 ðs rÞ; Y 0 ðs rÞ Bessel functions of the first and second kinds, respectively n direction normal to the surface ri ; ro inner, outer  1 radius, respectively sn constant m T absolute temperature Tf fluid temperature in the ðr; zÞ coordinates Tm average temperature of the fluid in the clearance To temperature of the surrounding fluid (generally assumed to be a constant)

equations were obtained whose solutions are presented in the form of Bessel series. Such results may be obtained if relevant boundary conditions are known. In the case when the equation governing the process and the boundary conditions are known it is a direct problem. The inverse problem may be classified into the following categories:     

a boundary inverse problem, a coefficient problem, an identification of sources, a geometric inverse problem, an identification of an initial condition.

In the paper the boundary inverse heat conduction problem was solved. In this problem one of the boundary conditions is not known. Such a situation occurs for example when on one boundary of the area, the temperature measurement is not possible and thus, it is impossible to determine the heat flux. That is why in this paper another method of specifying the temperature of the field was suggested which allows both direct and inverse heat transfer to be solved. The method is based on Trefftz functions for the differential equation under consideration. The method is presently quite well recognised for a wide range of linear partial differential equations in various systems of coordinates. For a given differential equation, a complete set of functions, which strictly fulfil that equation (Trefftz functions) is determined and the solution is approximated with the linear combination of Trefftz functions. The coefficients of the linear combination are determined in such a way so that the boundary conditions would be fulfilled in the best way (usually in the method of least squares). Trefftz method was first described in the paper of [10]. Then, many authors developed that method. Herrera, Jirousek, Kupradze, Leon, Sabina, Zielin´ski and Zienkiewicz should be mentioned herein [11–15]. In the papers mentioned above, stationary problems or problems brought to be stationary through time discretisation are considered. Rosenbloom and Widder in the paper of [16] obtained Trefftz functions for 1D non-stationary heat conduction equation, in which ‘time’ occurred as one of the variables. A number of papers have been published in this trend. In the papers of [17–21] Trefftz functions were applied to the solutions of problems of linear direct and inverse heat conductivity in various systems of coordinates. The nonlinear issue of heat conductivity was solved in the paper of [22]. Trefftz functions for the wave equation were presented in the papers of [23–26]. Direct and inverse problems of theroelasticity were solved with the use of Trefftz functions in the papers of

Vn Greek

Trefftz functions satisfying Laplace equation

a

convection coefficient b taper angle kf fluid thermal conductivity ki  ks ; kr thermal conductivity for the stator and the rotor, respectively lo dynamic viscosity at T o m/ distribution of the flow velocity of the fluid in the clearance x angular velocity hðr; zÞ ¼ T  T o change in the temperature in the ðr; zÞ coordinates hi  hs ; hr changes in the temperature of the stator and the rotor, respectively

[27–29]. Trefftz functions related to the problem of beam vibrations were described in [30,31], while to plate vibrations in the papers of [31,32]. Several monographs were written on Trefftz functions which among others include [20,33–35]. Experiences of authors of the above mentioned papers show high effectiveness of Trefftz method in solving the boundary inverse problems described by differential equations. The use of Trefftz functions in this paper and the results obtained with that method allow to determine the temperature distributions in the sealing rings. The results have been compared with the results obtained on the basis of the solution of the direct heat transfer under specified geometrical and operating parameters of a non-contacting face seal (type FMR).

2. Seal model The main structural concept related to non-contacting face seals is to maintain a clearance separating the co-operating rings at the level of a few micrometres during operation. The nominal height of the clearance hi results directly from the balance of forces affecting the sealing rings, mainly the pressure force coming from the elastic element and the hydrostatic force which depends on the distribution of the pressure generated in the clearance between the parallel rings. An overall scheme of a non-contacting face seal was shown in Fig. 1. This type of seal consists (as already mentioned) of two cooperating rings – one of them, the stator (1) which is rigidly mounted in the housing, while the rotor (2) rotates together with the shaft (6) of the rotor machine and is pressed to the stator with a spring (3).

Fig. 1. The scheme of a non-contacting seal. 1 – stator, 2 – rotor, 3 – spring, 4 – housing, 5 – O-ring, 6 – shaft, 7 – locator.

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In the case under consideration, it was assumed that high pressure is on the process side, i.e. on the external radius ro of the system of rings, while the pressure on the internal radius ri is practically equal to the atmospheric pressure. The mathematical model and the solution of the direct problem were presented and described in detail in the paper of [36,37], that is why the main assumptions related to the boundary conditions and the final dependencies describing the temperature distribution in sealing rings were quoted here. 3. Steady-state temperature The heat flux generated in the fluid film is transferred to the surroundings, first by conduction through the seal rings and then by convection to the surrounding fluid on the process side. The distribution of temperature in the seal rings can be written in a general form using the heat conduction equation:

1 @h @ 2 h @ 2 h þ þ ¼ 0: r @r @r2 @z2

ð1Þ

The same heat conduction equation will be used to describe the distributions of temperature in the stator and the rotor. It will be necessary to use upper indices, e.g., hs and hr , respectively. Fig. 2 shows the model of the heat transfer for a non-contacting face seal with the boundary conditions marked therein, which were adopted for calculations of the direct heat transfer. It was adopted that the surfaces of the stator and the rotor on the internal radius r i (Fig. 2) and the rear surfaces of rings that do not have a direct contact with the surrounding fluid are com@h ¼ 0, where n pletely insulated. It was described in the form of @n – is a normal direction to the surface. On the face surfaces of the rings (the stator and the rotor) limiting the radial clearance, which are in a direct contact with the layer of the medium, the condition constituting the case of the heat transfer through conductivity is fulfilled. The values of heat fluxes on the surface of the element under consideration and the medium are as follows: on the surface of the stator:

qsv ðrÞ ¼ ks

@hs @h f ¼ kf @z @z

and hs ¼ h f ;

ð2Þ

on the surface of the rotor:

@h f @hr kf ¼ kr ¼ qrv ðrÞ and h f ¼ hr : @z @z

between the fluid and the motionless surface of the ring. Hence, practically the whole heat flux is channelled through the second ring, namely the rotor. On the cylindrical surface of the stator (Fig. 2) which remains in contact with the surrounding medium, free convection takes place, while in the rotor case forced convection occurs. It is generally written as:

ki

 @hi   @r 

  ¼ ai hi  r¼r o

r¼r o

ð4Þ

:

In the literature regarding the heat transfer in non-contacting face seals, many models describing the heat transfer with the surrounding fluid with the use of forced convection are presented. The authors have chosen one of the most frequently quoted mathematical descriptions of that phenomenon that was used by Lebeck [2]. The convective heat transfer coefficient (convection coefficient) a is calculated with the use of Becker correlation: 1=3 ar ¼ 0:133 Re2=3 D Pr

kf ; D

where D denotes the outer diameter of the seal and ReD is the Reynolds number based on this diameter. The parameters k f and Pr are the thermal conductivity and the Prandtl number of the fluid, respectively. 2

The Reynolds and Prandtl numbers are described as: ReD ¼ x Dl and Pr ¼

Cp l , kf

It means that the maximum temperature occurs on the border of the fluid film and the stator, where there is the biggest friction

q

where x is the angular velocity and q l Cp are the den-

sity, dynamic viscosity and specific heat of the fluid, respectively. 4. Analytical solution of direct problem The mathematical model formulated in such a way was solved analytically. In the first step of the analysed problem, the temperature distributions in sealing rings were determined by specifying the general form of functions fulfilling Laplace’s equation (1), both for the stationary ring (stator) and the rotor, with specific boundary conditions taken into consideration (2)–(4). The method of separating variables was used by introducing two harmonic functions RðrÞ and ZðzÞ each of them was the function of only one variable and the temperature distribution to be found hðr; zÞ, was presented as:

hðr; zÞ ¼ RðrÞ  ZðzÞ: ð3Þ

ð5Þ

ð6Þ

After substituting those functions to the differential equation (1) one received:

1 R00 ðrÞZðzÞ þ R0 ðrÞZðzÞ þ RðrÞZ 00 ðzÞ ¼ 0: r After ordering: 00

R ðrÞ 1 R0 ðrÞ Z 00 ðzÞ þ ¼ ¼ s2 : RðrÞ r RðrÞ ZðzÞ

ð7Þ

The equation mentioned above may be fulfilled for any two variables ðr; zÞ only if both sides of the equation have the same value s2 . Comparing individual components of the equation with the constant s2 , the conductivity equation (1) may be brought down to two ordinary differential equations:

1 2 2 R00 ðrÞ þ R0 ðrÞ þ k RðrÞ ¼ 0 and Z 00 ðzÞ  k ZðzÞ ¼ 0: r With general solutions: RðrÞ ¼ C 1 J 0 ðsn  rÞ þ C 2 Y 0 ðsn  rÞ and ZðzÞ ¼ C 3 coshðsn  zÞ þ C 4 sinhðsn  zÞ, whose product amounts to: hðr; zÞ ¼

1 X ðC 1  J 0 ðsn  rÞ þ C 2  Y 0 ðsn  rÞÞ  ðC 3  coshðsn  zÞ þ C 4  sinhðsn  zÞÞ: n¼1

Fig. 2. Conditions of the heat transfer in a non-contacting face seal.

ð8Þ

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Taking the boundary conditions (2)–(4) into consideration, the values of the constants of integration for the rotor were determined. A similar course of procedure was conducted in order to determine the constants of integration for the second ring – the stator. Having conducted the algebraic calculations, the equation describing the temperature distribution in the rotor and the stator were written as: for the rotor

The complete solution of the model required also solving the energy equation which described the temperature changes in the clearance and determining its average value, what was described in detail in the paper of [37]. The presented mathematical model of a non-contacting face seal allows to determine the temperature distributions both in the sealing rings and in the fluid film separating them. The dependence describing the stream of the heat generated in the fluid film is presented in a simplified form with the energy equation

(

!   1 X   J sr  r i      Y 0 srn  r  cosh srn  z Tr ¼ T0 þ Brn J 0 srn  r  1  nr Y 1 sn  r i n¼1

ð9Þ

qC p

@T f v / @T f @T f vr þ þ vz @r r @h @z

!   1 X  s  J 1 ssn  r i  s    s  Y 0 sn  r  cosh ssn  z T ¼ T0 þ B n J 0 sn  r   s Y 1 sn  r i n¼1 ð10Þ By introducing the boundary condition for free or forced convection (4), when the sealing ring is surrounded by the medium with the temperature T 0 on the external radius r o , the equations for i meaning i ¼ r; s were determined:

        J si  r i   Y 1 sin  r o þ ai  J 0 sin  r o  sin  ki  J 1 sin  ro þ sin  ki  1  ni Y 1 sn  r i i    J s  r i 1  Y 0 sin  r o ¼ 0:  ai   ni ð11Þ Y 1 sn  r i The solution for the above equations allows the subsequent values of the coefficient sin (where n ¼ 1; 2; 3; . . .), which are the elements of Eq. (11) both for the rotor and the stator, to be determined. In order finally establish the function describing the temperature distribution in the rotating ring, one more constant value Bn needs to be determined. By applying the dependence (3), which constitutes the last of the boundary conditions, it can be written:

n¼1

!   r  J 1 srn  ri r   J 0 sn  r  Y 0 sn  r Y 1 srn  r i 

  qr ðrÞ  srn  sinh srn  ðLr Þ ¼  v f : k

ð12Þ

By multiplying both sides of the equation by the expression   r  J1 ðsrn ri Þ   and integrating both sides of the J 0 sn  r  Y sr r Y 0 srn  r 1ð n iÞ equation in relation to the radial coordinate r 2 hr i ; r o i, it can be written: 

   Y 0 srn  r dr ð Þ Brn ¼  :     J1 ðsrn ri Þ   2   R ro r r r r r r  sn  J 0 sn  r  Y sr r Y 0 sn  r sinh sn  ðL Þ dr ri 1ð n iÞ R ro ri

qr kr

    r J 0 srn  r 

@2T f ; @z2

which for the generally adopted simplifying conditions for non-contacting face seals was written as [3]:

s



( 2  2 ) @v r @v / ¼l þk þ @z @z 

for the stator

1 X Brn 

)

J 1 ðsrn r i Þ

 2 @ m/ @2T f þ kf ¼ 0: @z @z2

l

ð15Þ

Eq. (15) describes the linear change of the temperature along the height of the clearance, while some individual components constitute respectively: the kinetic energy dissipation in the fluid film and the thermal energy conduction in accordance with Fourier’s law. In the case under consideration, similarly as in the paper of Pascovici and Etsiona [3], the linear distribution of speed was assumed m/ (Couette flow) ranging from zero on the surface of the stationary ring (stator) to the value of speed amounting to ðx rÞ on the surface of the rotating ring (rotor), what was written as:

@ m/ xr ¼ : h @z

ð16Þ

The dependence (16) describes the changes of the fluid speed along the height of the radial clearance which was presented as:

h ¼ hðrÞ ¼ ho þ ðr  r i Þ  b:

ð17Þ

By integrating twice Eq. (15) and using the boundary conditions (2) and (3) the temperature distribution in the fluid film was determined – similarly as in the paper of [3]: 1 X 1 l x2  r 2 2  f  ðh  z2 Þ þ Brn 2 2 k h n¼1 " # r      J s r i 1 n  coshðsrn zÞ J 0 srn r   r  Y 0 srn r : Y 1 sn r i

T f ¼ T0 þ

ð18Þ

The above dependences (9), (10) and (18) represent the solution of the mathematical model for the direct heat transfer problem and describe the temperature distributions in the system of the rings and the fluid film.

Y 1 srn r i

5. Solving the inverse problem with the use of the Trefftz functions

ð13Þ x2 r 2

P1



For the stator, when hs ¼ h f and h f ¼ l2 k f þ n¼1 Brn cosh srn z     J ðsr r Þ   J 0 srn r  Y1 snr ri Y 0 srn r (see Eq. (18)), the constant Bsn is: ð Þ 1 n i

 s  Y s  r dr ri Y 1 ðssn r i Þ 0 n Bsn ¼  :  2   J ss r     R ro s  r  1 ð n i Þ Y ss  r s  Ls dr r  J s cosh s 0 n n ri Y 1 ðssn ri Þ 0 n R ro

   h f  r J 0 ssn  r 



J 1 ðssn r i Þ

ð14Þ

The use of Trefftz functions allows formulating a simpler model of the heat transfer for the system: stator-film-rotor presented in Fig. 2 – then the one presented in chapter 2. Similarly as assumed earlier that Laplace’s equation is the equation governing the process for the stator and the rotor what was described with the dependence (1). The temperature distribution in the stator was determined at first. Temperature measurement at the point of contact of the stator with the fluid film is not possible. That is why in the further considerations we assume that the boundary condition for the stator at the boundary z ¼ 0 is not known. In exchange for

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that, the temperatures that are at a certain distance from that boundary (internal temperature responses) are known and were determined e.g. on the basis of the temperature measurement at a specific distance from the surface which was in contact with the fluid film in the radial clearance, what is shown in Fig. 3. In this case, the temperature distribution in the stator was to be determined without knowing all boundary conditions. It is an example of the boundary inverse problem where one of the conditions was replaced with the internal temperature responses known for z ¼ 0:002 ðmÞ, and the remaining known boundary conditions of the stator were adopted as in Fig. 2:

@T s ðr; Ls Þ ¼ 0; @z

ð19Þ

@T s ðri ; zÞ ¼ 0; @r

ð20Þ

ks

@T s ðr o ; zÞ ¼ as ðT s ðr o ; zÞ  T 0 Þ; @r

T s ðr j ; z ¼ 0:002Þ ¼ T sj :

ð21Þ

Fig. 4. A boundary conditions schema.

ð22Þ

In order to obtain the coefficients of the linear combination (24), the functional taking the conditions under consideration (19)–(22) is minimized:

The temperatures (22) upon testing Trefftz methods were simulated from the accurate solution presented with the formula:

" #   1 X  s  J 1 ssn r i  s  s s s T ¼ T0 þ Bn coshðsn zÞ J 0 sn r   s  Y 0 sn r : Y 1 sn r i n¼1

ð23Þ

A schema of boundary conditions that are determined in (19)–(22) was presented on Fig. 4. A boundary conditions schema. In Trefftz method, as the approximation of the exact solution is adopted:

T s ðr; zÞ  T s ðr; zÞ ¼

N X

csn V n

ð24Þ

n¼1

Trefftz functions V n for Laplace equation in cylindrical coordinates were presented in [18]:

V 0 ðr; zÞ ¼ 1; V 1 ðr; zÞ ¼ z  V 0 ðr; zÞ ¼ z; V ðkþ1Þ ðr; zÞ ¼

ðð2k þ 1Þ  z  V k ðr; zÞ  ðr2 þ z2 Þ  V ðk1Þ ðr; zÞÞ 2

ðk þ 1Þ

for

k ¼ 1; 2; . . . : For example:

V 0 ¼ 1; V4 ¼

V 1 ¼ z;

z2 r 2 z3 zr 2  ; V3 ¼  ; 2 4 6 4 5 3 2 4 z z r zr V5 ¼  þ ;...: 120 24 64

V2 ¼

z4 z2 r 2 r4  þ ; 24 8 64

I¼

 s 2 2 Z 0  s @ T ðr; Ls Þ @ T ðri ; zÞ dr þ dz @z @r Ls ri  Z 0    2 @ T s ðr o ; zÞ þ ks  as T s ðr o ; zÞ  T 0 dz @r Ls 11

2 X T s ðr j ; 0:002Þ  T sj : þ

Z

ro

ð25Þ

j¼1

The condition necessary to minimize the functional (25) is:

@I @I @I ¼ ¼    ¼ s ¼ 0: @cs1 @cs2 @cN

ð26Þ

The solution of the linear system of Eq. (26) leads to specifying the coefficients of the linear combination (24). Approximation (24) allows to specify the temperature distribution for the whole stator. In particular, the temperature distribution is obtained for z ¼ 0 (solution of the boundary inverse problem). 6. Results and discussion The parameters used for doing the calculations for the non-contacting face seal are presented in Table 1. Water at the temperature T o of 20 °C was adopted as a working medium. The mechanical seal with the rings with the external radius of 45 mm and the internal radius of 40 mm and the thickness of 10 mm for both cooperating rings was considered. It was also assumed that the rotor make a rotary movement with angular velocity of 1500 rad/s, and the height of the radial clearance amounts to

Table 1 Geometry- and performance-related parameters.

Fig. 3. The diagram of a non-contacting face seal with thermocouples location.

Inner radius ri Outer radius ro

0.040 (m) 0.045 (m)

Angular velocity x Dynamic viscosity lo

Thickness of the rings Ls and Lr Fluid Thermal conductivity kf Taper angle b

0.01 (m)

Fluid temperature To

0.65 (W/ m K) 1  104 (rad) 2  106 (m)

Convection coefficient (water) a Rotor Thermal conductivity kr Stator Thermal conductivity ks

Nominal clearance height ho

1500 (rad/s) 1  103 (Pa s) 20 (°C) 18 000 (W/ m2 K) 130 (W/ m K) 16 (W/m K)

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2 lm. Moreover, the cooperating face surfaces of the working rings are not parallel as the surface of one of the rings is situated at the angle of b ¼ 1  104 ðradÞ in relation to the counter-ring. Materials for rings in non-contacting face seals are selected in such a way so that one of the rings was made of a material with high rigidity, e.g. silicon carbide, tungsten carbide, stainless steel, whereas the counter-ring of a soft material, e.g. impregnated graphite. In the conducted calculations, two materials were combined assuming that the rotating ring was made of SiC (Silicon carbide), and the stator of the resin-impregnated carbon. Resin-impregnated carbon is a material that is perfect for sealing rings due to its high resistance to the influence of high pressure fluids. Moreover, the material is dimensionally stable which means that the faces surfaces of the sealing rings can be processed in order to obtain very low surface roughness. Resin-impregnated carbon is highly more chemically resistant than almost all chemical substances, except for very strong ones, such as oxidising acids and alkalis. Due to those features, this material is used for sealing rings in non-contacting face seals which are applied in many branches of the industry. Its application in devices operating in the conditions of a broad range of operating temperatures or in extremely low and high temperatures, in the chemical, petrochemical, food, pharmaceutical and cosmetic industries, in automotive applications and in nuclear technology should be noted as well. Fig. 5 shows the determined temperature distribution on the boundary z ¼ 0, i.e. the missing boundary condition for the stator as the approximation with the respective use of: 40, 44, 51 and 62 of Trefftz functions [38]. The results were compared with the temperature distribution that resulted from the accurate solution.

(a)

50

40

30

Exect Trefftz functions

0.040

ho ¼ 2  106 ðmÞ, it was assumed that the temperature distribution on the boundary z ¼ 0 for the stator is the same as the temperature distribution on the boundary z ¼ ho for the rotor. It results from the fact that the balance of the heat flux in fluid film qrv jh¼ho þ ðqsv jh¼0 Þ ¼ q f must be fulfilled. In order to determine the temperature distribution, the following boundary conditions have been taken into account:

@T r ðr; ho þ Lr Þ ¼ 0; @z

ð27Þ

@T r ðr i ; zÞ ¼ 0; @r

ð28Þ

Stator temperature Ts (oC)

Stator temperature Ts (oC)

60

Comparing the obtained results, it can be stated that it is possible to achieve a good approximation of the boundary conditions for z ¼ 0 with the suitable number of Trefftz functions. In this place a somewhat specific analogy between the presented computational method and the finite element method comes to mind. In the latter, considering the way in which we assess the approximation error, there is a possibility of increasing the precision of the method by decreasing the sizes of finite elements and increasing the approximation degree. As already mentioned, with the use of Trefftz method increasing the approximation degree has a similar effect. Both in the first and in the second method, it is of course connected with a greater demand for computational output. In the further part of the computational process, the obtained temperature distribution in the stator for z ¼ 0 was used for determining the temperature of the rotor. Because the fluid film between the stator and the rotor has a set height

0.042

0.044

60

50

40

30

0.046

(b)

Exect Trefftz functions

0.040

0.042

60

(c)

50

40

30

Exect Trefftz functions

0.040

0.042

r (m)

0.044

0.046

0.044

0.046

r (m) Stator temperature Ts (oC)

Stator temperature Ts (oC)

r (m)

0.044

0.046

60

(d)

50

40

30

Exect Trefftz functions

0.040

0.042

r (m)

Fig. 5. The temperature distribution on the surface of the stator for (a) 40, (b) 44, (c) 51 and (d) 62 of Trefftz functions.

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@T r ðr o ; zÞ ¼ ar ðT r ðr o ; zÞ  T 0 Þ; @r

T r ðr; ho Þ ¼ T s ðr; 0Þ:

ð29Þ

the approximation (24) and (31) with the use of N ¼ 70 of Trefftz functions.

ð30Þ

7. Analysis of errors

ð31Þ

Figs. 4 and 5 show a good approximation of the solution with the linear combination of Trefftz functions. In order to verify the quality of the approximated solution, an average relative approximation error was determined in a hundred points that ware distributed evenly in the rotor in accordance with the formula:

Approximation of the solution for the rotor was written as:

T r ðr; zÞ  T r ðr; zÞ ¼

N X

crn V n :

n¼1

In order to obtain the coefficients of the linear combination (31), the functional taking the conditions under consideration (27)–(30) is minimized:

I¼

 r 2 2 Z ho þLr   r @ T ðr; ho þ Lr Þ @ T ðr i ; zÞ dr þ dz @z @r ri ho 2 Z ho þLr  @ T r ðr o ; zÞ þ kr  ar ðT r ðr o ; zÞ  T 0 Þ dz @r ho Z ro  r 2 T ðr; ho Þ  T s ðr; 0Þ dr: þ

Z

ro

ð34Þ

ð32Þ

ri

The condition necessary to minimize the functional (32) is:

@I @I @I ¼ ¼    ¼ r ¼ 0: @cr1 @cr2 @cN

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



2 uP P u 10 10 r j1 j1 k1 k1 ; h0 þ 900 ; h0 þ 900  T r r i þ 1800 u k¼1 j¼1 T r i þ 1800 u  100%:

2 t P10 P10 r j1 k1 r i þ 1800 ; h0 þ 900 k¼1 j¼1 T

ð33Þ

As a result of solving the mathematical model of the heat transfer for the direct and the inverse problem in non-contacting face seals with the flexibly mounted stator, the obtained results have been compared. Fig. 6 shows two-dimensional temperature distributions for the stator and the rotor that were, determined on the basic of the accurate solution and the approximation with the use of Trefftz functions. Fig. 6(a) shows the temperature distributions determined on the basis of the dependences (9) and (10) while Fig. 6(b) shows

The measurement points were located evenly on the boundary and inside the area. A relative difference of the accurate and the approximated temperature was determined in those points in relation to the accurate value. The error (34) is expressed as a percentage. Approximation accuracy depends on the number of Trefftz functions used for approximation. Fig. 7 shows the values of error (34) depending on the number of Trefftz functions. For N = 20 a relative error amounts to 3.51% and it decreases together with the increase of the number of Trefftz functions. That confirms the convergence of the method. For N > 30 the error becomes smaller than 1%. Such a result should be considered as very good as the solution for the rotor was obtained with the use of the boundary solution of the inverse problem for the stator. Increasing the number of Trefftz functions over N = 50 results in a slight decrease of the error which remains approx. at the level of 0.15%. Generally, inverse problems are ill-posed, which results in a great sensitivity of the solutions to the disturbances in the input data. Therefore, each method’s sensitivity to these disturbances

(a)

(b)

Fig. 6. The temperature distribution in the sealing rings (a) exact, (b) approximation.

S. Blasiak, A. Pawinska / International Journal of Heat and Mass Transfer 90 (2015) 710–718

4

Approximation error (%)

3.51 3

2

0.865

1

0.287 0

20

30

40

0.145

0.145

0.14

50

60

70

Number of Trefftz functions Fig. 7. An approximation error depending on the number of Trefftz functions.

7 5.94

Relative error (%)

6

717

the mathematical models which describe the mentioned phenomena allow obtaining more and more accurate results of the analyses. The specificity of the operation of non-contacting seals, i.e. large angular speeds of the rotating elements, a slight clearance at the level of 1–5 lm, roughness of surfaces of the sealing rings or the process of undergoing thermoelastic deformities, do not allow the heat stream generated in the fluid film to be accurately specified. Solving the problem of the inverse heat transfer in the system: fluid film – sealing rings enables determining the temperature distributions in the sealing rings on the basis of the data collected in the experiment. There is a strictly defined correlation between such parameters as: the geometry of the radial clearance, the balance of the forces working on the rings, the leak and the power losses in non-contacting face seals. Any disruption of those parameters may lead to faulty operation of the whole sealing node which can result in the failure of the device in which such seal was installed. Selecting suitable materials for sealing rings which are adjusted to the operating conditions and determining, at the designing stage or during experimental tests, the values of the temperature and thermal deformities, will allow excessive wear of the surfaces of the sealing rings during the operation time to be avoided and prevent any uncontrolled increase of leaks.

5 4

References

3.51 2.77

3 2 0.872

1 0

0.404 20

30

40

0.733

50

60

70

Number of Trefftz functions Fig. 8. A mean relative error of approximation depending on the number of Trefftz functions for the disturbed internal responses.

needs to be tested. To this end, the internal responses (22) were disturbed by a random error, according to the following formula:

T~ sj ¼ T sj  ð1 þ dj Þ;

for j ¼ 1; . . . ; 11;

ð35Þ

where dj has a normal distribution with mean 0 and standard deviation 0.02. Fig. 8 shows the values of error (34) depending on the number of Trefftz functions for the disturbed internal responses. The error decreases along with the increasing number of Trefftz functions. The best results were obtained if the number of functions were between 30 and 60. In that case, the disturbance of the measurements results only in a slight increase of the error of approximation. This proves the method’s invulnerability to the random disturbance of the data. 8. Conclusion Non-contacting face seals are used in high volume devices in which safety and reliability are of utmost importance. Meeting these requirements is possible thanks to the detailed tests and the analysis of the physical processes connected among others with the fluid flow through the radial clearance, the heat transfer occurring in the sealing node or the thermo-elastic deformations that are undergone by the working rings. Preparing and developing

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