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W~irm e- und S to f f t ibe r t ragung 22 , 55 -6 0 (1988) W ~ r m e -u n d S t o f f' t ib e r t r a g u n g9 Spr inger -Ver lag 1988

Tu rb u len t con vect io n in h e l ico id a l tu b esM . C h fi ve z, W . Z h i x u e , M e x i c o C i t y a n d M . S e n , N o t r e D a m e , I N , U S A

Abstract. T h e r e s u lt s o f e x p e r i m e n t s c a r r i e d o u t w i t h f o r c e d c o n -v e c t i on i n h e l i c o i d a l t u b e s a r e r e p o r t e d . W a t e r w a s m a d e t o f l o wi n s i d e t h e t u b e k e p t w i t h i n a s t e a m c h a m b e r . T h e f l o w r a t e a sw e l l a s t h e b u l k i n l e t a n d o u t l e t t e m p e r a t u r e s w e r e m e a s u r e d .N ine co i l s o f d i f fe ren t geom etr ies were tes ted , each one fo r tend i f f e r e n t m a s s f l o w r a te s . T h e e x p e r i m e n t a l d a t a f o r t h e i n t e r n a lconvec t ion hea t t ransfe r coef f ic ien t can be cor re la ted to w i th in9 .5% in te rm s o f the p ower- law re la t ion :Nu = 0.00032 Re T M D 0"29p0.67,w h e r e Nu a n d Re a r e t h e N u s s e l t a n d R e y n o l d s n u m b e r s r e s p e c -t i v el y ; D i s t h e t u b e d i a m e t e r a n d P i s th e p i t c h , b o t h n o n d i m e n -s i o n a l i z e d w i t h r e s p e c t t o t h e m e a n c o i l d ia m e t e r . T h e p r o p e r t i e so f w a t e r i n t h i s c o r r e l a t io n a r e c o n s i d e r e d a t t h e a v e r a g e o f t h ei n l e t an d o u t l e t b u l k t e m p e r a t u r e s .Turbulente Konvektion in gewendelten RohrenZusammenfassung. Es w ird t ibe r Versuche m i t e rzwungener Kon-v e k t i o n i n g e w e n d e l t e n R o h r e n b e r i c h t e t . D a b e i f l o B W a s s e r i nd e n R o h r e n , d i e i n e i n e r m i t D a m p f b e a u f s c h l a g t e n K a m m e ru n t e r g e b r a c h t w a r e n . G e m e s s e n w u r d e n d i e z u s t r 6 m e n d e M e n g eu n d d i e m i t t l e r e n E i n - u n d A u s l a B t e m p e r a t u r e n d e s W a s s e rs . E sw u r d e n n e u n v e r s c h i e d e ne W e n d e l n u n t e r s u c h t, j e d e m i t 1 0 v e r -s c h i e d e ne n M e n g e n s tr 6 m e n . D i e e x p e r i m e n t e l l e n E r g e b n is s e f t irden W ~irm et ibergangskoeff iz ien ten be i d iese r K onvek t ion in denR o h r e n k 6 n n e n i n n e r h a l b v o n 9 ,5 % G e n a u i g k e i t m i t d e m A n s a t zNu : 0,00032 Re TM D 0"29p0,67berechne t werden . H ie r in s ind Nu d i e N u s s e l t - u n d Re d i e R e y -n o l d s - Z a h l, D d e r R o h r d u r c h m e s s e r u n d P d e r A b s t a n d d e rW e n d e l n b e i d e d i m e n s i o n s l o s g e m a c h t m i t d e m m i t t l e r e n W e n -d e l d u r c hm e s s e r . D i e Z u s t a n d s e i g e n s c h a f te n d e s W a s s e r s i n d i e s e rG l e i c h u n g s i n d m i t d e m M i t t e l w e r t zw i s c h e n E i n - u n d A u s t r i t ts -t e m p e r a t u r a n z u se t z en .

1 I n t r o duc t i o nT h e s t u d y o f f l o w in a c u r v e d t u b e h a d a s it s s t a r t i n gp o i n t t h e w o r k o f D e a n [ 1, 2 ] n e a r l y s i x t y y e a r s a g o . S i n c et h e n t h e r e h a ve b e e n a c o n s i d e r a b l e n u m b e r o f p u b l i c a -t i o n s i n t h e l i t e r a t u r e r e l a t e d t o f l u i d m e c h a n i c s i n c u r v e da n d h e l i c o i d a l t u b e s . A w i d e v a r i e t y o f n u m e r i c a l a n dt h e o r e t i c a l m e t h o d s h a v e b e e n u s e d , i n a d d i t i o n t o e x p e r i -m e n t s . T h e p r i n c i p a l d i f f e r e n c e w i t h r e s p e c t t o s t r a ig h t

p i p e s l i e s i n t h e p r e s e n c e o f s e c o n d a r y v o r t e x t y p e m o t i o ni n a t r a n s v e r s e c r o s s - se c t i o n . A s a d i r e c t c o n s e q u e n c e o ft h i s s e c o n d a r y f l o w th e r e i s a h i g h e r p r e s s u r e d r o p f o rs u c h t u b e s a s w e l l as h i g h e r h e a t a n d m a s s t r a n s p o r t . O u ri n t e r e s t h e r e i s in c o n v e c t i v e h e a t t r a n s f e r t o t h i s k i n d o ff l o w .

C u r v e d h e a t t r a n s f e r d u c t s a r e f a i r l y c o m m o n i ne n g i n e er i n g a p p l i c a t i o n s w h e r e c o m p a c t n e s s a n d h i g hh e a t t r a n s f e r r a t e s a r e d e s i r a b l e . A s i s u s u a l i n s u c hc i r c u m s t a n c e s , th e a d v a n t a g e o f h i g h e r c o n v e c t i o n c o e f f i -c i e n ts i s o f f s e t b y t h e n e e d f o r l a r g e r p u m p i n g p o w e r .M o s t o f t h e s e c u r v e d d u c t s a r e i n t h e f o r m o f h e l i c o i d a lc o i ls m a d e u p o f t u b e s o f c i r c u l a r c r o s s - s ec t i o n . I t i si m p o r t a n t t h e n t o b e a b l e to r e l a t e th e h e a t t r a n s f e r f r o ms u c h c o i ls w i th t h e f l o w a n d g e o m e t r i c a l c h a r a c t e r i s t i c s .M o r e o v e r , t h e s e c o i l s a r e o p e r a t e d n o t j u s t in t h e f u l l yd e v e l o p e d r e g i m e , b u t m o s t l y u n d e r e n t r a n c e c o n d i t i o n s .

L i t e r a t u r e d e a l in g w i t h t h e p u r e l y h y d r o d y n a m i c a la s p e c t o f t h e p r o b l e m i s p l e n t i f u l a n d w i l l n o t b e d i s -c u s s e d h e re . H e a t t r a n s f e r s t u d i e s h a v e b e e n m a d e b y an u m b e r o f a ut h o rs . M o r i a n d N a k a y a m a [3 , 4 ] a n a l y z e da n d e x p e r i m e n t e d w i t h l a m i n a r a n d t u r b u l e n t f l o w i n ac o i l c o n s i s ti n g o f a si n g l e t ur n . O z i s i k a n d T o p a k o g l u [5 ]u s e d a s e ri e s e x p a n s i o n m e t h o d i n a n a n a l y t i c a l s t u d y .C h e n g a n d A k i y a m a [6 ] f i r s t c o n s i d e r e d d u c t s o f r e c t a n g u -l a r c r o s s - se c t i o n , a n d t h e n e m p l o y e d [7 ] f i n i t e d i f f e r e n c em e t h o d s t o s t u d y t h e e ff e c ts o f P r a n d t l n u m b e r o n h e a tt r a n s fe r . P a t a n k a r e t a l . [8 ] u s e d a c a l c u l a t i o n p r o c e d u r ef o r th r e e - d i m e n s i o n a l p a r a b o l i c f l o w s to d e t e r m i n e t h ev e l o c i t y a n d t e m p e r a t u r e f i e ld s . T y a g i a n d S h a r m a [ 9]c o n s i d e r e d t h e e f f e c t o f h e a t g e n e r a t i o n a t t h e w a l l o f t h et u b e . Z a p r y a n o v e t a l . [ 10 ] u s e d n u m e r i c a l m e t h o d s t od e t e r m i n e N u s s e l t n u m b e r s i n c u rv e d t u b e s. Y a o a n dB e r g e r [1 l ] a n d P r u s a a n d Y a o [ 12 ] r e p o r t o n r e s u l t s f o r ac o n s ta n t a x i a l t e m p e r a t u r e g r a d i e n t i n c l u d i n g b o t h b u o y -a n c y a n d c e n t r i f u g a l f o r c e s.

N u m e r i c a l r e s u l ts a r e o f l i m i t e d u s e f o r t u r b u l e n t f l o w su n l e ss c a r e f u l l y c h e c k e d a g a i n s t e x p e r i m e n t a l d a t a . T h el a t t e r a r e f e w , a n d s o a r e c o r r e l a t i o n s w h i c h t a k e i n t oa c c o u n t t h e d i f f e r e n t g e o m e t r i c a l p a r a m e t e r s f o r t h e

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56 W/irm e- und Stoffiibertragung 22 (1988)h e l i c o id a l c o i l . S e b a n a n d Mc L a u g h l in [1 3 ] u s e d e l e c t r i c a lh e a t i n g t o p r o v i d e a c o n s t a n t h e a t f l u x a l o n g t h e t u b e i no r d e r t o m e a s u r e t h e c o n v e c t i o n h e a t t r a n s f e r c o e f f i c i e n t .R o g e r s an d M a y h e w [ 1 4 ] h a d a s t e a m c h a m b e r t o m a i n -t a i n a c o n s t a n t o u t e r t e m p e r a t u r e . B u l k t e m p e r a t u r e s w e r ei n f e r r e d f r o m t h e r m o c o u p l e r e a d i n g s a n d a N u s s e l t n u m -b e r c o r r e l a t i o n o b t a i n e d f o r t h r e e d i f f e r e n t c o i l s . D e t a i l e dt u r b u l e n t f l o w r e s u l t s w e r e a l s o r e p o r t e d b y S c h m i d t [ 1 5 ](a n d e r ro n e o u s ly r e p ro d u c e d in [1 6 ] ) fo r f r i c t io n f a c to r sa n d N u s s e l t n u m b e r s i n h e l i c o i d a l t u b e s . S o m e a d d i t i o n a li n f o r m a t i o n f r o m t h e e x p e r i m e n t s w a s re p o r t e d b y M i r o -p o l s k i i [ 17 ] . T h e e x p e r i m e n t s o f J a n s se n a n d H o o g e n d o r n[ 1 8 ] w e r e f o r R e y n o l d s n u m b e r s l o w e r t h a n t h o s e i n t h ep r e s e n t s t u d y . A r e c e n t s e t o f e x p e r i m e n t s h a s b e e nc a r r i e d o u t b y N a g e t a l . [19 ] w h o p re s e n te d th e i r r e s u l t sa s a p o w e r - l a w c o r r e l a t i o n .

I n n o n e o f t h e s e e x p e r i m e n t s w a s t h e e f f e c t o f t h e p i t c ho f t h e c 0 il i n c l u d e d a s a p a r a m e t e r . I n t h e p r e s e n t i n v e s -t i g a t i o n w e o b t a i n d a t a u n d e r t u r b u l e n t c o n d i t i o n s i no r d e r t o c o r r e l a t e t h e i n t e r n a l h e a t t r a n s f e r c o e f f i c i e n tw i t h f l o w a n d a l l g e o m e t r i c a l p a r a m e t e r s o f th e c o i l. O n l yw a t e r i s u se d a s t h e w o r k i n g f l u i d s o t h a t P r a n d t l n u m b e re f fe c t s a re n o t i n c lu d e d .

2 Experimental procedureF i g u r e 1 s h o w s s c h e m a t i c a l l y t h e e q u i p m e n t t h a t w a se m p l o y e d i n t h e e x p e r i m e n t s . S t e a m w a s s u p p l i e d t o t h ec h a m b e r f r o m a b o il e r. T h e t e m p e r a t u r e o f th e c h a m b e rw a s a lw a y s a r o u n d 8 5 ~ a n d , n o t b e i n g a ir - t i g h t, w a s a ta t m o s p h e r i c p r e s s u r e . T h i s p r e s s u r e i s n o r m a l l y a r o u n d7 8 k P a a n d t h e c o r r e s p o n d i n g s a t u r a t i o n t e m p e r a t u r e o fw a t e r s u b s t a n c e i s 9 3 ~ C o n d e n s a t e f r o m th e c h a m b e rw a s c o n t i n u o u s l y r e m o v e d t h r o u g h a l o w e r d r a i n . T h eh e l i c o i d al p a r t o f th e t u b e h a d a l on g i s o t h e r m a l e n t r a n c er e g i o n t o p r o v i d e f l o w a t t h e i n l e t w h i c h w a s f u l l y d e v e l -o p e d h y d r o d y n a m i c a l l y . H o w e v e r , th e p r e s e n c e o f t h e

S t e a m i n l e tW a t e r ~ Eo u t l e t

O p e n t o a t m o s p h e r e

T h e r m o c o u p l e 7~

T h e r m o m e t e r, ~ W a t e ri n l e t

m i x i n g c u p ju s t b e f o r e e n t r y to t h e s t e a m c h a m b e r s h o u l dh a v e th e e f f e c t o f i n c re a s in g lo c a l t u rb u le n c e . T h e c o i l w a ss u p p o r t e d i n s i d e t h e c h a m b e r , a n d w i t h i n t h i s c o i l t a pw a t e r w a s m a d e t o f l o w . T h e b u l k t e m p e r a t u r e s a t t h ew a t e r i n l e t a n d o u t l e t w e r e m e a s u r e d w i t h m e r c u r y i ng l a s s t h e r m o m e t e r s d i p p e d i n t o m i x i n g c u p s l o c a t e d a tt h e s e p o s i t i o n s . A l l o t h e r t e m p e r a t u r e s w e r e m e a s u r e dw i t h t y p e J , g a u g e 3 0 t h e r m o c o u p l e s . T h e t e m p e r a t u r ed i s t r i b u t io n a l o n g th e w a l l o f t h e t u b e w a s m e a s u r e d b y as e r ie s o f ( N + 3) t h e r m o c o u p l e s t i g h t l y a t t a c h e d t o t h eo u t s id e , w h e re N i s t h e to t a l n u m b e r o f t u rn s in th e c o i l.T h e s e w e r e c o n n e c t e d t o a D a t a A c q u i s i t i o n S y s t e mY O D A C - 8 T y p e 3 8 7 3 w i t h p e r i o d i c o u t p u t a n d d i s p l a yd i r e c t l y i n d e g r e e s C e l s i u s . A n o t h e r t h e r m o c o u p l e c o n -n e c t e d t o a Y E W 2 5 7 2 D i g i t a l T h e r m o m e t e r w a s p l a c e dw i t h in t h e c h a m b e r t o m o n i t o r t h e t e m p e r a t u r e o f th es t e a m . T h e p r e s s u r e d i f f e r e n c e b e t w e e n t h e i n l e t a n do u t le t w a s m e a s u r e d b y a d i f f e r e n ti a l m e r c u r y m a n o m e t e rc o n n e c t e d t o t h e s e p o in t s . W a t e r f l o w i n g o u t i n a n i n t e r v a lo f t i m e , m e a s u r e d w i t h a s t o p - w a t c h , w a s w e i g h e d t od e t e r m i n e t h e m a s s f l o w r a t e . W a t e r i n l e t w a s a t a c o n -s t a n t t e m p e r a t u r e o f 1 9 ~ a n d t h e o u t l e t w a s b e t w e e n3 0 o a n d 4 0 ~ d e p e n d i n g o n th e c o i l u s e d .

I n o r d e r t o c o r r e l a t e t h e h e a t t r a n s f e r w i t h t h e g e o m e t r yo f th e c o i l , d i f f e re n t c o i l s w e re u s e d in th e e x p e r im e n t .T h e s e w e r e a ll m a d e f r o m c o m m e r c i a l l y a v ai l a b l e c o p p e rtu b in g o f i n t e rn a l a n d e x te rn a l d i a m e te r s 7 . 5 a n d 9 .5 m mr e s p e c t i v e l y , f o r m e d b y b e i n g w o u n d a r o u n d c y l i n d e r s o fd i f f e r e n t d i a m e t e r s . T h i s w i n d i n g p r o c e s s p r o d u c e d ac h a n g e i n t h e t u b e l e n g th . T h i s l e n g t h w a s m e a s u r e d a c c u -r a t e l y u s i n g a p l a s t i c t u b e o f t h e s a m e d i a m e t e r w h i c hc o u l d b e c o i l e d s i m i l a r l y a n d t h e n u n c o i l e d t o e n a b l e i t sl e n g t h t o b e d e t e r m i n e d . S i n c e t h e i n n e r s u r f a c e f i n i s h o fth e tu b e s i s c ru c ia l i n th e h e a t t r a n s fe r p ro c e s s , t h e s a m etu b e s w e re u s e d fo r a se r i e s o f ru n s w i th d i f f e re n t p i t c h e s .

B e f o r e s t a rt i n g e x p e r i m e n t a t i o n w i t h e a c h c o i l , a l l te m -p e r a t u r e r e a d i n g s w e r e f i r s t c h e c k e d t o b e a t r o o m t e m -p e r a t u r e . T h e s t e a m a n d w a t e r w e r e t h e n t u r n e d o n a n d ,a f t e r a b o u t 1 0 m i n u t e s , t h e t e m p e r a t u r e s r e a c h e d t h e i rs t e a d y - s t a t e v a lu e s w i th o n ly s m a l l f l u c tu a t io n s . A t th i st i m e t h e m a s s f l o w r a t e , t h e p r e s s u r e d r o p a n d t e m p e r a -t u r e d a t a w e r e r e c o r d e d . T h e m a s s f l o w r a t e w a s c h a n g e da n d m e a s u r e m e n t s w e r e t a k e n a f t e r w a i t i n g f o r s t e a d y -s t a t e . T h i s w a s r e p e a te d a s m a n y t im e s a s n e c e s s a ry .R e p r o d u c i b i l i t y o f d a t a w a s c o n f i r m e d a n d o n l y n e wc o p p e r t u b e s w e r e u s e d , a s t h e h e a t t r a n s f e r s u r f a c e a n dc h a r a ct e r is t i cs d o c h a n g e o v e r l o n g p e r i o d s o f t i m e .

I DataoequisiiionsysterEXPERIMENTALET-UP

Fig. 1. Schematic o f experimental set-up

3 Hea t transfer calculat ion sH e a t t r a n s f e r b e t w e e n t h e w a t e r a n d t h e o u t e r s u r f a c e o ft h e t u b e c a n b e r e p r e s e n t e d b y a n o v e r a l l h e a t t r a n s f e rc o e f f i ci e n t U o f t h e f o r mU = 1 / ( d2 / d ~ h + d 2 1 n ( d 2 / d l ) / 2 k w ) , (1 )

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M. Chfivez et al. : Turbulent convection in helicoidal tubes 57defined on the basis of the outer surface area and tem-perature difference between the fluid and the outersurface of the tube; d] and d2 are the inner and outerdiameters respectively of the tube and kw the thermal con-ductivity of the material of the tube. The convective heattransfer coefficient h between the water and the inner wallof the tube is taken to be a constant. The values deter-mined from the present study will thus be an average overthe entire length of the coil.

We take x to be the longitudinal coordinate along thehelicoidal tube and T(x) the local bulk temperature of thewater. Energy balance for the water givesmcd T/d x = gd 2 U (T2 - T) , (2)where m and c are the mass flow rate and specific heatrespectively of the water. Tz(x) is the temperature of theouter surface of the tube. Integrating this equation be-tween the i n l e t and outlet, we have

LH[To exp (L/H) - T/] = ~ exp (x /H) Tz(x) dx, (3)

0

where L is the total length of the tube, Ti and To are theinlet and outlet temperatures. FurthermoreH = rn c/Tc d 2 U. (4)

In Eq. (3), T/, To and T2 (x) are known f rom m easure-ments for any particular test run. The integral is solvediteratively for H, with a trapezoidal rule being used toperform the integration on the right hand side. It is im-portant to mention that an accurate measurement of L isnecessary since a 1% error in L produces an error of ab out15% in H. This error in H was kept to be within 0.8% in thepresent experiments. Equation (4) then gives the overallheat transfer coefficient U, and Eq. (1) the mean value ofthe internal heat transfer coefficient h. The mass flow ratern is needed for these calculations as well as materialproperties such as the specific heat of water c and thethermal conductivity of copper kw, apart from measure-ments of the inner and outer diameters.

The internal convection coefficient is nondimensional-ized as a Nusselt number, using the tube inner diameter asfollowsNu = h dJk f , (5)where kf is the thermal conductivity of water taken at theaverage bulk temperature between the inlet and the outlet.This Nusselt number is also a mean value over the lengthof the coil.

The Nusselt numbers obtained are to be correlatedwith the Reynolds number defined asRe = 4rh/Tr# dl , (6)where the dynamic viscosity of water /~ is also at theaverage of the inlet and outlet bulk temperatures.

4 E x p e r i m e n t a l r e s u l t s a n d c o r r e l a t i o nNine different coils were employed (indicated by theletters A-I), the geometrical characteristics of which aregiven in Table 1. These are represented by the n o n d i m e n -sional tube diame terD = d/I, (7)and the nondimensional pitchP =p/t. ( 8 )Here d and p are the dimensional tube diameter and pitchrespectively and / the mean diameter of the coil. Due tothe analysis of Dean [1, 2], it might be tempting to use theDean number Re ~/D instead of Re and D separately.Experience however shows that such a dependence on theDean number alone is not possible, at least for turbulentflows.

For each one of these coils ten different mass flow ratesof water were used, corresponding to a total of ninetydifferent test runs. Only in one of these was the flowlaminar as determined from the transition Reynolds num-ber criterion of Ito [20]. Data for this run were remo ved~before determining the final correlation.

The temperature of the exterior surface of the tube wasfound to vary approximately in a linear manner along thetube with the average standard deviation from a straightline for all the test runs being only 1.6 ~ As an example,Fig. 2 shows this temperature distribution for four differ-ent mass flow rates. The letters i and o in the abscissarepresent the inlet and outlet of the coil respectively. Theposition of the thermocouples are indicated by the num-ber of the turn.

Nusselt numbers determined in this experiment areshown in Fig. 3 as functions of the Reynolds number ofcoils A - E all with D = 0.0441. Simila r inf ormatio n is

80

7 0

6 0

5 0

30

2O

10

0

ooo t,o

ooo

ooooo o0

" ooo

o ~ = 0 . 0 9 1 1 k g / s9 m : 0 . 0 7 9 4 , ,

~ : 0 . 0 6 9 2 , ,o ~ = 0 . 0 5 6 8 , ,

I I I I I I I I I 1101 2 3 4 5 6 7 8 9 1T H E R M O C O U P L E L O C A T I O N

Fig. 2. Temperature distribution at exterior of wall for coil D

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5 8

2 5 0 12 2 0 :1 9 01 6 0

1 5 0

= 1 0 0z

7 0

4 0

a =a 9

o . o ~ x e l 9g x . o o o

2 s (1 4 )

1~ 1~5R e x 1 0 - s

o Ex D D = 0 . 0 4 4 1o C9 B~ A

115 1 7 19

Fig . 3 . N u - R e d a t a f o r c o i l s A - E . T h e l i n e s a r e f r o m p u b l i s h e dco rre la t io n s

W g rm e- u n d S t o ff t ~b er tra g u ng 2 2 ( 1 9 8 8 )( 2 0 ) - ~ .

S t r o i g h t p i p e( s m o o t h )

1 o Ex Do C9 Bo A

0 I9 111 lt 3R e x l O - 3

Fig . 6 . f - R e d a t a f or c o i l s A - E . T h e b r o k e n l i n e s a r e f r o m p u b -l i s h ed co rre la t io n s

D = 0 . 0 4 4 1

I115 17 19

2 5 0

2 2 O1 9 01 6 0

1 3 0

z l O 0

7 0 o Xo X o

4 0 i9 l i l

F i g . 4 . N u - R e d a t a f o r c o i ls E - I

o eo

" o x 9o

x Eo Fo G9 Hm I

J15

R e x 10 3

ao" o Io

o X O xo x a

e o l

P = 0 . 2 5 4

o

I I15 17 19

1 2 5 0 + 5 % / /1 1 5 0 o / / ~

J / O ~1 0 5 0 / ' o / / oo / ~ o / // o o / o o, , / o / / / o

Y 5 0 ~/ / / z / o / / o

o r o / ~y , ~ / o o

55 O

4 5 O * i3 5 0 I7 8 9 10 11 12 15 14 15 16 17 18 19

R e x 10 -~Fig . 5 . A l l ex p er im en t a l d a t a p o in t s ; s t ra ig h t l in e i s N u / D 0"29p 0 . 6 7a s a f u n ct io n o f R e; bro ken l ines repre sent ___ 5% error bo und s

9 5 02"~ _ 8 5 0mN~ 7 5 0=z 6 5 0

s h o w n i n F ig . 4 f o r c o i l s E - I w i t h P = 0 . 2 5 4 . F o r t h e r e l a-t i o n b e t w e e n t h e N u s s e l t n u m b e r , t h e R e y n o l d s n u m b e r ,t h e n o n - d i m e n s i o n a l t u b e d i a m e t e r a n d p it c h a p o w e r - l a wc o r r e l a t i o n i s p r o p o s e d :N u = q R e ' D s P . (9 )A l e a s t sq u a r e t e c h n i q u e w a s u s e d o n t h e lo g a r i t h m s o ft h e n o n d i m e n s i o n a l g r o u p s i n v o l v e d i n o r d e r t o d e t e r m i n et h e b e s t f i t v a l u e s o f t h e c o e f f i c i e n t q , a n d o f t h e e x p o -n e n t s r , s a n d t . T h e s t a n d a r d d e v i a t i o n o f t h e p e r c e n t a g ee r r o r f r o m t h i s p o w e r - l a w w a s 9 .9 % . T w o t e s t r u n s w h i c hh a d e r r o rs l ar g e r t h a n 1 8 % , a n d t h e l a m i n a r f l o w d a t a ,w e r e e l i m i n a t e d . R e p e a t i n g t h e r e g r e s s i o n p r o c e s s w i t ht h e r e m a i n i n g e i g h t y s e v e n t e s t ru n s , t h e f o l l o w i n g v a l u e sw e r e o b t a i n e d :q = 0 . 0 0 0 3 2r = 1 . 5 4s = 0 . 2 9t = 0 . 6 7 . ( 1 0 )S t a n d ar d d e v i a t i o n o f t h e d a t a p o i n t s f r o m t h i s r e l a t i o nw a s o n l y 9 .5 % . F i g u r e 5 s h o w s t h e v a l u e s o f Nu /D 0"29p 0 . 6 7a s a f u n c t i o n o f R e f o r a ll n i n e t y t e s t r u n s. C o r r e l a t i o n ( 9 )i s r e p r e s e n t e d b y a s t r a i g h t l i n e i n t h i s g r a p h a n d i s s e e nt o g o t h r o u g h th e e x p e r i m e n t a l l y d e t e r m i n e d d a t a p o i n t s .T h e + 5% a n d - 5% e rr o r b o u n d s a r e in d i c a t e d b y b r o k e nl i n e s a n d m o s t o f t h e d a t a p o i n t s a r e w i t h i n t h i s r e g i o n .

A l t h o u g h t h is e x p e r i m e n t w a s d i r e c t e d t o w a r d s t h eh e a t t r a n s fe r a s p e c t o f h e l i c o i d a l t u b e s , p r e s s u r e d r o pi n f o r m a t i o n w a s a l s o t a k e n . A v e r a g e f r i c t io n f a c t o r s f c a nb e c a l c u l a t e d f r o mf = 3 2 ~ A / ~ m 2 L d 3 , ( 1 1 )w h e r e ~ i s t h e d e n s i t y o f w a t e r a n d A t h e p r e s s u r e d r o pb e t w e e n t h e i n l e t a n d t h e o u t l e t . I n F i g s . 6 a n d 7 , f i ss h o w n a s a f u n c t i o n o f t h e R e y n o l d s n u m b e r f o r t u b e s

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M. Ch~tvez et al.: Turbulent convection in helicoidal tubes 59

o0 2

n

on

a u~ o o o ~ . . . . ~ o

I9 I11

R e x I 0 - 3

Fig. 7. f - R e data for coils E- I

S t r o i g h t p i p e( s m o o t h )

x Eo F P : 0 . 2 5 4< > G9 Ho I

1 3 1 5 I

T able 1. Geometrical characteristics of the coils usedCoil l d N p D PA 0.170 0.0075 7 0.061 0.044 0.359B 0.170 0.0075 8 0.055 0.044 0.324C 0.170 0.0075 9 0.050 0.044 0.294D 0.170 0.0075 10 0.047 0.044 0.277E 0.170 0.0075 11 0.043 0.044 0.255F 0.230 0.0075 7 0.059 0.033 0.255G 0.215 0.0075 7 0.055 0.035 0.255H 0.150 0.0075 11 0.038 0.050 0.255I 0.123 0.0075 14 0.031 0.061 0.255

1 9

A- E with D = 0.441 and E - I with P = 0.255 respectively.The friction factor for a straight tube is also representedfor comparison.

There are many experiments related to the measure-ment of friction factors in helicoidal tubes. Comparisonwas made with the correlations of Ito [20] and Mishra andGupta [21]. These are included in Fig. 6 for constant D.The scatter in the present data is about 4% and lies inbetween the two. Friction factors determined by Sebanand McLaughlin [ 13] were also lower than Ito's forD = 0.059. Due to the intimate relation between the hydro-dynamics and the heat transfer in a coiled tube this com-parison gives an estimate of the veracity of the presentconvection results.

5 D i s c u s s i o n

Heat transfer measurements typically involve some scatterin the results so such that empirical correla tions can beused only with the knowledge that var iat ions from thecalculated values may occur. The purpose behind taking alarge number of data is precisely to reduce these uncer-

tainties to a minimum. As an example the correlationpresented by Rogers and Ma yhew [14] is determined b ythem to be correct only within 10%. Such an error is quitereasonable in the heat transfer literature. Another factorthat must be taken into account in the application of thesecorrelations is the difference between the results obtainedby different groups of investigators. Provided all reason-able care has been taken in carrying out the relevantexperiments, it is difficult to decide between correlationswhich provide radically different values of the heattransfer coefficient. Fortunately, more often than not thedifferent correlations provide figures which are verysimilar even though the correlations themselves may lookentirely different.

In the present case, the basic reason for experimentingwith different coil geometries was to be able to correlatethe Nusselt number not only with the Reynolds numberand tube diameter as had been done, but also to includethe effects of the coil pitch. That the hydrodynamics aswell as heat transfer characteristics of the helicoidal tubeshould depend on the coil pitch also, is evident if onethinks of the extreme limits of a tightly coiled tube ascompared to a very loosely coiled one. In fact the correla-tion deduced here points to a fair degree of dependence ofthe overall Nusselt num ber on the pitch of the coil.

The existence of secondary motion which enhances theheat transfer also makes it difficult to estimate the bulkfluid temperature from a local temperature reading. Thebulk temperature can be easily measured only at the inletand at the outlet. As a consequence it is very difficult toobtain quantitative results for local Nusselt numbers. Fur-thermore, as the secondary motion is related to the flowReynolds number, the difference between the coil Nusseltnumber and a straight tube Nusselt number shouldincrease with Reynolds number. In other words, the finalpower-law correlation should have an exponent which islarger than its value for a straight tube.

Normally the Nusselt number of a forced convectionprocess is also correlated with respect to the Prandtlnumber. In this case this was not done, as only water wasused as the working fluid. The Prandtl number of water isextremely sensitive to temper ature, being 61.4% higher at20 ~ than at 40 ~ The ma jor part of this variation is dueto the kinematic viscosity which is 52.4% higher at thelower temperature. A correlation with Prandtl numberwould be confusing to the user and would not strictly bevalid for fluids with other Prandtl numbers. The variationof the kinematic viscosity also presents a problem withrespect to the Reynolds number. The local Reynoldsnumber at the inlet to the coil may be 10,000 while that atthe outlet may be 15,240, simply due to a variation inviscosity as the water t empera ture changes fr om 20 ~ to40 ~ In the present correlation the Reynolds numbe r ismeasured at a mean bulk temperature between the inletand outlet. However, it is obvious that experimentscarried out at other temperature ranges would give differ-

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60 W~rrne- und S to f f i ibe r t ragung 22 (1988)e n t q u a n t i t a t i v e r e s u l t s . T h i s f a c t c o u l d a c c o u n t f o r m a n yo f t h e s e e m i n g l y d i f f e r e n t c o r r e l a t io n s a v a i l a b l e f o r m o s tt u r b u l e n t f l o w i n d u c t s .

C o m p a r i s o n o f t h e p r e s e n t d a t a w i th p u b l i s h e d c o r r e-l a t i o n s c a n b e m a d e i f o n e a s s u m e s a c o n s t a n t P r a n d t ln u m b e r . T a k i n g i t s v a l u e f o r 2 0 ~ s o m e o f t h e s e a r ec o m p a r e d i n F i g . 3 to t h e p r e s e n t m e a s u r e m e n t s , f o r as e r i e s o f c o i ls w i t h t h e s a m e D . P r a t t ' s c o r r e l a t i o n [ 2 2] isg i v e n i n [4 ]. T h e q u a n t i t a t i v e v a l u e s a r e s e e n t o b e s i m i l a re v e n t h o u g h t h e t r e n d c a n b e i d e n t i f i e d t o b e d i f f e r e n t i ne a c h c a se . C o m p a r i s o n s o f t h e i n d i c a t e d c o r r e l a t i o n s w i t ho t h e r p r e v i o u s d a t a a r e g i v e n i n t h e s o ur c e s .6 ConclusionsT h e p r e s e n t s tu d y c o r r e s p o n d s t o t u r b u l e n t f o r c e d c o n v ec -t i o n f r o m a w a l l w h i c h h a d a l i n e a r t e m p e r a t u r e d i s t r i b u -t i on . R e s u lt s f r o m t h e e x p e r i m e n t c a n b e s u m m a r i z e d i nt h e f o r m o f a c o r r e l a t i o n r e p r e s e n t e d b y E q . (9 ) w i t h t h ec o n s t a n t s g i v e n i n (1 0 ). T h e p r o p e r t y v a l u e s o f w a t e r a r ec o n s i d e r e d a t th e a v e r a g e i n l e t a n d o u t l e t b u l k t e m p e r a -t u r es . T h e c o r r e l a t i o n i s v a l i d f o r7 , 1 0 0 < R e < 19 ,0000 .033 < D < 0 .0610 . 2 5 5 < P < 0 . 3 5 9 .

AcknowledgementO n e o f t h e a u t h o r s ( M . S . ) a c k n o w l e d g e s t h e s u p p o r t o f C o r n e l lU n i v e r s i ty w h e r e r e v i s io n s to t h e m a n u s c r i p t w e r e m a d e . T h ee x p e r i m e n t s w e r e c a r r i e d o u t a t t h e Laboratorio de Termofluidoso f t h e Facultad de Ingenieria o f t h e U . N . A . M . , M e x i c o C i t y .

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Rece ived Decem ber 18 , 1985