ML20010G157
| ML20010G157 | |
| Person / Time | |
|---|---|
| Site: | Wolf Creek, Callaway  |
| Issue date: | 09/11/1981 |
| From: | Petrick N STANDARDIZED NUCLEAR UNIT POWER PLANT SYSTEM |
| To: | Harold Denton Office of Nuclear Reactor Regulation |
| References | |
| SLNRC-81-98, NUDOCS 8109150391 | |
| Download: ML20010G157 (28) | |
Text
~
09 e- 0)
SNUPPS Standardaad Nuclear Unit Power Plant System '
g'-
1 gS[ gg i I
M, 'r -
scyy ed m ,o q ,,s. Nicholas A. Petrick 7p g ,,c,t;,,o;,,,go,
/ ptember 11, 1981 SLNRC 81- 98 RILE: 0541 SUBJ: Turbine Missile Analysis
/
dir. Harold R. Denton, Director Office of Nuclear Reactor Regulaaon U.S. Nuclear Regulatory Commission Washington, D. C. 20555 Docket Nos. STil 50-482, STN 50-483, and STN 50-486
Dear Hr. Denton:
In discussions witn Dr. Gordon Edison, NRC poject manager for the DiUPPS applications, it was determined that the itRC Staff required additional in-formation in order to complete the review of turbine missile analysis.
Enclosure A to this letter is a group of taAR changes that will be incer-porated in the next revision to the SNUPPS isAR and the Callaway and Wolf Creek Site Addenda. Enclosure G :s a detailed discussion of the calcu'ation of turbine missile strike probability.
Very 'truly yours, CMV(v Nicholas A. Petrick RS/jdk Enclosure A cc: J. K. Bryan UE G. L. Koester KGE D. T. McPhee KCPL D. F. Schneli UE W. A. Hanser NRC/ CAL T. E. Vandel NRC/WC I
8 0$
Gu l 1 ,
l 0109150391 810911 PDR ADOCK 05000482 l
A PDR
I _m O
A e
OCIoSuw k suutPS STAMMLb Ttur L___ _ _
SNUPPS The main steam turbine inlet valves are provided in series ar-rangements: a group of stop valves actuated by either of two overspeed-t' rip signals, followed by a group of control valves medulated by tne speed-governing system, and tripped by either overspeed-trip signal. These systems are described in Section 10.2.2.3.2.
The intermediate valves are arranged in series-pairs, with an intermediace stop valve and intercept valve in one casing. The closure of either one of the two valves will close off the cor-responding steam line. Thus, a single failure of any r,omponent will not Jead to destructive overspeed. A multiple failure at the instant of load loss would be requireo, involving combina-tions of undetected electronic faults and/or rechanically stuck valves and/or hydraulic fluid contamination. The probability of such joint occurrences is extremely low, due both to the inherently high reliability of the design of the components and frequent inservice testing. For further description and functioning of intercept valves, refer to Section 10.2.2.3.2.
The LP section wheels would fail by general ductile yielding at ,
about 180 percent of the normal operating speed. The attainment of thic runaway speed is unlikely since a progression of failures wo.Ild act to disrupt the steam path, limiting the ability to further accelerate the nachine and to stop further acceleration by performing work upon the entrained debris. Such a failure is solely dapcndent on the failure of the cor. trol systems. This failure rate has bcen cstimated from actual performance records (see Pages 10 and 19, Ref. 1). The lifetime probability of a missile occurring at runaway speed (127-180 percent) has been estimated to be 1.a x 10 ~7 3.5.1.3.3 Missile Data The hypothetical missile data for the 38-inch last-stage bucket, 1,800 rpm low-pressure turbine are gr /en in Table 1 of Reference
- 2. ,
, _ . _ _ . ( IU S NT k j
' 3.5.1.3.4 Probability of Damage The probability of significant damage (?4) to critical components in the plant due to turbine failure has been assessed by first determining the separate probabi]ities of turbine failure and missile ejection (P3), such a missile striking a critical component or entire structt.re of safety significance (P2 ), and significant damage occurring to the component (P3). Then the overall probability P 4 =Pi xP2 xP3 The probabilistic rates for P for turbine-generator failures, which are based upon detailed knowledge of the characteristics and properties of critical components and modeling the event as a sequence of simple events, are soundly based because they 3.5-6
a -. - - .
T NSEET A.
C 4o P35-G)
% Ta & C. s. g - s p m d & p f f 1 5~&&%M M.
l I
l
.. ,sa,- -ww.,w, e- - , -e +-,n -,----.,-----..-------~<-------c - . - - ,e--,-,-y-.---,-----,m . ------- r-. r- .. , ---- -_ --- -- ---- -
9 e
TaAA.e 3-C - S -
f g8 A Ywk&W /
M pM'
'Otu wE WMfL 626uAs (n) 7 TF E l m. +. m. &. m +.
%m TIO 393 7Sb 413 (io 400 m
b b?-o 484 76D 52.) 7SO 998 l cc I
Yc 93o 615~ 8Bo (,2 5~ 9 10 bzS-(1)
YJ (3) IO N (.1) 1b6'$ (3) 994 (l) *fle- s*1on & Amxceuu Nhob '
36ee 9 qf aQ cz) V E A s L -
KL M "ys a,A
^t"'";$ /"~ W *'
. - . a wn - l _ . . --
6
. ia%. 3.s-3 wores ccoc,r.3 riadA CD L ,m ssi w s/'
s JL m
a pWa~d) f gA %.
y, &%,# Mp e
, . , . - - - - .,.%-, ,.m- _,.,-._-----,-..,.w.. , ,..,,,,,m -.,,---,-w -g,.. y - ,,, . .,c. -- - ----
i S WPPS
, reflect pertinent material, stress, and environmental parameters
- ') and present techniques of analysis, Appendix A, of Reference 1.
V Pi is given in Table 1 of Reference 2 as 1.5 x 107 However, a constant value for P 3 = 1 x 104 , without regard to historical data, has been conservatively assumed for the probability analysis.
From the standpoint of reactor safety, it is necessary to consider Q ,
the P 2 probability that, given a turbine-failure missile, the missile will impact a seismic Category I structure, system, or i "4 To7ponenC3 iPIhe product of the probability of missile ejection g d
2 f(P ) ano e probability of the missile strike (P2 ) yield the 3
probability that a turbine missile will impact the component.
b Orientation of the seismic Category I structures with respect to the missile origin is shown in Section 3.5 of each Site Addendum; The probability of a turbine missile striking a crit 4calmomponent-- !
-er a structure housing a critical component (P2 ) was calculated I for both sites and is given in each Site Addendum. t rD +
- Missiles due to low energy burst (130-percent overspeed) are of cf no consequence to the total probability P 4 , since probability P3 o = 0 for low-energy missiles. Refer to each Site Addendum for low 2 trajectory missiles' strike zones.
Probability P 3 , the probability that, given a turbine failure, a missile has struck a seismic Category I structure and has per-O forated through, is assumed as unity for high-energy rnissiles.
Using the above data, the annual probability P. of a turbine missile damaging a critical ccmponent is Icss than 10 ?, as shown in each Site Addendum. This value is sufficiently low that no specific protective measures are required.
3.5.1.4 Missiles Generated by Natural Phenomena Tolnado-generated nissiles were considered as the limiting ,atural-phenomena hazard in the design of all structules which are re-quired for safe shutdown. The missiles considered in design are as listed in Table 3.5-1.
Vertical velocities of 70 percent of the indicated horizontal velocities are considered for all missiles, except the 1-inch-
- diameter steel rod which is critical for penetration and is assumed to have a vertical velocity equal to the horizontal velocity. These design basis missiles are in accordance with Standard Review Plan 3.5.1.4, Revision 1 (Draft).
O /
3.5-7
LLsePTA G To f 3 5-7 )
Fn 46 adp , R A adcas hA m L 4*$ ~$ & *p$Mr J6LJA ad f xAtt A M - p z A w .i
~ & - +4 9
0 m
F
TM SEV.Tb
( 70 P 3.5-7 )
The gravity parabola gives.two solutions for an angle of departure for a given initial velocity and rang;. One solution corresponds the high trajec-tory paraboja tLe other to the low trajectory parahala.
Horizontal targets, such as roofs are considered in connection with the high trajectory, while vertical targets such ac walls, in connection wi h the low trajecto.y. In both cases the impset is .:onsidered to be normal to the target areas, that is, no reduction of impact velocity is considered due to angle of impact. The probability factor P2 is not appreciably changed by the above consideration as known by parametric studies ~.
Jince the target area is fix1d in space and the possible planes of missile departure have been predetermined by the turbine manufacturer, the probability of a given taissile to be "on-target" becomes a " mathematical expec,.ancy".
A numerical process of integration is followed since a closed form solution is impractical due to the many variables involved.
Wheels have been divided into three groupa by the manufacturer and within each group e failure may occur in four different ways. There are three wheels on the first two groups and one wheel on the third for each half of the turbine.
A life-time probability of 1.5 x 10-7 has been established by the manufactures for any one group. As a result an equal probability of failure for any one group exists.
Twv mathematical relationships among four de ;criptive angles fcr the target exist. These mathematical expressions are combined with the gravatational parabola to determine the probability of all of them simultaneously concurring to produce a missile "on target".
The steps of the computational procedure are performed in the following sequence.
The minimum and maximum values of angle for the horizont'al circular sector that includes the target area, the minitum and maximum distances of the target area, the size of the target area, and the minimum and maximum values of departure velocity are determined. Next the limits of the departure angles for the limita-tions of distance to tar pt and velocities are found. After determination of the limits a double summation is performed over the ranges that jointly satisify the two relationships among the four descriptive angles. This summatNn is followed by an adjustment of the summation results to account for the condi-tional probability of distance due to the velocity of departure and adjustment for the probability of occurence of those angles, and finally adjustment of the probability that a missile will strike in the sector to the probability that it will strike "on target".
SutAt?S C A LLA W A y S'TE O
l
I SNUPPS-C 3.5 MISSILE PROTECTION
?.5.1 HISSILE SELECTION AND DESCRIPTION 3.5.1.5 Missiles Generated by Events Near the Site Refer to Site Addendum Section 2.2.3. '
3.5.1.6 Aircraft Hazards Sections 2.2.1.3.1 and 2.2.1.3.2 describe the locations of airports and air routes in the vicinity of the Callaway plant site. Aircraft movements at thece airports and on the air routes do not pose any undue risk to the safe operation of the Plant. The location of the ,
Plant with respect to airports and air routes meets the criteria set i forth in the Standard Format Guide (R.G. 1.70), section 3.5.1.6 as explained in the following:
- a. There are no airports within 10 miles of the plant site.
- b. The nearest commerical airport, Fulton Memorial, and two private airstrips are located 12 miles beyond the site. The annual number of operations at these airports is less than 1,000 d , where d is the distance from the airport to the ,
site in miles.
- c. There are four low-altitude airways passing 5 miles beyond, the plant and their annual movements are less than 1,000 d .
- d. No military routes pass within 20 miles of the plant t. i t e .
Since airaraft movements at airports and on air routes do not pose any undue risk to the safe operation of Callaway Plant, no design-basis aircraft impact is postulated.
(
3.5.2 STRUCTURES, SYSTEMS, AND COMPONENTS TO BE PROTECTED FROM EXTERNALLY GENERATED MISSILES The turbine and tornado missiles which, if generaced, could affect
- he safety of the plant are discussed in Standard Plant FSAR Eection
. 2.L.
The probability of significant damage (P to critical components in the plant due to turbine f ailure has been,) assessed by first determining thE separate probabilities of turbine failure and missile ejection (P , Refer to Standard Plant FSAR Section 3.5.1.2.4), such a missile strlking a critical component or entire structure of safety significance (P 2) , and significant damage occurring to the component (P . , Refer to Standard Plant FSAR Section 3.5.1.2.4). Tnen the overa11' annual probability P, = P x P, x P 3 3
The probability of a high or low trajectory, turbine missile striking a critical ccaponent -er a is found to be 3.89 x 10 ,structura housingPlant.
at the Callaway a criticalRefer component to Table (P2)
' .5-1
-- _ - ______ _ _ ._ _- m m- -
SNUPPS-C I
3.5-1. (ID32 6 )
The annual probability (P S) of a turbine missile damaging"a critical component at the Callaway Plant is found to be 3.89 x 10 . This value is less than 10- and is sufficiently low that no specific protective measures are required for turbine missiles, refer to Table 3.5-1.
Figures 3.5-1 and 3.5-2 idantify the safety-related structurea, including those outside the Power Block, within the turbine missile trajectory. Figures 3.5-3, 3.5-4 and 3.5-5 illustrate low trajectery missile strike zones.
Protective measures are provided to minimize the effect of rotential tornado-generated missiles. The ptstective structures, shi lds, and barriers are designed utilizing the procedures given in Standard Plant FSAR Section 3.5.3.
The portions of the Essential Service Water System (ESWS) located outside the Power Block, the ultimate heat sink (UHS), and their associated protectivt structures, shields and barriers are discussed below.
3.5.2.1 Essential Service Water System (ESWS) Pumphouse The two-unit ESWS pumphouse is a tornado-resistant, reinforced concrete structure on a common foundation having redundant operating floors at Elevation 2000'-0". The separation of trains of the ESWS and the separation of units is provided by three interior barrier walls. A tornado-resistant skimmer wall at the UHS retention pcod interface provides protection for the ESWS pumps, whose suction ends are located 25 feet below the normal surface of the pond. Torna'o-resistant sh; elds protect the inlets and outlets of the ventilation system at the roof elevation and protect the personnel doors at grade level. Tornado-recistant covers protect the roof openings.
Figures 3.8-1 through 3.8-3 show the tornado missile protection for the safety-related penetrations in the ESWS pumphouse.
l 3.5.2.2 ESWS Pipes, Electrical Duct Banks and Manholes All ESWS pipes are buried a minimum depth of 4.5 feet to resist the I . effects of tcrnado-generated missiles and frost penetration. All l
ESWS electrical duct banks are reinforced concrete structures which are buried a minimum depth of 3.5 feet to resist the effects of tornado-generated missiles.
The buried ESWS electrical manholes are tornado-resistant, reinforced concrete structures with missile-resistant manway covers and roofs.
l Figure 3.8-11 shows the tornado missile protection for the ESWS l electrical manholes.
l 3.5-2
i
- (vo CA U.A w A g g .r..r - 2. )
h , M MMV 1 xL ya LkyL9%
aM.
,A -
~
h# q b k t.97elX.*O *s & A
~
& et a Na0 acre <AsA& fqdWg ra, ze7 pf.
1 e
, ,---,me--- ,,- _. , - ,.,
e t00cf CLEEK stTE
,q y . ,-v
. g -- -- -
---= -
_ ~_v" v a_ , - -
SNUPPS-WC 3.5.2 STRUCTURES, SY.CT EMS , AND COMPONENTS TO BE PROTECTED FROM EXTERNALLY GENERATED MISSILES The turbine and tornado missiles which, if generated, could effect the safety of the plant are discussed in Standa:d Plant FSAR Section 3.5.1.
The probability of significant damage (F 4 j to critical components in the plant due to turbine fa11ure has been assessed by first determining the separate probabilities of turbine failure and missile ejection (P , Refer to Standard such a lmissi?.e striking -e-- am- I Plant FSAR Section 3.5.1.2.4),
m4t4 cal-component -or- entire structure of safety signifi-cance (P2), and significant damage occurring to the component Refer to Standard Plant FSAR Section 3.5.1.2.4). Then l (P
! th$,overall annual probability P4=Py xP 2 xP 3 The probability of a high or low trajectory, turbinemissileh striking -a -or4t-icEl comp ^nent er a structure hgusing a 2
${
critical component (P2) is found to be -3rW x 10 at the Wolf Creek site. Refer to Table 3.5-1. 3 80 p
-1 The annual probability (P of a turbine missile damaging a #j i
critical ~pmponent at thk) Wolf Creek site -j is found to be i
l 3.30 --h46 x 10 . This value is less than 10 and is sufficient- I ly low so that no specific protective measures are required 7
for turbine missiles. Refer to Table 3.5-1.
identifies the safety-related structures, Figure 3.5-1 including those outside the power block, within the turbine missile trajectory.
Protective measures are provided to minimize the ef fect of potential tornado-generated missiles. The protective structures, shields, and barriers are designed utilizing the procedures given in Standard Plant PSAR Section 3.5.3.
The portions of the essential service water system (ESWS) located outside the power block requiring protective struc-tures, shields, and barriers are discussed below.
3.5.2.1 Essential Service Water System Pumphouse 1
pumphouse is a tornado-resistant, reinforced The ESWS concrete structure with an operating floor at elevation 2000 ft. The separation of the tr.ains of the ESWS is provided by an interior barrier wall. A tornado-resistant skimmer wall at the ultimate heat sink (UHS) interface provides i
protection for the ESWS traveling water screens and the ESWS pumps, whose suction ends are located 28 feet below the O
3.5-4 I
I 5.5-q)
~
U$(Aa$ f h Yw, dg -
AA9 n
/ Res sh-q ma A' 2 72 x10 ~*&L Q , 4 g
" V"~ y co, w y e
- - _ - - - - - . ,---.--,,.,-_..,.,,_,_.,__m,_ _
SNUPPS-WC TABLE 3.5-1 TURBINE MISSILE PROBABILITIES HIGH ENERGY, HIGH AND LOW TFAJECTORY Missile Source Target Structure Striking Probability, P 2
Turbine Power 21ock 1.30 x 10-4
~
Turbine ESWS Pumphouse 0.17 x 10 (High Trajectory)
Turbine ESWS Pumphouse 4
1.08 x 10 (Low Trajectory on Vertical Wall)
- 1. Rar Turbine Buried ESWS Valve -bHEF x 2 0 4
1 -
House, Pipes and Duct Banks 3.8 o -
4 TOTAL P 2= x 10 %
P = P y xP xP 4 2 3 P y = 1.0 x 10~4 (Refer to Standard Plant FSAR Section 3.5.1.2. 4 )
-4 P
2
- l*?? x 10 P3 = 1.0 (Refer to Standard Plant PSAR Section 3.5.1.2.4 )
3.80 -6 4 1.0 x 10
~
P4 = 9794 x 10 ,
w' -+f +-w--w-i-*---e--*m-=y Te+w- ? -~--T -+t - ~ - + - - + - - + - - - - - , - - - - ----*wy m -- -w %-,e e,e er-- - - + - - - - m - - - - y==
Enc 3.a B APPENDIX TO DESCRIPTION OF TURBINE MISSILE STRIKE PROBABILITY (P2) CALCULATION FOR SNUPPS TURBINE MISSILE ANALYSIS i
(' 1. INTRODUCTION This appendix describes the procedure used in computing the prob .
ability factor (P 2) f a missile strike due to a turbine failure.
The system of descriptive coordinates shown in Figure i is used to formulate the turbine missile trajectories. From inspec-tion of these coordinates, it is apparent that a successful com-bination of the angles a, #,# , and H must occur for a missile to be on target.
The range of the angle $ usually is specified by the turbine manufacturer. For computational convenience, the value of B used is measured from the horizontal plane of the turbine shaft. Thus, the O $o 15 degree range is used as a 65 E 10 degree range.
The range of the angle a is the full 360 degree circle of wheel rotation. For computational convenience, the range of a is limited to a single quadrant (i.e. O to 90 degrees) to avoid negative values.of the cosine function. Limiting the range of the angle to be considered at one time does not decrease the probability range, but instead means that the trajectories under consideration at any one time fall within a single quadrant.
(' The angle 6 , which describes the orientation of the target in the horizontal plane with respect to the vertical plane through the turbine shaft, also is limited to a single quadrant. Thus, only angles from 0 to 90 degrees are considered at any one time.
The angle of departure H required to strike a specific target is a function of the departure velocity V of the missile and the distance L of the target from the wheel.
L. FORMULATION OF PROCEDURE Trajectory of the Missile It is assumed that the missile traversos a trajectory described by the gravity parabola, 2 (Eq. t Y= (tan H)L - g(L /2V cos H) )
l where Y = elevation above (+) or below (-) the departure point
. H = vertical plane angle of departure L = horizontal distance from point of departure V = translational velocity of departure g = gravitaticaal acceleration (32.2 ft/sec/sec) l
. _ , _ _ . ~_ _ _ _ _. -_ _ _ _ _ _ _ . _ . _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Only gravitational forces are considered to be acting on the mis-sile from the point of departure onward. While air drag doubt-lessly will decrease the impactive velocity, especially on dis- -
tant targets, numerous computations have shown that it has a
! negligible effect on probabilities. For this reason, air drag on l the missile is not considered.
i Relationships of the Four Angles (a,p ,# , and B)
The mathematical relationships among the four descriptive angles a, 4 , # , and H are, tan # cos a tan H = 2 2 I (Eq. 2, )
p'1 + tan $ sin a and tan O = tan $ sin a (Eq . ) )
For convenience in the computation of the limits for the combi-nation of angles, the expression for 8, derived from Eq. & , is 2
tan H 2
) (Eq . t) )
tan # =(l - sin 2 a (1 + tan H) 2 I( )
l Ther; mathematical expressions are combined with the gravita-
)
tional parabola (Eq. l ) to determine the probability of all of them occurring simultaneously to produce a missile on target.
Computational Sequence The steps of the computational procedure are performed in the following sequence:
- a. Input
- 1. Minimum and maximus distances of the target area,
- L g and %2, respectively
- 2. Minimum and maximum values of angle 8 for the horizontal circular sector that includes the tar-get area (#3 and #2' respectively) l
- 3. Size of the target area
- 4. Minimum and maximum values of departure velocity (V 3 and V 2, respectively) 7-
(
Determination of the limits (H g and H2) b . ,- f the depar-ture angle for the conditions Lj<L <L 2 and V3< V<V 2
- c. Determination of the limits of the angles a and #
- 1. One limit from Eq. i
- 2. Two limits from Eq. J
- d. Double summation of a and 6 over the ranges of a and $
that jointly satisfy Eqs. L and J
- e. Adjustment of the summation result to account for the conditional probability of distance due to the veloc- '
! ity of departure. This is accomplished by introduction
! of a velocity range factor (VRF) l f. Adjustment of the result, which includes all possible
- angles, to account for the probability of occurrence of those angles (i.e., 26 degrees for # and 360 degrees for a )
Adjustment of the probability that a missile will I
(~ g.
strike in the sector to the probability that it will strike on target. This adjustment is accomplished '
through introduction of the area factor, A f, where Af= (target a rea)/(sector area) ,
The final result of this computational procedure is the probabil-ity factor P2 that a missile will be on target in the target area under consideration.
j, DETAILED COMPUTATIONAL PROCEDURE Input Data l The physical arrangement data are determined readily from a scale I drawing. For determination of the sector under cot..ideration, it is sufficient to measure the minimum and maximum distances to the target area to the nearest foot, and the minimum and maximum values of 6 are measured either to the left or to the right of the turbine shaft. But as indicated previously, it is computa-tionally. inconvenient to include angles both to the left and to the right of the turbine shaft in the same calculation. There-fore,~each quadrant about the shaft-wheel origin is treated sepa-rately. ~The net area of the target is determined either from a layout drawing or from a construction drawing.
(;?
3
t l
l
~
~.
The range of . maximum missile velocities is specified by the nanu-facturer of the turbine, which is approximately from 332 to 488 feet per second. It should be noted that the weights of missiles -
I are not considered, since the missile energy is not used in the computation of the probability factor P2*
Determination of Departure Angle Limits
! Setting the gravitational parabola equation (Eq. l ) equal to l
zero produces:
sin 2H = gL/V (Eq. S' )
And considering the velocity and distance limits, the limits of l angle H are:
j sin 2H = gL /V (Eq. 64 .)
2 and 2
sin 2H = gLg /V 2 (Eq. Gb )
Each of these equations produces two positive values of sin 2H.
One value is for a high trajectory parabola and the other is for %-
a low trajectory parabola. The high trajectory parabolas have 1 values of H from 45 to 90 degrees, while the low trajectory parab- ,
olas have values of H from 0 to 45 degrees.
A tremendous amount of work can be saved by treating the two types of parabolas separately, since the probability is very remote that a missile traversing a low trajectory parabola will hit a target at an elevation below the turbine wheel. Because of this, many targets are exc7aded from the low trajectory computa- ;
tions on the basis of probability, Stated differently, many tar-gets will be found to have ztxo probability for the low trajec-tory simply by inspection of the equation of the parabola. The inspection proceeds es foll.ows:
l For targets below wheel elevation, the negative values of t
Y from Eq. I are of interest. The most negative values for a given distance L are attained when H = 0 and Eq. l becomes, 2
Y = -gL /2V (Eq. 7) i
(
- t. By choosing the maximum distance L2 and the minimum veloc-ity V 1 , the most negative elevation for a target situation is attained. If the target lies below the value Y computed (i.e., if the difference in elevation between the wheel and -
the target is greater than the value of Y calculated from
), a missile on a low trajectory cannot strike the Eq.
targe ] t, and the low trajectory is excluded from consideration.
Further, because of the usual arrangement of installing con-densers below the turbine, the shaft elevation is usually much higher than other structures.
There are other cases, called " direct hits" (see Section' 4 ),
where the Y values of the target have a range extending both above and below zero elevation. These must be treated as special cases of low trajectory, and in actual layouts these cases are avoided as they engender high probabilities.
Determination of the Limits of a and #
From inspection of the coordinate system used, the limits of a and # with respect to H are H
min #< 900 mm 00 <a < 9 00- Hmin Further restriction of these ranges is feasible by refining the definitions to include the relationship of a , # , and 8 as given in Eq. 3B-3 below:
tan d = tan # sin a (Eq. 3)
By rearrangement and selection of the variables to deduce # min' equation 3 becomes:
tano min tang ,jn = ,
sin (90 - Emin }
If the value of 8 min is less than Hmin*0 mi may asthelowerlimitofSinthedefinitionabove,besubstituted and the lange of
- is reduced accordingly.
l By similar rc "rangement and selection, the value of #,,x may be deduced using ,
tan 8 max tan S ,,x
=s in (90 - H ,,x It should be noted that when a = 0,S ,,,= Hmax*
{
^
.The larger crf these two values of S max may then be substituted '
for the 90 degree upper limit for p , thereby reducing the range of g which must be considered. .
When the value of # max has be. .n determined, the value of a min "*Y be determined from:
tan 8 min sin a min " tan 4 max The value of a max may be determined by rearrangement and substi-tution in Eq 1 ' so that:
?
1- tan H sina max = min / tan $ max 2
1 + tan H min Double Summation Over the Ranges of a and S
~
To determine the total angular space %nt must be considered as a source of possible on-target missiles, it is necessary to cal- _
culate the double summation of the intervals of a and # tha.t )
I jointly satisfy the conditions of Eqs. & and 3 . This summa-
- tion is represented by
i I 0 1 *1 D= f (a,$) dad S 0
2 "2 This double summation satisfies Eqs. 2 and f when Hy<
H<H2 "" #14 ## # 2*
Because the conditions of the double summation do not constrain the velocity, the results represent a situation in which all l velocities between V1 and V2 may occur simultaneously. Since 1
only one velocity can actually occur for any given case, the result of the double surmnation must be adjusted accordingly.
Velocity Range Factor (VRF)
The ranges of missile departure velocity and target distance are specified as:
l V3<V<V2 and L3<L<L2
)
(- The limits of the angle of departure that satisfy the conditions
- are calculated by
I 2 -
. sin 2H y = gL l
2 1 l and sin 2H2 " 9L1V 2 These values represent the limits of the angle of departure, out-side of which it is not possible for the missile to hit the target for any velocity between V and V , However, this does n6t mean that between the angl s H, a$d H all velocities between V1 and V2 Will produce missile 8 on ta$get. In fact, for angle H,
1 any velocity greater than V1 will overshoot the target, and for angle Hg, any velocity less than V2 will undershoot the target. This situa' tion gives rise to what is known as "the condi-tional probability of the departure velocity."
l To solve this problem, the velocity range may be divided into l groups on the basis of the kinetic energy of the ciasiles. Each l
repr'esentative velocity Vn of each group produces an on-target ndssile within a specific range of angle H, which must be deter-mined for each Vn. The cumulative range of all velocities is then divided by n, the number of the velocity group used, to
[- -
obtain the range of the conditional angle of departure. This con-ditional range is in turn divided by the total angle of departure range H ~E to obtain the velocity range factor (VRF).
2 i The computational procedure can be expressed as:
H),L2 V dH n
M=
n M
then, VRF = H g -S{
The relationship between L and H is evaluated by applying the gravitational parabola equation, 2 2 (tanH) 2V cos H L=
9 I' The calculated velocity range factor is applied to the probability.
\ -
of a successful combination of the angles a, B , 8, and H when H is taken between H 3 and H 2*
3
~
)
Adjustment for tne Probability of Occurrence of Possible Angles The range of occurrence cf the angles a and # is not the same for cll wheels and manufacturers and so must be determined accord-ingly. For example, the range of occurrence of $ may be from 71 to 97 degrees, a total range of 26 degrees. To adjust the double integration to account for this range of # , the integration result must be divided by 0.454 radians. Similarly, the integration result must be divided by 2r to account for the total range of n .
This normalization operation is expressed mathematically as:
p , D (VRF) s 2rio.454) where P = the probability of a missile strike in the sector s bracketing the target area ,
l 3djustment for Actual Target Area, The probability Ps that the missile will strike in the bracket-ing sector is not the same as the probability that it will strike the target, due to the difference in the two areas. Adjus tment of the probability Pc to account for this difference is accom-plished most readily by dividing the target area by the sector crea. The division produces the area adjustment factor,
'-)
Af = area of target / area cf sector The final probability of a missile strike on target, P2, may be stated by the cumulative expression:
D (VRF) A f P
2 "2 r(0.454)
- 4. DIRECT H1T - A SPECIAL CASE OF THE Lni TRAJECTORY gmputat!onal Procedure A direct hit ir an instance when a wissile strikes a vertical aurface. This is a special case of the low trajectory parabole wherein values for the angle of departure may not be computed by setting the gravitational parabola equation equal to zero.
For this casd, there is only one distance L from the wheel tu the vertical surface, and the limits of the angle of departure H must be evaluated from the gravitational parabola, 2
Y= (tan H)L - g (L 2V2 cos H) \
l
( by solving for H at the limits of Y1 and Y2 of the vertical surface. 'These two values of Y are respectively, the relative elevations of the bottom and top of the vertical surface with .
respect to the elevatica of the turbine wh6el.
Evaluation for the limits of H may be simplified by taking cos H = 1, which is tne limit of the low trajectory pa::abola, to produce:
2 gL 2V +Y tan H =
L The limits of H are then obtained by solving for !! at V,,x and V
min
- The remain 6er of the procedure is similar to that in Section 3 , with the following modifications:
- a. Double summation of the ranges of a and g is not applicable.
- b. The velocity range factor VRi is computed for the single range value L between the limits Y y and Y 2'
- c. The area factor is not applicable.
l kg ,J 9
A
' o as _1 o's risSHAFT
[
1 =
p, t-(TuRSINE CASINGh '
, -s
-- TUESINE CASNG 6
i l LAST STAGE DISC)S o a
CN. .(._ ., __
f _... ( - '\b_*A'T
. ' [ f9J g 1 U j j W U l ;o A -TumNE DISC l : )1 SE CTION A-A_ CROSS SECTIOM (A HUE PLANE For. AWGLE p) ( A TeuE PLANE FOR ANGLE QC', ,
I ff0 ,\
\, p./
\2r A j j
'f...
b
~
o, o PLAh!
g (A TRUE PLANE FOR A%).E G)
O n@O) d cfc.C9yks. f.
it go@* g EQUATIONS RELATING VARIABLES:
icn $ cosa:
tan R :
I l i tan1 ) l**C SNUPPS '
FIGURE 1 tan @ s fan @ Sin % w a e N E m issit.E r
( .,
waaoaits l