ML20079P200
| ML20079P200 | |
| Person / Time | |
|---|---|
| Site: | Pilgrim |
| Issue date: | 10/31/1982 |
| From: | Delaurentis L, Ranganath S, Skogg E GENERAL ELECTRIC CO. |
| To: | |
| Shared Package | |
| ML20079P172 | List: |
| References | |
| RTR-NUREG-0612, RTR-NUREG-612, TASK-A-36, TASK-OR NUDOCS 8303040555 | |
| Download: ML20079P200 (37) | |
Text
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ATTACHMENT STRUCTURAL ANALYSIS OF PILGRIM / BOSTON EDISON COMPANY VESSEL BEAD DROP, SHROUD HEAD ASSEMBLY DROP, AND STEAM DRYER ASSEMBLY DROP CONDITIONS OCTOBER, 1982 PREPARED BY: b Awd ke I 0/n/ C 2.
E.E. SK0GG DATE REVIEWED BY: [ ~
as /d 82, L.E. DELAURENTIS DATE APPROVED BY: ^~-3 /0h)U S. BANGANATH DATE W l ll U v J. DATES DATE NUCLEAR SERVICES ENGINEERING OPERATION GENERAL ELECTRIC COMPANY, SAN JOSE, CALIFORNIA 8303040555 830228 PDR ADOCK 05000293 P ppg
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e IMPORTANT NOTICE REGARDING CONTENTS OF THIS REPORT i
PLEASE READ CAREFUILY The only undertakings of General Electric Company with respect to information in this document are contained in the contract between Boston Edison Company and General Electric Company (ret'erence GE Proposal No. 424-TY-683-HK1) and nothing contained in tais document shall be construed as changing the contract. The use of tais intornation by anyone other than Boston Edison Company, or for any other purposes other taan tnat for which it is intended, is not authorized: and with respect to any unauthorized use, General Electric Cor.ipany makes no presentation or warranty, and assumes no liability as to the completeness, accuracy, or userniness of tne information contained in tais document.
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ABSTRACT A detailed finite element elastic plastic analysis was performed to evaluate the effects of the vessel head drop, shroud head drop and steam dryer assembly drop onto tae reactor pressure vessel for the Pilgrim Nuclear Power Station. The results of tae analysis show that structural integrity of the vessel and shroud is maintained, altnough some yielding would occur in the vicinity of impact.
Based on tasse resnits, loss of coolant would not be expected.
Furthermore, damsge to the fuel von 1d be precluded due to component geometry.
In view of tne f act tant structural integrity of the vessel and shroud would be maintained and tae fuel would not be damaged, neither critically nor release of radioactivity l's predicted as a result of the postulated equipment drops.
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TABLE OF C0fm NTS P_4G.E.,
1.0 IN110 DUCTION I 2.0 ASSUMPTIONS AND VERIFIED DATA INPtTT 2 3.0 RESULTs AnD CuNCLUSIONS 3 4.0 VESSEL HEAD DROP ANALYSIS 4 4.1 MASS AND VELuCITY OF VESSEL HEAD 4 4.2 VESSEL HEAD DROP RESULTS 7 5.0 SHROUD HEAD AsSEhBLY DROP ANALYSIS 10 5.1 MASS OF THE SHROUD HEAD 11 5.2 VELucITY OF THE SHROUD HEAD 16 5.3 SHROUD HEAD DROP RESULIS 17 6.0 STEAM DRYER ASSEMBLY DROP 19 6.1 VELucITY OF THE STEAM DRYER ASSEMBLY 19 6.2 SIEAN DRYER DROP RESULis 21
7.0 REFERENCES
22 APPEND 11 A VESSEL SUPPORT SKIET STABILITY ANALYSIS APPEND 11 B SHROUD HEAD VELOCITY UNDER SUBMERGED CONDITIONS
-111-
1.0 INIBODUCTION The structural conseguences of dropping the Pilgrim Noelear Power Station roactor pressure vessel head, steam dryer, and steam separator (shroud head assembly) during maintenance operations are investigated.
The vessel head is assumed to be dropped from a height of 65.1 feet above tae reactor vessel pressure (RPV) flange. During the drop, the head is assumed to rotate 90', producing a local point impact between the vessel head and the RPV as illustrated in Figure 1 and Figure 2.
The saroud head and dryer are assumed to drop from an olevation of 98.0 feet, 66.75 feet above the shroud flange. During the latter drop conditions, the water level in the vessel is assumed to be at the centerline of the main steam line (ELEV = 50.38 feet). The 229.5 inches of water that the shroud head and dryer assembly must traverse is of sufficient depth for fluid drag forces to establish a steady state terminal velocity of the bodies. Due to geometrical considerations which prevent rotation of the assemblies inside the vessel, an axisymmetric Lapact of the shroud head and dryer assembly on the shroud flange is assumed.
Using the finite-element code ANSYS (Reference 1), an elastic plastic dynamic analysis is performed for both the vessel head drop and shroud head drop. Stresses and displacements following the impact are plotted.
Since the shroud head weighs more than the steam dryer assembly and postulated drop conditions are identical for the two bodies, analysis of the steam dryer asembly drop is not specifically performed. The conclusions from the shroud head drop analysis can be conservatively applied to the steam dryer assembly drop.
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2.0 ASSUMPTIONS AND VERIFIED DATA INPUT General Assuantions to the Analysis
- 1) The strongback and crane lifting block weights are added to assembly verghts to comply with Appendix A of NUREG 0612. (Reference 4)
- 2) The crane lif ting block weighs 4,000 lbs.
Vessel Head Dron
- 1) The mariana lif t height is 3.5 feet above the refueling floor.
(Reineling floor elev. = 117.0 feet)
- 2) The vessel head weight is 80.25 tons.
- 3) The vessel head strongback and crane block weigh 17 tons.
- 4) The vessel head rotates 90 while falling through air resulting in a point impact on the RPV flange.
Shroud Head and Steam Drver Assemb1v
- 1) The sarond head weighs 47.5 tons.
- 2) The steam dryer assembly weighs 27 tons.
- 3) The sarond head strongback weighs 4000 lbs.
- 4) The maximum lif t elevation is 98.0 feet, 1 foot above the wall separating the reactor vessel region and the shroud head / steam dryer refueling storage area.
- 5) An axisymmetric impact with the shroud flange occurs.
- 6) Flow through the moisture separator stand pipes is negligible.
- 7) The water level in the vessel is at the centerline of the main steam line at the time of the postnisted drop.
- 8) The sarond ring thickness is reduced by 1/2 to account for jet pump hole cut-outs.
, i 3.0 4ESULTS AND CONCLUSIONS A detailed finite element elastic plastic analysis was perforned to evaluate the effects of the vessel head drop, shroud head drop and steam dryer assembly drop for the Pilgria Nuclear Power Station.
Structural integrity of the vessel and shroud is maintained even though some local yselding occurs following impact. Specifically, yielding occurs in tne vessel flange to vessel junction area following a vessel
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head drop. The maximum deflection of the RPV flange is 3.2 inches. The maximum deflection of the shroud is 0.86 inches with no localized yielding occuring in the region of impact.
Damage to the fuel is not predicted as the component geometry precludes any direct impact on the fuel rods. In view of the fact that structural integrity of the vessel and shroud is maintained and the fuel would not be damaged, neither rolesse of radioactivity nor criticality is expected as a result of the drop.
Loss of reactor coolant is also not expected since yielding is localized and no gross deflections occur.
l 4.0 YESSEL HEAD DROP ANALYSIS A fantte element model is used to determine the vessel response when impacted by the vessel head. The ANSYS (Reference 1) computer code is used for tae analysis. Due to symmetry, c..., a 180 segment of the vessel body between the vessel-head flange and the bottom of the support skirt is modeled. The plane stress isoparametric quadrilateral element from tae ANSYS element library is used to model the reactor vessel body. The model reflects the longitudinal and lateral, extensional and inextensional effects. Transverse effects are considered negligible and therefore the use of the plane stress option is justified. The model contains 50 nodes and 57 elements. To simulate the impact of the vessel head with the vessel, the ANSYS gap element is used. Figure 3 shows the modeling of the vessel and vessel head used in the analysis. The bottom row of elements represents the reactor vessel skirt. Nodes 7 and 8 dellne tne gap element.
4.1 Mass and Velocity of Fallina Head The RPV flange elevation is 664.56 inches (Reference 3). The weight of the Pilgrim vessel head is 80.25 tons. To comply with Appendix A of NURMI 0612, the strongback weight is also sonsidered.
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97.25 tog
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65.1 f" .
g 115.56in c = c .
n Df a =231 in V J V 2 Figure 1 - Drop Height Figure 2 - Impact Configuration
. VESSEL HEAD N0DE 8
" m = 1/2 head weight
- - Includes strong-back weight to comply with NUREG 0612 - Appendix A ANSYS element 40 1 (gapelement) 3 l k = infinite stiffness
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" /q IMPACT POINT t = 6.5" g t = 5.53'5
== g N - @
@ g g @= elements t
- 5.53 "2 h .
j n20
@ SA533 Gr B MATERIALPROPgRTIES:
h V / E = 26.4 x 10 PSI Q Oy = 70 KSI G = 10.2 x 106p37 h O /
h h h t = 3.25"
- =
o 12
, / p\
t = 2.06" t" h g g NODE 2
\g @
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FIGURE 3 - PILGRIM ANSYS VESSEL MODEL Two Dimensional Isoparametric Element - plane stress
The estimated weight for the Pilgrim vessel strongback and crane block ts 17 tons. The impacting mass due to a vessel head drop is determined to be, M = (80.25 + 17) x 2000 = 467.13 lbf - sec /in. (4-1) 386.4 Since only half of the vessel is being modeled due to symmetry, a value of N/2 is used in the analysis.
The maximum elevation to which the vessel head is lif ted is 120.5 ft. (3.5 feet above refueling floor to clear handrails). The drop height, assuming the head rotates 90 so that a point impact occurs, is therefore, h = (120.5 - 664.56/12) - 115.5/12 = 55.5 ft. (4-2)
The impact configuration is illustrated in Figure 2. At impact, the vessel head velocity is given by, V, = g[ 23h= /2 (386.4)(666) (4-3)
V, = 717.42 in/sec. (4-4)
In sinnlating the head drop, node 8 of the finite element model is given a velocity of 717.4 in/sec just before closure of the gap.
At the same time, mode 8 is subjected to a downward force equivalent to the venght of the head.
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The material behavior is taken to be perfectly plastic with a yield strength of 70 Esi (Reference 2) .
4.2 Vessel Head Dron Resnits Figures 4 and 5 show the axial stress at elements 1 and 31 and deflections at nodes 2 and 7 respectively. The impact wave propagation is clearly evident from the figures, taking approximately 0.0035 seconds to reach the top of the support skirt. The maximum deflection at the point of impact and top of the support skirt is 3.18 and 0.127 inches respectively. Yielding ocents rapidly at element 31 which is the location of the impact, however, no yielding occurs in the skirt. Buckling of the skirt does not occur as shown by calculations presented in Appendix A.
Similar analyses using more refined modeling indicate that the extensive inelastic yielding in the immediate element at the point of impact is extremely localized.
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5.0 SIROUD READ ASSEMBLY DROP ANALYSIS The dynamic response of the shroud and shroud support ring to an accidental drop of the shroud head assembly is found by a finite element analysis. The dropped shroud head is assumed to impact amitymmetrically on the apper flange of the top guide shrosd. As a resnit, the loads are transferred through the top guide shroud to the lower shrond, and subsequently to the vessel shell through the shroud attachment ring. The moments developed due to eccentricities in the load path are assumed to be carried by the top guide assembly and core support plate, where the eccentricitses exist.
Figure 6 shows the model used in the finite element analysis. The model is axisymmetric using isoparametric quadrilateral elements. Rotation of the shroud head inside the vessel is precinded by geometrical considerations. The core plate and top guide are simulated in the analysis by modifying the density of the representative elements. The structure is fixed at the vessel attachments. Since the core plate is relatively staff, the element representing it is not allowed to move radially. The sarond support ring is 1 inch instead of the actual 2 inches taick to compensate for the stiffness reduction due to the jet pump cutouts. The reinforcement supplied by the twenty two gussets is conservatively neglected.
To sioniate the drop, two gap elements are used as shown in Figure 6. The mass of the falling head is divided evenly between the two point masses shown.
e The material behavior is taken to be perfectly plastic with a yield strength or 30 Esi (Reference 2).
5.1 Mass of shroud head The weight of the falling shroud head is, l
l M = 95000 lbs. (5-1) l To satisfy the guidelines of Appendix A of NURBG-0612), 8,000 lbs is added to account for the shroud head strongback and crane lif t-ing block. The mass of the falling shroud head is taerefore, m = 103000 = 266.6 lb-sec /In (5-2) 386.4 Half of this mass is applied at each gap element.
-11
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- The water l evel in the vessel is assaned to be at the centerline of the main steam line (MSL) for refueling operations. The elevation difference between the RPV flange and the centerline of the MSL is 60.1 inches. The sarond head therefore drops 571.5 inches in air before contactsag water. The velocity of the shroud head upon striking the water is, V, = /2 g h = /2 (386.4) (571.5) = 665 in/sec (5-3)
The elevation difference between the centerline of the MSL and the shroud flange is 229.5 inches (see Figure 7). With the initial velocity V,, the shroud head drops 229.5 inches through water before inpacting the shroud flange. The fornala j derived in Appendix B to determine the drop velocity through l
l water assaning a zero initial velocity can be nodified to l
account for tae initial velocity V,. The result is, V =s/a - (a-V,2), - pA,Cd 8 " (5-4) where, a = 2nm (1- R/T ) . (5-6)
- A,Cd
,y , - - - -
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I 5.2 Velocity of Shroud Head at Ianact The following data is used to determine the shroud head velocity:
Vessel I.D. = 226 inches Shroud Head 0.D. = 195.5 Shroud Head Weight = 103,000 lbs At the time of the postulated drop, the shroud head and moisture separators are located 1 ft above the wall separating the reactor vessel region and the shroud head / steam dryer refueling storage area (See figure 7).
h
), ELEV = 1176" - Maximim lift height f the shroud head anc steam Refueling Storage dryer assembly Area e-y ELEV = 664.6" - RPV Flange MainSteamLine$. -
ELEV = 604.5" i
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FIGURE 7 Shroud Head and Steam Dryer ,
ELEV = 375.0" l Drop Elevation Sumary Shroud Flange b
o Refer to Appendix B for definitions of the above terms.
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Notice tant if the term e #^vd is approximately zero, i.e., if tae drop height through water, "S " is sufficiently large, then V = 0 . This implies the shroud head has 1
reached a steady state terminal velocity which is independent of l
the initial velocity. H e dimensionless coefficient C
- d Equations (5-4) and (5-5) is given by, Cp" Ar-- -2 (5-6)
-dpA .
He snroud head proj ected area "A" is defined as A= w (195.5)2 = 30020 in2 (5-7) 4 In Equation 5-7, the stand pipe area is included in the shroud head proj ected area. Justification of this assumption of limited free flow through tne stand pipe is that fluid would have to travel a minimum of 150 inches of tortuous path before it emerges on the other side. he path includes stationary vanes, which impart a spiral velocity to the flow, and obstructions that provide steam separating functions. Rus , it is reasonable to include the stand pipe area in the shroud head proj ected area calculation.
He vessel projected area is, A
= n (226) = 40115 in2 (5-8) 4 h e annulus area is, Ag= A -A = 40115 - 30020 = 10100 in2 (5-9) s _ _ _ _ _ _ _ _ _ _ _ _
The discharge coefficient of 0.74 is obtained from Figure B4 of Appendix B for a diameter ratio of 195.5/226.
D e drag coefficient Cp for the geometry considered becomes, C, = 40115 -1 = 19 (5-10) 0.74(10100) .
Evaluating e- PA,Cd , where, p = 0.036 lbs/in , S = 229.5 inches, M = 103000 lbs gy,,, , - 0.036 x 4015 x 19 x 230 , , - 61 0.
Equation (5-11) indicates tne shroud head assembly will reach a steady state terminal velocity by the time it impacts the shroud flange. He steady state terminal velocity (derived in l Appendix B) is 2em (1- 0/T) 1/2 y ,, , _
pA C y d where, p = density of water = 0.036 lb/in .
7 = density of steel = 0.280 lb/in .
Substituting into Equation (5-12) yields, 0.036 V* = 2(0.036 (103000) (386) (1 - 0.280)
(40115) (19) = 50.2 in/sec(5-13) .
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n SHROUD HEAD DROP is AXISYtt1ETRIC FINITE d
ELEMENT MODEL
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u 21 se 31 n
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= SHROUD SUPPORT RING
, a j es d in (
45 s s VESSEL
. o y 47 e 1411se l si l se l s31s41st Q ELEV = 112" "
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5.3 Shroud Road Dron Results Resnits of the analysis indicate no laelastic behavior in the shroud support ring and shroud except for some localized yielding in the shroud flange. Figure 8 shows the deflection at the point of impact and shroud support ring junction up to the time when peak deflections are reached. The maximum deflection at both locations is approximately 0.86 inches. The significance of this analysis is that the integrity of the shroud can be proven in a shroud head drop analysis without taking credit for the load carrying capability of the shroud ring gassets.
The conclusions from the shroud head drop analysis are that the associated deformations are not excessive and that the structural int'ogrity and stability of the shroud are maintained. Damage to the fuel by direct impact of the shroud head is precinded by component geometric considerations. Thus, no release of the coolant, release of radioactivity, or criticality is expected.
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6.0 STEAN DITER ASSEMBLY DROP ANALYSIS An accidental drop of the steam dryer assembly is assumed to produce an impact on the shroud flange. The dryer assembly would first impact the steam dryer support brackets. If it is conservatively assumed that these brackets do not impede the motion of the falling dryer assembly, the assembly wonid then impact the shroud head flange. The impact would be absorbed by the same structure that would absorb the shroud head assembly drop. If it can be shown that the mass and kinetic energy of the steam dryer assembly are less than those of the shroud head, then no additzonal analysis is required since the concInsions from the shroud head drop analysis can be applied to the steam dryer assembly drop.
The weight of the steam dryer assembly is 31* tons which is less than tne weight of the shroud head assembly.
To determine if the kinetic energy of the steam dryer is less than the shroud head upon impact, the velocities must be compared.
6.1 Velocity of the Steam Drver Asssemb1v At the time of the postulated drop, the drop height and vessel water level are identical to the shroud head drop conditions.
The dras coefficient expression for the steam dryer can be expressed as,
' Includes the strongback and crane lif ting block weight to comply with l Appendix A of NUREG-0612.
e
=
C D
A (6-1) cad d + 0.088 Ad-where A g is the area of the vanes. This area is approximately 66000 in .
The diameter of the steam dryer assembly is 220.25 inches.
'Ihe steam dryer proj ected area "A" is, A= n (220.25)2 = 38100 in2 (6-2) 4 Using the proj ected vessel area calculated in Equation (5-8),
the annnins or flow area A g is, A =A 2 f v - A = 40115 - 38100 = 2015 in . (6-3)
Snestitutang these vaines into Equation 6-1 results in, 2
C = 38100 D (6-4)
. 0.85 (2015) + 0.088 (66000) ,
or, C,= 25.:.
The drop height tarough water is 229.5 inches, therefore Equation (5-11) becomes,
,pACd y =e .03 0 40119 (25.M M2 9.0 (6-5) 62000
,-137.0 0 t _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - _ _ _ _ _ - - _
From Equation (6-5), it is apparent that the steam dryer assembly also reaches a steady state terminal velocity before i impact. From Equation (5-12), the terminsi velocity is, I
V* =
2ma(1- p/T ) , (6-6) pA, C, or, V*
= 2 (62000)(386) (0.87)
(0.036) (40115) (25.1) * -'
giving, Vt = 33.8 in/sec. (6-8) 6.2 Steam Drver Assembly Dron Results The steam dryer assembly impact velocity of 33.8 in/sec is less than tne terminal velocity of the shroud head (50.2 in/sec).
Likewise, the steam dryer mass is less than the mass of the shroud head. Therefore, since the mass and velocity of the steam dryer assembly are less than that of the shroud head assembly and recognizing that both units impact the shroud flange axisymmetrically, the conclusions from the shroud head assembly bound the consequences or a steam dryer assembly drop.
L. .
7.0 REFERENCES
l
- 1. ANSYS Engineering Analysis System, Swanson Systems, Inc., I March 1973
- 2. ASME Boiler and Pressure Vessel Code,Section III, Sunsection NA,1980 Edition.
- 3. Final Design Document for General Electric - NED, Analytical Report for Pilgrim Reactor Vessel, Combustion Engineering Inc.
VPF 1979-308-1.
- 4. NUREG-0612, Control of Heavy Loads at Nuclear Power Plants, July 1980.
I L ___ _ _ _ _ _ _ _ _ _ _ _ _ _ . - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . -
APPBaD11 A - Yessel Sannort Skirt Bucklina Angl.Yais A conservative clastic buckling analysis is performed to evaluate the stability of tae stirt under the impact loading following the head drop. The dynamic I stress in tae shirt is assumed to be uniform around the circumference of the skirt, and tae stress is applied statically. Since the inertia of the material l tends to help prevent buckling at tas beginning stage, it is conservative to ignore tne inertie in the estimate of buckling load. Some experimental data have been published showing that cylinders buckle at higher stresses under dynamic taan under static loads.
To determine tne buckling stress, it is necessary to first identify the governing mechanisms. TWo geometric paraneters are particularly helpful in determining the buckling load of circular cylinder under axial load: 1.e., R/t ana (1-v ) L /Rt, where R is the radius, t is the thickness, and L is the length (or height) of' the cylinder: p is the Poisson's ratio of the material. For the stirt, R/t = 99 = 48, (A-1) 2.06 and, Z = (1 -v ) L = (1-0.32 )(52.6) = 12.3 (A-2)
Rt (99)(2.06)
-Al-L l
, i, The R/t value indicates tant the skirt is a moderately thick cylinder whose backlings stress is not very sensitive to randomly-distributed geometric imperfections, if it buckles at a stress above the proportional limit. The valse (of tae sairt snows that it is an " intermediate-length" whose buckling stress and datornation are strongly affected by the length and end condition of tae cylinaer. An analytical prediction of the backling stress is difficult.
Gerard (Rererence A1) provided a conservative empirical formula for the predictson of tas buckling stress for cylinders with clamped ends:
2 2 a,,= r, r Et 12 (1 - F )L ^~
casre K, for moderate length cylinders under compression is, E = 0.702 ( (A-4) leaving, K, = 0.702(12.3) = 8.63. (A-5)
Snostitutang into Equation A-3 yields, a
cr
= (8.63) 7 (26.4 x 10 ) (2.06) (A-6) 12 (1-0.3 ) (52.6) cr, a = 3.16 x 10' ps1 (A-7)
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Egastion A-7 implies that yle1 ding will occur before the skirt buckles.
Yielding did not appear in the skirt due to a vessel head drop, therefore, the starts stability is not jeopardized.
REFERENCES A1. Gerard, G.,
and Becker, H. ' Handbook of Structural Stability, Part III -
Buckling of Curved Plates and Shell', NACA 'IN 3783, August 1957.
1
- A3 -
.. 's APPENDIX B SHROUD HEAD VELOCITT UNDER SUBMERGED CONDITIWS
)
f REACTOR " % #
Il PRESSURE VESSEL
'~"*"k*b v
U H
l F&re B-1. Schenstic of Shroud Heed Drop Define:
V = shroud head velocity a = shroud head mass A = projected area of the shroud head normal to V V
f
=
flow velocity relative to the shroud head (neglect friction and assume that velocity through the steam separators is negligible).
A f
=
total flow area = A y -
A Ay = vessel cross-section area p = mass density of water
= mass density of steel 6
S = displacement coordinate with S = 0 when V = 0 Cp =
drag coefficient in equation drag force = CD*p- . area 2
-B1-C -.
l
~.. *e CALCULATION OF THE DRAC COEFFICIENT CgAND THE DRAG FORCE ODetTROL VOLUME p2 g
RELATIVE TO g l STATIONARY ly RELATIVE TO VESSE L g ;
PLATE l l 5 I v n -v,-V V Vi f
u '
a a ,
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I PLATE I q ] l STATIONARY l (
s N I I s k '
s s N I ! ks C A ;
v l Il l l v i L*' J
. Figure B 2. flotative Velocities in the Shroud Head Drop The actual situation is shown in the figure above on the left hand side. A plate with projected area A (representing the shroud head) moves c'ownwards with velocity V. Shown on the right hand side above is the same system with the plate made stationary by adding a velocity component V, to the vessel.
Now, consider a control volume of the fluid as shown by the dotted line. At the bottom, the pressure is Py and the velocity V. For the time being, neglect gravitational considerations. The fluid passes through the annular area, Af , with velocity V .f At some distance, the velocity V is regained, but due to enlargement of area, there is some head loss. Therefore, the pressure is only P at the upper end of the control volume.
2 The head loss due to the enlargement of the cross-sectional area to A is y
=
(Vf - V)2 ( B-l) 28 l
The velocity V is at the vena contracta.
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VENA CONTRACTA figure 83. Vene Contracts for Fluid Flow The cross-sectional area at the vena co'ntracta is Cg Af, where Ce s the coefficient of contraction. Values of Cc in the literature range from 0.5 to 1.0. For ideal flow through a rectangular sharp edged orifice classical hydrodynamics yieldc a value of 0.61.
Figure B-4 shows Cgy C for orifices with diameter ratios Dy /D .
Since Cy is very close to 1.0, the value of C C d c*
y ,
[Ay \ y C=C y f
NCAg }
A
-2 h =V v -1 ( B-2)
L
^f .
- Hydraulics and Fluid Mechanics by E. H. Lewitt, Publisher, Sir Isaac Pitman, London,1958 edition, p.112.
-B3-L ..-. .
! . now l
l 2 '. ~
2
- 2 Py -P2 " Pk " D V23 *
- CA
~1 "
PL 2 CA
-1 ( B-3)
, f , f This is now in mass density units rather than weight units.
The free body diagram of the control volume shows the net reaction force on the fluid = (P y -P)*
2 A.
y This force is equal to the drag force on the shroud head.
e
-B4-L' -
FROM: FLUID MECHANICS WITH ENGINEERING APPLICATIONS,BY
.' ; DAUGHET.TY AND FRANZINEl, McGRAS HILL,1985, EDITION p 358 1D 3 I
- 1 ) *
/ /
02 b c
, r
. f /
- (< C, - c,9 g
o QS G T f Os %g NET HEAD LOSS (ORIFICE) i 100 h, }
B0 g
0.4 x'x xx y
s s
" NN $
5 40 b 0.2 r c
- NET HEAD LOSS (FLOW NOZZLEi X,\ - -
W 20 0.1
.]
0 $
0 #
0 0.2 0.4 0.6 02 1.0 DI AMETER RATIO, D0/D3 Figure 84. Coefficients for Sharp Eciped Orifice with Pressure Differential Measured Either at the Flanges -
or at the Vens Contracta. Also Head Loss Across Orifice and Flow Nozzle. All Curns are for Ng(D,V, p,/p i> 1W.
g l.
v 2/2e IL ,t
__ M ,
From: FLUID MECHANICS , ,
> 1. v2/2e With Engineering App-e sat lications, By Daugherty 4
and Franzinei,McGRAW l "1/7 a Pa '7 Hitt. 1965, Edition p 358. i e a
\ I li y
.H_ n Vldyh' f l
c, ik DO D 2 u .--'uunn.w l l
Figure 85. Thinnate Orifice in a Pipe t
I
-B5-
. . .2
Tha Dreg ferca = 0 . A, . V'
- h -1 ( B-5) 2 '. ,
7 f ,
. s p 2 If the drag force is defined *as C *
- V . A
- then:
'2 V C
D
= - 1 ( B-6)
, f ,
Determination of Velocity V:
The equation of motion is as follows:
F=mv (B-7)
Two forces on the decending body are:
Buoycncy force = ag (1 - p/6 ) (B-8) 29 Drag force =C D' 2
- b*E (B-9)
Therefore we have:
mg (1 p/6)-C3 'v2
( B-10) 2 g*p = mv Since v = dv and V = ds dt dt Then dt = ds and v =V dv V ds Combining the above equations and solving for ds.
2 sg (1 9/6 ) -
C *p* V
- A = mV
- dv
. D p .
V
. da .
O or ds = mV dv .
mg (1- p / 6 ) - C
- b D
- [2 ,
- B6 -
- C
. r'ds - (2-V/D D b) dv (3-11) 2ag *
(1 8/6 ) -V p C, Ay Define a = 2ag *
(1 8/6 )
P= C,* A 9
hen.
S V ds = 2a V dv (B-12) o /* PCDb o a-[
From standard integration formulas, we have V dy = -
1_ En (a-V ) (B-13) a-V S= -
a in (a - V )
pC A .
o D y ,
=- a
- An (1 - V ) ( B-14) pCD ^V Solving for V ,
~
C VD S/a ( B-15 )
2 V = a 1-e -
An inspection of above equation shows that if the f alling height, S through the liquid is large enough, then, the second term is essentially zero, therefore, Terminal steady state velocity Vt " Y"
- jag (1 0 /6 ) (B-16) pC A D V
-B7-
Conversely, we can also determine height, S, through which a body has to fall before it essentially attains steady state terminal velocity. For example,
'S' when shroud head attains 90%
let us determine the formula for drop height of steady state velocity.
0.9 2 , y,, -%C D g ,_y7) i or 9A y C D
S/m = 1.66 5 = 1.66 m (B-18) 90
- AVCD Also, given a drop height, we can find what percentage of steady state velocity it will reach at the end of this drop height.
9
- B8 -
_ _ . .