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Friction factor characteristics for flow regime transition in concentric annuli Foster, Allan Wilson 1965

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FRICTION FACTOR CHARACTERISTICS FOR FLOW REGIME TRANSITION IN CONCENTRIC ANNULI by ALLAN WILSON FOSTER B.A.Sc, U n i v e r s i t y of Toronto, 1963 A THESIS SUBMITTED IN PARTIAL FUI^DffiKT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CHEMICAL ENGINEERING We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, I96U In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada , ABSTRACT F r i c t i o n f a c t o r s have been determined experimentally f o r flow of water and various aqueous solutions of polyethylene g l y c o l i n four d i f f e r e n t concentric annuli. The annuli studied covered a diameter r a t i o range of 0 . 0 U 0 6 to O . 6 3 3 1 . The annular entrance was sharp-edged and no spacers were used within the system. The Reynolds numberjrange investigated was approximately 2 0 0 to 2 6 , 0 0 0 , based on equivalent diameter equal to four times the hydraulic radius. The f u l l y developed f r i c t i o n f a c t o r s f o r a l l four diameter r a t i o s were co r r e l a t e d by the Nikuradse equation f o r a smooth pipe when the Reynolds number exceeded 3 5 0 0 and by the Knudsen and Kata theory f o r laminar flow when the Reynolds number was l e s s than 2 2 0 0 . Deviations from the theory of Hanks fo r laminar - turbulent t r a n s i t i o n i n the well-developed flow region of concentric annuli could be t e n t a t i v e l y accounted f o r by the extra turbulence a r i s i n g from the sharp-edged entrance i n the present i n v e s t i g a t i o n . For well-developed flow the c r i t i c a l Reynolds numbers f o r the various diameter r a t i o s ranged between 2 6 5 O and 3 7 0 0 , and the mode of t r a n s i t i o n was sharp. However, f o r low, intermediate and high values of entrance length, r e s p e c t i v e l y , three d i f f e r e n t modes of t r a n s i t i o n were found to e x i s t i n the annuli studied. Local f r i c t i o n f a c t o r s were based on pressure gradient, uncorrected f o r changes i n k i n e t i c energy due to the developing v e l o c i t y p r o f i l e . AC^WLEDGEMENTS i i i The author i s greatly indebted to Dr. Norman Epstein f o r h i s guidance and help throughout the course of t h i s work. The author would also l i k e to thank the Un i v e r s i t y of B r i t i s h Columbia f o r the Graduate Fellowship and the National Research Council of Canada f o r t h e i r f i n a n c i a l support of t h i s p r o ject. TABLE OF CONTENTS Page INTRODUCTION 1 HANKS* THEORY 12 APPARATUS 15 EXPERIMENTAL PROCEDURE 19 CALCULATIONS 20 RESULTS 21 DISCUSSION 36 Laminar Flow 36 T r a n s i t i o n a l Flow 1+3 Turbulent Flow k9 SUMMARY 51 NOMENCLATURE 53 REFERENCES 55 APPENDIX I 1-1 APPENDIX I I I I - l APPENDIX I I I I I I - l LIST OF TABLES Table Page 1. A Comparison of fRe Versus Re C h a r a c t e r i s t i c s f o r Previous Works on Annuli 9 2 . Annuli Dimensions 1 7 3 - Estimated Deviations i n E c c e n t r i c i t y f o r the Annuli . . . . 1 7 h. T r a n s i t i o n Reynolds Number Range and Turbulent Slope as a Function of Entrance Length 3^-5 . Comparison of Entrance Length Using the Pressure Gradient C r i t e r i a k2 6 . Comparison of C r i t i c a l Reynolds Numbers f o r Various Diameter Ratios 1+5 I I - l Hanks' C r i t i c a l Reynolds Number as a Function of Diameter Ratio I I - 2 I I - 2 Estimated Deviations of f.Re Product f o r Laminar Flow . . I I - 3 I I - 3 Measurements of Annular Gap at the Top and Bottom of the Test Section Il-k I I I - l a Data f o r A-Runs, CT = O . 6 3 3 I . . . I I I - 2 I l l - l b Data f o r B-Runs, CT = O.I+63I I I I - 2 I I I - l c Data f o r C-Runs, <j = 0 . 3 1 * 0 3 I I I - 3 I l l - I d Data f o r D-Runs, <T = 0 . 0 U 0 6 I I I - 3 I I I - 2 a Data f o r A-Runs, Well-developed Region, CT = O . 6 3 3 I . . III-U I I I - 2 b Data f o r B-Runs, Well-developed Region, CT = O . U 6 3 I . . III-1+ I I I - 2 c Data f o r C-Runs, Well-developed Region, <T = 0 . 3 ^ 0 3 . . I I I - 5 I I I - 2 d Data f o r D-Runs, Well-developed Region, CT - 0 . 0 U 0 6 . . I I I - 5 LIST OF FIGURES Figure Page 1. C h a r a c t e r i s t i c f.Re Curves 2 2. F r i c t i o n Factors f o r Galloway's Open Tube Data at Various Entrance lengths . . k 3. Various fRe C h a r a c t e r i s t i c s from Galloway'B Open Tube Data 5 k.k. Galloway's Round Tube F r i c t i o n Factor P r o f i l e s i n Entrance and F u l l y Developed Regions 6 k A . Sketch Showing the Types of Annular Entrance Used i n the Present Investigation 16 5- Local F r i c t i o n Factor P r o f i l e s at Various Reynolds Numbers, CT = O.633I 22 • ';.6. Local F r i c t i o n Factor P r o f i l e s at Various Reynolds Numbers, CT = O.U63I 23 7. Local F r i c t i o n Factor P r o f i l e s at Various Reynolds Numbers, CT = O.3I1O3 2k 8. Local F r i c t i o n Factor P r o f i l e s at Various Reynolds Numbers, CT = 0.0U06 . 25 9. Local F r i c t i o n Factor - Reynolds Number Product f o r CT = O.633I Based on Equivalent Diameter 26 10. Local F r i c t i o n Factor - Reynolds Number Product f o r CT = OA63I Based on Equivalent Diameter 27 11. Local F r i c t i o n Factor - Reynolds Number Product f o r <T = O.3I+O3 Based on Equivalent Diameter 28 12. Local F r i c t i o n Factor - Reynolds Number Product f o r CT = 0.0U06 Based on Equivalent Diameter 29 13• Well-developed F r i c t i o n Factor-Reynolds Number Product f o r CT = O.633I Based on Equivalent Diameter . . 30 Ik. Well-developed F r i c t i o n Factor-Reynolds Number-Product f o r CT = O .U63I Based on Equivalent Diameter . . 31 15» Well-developed F r i c t i o n Factor-Reynolds Number Product f o r CT * 0.3^03 Based on Equivalent Diameter 32 16. Well-developed F r i c t i o n Factor-Reynolds Number Product f o r CT = 0.01+06 Based on Equivalent Diameter . . 33 • i i LIST OF FIGURES (cont'd) Figure Page 17. Laminar Flow F r i c t i o n Factors f o r CT = O.633I 38 18. Laminar Flow F r i c t i o n Factors f o r CT = O.I+63I 39 19. Laminar Flow F r i c t i o n Factors f o r CT - 0. 3J+O3 kO 20. Laminar Flow F r i c t i o n Factors f o r CT = 0.0h06 kl 21. A comparison of Hanks' predicted values of c r i t i c a l Reynolds number with those found experimentally, based on d e „ ^ ( € ) , kk 22. Pressure development i n an annulus using: ( i ) Spacing pins, ( i i ) Faulty pressure taps, ( i i i ) Good pressure taps. 1-3 1 INTRODUCTION In a recent paper on flow through concentric annuli Hanks ( l ) noticed from h i s l i t e r a t u r e survey that e a r l y data f o r t r a n s i t i o n a l flow i n annuli seemed to represent three d i s t i n c t types of c h a r a c t e r i s t i c behaviour. Some flows exhibited the required c h a r a c t e r i s t i c s of r e c t i l i n e a r i t y . This condition i s e a s i l y tested from the data by a log-log p l o t of the product of f and Re as ordinate with Re as abscissa. Here f i s the Fanning f r i c t i o n f a c t o r given by and Re i s the Reynolds numbers-given by Re = De V (2) V Truly r e c t i l i n e a r flow i s characterized by a unique constant value of the f Re product f o r the e n t i r e laminar region. The laminar-turbulent t r a n s i t i o n i s marked by a sharp departure of the data from t h i s constant value along some well-defined curve. This i s the Type I c h a r a c t e r i s t i c shown i n Figure 1. The second type of c h a r a c t e r i s t i c observed was the existence of two d i s t i n c t constant values of the product, f.Re, i n the laminar range. The data seemed to be well represented by the t h e o r e t i c a l value of f Re f o r low Reynolds numbers. A somewhat higher but constant value.,was',however, obtained f o r intermediate Reynolds numbers, up to a well-defined point of t r a n s i t i o n to a type of flow which was c l e a r l y non-laminar. This c h a r a c t e r i s t i c i s shown as Type II i n Figure 1. The t h i r d c h a r a c t e r i s t i c behaviour suggested by some l i t e r a t u r e data was the complete absence of a unique constant f.Re product at a l l measured values of Re, thus making i t impossible to assign any d e f i n i t e t r a n s i t i o n Reynolds number. This i s shown by the Type I I I curve of Figure 1. 2 l o g R e FIGURE -.1: CHARACTERISTIC f'Re CURVES 3 Before studying the l i t e r a t u r e data on concentric annuli, the author analysed f r i c t i o n f a c t o r and Reynolds number data c o l l e c t e d by Galloway (2) f o r an open tube. From these data, p l o t s of f versus Re and fRe versus Re were made.. Figure 2 shows some f versus Re curves with X/De, the dimensionless "entrance length", as a parameter.. The fRe versus Re curves, given by Figure 3, show that the three c h a r a c t e r i s t i c s mentioned by Hanks are a function of the parameter X/De. I t can be observed i n these p l o t s that v a r i a t i o n of the "entrance length" causes the c h a r a c t e r i s t i c curve to vary also. I t appears that the three types of c h a r a c t e r i s t i c curves which Hanks thought to represent three types of flows i n separate systems-, do i n f a c t e x i s t i n the same system, but at various distances from the entrance of the section. I f the f r i c t i o n f a c t o r s are measured very close to the entrance, the fRe versus Re p l o t (Figure 3) appears f o r the open tube to be that of the t h i r d c h a r a c t e r i s t i c . I f , however, the f r i c t i o n f a c t o r s are measured at some intermediate length from the entrance f o r the open tube, a Type II c h a r a c t e r i s t i c appears. As the "entrance length" increases to s t i l l l a r g e r values, one constant value of fRe versus Re (Figure 3) appears with a sharp c r i t i c a l point, i n d i c a t i n g the c h a r a c t e r i s t i c of Type I. It appears thus that Hanks was wrong i n categorizing:; these three behaviours as types of flow. Instead these curves are c h a r a c t e r i s t i c of the f versus Re or fRe versus Re p l o t s at varying values of X/De. p The l a t t e r point may be emphasized by considering a p l o t of f.Re versus X/De, with Re as a parameter,. For Galloway's (2) open tube data a p l o t of t h i s nature was made (Figure h). I f three v e r t i c a l l i n e s are drawn at low, intermediate and high values of X/De, i t i s po s s i b l e , i f the various curves are c r o s s - p l o t t e d on fRe versus Re coordinates, to obtain the three c h a r a c t e r i s t i c s of Figure 1. In f a c t , the points of i n t e r s e c t i o n of l i n e s A, B and G i n Figure k form the curves A, B and C r e s p e c t i v e l y i n Figure 3. The term X/De or "entrance length" r e f e r s to the distance allowed from the entrance to the section of the tube or annulus where the t e s t s are being run, P a r a m e t e r * x / D e l o g R e FIGURE 2t FRICTION FACTORS FOR GALLOWAY'S OPEN TUBE DATA AT VARIOUS ENTRANCE LENGTHS. 5 a> a: > / l 2 8 c — ' B 1 6 A P a r a m e t e r 1 x / D e l o g R e FIGURE 3» VARIOUS f»Re CHARACTERISTICS FROM GALLOWAY'S OPEN TUBE DATA. 6 A B C Parameter: Re x T D e FIGURE 4t GALLOWAY'S ROUND TUBE FRICTION FACTOR PROFILES IN ENTRANCE AND FULLY DEVELOPED REGIONS. expressed as number of equivalent diameters of the t e s t section being used. There are two papers which study the entrance e f f e c t s i n concentric annuli, and these are i n disagreement. According to Rothfus et a l (3), w e l l established turbulent flow i n a concentric annulus requires that the "entrance length" be 250 equivalent diameters or greater. The term "well established flow" r e f e r s i n t h i s case to a flow whose shear stress at the inner w a l l i s within 5$ of i t s ultimate value. Sparrow and Olson {h), on the other hand, when studying turbulent flow development i n tubes and concentric annuli with square or rounded entrances, found that the "entrance length" needed to reduce pressure gradient to within 5$ of i t s ultimate value was l e s s than 25 equivalent diameters. The analysis of Galloway's tube data l e d to a survey of f r i c t i o n f a c t o r versus Reynolds number data f o r concentric annuli from the year 1923 onward. P a r t i c u l a r notice i n t h i s survey was given to equipment design and the "entrance lengths" preceding the t e s t sections. Also the diameter r a t i o s f o r the annuli were c a r e f u l l y noted. The data c o l l e c t e d f o r annuli showed a s u r p r i s i n g trend s i m i l a r to that i n d i c a t e d by Galloway's open tube data. The best set of data i n annuli showing t h i s trend i s given i n the paper by Walker et a l (5)- The diameter r a t i o s f o r these data extended from zero f o r open tubes to one f o r p a r a l l e l p l a t e s , with s i x diameter r a t i o s i n between. Since the measurements of f r i c t i o n f a c t o r were made at a constant distance from the entrance, by varying the diameter r a t i o Walker et a l (5) e f f e c t i v e l y v a r i e d the entrance X/De r a t i o . Their p l o t s of f versus Re show that changes i n the c h a r a c t e r i s t i c s of these curves occur with entrance r a t i o i n a manner p a r a l l e l i n g Galloway's data. The data c o l l e c t e d by e a r l i e r workers (6, 7, 8, 10, 11, 12, 13, lU) generally are quite scattered. Because the differences i n diameter r a t i o s used and design of equipment were so great f o r most of the e a r l y data c o l l e c t e d , 8 the r e s u l t s must be viewed with a cautious eye, e s p e c i a l l y i n making comparisons. However, i f one looks at i n d i v i d u a l workers' data, i t i s observed that the three f.Re versus Re c h a r a c t e r i s t i c s have been recorded over the years, but without explanation as to t h e i r occurrence. Becker's (8) data f o r two diameter•ratios ind i c a t e that the c h a r a c t e r i s t i c curve f o r flow regime t r a n s i t i o n i s described by two d i s t i n c t , constant values of fRe i n the laminar region, followed by a well-defined point of t r a n s i t i o n . Becker used an "entrance length" of from 11 to 16 equivalent diameters. Kratz et a l (10) found the same r e s u l t s as Becker, but the second constant value of fRe was more pronounced and an "entrance length" of 70 equivalent diameters was used. Lonsdale (6), on the other hand, found that using an "entrance length" of 100 equivalent diameters, he achieved one constant value of fRe followed by a sharp t r a n s i t i o n , i n d i c a t i n g a Type I c h a r a c t e r i s t i c . Carpenter et a l ( l l ) found, using an "entrance length" of 230 equivalent diameters, that the f versus Re p l d t was i n d i c a t i v e of a Type III c h a r a c t e r i s t i c , i n which there was an absence of any constant fRe value. However, the t a n g e n t i a l entry design used by Carpenter et a l was r a d i c a l l y d i f f e r e n t from that employed by the other workers and hence l i t t l e weight should be applied to h i s r e s u l t , i n comparing entrance lengths required to produce a.given f versus Re t r a n s i t i o n c h a r a c t e r i s t i c . Table 1 l i s t s the early investigators;:who measured f r i c t i o n f a c t o r s i n concentric annuli, the diameter r a t i o s of annuli used, the "entrance lengths" and the fRe c h a r a c t e r i s t i c s near the t r a n s i t i o n region. TABLE 1. A Comparison of f»Re versus Re C h a r a c t e r i s t i c s "for Previous Worst's on AnriuM. Investigator Ref. No. o~ Entrance length. No. of equiv. diameters C h a r a c t e r i s t i c s of the fRe versus Re curve. Lonsdale 6 0.1+19 100 One constant fRe value. Cornish 7 0.990 0 to 510 Results too scattered. Becker 8 0.932 11 Two constant fRe values. n 8 0.962 16 it n it it Atherton 9 0.800 1+5 Results too scattered. Kratz et a l 10 0.514 5^ Turbulent flow data only. ti ti ti 10 0.628 70 Two constant fRe values. II . 'i I I it II 10 0.805 135 it ti 11 11 Carpenter et a l ..111 0.71+9 230 No constant fRe values. Schneckenberg 12 0.9U5 6 As a whole these data it 12 0.985 11 indi c a t e that- two 11 12 0.9903 16 constant values of fF,3 it i 12 0.9937 25 occurred. Winkel 13 0.833 not given Two constant fRe values. Walker et a l 5 0.0260 171 tt ti 11 11 II ti I I 5 0.0667 179 I I 11 11 it •• II ti 5 0.1251 190 I I 11 11 ti n ii I I 5 0.1653 200 One constant fRe value. it I I II 5 0.3312 21+9 11 v 11 11 11 II il I I 5 0.1+987 332 11 11 11 11 Rothfus et a l Ik 0.162 93 Two constant fRe values. it ir il Ik 0.650 223 11 11 11 11 Croop et a l 15 0.062 725 ti 11 it ti i ti H I I 15 0.197 725 it ir 11 11 II I I n 15 0.331 725 11 11 11 tt it I I II 15 0.500 725 11 tt ti 11 Lea and Tadros 16 0.1+99 not given One constant f.Re value. A f t e r studying the c o l l e c t e d tube and annuli data the author f e l t that deliberate i n v e s t i g a t i o n of the c h a r a c t e r i s t i c s of flow regime t r a n s i t i o n i n concentric annuli would be of value. I t was proposed, therefore, to study the pressure development i n concentric annuli as a function of diameter r a t i o . I t was a n t i c i p a t e d that t h i s study, added to already e x i s t i n g data, would show c o n c l u s i v e l y that Hanks ( l ) was wrong i n categ o r i z i n g the three t r a n s i t i o n c h a r a c t e r i s t i c s which he noted as types of flow behaviour, and that they a r i s e f o r t u i t o u s l y out of the d i f f e r e n t shapes of low, intermediate and high Reynolds number curves of f versus x/j)e. The present t h e s i s i s a r e s u l t of t h i s proposed i n v e s t i g a t i o n . For each diameter r a t i o the parameter of "entrance length" was studied, and i n t h i s way the e f f e c t s of "entrance length" and of varying diameter r a t i o s were observed. The present study had a d d i t i o n a l and equally important objectives. One was to t e s t a theory proposed by Hanks ( l ) f o r p r e d i c t i n g the c r i t i c a l Reynolds number f o r t r a n s i t i o n from laminar to turbulent flow i n the "well-established" flow region of concentric annuli. Another was to attempt c o r r e l a t i o n of pressure gradient data i n both the entrance and "well-established" regions, with p a r t i c u l a r emphasis on entry length necessary to achieve "well-established" flow, under laminar and t r a n s i t i o n a l regimes, as w e l l as fo r the disputed (U) turbulent regime. A f i n a l objective was to investigate the p o s s i b i l i t y that d i f f e r e n t modes of t r a n s i t i o n (e.g. sharp versus gradual) from laminar to turbulent flow f o r d i f f e r e n t diameter r a t i o s p e r s i s t e d even i n the "well-established" region of the annuli, as suggested by s i m i l a r v a r i a t i o n s i n t r a n s i t i o n c h a r a c t e r i s t i c s observed by Galloway (2) i n rod bundles having various, pitch-to-diameter r a t i o s . The experimental work consisted of f i n d i n g l o c a l f r i c t i o n f a c t o r s f o r various diameter r a t i o s of v e r t i c a l concentric annuli.. by measuring pressure drops over incremental lengths of the annuli from the entrance to the e x i t . The diameter r a t i o was varied by varying the outside diameter of the inside core, and the range of hydraulic Reynolds number covered was approximately 200 to 26,000. HANKS THEORY Hanks ( l ) i n h i s recent paper proposes a generalized s t a b i l i t y parameter which i s independent of geometry of the flow system, i s proportional to the Reynolds number f o r Newtonian flow, and contains Ryan and Johnson's (17) r e s u l t s f o r pipe flows as a s p e c i a l case. Using the equations which describe the motion of a f l u i d d i v = - i £ (3) and _ £ a ) p _|I + i p grad (V.V) - pV x £ (b) = T - grad p - div T (k) Hanks suggests that when the magnitude of the a c c e l e r a t i o n force (a) reaches a c e r t a i n multiple of the magnitude of the viscous force (b), the f l u i d motion w i l l be unstable to c e r t a i n types of disturbances and stable laminar flow w i l l no longer e x i s t . Mathematically t h i s suggestion i s expressed as V x f = K div T (5) where K i s a l o c a l s t a b i l i t y parameter which i s a function of p o s i t i o n i n the flow f i e l d . Since V x £ vanishes at a l l s o l i d boundaries and div T does not, K must also vanish on s o l i d boundaries. At some point i n the flow region K acquires a maximum value, which i s designated by K and i s proportional to the bulk Reynolds number. It i s postulated by Hanks ( l ) that when K reaches a s u f f i c i e n t l y large constant c r i t i c a l value k at some point i n the flow f i e l d , c e r t a i n types of disturbances, i f introduced at t h i s point, w i l l be able to grow and spread to the s o l i d boundary surfaces. Here, they become self-maintaining and give r i s e to a general t r a n s i t i o n to turbulence throughout the region of flow. Hanks then considers the s p e c i a l case of steady state flow, f o r which equation (5) s i m p l i f i e s to \ p \ s r a d <™> 1 - K (6) I ¥ - grad p I The Reynolds number and f r i c t i o n f a c t o r are s t i l l defined by equations ( l ) and (2), but a new equivalent diameter de i s substituted f o r De. Hanks' equivalent diameter i s expressed as de = 8 ( € ) (7) where § = r' 2 - v\ = \ De ^ \l/(€ ) = [ l + ( l + € ) 2 ] l n ( l + € ) + l - ( l + € ) 2 (8) 2 €2 l n ( l + € ) For pipe flow, using a dimensionless coordinate system, Hanks f i n d s that K = y V 2 7 - Re (9) He then takes Re f o r pipe flow equal to the c r i t i c a l value of 2100 and l e t s "K = k, f o r which he obtains from equation (9) that k = kOk (10) This value of k i s assumed to be constant f o r a l l geometries. Using dimensionless coordinates s i m i l a r to those f o r pipe flow, Hanks fin d s that f o r concentric annuli K = - (Re)based on de V 2 ^ ( € ) € ( i 2 + € \ l n ( l + C x > l - ( l + C x ) ^ [ € [ 2 + € \ - 2 ( l - € x ) 2 l l n ( l + € ) J L l n ( l + € ) J ( l l ) €3.*/>( € ).(l+€x) where x i s the dimensionless coordinate f o r annular flow, (r-r^)/ (i^-t^) Using equations (lO), ( l l ) and a derived r e l a t i o n s h i p f o r x^-j^ as a function of € (Hanks' equation (26)), Hanks p l o t s R eCRiTICAL a s a function of CT (See Figure 21). This p l o t was c a l c u l a t e d by a hand c a l c u l a t i o n i n Appendix II (b). 15 APPARATUS The apparatus vas e s s e n t i a l l y the one constructed by Galloway (2), with a few minor modifications. The outside tube f o r the annuli was made of 0.375-inch + 0.003 O.D. Admiralty brass tubing having 0.0685-inch w a l l thickness (ASTM number B l l l - 6 2 ) . The 12-foot long brass tube was i n s t a l l e d i n Galloway's apparatus using the same top and bottom p o s i t i o n i n g p l a t e s . The pressure taps used were s i m i l a r to those used iby Galloway f o r h i s open tube. They were located at 6-inch i n t e r v a l s along the annulus, the f i r s t tap being s i x inches from the entrance and the l a s t tap being s i x inches from the e x i t . Each pressure tap was connected to a solenoid valve which i n turn was connected to one of two v e r t i c a l headers running the length of the t e s t section. The solenoid valves, which were connected a l t e r n a t e l y to the l e f t -and"right-hand headers, were mounted one above another on a v e r t i c a l framework alongside the t e s t section. Tne headers were connected by 3/8-inch O.D. copper tubing to a 150-cm inverted a i r manometer mounted on the panel board. The switches which operated the valves were also mounted on t h i s panel. The tubes and cores used i n t h i s i n v e s t i g a t i o n have the dimensions shown i n Table 2. These c y l i n d e r s were held v e r t i c a l l y i n p o s i t i o n using a p o s i t i o n i n g p l a t e at the top and bottom of the column. These p o s i t i o n i n g p l a t e s were the::same as those used by Galloway f o r h i s square array with an a d d i t i o n a l 1/8-inch diameter hole d r i l l e d i n the centre of t h i s p l a t e . (See reference (2), page 1-57* Figure I-B). The annular entrance f o r runs A, B and C i s shown by Section A-A of Figure k A . For these runs there i s a sudden enlargement from l / 8 - i n c h to the diameter of the c y l i n d e r being used. Section B-B of Figure kA shows the type of entrance f o r D-run. The brass cores used were i d e n t i c a l to three of Galloway's rods. A SECTION A-A BRASS CORE B B w / / / / / / / / A -/ / / / / SECTION B-B STAINLESS STEEL CORE FIGURE 4A: SKETCH SHOWING THE TYPES OF ANNULAR ENTRANCE USED IN THE PRESENT INVESTIGATION. 17 TABLE 2 Annuli Dimensions Run I.D. Outer Tube inches O.D. Inner Core inches Equivalent Diameter . De. mcnes Radius Ratio cr A O.738O O.I+672 0.2708 0.6331 B O.7380 0.31+177 0.3962 0X631 C O.738O 0.25116 0.U868 0.31+03 D O.738O 0.0300 O.7080 0.01+06 The outer column was aligned using a f i n e s t a i n l e s s s t e e l plumb-line along two outer edges of the outer tube 90 degrees apart. The gap between the core and the outer tube was measured at four p o s i t i o n s 90° apart at the top and bottom of the column, using c a l i p e r s . The percent deviation from concentric geometry was then taken as the center-to-center deviation divided by the inside radius of the outer tube. The r e s u l t s are shown i n Table 3 and a sample c a l c u l a t i o n i s found i n Appendix I l ( d ) . TABLE 3 Estimated Deviations i n E c c e n t r i c i t y f o r the Annuli Run De Nominal inches De(AVG) Measured inches Percent Center-to-Center Deviation. = Center-to-Center Deviation Radius of Core Percent Deviation between D e ^ ^ ^ and DeMEASTJRED = ^MEASURED" ^NOM ^NOMINAL A 0.2708 0.2688 1.10 -O.738 B O.3962 0.3975 1.72 +0.321 C 0.U868 O.I+875 1-31 +O.I36 D O.7080 0.7062 0.60 -0.251+ I n i t i a l l y , spacing pins made of 0.0255-inch O.D. hypodermic needle were i n s t a l l e d at points h and 8 f e e t from the entrance of the t e s t section. The pins, however, were not used on the four diameter r a t i o s of annuli studied because they were found to cause interference with the pressure development (See Figure 2 2 ( i ) ) . Sparrow and Olson (1+J, who used supporting s t r u t s 18 positioned at + hj degrees from the pressure tap l o c a t i o n s , found that the str u t s increased the pressure gradient f o r at l e a s t 150 s t r u t diameters downstream. Redberger and Charles ( l 8 ) studied a x i a l laminar flow i n a c i r c u l a r pipe containing a f i x e d eccentric core, and found that the e f f e c t of d i s p l a c i n g the inner surface from a concentric p o s i t i o n was to increase the volumetric flow rate f o r a given pressure gradient. Using t h e i r t h e o r e t i c a l r e s u l t s , the author found that f o r the small e c c e n t r i c i t i e s which were prevalent i n the present i n v e s t i g a t i o n (Table 3), the increase -of volumetric flow rate was n e g l i g i b l e . The pressure taps and outside tube i n i t i a l l y used were rejected because of systematic c y c l i n g deviations i n pressure development along the annular t e s t section. During t h i s i n i t i a l p eriod the equipment used by Galloway (2) was checked from the pressure taps to the manometers. These t e s t s (See Appendix i ) showed that the r e p r o d u c i b i l i t y of Galloway's apparatus had not a l t e r e d appreciably, and where differences d i d occur, the apparatus was appropriately repaired. 19 EXPERIMENTAL PROCEDURE The procedure used was e s s e n t i a l l y that employed by Galloway ( 2 ) , with a few minor abbreviations. These abbreviations arose out of the f a c t that, whereas f o r Galloway the incremental drop measurements were a preliminary to measurements over the region of constant pressure gradient, i n the present study a run was confined to measuring the incremental pressure drops. Before s t a r t i n g a run, the f l u i d being used was c i r c u l a t e d within the system to remove any entrained a i r . While t h i s was done, the system was allowed to come to room temperature, and was held at t h i s temperature using a heat exchanger with water as the coolant. The flow through the annular t e s t section was set using e i t h e r c a l i b r a t e d rotameters or a ljy-inch o r i f i c e . With a l l solenoid valves closed, and the by-pass valve on the 150-cm inverted a i r manometer open, the manometer l e v e l s were checked and recorded f o r balance at the beginning and end of each run. Only r a r e l y d i d the manometer l e v e l s not balance to within 0 . 5 mm., i n which case the manometer l i n e s and pressure headers were vented u n t i l balance was achieved. Flow settings, temperature readings at various points on the apparatus and panel board temperatures were recorded at the s t a r t , at the middle and at the end of each run. Between the temperature measurements, f r i c t i o n a l pressure drops were recorded f o r 6-inch i n t e r v a l s over the whole t e s t section. The f i r s t eleven readings were taken by opening solenoid valves l - * 2 , 3-4 , 5-6, etc., up to valves 21-22, and the second eleven readings were taken by opening valves 22-23> 2 0 - 2 1 , 18-19, e t c , down to valves 2-3-The f l u i d s used i n the system were water, 3 centistoke aqueous polyethylene g l y c o l (PEG) and 15 c s . aqueous PEG. The samples of l i q u i d used to measure v i s c o s i t y and density were taken before and a f t e r changing from one v i s c o s i t y to another. These samples were taken from the vent on the 1^-inch o r i f i c e located immediately downstream from the rotameters. 20 CALCULATIONS The method used to c a l c u l a t e f r i c t i o n f a c t o r and Reynolds number from the data c o l l e c t e d was v i r t u a l l y i n d e n t i c a l to the method used by Galloway (2) f o r h i s round tube i n v e s t i g a t i o n . The nominal equivalent diameter l i s t e d i n Table 2 was used i n the c a l c u l a t i o n of Re and f. This was done f o r two reasons. F i r s t , the average deviation between the nominal and measured diameter was very small f o r a l l four diameter r a t i o s of concentric annuli investigated (See Table 3). Second, the measurement of the equivalent diameter was made with the annuli t e s t section i n s t a l l e d , which meant that the top and bottom p o s i t i o n i n g brackets made precise measurements^difficult using the univ e r s a l c a l i p e r s . For t h e : f u l l y developed f r i c t i o n f a c t o r s and Reynolds number, the average pressure drop f o r a 6-inch increment was found over the region of near constant pressure gradient. Again, the nominal equivalent diameter was used. More d e t a i l s of the c a l c u l a t i o n s are presented i n Appendix II and are i l l u s t r a t e d by some sample c a l c u l a t i o n s . The data c o l l e c t e d on a l l runs, from which the r e s u l t s were ca l c u l a t e d , are tabulated i n Appendix I I I . The c a l c u l a t i o n s were made on an IBM-1620 computer. The r e s u l t s are summarized i n the f o l l owing section. 21 RESULTS The l o n g i t u d i n a l p r o f i l e of the l o c a l f r i c t i o n f a c t o r at various Reynolds numbers i s given i n Figure 5 to 8 f o r the four d i f f e r e n t diameter o r a t i o s investigated. The r e s u l t s are p l o t t e d as f.Re versus the number of t o t a l equivalent diameters from the entrance. The f r i c t i o n f a c t o r s and Reynolds numbers are based on an equivalent diameter equal to four times the hydraulic radius. The l o c a l f r i c t i o n f a c t o r r e s u l t s are presented i n Figures 9 to 12 f o r the four diameter r a t i o s investigated. The r e s u l t s are p l o t t e d as f.Re versus Reynolds number with entrance length expressed i n equivalent diameters as a parameter. Again the f r i c t i o n f a c t o r s and Reynolds numbers are based on the equivalent diameter. The f u l l y developed f r i c t i o n f a c t o r r e s u l t s are presented i n Figures 13 to 16. These r e s u l t s are p l o t t e d i n two ways. The upper points are p l o t t e d as f.Re versus Re, where the f r i c t i o n f a c t o r s and Reynolds numbers are ca l c u l a t e d using equations ( i l - l ) and ( I I - 2 ) and the average pressure gradient over the well-developed flow region. The lower points are p l o t t e d as i • 1i f . R e / ^ / d ^ ) versus Re, where the f r i c t i o n f a c t o r s and Reynolds numbers are again given by equations ( i l - l ) and I I - 2 ) , and 0 ( < ^ ' /d2) i s given by equation (13) -Figure 21 shows Hanks' proposed theory g i v i n g the c r i t i c a l Reynolds number f o r t r a n s i t i o n from laminar to turbulent flow i n the well-developed region, as a function of diameter r a t i o . Hanks' equivalent diameter (equation 7) is'used i n the c a l c u l a t i o n of the Reynolds numbers. This p l o t also includes values from Galloway ( 2 ) , Walker et a l ( 5 ) , Whan and Rothfus ( l 8 ) , Cornish (7) and Davies and White ( l9)> The r e s u l t s f o r these references have been adjusted by mul t i p l y i n g Re c by s/2. \j/ (€) to make them comparable to Hanks' predicted values. 1 0 0 L o S f c f 10 ^ A - R U N S <r = 0 . 6 3 3 1 o 0 o 0 o o o o 0 o ° 0 ° o 0 0 o o o o 1 2 , 2 4 4 o ° o o o 0 o 0 0 o 0 o G o o 0 ° o o o ° o o | 0 , 7 9 6 o o o o 0 0 0 o o o 0 0 ° 0 0 0 0 0 ° ° ° o 8 8 7 l ° o 0 0 O o o 0 o o 0 0 0 o o o o 0 o o ° o 7 0 7 7 o o o 0 o o o 0 o o ° ° o o o o 0 o o o ° 4 6 8 1 0 O O 0 O O O 0 O O O 0 G O O O O O O O O O 4 0 4 2 o o o o o o o o o o 0 0 o 0 o o o o o o o 0 3 5 4 8 o 8 8 S 8 8 g 8 8 8 8 8 8 8 8 § 8 3 g o ° n O O o o o o o o o o o O o O o o o 2 9 0 8 o o ° o O o o 2 4 3 5 o D o ° . 0 2 2 2 7 ° ° 0 o 0 0 o o ° ° ° 2 1 9 9 ° o O o o m . « ° [_ O O G o o 0 0 o o o 0 o o o o o o o o 0 o G 1 7 3 7 G O G o o o o o O o 0 o 0 o 0 o o ° o 0 ° o l 4 5 4 o o ° o o o 0 o o o O o 0 0 0 0 0 o 0 o ° o 9 9 l ° ° ° o o o o o o o o o o o o o o o o o 0 6 2 7 ° ° o 0 o . o o o o o o O Q O O O O O O O O 4 3 8 o O o o G O o 0 ^ O o O o ° o O o O o O O 3 i | ° o ° 0 o o o o o o o o O o o o o o o 0 o ° P a r a m e t e r R e 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 E N T R A N C E L E N G T H — x / D e FIGURE 5t LOCAL FRICTION FACTOR PROFILES AT VARIOUS REYNOLDS NUMBERS, OT = 0.6331. B - RUNS cr= 0 . 4 6 3 1 0 o o 0 o ° o o 0 o o O O O Q O O O O O O 1 9 , 6 3 2 o ° 0 0 0 G 0 0 0 o 0 0 O O O 0 O 0 0 O O 0 | 4 2 8 4 o ' 1 0 0 o 0 o 0 o O o 0 o o ° o o o o o 0 o ° o ° 10,155 o o o 0 O o 0 0 o o O o 0 o o o o o 0 O o o 8 4 4 7 o o o 0 o o o 0 o o o o o o o o o o o o o 7 2 2 6 O O O 0 O G O 0 O O ° 0 O O O O O O O O O 0 5 6 4 6 x | o o O o O o o 0 o 0 0 0 o . 0 o 0 o 0 o 0 O o 5288 o o ° o 0 o 0 o 0 o 0 0 ° o o o o o o o o o 4 8 7 7 & f | o o ° o 0 0 0 o o o o O Q O O O O O O O O 4 4 7 6 II - I o ° o 0 o o0 o O o 0 o ° o o o O o o o O o 3 9 4 0 10 0 0 o o 0 o o o 0 o 0 o 0 o 0 0 o o 0 3 6 2 4 O o o o 0 o 0 o 0 o 0 o 0 0 0 o 3 2 9 7 0 O O Q O O 3 0 3 4 w ^ y v - - v ^ y a 2 1 4 8 o o o o 0 o 0 o 0 o o o o o o o o o o o o 1 7 9 8 ° 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 5 4 0 o ° o o 0 O o o o o o 0 o o o o o o o o o 7 8 8 O o o o o 0 o o o o o o ° o ° o O o o o O o 5 3 1 o o O o o 0 o o o o ° o ° o O o o o o o o o 3 5 4 Parameter! Re 2 0 0 1 0 0 3 0 0 ENTRANCE LENGTH — x/De FIGURE 6: LOCAL FRICTION FACTOR PROFILES AT VARIOUS REYNOLDS NUMBERS, CT = 0.4631. <r* 0 . 3 4 0 3 o o o o 0 0 0 o O O O ° O O o 0 o ° 0 0 26,080 o 100 fl M •o CM 3 u o O O A O O O n o o - o ^ o O ° O O ° O O O o 0 | 8 , 2 8 3 o ° o o o O ° o 0 0 o ° o 0 o o O o o o 0 0 l 2 , 8 7 5 o o 0 o 0 o o o 0 0 0 0 Q 0 o o o O Q O O O O 7152 O Q O O O O O O O O O Q O O O O O O O O O O 5 0 2 5 ° o 0 o 0 o ° o 0 0 o o o 0 o 0 o 0 o o o o 4 7 6 , 10 H ^ P o o o o o o o o o o 0 o o o o o o o o 4 2 5 4 o o o o o o o o o o o o o o O o o O 3 8 4 0 0 o o o o o 0 o o o o 0 o o o 0 0 3518 o o o 0 o o o o 0 o 0 o ° o 3 3 5 7 3128 2 9 4 4 2771 2 6 3 3 2 3 0 6 o 8 § 8 8 8 8 8 § 8 8 8 8 8 8 8 8 8 8 8 8 8 f 9 2 l o ° o ^ o o o o o o 0 o 0 o o o o o o o o o o 1683 u ° o O o o o o o o 0 o o o o o o o o o o 1570 o o o o o o o o o o o o o o o o o o o o o o 6 9 9 o o o 0 o o o o o o o o O o O o O o o o o o 5 0 5 Parameter 1 Re 100 2 0 0 3 0 0 E N T R A N C E LENGTH — x/De FIGURE 7 t LOCAL FRICTION FACTOR;! PROFILES AT VARIOUS REYNOLDS NUMBERS, 0" = 0.3403o D - R U N S <r = 0 . 0 4 0 6 o 0 o o o 0 o 0 o o 0 o 0 0 o 0 o o o 0 o 0 2 5 t , 7 6 1* a CM II 10 o 0 0 P o o 0 o 0 o o ° o O Q O Q O O O O 8 0 8 6 0 O 0 O O O 0 O O O 0 O O O G O O O 0 O 0 6 8 1 3 o o o o o o o ° o ° o ° o O o o o O o 4 8 5 8 O o O O O O O ° o 0 0 0 G G o 0 0 0 o o 4 2 6 5 3 7 4 6 « f t « 8 « t « i t « t « i 8 8 « ^ 3 7 0 3 ° o ° o 0 0 0 0 o 0 o 0 o 0 o 3 5 3 7 o 3 1 6 7 3 1 7 5 o o o o O o o o ° o o n ° 3 0 2 6 O O o o o 0 O o 0 o 0 o 0 o ° o o 0 o 0 o 0 o 0 o O ° o ° o o 0 o 0 o 0 o O o u ° o ° ° , o o o U ° 0 o ° o 0 o o ° ° o o 0 o 0 o o ° o o 0 o 0 o 0 0 0 o 0 o 0 o o o O o o ° 0 o o 0 o o 0 o 0 o 0 o 0 o 0 o o o o o „ 2 8 9 1 2 6 7 9 2 5 3 4 2 2 8 9 2 1 1 8 1 8 4 5 0 0 o o o o 0 o 0 o 0 o 0 o 0 o o o o 1 5 7 6 o ° ~ o o o o ^ o o o o o o Q o o o o o Q o o o o 1 2 6 2 o ° o o o o o o 0 o o ° o ° o o o ° o 1 0 6 0 P a r a m e t e r : R e 5 0 1 0 0 1 5 0 2 0 0 E N T R A N C E L E N G T H - x / D e FIGURE 8 i LOCAL FRICTION FACTOR PROFILES AT VARIOUS REYNOLDS NUMBERS, 0" » 0.0406. A - RUNS o 0 o 498.5 X II Q) o ° o o o o o 0 0 ° o 0 o 476.3 *o o o o 454.2 6 ° ° o o o o o o o - 4 3 2 X ) O O G O O OO & ° 4 Q a 9 o ° & o 0 o o o ° o o O O O © _ M „ o Q o 387.7 o o o o o o o o o d o o o o o o o < ^ o o o o o o o d P o o o o o o o (3d o ° o o o OOQP 0 ° 0 0 O O O O D o O o 365.6 ^o ° & o o o o ^o 343.4 o o & o o o o o 321.3 o o e9 o o u o Q o 299.1 <£> G o o o ° 2 7 7 0 & o o o o o O O O Q ) o o ° o 254.8 O ° O O O 0 0 ( J ) o o o o o o o oo o o o o O O O Q D o ^o 232.6 o ° o ° o o 210.5 o o G O o G o 188.3 6° ° o o O O O O O- O O C D O o o 166.2 ,o 144.0 6° o o o o o o o d P o O o 121.9 o ° 6 ° ° Q O O O O 0 0 ( 5 ) Q O O O O O O C D G Q o 99.7 o o o o & o o 77.5 o o o o o o o o»o O n 55 4 O O 6 ° o o ° 0 0 0 0 O O O O D o ° o G 3 3 2 o o o o o o o o o o o ° o o o o o o o o Parameter* x/De J I 1 1 1 L J 1 L 10 Re x I02 100 FIGURE 9: LOCAL FRICTION-FACTOR-REYNOLDS NUMBER PRODUCT FOR O* =0.6331 BASED ON EQUIVALENT DIAMETER. B - RUNS o 340.7 o o' o 325.6 o o o o o<S$° o o o ° 310.4 4? o o o o o o o o ° 2 9 5 3 o o o o oC$? o . o o 280.1 o o o o o o<#5° c° ° 6° 0 ° o 265 .0 <P ° ~o o o o o 2 4 9 . 8 O O O O OQf i P G ° o 2 3 4 . 7 o o o o oaSS> /O o o ° o 219.6 o o o o ocRS> cS> o Q O 0 ° . o 2 0 4 . 4 O O O O OdSQD jO o o ° o° o o 189.2 o o o o o<38& ^S> o o6* o ° o 174.1 o o o o o o o a j © j o o * r o ~o^ o o o 159.0 o O O O O O<505) r p CP ° Q ° 0 ° O 143.8 O O O O O0538 nP o ^ n o 128.7 c r o o o ° ° ° O o Q ® JD o ° ° ° o o a S ) 0 « ° o 113.6 cP o O 9 8 . 4 o o o o oaS© o o o o o o o ° o o o 8 3 . 3 <°" o ° o o o 68.1 o o o o o O ' o O 5 3 . 0 o o o o o<jns> o° 6 ? J£> O o o o o oacSS^ O o 37.8 o cP o cP ° o o o o o$®> ^ 9 o ^ o o 22.7 o o o o o o ( Parameter : x/De J I L J L 10 100 Re x 10 -2 FIGURE 10: LOCAL FRICTION FACTOR-REYNOLDS NUMBER PRODUCT FOR CTsO.4631 BASED ON EQUIVALENT DIAMETER. C - RUNS o 49 o O O O CDOo fi G O O C D O Q OCS2& o 6° a (9 oaa> O O O CDCo -OCXEP O O O QDCfc ^ o 6° O O O C D O Q >a6> 49 O O O CD0 0 0dB° , p f O O O O0O-P6D 6* o o o axxpoJ? <9° o 49 o o o ooo^as? 0 o o o ooqpo^ x) O O O C D C j ^ C ^ ^ / O O O CDC&dfi o -O O O O C&CbOofP o o O O O CDOQOdS? ^ o° o O O QDOQOCSS? ~ o^  o c o o o Qx^od33* o o coqp <^5> o* O O O C D C ^ J9 O O CDOo o o <S>°o o 277.3 o o 265.0 o 252.6 o o 240.3 o o 228.0 o 215.7 o o 203.4 o 191.0 o 178.7 o 166.4 o 154.0 o 141.7 o 129.4 o o o 117.1 o 104.8 o o 92.4 o 80.1 o 67.8 o o 55.5 o 43.1 o o 30.8 o 18.5 Parameter* x/De I I I I I L J L 10 100 Re x 10 -2 FIGURE H i LOCAL FRICTION FACTOR-REYNOLDS NUMBER PRODUCT FOR <T =0.3403 BASED ON EQUIVALENT DIAMETER. D - RUNS o 190.7 o 182.2 o 6+ 8 ° o ° O O -rP o ° o o oo<*w° Q o 6? 0 o ° O O o o / o ° o G O ooo^or 0 ° 9 o ° o o o o t P o P _p &° OOOOC5X3? 0 ° e Q o o o 0 o a x $ n o n 0 - ° o O O O O 0 E X 3 ? Q O u 0 o O O OOCDOP Q O o * 9 o o O O o OC0C5? N O 6 » ° 0 o ° O O o o o Q ° o o o o c c c e 0 ° <9 0 ^ ° O O OOCECw fto <5> ° o ° O O O O C D C V N 0 9 o ° O O O O o o O O C C K n o 9 J 0 o o o o c c K O / 0 ^ 0 o o o o c P Q o o o o o c v Q o o o o o c P 0 o o 0 ° ^ r , o o o O < ° o 173.7 o 165.2 o 156.8 o 148.3 o 139.8 o 131.4 o 122.9 o 114.4 o 105.9 o 97.5 o 8 9 . 0 o 8 0 . 5 o 7 2 . 0 o 6 3 . 6 o 55.1 o 4 6 . 6 o 38.1 o 29.7 o 21.2 o 12.7 Parameter ! x/De J L 10 100 Re x 10 r2 F I G U R E 12l LOCAL F R I C T I O N FACTOR-REYNOLDS NUMBER PRODUCT FOR IT=0.0406 BASED OH EQUIVALENT DIAMETER. 30 1000 VIOO ^ 10 100 A - RUNS NIKURADSE EQUATION EQN (12) e © e e O C K 2 > ^ e e e o - f Re e - f •Re/^[d l/d t] J L 1000 10000 Re FIGURE 13: WELL-DEVELOPED FRICTION FACTOR:. - REYNOLDS NUMBER PRODUCT FOR O" =0.6331 BASED ON EQUIVALENT DIAMETER. 31 1000 CP CD } B - RUNS — NIKURADSE EQUATION cr e EON (12) . e e e e e — o-f-Re e-f-Re/<£[d./d2] 1 — i . . . i 1 i i 1000 10000 Re FIGURE 141 WELL-DEVELOPED FRICTION FACTOR - REYNOLDS NUMBER PRODUCT FOR CT =0.4631 BASED ON EQUIVALENT DIAMETER. 32 IOOOI C - RUNS - I O O | CD NIKURADSE EQUATION EQN (12) CD e e e - 10 o - f-Re e - f - R e / f r d / d J • ' L 1000 J I L Re 10000 F IGURE 15: TOLL-DEVELOPED FRICTION FACTOR - REYNOLDS NUMBER PRODUCT FOR CT = 0.3403 BASED ON EQUIVALENT DIAMETER. 33 1000 HOO CO XXL - 10 D -RUNS -e NIKURADSE EQUATION--^EQN(I2), w e e e e ^ d o-f-Re e - f Re/<tfd,/d2] i l l i i i i I i 1 1000 10000 Re FIGURE 16t WELL-DEVELOPED FRICTION FACTOR - REYNOLDS NUMBER PRODUCT FOR <T=0.0406 BASED ON EQUIVALENT DIAMETER. Table;k gives a tabulation of the span of Reynolds numbers f o r the tr a n s i t i o n . r e g i o n as a function of entrance length f o r the four diameter r a t i o s of annuli studied plus Gallovay's open tube. For the four sizes of annuli studied, the slope of the turbulent s t r a i g h t l i n e (T.S.L.) i s tabulated as a function of entrance length. The slope of the T.S.L. was ca l c u l a t e d by a l e a s t mean squares method. Table k T r a n s i t i o n Reynolds number range and turbulent slope as a function of entrance length. or = 0 .6311 cr = 0.1+631 X/De Re L Reu Re Span T.S.L. Slope ReL Re u Re Span T.S.L. Slope 33 Smooth T r a n s i t i o n 0.876 23 ; Smooth T r a n s i t i o n 0.81+2 55 2720 3500 780 O.91I+ 38 Smooth T r a n s i t i o n O.887 78 2780 3560 780 O.896 53 2750 lf300 1550 0.880 100 2780 3560 780 0.861+ 68 2750 3600 850 O.87I 122 2730 3350 620 O.859 83 2920 36OO 680 0.861 Ihk 2700 3350 650 0.852 98 2920 > 365O 730 O.872 166 2650 3350 700 O.855 lll+ 2900* 3600 700 O.859 188 2650 3350 700 O.83I+ 120 287O 3600 730 0.81+8 210 2650 3350 700 0.81+1+ 11+1+ 2800 3600 800 O.851 233 2650 3250 600 0.851 159 2800 > 36OO 800 O.855 255 2650 3250 600 O.859 17k 2800 36OO 800 O.859 277 2660 3250 590 O.813 189 2800 3270 1+70 O.813 299 2670 33do ' 630 0.833 201+ 285O 3270 1+20 0.81+0 321 2680 3350 670 0.852 220 2820 3270 1+50 O.85I+ 3^3 21+00 325Q 850 O.838 235 2800 3270 1+70 0.81+0 366 2350 3250 900 0.81+9 250 2800 3270 1+70 0.81+1+ 388 2200 3250 1050 0.81+1 265 2810 3270 1+60 O.833 i+io 2250 3350 1100 O.839 280 2550 3270 720 O.836 U.32 2200 3250 1050 O.8I+3 295 2250 3270 1020 O.833 k$k 2150 3250 1100 0.81+1+ 310 2200 3100 900 0.81+8 1+76 2150 3250 1100 0.8J+9 326 2300 3200 900 O.83I 1+99 2050 3250 1200 0.821 3hl 2200 3270 1070 0.81+1 Table k (cont'd) <T = O.3U03 a = 0.01+06 X/De Re T.S.L. V D e Re T.S.L. Re L Span Slope Re L Span Slope 18 Smooth T r a n s i t i o n O.837 13 Smooth T r a n s i t i o n 0.752 31 Smooth T r a n s i t i o n 0.852 21 Smooth T r a n s i t i o n 0.808 ^3 2000 U300 2300 O.837 30 Smooth Tr a n s i t i o n O.798 55 2100 ^300 2200 0.821 38 Smooth T r a n s i t i o n 0.776 68 2650 1+000 165P 0.828 hi 1900 3700 1800 O.76U 80 2800 385O 1050 O.832 55 2150 3700 1550 O.78O 92 2800 38OO 1000 O.83O 61+ 2350 3700 1350 0.77!+ 105 285O 3500 '65Q. 0.816 72 2350 3700 1350 0.758 117 281+0 3500 •'•660 0.822 80 2600 3700 1100 0.770 129 2800 3500 700 0.829 89 2650 3700 1050 O.78O li+2 2750 3500 750 0.81+6 97 2780 3700 920 O.789 15k 2820 3500 680 0.783 106 2800 3700 880 O.77I+ 166 2880 3500 620 0.821 111+ 2820 3700 880 0.77)+ 179 2900 3500 600 O.829 123 2820 3700 880 O.77I+ 191 29U0 3500 560 0.822 131 2820 3700 880 0-773 203 291+0 3500 56O 0.826 11+0 2820 3700 . 880 O.77I+ 216 2700 3500 800 O.82I+ 11+8 2250 3700 11+50 0.770 228 2200 3500 1300 0.828 157 2200 3700 1500 O.766 21+0 2120 3500 1380 0.825 165 2150 3700 1550 O.77I+ 253 2120 3500 1380 0.827 171+ 2120 3700 1580 O.769 265 2120 3500 1380 0.820 182 2150 3700 1550 O.76I 277 2130 3500 1370 0.866 191 2150 3700 1550 0.771 Table k jfrcoHtfd) <T = 0.0 (Ref.2) Re L Re u Re Span 16 Smooth T r a n s i t i o n 32 2200 1+1+00 2200 1+8 2300 3500 1200 61+ 2200 3100 900 80 2300 3000 700 96 2300 2900 600 112 2300 2900 600 128 2300 2900 600 11+1+ 2300 2900 600 160 2300 2600 300 176 2300 2550 250 36 DISCUSSION Generally the r e s u l t s appear to be consistent and r e l i a b l e . In the l i t e r a t u r e on concentric annuli, i t i s s u r p r i s i n g to note the s c a r c i t y of data on l o c a l f r i c t i o n f a c t o r measurements. U n t i l recently (k), most of the f r i c t i o n f a c t o r s had been found e i t h e r by o v e r a l l pressure drop measurements or v e l o c i t y p r o f i l e measurements, and hence d i r e c t comparisons between t h i s i n v e s t i g a t i o n and the l i t e r a t u r e are l i m i t e d . LAMINAR FLOW In well-developed laminar flow, the average values of f.Re f o r the four runs compare favourably with those predicted t h e o r e t i c a l l y by Knudsen and Katz ( 2 l ) . Using the equivalent diameter as four times the hydraulic radius, Knudsen and Katz express the f.Re product as f.Re = 16.0 0 ( d l / d 2 ) (12) where M a 1 / d , . 1 , (1 - V a ) 2 (13) P 2 i*<«Va2)2 • [ 1 - < dVa/] /m<ai/a2) On the average, the value of f.Re found experimentally was 0.80$> lower f o r runs A and B, and 2.6$ higher f o r runs C and D. Equation (12) i s equivalent to saying that f ^ e Re^ e = 16, where the subscript de s i g n i f i e s that Hanks' d e f i n i t i o n of equivalent diameter (equation (7)) i s used i n both the f r i c t i o n f a c t o r and. the Reynolds number. Table II - 2 compares experiment with theory. Several works are reported i n the l i t e r a t u r e i n which attempts have been made to p r e d i c t laminar flow v e l o c i t y p r o f i l e s i n the entrance region of open tubes. Only two works ""have appeared on the entrance regions i n annuli (3>M> but these are i n the turbulent flow regime. Galloway, i n analysing h i s round 37 tube data, found that the coefficient'C which i s frequently reported i n the form v/here x i s the distance from the entrance, was equal to 0.115 when the pressure gradient had decreased to i t s f u l l y developed value. Galloway's open tube data agreed very w e l l with Langhaar's (22) p r e d i c t i o n s . In order to obtain a good estimate of the entrance length corresponding to f u l l development of the f r i c t i o n f a c t o r from the data c o l l e c t e d here i t was speculated from equation (ik) that the l o c a l f r i c t i o n f a c t o r might equal the f u l l y developed value i n each of the laminar flow runs at the same value of x/DeRe. The data were p l o t t e d f o r the four diameter r a t i o s as f.Re/0( dl/d2) versus x/DeRe, with the r e s u l t s shown i n Figures 17 to 20. The value of 0(^l/d2) f o r each run i s given by equation (13)- A tabula t i o n of Langhaar's (22) t h e o r e t i c a l l i n e i s given by Eng (23) and i s shown on Figures 17 to 20 as a dotted l i n e followed by a f i n e s o l i d l i n e . The best empirical curve through the experimental points i s shown as a s o l i d black l i n e . Although the r e s u l t s below the x/DeRe value of 0.1 are quite scattered, i t appears that as the diameter r a t i o approaches zero, Langhaar's p r e d i c t i o n becomes better. This i s to be expected,since Langhaar's equation i s a t h e o r e t i c a l r e l a t i o n s h i p which has been derived f o r boundary layer development i n a c i r c u l a r pipe without any c e n t r a l s o l i d core. I t i s therefore not su r p r i s i n g that i t does not work f o r annuli, where a boundary layer develops and grows along both the inner and outer w a l l and hence achieves i t s ultimate growth more quickly. As the diameter r a t i o increases i n annuli the t o t a l i n t e r n a l surface area per u n i t , f r e e volume increases and the gap between surfaces decreases. This would mean that the v e l o c i t y p r o f i l e and hence the wall shear should become constant more quickly f o r the la r g e r diameter r a t i o s than f o r the smaller ones, and hence the entrance length required to achieve w e l l -developed laminar flow should be greater f o r the smaller diameter r a t i o s than 20i \ \ cr = 0.6331 \ 19 \ \ LAN6HAAR PREDICTION FOR OPEN TUBE J L i ' 1 L £ I L o q >.0I5 0.1 x / D e R e FIGURE 17: LAMINAR FLOW FRICTION FACTORS FOR <T =0.6331. 1.0 oo x / D e - R e FIGURE 18j LAMINAR FLOW FRICTION FACTORS FOR CT=0.4631. 20i 19 151 \ <r = 0.3403 \ \ \ LAN6HAAR PREDICTION FOR OPEN TUBE O Q O n Q n ^ n ^ J L J L J L o x y r 0.1 x/De Re 0.6 FIGURE 19: LAMINAR FLOW FRICTION FACTORS FOR <T =0.3403. o 211 \ 2 0 \ or = 0 . 0 4 0 6 LANGHAAR PREDICTION FOR OPEN TUBE o o I 5 L _ 0.01 J _ _ l I L OA x / D e • R e FIGURE 20: LAMINAR FLOW FRICTION FACTORS FOR O" =0.0406. 0.3 1+2 f o r the l a r g e r ones. From Table 5> i t appears that the values of C decrease with cr to a minimum value at an approximate diameter r a t i o of 0.15 and increase as the diameter r a t i o increases beyond t h i s value. This increase i s more pronounced f o r the 1$ c r i t e r i o n than f o r the 5$ c r i t e r i o n . These c r i t e r i a s i g n i f y that the pressure gradient has decreased to e i t h e r Vfo or 5$ of "the well-developed value. Table 5 Comparison of entrance length using the pressure gradient c r i t e r i a . . c 1$ C r i t e r i o n C 5$ C r i t e r i o n 0.000 0.075 0.01+5 0.01+06 O.O36 0.022 0.3403 0.035 0.017 0.1+631 O.O63 0.021 0.6331 O.O79 0.021 A possible explanation f o r the l a t t e r trend may be given i n the entrance geometry. For O" = 0.01+06 a s t a i n l e s s s t e e l wire was used as the core. For the other cores, l a r g e r tubes (see Table 2) were used and connected to a l / 8 - i n c h brass rod at the entrance (see Figure 1+A). As the core sizes increased, the perimeter of the inner sharp edge at the entrance of the t e s t section increased and hence a possible source of entrance disturbance. In laminar flow, t h i s increased disturbance would cause an increase of entrance length, assuming the disturbances would have to d i s s i p a t e p r i o r to the development of the laminar v e l o c i t y p r o f i l e . h3 TRANSITIONAL FLOW It can be discerned from a c a r e f u l examination of Figures 9 "to 12 that the three types of c h a r a c t e r i s t i c s which Hanks thought represented three types of flow do i n f a c t e x i s t i n the same system. Here, the Type I I I , I and II c h a r a c t e r i s t i c s occur at low, intermediate and high entrance lengths r e s p e c t i v e l y , along the annular t e s t section. Note, however, that Galloway's data show Types I I I , II and I as corresponding to low, intermediate and high values of X/De r e s p e c t i v e l y . The present r e s u l t s along with those of Galloway (2) confirm the f a c t that Hanks was mislead i n c a t e g o r i z i n g these three modes of behaviour as types of flow. Instead these curves are c h a r a c t e r i s t i c of the f versus Re of f.Re versus Re p l o t s at various values of X/De. The t h e o r e t i c a l curve proposed by Hanks f o r p r e d i c t i n g the c r i t i c a l Reynolds number f o r t r a n s i t i o n from laminar to turbulent flow i n the "well-established" flow region of concentric annuli i s shown i n Figure 21. A l l the values of c r i t i c a l Reynolds number p l o t t e d are c a l c u l a t e d using a modified equivalent diameter (equation (7))- The values of R e c R I T I C A L f o r diameter r a t i o s of 0.0k06 and O.3I1-O3 l i e within 1.8$ of the values predicted by Hanks. However, the values of R e c R i i ' i c A L °^^ i e r ^ w o diameter r a t i o s f a l l w e l l below the predicted values. Table 6 i s a t a b u l a t i o n g i v i n g a comparison of the l i t e r a t u r e values f o r R e c R I T I C A L a s a function of diameter r a t i o with those found by the present i n v e s t i g a t i o n . Here, the Reynolds number i s based on the conventional equivalent diameter, namely four times the hydraulic radius. The c r i t i c a l Reynolds numbers of Walker et a l (5) and Croop and Rothfus (15) f a l l below those of the present i n v e s t i g a t i o n . A possible explanation i s that the 2.5 o r 2.1 2 . 0 f 1.91 0.0 o HANKS PREDICTION 0.2 0 . 4 0.6 LEGEND SYM REF Q 2 A 5 • 18 7 • 19 o THIS WORK 0.8 1.0 FIGURE 21: A COMPARISON OF HANKS' PREDICTED VALUES OF CRITICAL REYNOLDS NUMBER WITH THOSE FOUND EXPERIMENTALLY, BASED ON d e = D e y 2\j/{€) . 45 Table 6 Comparison of c r i t i c a l Reynolds numbers (De=4r$f.) f o r various diameter r a t i o s . Ref Diameter Ratio C r i t i c a l Reynolds No. No. This work 0.6331 2720 This work 0.U631 2820 This work 0.3^03 2920 This work O.OU06 2820 2 0 .000 , 2300 1U 0.000 2100 5 0 .000 2020 5 0.0250 2U35 5 O.O667 2595 5 0.1251 2550 5 0.1653 2580 5 0.3312 2690 5 O.4987 2660 5 1.000 27ho 15 0.062 2650 15 O.197 2600 15 O .33I 2590 15 0.500 2600 7 1.000 2120 18 1.000 2U00 17 1.000 2700 turbulence l e v e l was d i f f e r e n t i n these various i n v e s t i g a t i o n s on annuli. I t i s shown f o r a f l a t p l a t e (21a) , f o r example, that as the percentage turbulence increases the l o c a l Reynolds number at which the laminar boundary layer becomes turbulent decreases. This could indicate that turbulence l e v e l within the t e s t sections used i n t h i s i n v e s t i g a t i o n was lower than those f o r Walker et a l and Croop and Rothfus. It appears that Hanks has based h i s proposed equation upon the turbulence l e v e l associated with a c r i t i c a l Reynolds number of 2100 f o r open tube. This could p o s s i b l y be the reason f o r the deviation from h i s predicted values. As the diameter r a t i o of the annulus increases the accidental turbulence l e v e l , caused by the increased perimeter of the sharp-edged entrance, increases and k6 hence the c r i t i c a l Reynolds number decreases. Reynolds (2h) found that under s p e c i f i e d conditions and f o r an open tube, t r a n s i t i o n from laminar to turbulent flow occurred at a d e f i n i t e value of the Reynolds number, i r r e s p e c t i v e of values of the speed, pipe diameter, or v i s c o s i t y separately. The value of c r i t i c a l Reynolds number was, however, found to depend on the accidental turbulence of the incoming flow. The greater the a c c i d e n t a l turbulence, the lower the c r i t i c a l Reynolds number, u n t i l a c e r t a i n l i m i t was reached beyond which fur t h e r increase of accidental turbulence had l i t t l e e f f e c t . The turbulence i n the incoming flow was produced by objects placed near the entrance of the pipe, by honeycombs across the pipe, or by the shape of the entrance i t s e l f . The turbulence l e v e l w i t h i n a flow system i s equal to the free turbulence plus the a c c i d e n t a l turbulence. The free turbulence i s caused by the flow of the turbulent core past the laminar sublayer. Hanks' proposed theory f o r p r e d i c t i n g the c r i t i c a l Reynolds number f o r t r a n s i t i o n from laminar to turbulent flow i n the "well-established" flow region of concentric annuli was found to p r e d i c t values which were higher than the experimental values, f o r diameter r a t i o s greater than 0.3- A po s s i b l e explanation f o r the discrepancy i s that the entrance structure of the annuli with diameter r a t i o s greater than 0.3 was such (see F i g . UA) that the sudden enlargement of the core at the entrance could cause accidental turbulence. Such accidental turbulence has not been accounted f o r i n Hanks' proposed theory. Rothfus, Monrad and Senecal ( l ^ ) suggested that the extent of the t r a n s i t i o n region was a function of the diameter r a t i o , being longer i n the annulus with the smaller diameter r a t i o . They found f o r diameter r a t i o s of 0.162 and O .65O that the t r a n s i t i o n Reynolds number spans were 1250 and 600 r e s p e c t i v e l y . The t a b u l a t i o n of t r a n s i t i o n Reynolds number ranges, given i n Table k, indicates that f o r the e x i t and entrance regions of the t e s t 47 section the t r a n s i t i o n spans are dependent upon the diameter r a t i o . However, f o r the well-developed regions of the t e s t section, these spans appear to be p r a c t i c a l l y independent of diameter r a t i o , and average about 700. The e f f e c t of entrance on f r i c t i o n f a c t o r s measured i n t h i s i n v e s t i g a t i o n i s i l l u s t r a t e d i n Figures 5 through 8. In the Reynolds number range of approximately 3000 to 5000 the l o c a l f r i c t i o n f a c t o r was found to decrease gradually f o r a short distance from the entrance, then suddenly r i s e to a higher value and remain at t h i s higher value f o r the remaining length of the tube. I t appears that the laminar boundary layer undergoes a t r a n s i t i o n to turbulence at the minimum point on the curve. This t r a n s i t i o n i s accompanied by an increase i n w a l l shear, and i n addition the boundary la y e r increases i n thickness much more r a p i d l y . These r e s u l t s are s i m i l a r to those found by Galloway (2) f o r h i s rod bundles and open tubes. For has open tube, he found that a dip i n the pressure gradient curve occurred over a Reynolds number range from 2900 to 4-000, and f o r h i s arrays of rod bundles over a Reynolds number range from 400 to 1500. Even though a square-edged entrance was used by Galloway and i n the present i n v e s t i g a t i o n , i t i s i n t e r e s t i n g to note that the dip i n the pressure development curve i n or near the t r a n s i t i o n region i s s i m i l a r to that found by Sparrow and Olson (4) using a rounded entrance and turbulent flow. However, i n the turbulent flow region the entrance e f f e c t found by t h i s i n v e s t i g a t i o n and f o r Galloway's open tube was s i m i l a r to that found i n turbulent flow by Sparrow and Olson using a sharp-edged (square-edged) entrance. There were f l u c t u a t i o n s i n flow, i n the t r a n s i t i o n region, which were manifested as f l u c t u a t i o n s i n the height of f l u i d i n the manometer connected to the pressure taps. I t seems p l a u s i b l e that t h i s was due to a s h i f t i n g back and f o r t h of the l o c a t i o n of the boundary layer t r a n s i t i o n near the conduit entrance. This has also been noticed by Sparrow and Olson f o r studies U8 on annuli with rounded entrance and by Dryden (25), who observed t r a n s i t i o n wandering i n external flow i n the boundary la y e r along a p l a t e . The s h i f t i n g of the t r a n s i t i o n point from a laminar to a turbulent boundary layer i s a poss i b l e reason f o r the i r r e g u l a r behaviour of the pressure development curves i n the t r a n s i t i o n flow regime. Ov e r a l l pressure drops f o r the f r i c t i o n f a c t o r measurements were found by averaging the pressure gradient f o r a si x - i n c h increment over what appeared, from Figures 5 "to 8, to be the "well-developed" region of flow. Figures 13 to 16 show that the r e s u l t i n g f.Re versus Re curves follow a Type I c h a r a c t e r i s t i c . I f , however, the average was taken over the section of the pressure gradient p l o t nearer the e x i t , the Type II c h a r a c t e r i s t i c appeared to represent the flow behaviour. On the other hand, when the average pressure drop was taken" nearer the entrance section, a Type I I I t r a n s i t i o n c h a r a c t e r i s t i c was indicated. Figures 9-12 i l l u s t r a t e both Type II and Type I I I c h a r a c t e r i s t i c s . When the diameter r a t i o s i n the annular t e s t section are changed i t may thus be po s s i b l e , providing the o v e r a l l t e s t section remains i n the same absolute p o s i t i o n with respect to the entrance, to change a Type I f.Re versus Re p l o t to e i t h e r a Type II or Type I I I c h a r a c t e r i s t i c . The workers (5> 8, 10, 12, 13, lh, 15) "who found that there were two di s t i n c t , c o n s t a n t values of fRe i n the apparently laminar flow region have probably taken t h e i r o v e r a l l measurements nearer the e x i t of the t e s t section. This i n v e s t i g a t i o n showed that i f the rggion of o v e r a l l pressure drop measurements was selected from the pressure development p l o t f o r an annulus so that a l l end e f f e c t s were eliminated, then only one constant value of f.Re occurred i n laminar flow f o r a l l the diameter r a t i o s studied, and there was a sharp t r a n s i t i o n to turbulent flow. Galloway's data f o r the open tube also indicated one constant f.Re value k9 and sharp t r a n s i t i o n . However, h i s data f o r l o n g i t u d i n a l flow i n square arrays of c y l i n d e r s i n d i c a t e d that d i f f e r e n t modes of t r a n s i t i o n from laminar to turbulent flow f o r d i f f e r e n t pitch-to-diameter r a t i o s p e r s i s t e d even i n the "well-developed" region of these arrays. TURBULENT FLOW The o v e r a l l f r i c t i o n f a c t o r r e s u l t s f o r the f u l l y developed turbulent region are consistent (Figures 13 to 16). For Reynolds numbers of 3 -^00 or greater (based on the equivalent diameter equal to four times the hydraulic radius) the experimental values of f.Re f o r runs' A, B, and C l i e , on the average, 1.6$, 0.8$ and 1-3$; r e s p e c t i v e l y , below the values predicted by the Nikuradse equation: 'while the experimental values of f.Re f o r D run l i e on the average, 3-2$ above the values predicted by equation 15• For a l l runs, the experimental values of f.Re based on Hanks' d e f i n i t i o n of equivalent diameter (equation 7) are, on the average, 32$ lower than those predicted by equation (l5)> when p l o t t e d against Re based on the conventional equivalent diameter, as i n Figures 13 to 16. Figures 5 to 8 show a dip> i n the pressure development p l o t at the center of the t e s t section. This dip becomes more pronounced as the Reynolds number increases. Gallowayls pressure development curves also showed t h i s regular deviation. I t i s believed, a f t e r performing various t e s t s on the apparatus (Appendix i ) , that t h i s dip was due to the equipment design and not to any c h a r a c t e r i s t i c mechanism of the flow i n the annular t e s t section. Various methods have been studied to c o r r e l a t e f r i c t i o n f a c t o r s i n turbulent flow. Rothfus and co-workers (5> 14) applied the mean-hydraulic-radius concept to the f l u i d i n the region l y i n g between the radius of maximum h.O l o g (Rev^ T) - 0.40 (15) 50 v e l o c i t y and the radius of the outer tube. Their empirical method seems to represent the experimental data very well. Meter and B i r d (26) used a Prandtl mixing length approach to give a f r i c t i o n f a c t o r versus Reynolds number expression f o r annuli. Their expression describes tube flow and sliib flow as s p e c i a l cases. The r e s u l t s of the l a t t e r treatment are more conservative i n that they p r e d i c t very s l i g h t l y higher f r i c t i o n f a c t o r s than does the Rothfus method. The present i n v e s t i g a t i o n shows that the Nikuradse equation (15); based on the equivalent diameter equal to four times the hydraulic radius, p r e d i c t s the o v e r a l l f r i c t i o n f a c t o r s i n turbulent flow f o r the well-developed region i n annuli with the same confidence as do the Rothfus and Meter methods. SUMMARY 1. F u l l y developed f r i c t i o n f a c t o r s have been measured i n four diameter r a t i o s of concentric annuli f o r laminar, t r a n s i t i o n and turbulent flow. The r e s u l t s are i n good agreement with the theory of Knudsen and Katz (21) f o r laminar flow and with Nikuradse's round tube equation (lk) i n turbulent flow. 2. The fRe versus Re p l o t s with entrance length as parameter indicate that Hanks (l)was mislead i n categorizing the three t r a n s i t i o n c h a r a c t e r i s t i c s as types of flow behaviour, and that they a r i s e f o r t u i t o u s l y out of the curves of f versus Re at low, intermediate and high values of X/De. 3. The theory proposed by Hanks f o r p r e d i c t i n g the c r i t i c a l Reynolds number f o r t r a n s i t i o n from laminar to turbulent flow i n the "well-established" flow region of concentric annuli was found to p r e d i c t values which were higher than experimental values f o r diameter r a t i o s greater than 0.3- A p o s s i b l e omitted parameter i n the theory i s thought to be turbulence l e v e l and i t s function of entrance geometry. k. For the e x i t and entrance regions of the concentric annuli, the Reynolds number span of the t r a n s i t i o n range appeared to be a function of diameter r a t i o . For the w e l l developed region i n annuli, the t r a n s i t i o n Reynolds number span was found to be f a i r l y constant at about 700. 5. The three modes of t r a n s i t i o n noted by Hanks are shown to be p o s s i b l e i n making o v e r a l l f r i c t i o n f a c t o r measurements on a given annular t e s t section. The influence of the d i f f e r e n t modes of t r a n s i t i o n i n d i f f e r e n t sections of a given concentric annulus are seen to be a possible explanation to s l i g h t deviations of other workers* r e s u l t s (5, 8, 10, 12, 13, lk, 15) from the values predicted t h e o r e t i c a l l y f o r laminar flow. This i n v e s t i g a t i o n showed that i f the region of o v e r a l l pressure drop measurements was selected from \ 52 the pressure development pl o t f o r an annulus so as to eliminate a l l end effects, then only one constant value of fRe occurred i n laminar flow for each of the diameter r a t i o s studied. 53 NOMENCLATURE G = c o e f f i c i e n t defined by equation (lU)> dimensionless de = Hanks' equivalent diameter defined by equation (7), inches. De = equivalent diameter, -^D-^ , inches D-j_ = outside diameter of insi d e core, inches D2 = inside diameter of outside pipe, inches f = f r i c t i o n f a c t o r defined by equation ( l ) , dimensionless g c = g r a v i t a t i o n a l constant, ( l b . j ^ ) ( f t . )/(lb.p)(sec ) 2 h~2_ = pressure at pressure tap number 1, l b . p / f t . bn - pressure at pressure tap number n, l b . p / f t . A h = pressure gradient i n t e s t section, (cm. of l i q u i d ) / ( s i x inches) k = c r i t i c a l value of s t a b i l i t y parameter = hOh, dimensionless K >» Hanks' l o c a l s t a b i l i t y parameter defined by equation ( 5 ) , dimensionless K = maximum value of K i n the cro s s - s e c t i o n of flow, dimensionless dp^ x = a x i a l pressure gradient, ( l b . F ) / ( f t . ^ ) ( f t . ) r = a x i a l coordinates r-j_ = radius of the inner c y l i n d e r i n the concentric annulus, inches. r 2 = radius of outer c y l i n d e r i n the concentric annulus, inches. rH = hydraulic radius = 8 / 2 f o r annuli Re = Reynolds number defined by equation ( 2 ) , dimensionless Re c = c r i t i c a l Reynolds number based on cLg, dimensionless Re^ = lower c r i t i c a l Reynolds number based on D e,dimensionless Re u = upper c r i t i c a l Reynolds number based on De, dimensionless t = time, seconds V = bulk f l u i d v e l o c i t y , f t . / s e c . W = mass flow rate, lb./sec. x = a x i a l distance from the entrance of a conduit to any point within the conduit, inches X = dimensionless coordinate f o r annular flow, ( r - r - ^ / ^ g - r ^ ) . GREEK SYMBOLS 8 = annular gap = * r i > inches € = 8 / , dimensionless kinematic v i s c o s i t y of f l u i d i n t e s t section, c.s. P = density of f l u i d i n te s t section, ^ ' M / f t .3 a = diameter r a t i o of the concentric annulus, D-J_/D2> dimensionless "a function of" function defined by equation ( 8 ) , dimensionless VECTOR AND TENSOR QUANTITIES F = external body force a c t i n g on f l u i d element P = pressure tensor 5 • v o r t i c i t y V = v e l o c i t y tensor T = stress deviation tensor 55 REFERENCES 1. Hanks, W. Richard., A.I.Ch.E. Journ., % 45 (1963). 2. • Galloway, L.R., Ph.D. Thesis, Chem. Eng. Dept., University of B r i t i s h Columbia, I963. 3. Rothfus, R.R., Monrad, C C , Sikchi, K.G., and Heideger, W. J. , Ihd. Eng. Chem., 4_7, 913 (1955)-4. Sparrow, E.M. and Olson, R.M., A.I.Ch.E. Journ., £, 766 (1963). 5-. Walker, J.E., Whan, G.A. and Rothfus, R.R.> A.I.Ch.E. Journ., 3_, 484 (1957). 6. Lonsdale, T., P h i l . Mag., (6) 46, 163 (1923). 7. Cornish, R.J.., P h i l . Mag., (7) 16, 897 (1933). £ 8. Becker, E.Z.,vVer. deut. Ing., 51, I I 3 3 (1907). 9- Atherton, D.H., Tirana A.S.M.E., 48, 145 (1926). 10. Kratz, A.P., et a l . , Univ. I l l i n o i s Eng. Expt. Sta., B u l l . 222 (1931)-11. Carpenter, F.G., et a l . , Trans. Am. Inst. Chem. Engrs., 4_2, 165 (1946). 12. Schneckenberg, E.Z., angew Math. Mech., 11, 27 ( l 9 3 l ) -13. Winkel, R.Z. , -angew Math. Mech., _3_, 251 (1923)-14. Rothfus, R.R., Monrad, C.C and Senecal, V.E.-', Ind. Eng. Chem. 42, 2511 (1950). 15. Croop, E.J. and Rothfus, R.R., A.I.Ch.E. Journal, 8, 26 (1962). 16. Lea, F.C., and Tadros, A.G.; P h i l . Mag., 11A, 1235 ( l 9 3 l ) -17. Ryan, N.W., and Johnson, M.M., A.I.Ch.E. Journal, 5, 433 (1959)-18. Redberger, P.J. and Charles, M.E., Canadian Journ. Chem. Eng., 40, 148 (1962). 19. Whan, G.A. and Rothfus, R.R., A.I.Ch.E. Journal, 3, 204 (1959)-20. Davies, S.J. and White, CM., Proc. Roy. Soc. (London), 119A, 92 (1928). 21. Knudsen, J.G. and Katz, L.K. , F l u i d Dynamics and Heat Transfer, McGraw-Hill Book Company Inc., New York, 1958, pp.95-97-21a. i b i d , p.270. 22. Langhaar, H.L., Trans. A.S.M.E., 64, A-55' (1942). 56 23- Eng, J.,, ;B.A.Sc. Thesis, Department of Chemical Engineering, U n i v e r s i t y of B r i t i s h Columbia, I96U. 2k. Reynolds, 0., P h i l . Trans., V]k_, 111:935 (1883). 25. Dryden, H.L., Nat'l. Advisory Comm. Aeronaut. Report 562 (1936). 26. Meter, D.M. and B i r d , R.R., A.f.Ch.E. Journal, J_, hi (1961). \ 1-1 APPENDIX I Various t e s t s on Galloway's ( 2 ) equipment werepperformed i n order to check a systematic c y c l i c deviation which occurred i n measuring A h as a function of distance from the entrance (See Figure 2 2 , ( i ) and ( i i ) ) . A. Procedure The following t e s t s were employed on the equipment: 1. Back pressure t e s t on solenoid values. The analysis was performed i n order to check the back pressure which would cause leakage past the seats of the solenoid valves. The back pressure was applied by applying compressed a i r pressure on water i n the pressure header. The pressure was increased slowly u n t i l a valve leaked, at which point the pressure readings on the low-pressure regulator gauge were taken. 2 . Testing f o r improper measuring procedures. This t e s t was done to analyse various measuring techniques, to see i f any d i f f e r e n c e s i n the methods was evident. The following t e s t s were a l l c a r r i e d out at a Reynolds number of 1+300, using 3 c s . . BEG s o l u t i o n . (a) S p l i t t i n g the pressure header and measuring the pressure drops over 6-inch i n t e r v a l s along the column. ( S p l i t t i n g the header e n t a i l s c l o s i n g two valves located i n the center of the pressure headers. ) (b) Measuring pressure drops at 6-inch i n t e r v a l s without s p l i t t i n g the pressure header. (c) Measuring pressure drops at \\ - foot i n t e r v a l s without s p l i t t i n g the pressure header. 1-2 (d) From runs i n 2(b), making the following p l o t s : ( i ) A h versus x ( i i ) A h versus x f o r the even-numbered pressure taps and then f o r the odd-numbered pressure taps. ( i i i ) P l o t t i n g (h;i_-hn) versus x f o r n=l, 2, 23. 3. Test f o r change i n c h a r a c t e r i s t i c s of the pressure taps. Before i n s t a l l i n g the brass tube as the outer tube f o r the annular section, the author i n s t a l l e d extra pressure taps on Galloway's (2) open tube. The author's pressure taps then became the odd-numbered ones and Galloway's the even-numbered taps- In order to check only the pressure taps, the header and solenoid valves were bypassed and the pressure drop measurements were made d i r e c t l y . This was done by connecting p o l y v i n y l tubing from pressure taps d i r e c t l y to the inverted a i r manometer B. Observations. 1. A l l the solenoid valves but two were found to withstand greater back pressures than Galloway (2) had found. The two f a u l t y taps were repaired. 2. The measuring techniques used were found to be s u f f i c i e n t l y good to give the r e p r o d u c i b i l i t y of r e s u l t s found by Galloway. Taking pressure measurements over l a r g e r increments merely reduced the percentage deviation of the pressure drops. P l o t t i n g Ah versus x f o r both the odd-and the even-numbered pressure taps showed that the author's pressure taps alone were s e l f - c o n s i s t e n t as were Galloway's. The l a t t e r observation i n d i c a t e d a difference i n the c h a r a c t e r i s t i c s of the odd-and even-numbered pressure taps. £ o I < r -6 S c a l e = 6 c m . o f f l u i d . .*o o o . o 0 o o o 0 o O O o o 0 o 0 o o ° 3 1 2 6 ° ° ( i i ) o o 0 o o 0 o o O o o 0 0 ° o 0 o 0 o 0 o 3 0 3 4 o ( i ) o o o o 0 0 o O ° o ° ° o % ° o 3 2 6 6 o o 0 P a r a m e t e r : R e 0 / 3 1 / 9 3 / 9 5 / 9 7 / 9 9 / 9 11/9 x - F T / I N . FIGURE 22: PRESSURE DEVELOPMENT IN AN ANNULUS USING: (i) SPACING PINS, ( i i ) FAULTY PRESSURE TAPS, ( i i i ) GOOD PRESSURE TAPS. 3- When the d i r e c t pressure drops were measured between even- and odd-numbered pressure taps, the systematic c y c l i c deviation was s t i l l evident. It was concluded, therefore, that the difference i n c h a r a c t e r i s t i c s between the even- and the odd-numbered pressure taps had caused the systematic c y c l i c d e v i a t i o n s . i n the pressure development data shown by Figure 2 2 ( i i ) . C. Resolution of Problem. The open tube used by Galloway and modified by the author was f i n a l l y removed, and cut apart f o r v i s u a l observation. It was found that the burr caused by d r i l l i n g the pressure tap holes through the outside tube w a l l had been removed d i f f e r e n t l y f o r the even-and the odd-numbered pressure taps. It i s suggested, therefore, that a standard method of d r i l l i n g pressure tap holes be adopted. The author achieved t h i s standard by f o r c i n g a c y l i n d r i c a l , s p l i t brass rod against the inside of the tube w a l l before d r i l l i n g . This operation removed burr formation completely. The r e s u l t of doing t h i s i s shown i n Figure 2 2 ( i i i ) . I I - l APPENDIX II - CALCULATIONS a) Local F r i c t i o n Factors and Reynolds number. The treatment of the i n i t i a l data was s i m i l a r to that used by Galloway (2) i n h i s Appendix V. I f the o r i f i c e s were used to c o n t r o l flow, the mass flow rates were c a l c u l a t e d using Galloway's (2) equations ( i l - l l ) , (11-12) and ( H - 1 3 ) . I f rotameters were used f o r flow c o n t r o l , mass flow rates were read from c a l i b r a t i o n curves. The c a l c u l a t i o n of l o c a l f r i c t i o n f a c t o r s and Reynolds number are made using equations 1 and 2 . These equations are put i n a form to accept the raw experimental data d i r e c t l y and s t i l l remain dimensionally consistent. Equations ( l ) and (2) are therefore written, respectively, as follows: f = 1 0 . 1 9 6 U 8 x l O " 3 x De Ah ^ o(Dg 2-D^) 2 ( I I - l ) W2 Re = 2.27U795xlo' 4 x De W ( I I - 2 ) / 5 l / ( D 2 2 - D l 2 ) where Ah = l o c a l pressure gradient (cm. of l i q u i d / s i x inches) p = density of l i q u i d - gms./c.c. I>2 = inside diameter of outside tube - inches D^ = outside diameter of i n s i d e tube - inches = kinematic v i s c o s i t y of l i q u i d i n the tube, centistokes De = equivalent diameter (D2-D-^) - inches W = mass flow rate of l i q u i d - ^'/sec. b) Check on Hanks' ( l ) p l o t . Figure 21 shows Hanks' p l o t of Re c as a function of diameter r a t i o . Using equations (8) and ( l l ) , values of \j/(€), Re c and O " were ca l c u l a t e d -on a-' Sto^Lden' fcandr c&cufalSbr i n order to check h i s p l o t . II-2 The following i s a sample c a l c u l a t i o n f o r O" = 0.2, r2 ~ 0 ' 7 5 0 0 and r x = 0.1500: €= 4.000 from the d e f i n i t i o n of €. K=K at x .= 0.741 from Hanks' equation (26). l//( « ) = 0.3465 from equation (8). Taking K=k=404, then from equation ( l l ) , Re = Re c = 246l This value of Re c agx-ees with Hanks' p l o t , within 0.2$. The following table gives the values of Re c as a function of diameter r a t i o . Table I I - l Hanks' c r i t i c a l Reynolds number as a function of diameter r a t i o . CT R eCRITICAL 0.0 2100 0.1 2454 * 0.2 2461 * 0.3 2448 0.4 2430 0.5 2404 * 0.6 2308 0.7" 2357 0.8 2333 * 0.9 2310 1.0 2288 * * These values were checked by hand c a l c u l a t i o n . (c) Deviations of f r i c t i o n factor-Reynolds number product from predicted values f o r well-developed laminar flow. Knudsen and Katz (2l) derived equations (12) and (13) p r e d i c t i n g the f.Re product i n w e l l developed laminar flow as a function of diameter r a t i o . The equations are f.Re = 16 . jZ>(dl/d2) (12) 0 ( d V d 2 ) - (1 - d l A 2 ) ( 1 3 ) l+( d Vd 2 ) ' d + [ l - ( d l / d 2 ) ^ ] / l n ( < i l / d 2 ) II - 3 As a sample c a l c u l a t i o n , consider the r e s u l t s of B-runs f o r the well-developed laminar region. For t h i s run, d 2 = O.738O inch d^ = 0.34177 inch From equation (l3)> 0 ( d V d 2 ) = 1.48565 From equation (12), f' ReTHE0RETICAL = 23-770 Averaging f.Re product i n the laminar region f'^EXPERIMENTAL = 23-664 $ Deviation = f - ^SXPT. ~ f • ReTHE0R. xlOO = f-Re n -0.45$ 'THEORETICAL Table I I - 2 gives a tabulation of the deviations f o r the four runs completed. Table I I - 2 Estimated deviations of f.Re product f o r Laminar flow Set of Runs f.Re (Experimental) f.Re (Theoretical) $ Deviation = f• R e ExPT." f- R eTHE0R. x 100 ^"ReTHE0R. A 23.620 23.920 - 1-25 B 23.664 23.770 - 0.45 C 23.923 23.534 + I . 6 5 D 21 .760 20.756 + 4 . 8 0 d) E c c e n t r i c i t i e s i n annuli. The equivalent radius was measured with u n i v e r s a l c a l i p e r s at four p o s i t i o n s £0 degrees apart, at the top and bottom of the annular t e s t section. These measurements are tabulated i n Table II-3-II - 4 Table II - 3 Measurement of annular gap at the top and bottom of the t e s t section. Pain P o s i t i o n Equivalent radius measured with u n i v e r s a l c a l i p e r s inches 1 2 3 4 A Top Bottom 0.131 0.135 O.lkO 0.13k 0.134 , 0.135 ffi.128 O.138 B Top Bottom 0.200 0.19k 0.197 0.202 0.194 0.205 0.206 0.192 C Top Bottom 0.247 0.21+5 0.244 0.250 0.235 0.245 O.250 0.244 D Top Bottom 0.355 • 0.355 0.351 0.354 0.351 0.352 0.354 '0.353 I f the four columns i n Table I I - 3 are denoted by Cl,. C2, C3, and C4 r e s p e c t i v e l y , i t can be seen that the maximum deviation between centers of the core and outer -tube i s given by DEVIATION C3-C1 * • C4-C2 2 2 2 II - 3 and hence the percent deviations f o r e i t h e r the top or the bottom of the annular section i s given by PERCENT DEVIATION = 100x^ Vfc3-ci |C4-C2 II - 4 INSIDE TUBE RADIUS Using equations I I - 4 , the equivalent r a d i i from Table I I - 3 , and averaging the percent deviations at the top and bottom of the annular section r e s u l t s i n the percent center-to-center deviations given i n Table 3« The average measured equivalent diameters given i n Table I I I were c a l c u l a t e d by summing the eight equivalent r a d i i measured f o r each diameter r a t i o of annulus (See Table II - 3 ) and d i v i d i n g t h i s r e s u l t by four. APPENDIX I I I The data f o r each set of runs i s given i n the following tables. The left-hand part of tables I l l - l a to I l l - I d gives f l u i d p roperties f o r each run. The right-hand part of the table gives the l o c a l pressure gradients f o r each run. Tables III-2a to III-2d give the data f o r the well-developed region of flow. 111-2 e I I i 1 ! i i i i i l i l i i l i ini i i i i l i mmm mmm mmm mmm -mmmmmm mmmmmm ~mmmmmmm ^mmmmmm immmmmmm mmmmm 4 . i l i i l i i i i i i i i i i i i i i H ! i ! ! ! ! ! ! ! ! ! ! ! I i i ! l i mmmmmm mmmmmm -A mmmimmm mmmmmm ^mmmm 1 l l l l l l l l l l l l l l l l l l l l l l l l l l « l i l l l l l l l f l l l l f l f l l l l l l l l f f l i i i i i i i i l Toble l l l - l e Doto for C-RUNS, o-«0.3403 w HxAec. P gm/x, failure Drop Acroi* Eoch 6 — Inch Interval , em. 6 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 o . n a , l . i lSO J.JT06 0 . 1 ' » 5 Q.+}•>•> a ).*7*5 9 . * « 2 0 0.6221 0.685S O.T6*9 0*9*01 0 . 6 O * o.ois 0.1111 1.2i*6 1.0691 1.2112 1.6 n o 3.6620 1.6620 1 1.6710 I J.6OS0 13.8000 11.6000 11.5800 i*.2oso I*.2*50 •O.00 MOO >T.oO 6 1.00 • 4.00 n.oo • 1 .00 sr.oo 91.00 121.00 i2».oa I 3 5.00 1.57 1.56 l.*5 1.*2 I.SO 2.i-i 2.'0 Toble III-Id Dolo for D-RUNS, er«0.0406 IkAee. .51»6 i .T* 10 «..5 ieo 4 . S M 0 *.".50O 4.1*160 4.5T9Q *.6 '00 S.t- '00 ».*: oo 1.4100 1.65?o 1.6S20 1.6100 1.6^00 1.6200 jm/cc. .0292 • 0281 .0281 11.00 39.00 t.5.00 51.00 i T . 00 61.00 69.00 11.00 SI .00 • 1.00 •11. 00 99.00 1/3.00 ]29.00 IJ).00 Pr«Mur» Drop Acrott Eoch 6—Inch Interval t cm. 2.81 2.60 2.6? 2.66 2.68 2 . TO 2.TO 2.66 2.66 2.65 2.61 2.TO 2.65 ; . 6 « 2.65 2 . TO 6 8 9 IS. TO 11.12 12.60 11.96 12.8* 17.80 11.10 12.90 11.12 W.86 11.00 13.00 11.00 12.82 11.0* 12.S3 11.10 12.82 12.96 12.TI 1 J.C? 10 II 11.16 IB.10 9.4.4 8. 02 a.oo T.«2 8.000 T.91 a. IO 12 13 14 15 16 17 18 l . ? l 1.11 1.11 19 20 M M M 1 III - 4 Tabid III-2a« Data for A - RUfcSS , Wel-Developed Region, o*80.6331 • - o 2 w V P Ah fi! lb./sec. cs. gm./cc. cm. 1 0.5987 0.9253 0.9974 10.009 2 0.4338 0.9253 0.9974 5.654 3 0.3404 0.9102 0.9973 3.655 4 0.9821 3.9080 1.0133 30.640 5 0.6784 3.8480 1.0132 15.465 6 0.4540 3.8460 i . o n ? 7.307 7 0.8456 3.8968 i . o n 2 23. 138 8 0.7585 3.9810 1.0135 19.171 9 0.7072 3.9430 1.0134 16.824 10 0.6235 3.9440 1 .0134 12.577 1 1 0.5246 4.0120 1.0136 8.759 12 0.3749 4.0200 1.0136 6.334 .13 0.4816 4.0270 1.0136 8.212 14 0.3120 3.9970 1.0136 5. 197 15 0.2125 3.9930 1.0135 3.581 16 0.5163 15.1100 1.0292 32.330 17 0.3590 15.0350 1.0291 22.543 18 0.2456 14.5100 1.0288 15.043 19 0.1775 14.6250 1.0287 30.353 , 2 0 0.5430 0.9515 0.9977 8.420 Table lll-2b' Data for B-RUNS, Well-Developed Region, <r=0.4631 • o z z V P Ah RU IbVsec. cs. gmycc. cm. i 0.8436 0.9078 0.9972 7.454 2 0.6074 0.8985 0.9970 4.013 3 0.4342 0.9033 0.9972 2. 199 4 0.3593 0.8988 0.9970 1.565 . 5 0.9747 3.5950 1.0117 11.692 6 0.6787 . 3.5875 1.0117 6.030 7 0.4650 3.5630 1.0116 2.558 --. 7 0.4650 3.5630 1.0116 2.672 8 0.7632 3.5505 1.0116 7.400 9 0.8344 3.5630 1.0116 8 .846 10 0.9047 3.5630 1.0116 10.214 11 0.6191 3.5580 1.0116 5. 153 12 0.5619 3.5490 1.0116 4.271 13 0.5183 . 3.5580 1.0116 3.474 14 1 .2338 3.5560 1.0116 17.849 15 0.3171' i 3.6710 1.0120 1.745 . 16 0.3761 3.6450 1.0119 2.052 17 0.7829 13.8950 1.0282 15.977 18 0.5298 13.7780 1.0281 10.815 19 0.3568 13.7730 .1.0281 7.353 20 0.2512 14.5320 1.0288 5.358 21 0.4204 3.8110 1.0120 2.434 22 0.3895 3.8030 i 1.0120 2.237 23 0.4541 3.8142 1.0120 2.593 24 0.4913 1 3.7915, 1.0124 2.809 25 0.4205 3.6390 1.0122 2.333 26 0.4449 , 3.6025 1*0121 2.428. Table III — 2c * Data for C -RUNS, Well - Developed Region, o*=0.3403 • • o z z w V P Ah RU lb./sec. cs. gra/cc. cm. i 1 .0466 0.9253 0.9974 6 . 9 7 7. • ' 2 0.7375 0.9300 0.9975 3.5 38 3 0.5168: 0.9155 0.9974 l .e58 4 1 . 1698 3.7150 1.0125 ] 1 . 0 3 9 : 5 0.8180 • 3.698 0 1.0124 5 .442 6 0.5 706 . 3.6840 1.0124 2.801 7 0.3735 3.6785 1 .' 0 1 2 3 1.242 8 0.4255 - 3.6710 1.0123' 1.363 •9 0.4474 \ 3.6675 1.012^ 1 .446 10 0.4745 3.6620 1.0123 1 1 .590 •• 11 0.5042 3.6620 1.0123 1.910 1 2 0.5420 3.6675 1.0123 2.443 , 13 0.6221 3.680q 1.0123 3.231 u 0.6858 3.6620 1.0123 3.950 ' 15 0 . 7 6 4 9 . ' 3.6505 1.0122 4.727 16 0.9601 1 3 . 6 7 1 0 ] .0282 ] 1 .572 17 0 . 6 4 3 4 1 3.6050 1.0281 7.782 18 0 . 4 3 1 5 13.8000 1.0281 5.346 19 0 . 3 1 1 3 13.8000 ,1 .028 1 3.737 2 0 1 . 2886 13.5800 1.0281 ' 1,4.'l 18 21 1 . 0 6 9 7 14.2080 1.0285 13.[2 2 7 22 1 . 2182 14.2450 1.0285 '.4.962 . Table III-2d* Data for D-RUNS, Well-Developed Region, o-*0.0406 • o i z z w V P A h oc Ib./sec. c s . gra/cc. cm. 1 0.7565 - 0.8927 0.9970 1 .902 2 . 1.2568, 4.5380 1.0145 6.498 ,3 1.0596 4.5410 1.0145 4.815 4 0.7570 4.5500 1.0145 2.673 5 0.5778 4.5560 1.0145 1.676 6 0.6689 4.5790 l .o ' l4 5 2.132 7 0.7296 5.6700 1.0172 2.607 . 8 0.6171 5.6700 1.0172 1.752 . 9 1.6565 13.4900 1.0282 12.916 10 1 .4838 13.4600 1.0281 9.846 1 1 1.4116 13.4400 • 1.0281 7.949 .. 12 1 .3489 13.4400 1.0281 6.773 1.3 1.2472 13.4100 . 1.0281 5.729 14 1.1798 13.4100 1.0281 5.372 15 1 .085 1 13.6550 1.0283 4.989 16 1.0039 13.6520 1.0283 4.563 17 0.8717 13.6100 1.0283 • 3.962 18 0.7456 • 13.6300 1.0283 3.326 19 0.5966 13.6200 1.0283 2.743 20 0.5008 13.6100 1.0283 2.317 

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