Use of pipe flow characteristics in checking fluid flowmeters

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Use of pipe flow characteristics in checking fluid flowmeters Racine, J. G.; Sasaki, J. R.

NATIONAL RESEARCH COUNCIL CANADA

D I V I S I O N O F B U I L D I N G RFrSEARCH

T H E U S E O F PIPE PLOW C H A R A C T E R I S T I C S II? CHECKING FLUID FLOWiIETERS

by J . G. R a c i n e and J. R. S a s a k i I n t e r n a l R e p o r t N o . 217 o f t h e D i v i s i o n o f B u i l d i n g R e s e a r c h OTTAWA February 1961

PREFACE

The a c c u r a t e c a l i b r a t i o n of f l u i d metering devices poses many problems, y e t t h e r e a r e many kinds of l a b o r a t o r y s t u d i e s which depend on t h e a b i l i t y t o measure f l u i d flow a c c u r a t e l y . The Division has encountered such a requirement r e c e n t l y i n t h e c a l i b r a t i o n of simple p l a t e o r i f i c e meters f o r use

i n t h e measurement of window i n f i l t r a t i o n . Recognizing t h a t t h e establishment of a s u i t a b l e primary s t a n d a r d could be both d i f f i c u l t and time-consuming it was decided t o a c c e p t a s a n i n t e r i m l a b o r a t o r y standard a s e r i e s of commercial flowmeters o f f e r e d w i t h s p e c i a l c a l i b r a t i o n . These were n o t found t o be a c c u r a t e and i n t h e course of checking them

it

was p o s s i b l e t o examine t h e flow c h a r a c t e r i s t i c s of s t e e l pipes w i t h

a view t o u s i n g them a s a c a l i b r a t i o n device. The r e s u l t s obtained a r e now r e p o r t e d .

The f i r s t a u t h o r , M r . Racine, c a r r i s d o u t much of t h e measurement work a s a summer s t u d e n t w i t h t h e Division. M r . Sasaki,a r e s e a r c h o f f i c e r w i t h t h e Building Services S e c t i o n , i s now engaged. i n t h e s t u d y

of window i n f i l t r a t i o n .

Ottawa

THE USE OF FIFE FLO'JV CHARACTER1STIC:S FLUID

J . G. Racine and J . R. S a s a k i

Five commercially a v a i l a b l e c a l i b r a t e d v a r i a b l e - a r e a flowmeters were purchased by t h e D i v i s i o n of B u i l d i n g Research Lo s e r v e a s a l a b o r a t o r y s t a n d a r d f o r a i r flow measurement. The r o t a m e t e r s were i n t e n d e d f o r use i n t h e c a l i b r a t i o n of sharp-edged o r i f i c e meters. These f lowmeters werc s e l e c t e d t o cover a range of f l o w s and t h e r e was some

o v e r l a p i n range between each p a i r -taken i n o r d e r of r a n g e s . It was -thus p o s s i b l e t o compare t h e r e a d i n g given by one meter a t -the t o p of i t s range w i t h t h a t g i v e n by t h e n e x t l a r g e r rneter o p e r a t i n g n e a r t h e bottom of i t s r a n g e , D i s - c r e p a n c i e s of up t o

5

p e r c e n t were found, b u t it could n o t be determined how much d i s c r e p a n c y coulcl be a t t r i b u t e d t o e a c h meter of t h e p a i r .

The poor correspondence of t h e r e a d i n g s from two s u c c e s s i v e f l o w n e t e r s was a t t r i b u t e d t o two p o s s i b l e s o u r c e s :

a t h a t t h e a c c u r a c y of t h e purchased r o t a m e t e r s was no-t +1 p e r

ten-t,

a s g$arantced by t h e manufacturer; o r

b ) t h e d e n s i t y c o r r e c t i o n s a p p l i e d t o t h e metered f l o w were i n c o r r e c t ,

Subsequent t e s t s showed t h e d e n s i t y c o r r e c t i o n s t o be v a l i d and t h e c o n c l u s i o n was, t h e r e f o r e , t h a t -the r o t a m e t e r s d i d n o t a c h i e v e a n a c c u r a c y of

-

+1 p e r c e n t ,

The ob jec-Live i n d e t e r m i n i n g t h e f r i c - t i o n f a c t o r Reynolds' number c h a r a c t e r i s t i c of t h e t h r e e s t e e l p i p e s u s i n g t h e r o t a m e t e r s was t h r e e f o l d . The f i r s t was t o check t h e

d i s c r e p a n c i e s between t h e i n d i v i d u a l r o t a m e t e r s by o b t a i n i n g t h e c h a r a c t e r i s t i c f o r a p a r t i c u l a r pipe o v e r a range of Reynolds1 number common t o two meters w i t h a n o v e r l a p p i n g flow r a n g e , and comparing t h e f r i c t i o n f a c t o r v a l u e s o b t a i n e d w i t h t h e d i f f e r e n t m e t e r s , The second was t o compare t h e pipe c h a r a c - t e r i s t i c s o b t a i n e d w i t h t h e r o t a m e t e r s w i t h published pipe c h a r a c t e r i s t i c s such a s t h o s e p r e s e n t e d by Moody(2). This would g i v e some i d e a of t h e r e l a t i v e a c c u r a c y

of t h e whole s e t of r o t a m e t e r s i n a d d i t i o n t o t h e r e g i o n s of o v e r l a p p i n g flow. The t h i r d purpose was t o determine whether t h e f lovr r a t e s o b t a i n e d by a p p l y i n g t h e g e n e r a l l y - a c c e p t e d f r i c t i o n f a c t o r c h a r a c t e r i s t i c f o r smooth pipe t o t h e g a l - vanized i r o n c o n d u i t s a v a i l a b l e were s u f f i c i e n t l y a c c u r a t e f o r c a l i b r a t i o n purposes.

D e s c r i p t i o n of t h e Apparatus

The a p p a r a t u s f o r %he t e s t s was a r r a n g e d a s shown i n Fig. 1. A i r flow was o b t a i n e d from t h e l a b o r a t o r y -

compressed a i r supply main and was reduced from 100 p s i i n t h e main t o between 30 and 80 p s i a t t h e i n l e t s i d e of t h e t h r o t t l i n g v a l v e s . Immediately ahead of t h e p r e s s u r e -

r e d u c i n g v a l v e t h e a i r passed through a Logan a r i d i f i e r which e l i m i n a t e d any d r o p l e t s of w a t e r o r o i l , a s w e l l a s p a r t i c l e s of pipe s c a l e o r r u s t e n t r a i n e d i n t h e compressed a i r . The moisture c o n t e n t of t h e a i r which passed through t h e t e s 3 c o n d u i t was measured u s i n g a Burton clew p o i n t a p p a r a t u s . The q u a n t i t y of a i r f l o w i n g through t h e sys-tern was c o n t r o l l e d by t h e t h r o t t l i n g v a l v e s on t h e low-pressure s i d e of t h e p r e s s u r e - r e d.uc i n g valve

.

Prom t h e f l o w c o n t r o l p a n e l t h e a i r passed through a 2 0 - f t l e n g t h of l $ - i n . galvanized i r o n pipe i n t o a l k - i n . h e a d e r pipe and thence through any one of t h e t h r e e 4 0 - f t r u n s of t e s t c o n d u i t of g a l v a n i z e d i r o n . The t e s t c o n d u i t s were t h o s e used i n a p r e v i o u s s t u d y , d e s c r i b e d i n DAR I n t e r n a l Report No.

93 ( 3 ) . From t h e c o n d u i t t h e a i r passed t h r o u g h

a r o t a m e t e r t o atmosphere.

S i x s t a t i c p r e s s u r e h o l e s 1/'16 i n . i n d i a m e t e r had been d r i l l e d a t 60' i n t e r v a l s around t h e condui-t a p p r o x i m a t e l y

4 i n . from one end of each 1 0 - f t s e c t i o n . These h o l e s were covered by b r a s s piezometer r i n g s , s o l d e r e d t o t h e o u t s i d e of t h e c o n d u i t t o make a n a i r t i g h t s e a l .

The i n s i d e of t h e c o n d u i t had been reamed f o r approximately 6 i n . a t t h e end where t h e piezometer r i n g was f i t t e d and 2 i n . a t t h e o t h e r end. This reaming ensured t h a t t h e i n t e r n a l a r e a of t h e f l o w passage was t h e same a t each p r e s s u r e t a p p i n g and t h a t t h e i n s i d e d i a m e t e r of t h e c o n d u i t s were t h e same a t t h e j o i n t s . The d i f f e r e n t i a l p r e s s u r e was measured w i t h a 250 mrn Aetz w a t e r manometer and when n e c e s s a r y w i t h a 100-in. v e r t i c a l Meriam manometer u s i n g coloured w a t e r . Gauge p r e s s u r e s throug!~ t h e system were measured u s i n g a n o t h e r 100-in. v e r t i c a l Meriam manometer and when n e c e s s a r y w i t h a Bourdon gauge. A p r e s s u r e - s w i t c h i n g p a n e l was used t o connect t h e manometers t o any of t h e twelve piezorneter r i n g s . With t h i s arrangement, t h e Betz o r Neriam manometers could r e a d i l y be made t o read t h e d i f f e r e n t i a l p r e s s u r e between any two piezometer r i n g s .

Before assembly of t h e p i p i n g , t h e i n t e r n a l c r o s s - sec-bional a r e a of each l e n g t h of c o n d u i t had been determined. The pipe d i a m e t e r s a r e given i n Table 1.

- 3 -

TABLE 1 T e s t Conduit S i z e s

The t e m p e r a t u r e s i n s i d e t h e p i p e were measured w i t h thermocouples r e a d on a n e l e c t r o n i c t e m p e r a t u r e i n d i c a t o r . Thermocouples were p l a c e d i n t h e h e a d e r p i p e , a t t h e e n d s o f e a c h p i p e , and a t t h e o u t l e t o f each r o t a m e t e r . Conduit Number 3/4 i n .

I

I1 I11 IV 1 i n . I I1 I11 IV 1 1/4 i n . I I1 111: IV

The a i r f l o w s t h r o u g h t h e p i p e were measured u s i n g f o u r r o t a m e t e r s . The f l o w r a n g e s a r e shown i n Table 2.

~ i a m e t e r ( 3 ) (ft) 6.908

x

6.869

x

101; 6.828 x 6.319

x

1 0 8.744

x

101; 8 . 6 7 1 x 8.712 x 8.728

x

1 0 11.575

x

101; 11.413 x 11.578

x

11.586 x 1 0 I TABLE 2

Rotame t e r Blow Ranges M e t e r 1 2

3

4 1 Flovr Range(cfm) 0.29

-

1 . 6 0 1.15

-

6 . 3 4.0

-

22.5 18.0

-

98.0

Theory of Flow througli a E p e w i t h F r i c t i o n

A i r f l o w i n g t h r o u g h a p i p e , i n which f r i c t i o n o c c u r s , can be d e a l t w i t h a s e i t h e r a c o m p r e s s i b l e o r i n c o m p r e s s i b l e f l u i d depending on th.e magnitude o f f l o w , t h e l e n g t h of t h e p i p e , and t h e pipe d i a m e t e r . '.men t h e f l o w i s s m a l l , t h e

p i p e l e n g t h s h o r t , and t h e p i p e d i a m e t e r r e l a t i v e l y l a r g e , t h e p r e s s u r e d r o p and c o n s e q u e n t l y t h e d e n s i t y change i s v e r y s m a l l and i n c o m p r e s s i b l e f l o w t h e o r y can be a p p l i e d . I f , hoviever, t h e f l o w i s l a r g e , t h e p i p e l e n g t h l o n g and t h e d i a m e t e r of t h e p i p e r e l a t i v e l y s m a l l , t h e d e n s i t y change c a n no l o n g e r be i g n o r e d , and t h e f l o w must be c o n s i d e r e d c o m p r e s s i b l e .

Reference No. 1 s t a t e s t h a t i n v e s t i g a t o r s have

found t h e f r i c t i o n f a c t o r , f

,

t o be a p h y s i c a l c h a r a c t e r i s t i c of a p i p e , independent of t h e Mach number and dependent o n l y on t h e Reynolds' number and t h e p i p e roughness, no m a t t e r what t y p e of flow e x i s t s i n t h e p i p e . The problem i s t o o b t a i n a n e x p r e s s i o n f o r f l o w t h a t r e p r e s e n t s t h e flov? c o n d i t i o n s a c t u a l l y e x i s t i n g i n t h e p i p e . A c h o i c e of t h r e e e x p r e s s i o n s p r e s e n t e d -themselves; t h e i n c o m p r e s s i b l e e q u a t i o n and t h e i s o t h e r m a l and a d i a b a t i c e q u a t i o n s f o r compressible flow. T h e i r developments a r e shown i n

Appendix A . C o n s i d e r i n g t h e maximum flow and t n e minimum p i p e d i a m e t e r t o be encountered i n t h e s t u d y it was decided t h a t t h e p r e s s u r e r a t i o , '2/-pl, would be s u f f i c i e n t l y low t o n e c e s s i t a t e t h e u s e of t h e c o m p r e s s i b l e f l o w e q u a t i o n . Reference No. 1 s t a t e s f u r t h e r , t h a t o n l y i n v e r y l o n g p i p e s w i t h a l a r g e t e m p e r a t u r e d i f f e r e n c e between t h e i n s i d e and o u t s i d e of t h e p i p e w i l l i s o t h e r m a l f l o w c o n d i t i o n s be approached. I n s h o r t p i p e s where t h e g a s t e m p e r a t u r e i s n e a r -the room t e m p e r a t u r e , a d i a b a - t i c c:ond.itions a r e more n e a r l y approached. The c o n d i t i o n s d u r i n g t h e t e s t s on t h e p i p e viere s u c h t h a t t n e a c t u a l t e s t c o n d i t i o n s p r o b a b l y approached a d i a b a t i c more c l o s e l y t h a n iso-iiherrnal, A s a n p l e c a l c u l a t i o n of t h e f r i c t i o n f a c t o r i s shovm i n Appendix

A

u s i n g a l l t h r e e e x p r e s s i o n e . It was fouiid t h a t t h e r e s u l t s o b t a i n e d w i t h t h e i s o t h e r m a l e x p r e s s i o n variecl v e r y l i - t ; t l e from t h a t o b t a i n e d n i t h t h e a d i a b a t i c e x p r e s s i o n , Theref o r e , a l t h o u g h r e a l i z i n g t h a t t h e t e s t c o n d i t i o n s more c l o s e l y approached a d i a b a t i c , t h e i s o t h e r m a l c o m p r e s s i b l e flow e x p r e s s i o n was used t o d e - l - e r m i ~ e t h e f r i c t i o n f a c t o r from t h e t e s t r e s u l t s f o r i t s s i m p l i c i t y . The i s o t h e r m a l c o m p r e s s i b l e f l o w e x p r e s s i o n f o r f r i c t i o n f a c t o r i s

The c o r - r e s p o n d i n g p i p e Reynolc?st number i s

The above symbols a r e d e f i n e d i n i'A~penc?ix !Z.

T e s t Procedure

The ro-l-arneter was c o n n e c t e d t o t h e p a r t i c u l a r p i p e t o be t e s t e d . The a i r from t h e l a b o r a t o r y s u p p l y was t u r n e d on and governed by t h e c o n t r o l v a l v e u n t i l t h e a p p r o p l - i a t e f l o w was i n d i c a t e d . on t h e r o t a m e t e r . F o r t h e g i v e n f l o w , t h e p r e s s u r e d r o p a c r o s s o n e , two o r t h r e e l e n g t h s of t h e c o n d u i t bei-ng t e s t e d was measured u s i n g t h e B e t z m a ~ o m e t e r f o r . t h e s m a l l v a l u e s , and t h e 1 0 0 - i n . Tderiam manometer f o r t h e l a r g e r v a l u e s . ;'hen t h e p r e s s u r e d r o p was small., i t tvas measureci o v e r t h e l o n g e s t a v a i l a b l e l e n g t h , i . e . t h r e e 1 0 - f t s e c t i o n s t o g e t h e r . The gauge p r e s s u r e a t t h e p i p e i n l e t was measured w i t h t h e 1 0 0 - i n . manometer and w i t h t h e nourdon gauge a t t h e l a r g e r pressures, and t h e p i p e i n l e t and o u t l e t and. r o t a m e t e r o u t l e t t e m p e r a t u r e s were measured w i t h thern!ocouples. The above read.ings were made a t e a c h f l o v ~ r e a d i n g .

The b a r o m e t r i c p r e s s u r e , room t e m p e r a t u r e and t h e l i n e a i r de-m-point t e m p e r a t u r e were measured a t t h e s t a r t and f i n i s h of each t e s t . Four t e s t s were made u s i n g m e t e r 1 and t h r e e tes-t-s each were made with m e t e r s 2 ,

3

and 4 on t h e

3/4 i n . p i p e . Three t e s t s e a c h were made w i t h m e t e r s 2 , 3 and 4 on t h e 1 i n . p i y e , and t h r e e t e s t s e a c h were made w i t h

m e t e r s 2 ,

3

and 4 on t h e 1& i n . p i p e . Data process in^

The d a t a p r o c e s s i n g was performed by t h e G-15 Dendix computer f o r rvhich a program was w r i t t e n . The program

computed v a l u e s of f r i c L i o n f a c t o r and t h e c o r r e s p o n d i n g

Eieynolds' numbers. 11 p l o t t e r s u b r o u t i n e was w r i t t e n i n t o t h e program t o p r o v i d e two forms of o u t p u t : t y p e - o u t i n f i x e d p o i n t n o t a t i o n and punched p a p e r t a p e . The punched p a p e r t a p e r e s u l t was i n p u t t e d t o a IiIosely p l o t t e r which y i e l d e d a p l o t of l o g ( f ) v e r s u s l o g ( R , ) . The p r o c e s s e d r e s u l - t s o f t h e t e s t s a p p e a r i n F i g s . 2 t o 8 , i n c l u s i v e . I n a d d i t i o n , s e v e r a l t e s t p o i n t s were r e - c a l c u l a t e d w i t h t h e v a l u e of Q, m o d i f i e d by

-

+

l p e r c e n t t o

-

+

5

p e r c e n t . The i n a c c u r a c y i n t r o d u c e d int;o t h e f i n a l r e s u l t s due t o i n s t r u m e n t e r r o r s i s o u t l i n e d i n b-ppendix B.

Discussion of R e s u l t s

I n t h e subsequent d i s c u s s i o n , a l l d i s c r e p a n c i e s i n

f

a r e expressed a s percentage d i f f e r e n c e s i n flow.

I. Pi-oe F r i c t i o n Fac-tor C h a r a c t e r i s t i c

( a ) 3/4 i n . @ Conduit ( P i g s . 2 and 3 )

The f i g u r e s show v e r y poor c o r r e l a t i o n between t h e c h a r a c t e r i s t i c s a s determined by d i f f e r e n t meters. The poor c o r r e l a t i o n between t h e c h a r a c t e r i s t i c s was a t t r i b u t e d t o t h e c a l i b r a t i o n of t h e meters. A t R = 2240, t h e d i f f e r e n c e i s e q u i v a l e n t t o a d i f f e r e n c e of

4&

Eer c e n t i n t h e v a l u e s of flow a s measured by meters 1 and 2. A t Re = 12,000, t h e d i f f e r e n c e i n terms of flow i s 2Q pe? c e n t . A t Re = 42,600, t h e d i f f e r e n c e i n terms of flow i s 2 2 p e r c e n t .

Prom Pig. 2 it can be s e e n t h a t i n t h e t u r b u l e n t and t r a n s i t i o n r e g i o n s a d i f f e r e n - c e of 1 p e r c e n t i n Q produces a 1 p e r c e n t i n c r e a s e i n R and a 2 p e r c e n t decrease i n

f

which can be seen qui%e e a s i l y . However, a change i n

l;m

of 1 p e r c e n t i n t h e l a m i n a r r e g i o n can

s c a r c e l y be d i s c e r n e d a s t h e i n c r e a s e i n R and t h e d e c r e a s e i n f follolvs c l o s e l y t h e s l o p e of t h e lamifiar flow l i n e .

I n Pig.

3

published pipe f r i c t i o n f a c t o r s a r e p l o t t e d f o r comparison: curve 2 f a l l s from t h e smooth p i p e c h a r a c t e r i s t i c by 2 t o

5

p e r c e n t of flow, curve

3

by 2 t o

3

p e r c e n t of f l o w , and curve 4 by 1 t o

4

p e r c e n t of flow. If t h e c h a r a c t e r i s t i c of a pipe w i t h roughness 0.0004 i s t a k e n a s t h e mean, curve 2 f a l l s o f f from 0 t o 4

p e r c e n t , curve 3 f a l l s off from 1 t o 0 p e r c e n t and curve

4

f a l l s o f f by 4 p e r c e n t . T h e r e f o r e , i f t h e published pipe c h a r a c t e r i s t i c s a r e assumed c o r r e c - t , and t h e method of c a l c u l a t i n g

f

_{and Re a r e assumed c o r r e c t , t h e r e s u l t s show}

a t l e a s t 2 of t h e 3 meters a s having e r r o r s i n t h e c a l i b r a t i o n g r e a t e r t h a n 2 p e r c e n t .

( b ) 1 i n . @ Co11d.ui-t ( F i g s . 4 and

5 )

The poor c o r r e l a t i o n between t h e c h a r a c t e r i s t i c s , a s determined by meters 2 ,

3

and

4 ,

was a g a i n e v i d e n t , and i n t h e same manner. The d i f f e r e n c e between c u r v e s 2 and 3 i n c r e a s e d a s m a l l amount and t h e d i f f e r e n c e between c u r v e s 3 and 4 decreased.

I n Pig.

5

it

i s seen t h a t t h e c h a r a c t e r i s t i c f o r a r e l a t i v e roughness of .0004 i s a good approximation f o r curves

3

and

4

which a r e only about +1& and -2 p e r c e n t o f f . However, curve 2 i s o f f by a s much a s 2 t o 4 p e r c e n t .

( c ) 1$ i n .

jd

Conduit ( F i g s . 6 and

7 )

Very s i m i l a r i n c l ? a r a c t e r i s t i c t o tlie

3/4-

a n d 1 i n . p i p e s , e x c e p t t h a t t h e d i f f e r e n c e between c u r v e 2 a n d

3 i s q u i t e

s m a l l . The p r e s e n c e o f t h e lower p o i n t s on c u r v e 3 i s due t o a n e r r o r of 0.45 p e r c e n t i n t h e d i a m e t e r t h a t w a s u s e d i n t h e c a l c u l a t i o n . F i g u r e

7

compares t h e t e s t r e s u l t s ~ v i t h t h e curve f o r .0004 r e l a t i v e r o u g h n e s s . Curve 4 f a l l s below by 1 p e r c e n t , curve

3

l i e s above by 2 p e r c e n t and curve 2 l i e s above by

3

t o

5

p e r c e n t .

11. F i g u r e 8 shows a l l tlie measured c h a r a c t e r i s t i c s p l o t t e d on a s i n g l e g r a p h . Lkcepl; f o r t h e t r a n s i t i o n and h i g h t u r b u l e n c e r e g i o n s , a l l t h e c u r v e s f a l l v e r y n e a r l y one on t o p o f t h e o t h e r , a n i n d i c a t i o n t h a t a l l t h r e e conciuits t e s t e d h a d s i m i l a r r e l a t i v e r o u g h n e s s . C o n c l u s i o n s I . The p i p e f r i c t i o ~ ? f a c t o r c h a l - a c t e r i s t i c s a s d e t e r m i n e d w i t h t h e f o u r r o t a m e t e r s were a t v a r i a n c e w i t h one a n o t h e r i n t h e o v e r l a p p i n g r a n g e s of f l o w . The c a l i b r a t i o n of m e t e r 2 was lower t h a n t h a t of m e t e r 1 by a s much a s 4; p e r c e n t o f t h e flow v a l u e . The c a l i b r a t i o n o f m e t e r 3 v r a s h i g h e r t h a n t h a t of m e t e r 2 by 23 p e r c e n t o f t h e flow!, and t h s c a l i b r a - t i o n of m e t e r 4 was h i g h e r tlian t h a t o f m e t e r 3 by 2$ p e r c e n t of t h e flovr.

11. The p i p e f r i c t i o n f a c t o r c h a r a c t e r i s t i c s p r e s e n t e d by Moody were t a k e n a s a r e f e r e n c e . Vihen t h e e x p e r i m e n t a l c h a r a c t e r i s t i c s were superimposed, t h e y d i d n o t l i e on any one l i n e of a g i v e n r o u g h n e s s b u t c r o s s e d s e v e r a l . Die r e s u l t s o b t a i n e d f o r t h e 3/4 i n .

jd

p i p e can be t a k e n a s a n example. Curve 3 l a y a l o n g t h e g e n e r a l c h a r a c t e r i s t i c f o r a p i p e w i t h a r e l a t i v e rougllness o f 0.0004. Curve 2 l a y above t h i s l i n e , i n d i c a t i n g t h a t t h e whole c a l i b r a t i o n on m e t e r 2 may be low by u p t o 4 p e r c e n t . Curve 4 l a y below t h e l i n e , a n i n d i c a t i o n t h a t t h e whole c a l i b r a t i o n of m e t e r 4 may be h i g h by 4 p e r c e n t . 111. The s t u d y sllovlrs t h a t t l i e r e i s a d i s t i n c t p o s s i b i l i t y t h a t f o r

R

) GOO0 t h e p u b l i s h e d p i p e f r i c t i o n f a c t o r c h a r a c - t e r i s t i c s !!!an be u s e d a s a c a l i b r a t i n g means, p r o v i d e d t h e g e n e r a l c u r v e s a r e a c c u r a t e a n d t h e p i p e r o u g h n e s s can be c l o s e l y e s t i m a t e d . The f r i c t i o n f a c t o r i s a n i n v e r s e f u n c t i o n of t h e s q u a r e o f t h e f l o w , a n d a n e r r o r of 2 p e r c e n t i n d e t e r m i n i n g t h e f r i c t i o n f a c t o r w i l l r e s u l t

i n

o n l y 1 p e r c e n t e r r o r i n flow. T h e r e f o r e . i f one i s a b l e t o choose a m b l i s h e d p i p e f r i c t i o n f a c t o r c h a r a c t e r i s t i c t o w t h e t r u e c h a r a c t e r i s t i c , by assuming no f l o w v a l u e s a c c u r a t e t o

I

p e r c e n t can b i t h i n 2 p e r c e n t of e x p e r i m e n t a l e r r o r , e o b t a i n e d .

I n t h e l a m i n a r r e g i o n , t h e r e i s o n l y one c h a r a c t e r i s t i c and t h i s i s i n d e p e n d e n t of r o u g h n e s s . However, t h i s c h a r a c t e r i s t i c must be v e r y a c c u r a t e s i n c e a srnall e r r o r i n

f

i n t r o d u c e s a v e r y l a r g e e r r o r i n t h e d e t e r m i n e d f l o w . Acknowledgements The a u t h o r s w i s h t o e x p r e s s t h e i r t h a n k s t o D. G. Stenhenson f o r h i s a s s i s t a n c e i n t h e s e l e c t i o n of t h e f l o w e q u a % i o n s , a n d t o C . J. S h i r t l i f f e f o r h i s a s s i s t a n c e i n programing t h e t e s t c a l c u l a t i o n s . R e f e r e n c e s 1. I a p p l e , C .

E.

I s o t h e r m a l and a d i a b a t i c f l o w o f c o m p r e s s i b l e f l u i d s . American I n s t i t u t e o f Chemical E n g i n e e r s , T r a n s . , 1943, v. 39, p. 385-432. 2 . Moody, L. F. F r i c t i o n f a c t o r s f o r p i p e f l o w . American S o c i e t y o f Mechanical E n g i n e e r s , Nov. 1944, T r a n s . , v. 66, n.8, p. 671-8, d i s c u s s i o n p. 678-81. 3. Stephenson, D. G. P r e s s u r e l o s s a s s o c i a t e d w i t h a i r f l o w i n e l e c t r i c a l c o n d u i t s . N a t i o n a l Research C o u n c i l , D i v i s i o n o f B u i l d i n g R e s e a r c h , I n t e r n a l R e p o r t No. 9 9 , August 1956, 4 1 p . ,

1 5

P i g s .

MODIFIED UNION

CONNECTION FLOW CONTROL

PRESSURE

SWITCHING PANEL

13 TO LOW PRESSURE FLOW METER

SlDE OF MANOMETER

14 TO HIGH PRESSURE

SlDE OF MANOMETER LOGAN ARlDlFlER

L L G L N V :

---- T E S T CHARACTERISTICS

pU6LlSHEU CHAQACTEqlSTlCS

x _VARIATIONS IN Q M of t 1,2,3,4,5,%

I

F l G U R L

5

F R I C T I O N FACTOP, vs. REYNOLDS NUMBED

in

C O N V U I T S E C T I O N S C O M P A Q I S O N of T E S T A N 9 PUBLISHLV C U A q A C T L Q I S T l C S

, I I I I I I I I I

-

s

-

In

*-

*:

-

_'?!

_-

_-

+

t %

-

25

-

d Z 0

-

-

w c4 -4-: C r c r K G

-

_u.ldu.ld

_-

+ + b - s

U U J d d

Z E Z >

..

-

b

-

z x o x

d

* v e v .

-

" ' 0 0 0

-

A

-

-

-

-

-

-

h

C

-

4

-

_-

I I I I I I I I I 0 0 0 0 0 0 0 0 0 C I 2 0 0 m og 9 u\ -4- ccr rJ 0 I-

(*)'

Ol

so;

I r a

STEADY PLOB OF A I R !THROUGH A PIPE T/I!ClI PEICTIOI\T

An e x p l a n a t i o n o f t h e symbols u s e d i n t h e e q u a t i o n s t o follokv m i l l be f o u a d a t t h e end o f Appendix A.

F o r t h e s t e a d y flovr o f a i r t h r o u c h a p i p e , t h e f o l l o w i n g r e l a t i o n s w i l l be assumed t o h o l d : ( i ) C o n t i n u i t y e q u a t i o n c o n s t a n t f o r a c o n s t a n t a r e a p i p e ;

ng

=

>=

( i i ) P e r f e c t g a s l a w The s t e a d y f l o w o f a f l u i d t h r o u g h a p i p e c a n be c o m p l e t e l y d e s c r i b e d b y t h e f o l l o v i i n g e q u a t i o n s : ( a ) t h e e q u a t i o n f o r t h e c o n d i t i o n o f s t a t e ; ( b ) -the dXnergy E q u a t i o n o f f l o w . The G e n e r a l Energy J q u a t i o n i s t h e a l g e b r a i c e x p r e s s i o n o f t h e f i r s t l a w o f t h e n n o d y n a m i c s which s t a t e s t h a t i n a c o n v e r s i o n o f therrflal e n e r g y t o m e c h a n i c a l e n e r g y , t h e amount o f m e c h a n i c a l e n e r g y d e v e l o p e d i s e q u a l t o t h e amount o f t h e r m a l e n e r g y v ~ h i c h d i s a p p e a r s a n d v i c e v e r s a . Ttle e q u a t i o n c a n be e x p r e s s e d a s f o l l o r v s , f o r h o r i z o n t a l f l o w , i n which no work i s done :

J d q

-

J d u = d($+ d

( P N )

Vhen f r i c t i o n i s p r e s e n t i n t h e p i p e , p a r t o f t h e n l e c h a n i c a l e n e r g y d e v e l o p e d i s r e d u c e d b y t h e f r i c t i o n t o h e a t . An e n e r g y e q u a t i o n i n v o l v i n g tile f r i c t i o n can be d e v e l o p e d from t h e dynamic e q u a t i o n . C o n s i d e r a n e l e m e n t o f f l u i d , dm, moving from p o i n t 1 t o p o i n t 2 a l o n g t h e p i p e . A c t i n g on t h e u p s t r e a m f a c e o f t h e e l e m e n t i s a p r e s s u r e ( p ) a n d on t h e downstream f a c e , ( p d p )

.

The v e l o c i t y a t t h e u p s t r e a m f a c e i s V a n d a t t h e dorvnstream f a c e V -t- dV.

R e s i s t i n g t h e movement of t h e element i s t h e f r i c t i o n s h e a r s t r e s s , T o , where fo =

ff

v2

8 g

If t h e p i p e h a s a d i a m e t e r , do, and t h e element h a s l e n g t h , dL, The dynamic e q u a t i o n s t a t e s t h a t t h e s u m of a l l t h e f o r c e s a c t i n g on t h e e l e m e n t c a u s e s t h e e l e m e n t t o a c c e l e r a t e . A c c e l e r a t i o n i s a = dV

_fi

S i n c e t h e f l o w i s s t e a d y , t h e a c c e l e r a t i o n i s a f u n c t i o n o f l e n g t h o n l y . dV

-

dL VdV

- -

- = - d t dL ' d t dL VdV

[P

-

( p d p ) ] do2

-

% .

d .

do. dL = dm.

-

_dL which i s

9

,

f v 2

.dL =

-

VdV

P

2gdo i5 T n i s i s t h e energy e q u a t i o n , i n c l u d i n g f r i c t i o n . The e q u a t i o n s f o r i n c o m p r e s s i b l e f l o w and i s o t h e r m a l and a d i a b a t i c c o m p r c s s i b l e f l o w s c a n be developed from t h e e n e r g y e q u a t i o n ( 2 ) by i n c o r p o r z t i n g t h e p a r t i c u l a r c o n d i t i o n s of s t a t e . ( a ) I n c o m p r e s s i b l e Plow The c o n d i t i o n of s t a t e f o r i n c o m p r e s s i b l e f l o w i s

N=

c o n s t a n t and s i n c e = c o n s t a n t , V = c o n s t a n t

2

Xquation ( 2 ) r e d u c e s t o

Upon i n t e g r a t i o n , t h e e x p r e s s i o n f o r f r i c t i o n f a c t o r i s g i v e n a s where Afm f o r t r u e i n c o m p r e s s i b l e f l o w i s e q u a l t o and

/Lf,

b u t f o r a i r f l o w i s e q u a l t o t h e 1 2 mean.

The correspond i n g p i p e Reynolds number i s

s i n c e d m i s c o n s t a n t , and f u r t h e r assuming t h a t t h e t e m p e r a t u r e from p o i n t 1 t o p o i n t 2 i s n e a r l y c o n s t a n t , A i s c o n s t a n t and t h e r e - f o r e

Re

i s c o n s t a n t a l o n g t h e p i p e l e n g t h . ( b ) I s o t h e r m a l Compressible Plow The condition of s t a t e f o r i s o t h e r m a l f l o w i s dT = 0 The P e r f e c t Gas R e l a t i o n d (p/vj = RdT = 0

-

c o n s t a n t From t h e c o n t i n u i t y e q u a t i o n ,

- -

/Ir E q u a t i o n ( 2 ) , r e - a r r a n g e d , g i v e s I n t e g r a t i o n g i v e s ,

( c ) A d i a b a t i c C o m p r e s s i b l e Flow The c o n d i t i o n of s t a t e i s dq = 0 and E q u a t i o n (1) becomes 2 ( V ) + J d . u = O d ( p W ) + i& 1 S i n c e Jdu = J c V dT =

~mr

.d ( P M ) The e q u a t i o n of t h e c o n d i t i o n of s t a t e becomes: ( h - 1 ) d V*

-

d ( p n d + k 0 - -2g

( 5 )

The e n e r g y e q u a t i o n i s E q u a t i o n ( 2 ) S o l v i n g E q u a t i o n s

( 5 )

and ( 6 ) s i m u l t a n e o u s l y and o b t a i n i n g a n e x p r e s s i o n f o r

,.

P i n t e r m s of pl and p would. 2 g i v e p r o p e r e x p r e s s i o n f o r

f

~.

_{However, t h e r e s u l t i n g e q u a t i o n} i s d i f f i c u l t t o s o l v e and n o r m a l l y t h e E q u a t i o n s

( 5 )

a n d ( 6 ) a r e e x p r e s s e d s e p a r a t e l y . E q u a t i o n

( 5 )

i n t e g r a t e s t o - )

vL

-

c o n s t a n t P&+

7 - 3 -

v

S i n c e

I

7

4

=

g P M of s t a t e . E q u a t i o n

( 6 )

c a n be r e - w r i t t e n a s

S u b s t i t u t i n g n i n t o ( 6 a ) and i n t e g r a t i n g , g i v e s t h e r e s u l t i n g energy equation: By s o l v i n g t h e e q u a t i o n s of s t a t e and energy t o g e t h e r f o r a p a r t i c u l a r value of '2/pl, t h e value o f f r i c t i o n f a c t o r i s obtained.

SAbTFLE CALCULATION U S I N G THE THIBE EXPFBSSIOMS

T e s t Condition

-

10 f t of uninsulated. i r o n pipe do = 6.828

x

f t (3/4 i n .

9)

Ba = 29.67 in.M.C. tdp = 17'3' Bc = 29.54 5n.M.C. hl-h2 = 63.6 i n . W.C. = 2.30 p s i P1 = 13.25 p s i g Qm = 97.1 cfm Ti = 541.2OR To = 538.6OR

Tr

= 538.7OR Ta = 536.6OR S o l u t i o n

P m

-

-

1i1:17

-

-378 p

]

= C v le317 538.3 x 29.51 = 0.07220 pcf

Pa = 2085 psf p 1 = 3993 psf p2 = 3661 psf

fi

= 7.210 c f p ( a ) I n c o m p r e s s i b l e Plow ( b I s o t h e r m a l C o r n ~ r e s s i b l e P l o v ~ ( c ) A d i a b a t i c Compressible Plom

p2/p1 = 0.9144;

B y i t e r a t i o n

4

_z,-

-

1 , 0 8 8 3

,

and

-

Checking t h e I s o t h e r m a l C o n d i t i o n Prom Heat T r a n s f e r C o n s i d e r a t i o n

-

The g e n e r a l e n e r g y e q u a t i o n f o r t h e i s o t h e r m a l c o n d i t i o n i s For i s o t h e r m a l , L - A =

37

-

Pf, 1.0907 from p r e v i o u s example. Thus, f o r a f l o w of 7 - 1 6 lb/min, t h e h e a t f l o w r e q u i r e d t o m a i n t a i n i s o t h e r m a l c o n d i t i o n s i s = 7.16 x 6 0 x 0.21 90 ~ t u / h r . T h e r e f o r e , i s o t h e r n a l c o n d i - t i o n s would have r e q u i r e d a h e a t f l o w from t h e s u r r o u n d i n g s i n t o t h e p i p e . During -the t e s t , t h e room t e m p e r a t u r e was l e s s t h a n t h e t e m p e r a t u r e i n t h e p i p e , making it v i r t u a l l y i m p o s s i b l e f o r h e a t t o f l o w i n , and w i t h t h e s m a l l t e m p e r a t u r e d i f f e r e n . c e e x i s t i n g , t h e p i p e was e f f e c t i v e l y i n s u l a t e d , a p p r o a c h i n g t h e a d i a b a t i c c o n d i t i o n r a t h e r t h a n t h e i s o t h e r m a l .

Symbols Used .- i n Appendix .-

-

A

p=

a i r s p e c i f i c w e i g h t = a i r s p e c i f i c volume

do = pipe d i a m e t e r ft V = a i r v e l o c i t y f t / s e c \rJ = a i r weight flow lb/rnin

T

_{= a b s o l u t e t e m p e r a t u r e} O R q = h e a t flow Bt-u/lb u = i n t e r n a l energy Btu./lb J = mechanical e q u i v a l e n t of h e a t =

778

f t

l b h t u P = f r i c t i o n energy l o s s = 0

f

= pipe f r i c t i o n f a c t o r

A

= a i r dynamic v i s c o s i t y l b s e c / f t 2 C

I?

k = r a t i o of s p e c i f i c h e a t s = -- = 1.4 f o r a i r

Cv

C~ = s p e c i f i c h e a t a t c o n s t a n t volume = s p e c i f i c h e a t a t c o n s t a n t p r e s s u r e L = l e n g t h of pipe f t d = d i f f e r e n t i a l o f , R = gas c o n t a c t f o r a i r =

53.7

Re = Reynoldsf number P = gauge p r e s s u r e P S ~ E p = a b s o l u t e p r e s s u r e p s f a g = a c c e l e r a t i o n due t o g r a v i t y = 32.2 f t / s e c 2 I n = n a t u r a l logari-thtn o f , a = a c c e l e r a t i o n f t / s e c pv = vapour p r e s s u r e i n . 1iI.C.

d-)

Jo = f r i c t i o n s h e a r s t r e s s Q, = volunle flow cftn

= dew p o i n t t e m p e r a t u r e o ;$ t d ~ I

M

= hlach number Ti = a b s o l u t e a i r t e m p e r a t u r e a t p i p e i n l e t O R = a b s o l u t e a i r t e m p e r a t u r e Tr a t f l o v m e t e r O R = a b s o l u t e mean a i r ICm t e m p e r a t u r e O R hl-h2 = p r e s s u r e d i f f e r e n c e between p o s i t i o n s 1 and 2 p s i g

p,

= mean a i r s p e c i f i c w e i g h t _pcf

Ys

= s t a n d a r d a i r s p e c i f i c w e i g h t = .075 pcf

APPENDIX B ACCURACY OF THE APPARATUS

A l l t h e measuring a p p a r a t u s showed f l u c t u a t i o n s i n r e a d i n g s , and c e r t a i n i n h e r e n t e r r o r s .

( a ) Rotameters

Readings made on t h e r o t a m e t e r s were g e n e r a l l y s t a b l e . Meter 1 mas r e a d a b l e t o

+

0.001 cfrn, meter 2 t o

+

0.005 cfm, meter

3

t o

+

0.01 cfm, and meter 4 t o

+

0.10 cfm.

-

-

-

The u n c e r t a i n t y i n t h e r e a d i n g s averaged

+

$

p e r c e n t of t h e l o v ~ e s t flow measured by each meter. HoGever, s u f f i c i e n t r e a d i n g s were talcen t o v i r t u a l l y e l i ~ n i n a t e t h i s u n c e r t a i n t y . ( b ) Thermocou>le

-

I n d i c a t o r Combination However, t h e e r r o r when w i g e t h e r o r w h e i n s t a l l e d i n a c a l i b r a t e d

The i n d i c a t o r s c a l e could be r e a d t o

+ O.l°F.

t2.lermocouples themselves were s u s c e p t i b r e t o . r e s o f d i s s i m i l a r d i a m e t e r s were joined t o - In t h e thermocouple junct-ions were improperly

t h e p i p e s . The thermocouples were checked with thermometer and found t o be a c c u r a t e t o

-

+

1°P. ( c ) Manometers and P r e s s u r e Gauge

( i ) Betz.

-

A t t h e l o w f l o w s , t h e r e a d i n g s were v e r y s t a b l e and could be read t o + 0.02 mm VI.C. b u t under t u r b u l e n t f l o w , t h e readings were u n s t a b l e . Small amplitude, high-f requency f l u c t u a t i o n s werc superimposed on a l a r g e a m p l i t u d e , l o w - f requency f l u c . t u a t i o n , ma k i n g t h e r e a d i n g s d i f f i c u l t t o t a k e . The low-frequency f l u c t u a t i o n appeared t o f o l l o w t h e f l u c t u a t i o n i n t h e r o t a m e t e r r e a d i n g .

heref fore,

t h e t ~ ~ o r e a d i n g s were made a s n e a r l y simultaneous a s p o s s i b l e , which l e f t only t h e s m a l l amplitude h i g h -

frequency f h ~ c t u a t i o n : ; a s e r r o r i n r e a d i n g . This e r r o r reached a magnitude of approximately

-

+ 0.7 p e r c e n t .

( i i ) 100 i n . Manometers.

-

These manometers were normally s t a b l e , b u t a t t h e h i g h f l o w s , reached

f h l c t u a t i o n s of n e a r l y + 1 - i n . w a t e r column, o r approached a n u n c e r t a i n t y o f

-

+

1 p e r c e n t i n r e a d i n g .

( i i i ) Bourdon Gauge.

-

The Bourdon gauge r e a d i n g s were g e n e r a l l y s t a b l e and were r e a d a b l e t o

+

1/8 p s i g i n t r o d u c i n c .t maximum u n c e r t a i n t y of

-

+ 1.2 p e r c e n t .

( d ) Dew P o i n t Apparatus

The dew p o i n t temperature was determined t o w i t h i n

-

+ rj°F, bu-b s i n c e i t s e f f e c t on d e n s i t y i s v e r y s m a l l ,

t h i s e r r o r was c o n s i d e r e d n e g l i g i b l e .

Extraneous l e a k a g e s from t h e a p p a r a t u s were reduced u n t i l t h e y were l e s s t h n n a p p r o x i m a t e l y 1

x

10-3 cfm,

The r e a d i n g s t h a t were made d u r i n g t h e t e s t were a l l s u b j e c t t o s y s t e m a t i c and random e r r o r s .

The measured f l o w was c o n s i d e r e d a s h a v i n g a s y s - t e m a t i c e r r o r , and t h e random e r r o r was assumed n e g l i g i b l e . The p r e s s u r e measurements were c o n s i d e r e d t o be s u b j e c t t o o n l y random e r r o r s .

The random e r r o r s could be c o n s i d e r e d s e l f - e l i m i n a t i n g when a s u f f i c i e n t l y l a r g e number of p o i n t s were talcen, On

t h e o t h e r hand, t h e s y s t e m a t i c e r r o r s would p e r s i s t no m a t t e r how many r e a d i n ~ s were t a k e n .

The f l o w was c o n s i d e r e d a s h a v i n g a s y s t e m a t i c e r r o r of

+

1 p e r c e n t ; t h e t e m p e r a t u r e s , a s h a v i n g a s y s t e m a t i c

e r r o r of

+

0 . 2 p e r c e n t , S i n c e t h e p r e s s u r e measurement e r r o r s weFe c o n s i d e r e d random, even t h o u g h l a r g e , t h e i r

e f f e c t s may be i g n o r e d , provided a s u f f i c i e n t l y l a r g e number of r e a d i n g s were made.

The s p e c i f i c w e i g h t o f t h e a i r , which i s a f u n c t i o n of t e m p e r a t u r e and p r e s s u r e , w i l l have a maximum p o s s i b l e s y s t e m a t i c e r r o r of

-

+ .2 p e r c e n t .

The weight flovt of a i r

By assuming t h e e r r o r

in^

and

-

d a s b e i n g n e g l i g i b l e ,

P1

The e r r o r i n I n

-

i s n e g l e c t e d , and P2

T h e r e f o r e , when t h e e r r o r s i n t h e p r e s s u r e measure- ment a r e c o n s i d e r e d a s b e i n g o n l y random, t h e maximum p o s s i b l e e r r o r i n Re i s + 1.1 p e r c e n t , and t h e maximum p o s s i b l e e r r o r

*

-

i n

+.

i s

+

2.4 p e r c e n t . However, i f t h e r e were s y s t e m a t i c e r r o r s in t h e p r e s s u r e measurements a s w e l l , t h e maximum p o s s i b l e e r r o r s would be g r e a t e r . I t can be assumed t h a t t h e p r o b a b l e e r r o r w i l l be l e s s t h a n t h e maximum v a l u e s d e t e r m i n e d , t h e d e t e r m i n i n g f a c t o r b e i n g t h e signs of t h e flow and t e m p e r a t u r e e r r o r s .