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Paper XIII(ii) Frictional losses in turbulent flow between rotating concentric cylinders
403
Paper Xlll(ii)
Frictional losses in turbulent flow between rotating concentric cylinders C.G. Floyd
The f r i c t i o n a l losses a r i s i n g in t h e t u r b u l e n t flow between a r o t a t i n g c y l i n d e r a n d a s t a t i o n a r y concentric o u t e r c y l i n d e r have been s t u d i e d experimentally. Tests have been c a r r i e d o u t u s i n g c i r c u l a r c y l i n d e r s a n d o t h e r geometries i n c l u d i n g c i r c u m f e r e n t i a l g r o o v e elements a n d r a d i a l disc elements. It has been f o u n d t h a t t h e f r i c t i o n a l losses c a n b e a c c u r a t e l y p r e d i c t e d by t h e simple addition o f t h e effects o f t h e r o t a t i n g s u r f a c e elements, a n d by assuming t h a t t h e dependence o f the s k i n f r i c t i o n coefficient on Reynolds number i s t h e same for a l l p o i n t s o n t h e r o t o r surface. 1
INTRODUCTION
A t v e r y high Reynolds numbers, t y p i c a l l y greater t h a n l o 6 , t h e flow o f a fluid between a rotating cylinder and a stationary concentric outer c y l i n d e r becomes fully t u r b u l e n t , as t h e superlaminar T a y l o r v o r t e x flow s t r u c t u r e breaks down. T u r b u l e n t flows o f t h i s t y p e occur in t h e a n n u l a r spaces a r o u n d t h e s h a f t s o f c e n t r i f u g a l pumps, in t h e a n n u l i between r o t o r s a n d s t a t o r s o n high speed e l e c t r i c motors, a n d c a n also o c c u r in l a r g e high speed journal bearings. T h e f r i c t i o n a l losses in s u c h systems d u e t o t h e s h e a r i n g o f t h e fluid f i l m a r e s i g n i f i c a n t a n d c a n g i v e r i s e t o excessive localised heat generation. In a l a r g e b o i l e r feed pump, f o r example, heat generation levels o f t h e o r d e r o f 40 k W have been r e c o r d e d in the a n n u l u s a r o u n d t h e mechanical seal i n s t a l l ation. Turbomachinery components s u c h as mechanical seals a r e sensitive t o fluid temperature, a n d t h e r e i s t h e r e f o r e a clear need f o r an accurate heat generation p r e d i c t i o n method f o r s u c h components t o p e r m i t d e s i g n optimisation. Considerable p u b l i s h e d w o r k i s available o n t h e subject o f superlaminar flow between r o t a t i n g c y l i n d e r s but l i t t l e f r i c t i o n a l loss data i s available, a n d t h e n n o t a t Reynolds numbers g r e a t e r t h a n 106.5. F u r t h e r m o r e t h e published w o r k i s limited t o e i t h e r p l a i n c i r c u l a r c y l i n d e r s o r t o discs, a n d t h e r e i s n o data available f o r more complex shapes i n c o r p o r a t i n g both c y l i n d r i c a l s u r f a c e a n d r a d i a l d i s c elements such as make up t h e m a j o r i t y o f t u r b o m a c h i n e r y components. T h e w o r k p r e s e n t e d in t h i s p a p e r i s a s t u d y o f these more complex s h a f t a n d h o u s i n g geometries, a n d i s a n attempt t o determine t h e importance o f t h e v a r i o u s parameters a f f e c t i n g heat generation a n d t o p r o v i d e empirical guidelines f o r p r e d i c t i n g t h e magnitude o f t h e heat generation. T h i s s t u d y i s p a r t o f a wider i n v e s t i g a t i o n i n t o flow problems in turbomachine r y a n n u l i (1).
Nomenclature
1.1
C
T o r q u e e x e r t e d o n t h e r o t o r by t h e fluid
R
Radius o f t h e r o t o r
I
Axial length o f the rotor
r
Local r a d i u s o f a p o i n t o f i n t e r e s t
t
Width o f a n n u l a r g a p between t h e r o t o r a n d t h e stator
v
Local v e l o c i t y a t a p o i n t o f r a d i u s r
v
Kinematic v i s c o s i t y o f t h e fluid
p
D e n s i t y o f t h e fluid
T
Shear s t r e s s o n t h e r o t a t i n g surface
w
Angular velocity o f t h e r o t o r
Non-dimensional Coefficients F Non-dimensional t o r q u e F = Re
R
C /p v 2 R
Rotor Reynolds number ReR = w R 2 / v
Cf S k i n f r i c t i o n c o e f f i c i e n t CF = o r local coefficient 2
cf
‘I / 0 . 5
pw2R
= T 10.5 pw2 r 2
EX PER IMENTAL A R RAN C EMEN T
A p u r p o s e built t e s t rig was developed f o r t h i s work, a n d a schematic arrangement o f t h e r o t o r a n d h o u s i n g assembly i s i l l u s t r a t e d in F i g u r e 1. As c a n b e seen f r o m t h e f i g u r e , t h e r o t o r s h a f t a n d h o u s i n g assembly were so designed t h a t d i f f e r e n t r o t o r s a n d housings c o u l d b e f i t t e d , t o g i v e a wide r a n g e o f a x i a l lengths, s u r f a c e finishes a n d r o t o r t o h o u s i n g g a p sizes. Representative t e s t a n n u l i a r e i l l u s t r a t e d in F i g u r e 2.
404
Fluid Annulus
water by a n e x t e r n a l heat exchanger, a n d t h e water was p r e s s u r i s e d t o between 2 a n d 8 b a r t o p r e v e n t c a v i t a t i o n a n d a i r entrainment t h r o u g h t h e seals from o c c u r r i n g . One o f t h e major problems in developing t h e experimental equipment was t h e d e s i g n o f suitable seals between t h e r o t o r a n d t h e housing. These seals were r e q u i r e d t o operate o v e r a r a n g e o f speeds a n d fluid p r e s s u r e s but w i t h minimal leakage a n d w i t h t h e minimum possible f r i c t i o n a l losses. T h e f i n a l arrangement u s e d consisted o f a c a r b o n seal ring running against a t u n g s t e n c a r b i d e seal ring w i t h t h e axial c l o s i n g f o r c e p r o v i d e d by compressed a i r . T h e c l o s i n g f o r c e between t h e t w o seal r i n g s c o u l d t h e r e f o r e b e regulat e d during t e s t s t o c o n t r o l t h e leakage.
Shaft
T h e use o f stainless steel f o r t h e r o t o r a n d h o u s i n g components a n d t h e use o f water as t h e w o r k i n g fluid were d i c t a t e d by t h e d e s i r e t o r e p r e s e n t t h e conditions in a n actual pump f o r w h i c h some experimental data T h e size o f t h e equipment were available. was also f i x e d f o r t h e same reason. T h e possibilities o f u s i n g a scale model o r o f u s i n g a l t e r n a t i v e f l u i d s were considered, but as t h e importance o f t h e v a r i o u s parameters a f f e c t i n g heat g e n e r a t i o n were n o t k n o w n it was t h o u g h t desirable t o eliminate a n y scale effects.
Figure 1 T e s t riq arrangement
I
/,
I'
J
A
rotor seal
T h e p r i n c i p a l parameters measured were:
seal
(i)
S h a f t speed a n d torque, measured by a s h a f t mounted optical s l i t t e d d i s c system, p r o v i d i n g f i l t e r e d analogue voltage o u t p u t s . F l u i d i n l e t a n d o u t l e t temperatures, measured by p l a t i n u m resistance thermometers. F l u i d pressures, measured by analogue p r e s s u r e gauges.
(ii)
F l u i d c i r c u l a t i o n flow rate, measured by a p a i r o f Rotameter u n i t s . Seal leakage flow rates, measured by t h e time t a k e n t o fill a g r a d u a t e d container
.
0
(iii) Fiqure 2 T y p i c a l a n n u l a r t e s t qeometries T h e r o t o r a n d h o u s i n g components were manufactured f r o m stainless steel, a n d t y p i c a l r o t o r r a d i i were in t h e r a n g e 107mm to 12lmm. T h e s h a f t was d r i v e n by a 75 k W motor t h r o u g h a b e l t d r i v e system t o g i v e a rotational speed r a n g e o f 2100 r p m t o 9400 rpm. T h e o p e r a t i n g fluid was water, a n d t h i s was c i r c u l a t e d t h r o u g h t h e t e s t cell a t a r a t e o f between 15 a n d 50 l i t r e s / m i n u t e by a n e x t e r n a l pump. Heat was e x t r a c t e d from t h e
S u b s i d i a r y measurements o f t h e compressed a i r s u p p l y p r e s s u r e s t o t h e seals a n d t h e b e a r i n g temperatures were also made. T h e heat g e n e r a t i o n in t h e system was determined by t h e s h a f t speed a n d t h e t o r q u e measurements. Determination o f t h e heat generation from t h e r i s e in fluid temperature a n d t h e fluid c i r c u l a t i o n f l o w r a t e g a v e o n l y a b o u t 709, o f t h e power loss, o w i n g t o t h e additional heat t r a n s f e r i n t o t h e b o d y of t h e housing.
3
CYLINDER TESTS
F o r fully t u r b u l e n t flow w i t h Reynolds numbers g r e a t e r t h a n lo6, t h e flow between t h e r o t o r a n d t h e h o u s i n g has been shown by Pai ( 2 ) a n d Ustemenko ( 3 ) t o consist o f t w o thin s u r f a c e shear l a y e r s a n d a c e n t r a l flow r e g i o n
405 over 80% o f th e gap f o r which v r i s a constant. The s k i n f r i c t i o n coefficient o n t h e r o t o r surface, a n d hence t h e heat generation ,is dependent o n th e velocity g r a d i e n t in t h e It w i l l t h e r e f o r e b e o n l y surface shear layer. weakly dependent o n t h e Reynolds number a n d on the housing geometry. It also follows t h a t the flow in th e a nnular g a p i s essentially tw o dimensional. T h e influence o f t h e helical flow a r i sing from t h e c i r c u l a t i o n of t h e fluid by t h e external pump was considered t o b e small. The axial Reynolds number was less t h a n 0.1% o f the rotational Reynolds numbers a n d t h e work o f Yamada (4) a n d o t h e r s suggests t h a t for these conditions helical flow effects can b e neglected. T h e r e s u l t s obtained in t h i s s t u d y showed no dependence o n fluid c i r c u l a t i o n rate. Sig nificant t h r e e dimensional e n d e f f e c t s did occur, however, b o t h d u e t o t h e additional surfaces o f t h e seals a n d t h e c y l i nder ends, a n d d u e t o t h r e e dimensional flow e ffects in t h e flow caused by these additional surfaces. T h e t o r q u e d u e t o these e n d effects comprised up t o 50% o f t h e total t o r q u e measured, a n d in p a r t i c u l a r t h e f r i c t i o n a t t h e seals was high. T h e t o r q u e d u e t o all t h e e n d effects co uld n o t b e measured separately fro m the t o r q u e d u e t o t h e cylinders, a n d it was t h e r efore necessary t o t e s t t w o c y l i n d e r s of d i f f e r e n t axial lengths, t o eliminate t h e e n d effects by s u b t r a c t i n g t h e two results. Problems have been encountered w i t h t h i s technique in t h e past (5) because o f t h e size o f t h e e n d effects, a n d so f o u r d i f f e r e n t c y l i n d e r l e n g t h s were tested, t o p e r m i t t h e s u b traction calculations t o be c r o s s checked. T h e shortest axial l e n g t h chosen was 40mm. and t h i s l e n g t h was also chosen f o r most o f the t e s t i n g o f more complex r o t o r a n d stator geometries. T h e o t h e r l e n g t h s chosen f o r t h e c y l i nder te sts were 65mm, 80mm a n d 120mm. It was assumed t h a t f o r a g i v e n r o t o r and housing geometry t h e s k i n f r i c t i o n coefficient i s re lated t o t h e Reynolds number b y a power law relationship CF = aReRb, where t h e s k i n f r i c t i o n coefficient i s g i v e n by CF = G / ( n p I R 4 w 2 ) .
However, t h e s k i n f r i c t i o n coefficient cannot b e determined d i r e c t l y from t h e experimental data, as t h e total t o r q u e measured ( G T), includes a component d u e t o e n d e f f e c t s ( G E ) as well as t h e c y l i n d e r surface component ( G C). T h e r e f o r e t h e s k i n f r i c t i o n coefficient is g i v en as
where Ic i s t h e actual l e n g t h o f c y l i n d e r s u r face, a n d IE i s a n equivalent e x t r a l e n g t h o f c y l i n d e r surface t o allow f o r t h e e n d effects. T o analyse t h e data, a non-dimensional t o r q ue co efficient F was introduced, where: F = (GC + GE)/pu2R
F can t h e n b e d e f i n e d in terms o f Reynolds n u m b e r by :
a n d thence f r o m a series o f t e s t s w i t h diffe r e n t c y l i n d e r l e n g t h s , a, b a n d IE can b e found.
F has n o p h y s i c a l significance, but was chosen as t h e non-dimensional c o e f f i c i e n t t o g i v e a significant gradient o n a plot o f log F against l o g ReR t o f a c i l i t a t e t h e data processing. I m p l i c i t in t h i s a p p r o a c h was t h e assumption t h a t t h e e n d e f f e c t s c a n b e q u a n t i f i e d by a n a d d i t i o n a l l e n g t h o f c y l i n d r i c a l s u r f a c e ( IE) F o r t h i s assumption t o b e valid, a l l t h e e n d e f f e c t s must v a r y w i t h R e y n o l d s Number in a similar fashion t o t h e c y l i n d e r t o r q u e . The additional torque due t o the e n d effects can be s p l i t i n t o t w o components; the torque d u e t o t h e seals, a n d t h e t o r q u e d u e t o t h e end surfaces o f the cylinders. The torque d u e t o t h e b e a r i n g s c a n b e i g n o r e d as d r y r u n s o f t h e t e s t rig showed t h a t t h e b e a r i n g t o r q u e was n e g l i g i b l e .
.
T o assess t h e t o r q u e d u e t o t h e seals, it was assumed t h a t t h e flow a t t h e seal i n t e r f ac e was t u r b u l e n t , w i t h a seal Reynolds Number o f t h e o r d e r o f 105. T h e seal Reynolds Number was c a l c u l a t e d f r o m t h e s l i d i n g v e l o c i t y a n d a n estimate o f t h e i n t e r f a c e f i l m t h i c k n e s s based o n t h e measured leakage a n d p r e s s u r e differential. The c i r c u m f e r e n t i a l flow a t t h e seal i n t e r f a c e i s t h e n comparable to t h e flow between m o v i n g p a r a l l e l plates, f o r w h i c h t h e dependence o f s k i n f r i c t i o n o n Reynolds n u m b e r i s similar t o t h a t f o r c o n c e n t r i c r o t a t i n g c y l i n d e r s ( 6 ) . It i s t h e r e f o r e reasonable t o assume t h a t t h e t o r q u e d u e t o t h e seals can b e c o n s i d e r e d t o b e equal t o the torque due t o a n additional length o f c y l i n d e r surface. T h e t o r q u e d u e t o t h e e n d surfaces o f t h e r o t a t i n g c y l i n d e r c o u l d not b e d e t e r m i n e d theoretically. However, as t h e flow in t h e annular gap i s a basically irrotational turbulent flow a t a high R e y n o l d s number, t h e t u r b u l e n t b o u n d a r y l a y e r s o n t h e e n d surfaces o f t h e a n n u l a r g a p w i l l b e similar t o those o n t h e c y l i n d e r surfaces. From t h i s i t can be assumed t h a t t h e r e l a t i o n s h i p s between s k i n f r i c t i o n c o e f f i c i e n t a n d Reynolds n u m b e r w i l l also b e similar. It i s t h e r e f o r e reasonable t o assume t h a t a l l t h e e n d e f f e c t s w i l l v a r y w i t h Reynolds n u m b e r in a similar fashion t o t h e c y l i n d e r torque. T h e n f o r each a n n u l a r g a p w i d t h t h e r e s u l t s for a l l t h e a x i a l l e n g t h s c a n b e p l o t t e d in t e r m s o f l o g F a g a i n s t l o g ReR a n d processed simultaneously t o g i v e a least squares r e g r e s s i o n fit o f f o u r p a r a l l e l l i n e s t o t h e f o u r sets o f experimental data.
T h e r e g r e s s i o n a n a l y s i s was weighted t o t a k e account o f t h e estimated a c c u r a c y o f t h e measurements. F i g u r e 3 shows a plot of t h e data p o i n t s f o r a 5mm g a p f o r t h e t w o
406 It can b e seen t h a t t h e c o r r e l a t i o n between t h e data a n d t h e f i t t e d lines i s good, a n d in f a c t t h e c o r r e l a t i o n c o e f f i c i e n t s were t y p i c a l l y g r e a t e r t h a n 0 . 9 9 . T h e r e g r e s s i o n analysis d e t e r m i n e d n o t o n l y t h e slope o f t h e line, but also t h e m a g n i t u d e o f t h e e n d e f f e c t s in t e r m s o f the equivalent e x t r a length o f cylindrical surface. T h e e q u i v a l e n t e x t r a l e n g t h s were calculated t o b e 5 8 , 75 a n d 75mm f o r a n n u l a r g a p w i d t h s o f 5 , 10 a n d 20mm r e s p e c t i v e l y . The reduced magnitude o f the end effects f o r t h e 5mm g a p was n o t d u e t o experimental e r r o r , as r e p e a t t e s t s showed t h a t t h e a c c u r a c y o f d e t e r m i n a t i o n o f t h e e n d l e n g t h s was It i s t h e r e f o r e consida p p r o x i m a t e l y f 3mm. e r e d t h a t it was a r e s u l t o f some t h r e e dimensional flow e f f e c t o v e r t h e c y l i n d e r e n d surfaces.
extreme axial lengths o f 40 a n d 120mm togeth e r w i t h t h e f i t t e d lines. T h e r e s u l t s for t h e i n t e r mediate lengths have been omitted f o r c l a r i t y .
12 .o
I
I = 120
log
11.5
11
Al l o w i n g f o r t h e calculated i n f l u e n c e o f t h e e n d effects, t h e r e l a t i o n s h i p s between s k i n f r i c t i o n c o e f f i c i e n t a n d Reynolds n u m b e r were determined f o r the different gap widths.
.o
These were f o u n d t o be:
10.5 6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
log ReR
Fiqure 3
T h e r e appears t o b e o n l y a s l i g h t dependence o n t h e g a p r a t i o ( t / R ) , a n d t h i s is in agreement w i t h t h e r e s u l t s o f o t h e r workers. T h i s c a n b e seen in F i g u r e 4 w h i c h is a plot o f other published results and the present results.
T o r q u e aqainst Reynolds number f o r two a xial lenqths
1.3
1.2
1.1
1 .o
l-
i
P t/R
o
0.15-0.4
A
0.07-0.15
V
0.04-0.07
0
0.02-0.04
5.6
1
5.8
Fiqure 4
6.0
I
6.2
I
f
I
6.4
6.6
6.8
1
7.0 l o g ReR
Selected p u b l i s h e d data compared w i t h mean lines f r o m p r e s e n t r e s u l t s f o r cylinders.
Published r e s u l t s from B i l q e n ( 7 ) . G o r l a n d ( 8 ) & T a y l o r ( 9 )
7.2
407 It would appear from F i g u r e 4 t h a t t h e s k i n f r i c tion coefficient i n i t i a l l y decreases w i t h decreasing gap r a t i o but t h e n begins t o increase as t he g ap r a t i o f u r t h e r decreases. T h e minimum s k i n f r i c t i o n coefficient o c c u r s when t h e g a p r a t i o i s in th e approximate r a n g e o f 0 . 1 t o 0 . 2 . A more precise statement o n t h e e f f e c t o f g a p r a t i o o n s k i n f r i c t i o n coefficient i s n o t possible, as the scatter o f experimental data p o i n t s i s greater t h a n a n y g a p r a t i o effect.
4
assumption was based o n t h e f a c t t h a t t h e s k i n f r i c t i o n c o e f f i c i e n t data f o r complete discs p u b l i s h e d by o t h e r w o r k e r s ( 1 0 , 1 1 , 1 2 ) showed good agreement w i t h t h e p r e s e n t r e s u l t s f o r c y l i n d r i c a l surfaces, as shown in F i g u r e 6 .
COMPLEX GEOMETRY EFFECTS Influence o f Housinq Geometry
4.1
A series o f te sts were c a r r i e d o u t u s i n g p l a i n c y l i ndrical r o t o r s but w i t h changes t o t h e housing geometry. Two d i f f e r e n t housing geometry effects were tested. T h e f i r s t o f these was a change t o t h e i n l e t geometry from t h a t shown in Figure 2 ( i ) t o t h a t shown in F i g u r e 2 ( i i ) . Figure 5 shows t h e data f o r b o t h geometries plotted as log F against log ReR f o r a n 80mm axial l e n g t h c y l i n d e r . It can b e seen t h a t changing t h e i n l e t geometry h a d n o discerna b l e effect o n t h e r o t o r torque.
12.0
11.5
11.0
+
.7 .6 -
1
1 6.4
6.5
1
6.6
6.7
6.8
6.9
7.0
7.1
published radial disc results
8
+
closed inle t
Q
open i n l e t
+ I
6.6
Comparison o f c y l i n d e r r e s u l t s against
i+
1
6.5
log ReR
+
F
6.4
Fiqure 6
0
0
Present c y l i n d e r r e s u l t s lppen (10) Ketola & McGrew ( 1 1 ) D a i l y & Nece ( 1 2 )
6.3
++
-1
6.3
-
.8
.5
1
10.5
.9
I
I
1
6.7
6.8
6.9
1
7.0
The variation o f the total torque for the 121mm r o t o r w i t h r o t o r Reynolds n u m b e r c o u l d t h e n b e p r e d i c t e d by i n t e g r a t i n g t h e local s k i n f r i c t i o n c o e f f i c i e n t o v e r t h e complete r o t o r s u r face. T h e e n d e f f e c t s were i n c l u d e d as a n e x t r a axial length o f cylinder. The predicted total t o r q u e a n d t h e experimental data a r e p l o t t e d in F i g u r e 7 a n d show good agreement.
1
7.1
12.0
F
-
Ef Effect o f doubling
log ReR Figure 5 Influence o f i n l e t qeometry o n t o r q u e T h e second housing change i n v o l v e d t h e i n t r o d u c t i o n o f a circumferential groove, as shown in F i g u r e 2 ( i i ) . Again t h i s h a d n o discernable effect o n t h e r o t o r torque. T h i s i s n o t s u r p r i s i n g as one would expect t h e r o t o r t o r q u e to depend o n t h e velocity g r a d i e n t in t h e shear layer o n t h e rotor, a n d t h i s will b e l i t t l e affect e d by h ousing changes. 4.2
11.5-
11.0
-
10.5
I
Radial Elements
The n e x t series o f tests i n v o l v e d t h e i n t r o d u c t i o n o f r a d i a l elements o n t o t h e r o t o r . T w o r o t o r s were used, w i t h r a d i i o f 116mm a n d 1 2 l m m , t o g i v e radial surface elements a t t h e c y l i n d e r ends. It was assumed t h a t t h e relations h i p between local s k i n f r i c t i o n coefficient a n d local Reynolds number f o r b o t h disc a n d c y l i n d e r elements was t h e same as t h a t between the s k i n f r i c t i o n c oefficient a n d r o t o r Reynolds This number f o r t h e c y l i n d r i c a l surfaces.
6.3
1
,
6.4
I
I
6.5
6.6
Fiqure 7
I
6.7
1
6.8
I
1
6.9 7.0 log ReR
Pr e d i c t e d a n d measured t o r q u e f o r a r o t o r w i t h r a d i a l s u r f a c e elements
1
7.1
408
The proportion o f the total torque due to the r a d i a l surfaces i s small, t y p i c a l l y a r o u n d 15%, but t h e estimate o f t h e t o t a l t o r q u e i s s t i l l sensitive t o l a r g e e r r o r s in t h e estimate o f t h e F i g u r e 7 shows a r e radial surface torque. calculated p r e d i c t i o n assuming t h a t t h e local s k i n f r i c t i o n coefficient for the radial surfaces was double t h e value f o r t h e c y l i n d r i c a l surfaces. It can b e seen t h a t t h e p r e d i c t i o n d i f f e r s s i g n i f i c a n t l y f r o m t h e experimental data. It can be concluded t h a t , f o r t h i s t y p e o f geometry where t h e p r o p o r t i o n o f t o t a l t o r q u e d u e t o t h e r a d i a l surfaces i s small, t a k i n g t h e local r a d i a l s u r f a c e s k i n f r i c t i o n c o e f f i c i e n t t o b e equal t o t h e c y l i n d e r s u r f a c e s k i n f r i c t i o n coefficient i s a v a l i d a p p r o a c h t o p r e d i c t i n g t h e total torque.
T o t e s t f o r i n t e r a c t i o n s between t h e c y l i n d r i c a l a n d r a d i a l s u r f a c e elements, a stepped r o t o r was tested. T h i s r o t o r was o f 40mm a x i a l l e n g t h , w i t h lOmm a t a r a d i u s o f 107mm a n d 30mm a t a r a d i u s o f 116mm. T h e t o r q u e f o r t h e r o t o r was p r e d i c t e d b y t h e i n t e g r a t i o n o f t h e local s k i n f r i c t i o n c o e f f i c i e n t across t h e r o t o r surface, as before, a n d t h e p r e d i c t i o n was again f o u n d t o a g r e e v e r y closely w i t h t h e experimental data, w i t h n o detectable interaction effects. T h e above r e s u l t s show t h a t t h e t o r q u e f o r a n y complex r o t o r a n d h o u s i n g geometry w i t h i n t h e r a n g e t e s t e d can b e p r e d i c t e d by t h e simple a d d i t i o n o f t h e components o f t o r q u e d u e t o t h e r o t a t i n g s u r f a c e elements, t o g e t h e r w i t h a n e x t r a component o f t o r q u e t o allow f o r e n d effects. The influence o f the housing geometry o n t h e t o t a l t o r q u e o f t h e system i s minor a n d can be neglected. T h i s i s most c l e a r l y shown by t h e r e s u l t s f o r t h e system shown in F i g u r e 2 ( i i i ) w h i c h has b o t h a stepped r o t o r a n d a c i r c u m f e r e n t i a l g r o o v e in t h e housing. F i g u r e 8 shows v e r y close agreement between t h e measured t o r q u e r e s u l t s a n d t h e p r e d i c t e d values.
11.5
4
5
GENERAL A P P L I C A B I L I T Y
T o a p p l y these r e s u l t s g e n e r a l l y , it i s necessary t o know w h e t h e r a l l o f t h e e n d e f f e c t s a r e d u e t o t e s t rig effects, o r whether t h e y a r e in p a r t caused by flow e f f e c t s a r o u n d t h e edges o f t h e r o t a t i n g s u r f a c e . It i s clear t h a t a l l t h e e n d e f f e c t s a r e n o t caused b y t h e seal f r i c t i o n a n d c y l i n d e r e n d A 40mm l o n g r o t o r w i t h a 5mm s u r f a c e s alone. a n n u l a r g a p has a n e n d e f f e c t e q u i v a l e n t l e n g t h o f 58mm. A p p r o x i m a t e l y h a l f o f t h i s can b e e x p l a i n e d as d u e t o t h e r a d i a l s u r f a c e s a t t h e sides o f t h e r o t o r a n d t h e l i k e l y losses in t h e t h i c k fluid films o n t h e seal faces. T h e remaining e n d e f f e c t s a r e p r o b a b l y d u e t o t h r e e dimensional flow e f f e c t s w h i c h may be u n i q u e t o t h e t e s t r i g o r may b e g e n e r a l l y applicable
.
T h e o n l y way t o find o u t i s t o compare t h e t o r q u e p r e d i c t e d f o r a complex geometry w i t h measured t e s t r e s u l t s o b t a i n e d o n a completely d i f f e r e n t t e s t rig. T h i s comparison was made, u s i n g r e s u l t s f o r a piece o f t u r b o m a c h i n e r y made up o f t w o sections similar t o F i g u r e 2 ( i i i ) a n d a l o n g p l a i n c y l i n d e r o f 400mm a x i a l l e n g t h . It was f o u n d t h a t t h e p r e d i c t e d t o r q u e , a g r e e d w i t h t h e t e s t data t o w i t h i n 10% i f a l l t h e e n d e f f e c t s were assumed t o b e d u e t o seal losses a n d t e s t r i g e f f e c t s . However, t h i s level o f agreement between p r e d i c t i o n a n d experimental data i s n o t c o n c l u s i v e as t h e l o n g p l a i n c y l i n d e r compone n t swamped a n y i n f l u e n c e o f t h e e n d e f f e c t s T h e r e remains, o n the total torque. t h e r e f o r e , some e n d e f f e c t s w h i c h a r e a f u n c t i o n o f u n k n o w n e f f e c t s in t h e flow o v e r t h e r o t a t i n g surface, a n d it i s s t i l l n o t clear if t h e y a r e u n i q u e t o t h e t e s t r i g o r a r e g e n e r a l l y applicable. It i s t h e r e f o r e concluded t h a t t h e following g u i d e l i n e s w i l l p r o v i d e a reasonable a n d c o n s e r v a t i v e estimate o f t h e t o r q u e o n t h e r o t o r o f a piece o f t u r b o m a c h i n e r y .
(1)
O n l y t h e r o t a t i n g s u r f a c e s h o u l d be considered.
(2)
A mean v a l u e o f s k i n f r i c t i o n c o e f f i c i e n t should be applied to b o t h the radial and t h e c y l i n d r i c a l surfaces. T h e value Re-Oe2 s u g g e s t e d i s Cf = 3.16. w h e r e Re i s e i t h e r a f u n c t i o n o f R f o r t h e c y l i n d r i c a l surface, o r a v a r i a b l e f u n c t i o n o f r f o r t h e r a d i a l surface. These numerical values a r e mean values o f those d e t e r m i n e d f o r t h e d i f f e r e n t g a p r a t i o s a n d h a v e been t e s t e d a n d f o u n d t o g i v e good c o r r e l a t i o n .
(3)
It i s c o n s i d e r e d t h a t a n additional length o f cylindrical surface should be a d d e d as a n estimate o f t h e u n k n o w n effects, p r i n c i p a l l y d u e t o i n t e r f e r e n c e e f f e c t s adjacent t o t h e r o t o r c o r n e r s . F o r r o t o r s w i t h t w o major changes in r a d i u s a l o n g t h e i r a x i a l lengths, s u c h as those tested, a n additional l e n g t h o f 35mm a t a r a d i u s o f 107mm i s appropriate. For r o t o r s w i t h a
11.0
10.5
6.3 6 . 4 Fiqure 8
6.5
6.6
6.7
6.8
6.9
7.0 7.1 log ReR
Comparison of p r e d i c t e d a n d measured r e s u l t s f o r a stepped r o t o r a n d a q r o o v e d h o u s i n q
409
(4)
greater number o f step changes in radius then it is likely t h a t t h i s l e n g t h should be increased.
(9)
TAYLOR, G 1 . " F l u i d f r i c t i o n between r o t a t i n g c y l i n d e r s 1-torque measurements" Proc. Roy. SOC. Ser A., 1936, 157 p546
In addition, f o r the test rig used f o r the present work, it i s necessary t o add a f u r t h e r l e n g t h o f 35mm o f c y l i n d r i c a l s u r face t o allow f o r t h e f r i c t i o n due t o the seals and adjacent radial surfaces.
(10)
IPPEN, A T "Influence o f viscosity on c e n t r i f u g a l pump performance". T r a n s ASME, 1946, 68, p 823
(11)
KETOLA, H N McCrew, J M "Pressure, frictional resistance a n d flow characteristics o f t h e p a r t i a l l y wetted r o t a t i n g d i s k " Trans. ASME. Ser. F., 1968, 90, p 295 -
(12)
DAILY, J W, NECE, R E "Chamber dimension effects o n induced flow and frictional resistance o f enclosed r o t a t i n g disks". Trans. ASME. Ser D., 1969, 82, p 217
It is considered t h a t t h e e r r o r in t h e estimate o f torque obtained u s i n g the above procedure is unlikely t o be greater than 20%, f o r r o t o r s w i t h gap ratios in t h e range 0.05 t o 0.2, and w i t h similar ratios o f radial t o c y l i n d r i c a l r o t o r surface areas t o those o f t h e r o t o r s tested. 6
ACKNOWLEDGEMENTS
The author acknowledges t h e assistance o f the SERC and o f t h e T.I. Research Laboratories who sponsored t h i s work u n d e r a CASE Project. The author also wishes t o acknowledge the s u p p o r t a n d advice g i v e n by t h e s t a f f o f the Department o f Aeronautical Engineering at the U n i v e r s i t y o f Bristol.
References FLOYD, C G "A S t u d y o n frictional losses o f enclosed r o t o r s a t high Reynolds numbers" Ph. D. Thesis, U n i v e r s i t y o f Bristol, September 1982. PAI, S I " T u r b u l e n t flow between r o t a t i n g cylinders" NACA Tech. Note 892, 1943. USTIMENKO, B P e t al " T u r b u l e n t t r a n s f e r in r o t a r y flows o f an incompressible fluid". F l u i d Mech. Soviet Research, p 121 1972,
1,
YAMADA, Y "Torque resistance Of a flow between r o t a t i n g coaxial c y l i n d e r s h a v i n g axial flow". Bull. JSME, 1962, 5. p 635 SULMONT, P. BOURGET, P L "Mesure experimentale du couple de frottement d'un c y l i n d r e t o u r n a n t dans un c y l i n d r e f i x e p o u r des faible e n t r e f e r s e t des g r a n d s nombres du Reynolds". Annales de Mechanique, Ecole Nat. de Mech., Nantes, 1969. ROBERTSON, J M. "On t u r b u l e n t plane Couette flow", 6th Mid West Conference o n F l u i d Mech., Univ. o f Texas 1959. BILGEN, E, BOULOS, R "Functional dependence o f t o r q u e coefficient o f coaxial c y l i n d e r s o n gap w i d t h a n d Reynolds numbers". Trans. ASME Ser. I . 1973 95, p 122.
-
GORLAND, S H e t al "Experimental windage losses f o r close clearance r o t a t i n g c y l i n d e r s in t h e t u r b u l e n t flow regime". NASA TM X-52851, 1970
Paper Xlll(ii)
Frictional losses in turbulent flow between rotating concentric cylinders C.G. Floyd
The f r i c t i o n a l losses a r i s i n g in t h e t u r b u l e n t flow between a r o t a t i n g c y l i n d e r a n d a s t a t i o n a r y concentric o u t e r c y l i n d e r have been s t u d i e d experimentally. Tests have been c a r r i e d o u t u s i n g c i r c u l a r c y l i n d e r s a n d o t h e r geometries i n c l u d i n g c i r c u m f e r e n t i a l g r o o v e elements a n d r a d i a l disc elements. It has been f o u n d t h a t t h e f r i c t i o n a l losses c a n b e a c c u r a t e l y p r e d i c t e d by t h e simple addition o f t h e effects o f t h e r o t a t i n g s u r f a c e elements, a n d by assuming t h a t t h e dependence o f the s k i n f r i c t i o n coefficient on Reynolds number i s t h e same for a l l p o i n t s o n t h e r o t o r surface. 1
INTRODUCTION
A t v e r y high Reynolds numbers, t y p i c a l l y greater t h a n l o 6 , t h e flow o f a fluid between a rotating cylinder and a stationary concentric outer c y l i n d e r becomes fully t u r b u l e n t , as t h e superlaminar T a y l o r v o r t e x flow s t r u c t u r e breaks down. T u r b u l e n t flows o f t h i s t y p e occur in t h e a n n u l a r spaces a r o u n d t h e s h a f t s o f c e n t r i f u g a l pumps, in t h e a n n u l i between r o t o r s a n d s t a t o r s o n high speed e l e c t r i c motors, a n d c a n also o c c u r in l a r g e high speed journal bearings. T h e f r i c t i o n a l losses in s u c h systems d u e t o t h e s h e a r i n g o f t h e fluid f i l m a r e s i g n i f i c a n t a n d c a n g i v e r i s e t o excessive localised heat generation. In a l a r g e b o i l e r feed pump, f o r example, heat generation levels o f t h e o r d e r o f 40 k W have been r e c o r d e d in the a n n u l u s a r o u n d t h e mechanical seal i n s t a l l ation. Turbomachinery components s u c h as mechanical seals a r e sensitive t o fluid temperature, a n d t h e r e i s t h e r e f o r e a clear need f o r an accurate heat generation p r e d i c t i o n method f o r s u c h components t o p e r m i t d e s i g n optimisation. Considerable p u b l i s h e d w o r k i s available o n t h e subject o f superlaminar flow between r o t a t i n g c y l i n d e r s but l i t t l e f r i c t i o n a l loss data i s available, a n d t h e n n o t a t Reynolds numbers g r e a t e r t h a n 106.5. F u r t h e r m o r e t h e published w o r k i s limited t o e i t h e r p l a i n c i r c u l a r c y l i n d e r s o r t o discs, a n d t h e r e i s n o data available f o r more complex shapes i n c o r p o r a t i n g both c y l i n d r i c a l s u r f a c e a n d r a d i a l d i s c elements such as make up t h e m a j o r i t y o f t u r b o m a c h i n e r y components. T h e w o r k p r e s e n t e d in t h i s p a p e r i s a s t u d y o f these more complex s h a f t a n d h o u s i n g geometries, a n d i s a n attempt t o determine t h e importance o f t h e v a r i o u s parameters a f f e c t i n g heat generation a n d t o p r o v i d e empirical guidelines f o r p r e d i c t i n g t h e magnitude o f t h e heat generation. T h i s s t u d y i s p a r t o f a wider i n v e s t i g a t i o n i n t o flow problems in turbomachine r y a n n u l i (1).
Nomenclature
1.1
C
T o r q u e e x e r t e d o n t h e r o t o r by t h e fluid
R
Radius o f t h e r o t o r
I
Axial length o f the rotor
r
Local r a d i u s o f a p o i n t o f i n t e r e s t
t
Width o f a n n u l a r g a p between t h e r o t o r a n d t h e stator
v
Local v e l o c i t y a t a p o i n t o f r a d i u s r
v
Kinematic v i s c o s i t y o f t h e fluid
p
D e n s i t y o f t h e fluid
T
Shear s t r e s s o n t h e r o t a t i n g surface
w
Angular velocity o f t h e r o t o r
Non-dimensional Coefficients F Non-dimensional t o r q u e F = Re
R
C /p v 2 R
Rotor Reynolds number ReR = w R 2 / v
Cf S k i n f r i c t i o n c o e f f i c i e n t CF = o r local coefficient 2
cf
‘I / 0 . 5
pw2R
= T 10.5 pw2 r 2
EX PER IMENTAL A R RAN C EMEN T
A p u r p o s e built t e s t rig was developed f o r t h i s work, a n d a schematic arrangement o f t h e r o t o r a n d h o u s i n g assembly i s i l l u s t r a t e d in F i g u r e 1. As c a n b e seen f r o m t h e f i g u r e , t h e r o t o r s h a f t a n d h o u s i n g assembly were so designed t h a t d i f f e r e n t r o t o r s a n d housings c o u l d b e f i t t e d , t o g i v e a wide r a n g e o f a x i a l lengths, s u r f a c e finishes a n d r o t o r t o h o u s i n g g a p sizes. Representative t e s t a n n u l i a r e i l l u s t r a t e d in F i g u r e 2.
404
Fluid Annulus
water by a n e x t e r n a l heat exchanger, a n d t h e water was p r e s s u r i s e d t o between 2 a n d 8 b a r t o p r e v e n t c a v i t a t i o n a n d a i r entrainment t h r o u g h t h e seals from o c c u r r i n g . One o f t h e major problems in developing t h e experimental equipment was t h e d e s i g n o f suitable seals between t h e r o t o r a n d t h e housing. These seals were r e q u i r e d t o operate o v e r a r a n g e o f speeds a n d fluid p r e s s u r e s but w i t h minimal leakage a n d w i t h t h e minimum possible f r i c t i o n a l losses. T h e f i n a l arrangement u s e d consisted o f a c a r b o n seal ring running against a t u n g s t e n c a r b i d e seal ring w i t h t h e axial c l o s i n g f o r c e p r o v i d e d by compressed a i r . T h e c l o s i n g f o r c e between t h e t w o seal r i n g s c o u l d t h e r e f o r e b e regulat e d during t e s t s t o c o n t r o l t h e leakage.
Shaft
T h e use o f stainless steel f o r t h e r o t o r a n d h o u s i n g components a n d t h e use o f water as t h e w o r k i n g fluid were d i c t a t e d by t h e d e s i r e t o r e p r e s e n t t h e conditions in a n actual pump f o r w h i c h some experimental data T h e size o f t h e equipment were available. was also f i x e d f o r t h e same reason. T h e possibilities o f u s i n g a scale model o r o f u s i n g a l t e r n a t i v e f l u i d s were considered, but as t h e importance o f t h e v a r i o u s parameters a f f e c t i n g heat g e n e r a t i o n were n o t k n o w n it was t h o u g h t desirable t o eliminate a n y scale effects.
Figure 1 T e s t riq arrangement
I
/,
I'
J
A
rotor seal
T h e p r i n c i p a l parameters measured were:
seal
(i)
S h a f t speed a n d torque, measured by a s h a f t mounted optical s l i t t e d d i s c system, p r o v i d i n g f i l t e r e d analogue voltage o u t p u t s . F l u i d i n l e t a n d o u t l e t temperatures, measured by p l a t i n u m resistance thermometers. F l u i d pressures, measured by analogue p r e s s u r e gauges.
(ii)
F l u i d c i r c u l a t i o n flow rate, measured by a p a i r o f Rotameter u n i t s . Seal leakage flow rates, measured by t h e time t a k e n t o fill a g r a d u a t e d container
.
0
(iii) Fiqure 2 T y p i c a l a n n u l a r t e s t qeometries T h e r o t o r a n d h o u s i n g components were manufactured f r o m stainless steel, a n d t y p i c a l r o t o r r a d i i were in t h e r a n g e 107mm to 12lmm. T h e s h a f t was d r i v e n by a 75 k W motor t h r o u g h a b e l t d r i v e system t o g i v e a rotational speed r a n g e o f 2100 r p m t o 9400 rpm. T h e o p e r a t i n g fluid was water, a n d t h i s was c i r c u l a t e d t h r o u g h t h e t e s t cell a t a r a t e o f between 15 a n d 50 l i t r e s / m i n u t e by a n e x t e r n a l pump. Heat was e x t r a c t e d from t h e
S u b s i d i a r y measurements o f t h e compressed a i r s u p p l y p r e s s u r e s t o t h e seals a n d t h e b e a r i n g temperatures were also made. T h e heat g e n e r a t i o n in t h e system was determined by t h e s h a f t speed a n d t h e t o r q u e measurements. Determination o f t h e heat generation from t h e r i s e in fluid temperature a n d t h e fluid c i r c u l a t i o n f l o w r a t e g a v e o n l y a b o u t 709, o f t h e power loss, o w i n g t o t h e additional heat t r a n s f e r i n t o t h e b o d y of t h e housing.
3
CYLINDER TESTS
F o r fully t u r b u l e n t flow w i t h Reynolds numbers g r e a t e r t h a n lo6, t h e flow between t h e r o t o r a n d t h e h o u s i n g has been shown by Pai ( 2 ) a n d Ustemenko ( 3 ) t o consist o f t w o thin s u r f a c e shear l a y e r s a n d a c e n t r a l flow r e g i o n
405 over 80% o f th e gap f o r which v r i s a constant. The s k i n f r i c t i o n coefficient o n t h e r o t o r surface, a n d hence t h e heat generation ,is dependent o n th e velocity g r a d i e n t in t h e It w i l l t h e r e f o r e b e o n l y surface shear layer. weakly dependent o n t h e Reynolds number a n d on the housing geometry. It also follows t h a t the flow in th e a nnular g a p i s essentially tw o dimensional. T h e influence o f t h e helical flow a r i sing from t h e c i r c u l a t i o n of t h e fluid by t h e external pump was considered t o b e small. The axial Reynolds number was less t h a n 0.1% o f the rotational Reynolds numbers a n d t h e work o f Yamada (4) a n d o t h e r s suggests t h a t for these conditions helical flow effects can b e neglected. T h e r e s u l t s obtained in t h i s s t u d y showed no dependence o n fluid c i r c u l a t i o n rate. Sig nificant t h r e e dimensional e n d e f f e c t s did occur, however, b o t h d u e t o t h e additional surfaces o f t h e seals a n d t h e c y l i nder ends, a n d d u e t o t h r e e dimensional flow e ffects in t h e flow caused by these additional surfaces. T h e t o r q u e d u e t o these e n d effects comprised up t o 50% o f t h e total t o r q u e measured, a n d in p a r t i c u l a r t h e f r i c t i o n a t t h e seals was high. T h e t o r q u e d u e t o all t h e e n d effects co uld n o t b e measured separately fro m the t o r q u e d u e t o t h e cylinders, a n d it was t h e r efore necessary t o t e s t t w o c y l i n d e r s of d i f f e r e n t axial lengths, t o eliminate t h e e n d effects by s u b t r a c t i n g t h e two results. Problems have been encountered w i t h t h i s technique in t h e past (5) because o f t h e size o f t h e e n d effects, a n d so f o u r d i f f e r e n t c y l i n d e r l e n g t h s were tested, t o p e r m i t t h e s u b traction calculations t o be c r o s s checked. T h e shortest axial l e n g t h chosen was 40mm. and t h i s l e n g t h was also chosen f o r most o f the t e s t i n g o f more complex r o t o r a n d stator geometries. T h e o t h e r l e n g t h s chosen f o r t h e c y l i nder te sts were 65mm, 80mm a n d 120mm. It was assumed t h a t f o r a g i v e n r o t o r and housing geometry t h e s k i n f r i c t i o n coefficient i s re lated t o t h e Reynolds number b y a power law relationship CF = aReRb, where t h e s k i n f r i c t i o n coefficient i s g i v e n by CF = G / ( n p I R 4 w 2 ) .
However, t h e s k i n f r i c t i o n coefficient cannot b e determined d i r e c t l y from t h e experimental data, as t h e total t o r q u e measured ( G T), includes a component d u e t o e n d e f f e c t s ( G E ) as well as t h e c y l i n d e r surface component ( G C). T h e r e f o r e t h e s k i n f r i c t i o n coefficient is g i v en as
where Ic i s t h e actual l e n g t h o f c y l i n d e r s u r face, a n d IE i s a n equivalent e x t r a l e n g t h o f c y l i n d e r surface t o allow f o r t h e e n d effects. T o analyse t h e data, a non-dimensional t o r q ue co efficient F was introduced, where: F = (GC + GE)/pu2R
F can t h e n b e d e f i n e d in terms o f Reynolds n u m b e r by :
a n d thence f r o m a series o f t e s t s w i t h diffe r e n t c y l i n d e r l e n g t h s , a, b a n d IE can b e found.
F has n o p h y s i c a l significance, but was chosen as t h e non-dimensional c o e f f i c i e n t t o g i v e a significant gradient o n a plot o f log F against l o g ReR t o f a c i l i t a t e t h e data processing. I m p l i c i t in t h i s a p p r o a c h was t h e assumption t h a t t h e e n d e f f e c t s c a n b e q u a n t i f i e d by a n a d d i t i o n a l l e n g t h o f c y l i n d r i c a l s u r f a c e ( IE) F o r t h i s assumption t o b e valid, a l l t h e e n d e f f e c t s must v a r y w i t h R e y n o l d s Number in a similar fashion t o t h e c y l i n d e r t o r q u e . The additional torque due t o the e n d effects can be s p l i t i n t o t w o components; the torque d u e t o t h e seals, a n d t h e t o r q u e d u e t o t h e end surfaces o f the cylinders. The torque d u e t o t h e b e a r i n g s c a n b e i g n o r e d as d r y r u n s o f t h e t e s t rig showed t h a t t h e b e a r i n g t o r q u e was n e g l i g i b l e .
.
T o assess t h e t o r q u e d u e t o t h e seals, it was assumed t h a t t h e flow a t t h e seal i n t e r f ac e was t u r b u l e n t , w i t h a seal Reynolds Number o f t h e o r d e r o f 105. T h e seal Reynolds Number was c a l c u l a t e d f r o m t h e s l i d i n g v e l o c i t y a n d a n estimate o f t h e i n t e r f a c e f i l m t h i c k n e s s based o n t h e measured leakage a n d p r e s s u r e differential. The c i r c u m f e r e n t i a l flow a t t h e seal i n t e r f a c e i s t h e n comparable to t h e flow between m o v i n g p a r a l l e l plates, f o r w h i c h t h e dependence o f s k i n f r i c t i o n o n Reynolds n u m b e r i s similar t o t h a t f o r c o n c e n t r i c r o t a t i n g c y l i n d e r s ( 6 ) . It i s t h e r e f o r e reasonable t o assume t h a t t h e t o r q u e d u e t o t h e seals can b e c o n s i d e r e d t o b e equal t o the torque due t o a n additional length o f c y l i n d e r surface. T h e t o r q u e d u e t o t h e e n d surfaces o f t h e r o t a t i n g c y l i n d e r c o u l d not b e d e t e r m i n e d theoretically. However, as t h e flow in t h e annular gap i s a basically irrotational turbulent flow a t a high R e y n o l d s number, t h e t u r b u l e n t b o u n d a r y l a y e r s o n t h e e n d surfaces o f t h e a n n u l a r g a p w i l l b e similar t o those o n t h e c y l i n d e r surfaces. From t h i s i t can be assumed t h a t t h e r e l a t i o n s h i p s between s k i n f r i c t i o n c o e f f i c i e n t a n d Reynolds n u m b e r w i l l also b e similar. It i s t h e r e f o r e reasonable t o assume t h a t a l l t h e e n d e f f e c t s w i l l v a r y w i t h Reynolds n u m b e r in a similar fashion t o t h e c y l i n d e r torque. T h e n f o r each a n n u l a r g a p w i d t h t h e r e s u l t s for a l l t h e a x i a l l e n g t h s c a n b e p l o t t e d in t e r m s o f l o g F a g a i n s t l o g ReR a n d processed simultaneously t o g i v e a least squares r e g r e s s i o n fit o f f o u r p a r a l l e l l i n e s t o t h e f o u r sets o f experimental data.
T h e r e g r e s s i o n a n a l y s i s was weighted t o t a k e account o f t h e estimated a c c u r a c y o f t h e measurements. F i g u r e 3 shows a plot of t h e data p o i n t s f o r a 5mm g a p f o r t h e t w o
406 It can b e seen t h a t t h e c o r r e l a t i o n between t h e data a n d t h e f i t t e d lines i s good, a n d in f a c t t h e c o r r e l a t i o n c o e f f i c i e n t s were t y p i c a l l y g r e a t e r t h a n 0 . 9 9 . T h e r e g r e s s i o n analysis d e t e r m i n e d n o t o n l y t h e slope o f t h e line, but also t h e m a g n i t u d e o f t h e e n d e f f e c t s in t e r m s o f the equivalent e x t r a length o f cylindrical surface. T h e e q u i v a l e n t e x t r a l e n g t h s were calculated t o b e 5 8 , 75 a n d 75mm f o r a n n u l a r g a p w i d t h s o f 5 , 10 a n d 20mm r e s p e c t i v e l y . The reduced magnitude o f the end effects f o r t h e 5mm g a p was n o t d u e t o experimental e r r o r , as r e p e a t t e s t s showed t h a t t h e a c c u r a c y o f d e t e r m i n a t i o n o f t h e e n d l e n g t h s was It i s t h e r e f o r e consida p p r o x i m a t e l y f 3mm. e r e d t h a t it was a r e s u l t o f some t h r e e dimensional flow e f f e c t o v e r t h e c y l i n d e r e n d surfaces.
extreme axial lengths o f 40 a n d 120mm togeth e r w i t h t h e f i t t e d lines. T h e r e s u l t s for t h e i n t e r mediate lengths have been omitted f o r c l a r i t y .
12 .o
I
I = 120
log
11.5
11
Al l o w i n g f o r t h e calculated i n f l u e n c e o f t h e e n d effects, t h e r e l a t i o n s h i p s between s k i n f r i c t i o n c o e f f i c i e n t a n d Reynolds n u m b e r were determined f o r the different gap widths.
.o
These were f o u n d t o be:
10.5 6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
log ReR
Fiqure 3
T h e r e appears t o b e o n l y a s l i g h t dependence o n t h e g a p r a t i o ( t / R ) , a n d t h i s is in agreement w i t h t h e r e s u l t s o f o t h e r workers. T h i s c a n b e seen in F i g u r e 4 w h i c h is a plot o f other published results and the present results.
T o r q u e aqainst Reynolds number f o r two a xial lenqths
1.3
1.2
1.1
1 .o
l-
i
P t/R
o
0.15-0.4
A
0.07-0.15
V
0.04-0.07
0
0.02-0.04
5.6
1
5.8
Fiqure 4
6.0
I
6.2
I
f
I
6.4
6.6
6.8
1
7.0 l o g ReR
Selected p u b l i s h e d data compared w i t h mean lines f r o m p r e s e n t r e s u l t s f o r cylinders.
Published r e s u l t s from B i l q e n ( 7 ) . G o r l a n d ( 8 ) & T a y l o r ( 9 )
7.2
407 It would appear from F i g u r e 4 t h a t t h e s k i n f r i c tion coefficient i n i t i a l l y decreases w i t h decreasing gap r a t i o but t h e n begins t o increase as t he g ap r a t i o f u r t h e r decreases. T h e minimum s k i n f r i c t i o n coefficient o c c u r s when t h e g a p r a t i o i s in th e approximate r a n g e o f 0 . 1 t o 0 . 2 . A more precise statement o n t h e e f f e c t o f g a p r a t i o o n s k i n f r i c t i o n coefficient i s n o t possible, as the scatter o f experimental data p o i n t s i s greater t h a n a n y g a p r a t i o effect.
4
assumption was based o n t h e f a c t t h a t t h e s k i n f r i c t i o n c o e f f i c i e n t data f o r complete discs p u b l i s h e d by o t h e r w o r k e r s ( 1 0 , 1 1 , 1 2 ) showed good agreement w i t h t h e p r e s e n t r e s u l t s f o r c y l i n d r i c a l surfaces, as shown in F i g u r e 6 .
COMPLEX GEOMETRY EFFECTS Influence o f Housinq Geometry
4.1
A series o f te sts were c a r r i e d o u t u s i n g p l a i n c y l i ndrical r o t o r s but w i t h changes t o t h e housing geometry. Two d i f f e r e n t housing geometry effects were tested. T h e f i r s t o f these was a change t o t h e i n l e t geometry from t h a t shown in Figure 2 ( i ) t o t h a t shown in F i g u r e 2 ( i i ) . Figure 5 shows t h e data f o r b o t h geometries plotted as log F against log ReR f o r a n 80mm axial l e n g t h c y l i n d e r . It can b e seen t h a t changing t h e i n l e t geometry h a d n o discerna b l e effect o n t h e r o t o r torque.
12.0
11.5
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+
.7 .6 -
1
1 6.4
6.5
1
6.6
6.7
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6.9
7.0
7.1
published radial disc results
8
+
closed inle t
Q
open i n l e t
+ I
6.6
Comparison o f c y l i n d e r r e s u l t s against
i+
1
6.5
log ReR
+
F
6.4
Fiqure 6
0
0
Present c y l i n d e r r e s u l t s lppen (10) Ketola & McGrew ( 1 1 ) D a i l y & Nece ( 1 2 )
6.3
++
-1
6.3
-
.8
.5
1
10.5
.9
I
I
1
6.7
6.8
6.9
1
7.0
The variation o f the total torque for the 121mm r o t o r w i t h r o t o r Reynolds n u m b e r c o u l d t h e n b e p r e d i c t e d by i n t e g r a t i n g t h e local s k i n f r i c t i o n c o e f f i c i e n t o v e r t h e complete r o t o r s u r face. T h e e n d e f f e c t s were i n c l u d e d as a n e x t r a axial length o f cylinder. The predicted total t o r q u e a n d t h e experimental data a r e p l o t t e d in F i g u r e 7 a n d show good agreement.
1
7.1
12.0
F
-
Ef Effect o f doubling
log ReR Figure 5 Influence o f i n l e t qeometry o n t o r q u e T h e second housing change i n v o l v e d t h e i n t r o d u c t i o n o f a circumferential groove, as shown in F i g u r e 2 ( i i ) . Again t h i s h a d n o discernable effect o n t h e r o t o r torque. T h i s i s n o t s u r p r i s i n g as one would expect t h e r o t o r t o r q u e to depend o n t h e velocity g r a d i e n t in t h e shear layer o n t h e rotor, a n d t h i s will b e l i t t l e affect e d by h ousing changes. 4.2
11.5-
11.0
-
10.5
I
Radial Elements
The n e x t series o f tests i n v o l v e d t h e i n t r o d u c t i o n o f r a d i a l elements o n t o t h e r o t o r . T w o r o t o r s were used, w i t h r a d i i o f 116mm a n d 1 2 l m m , t o g i v e radial surface elements a t t h e c y l i n d e r ends. It was assumed t h a t t h e relations h i p between local s k i n f r i c t i o n coefficient a n d local Reynolds number f o r b o t h disc a n d c y l i n d e r elements was t h e same as t h a t between the s k i n f r i c t i o n c oefficient a n d r o t o r Reynolds This number f o r t h e c y l i n d r i c a l surfaces.
6.3
1
,
6.4
I
I
6.5
6.6
Fiqure 7
I
6.7
1
6.8
I
1
6.9 7.0 log ReR
Pr e d i c t e d a n d measured t o r q u e f o r a r o t o r w i t h r a d i a l s u r f a c e elements
1
7.1
408
The proportion o f the total torque due to the r a d i a l surfaces i s small, t y p i c a l l y a r o u n d 15%, but t h e estimate o f t h e t o t a l t o r q u e i s s t i l l sensitive t o l a r g e e r r o r s in t h e estimate o f t h e F i g u r e 7 shows a r e radial surface torque. calculated p r e d i c t i o n assuming t h a t t h e local s k i n f r i c t i o n coefficient for the radial surfaces was double t h e value f o r t h e c y l i n d r i c a l surfaces. It can b e seen t h a t t h e p r e d i c t i o n d i f f e r s s i g n i f i c a n t l y f r o m t h e experimental data. It can be concluded t h a t , f o r t h i s t y p e o f geometry where t h e p r o p o r t i o n o f t o t a l t o r q u e d u e t o t h e r a d i a l surfaces i s small, t a k i n g t h e local r a d i a l s u r f a c e s k i n f r i c t i o n c o e f f i c i e n t t o b e equal t o t h e c y l i n d e r s u r f a c e s k i n f r i c t i o n coefficient i s a v a l i d a p p r o a c h t o p r e d i c t i n g t h e total torque.
T o t e s t f o r i n t e r a c t i o n s between t h e c y l i n d r i c a l a n d r a d i a l s u r f a c e elements, a stepped r o t o r was tested. T h i s r o t o r was o f 40mm a x i a l l e n g t h , w i t h lOmm a t a r a d i u s o f 107mm a n d 30mm a t a r a d i u s o f 116mm. T h e t o r q u e f o r t h e r o t o r was p r e d i c t e d b y t h e i n t e g r a t i o n o f t h e local s k i n f r i c t i o n c o e f f i c i e n t across t h e r o t o r surface, as before, a n d t h e p r e d i c t i o n was again f o u n d t o a g r e e v e r y closely w i t h t h e experimental data, w i t h n o detectable interaction effects. T h e above r e s u l t s show t h a t t h e t o r q u e f o r a n y complex r o t o r a n d h o u s i n g geometry w i t h i n t h e r a n g e t e s t e d can b e p r e d i c t e d by t h e simple a d d i t i o n o f t h e components o f t o r q u e d u e t o t h e r o t a t i n g s u r f a c e elements, t o g e t h e r w i t h a n e x t r a component o f t o r q u e t o allow f o r e n d effects. The influence o f the housing geometry o n t h e t o t a l t o r q u e o f t h e system i s minor a n d can be neglected. T h i s i s most c l e a r l y shown by t h e r e s u l t s f o r t h e system shown in F i g u r e 2 ( i i i ) w h i c h has b o t h a stepped r o t o r a n d a c i r c u m f e r e n t i a l g r o o v e in t h e housing. F i g u r e 8 shows v e r y close agreement between t h e measured t o r q u e r e s u l t s a n d t h e p r e d i c t e d values.
11.5
4
5
GENERAL A P P L I C A B I L I T Y
T o a p p l y these r e s u l t s g e n e r a l l y , it i s necessary t o know w h e t h e r a l l o f t h e e n d e f f e c t s a r e d u e t o t e s t rig effects, o r whether t h e y a r e in p a r t caused by flow e f f e c t s a r o u n d t h e edges o f t h e r o t a t i n g s u r f a c e . It i s clear t h a t a l l t h e e n d e f f e c t s a r e n o t caused b y t h e seal f r i c t i o n a n d c y l i n d e r e n d A 40mm l o n g r o t o r w i t h a 5mm s u r f a c e s alone. a n n u l a r g a p has a n e n d e f f e c t e q u i v a l e n t l e n g t h o f 58mm. A p p r o x i m a t e l y h a l f o f t h i s can b e e x p l a i n e d as d u e t o t h e r a d i a l s u r f a c e s a t t h e sides o f t h e r o t o r a n d t h e l i k e l y losses in t h e t h i c k fluid films o n t h e seal faces. T h e remaining e n d e f f e c t s a r e p r o b a b l y d u e t o t h r e e dimensional flow e f f e c t s w h i c h may be u n i q u e t o t h e t e s t r i g o r may b e g e n e r a l l y applicable
.
T h e o n l y way t o find o u t i s t o compare t h e t o r q u e p r e d i c t e d f o r a complex geometry w i t h measured t e s t r e s u l t s o b t a i n e d o n a completely d i f f e r e n t t e s t rig. T h i s comparison was made, u s i n g r e s u l t s f o r a piece o f t u r b o m a c h i n e r y made up o f t w o sections similar t o F i g u r e 2 ( i i i ) a n d a l o n g p l a i n c y l i n d e r o f 400mm a x i a l l e n g t h . It was f o u n d t h a t t h e p r e d i c t e d t o r q u e , a g r e e d w i t h t h e t e s t data t o w i t h i n 10% i f a l l t h e e n d e f f e c t s were assumed t o b e d u e t o seal losses a n d t e s t r i g e f f e c t s . However, t h i s level o f agreement between p r e d i c t i o n a n d experimental data i s n o t c o n c l u s i v e as t h e l o n g p l a i n c y l i n d e r compone n t swamped a n y i n f l u e n c e o f t h e e n d e f f e c t s T h e r e remains, o n the total torque. t h e r e f o r e , some e n d e f f e c t s w h i c h a r e a f u n c t i o n o f u n k n o w n e f f e c t s in t h e flow o v e r t h e r o t a t i n g surface, a n d it i s s t i l l n o t clear if t h e y a r e u n i q u e t o t h e t e s t r i g o r a r e g e n e r a l l y applicable. It i s t h e r e f o r e concluded t h a t t h e following g u i d e l i n e s w i l l p r o v i d e a reasonable a n d c o n s e r v a t i v e estimate o f t h e t o r q u e o n t h e r o t o r o f a piece o f t u r b o m a c h i n e r y .
(1)
O n l y t h e r o t a t i n g s u r f a c e s h o u l d be considered.
(2)
A mean v a l u e o f s k i n f r i c t i o n c o e f f i c i e n t should be applied to b o t h the radial and t h e c y l i n d r i c a l surfaces. T h e value Re-Oe2 s u g g e s t e d i s Cf = 3.16. w h e r e Re i s e i t h e r a f u n c t i o n o f R f o r t h e c y l i n d r i c a l surface, o r a v a r i a b l e f u n c t i o n o f r f o r t h e r a d i a l surface. These numerical values a r e mean values o f those d e t e r m i n e d f o r t h e d i f f e r e n t g a p r a t i o s a n d h a v e been t e s t e d a n d f o u n d t o g i v e good c o r r e l a t i o n .
(3)
It i s c o n s i d e r e d t h a t a n additional length o f cylindrical surface should be a d d e d as a n estimate o f t h e u n k n o w n effects, p r i n c i p a l l y d u e t o i n t e r f e r e n c e e f f e c t s adjacent t o t h e r o t o r c o r n e r s . F o r r o t o r s w i t h t w o major changes in r a d i u s a l o n g t h e i r a x i a l lengths, s u c h as those tested, a n additional l e n g t h o f 35mm a t a r a d i u s o f 107mm i s appropriate. For r o t o r s w i t h a
11.0
10.5
6.3 6 . 4 Fiqure 8
6.5
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6.9
7.0 7.1 log ReR
Comparison of p r e d i c t e d a n d measured r e s u l t s f o r a stepped r o t o r a n d a q r o o v e d h o u s i n q
409
(4)
greater number o f step changes in radius then it is likely t h a t t h i s l e n g t h should be increased.
(9)
TAYLOR, G 1 . " F l u i d f r i c t i o n between r o t a t i n g c y l i n d e r s 1-torque measurements" Proc. Roy. SOC. Ser A., 1936, 157 p546
In addition, f o r the test rig used f o r the present work, it i s necessary t o add a f u r t h e r l e n g t h o f 35mm o f c y l i n d r i c a l s u r face t o allow f o r t h e f r i c t i o n due t o the seals and adjacent radial surfaces.
(10)
IPPEN, A T "Influence o f viscosity on c e n t r i f u g a l pump performance". T r a n s ASME, 1946, 68, p 823
(11)
KETOLA, H N McCrew, J M "Pressure, frictional resistance a n d flow characteristics o f t h e p a r t i a l l y wetted r o t a t i n g d i s k " Trans. ASME. Ser. F., 1968, 90, p 295 -
(12)
DAILY, J W, NECE, R E "Chamber dimension effects o n induced flow and frictional resistance o f enclosed r o t a t i n g disks". Trans. ASME. Ser D., 1969, 82, p 217
It is considered t h a t t h e e r r o r in t h e estimate o f torque obtained u s i n g the above procedure is unlikely t o be greater than 20%, f o r r o t o r s w i t h gap ratios in t h e range 0.05 t o 0.2, and w i t h similar ratios o f radial t o c y l i n d r i c a l r o t o r surface areas t o those o f t h e r o t o r s tested. 6
ACKNOWLEDGEMENTS
The author acknowledges t h e assistance o f the SERC and o f t h e T.I. Research Laboratories who sponsored t h i s work u n d e r a CASE Project. The author also wishes t o acknowledge the s u p p o r t a n d advice g i v e n by t h e s t a f f o f the Department o f Aeronautical Engineering at the U n i v e r s i t y o f Bristol.
References FLOYD, C G "A S t u d y o n frictional losses o f enclosed r o t o r s a t high Reynolds numbers" Ph. D. Thesis, U n i v e r s i t y o f Bristol, September 1982. PAI, S I " T u r b u l e n t flow between r o t a t i n g cylinders" NACA Tech. Note 892, 1943. USTIMENKO, B P e t al " T u r b u l e n t t r a n s f e r in r o t a r y flows o f an incompressible fluid". F l u i d Mech. Soviet Research, p 121 1972,
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YAMADA, Y "Torque resistance Of a flow between r o t a t i n g coaxial c y l i n d e r s h a v i n g axial flow". Bull. JSME, 1962, 5. p 635 SULMONT, P. BOURGET, P L "Mesure experimentale du couple de frottement d'un c y l i n d r e t o u r n a n t dans un c y l i n d r e f i x e p o u r des faible e n t r e f e r s e t des g r a n d s nombres du Reynolds". Annales de Mechanique, Ecole Nat. de Mech., Nantes, 1969. ROBERTSON, J M. "On t u r b u l e n t plane Couette flow", 6th Mid West Conference o n F l u i d Mech., Univ. o f Texas 1959. BILGEN, E, BOULOS, R "Functional dependence o f t o r q u e coefficient o f coaxial c y l i n d e r s o n gap w i d t h a n d Reynolds numbers". Trans. ASME Ser. I . 1973 95, p 122.
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GORLAND, S H e t al "Experimental windage losses f o r close clearance r o t a t i n g c y l i n d e r s in t h e t u r b u l e n t flow regime". NASA TM X-52851, 1970