ML031120595
| ML031120595 | |
| Person / Time | |
|---|---|
| Site: | Point Beach  |
| Issue date: | 03/27/2003 |
| From: | Cayia A Nuclear Management Co |
| To: | Document Control Desk, Office of Nuclear Reactor Regulation |
| References | |
| GL-96-006, NRC 2003-0025 | |
| Download: ML031120595 (119) | |
Text
NMC Committed to Nuclear Excellenc Point Beach Nuclear Plant Operated by Nuclear Management Company, LLC NRC 2003-0025 GL 96-06 10 CFR 50.54(f)
March 27, 2003 U.S. Nuclear Regulatory Commission ATTN: Document Control Desk Washington, D.C. 20555 POINT BEACH NUCLEAR PLANT DOCKETS 50-266 AND 50-301 SUPPLEMENT TO GENERIC LETTER 96-06 RESOLUTION The NRC staff issued Generic Letter (GL) 96-06 on September 30, 1996. Wisconsin Electric Power Company (WEPCO), then licensee for the Point Beach Nuclear Plant (PBNP), provided its assessment of the waterhammer and two-phase flow issues for PBNP in letters dated January 28, June 25, and December 18, 1997, and related submittals dated September 9, September 30, and October 30, 1996. Responses to NRC requests for additional information were provided on September 4, 1998, and October 12, 2000. With these submittals, the GL 96-06 two-phase flow issues were fully addressed.
Actions to fully address the waterhammer issues were deferred pending completion of the EPRI project and its review and approval by the NRC. EPRI Report TR-1 13594 was issued in December 2000, and NRC accepted it on April 3, 2002.
On July 30, 2002, Nuclear Management Company, LLC, (NMC) submitted updated information regarding actions to address the resolution of GL 96-06 waterhammer issues at PBNP.
On August 14, 2002, the NRC requested additional information regarding the July 30, 2002, submittal. During a conference call held on August 20, 2002, the NRC staff, PBNP plant staff, and Fauske & Associates (FAI) discussed the additional information requested by the NRC to support their review of Reference 5. During the conference call, PBNP proposed to provide sample cases and additional basis for the rationale that the FAI analyses for PBNP bound the EPRI methodology. On September 10, 2002, NRC staff agreed to review the additional information as proposed by PBNP.
NMC recently replaced all eight containment fan cooler (CFC) units at Point Beach Units 1 and 2. The two-phase flow issues discussed in GL 96-06 were factored into the CFC replacement project. System piping configuration was changed in the course of the system redesign. The analysis provided in the enclosure to this letter is based on the flow and system characteristics of the new CFC unit configurations.
The enclosure to this letter provides the FAI Calculation Note generated to calculate the waterhammer loads for the PBNP Containment Fan Coolers using the EPRI TBR methodology and comparing those results against the results generated previously using TREMOLO. As indicated in the comparison results of the enclosure, FAI concluded that the TREMOLO produced forcing functions used in the PBNP piping stress analyses generally bound the 6590 Nuclear Road
- Two Rivers, Wisconsin 54241 Telephone. 920.755.2321
NRC 2003-0025 Page 2 EPRI TBR forcing functions. NMC agrees with FAI's conclusions. The enclosure demonstrates that the PBNP analyses are conservative with respect to the EPRI methodology.
This letter contains no new commitments and no revision to existing commitments.
4t.L A. (.
ia Site P
nt LASm
Enclosure:
Transmittal of Fauske & Associates, Inc., Report FAI103-07, Revision 1: Comparison of Point Beach TREMOLO Calculated Waterhammer Loads with the EPRI TBR Methodology, dated March 10, 2003.
References:
- 1. NRC Generic Letter (GL) 96-06, Assurance of Equipment Operability and Containment Integrity During Design-Basis Accident Conditions, dated September 30,1996.
- 2. Letter from DF Johnson (WE) to Document Control Desk, FL 96-06 120-Day Response, dated January 28, 1997.
- 3. Letter from AJ Cayia (WE) to Document Control Desk, Revision to GL 96-06, 120-Day Response, dated June 25, 1997.
- 4. Letter from AJ Cayia (WE) to Document Control Desk, Information pertaining to Implementation of Modifications Associated with GL 96-06, dated December 18,1997.
- 5. Letter from B Link (WE) to Document Control Desk, Detailed Operability Evaluation of the Service Water System With Respect to Post-Accident Boiling in Containment Fan Coolers, dated September 9,1996.
- 6. Letter from B Link (WE) to Document Control Desk, Evaluation of Steady-State Service Water System Hydraulic Characteristics During A Design Basis Accident, dated September 30, 1996.
- 7. Letter from B Link (WE) to Document Control Desk, Assurance of Equipment Operability and Containment Integrity During Design Basis Accident Conditions, dated October 30,1996.
- 8. Letter from LL Gundrum (NRC) to M. Sellman (WE), Request for Additional Information Regarding Responses to GL 96-06, dated June 25, 1998.
- 9. Letter from VA Kaminskas (WE) to Document Control Desk, Reply to Request for Additional Information to GL 96-06, dated September 4,1998.
- 10. Letter from D Cole (NMC) to Document Control Desk, Reply to Request for Additional Information to GL 96-06, dated October 12, 2000.
- 11. FAI/97-60 Revision 5, Point Beach Containment Fan Cooler Analysis in Response to Generic Letter 96-06, dated August 8, 2001.
NRC 2003-0025 Page 3
- 12. EPRI Report TR-1 13594, Resolution of Generic Letter 96-06 Waterhammer Issues, Volumes 1 and 2, dated December 2000.
- 13. NRC Acceptance of EPRI Report TR-1 13594, Resolution of Generic Letter 96-06 Waterhammer Issues, dated April 3, 2002.
- 14. Letter from D. Spaulding (NRC) to M. Reddemann (NMC), Resolution of Generic Letter 96-06 Waterhammer Issues, dated May 3, 2002.
- 15. Letter from A. J. Cayia (NMC) to Document Control Desk (NRC), Electric Power Research Institute Report TR-1 13594, Resolution of Generic Letter 96-06 Waterhammer Issues, dated July 30, 2002.
cc:
(w/ enclosure)
Project Manager, Point Beach Nuclear Plant, NRR, USNRC (w/o enclosure)
Regional Administrator, Region III, USNRC NRC Resident Inspector - Point Beach Nuclear Plant PSCW
ENCLOSURE TO NRC 2003-0025 FAUSKE & ASSOCIATES, INC.
TRANSMITTAL OF COMPARISON OF POINT BEACH TREMOLO CALCULATED WATERHAMMER LOADS WITH THE EPRI TBR METHODOLOGY POINT BEACH NUCLEAR PLANT, UNITS 1 AND 2
FAI/03-07 Page 3 of 31 Rev. I Date. 03/06/03 FAI/03-07 COMPARISON OF POINT BEACH TREMOLO-CALCULATED WATERHAMMER LOADS WITH THE EPRI TBR METHODOLOGY Rev. 1 Prepared for Nuclear Management Co. LLC Prepared by Fauske & Associates, Inc.
16W070 West 83rd St.
Burr Ridge, IL 60521 March 2003
FAI/03-07 Page 4 of 31 Rev. I Date 03/06/03 TABLE OF CONTENTS 1.0 PURPOSE.......................................
7
2.0 INTRODUCTION
8
3.0 REFERENCES
9 4.0 DESIGN INPUTS......................................
10 4.1 Assumptions......................................
I 1 5.0 RESULTS......................................
17 5.1 EPRI TBR Waterhammer Calculations.......................................
17 5.2 TREMOLO Peak Force/Impulse Calculations.......................................
23
6.0 CONCLUSION
S.......................................
3 1 APPENDIX A:
APPENDIX B:
APPENDIX C:
APPENDIX D:
APPENDIX E:
APPENDIX F:
APPENDIX G:
APPENDIX H:
APPENDIX I:
Point Beach CFC IA EPRI TBR Waterhammer Calculations Using MathCad 2000..........................
A-1 Point Beach CFC IC EPRI TBR Waterhammer Calculations Using MathCad 2000..........................
B-1 Point Beach CFC 2B EPRI TBR Waterhammer Calculations Using MathCad 2000..........................
C-1 Point Beach CFC 2D EPRI TBR Waterhammer Calculations Using MathCad 2000..........................
D-1 Service Water Pump Curve Calculations Using Microsoft EXCEL 97..........................
E-I EPRI TBR Max Force/Impulse Calculations for Point Beach CFC 1A Using Microsoft EXCEL 97..........................
F-i EPRI TBR Max Force/Impulse Calculations for Point Beach CFC IC Using Microsoft EXCEL 97..........................
G-1 EPRI TBR Max Force/Impulse Calculations for Point Beach CFC 2B Using Microsoft EXCEL 97..........................
H-1 EPRI TBR Max Force/Impulse Calculations for Point Beach CFC 2D Using Microsoft EXCEL 97..........................
I-i
FAI/03-07 Page 5 of 31 Rev I
Date: 03/06/03 LIST OF FIGURES 4-1 Diagram of EPRI TBR CFC Configuration (Open System)...................................... 16 5-1 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC IA......................... 20 5-2 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC IC.......................... 20 5-3 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC 2B.......................... 21 5-4 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC 2D......................... 21 5-5 EPRI Pressure-Force Time History Schematic.................................................
22 5-6 Sample TREMOLO Pressure Profile for a Point Beach SW Pipe Element Following a LOOP + LOCA.29 5-7 Sample TREMOLO Force-Time History for a Point Beach SW Pipe Element Following a LOOP + LOCA.30
FAI/03-07 Page 6 of 31 Rev. I Date: 03/06/03 LIST OF TABLES 4-1 EPRI TBR Calculational Inputs Table for Point Beach CFC IA.............................. 12 4-2 EPRI TBR Calculational Inputs Table for Point Beach CFC IC.............................. 13 4-3 EPRI TBR Calculational Inputs Table for Point Beach CFC 2B.............................. 14 4-4 EPRI TBR Calculational Inputs Table for Point Beach CFC 2D.............................. 15 5-1 Results of EPRI TBR Waterhammer Calculations for Point Beach CFCs.........
....... l 9 5-2a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC IA......................................................
24 5-2b Comparison of TREMOLO - EPRI TBR Maximum Impulses for Point Beach CFC IA.....................................................
24 5-3a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC IC.....................................................
25 5-3b Comparison of TREMOLO - EPRI TBR Maximum Impulses for Point Beach CFC IC......................................................
25 5-4a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC 2B.....................................................
26 5-4b Comparison of TREMOLO - EPRI TBR Maximum Impulses for Point Beach CFC 2B..................................................................................................26 5-5a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC 2D.....................................................
27 5-5b Comparison of TREMOLO - EPRI TBR Maximum Impulses for Point Beach CFC 2D.....................................................
27
FAI/03-07 Page 7 of 31 Rev. 1 Date: 03/06/03 1.0 PURPOSE The purpose of this calculation is to calculate the waterhammer loads for the Point Beach Containment Fan Coolers (CFCs) using the EPRI Waterhammer Issues TBR (EPRI, 2002a)
Methodology. These calculations will be performed on an elevated fan cooler and a lower fan cooler (in terms of elevation within containment) for Point Beach Units 1 and 2. The results of these calculations will then be compared against the results generated for the previously performed TREMOLO analyses (FAI, 2000 and 2001a & b). This comparison is being performed to satisfy NRC requirements for Generic Letter 96-06 as discussed in Section 2.0
FAI/03-07 Page 8 of 31 Rev. 1 Date. 03/06/03
2.0 INTRODUCTION
In response to the requirements of NRC Generic Letter 96-06 (NRC, 1996), the waterharnmer loads associated with column separation and energy transfer to the service water system (including the containment fan coolers) were analyzed using the FAI computer code TREMOLO Revision 1.02 (FAI, 1997). The analyses were performed for all of the fan cooler piping arrangements in both units and were based on the design basis accident conditions of a loss of off-site power event (LOOP), as well as a loss of off-site power combined with a large break loss of coolant accident (LOCA) in the containment (LOOP + LOCA).
These conditions were evaluated with the TREMOLO code and the resulting waterhammer loads associated with both condensation induced waterhamner and column closure following separation were assessed for the entire length of the fan cooler piping. These time dependent loads were transmitted to Sargent & Lundy to be analyzed with respect to the piping response to determine the associated loads on the piping hangers. The net result of this integrated analysis was that all of the piping hangers remained within their design basis loadings for both of the service water transients investigated. In general, the LOOP + LOCA transient provided the greatest loads.
Since the TREMOLO code has not been generically reviewed by the Nuclear Regulatory Commission (NRC). closure of the issues identified in NRC( Generic LTetter 96-06 (NRC, 1996) requires either a review of the computer code by the NRC or a comparison between the results generated for the Point Beach units and a generic methodology which has been approved by the NRC.
Generic approval has been given to the EPRI methodology Generic Letter 96-06 Waterharnmer Issues Resolution Technical Basis Report (EPRI, 2002a) for evaluating loads resulting from column closure events which is intended to bound condensation induced waterhammer events.
This calculation compares the calculated loads using the EPRI TBR methodology with those produced using the TREMOLO code.
FAI/03-07 Page 9 of 31 Rev. 1 Date: 03/06/03
3.0 REFERENCES
EPRI, 2002a, "Generic Letter 96-06 Waterhammer Issues Resolution - Technical Basis Report -
Non Proprietary," EPRI Report # 1003097, May 2002.
EPRI, 2002b, "Generic Letter 96-06 Waterhammer Issues Resolution - User's Manual," EPRI Report # 1006456, April, 2002.
Fauske & Associates, Inc. 1997, FAI Q.A. File 5.17 (includes TREMOLO Revision 1 Test Plan, Test Documentation, and User Documentation, March 1997 and TREMOLO Revision 1.02 Software Change Specification and Test Documentation, August 1997).
Fauske & Associates, Inc., 2000, "Point Beach Containment Fan Cooler Analysis in Response to NRC Generic Letter 96-06," FAI/97-60, Rev. 2.
Fauske & Associates, Inc., 200 la, "Point Beach Containment Fan Cooler Analysis in Response to NRC Generic Letter 96-06," FAI/97-60, Rev. 3.
Fauske & Associates, Inc. 2001b, "Point Beach Containment Fan Cooler Analysis in Response to NRC Generic Letter 96-06," FAI/97-60, Rev. 5.
NRC, 1996, "Generic Letter 96-06: Assurance of Equipment Operability and Containment Integrity During Design Basis Accident Conditions," September 30, 1996.
WEPCo, 1999, Point Beach FSAR: Section 9.6 (Service Water System), Rev. 6. 1999.
WEPCo, 2003, E-mails from Chuck Richardson (WEPCo) to R. J. Hammersley (FAI) dated 1/27/03 and 1/28/03, "Unit 1 & 2 WATER model output."
FAI/03-07 Page lOof31 Rev. I Date: 03/06/03 4.0 DESIGN INPUTS The objective of this calculation is to compare the loads calculated by TREMOLO and EPRI TBR due to column closure waterhammer once steam bubbles have been generated due to energy addition and a pressure reduction. The TREMOLO loads for each of eight fan coolers at Point Beach have been previously calculated and the results are documented in (FAI, 2000,2001 a, 2001 b).
Tables 4-1 through 4-4 illustrate the design inputs required for the EPRI TBR methodology and the actual values used for the analyses performed on Point Beach CFCs IA, I C, 2B and 2D. Figure 4-1 illustrates a graphic representation of the EPRI TBR CFC model used to perform the pressure pulse calculations.
As shown in Figure 4-1, the EPRI TBR methodology does not model the 6" and 2 1/2" piping that branch off the 8"piping on the supply and return side of the CFC. This is due to the fact that the EPRI methodology does not model parallel flow paths, which is what occurs immediately before and after flow enters and exits the CFCs. However, since TREMOLO demonstrated that the peak forces occur in the 8" piping and the void collapse occurs in this piping as well, the 6" and 2 1/2' piping does not need to be modeled. The forces in the 6" and 2 V2" piping could be calculated using transmission coefficients calculated in the EPRI TBR methodology. Based on such transmission coefficients, the pressure pulses produced by void collapse in the 8" piping are reduced as they are transmitted to 6" piping and even further reduced when the pressure is then transmitted into the 2 I/2" piping. Therefore, due to these pressure reductions the 6" and 2 1/2'i2-- piping do not need to be modeled directly.
The approach for developing the comparison was completed as follows:
- Select two fan coolerunits per unit based on elevational differences within containment (high and low elevation).
- Assemble the information for the EPRI TBR calculation based on the piping configuration documented in the various TREMOLO parameter files for the selected CFCs for analysis.
FAI/03-07 Page I I of 31 Rev. I Date 03/06103
- Calculate the peak pressure pulse using the EPRI TBR methodology. The iMathCad 2000 spreadsheet used to calculate the peak pressure pulses for the four Point Beach CFCs was developed and tested by implementing the EPRI Open Loop Example Problem (EPRI, 2002b) and verifying that identical results were produced.
- Apply the EPRI TBR calculated pressure pulse using the methodology described in Figure 5-5 to determine the loads (peak force/impulse) from the TBR evaluations for the selected CFCs.
- Determnine the maximum forces from the previous TREMOLO analyses [FAI, 2000, 2001a, 2001b] and calculate the impulses associated with those forces.
- Compare the results generated from the TREMOLO analyses to the results generated from the EPRI TBR methodology.
4.1 Assumptions Several assumptions were made in the EPRI TBR calculations for the Point Beach CFCs.
Listed below is a summary of the assumptions made in this analysis:
- The SW design temperature is used to calculate the amount of non-condensable gases that comnes ut of so 1or, i e Er, "vvdate ailnilel Calculations. A conservatively high temperature of 950F was assumed. This is conservative since a higher water temperature results in smaller amounts of non-condensable gases, which leads to less "cushioning" during void collapse.
Figure 4-1 illustrates two "other system loads" in the EPRI CFC model. The upper branch (b to f) "other system loads" (Qabf) was assumed to be the second fan cooler that branches off the supply header. Its flow was assumed to be = 800 gpm. The lower branch (a to g) "other system load" (Qag) in the 24" line was assumed to be equal to the total flow out of a SW pump minus the two CFC flows.
FAI/03-07 Page 12 of 31 Rev. 1 Date 03/06/03 Table 4-1: EPRI TBR Calculational Inputs for Point Beach CFC 1A TBR Parameter Value Description/Reference Tvoid 224 F Average void temperature when pumps restart [FAI, 200 la]
Pvoid 18.3 psia Saturation pressure of T,0 id (steam table)
TpPIP 75 F Initial pipe temperature [FAI, 2001a] (not used in EPRI methodology)
Patm 14.7 psia Atmospheric pressure (absolute)
Ntube 240 Number of fan cooler tubes [FAI, 2001 a]
IDtube 0.527" ID of fan cooler tubes [FAI, 200 la]
Ltube 22 ft Length of fan cooler tubes [FAI, 200 la]
EL, 33.2 ft Elevation of node 1 [FAI, 2001a]
EL2*
82.3 ft Elevation of node 2 [FAI, 2001 a]
Lab 30.5 ft Length from node A to B [FAI, 200 la]
Lbc 87.5 ft Length from node B to C [FAI, 2001 a]
Lcd 61.4 ft Length from node C to D [FAI, 200 la]
Lde 78.5 ft Length from node D to E [FAI, 200 la]
Lef 4.1 ft Length from node E to F [FAI, 2001 a]
Lfe 87.6 ft Length from node F to G [FAI, 2001 a]
IDAbf 13.124 in ID of piping along path a-*b-÷f [FAI, 2001 a]
IDbCd 7.981 in ID of piping along path b-*c-.d [FAI, 2001 a]
IDag 22.624 in ID of piping along path a-+g [FAI, 2001 a]
ODbcd 8.625 in OD of piping along path b-*c->d FAI, 2001 a]
H, 240.8 ft Pump shutoff head [WEPCo, 20031 (See Appendix E)
Ai 0.2547 sec/ft2 15 order pump curve coefficient [WEPCo, 20031 (See Appendix E)
A2
-0.5783 sec2/ft' 2nd order pump curve coefficient [WEPCo, 2003] (See Appendix E)
(Qabr 800 gpm Flocw along path a ob ff during steady state (assumed)
Qbcd 917 gpm Flow along path b-*c->d during steady state [FAI, 2001a]
Qag 5100 gpm Flow along path a-*g during steady state [WEPCo, 1999]
Vwtr-fcu 0.0 ft Volume of water present in FCU when pump restarts [FAI, 2001a]
Kvalve 158.41 Throttle valve loss coefficient [FAI, 200 la]
Pwtr 62 lb/ft3 Water density Tdes 95 F Design temp of Service Water System (assumed)
Rgas 1717 ft2/sec 2. R Universal gas constant PSYS 19 psig Initial steady state system pressure [FAI, 200 la]
Note: *Slnce the Point Beach CFCs have check valves on the 8" supply piping to the CFC, the void elevation (EL2) illustrated in Figure 4-1 will not be the same on the supply and return side of the CFC piping. For these analyses, EL2 was calculated to be the elevation of the void front on the supply side of the CFC. This is appropriate since EL2 is only used to determine the water head the SW pump must overcome.
FAI/03-07 Page 13 of 31 Rev. I Date 03/06/03 Table 4-2: EPRI TBR Calcnlatinnnl Innnts for Pnint RBech CFC ir TBR Parameter Value Description/Reference Tvoid 223.0 F Average void temperature when pumps restart [FAI, 200 lb]
Ploid 18.3 psia Saturation pressure of Tvo0 d (steam table)
TpIpe 75 F Initial pipe temperature
[FAI, 2001b] (not used in EPRI methodology)
Patm 14.7 psia Atmospheric pressure (absolute)
Ntube 240 Number of fan cooler tubes [FAI, 200 lb]
IDtube 0.527" ID of fan cooler tubes [FAI, 200 Ib]
Ltube 22 ft Length of fan cooler tubes [FAI, 2001 b]
ELX 33.2 ft Elevation of node 1 [FAI, 200 lb]
EL2*
37.4 ft Elevation of node 2 [FAI, 2001b]
Lab 30.5 ft Length from node A to B [FAI, 200 lb]
LbC 67.8 ft Length from node B to C [FAI, 200 lb]
Lcd 32.3ft Length from node C to D [FAI, 2001b]
Lde 79.8 ft Length from node D to E [FAI, 2001 b]
Ler 2.2 ft Length from node E to F [FAI, 200 lb]
Lr; 86.4 ft Length from node F to G [FAI, 2001b]
IDabf 13.124 in ID of piping along path a->b--f [FAI, 2001b]
IDbrd 7.981 in ID of piping along path b->c->d [FAI, 200 lb]
IDag 22.624 in ID of piping along path a->g [FAI, 200 lb]
ODbcd 8.625 in OD of piping along path b->c->d FAI, 200 lb]
H_
_240.8 ft Pump shutoff head [WEPCo, 20031 (See Appendix E)
Al 0.2547 sec/ft2 1" order pump curve coefficient [WEPCo, 2003] (See Appendix E)
A2
-0.5783 secT/ft 2n order pump curve coefficient [WEPCo, 20031 (See Appendix E)
Qabf 800 gpm Flow along nath a-4h--4f during steady state (assumed)
Qbcd 851 gpm Flow along path b--cc-4d during steady state [FAI, 200 lb]
Qag 5200 gpm Flow along path a->g during steady state [WEPCo, 1999]
Vwtr-fcu 0.0 ft3 Volume of water present in FCU when pump restarts [FAI, 200 lb]
Kvalve 161.472 Throttle valve loss coefficient [FAI, 2001b]
Pwtr 62 lb/ft3 Water density Tdes 95 F Design temp of Service Water System (assumed)
Rgas 1717 ft2/sec2 - R Universal gas constant PSYS 19 psig Initial steady state system pressure [FAI, 2001b]
Note: *Since the Point Beach CFCs have check valves on the 8" supply piping to the CFC, the void elevation (EL 2) illustrated in Figure 4-1 will not be the same on the supply and return side of the CFC piping. For these analyses, EL2 was calculated to be the elevation of the void front on the supply side of the CFC. This is appropriate since EL2 is only used to determine the water head the SW pump must overcome.
FAI/03-07 Page 14 of 31 Rev. I Date 03/06/03 Table 4-3: EPRI TBR Calculational Inputs for Point Beach CFC 2B TBR Parameter Value Description/Reference Tvoid 217.1 F Average void temperature when pumps restart [FAI, 2000]
Pvoid 16.3 psia Saturation pressure of T, 0 id (steam table)
Tpipe 75 F Initial pipe temperature [FAI, 20001 (not used in EPRI methodology)
Patm 14.7 psia Atmospheric pressure (absolute)
Ntube 240 Number of fan cooler tubes [FAI, 2000]
IDtube 0.527" ID of fan cooler tubes [FAI, 2000]
Ltube 22 ft Length of fan cooler tubes [FAI, 2000]
EL, 33.2 ft Elevation of node I [FAI, 2000]
EL2*
72.0 ft Elevation of node 2 [FAI, 2000]
Lab 36.8 ft Length from node A to B [FAI, 2000]
Lbc 139.4 ft Length from node B to C [FAI, 2000]
Led 83.6 ft Length from node C to D [FAI, 2000]
Lde 129.2 ft Length from node D to E [FAT, 2000]
Lef 4.8 ft Length from node E to F [FAI, 2000]
Lrg 118.6 ft Length from node F to G [FAI, 20001 IDabf 13.124 in ID of piping along path a-+b---f [FAI, 20001 IDbcd 7.981 in ID of piping along path b-*c--d [FAI, 20001 IDag 22.624 in ID of piping along path a-*g [FAI, 2000]
ODbcd 8.625 in OD of piping along path b->c-+d FAI, 20001 H,
240.8 ft Pump shutoff head [WEPCo, 2003] (See Appendix E)
Al 0.2547 sec/ft2 1't order pump curve coefficient [WEPCo, 2003] (See Appendix E)
A2
-0.5783 sec2/ft5 2nd order pump curve coefficient [WEPCo, 2003] (See Appendix E)
Qabf 800 gpm Flow along path a->b--f during steady state (assumed)
Qbcd_
886 gpm Flow along path b->c--d during steady state [FAT. 20001 Qag 5200 gpm Flow along path a-+g during steady state [WEPCo, 1999]
Vwtr-fcu 0.0 ft3 Volume of water present in FCU when pump restarts [FAI, 2000]
Kvalve 165.447 Throttle valve loss coefficient [FAI, 2000]
Ptr 62 lb/ft3 Water density Tdes 95 F Design temp of Service Water System (assumed)
Rgas 1717 ft2/sec2 - R Universal gas constant P
19 psig Initial steady state system pressure [FAI, 2000]
Note: *Since the Point Beach CFCs have check valves on the 8" supply piping to the CFC, the void elevation (EL 2) illustrated in Figure 4-1 will not be the same on the supply and return side of the CFC piping. For these analyses, EL2 was calculated to be the elevation of the void front on the supply side of the CFC. This is appropriate since EL2 is only used to determine the water head the SW pump must overcome.
FAI/03-07 Page 15 of 31 Rev. 1 Date 03/06/03 Table 4-4: EPRI TBR Calculational Inputs for Point Beach CFC 2D TBR Parameter Value Description/Reference Toid 204.4 F Average void temperature when pumps restart [FAI, 2000]
Pvoid 12.7 psia Saturation pressure of T~oid (steam table)
Tpipe 75 F Initial pipe temperature [FAI, 2000] (not used in EPRI methodology)
Patm 14.7 psia Atmospheric pressure (absolute)
Ntube 240 Number of fan cooler tubes [FAI, 20001 IDwbe 0.527" ID of fan cooler tubes [FAI, 20001 Ltube 22 ft Length of fan cooler tubes [FAI, 2000]
ELI 33.2 ft Elevation of node I [FAI, 20001 EL2*
30.3 ft Elevation of node 2 [FAI, 2000]
Lab 36.8 ft Length from node A to B [FAI, 2000]
Lbc 161.8ft Length from node B to C [FAI, 2000]
Lcd 46.5 ft Length from node C to D [FAI, 2000]
Lde 161.2 ft Length from node D to E [FAI, 2000]
Ler 6.4 ft Length from node E to F [FAI, 20001 Lfp 86.8 ft Length from node F to G [FAI, 20001 IDabf 13.124 in ID of piping along path a-*b--f [FAI, 20001 IDbCd 7.981 in ID of piping along path b-*c-*d [FAI, 2000]
- IDa, 22.624 in ID of piping along path a->g [FAI, 2000]
ODbcd 8.625 in OD of piping along path b-÷c-d FAI, 2000]
H, 240.8 ft Pump shutoff head [WEPCo, 2003] (See Appendix E)
Al 0.2547 sec/ft 1" order pump curve coefficient [WEPCo, 2003] (See Appendix E)
A2
-0.5783 sec7/ft5 2"d order pump curve coefficient [WEPCo, 2003] (See Appendix E)
Qabf 800 gpm Flow along path a-*b-*f during steady state (assumed)
Qbcd 949 gpm Flow along path b->c->d during steady state [FAI, 20001 Qag 5100 gpm Flow along path a-*g during steady state [WEPCo, 1999]
Vwtrfcu 0.0 ft3 Volume of water present in FCU when pump restarts [FAI, 2000]
Kvalve 139.326 Throttle valve loss coefficient [FAI, 2000]
Pwtr 62 lb/ft3 Water density Tdes 95 F Design temp of Service Water System (assumed)
Rgas 1717 ft2/sec2
- R Universal gas constant PSYS 19 psig Initial steady state system pressure [FAI, 2000]
Note: *Since the Point Beach CFCs have check valves on the 8" supply piping to the CFC, the void elevation (EL2) illustrated in Figure 4-1 will not be the same on the supply and return side of the CFC piping. For these analyses, EL2 was calculated to be the elevation of the void front on the supply side of the CFC. This is appropriate since EL2 is only used to determine the water head the SW pump must overcome.
FAI/03-07 Page 16 of 31 Rev. I Date 03/06/03 Figure 4-1 Diagram of EPRI TBR CFC Configuration (Open System).
EL 1DED 8_
VOIDED 12"1l1 OTHER SYSTEM LOADS lf b
EL 1l TE STtLODl a
241
FAI/03-07 Page 17 of 31 Rev. 1 Date 03/06/03 5.0 RESULTS 5.1 EPRI TBR Waterhammer Calculations During a postulated LOCA (or MSLB) with a concurrent LOOP (Loss of Offsite Power), the service water pumps that supply cooling water to the CFCs and the fans that supply air to the CFCs will temporarily lose power. The cooling water will lose pressure and stop faster than the fans stop.
During the fan coastdown, the high temperature steam in the containment atmosphere will pass over the CFC tubing with no forced cooling water flowing through the tubing. Boiling may occur in the CFC tubing causing steam bubbles to form in the CFCs and pass into the attached piping creating steam voids. Prior to pump restart, the presence of steam and subcooled water presents the potential for waterhammer. As the service water pumps restart, the accumulated steam will condense and the pumped water can produce a waterhammer when the void collapses. The hydrodynamic loads introduced to the service water piping by such a waterhammer event could challenge the integrity and function of the CFCs and the Service Water (SW) System, as well as containment integrity, should the waterhamrnmer loads fail the Service Water piping supports.
Section 7.0 of the EPRI TBR Waterhammer Users Manual (EPRI, 2002b) provides a prescribed methodology to calculate the pressure pulse due to a SW system column closure waterhammer event. The analysis is performed in the following manner:
Calculate the initial closing velocity Calculate the lengths of the accelerating water column
- Calculate the mass of gas in the voided region Calculate a "cushioned" velocity based on initial velocity, pipe size, void and column length
- Calculate sonic velocity Calculate the waterhammer pressure pulse rise time
- Calculate the pulse duration
- Calculate the transmission coefficients
FAI/03-07 Page 18 of 31 Rev. 1 Date. 03/06/03 Calculate the pulse pressure with no clipping Calculate the pressure considering clipping
- Calculate the pressure pulse shape.
Using this methodology and the design information specified in Tables 4-1 through 4-4, the calculations for the Point Beach CFCs using the EPRI TBR methodology were performed and the calculation for each of the four Point Beach CFCs (1HX15A, 1 HX 1 5C, 2HX1 5B, and 2HX 1 5D) is attached as Appendices A through D. Table 5-1 summarizes the results and Figures 5-1 through 5-4 illustrate the EPRI TBR calculated pressure pulses for each of the four Point Beach CFCs.
The pressure pulse calculated (see Figures 5-1 through 5-4) for each of the four Point Beach CFCs were then used to calculate the force history and impulse loading on the SW piping upstream and downstream for each of the four CFCs analyzed. The void collapse location was determined from the corresponding TREMOLO results [FAI, 2000, 2001a, and 2001b]. Figure 5-5 illustrates the manner in which the forces/impulse will be calculated when applying the EPRI-calculated pressure pulse to the Point Beach SW piping. Since the void collapse occurs in the return line, the force and impulse calculations focused on the piping between the CFC outlet header which is upstream of the void collapse and the MOV throttling valve that is downstream of the void collapse.
As shown.. in Figure 5-5, the EPAI-calculated pressure pulse (b) can be applied Io a pipe network (a) to calculate the force on the two elements PI and P2 (c). The force on PI is simply equal to the pressure times the pipe area. Since the pipe diameter is the same at points P. and P2, the force magnitude on P2 is the same as PI, except it is in the opposite direction and delayed by the transient pressure pulse's transient time between the two points. The transient time equals the length of the pipe between P1 and P2 (L2) divided by the sonic velocity. Due to the delay in the pressure pulse reaching P2, the pipe section experiences an unbalanced force until the pressure pulse reaches P2.
Therefore, due to this time delay and the forces being in the opposite direction, the resulting force (d) on the pipe section between PI and P2 is used to determine the peak forces and impulses.
FAI/03-07 Page 19 of 31 Rev. 1 Date 03/06/03 Table 5-1 Results of EPRI TBR Waterhammer Calculations for Point Beach CFCs Unit 1 Unit 2 CFC 1A CFC IC CFC 2B CFC 2D Rise time (ms) 48 39 39 28 Duration (ms) 79.7 78.8 118.2 119.1 AP (psi) 191 202 223 286 APno clipping_174 222 203 260 Refill velocity (Vinta) ftS 7.9 8.8 8.5 10.9 Cushion velocity (Vcushion) ftls 6.1 7.1 7.1 9.1 Total duration (ms)*
127 118 157 147
FAI/03-07 Page 20 of 31 Rev. 1 Date. 03/06/03 Figure 5-1 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC 1A.
Pre e Pulse
,210 276,25O 200 00
- c. pressure 150 e
PSI El 100 so
.19.,
o -0
,0, time 127 492, Fs time (Ms) trace I Figure 5-2 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC IC.
Pressure Pulse lin
- o. pressure L
psi I
E 0
50 100
,0, time ms time (Ms) 150
,11 8 206, trace I
FAI/03-07 Page 21 of 31 Rev. I Date 03/06/03 Figure 5-3 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC 2B.
Pressure Pulse
,241 903.,300
_R 200 ra pressure E
psi C-100 i19.
0 50 100 150 200
,0, time
,157 424, ms time (ins) trace I Figure 5-4 EPRI TBR Waterhammer Pressure Pulse for Point Beach CFC 2D.
Pressure Pulse
,304 63.,400 300 v
pressure Ei psi 200 C.
100 19.,
150
,0, time ms tIme (Ms) trace I
,147461,
FAI/03-07 Page 22 of 31 Rev. 1 Date' 03/06/03 Figure 5-5 EPRI Pressure-Force Time History Schematic.
L L2 (a)
Rise Time Dwell Time P
Time (b)
Time F
Time Ii (P2) 4 I I I i
I::
(C)
- I I
£ F A
(d )
Time RR032001.CDR 2-28-2003
FAI/03-07 Page 23 of 31 Rev. 1 Date: 03/06/03 The resulting force (d) begins to rise once the pressure pulse reaches P1 and continues to rise until the pressure pulse reaches P2. When the pressure pulse reaches P2, the resulting force levels out until the pressure pulse at PI reaches its peak at which time the resultant force turns around and goes to zero when the pressure pulse at P2 reaches its peak. The force in the pipe remains balanced until the pressure pulse begins to exit PI. The resulting force then goes in the negative direction until the pressure pulse begins to exit P2. At this time the forces balance until the pressure pulse completely exits PI. The resultant force then goes to zero as the pressure pulse completely clears P2.
The maximum forces and impulses are tabulated in Tables 5-2b through 5-5b for the four Point Beach CFCs. The maximum force and impulses were calculated assuming a single pressure pulse calculated for each CFC (shown in Appendices A through D) is propagated through the SW piping. The point of collapse for the calculation was assumed to be at the same location of final void collapse calculated by TREMOLO for each CFC. The peak forces and impulse calculation for each of the fan CFCs analyzed are attached as Appendices F through I. As shown in these Appendices and Figure 5-5, the peak force is limited by the length of piping between two sequential elements (i.e., elbow). Once the pressure pulse reaches one elbow it begins to exert a force on the section of piping between the two elements. However, when the pressure pulse is transmitted to the next elbow, which is length of pipe divided by the sonic velocity of the pulse, the force on the second element counteracts the first force, thus limiting the peak force on the piping due to the relatively short length of piping between the various elements within the system (tvnirlly lies theIn2 ft).
The pipe section peak forces and corresponding impulses (calculated as the rise time of the pressure pulse x the peak resultant force) are summarized in Tables 5-2b through 5-Sb for each of the four CFCs analyzed.
5.2 TREMOLO Peak Force/Impulse Calculations The TREMOLO peak forces were taken from the previously performed TREMOLO analyses
[FAI, 2000, 2001a and 2001b] on the Point Beach CFCs. The corresponding impulse for these forces was not directly calculated in the TREMOLO analyses referenced above, rather they were determined separately by conservatively estimating the area under the peak force pulses.
FAI/03-07 Page 24 of 31 Rev. 1, Date: 03/06/03 Table 5-2a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC 1A.
TREMOLO-Calculated EPRI-Calculated Maximum Force (1b)
Maximum Force (lb,)
1858 907 1840 605 1796 592 1702 559 1633 488 1201 326 997 302 Table 5-2b Comparison of TREMOLO - EPRI TBR Maximum Impulses for Point Beach CFC 1A.
TREMOLO-Calculated EPRI-Calculated Impulses (lbf - s)
Impulses (Ibf
- s) 103.3 43.5 65.6 29.1 65.0 28.4 64.8 26.8 41.4 23.4 36.6 15.6 36.0 14.5
FAI/03-07 Page 25 of 31 Rev. l, Date: 03/06/03 Table 5-3a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC IC.
TREMOLO-Calculated EPRI-Calculated Maximum Force (lbf)
Maximum Force (lbr) 3119 1200 2917 732 2545 732 2422 732 1950 706 1599 580 1303 535 Table 5-3b Comparison of TREMOLO - EPRI TBR Maximum Impulses for Point Beach CFC iC.
TREMOLO-Calculated EPRI-Calculated Impulses (lb1 - s)
Impulses (lbr-s) 102.5 46.8 68.9 28.6 63.3 28.6 62.5 28.6 58.5 27.5 35.5 22.6 32.1 20.9
FAI/03-07 Page 26 of 31 Rev. 1, Date: 03/06/03 Table 5-4a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC 2B.
TREMOLO-Calculated EPRI-Calculated Maximum Forces (lbf)
Maximum Forces (lbr) 2619 2408 1664 1299 1248 1144 1039 943 898 870 889 678 822 613 Table 5-4b Comparison of TREMOLO - EPRI TBR Maximum Impulses for Point Beach CFC 2B.
I 5e, J.
_ I I11 TREMOLO-Calculated EPRI-Calculated Impulses (Ibf - s)
Impulses (Ibf - s) 104.3 93.9 75.0 50.6 65.3 44.6 63.9 36.8 49.9 33.9 40.6 26.4 37.5 23.9
FAI/03-07 Page 27 of 31 Rev. 1, Date: 03/06/03 Table 5-5a Comparison of TREMOLO - EPRI TBR Maximum Forces for Point Beach CFC 2D.
TREMOLO-Calculated EPRI-Calculated Maximum Forces (Ib,)
Maximum Forces (lbf) 3551 4779 3264 3596 2670 1541 1521 1075 1516 1075 1474 836 1330 818 Table 5-5b Comparison of TREMOLO - EPRI TBR Maximum Impulses ;
for Point Beach CFC 2D.
S
-. e1 hA) -
I
>Wr TREMOLO-Calculated EPRI-Calculated Impulses (lb1)
Impulses (lbf
- s) 283.5 133.8 254.8 100.7 166.9 43.2 90.2 30.1 88.5 30.1 73.4 23.4 63.5 22.9
FAI/03-07 Page 28 of 31 Rev. 1 Date. 03/06/03 The peak forces and impulses tabulated in Tables 5-2a through 5-5a for the four Point Beach CFCs analyzed were identified and tabulated independently.
As stated earlier in Section 2.0, TREMOLO is a transient code that models the fluid hydrodynamics within the SW piping system as well as performs pressure and force calculations in the piping network following a waterhammer event. TREMOLO also considers distributed voids in several pipe segments which upon collapse will transmit a pressure pulse. A sample illustration of these multiple pressure pulses is illustrated in Figure 5-6. As shown in Figure 5-6, the pressure response through this particular piping event is very dynamic. This is due to the fact that once TREMOLO calculates void collapse within a node, a pressure pulse is calculated and is transmitted throughout the system. In addition, TREMOLO models the pressure wave transmission and reflections. The net result is numerous pressure waves traveling through the SW system as the voids collapse throughout the piping system. Since the void collapse occurs in the CFC return line, the TREMOLO force and impulse calculations focused on the piping between the CFC outlet header which is upstream of the void collapse and the MOV throttling valve which is downstream of the void collapse.
The pressure pulses traveling throughout the system exert forces on the piping as shown in Figure 5-7. Figure 5-7 illustrates the force history for a typical pipe segment during the time interval when the pumps would restart and voids would begin to collapse. As shown in this figure, TREMOLO predicts that a pipe segment will undergo numerous force pulses throughout this time window. However, since TREMOLO calculates the multiple force pulses as a function of time and the EPRI TBR methodology yields a single pulse through the SW piping, the values listed in Tables 5-2a through 5-5a only consider the single maximum force pulse calculated by TREMOLO over a pipe element and its corresponding impulse. As shown in Figure 5-7, the SW piping forces are very dynamic and the piping forces are "pushing and pulling" the piping and pipe restraints for each period of time (i.e., tens of seconds) such that the TREMOLO analyses performed on the Point Beach CFCs provided a dynamic force-time history analysis.
POINT BEACH
.t0.
.E.
o OD TIME SECONDS at o
lIL POINT BEACH
- 2HX15D, LOOP WITH LOCA*
-4 4=
-J en a:
CD CD r-CD
_.to 0n
'-4 I-n _
.f3 5 35.5 36 36.5 TIME SECONDS 37
- o '
- >
X tv o
co IV
..R
FAI/03-07 Page 31 of 31 Rev. 1 Date: 03/06/03
6.0 CONCLUSION
S The results of the comparison of the EPRI TBR methodology versus the TREMOLO code for calculating the peak forces and impulse loading on the SW piping due to waterhamrnmer events following a LOOP + LOCA event are summarized in Tables 5-2 through 5-5. The impulses were included in this comparison because they provide a measure of the dynamic character of the forcing function when comparing the overall loads that pipe supports/restraints must overcome when pressure induced loads are calculated to occur within the piping. The impulse measures the integrated force over a period of time that the pipe supports must overcome. As shown in these tables, the peak forces and impulse loading calculated by TREMOLO are generally larger than those calculated using the EPRI TBR methodology. It should be noted, although it was not quantified in this comparison, that the TREMOLO force calculations include the effects of multiple pressure wave reflections and void collapses which would significantly add to the total impulse loadings on the SW piping. The dynamic TREMOLO forcing function histories were used in the piping and piping supports stress calculations. The simplified EPRI-calculated methodology only assumes a single pressure pulse propagated through the SW piping. Based on the comparison of the pipe section forcing functions provided in this assessment, it is concluded that the TREMOLO produced forcing functions generally bound the EPRI-TBR forcing functions for the Point Beach containment fan cooler cooling water supply and return piping.
FAI/03-07 Page A-1 of A-16 Rev. 1 Dale: 03/06/03 APPENDIX A Point Beach CFC 1A EPRI TBR Waterhammer Calculations Using MathCad 2000
POINT BEACH CFC 1 A VOIDED f
12"[
8"
+ VOIDED di t
EL2 a
Figure 1 Open Loop Configuration Pressure & Temperature Note, pressures listed as "psi" are absolute (psia) or differential (psid) unless otherwise stated PatM:= 14.7-psi Tmd := 224.0-F Tpipeitial := 75 F Pipe Geometry ELI := 33.2-ft EL,:= 82.3-ft L<< := 30.5 -ft Lb,:= 87.5 -ft L~d:= 61.4-ft Ld,:= 78.5 -ft Lef := 4.1-ft Lfg:= 87.6-ft Lug_,:= 400-ft IDabf 13.124-in lobcd 7.981-in IDag:= 22.624-in ODbd := 8.625 in Pressure above reservoir and above heat sink (absolute)
Temperature in the void when the pumps restart (i.e. surface temperature of piping) [Ref. FAI/97-60 Rev. 3] (Assumed average T in void at 25 sec)
Temperature of the fluid and piping when the transient starts [Ref. FAI/97-60 Rev. 3]
Elevation of node "1" [Ref. FAI/97-60 Rev. 3]
Elevation of node "2" [Ref. FAI/97-60 Rev. 3]
Length from node "a" to node Wb" [Ref. FAI/97-60 Rev. 3]
Length from node "b" to node "c" [Ref. FAI/97-60 Rev. 3]
Length from node "c" to node 'd" [Ref. FAI/97-60 Rev. 3]
Length from node "d" to node He' [Ref. FAI/97-60 Rev. 3]
Length from node "e" to node "f" [Ref. FAI/97-60 Rev. 3]
Length from node "fI to node 'g" [Ref. FAI/97-60 Rev. 3]
Length from node "g" to the ultimate heat sink [Ref. N/A -not used]
I.D. of piping along path from 'a" to "b" to "f" [Ref. FAI/97-60 Rev. 3]
I.D. of piping along path from Wba to "c" to "d" [Ref. FAI/97-60 Rev. 3]
I.D. of piping along remaining path from "a" to 'g" [Ref. FAI/97-60 Rev. 3]
O.D. of piping along path from "b" to 'c" to "d" [Ref. FA1197-60 Rev. 3]
Page 2
FAI/03-07 Page A-3 of A-16 Rev. 1 Date: 03/06/03 Flows
,ga]
min Flow along path from "a" to fb" to "fT during steady state condition without voiding [Ref. FAI197-60 Rev. 3]. Assume other CFC 800 gpm.
Qbcd = 917-gal Flow along path from 'b" to "c" to "d" during steady state condition without min voiding [Ref. FAI/97-60 Rev. 3]
min Flow along path from 'a" to "g during steady state condition without voiding
[Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]. Per FSAR nominal flow is 6800 gpm.
FCU Characteristics N~be := 240 IDube := 0.527-in Llbe := 22-ft Number of tubes in cooler [Ref. FAI/97-60 Rev. 3]
Internal diameter of tubes [Ref. FAI/97-60 Rev. 3]
Length of tubes [Ref. FAI/97-60 Rev. 3]
Pump Characteristics H, := 240 8.ft Al := 0.2547.-
ft2 A2 := -0.5783 ft5 Pump shutoff head [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
1st order pump curve coefficient [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
2nd order pump curve coefficient [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
Hpump(Qp) := A2-Qp2 + Al Qp + H, Pump curve equation Other Inputs KIV:= 158.41 Vw tr f := 0..0ft3 Vw,.2pie := 6-ft3 lb pAw:= 62-3 ft Tdes := 95.F ft2 Rga := 1717-sec-R Valve frictional flow coefficient for throttled globe valve [Ref. FAI/97-60 Rev. 3]
Volume of water that is left in the FCU when the pump restarts [Ref.
FAI/97-60 Rev. 3]
Volume of water that flows into the cooler after voiding has started and before the pumps restart. This volume of water is exposed to two phase flow conditions. [Ref. N/A -not used]
Water density Design temp of the system Gas constant Page 3
Pump Flow Rate Equation Qttnorta = Qag + Qbcd +
pbf H.orm:= Hpump(Qtot..rma)
Qtotn, 0,j 6.817x 103 gal HOrm=llIft min The total system flow rate is solved at any pump operating point using:
-Al.- 4AIl 2 -4.A2 (H1 - Hd)
Qpump(Hd):=
2-A2 Qpump(H,,m) = 6.817x mi ga 300.
200 0
100 0 o 2000 4000 6000 8000 GPM Pum.p Curve 00 Operating Point PUMP CURVE & OPERATING POINT Figure 2 SW Pump Curve Page 4
FAI/o3-07 Page A-5 of A-1 6 Rev. 1 Date 03/06/03 7.4.1 Initial Velocity & FLOW COEFFICIENT PREDICTION The water at the front of the void (point "d") is assumed to not move or simplification of this problem. More detailed hydraulic modeling may be performed to determine the reverse or forward flow at point "d". In many cases this flow is less than 10% of the incoming flow.
After combining parallel paths the system is then simplified to:
Figure 3 Simplified Open Loop Model In terms of the initial flow diagram (Figure 1), the flow area for each path is calculated:
Aabf := 7-1Dbf 4
Of.939f Abcd := ° 34D7 4
Abcd = 0 347 f t2 IC Ag g=
IDag 4
kg= 2.792 ft2 The velocity for each path is calculated:
Qabf Vf =Aabf Qbed Vbcd =bcd Abcd V
Qag Aag Vag= 4 1-Vabf = 1.9-S Vt-d = 5.9-S Calculate equivalent velocity for all other loads:
V Qabf + Qag e'
Aabf + Aag Veq = 3.523-S Page 5
FAI/03-07 Page A-6 of A-1 6 Rev. 1 Date 03/06103 The flow coefficient for each path is calculated.
The flow resistance from point "a" to point "b" and from point If to point "g" are assumed to have a negligible effect on the flow split to the different paths. In an actual plant system, the engineer may choose to use values from a previously qualified system hydraulic model to determine a more accurate initial velocity.
2-g 2-g hf v 2 2-g-H.,..
Kabf =
Vab 2-g-H...
Kbcd =
2 Vbcd Kag V= 2 Vag Kabl = 1.989 x 10 Kbcd = 207 Kag = 432 An equivalent flow coefficient for the "other loads" path (Figure 1) is calculated from:
(
Aabf Kilher.=
Aabf + Aag Kother = 37 Aether := Aabf IDother := IDabf An equivalent flow coefficient from all other loads is calculated from:
2-g-Hnorm Kother :=
2 ocq Aithcr := Aabf + Aag Kohu = 576.797 4 : AotherJ M~other :=
J IDothe, = 2.18 ft The flow coefficient for the path to the void is calculated by subtracting the flow coefficient downstream of the void along this path. To simplify this sample problem only the valve resistance downstream of the void is considered:
Kvwod := Kbcd K-vI Kvold = 49 The pressure in the void is assumed to correspond to the saturation pressure for the void temperature.
Pvoid := 18.6-psi Absolute pressure based on saturation pressure of Tvoid shown above.
Comes from TREMOLO output. [Ref. FA1197-60, Rev. 3]
Page 6
FAI/03-07 Page A-7 of A-16 Rev. 1 Date: 03/06/03 The pump total developed head (TDH) is written by using Bernoulli's equation:
H=tm + EL, + TDH = Hvod + EL, + Hf where the following terms are defined in terms of feet H20 Hatm
= atmospheric pressure head EL,
= elevation of node "1" TDH = total developed head from pump EL,
= elevation of node "2" Hf
= frictional losses from point '1" to "2" The frictional losses are written using Darcy's formula with an appropriate units conversion factor:
Q 2 Hf = 0 00259-K 1 0 5 5 4 where ID K,05,
= loss coefficient a
= flow rate in gpm ID
= pipe diameter in inches Two equations for the total developed head (TDH) by the pump are written with a corresponding flow balance and initial guesses for the simultaneous solution of these equations:
Qvod :=.1
- Lher
= *5 TDH := 300 Given TDH = 0.00259 Kother o
( IDn h) frictional losses along "other" path equal the total developed head Q~oid2 r
Pat.
Pvoid A
-I TDH = 0.00259-Kvd-d
+
EL-EL, --
+
ft bcd 4 pwtr'g Pw)wteg i n Bernoulli's along the 'void' path Qother -I-Qvoid = -Pulrl.ll 1
. ( gal )'
min) piumpn curve The solution to the simultaneous equations is solved and defined as "Results".
Results := Find(TDH, QotherQvoid)
TDH := Resultsn ft TDH = 105.207 ft Qothr= Results,.
7 Qothr = 5.741 x 1 gal min min Qoid := Results2 gal min The initial velocity is then:
Q,,,d = 1.231 x 3 gal min The total resistance for this path is:
-Qvod V
=itial
= 7.9-Kvold = 49 A~bcd Check: is the velocity within the RBM bounds?
Vtial
< 20 ft/sec
===> yes, velocity is within bounds of RBM runs Page 7
FAI/03-07 Page A-8 of A-16 Rev. 1 Date: 03/06/03 7.4.2 VOID & WATER COLUMN LENGTHS The volume of piping that is voided is calculated:
Vpipe-voided : Lcd'
- IDbcd Vpipe voided 21 ft 4
The void of the fan cooler unit is calculated:
Vfc
= NbeLtbbe-DljDtbe Vf" 8 ft 4
The equivalent void length is then:
Lao Vpipe-voided + Vfcu Lao = 84 ft Abed The initial water column length is assumed to be the distance from point "a" to point ace. The discussion that follows explains why point "a" was chosen.
Ignoring the FCU, the flow area changes from the closure point to node "a" are the same as the area changes from the closure point to node "g" on the return side. The transmission coefficients calculated for the return side demonstrate that less than 10% of the pressure pulse propagates to the header. Because of the similar flow area changes, less than 10% of any pressure would propagate into the supply header upstream of point "a". In general, this indicates that the header acts like a large pressurized reservoir during the void closure process and water in the supply header does not add to the inertia of the decelerating water column.
Note: if desired, a plant could select a length all the way back to the pumps. However, this is considered excessively conservative.
The length if the accelerating water column is then:
Lwo.= Lab + Lbc Lwo = 118 ft Check: are the lengths within the bounds of the RBM runs?
Lao < 100 ft Lwo < 400 ft
===>> yes lengths are within bounds of RBM runs Page 8
FAI/03-07 Page A-9 of A-16 Rev 1 Date. 03106/03 7.4.3 GAS RELEASE AND MASS OF AIR CONCENTRATED IN VOID The mass of air concentrated in the void during the void phase of the transient is calculated by assuming that the water that has experienced boiling and subsequent condensation releases its air as described in Section 5 of the User's Manual.
For this problem, the tube volume only will be credited, assuming a draining of the FCU in which the headers do not remain full. This mass of water will release 50% of its non-condensable gas.
Vf"' = 7.998 ft3 or Vf,. = 226 liter from 7.4.2 This represents the mass of water in the tubes which will lose 50% of its non-condensable gas The concentration of gas is obtained from Figure 5-3.
Tdes = 95 F Tdes - 32F =35 degC 1.8F CON,,j:= 18.5 g
liter From Figure 5-3 m.,=r 0.5 CON.r-Vfcu Mair = 2095 mg Check: is the mass of air within bounds of UM?
for void closure in 8" piping there should be at least 900 mg of air per Table 5.2
===> yes, mass if air is within RBM run bounds.
Page 9
7.4.4 Cushioned VELOCITY The graphs presented in Appendix A for the velocity ratios are solutions to the simultaneous differential equations that capture the acceleration of the advancing column and pressurization of the void.
In order to determine the cushioned velocity the following terms that are needed are repeated.
V.I.,., = 7 897-S Kvoid = 49 Lao = 84.422 ft Lwo = 118ft mar = 2.095 x 10 3mg Tvod = 224 F Check: is the temperature within the bounds of the RBM?
Tvoid > 200 F===> yes, the temperature is within the RBM run bounds 7.4.4.1 Air Cushioning If only credit for air cushioning is considered then Figure A-1 0 from Appendix A is selected. This figure corresponds to 10" piping while the sample problem has 8" piping. 10" piping bounds the 8" piping since the inertia modeled in the 10" piping runs is greater than that in the 8" piping runs and the velocity has reached a steady state until the final void closure occurs. This is apparent by comparison of the 4" and 10" RBM run results for the same gas mass; the velocity is reduced more in the smaller pipe case. If the pipe size at a given plant is not shown then the Velocity Ratio chart for the next larger size pipe will always be bounding.
Figure A-1 0 corresponds to an initial velocity of 10 fps. The initial velocity calculated in this sample pr ob-c.-
is less. The highervlocIitv Phart is selected because the higher momentum associated with the higher velocity bounds the lower velocity. If the initial velocity at a plant Is not shown then the Velocity Ratio chart for the next larger velocity will always be bounding.
For a K of 49 as calculated in the sample problem, from Figure A-1 0 the ratio of the second to initial velocity is:
c-=
82%l only air cushion credited Therefore, the final closure velocity will be reduced by 18% just considering air in this sample problem. Pressure "clipping" is not included here and is calculated later.
Page 10
7.4.4.2 Air and Steam Cushioning The velocity that results by considering steam cushioning is found using Figure A-37 from Appendix A. Note that the condensing surface temperature was verified being within the bounds of the RBM run limitations so steam condensation cushioning may be credited. The steam and air cushioning result in a ratio of cushioned to initial velocity of:
lVcusluon 7
l 777%
[ijualJ air and steam cushioning The cushioned velocity is then:
lVcushion := 0.77 Vinitiall VcusIUon = 6.1 1 s
7.4.5 SONIC VELOCITY The sonic velocity is calculated from Equation 5-1 and 5-2 in the main body of the User's Manual.
Pvoid = 18 6psi where B=bulk modulus of water E=Young's modulus for steel OD=outside diameter of pipe t=wall thickness B := 319000psi E := 28-10 6psi C
B C
I P-tr I I B.ODbcd l
l (ODbcd-IDbcd)
XL 2Ej C = 4274-f S
Page 1 1
7.4.6 PEAK PRESSURE PULSE WITH NO "CLIPPING" The peak waterhammer pressure is calculated using the Joukowski equation with a coefficient of 1/2 for a water on water closure:
APno.clipping I= PwtrCVcushzon APno.-cipping = 174 psi 7.4.7 RISE TIME The rise time is calculated by using equation 5-4 from the UM.
ms := O.OOls TR := O.5sec{ Vcu-;Ion 3
TR = 48 ms sec Page 12
FAI/03-07 Page A-13 of A-16 Rev 1 Date: 03106/03 7.4.8 TRANSMISSION COEFFICIENTS The pressure pulse may be affected by rarefaction waves as it is developing and the peak may be "clipped". In addition, the pressure may be attenuated as it propagates through the system as a result of area changes. In order to calculate each of these effects, the transmission coefficients at junctions is required. The transmission coefficients are calculated consistent with section 5.3 of the UM.
At points "f' and "g" the transmission coefficients are calculated using Equation 5-8 from the UM; for simplification here the sonic velocity is assumed to be constant up and downstream of the junction:
2 Auaczdent Aicident + E Aj J
TfAd f = 0.312
=> this fraction of the incident pulse Abcd + AW + Aabf continues past point "f" and the remainder of the incident pulse returns towards the initiation point.
2~ - abf T *=
+g
= 0 288
=> this fraction pulse that is incident upon 9
AabW + Aag + Aa, point 'g" continues past point "g" and the remainder of the incident pulse returns towards the initiation point.
r10tf :=
-T Tmoal =09 When the pressure pulse travels past point "g" only 10% of the pulse will continue on. 69% of the incident pulse was reflected as a negative pulse at point "f" and then 71% of the pulse that was incident upon point "g" was refcteind back as a negative pulse. The net refiection offect is:
lPref = P.c, (69%)
+ (31%-Pnc *(-71%) = P.c(69% -31%-71%) = 91%
This reflection travels back to the initiation point. The pulse at the initiation point is 9% of its original value when this reflection arrives. For simplicity, the compounding effect of the "f" node transmission coefficient on the reflected wave from node "g" is ignored.
The transmission coefficient evaluation needs to consider the control valve. The transmission coefficient at the control valve is calculated by assuming the valve acts like an orifice as the pressure pulse propagates through it. Equation 5-14 provides a simple relationship for an orifice flow coefficient in terms of its diameter ratio (13). This equation is used to back calculate an equivalent 1 ratio for the control valve knowing its coefficient and assuming Co=0.6.
1 = 0.5 Initial guess for the iteration below roo {(
I Kviv
=0.35 Page 13
For this P ratio and for the approximate waterhammer pressure already solved, the control valve will have a slight effect on the pressure pulse propagation by inspection of Figure 5-15. The reflection from this interaction will add approximately 10% to the incident pulse.
In general what this means is that 10% of the pulse magnitude is reflected in a positive sense back towards the initiation point. To account for this effect, the peak pressure pulse is conservatively increased by 10%.
7.4.9 DURATION The pressure pulse is reduced to approximately 10% of its peak value as a result of the reflections from the area changes at points 1f" and "g". As a result, the time that it takes the pressure pulse to travel to point "g" and back may be used to calculate the pressure pulse duration.
TDcg:=
(Lde + Lef + Lfg) -2 C
TDeg = 79.7 rns Time for pulse to travel to and from point "g'". Note that reflections from "a' and "b" are not credited.
The total duration is conservatively increased by adding the rise time.
TD := TDeg + TR TD = 127 rns 7.4.10 PRESSURE CLIPPING The peak pressure is checked for "clipping" using Table 5-3.
T
- =
+
t 1c--
1 dc +
ef +
g Le -170.2ft TZ-- = I12ft 2
X At= 0.09 This corresponds to the conditions in row two of the table referenced and no pressure clipping is expected.
lAP:
1.1 n-fi-pp-.1g 1.1 is from the control valve AP = 191 psi Page 14
7.4.11 PRESSURE PULSE SHAPE The pulse shape is then characterized by four points lPsys l=
9psi this is the steady state system pressure [Ref. FAI/97-60 Rev. 3]
Using an index, i=0,1,2,3
- = O. 3 time,
- =
'ID pressure, :=
Psys AP + Psys AP + Psys Psys This provides the following values, which are plotted below.
( 0 r
19 '
10.048 2101 time = 0.08 s
pressure = 210 psi
\\0.127)
L 19)
Pressure Pulse 250 200 150 VI c4 VI 100 0
50 tune (Ms) trace I 150 Calculate the area underneath the curve to get the pressure impulse:
integral := TR-AP + AP-(TDcg - TR) integral = 1 05 x 105 kg ms impulse := integral-Abd impulse = 762.175 Ibf s Page 15
7.4.12 FLOW AREA ATTENUATION To simplify the analysis of the SW structures, the approach suggested here is to take the initiating pressure pulse and propagate the pulse through the system. For this example problem, the duration of the pulse is assumed to remain unchanged as it travels. In reality, the duration of the pulse is shortened as it approaches negative reflection sites. Maintaining the duration conservatively increases the impulse.
As the pressure pulse propagates through the system it will be atenuated/amplified by flow area changes. For this example, only the downstream propagation is considered. The pulse will be attenuated by the increase in area at "f" and "g'. The transmission coefficients were previously calculated.
incident pulse transmitted pulse transmission pulse AP= 191 psi APf :=¶f-AP APf - 60psi APf = 60psi ApPgr *g-Pf APy =17psi Downstream of point " only the following pulse magnitude will remain:
AP, =17 psi Page 16
FAI/03-07 Page B-1 of B-16 Rev. 1 Date: 03/06/03 APPENDIX B Point Beach CFC 1C EPRI TBR Waterhammer Calculations Using MathCad 2000
FAI/03-07, Page B-2 of 8-16 Rev. 1 Date: 03/06/03 POINT BEACH CFC 1 C 86 VOIDED 12 7
a Figure 1 Open Loop Configuration Pressure & Temperature Note, pressures listed as apsis are absolute (psia) or differential (psid) unless otherwise stated Patm:= 14.7-psi Totd := 223-F Tpipeitijzuai : 75-F Pipe Geometry ELI := 33.2-ft EL, := 37A.ft Lab := 30.5-ft Lb, := 67.8-ft Ld := 32.3-ft L& := 79.8-ft Ler := 2.2ft Lfg := 86.4-ft Lgsink := 400-ft Mabf := 13.124-in IDbcd:= 7.981-in mag := 22.624-in ODb~d:= 8.625-in Pressure above reservoir and above heat sink (absolute)
Temperature in the void when the pumps restart (i.e. surface temperature of piping) [Ref. FAI/97-60 Rev. 5] (Assumed average T in void at 25 sec)
Temperature of the fluid and piping when the transient starts [Ref. FAI/97-60 Rev. 5]
Elevation of node "1" [Ref. FAI/97-60 Rev. 5]
Elevation of node "2" [Ref. FAV97-60 Rev. 5]
Length from node 'a' to node "b" [Ref. FAI/97-60 Rev. 5]
Length from node Ebb to node "c" [Ref. FAI/97-60 Rev. 5]
Length from node 'c' to node "d" [Ref. FAI/97-60 Rev. 5]
Length from node 'd" to node "e' [Ref. FAI/97-60 Rev. 5]
Length from node 'e" to node 'f" [Ref. FAI/97-60 Rev. 5]
Length from node f" to node "gm [Ref. FA1197-60 Rev. 5]
Length from node "g" to the ultimate heat sink [Ref. N/A -not used]
I.D. of piping along path from "a' to "b" to mf" [Ref. FAI/97-60 Rev. 5]
I.D. of piping along path from "b" to "c" to 'd" [Ref. FA1197-60 Rev. 5]
I.D. of piping along remaining path from "a" to "`" [Ref. FAI/97-60 Rev. 5]
O.D. of piping along path from 'b" to 'c" to "d" [Ref. FAI/97-60 Rev. 5]
Page 2
Flows Qbf := 800 gal min Qbcd 857-gal mnu g := 5200- gal min FCU Characteristics NmX
- = 240 IDube := 0.527-in Sbe := 22-ft Flow along path from "a' to "b" to "f" during steady state condition without voiding [Ref. FAI/97-60 Rev. 5]
Flow along path from 'b" to "cm to "d" during steady state condition without voiding [Ref. FAI/97-60 Rev. 5]
Flow along path from "a' to "g during steady state condition without voiding
[Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
Number of tubes in cooler [Ref. FAI/97-60 Rev. 5]
Internal diameter of tubes [Ref. FAI/97-60 Rev. 5]
Length of tubes [Ref. FAI/97-60 Rev. 5]
Pump Characteristics 1s := 240.8-ft Pump shutoff head [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
Al := 0.2547-1 st order pump curve coefficient [Ref. Chuck Richardson Emails dated ft2 1/27/03 & 1/28/03]
A2 := -0.5783-se 2nd order pump curve coefficient [Ref. Chuck Richardson Emails dated ft5 1/27/03 & 1/28/03]
Hpump(Qp):= A2-Qp + Al-Qp + Hs Pump curve equation Other Inputs K1,1 := 161.472 Vwirfcu := 0.0 ft3 Vtj_2phe := 6 ft3 lb pv,:= 62- - 3 Tdes := 95-F ft2 Rgas:= 1717-sec R Valve frictional flow coefficient for throttled globe valve [Ref. FAI/97-60 Rev. 5]
Volume of water that is left in the FCU when the pump restarts [Ref.
FAI/97-60 Rev. 5]
Volume of water that flows into the cooler after voiding has started and before the pumps restart. This volume of water is exposed to two phase flow conditions. [Ref. N/A -not used]
Water density Design temp of the system Gas constant Page 3
Pump Flow Rate Equation QtOtnormSa= Qag + Qd + QbfK
- = Hpump( QtOt0oi)
Qtot 0,,
= 6.857x 103 gal HOrm=I oft min The total system flow rate is solved at any pump operating point using:
-Al - 4A12 - 4.A2-(H, - Hd)
Qpump(Hd):=
2-A2 Qpump( HIo0 ll) = 6 857 x 103 gal min 300
=
I I
I 200_\\_
0 100 o
2000 4000 6000 8000 GPM 1 MITaj
- U.
00 Operating Point PUMP CURVE & OPERATING POINT Figure 2 SW Pump Curve Page 4
7.4.1 Initial Velocity & FLOW COEFFICIENT PREDICTION The water at the front of the void (point "d") is assumed to not move or simplification of this problem. More detailed hydraulic modeling may be performed to determine the reverse or forward flow at point 'd". In many cases this flow is less than 10% of the incoming flow.
After combining parallel paths the system is then simplified to:
Figure 3 Simplified Open Loop Model In terms of the initial flow diagram (Figure 1), the flow area for each path is calculated:
Aaf :=
Z IDWbr 4
A~bf = 0.93 9 f~t Abcd :=-IDbcd2 4
Abcd = 0.347 ft2 A~ig.= 7-IDag2 4
Aag = 2.792ft-The velocity for each path is calculated:
Qnbf Vabf:= -Aabf Qbcd V. MO-A Vag :=Q-g Aag Vag = 4.2-S ft V~bf = 1.9 S
Vbd = 5.5-S Calculate equivalent velocity for all other loads:
V Qabf + Qag AabW + Aag V~q = 3.583-S Page 5
FAI/03-07, Page B-6 of B-16 Rev. 1 Date: 03106/03 The flow coefficient for each path is calculated.
The flow resistance from point "a' to point 'b" and from point "I" to point "g" are assumed to have a negligible effect on the flow split to the different paths. In an actualplantsystem, the engineer may choose to use values from a previously qualified system hydraulic model to determine a more accurate initial velocity.
V2 hf= K-
=>
2-g K=~
V2 2*g*H.Om Vabf-2
- g-Hrn IIcd=
Vbcdd 2-g-.H..
Kag- :
Vag Kabf = 1.961 x 103 Kbcd = 234 Kag = 410 An equivalent flow coefficient for the "other loads" path (Figure 1) is calculated from:
el Aabft~
Kother :
Aabf 2
Aabf Aag s,
Wf+
9 Kother = 35 kther := Aabf IDother := IDabf An equivalent flow coefficient from all other loads is calculated from:
2-g-IH-4.
Kother =
2 K V= q Kvuh, = 549.974 Aother := AW + Ang (4 -Aothr J 05 M~other :=
IDother = 2.18 ft The flow coefficient for the path to the void is calculated by subtracting the flow coefficient downstream of the void along this path. To simplify this sample problem only the valve resistance downstream of the void is considered:
Kvoid := Ktd - Kvlv Kv.od = 72 The pressure in the void is assumed to correspond to the saturation pressure for the void temperature.
Pvoid := 18.3-psi Absolute pressure based on saturation pressure of Tvoid shown above.
Comes from TREMOLO output. [Ref. FAI/97-60, Rev. 5]
Page 6
FAI/03-07, Page B-7 of B-i 6 Rev. 1 Date: 03/06/03 The pump total developed head (TDH) is written by using Bernoulli's equation:
Hatm + EL, + TDH = H, 0 1d + EL, + Hi where the following terms are defined in terms of feet H20 11atm
= atmospheric pressure head EL,
= elevation of node "1" TDH = total developed head from pump EL,
= elevation of node "2" Hf
= frictional losses from point "1" to "2" The frictional losses are written using Darcy's formula with an appropriate units conversion factor:
Hf = 0.00259-Ki 0
where ID Koss
= loss coefficient 0
= flow rate in gpm ID
= pipe diameter in inches Two equations for the total developed head (TDH) by the pump are written with a corresponding flow balance and initial guesses for the simultaneous solution of these equations:
Qvojd =.1 Qth:
= *5 TDH := 300 Given TDH = 0.00259-Khe,-
t (Mother 4
frictional losses along "other" path equal the total developed head QvOid Patm Pvoid I
TDH = 0.00259-K,.id-4 2
- ELI - -
+
f Ibcd )
Pwtr g Pwteg in Bernoulli's along the "void" path Qother t void = QPuQtiP(T v"-'- -i) ne o
^n o.
inc F..u"j F
-W The solution to the simultaneous equations is solved and defined as "Results".
Results := Find(TDHQothervQvotd)
TDH := Resultsn-ft
%,he gal
,t:= Results1-m Qvoid:= Results= gal min The initial velocity is then:
TDH = 99.906 ft Qothr = 5.729 x 103 gal rnun Qvod = 1.376x 103 gal mnu The total resistance for this path is:
v-iual
=-
Vanitwai = 8.8-KvOid = 72 Abcd s
Check: is the velocity within the RBM bounds?
V~nitial
< 20 ft/sec
===> yes, velocity is within bounds of RBM runs Page 7
7.4.2 VOID & WATER COLUMN LENGTHS The volume of piping that is voided is calculated:
Vpipe voided Lcd IDbcd Vplpe-voided = 11 ft 4
The void of the fan cooler unit is calculated:
Vf.
- = Ntbe Ltmbe
'1be Vc
= 8 ft3 4
The equivalent void length is then:
Lao:= Vpipe voided + Vfcu Lao = 55 ft Abcd The initial water column length is assumed to be the distance from point 'a" to point "c". The discussion that follows explains why point Wag was chosen.
Ignoring the FCU, the flow area changes from the closure point to node "a" are the same as the area changes from the closure point to node "g" on the return side. The transmission coefficients calculated for the return side demonstrate that less than 10% of the pressure pulse propagates to the header. Because of the similar flow area changes, less than 10% of any pressure would propagate into the supply header upstream of point "a". In general, this indicates that the header acts like a large pressurized reservoir during the void closure process and water in the supply header does not add to the inertia of the decelerating water column.
Note: if desired, a plant could select a length all the way back to the pumps. However, this is considered excessively conservative.
The length if the accelerating water column is then:
Lwo := Lab + Lbc Lwo = 98.3 ft Check: are the lengths within the bounds of the RBM runs?
Lao < 100 ft Lwo < 400 ft
===>> yes lengths are within bounds of RBM runs Page 8
7.4.3 GAS RELEASE AND MASS OF AIR CONCENTRATED IN VOID The mass of air concentrated in the void during the void phase of the transient is calculated by assuming that the water that has experienced boiling and subsequent condensation releases its air as described in Section 5 of the User's Manual.
For this problem, the tube volume only will be credited, assuming a draining of the FCU in which the headers do not remain full. This mass of water will release 50% of its non-condensable gas.
Vf
= 7.998 ft3 or Vf,, = 226 liter from 7.4.2 This represents the mass of water in the tubes which will lose 50% of its non-condensable gas.
The concentration of gas is obtained from Figure 5-3.
Tdes = 95 F Td,- 32F
- 3
= 35 deg C 1.8F CON.,r:= 18.5 mg From Figure 5-3 liter mair := 0.5.CON,,-Vfc, mair = 2095 mg Check: is the mass of air within bounds of UM?
for void closure in 8" piping there should be at least 900 mg of air per Table 5.2
===> yes, mass if air is within RBM run bounds.
Page 9
7.4.4 Cushioned VELOCITY The graphs presented in Appendix A for the velocity ratios are solutions to the simultaneous differential equations that capture the acceleration of the advancing column and pressurization of the void.
In order to determine the cushioned velocity the following terms that are needed are repeated:
Vinitial = 8.826-S Kvo~d = 72 Lao = 55.322 ft Lwo = 98.3 ft mr = 2.095 x 103 mg Tv01d = 223 F Check: is the temperature within the bounds of the RBM?
Tvoid > 200 F===> yes, the temperature is within the RBM run bounds 7.4.4.1 Air Cushioning If only credit for air cushioning is considered then Figure A-1 0 from Appendix A is selected. This figure corresponds to 10" piping while the sample problem has 8" piping. 10" piping bounds the 8" piping since the inertia modeled in the 10" piping runs is greater than that in the 8" piping runs and the velocity has reached a steady state until the final void closure occurs. This is apparent by comparison of the 4" and 10" RBM run results for the same gas mass; the velocity is reduced more in the smaller pipe case. If the pipe size at a given plant is not shown then the Velocity Ratio chart for the next larger size pipe will always be bounding.
Figure A-1 0 corresponds to an initial velocity of 10 fps. The initial velocity calculated in this sample pobem_
is 1655. Thc hghe velo., cit chart IS Selected becrause the highor mnm entu. asscrPiate with the higher velocity bounds the lower velocity. If the initial velocity at a plant Is not shown then the Velocity Ratio chart for the next larger velocity will always be bounding.
For a K of 72 as calculated in the sample problem, from Figure A-1 0 the ratio of the second to initial velocity is:
lVrs3%on 83I only air cushion credited lVinittall Therefore, the final closure velocity will be reduced by 17% just considering air in this sample problem. Pressure "clipping" is not included here and is calculated later.
Page 10
FAI/03-07, Page B-1 1 of B-16 Rev. 1 Date: 03/06/03 7.4.4.2 Air and Steam Cushioning The velocity that results by considering steam cushioning is found using Figure A-37 from Appendix A. Note that the condensing surface temperature was verified being within the bounds of the RBM run limitations so steam condensation cushioning may be credited. The steam and air cushioning result in a ratio of cushioned to initial velocity of:
uisa
= 80%i lVinitallI air and steam cushioning The cushioned velocity is then:
lV.u~on := 0.80-VinitialI Vcushon = 7 S 7.4.5 SONIC VELOCITY The sonic velocity is calculated from Equation 5-1 and 5-2 in the main body of the User's Manual.
Pvoid = 18.3 psi where B=bulk modulus of water E=Young's modulus for steel OD=outside diameter of pipe t=wall thickness B := 319000psi E:= 28-10 6psi C
l B
B.ODbcd C I pe(wbcd
-Dd)
E-C = 4274-S Page 1 1
FAI/03-07, Page B-12 of B-16 Rev. 1 Date: 03/06/03 7.4.6 PEAK PRESSURE PULSE WITH NO "CLIPPING" The peak waterhammer pressure is calculated using the Joukowski equation with a coefficient of 1/2 for a water on water closure:
aPno-clpping I PWVu CVcushion AP'%_chpping = 202 psi 7.4.7 RISE TIME The rise time is calculated by using equation 5-4 from the UM.
ms := 0.OOls TR := 0.5sec.
cushuof TR = 39 ms ft sec)
Page 12
FAI/03-07, Page B-13 of B-16 Rev. 1 Date: 03/06/03 7.4.8 TRANSMISSION COEFFICIENTS The pressure pulse may be affected by rarefaction waves as it is developing and the peak may be
.clipped". In addition, the pressure may be attenuated as it propagates through the system as a result of area changes. In order to calculate each of these effects, the transmission coefficients at junctions is required. The transmission coefficients are calculated consistent with section 5.3 of the UM.
At points "f and *g" the transmission coefficients are calculated using Equation 5-8 from the UM; for simplification here the sonic velocity is assumed to be constant up and downstream of the junction:
2 Aincadent Amcident +
E Au Tr=
2 -Abcd If = 0.312
=> this fraction of the incident pulse Abcd + Aabf + AW continues past point "f" and the remainder of the incident pulse returns towards the initiation point.
2 'Aabf g :=
b
= 0.288
=> this fraction pulse that is incident upon AW + Aag + Aag point og" continues past point "g" and the remainder of the incident pulse returns towards the initiation point.
ttotal :=
fTg Ttota= 0.09 When the pressure pulse travels past point "g" only 10% of the pulse will continue on. 69% of the incident pulse was reflected as a negative pulse at point "f" and then 71% of the pulse that was incident upon point ig-was reflected back as a negative pulse. Tne net reflection effect is:
Pef = Pmc(-69%) + 31%-PmcJ -(-71%) = PmJ&9% - 31%-71%) = 91%
This reflection travels back to the initiation point. The pulse at the initiation point is 9% of its original value when this reflection arrives. For simplicity, the compounding effect of the 'f' node transmission coefficient on the reflected wave from node mg" is ignored.
The transmission coefficient evaluation needs to consider the control valve. The transmission coefficient at the control valve is calculated by assuming the valve acts like an orifice as the pressure pulse propagates through it. Equation 5-14 provides a simple relationship for an orifice flow coefficient in terms of its diameter ratio (1). This equation is used to back calculate an equivalent j3 ratio for the control valve knowing its coefficient and assuming Co=0.6.
D 0.5 Initial guess for the iteration below roo 0349 Page 13
For this D ratio and for the approximate waterhammer pressure already solved, the control valve will have a slight effect on the pressure pulse propagation by inspection of Figure 5-15. The reflection from this interaction will add approximately 10% to the incident pulse.
In general what this means is that 10% of the pulse magnitude is reflected in a positive sense back towards the initiation point. To account for this effect, the peak pressure pulse is conservatively increased by 10%.
7.4.9 DURATION The pressure pulse is reduced to approximately 10% of its peak value as a result of the reflections from the area changes at points 1f" and mg". As a result, the time that it takes the pressure pulse to travel to point "gt and back may be used to calculate the pressure pulse duration.
TD-g =
(Lde + Let + Lrg).2 C
TDeg = 78.8 ms Time for pulse to travel to and from point "g". Note that reflections from 'a' and 'b" are not credited.
The total duration is conservatively increased by adding the rise time.
TD := TDeg + TR TD = 118ms 7.4.10 PRESSURE CLIPPING The peak pressure is checked for "clipping' using Table 5-3.
L,:= L, + L.f + Lf0 L, = 168.4 ft TR-C =84ft 2
Xt,]o, = 0.09 This corresponds to the conditions in row two of the table referenced and no pressure clipping is expected.
1AP:= 1.1 APn0_ClippIng 1.1 is from the control valve AP = 222 psi Page 14
7.4.11 PRESSURE PULSE SHAPE The pulse shape is then characterized by four points.
lPsys = 19psi this is the steady state system pressure [Ref. FAI/97-60 Rev. 5]
Using an index, i=0,1,2,3 i := 0.. 3 time, :=
Omsl TD - TR TD 300 200 C.
C.
pressure, :=
Psys AP + Psys AP + Psys Psys This provides the following values, which are plotted below.
-0.039 12411 time =
s pressure =
1 psi
<0.118 19J Pressure Pulse 0
50 tune (Ms) trace I 100 150 Calculate the area underneath the curve to get the pressure impulse:
integral := TR-AP + AP. (TDeg - TR) integral = 1.207 x 105 kg ms impulse:= integral.Atcd impulse = 875.596 lbf *s Page 15
7.4.12 FLOW AREA ATTENUATION To simplify the analysis of the SW structures, the approach suggested here is to take the initiating pressure pulse and propagate the pulse through the system. For this example problem, the duration of the pulse is assumed to remain unchanged as it travels. In reality, the duration of the pulse is shortened as it approaches negative reflection sites. Maintaining the duration conservatively increases the impulse.
As the pressure pulse propagates through the system it will be atenuated/amplified by flow area changes. For this example, only the downstream propagation is considered. The pulse will be attenuated by the increase in area at f"f and "g". The transmission coefficients were previously calculated.
incident pulse transmitted pulse transmission pulse AP =222 psi tPf := Tf -P Pf = 69psi APf= 69 psi APg := Tg-APf APg = 20psi Downstream of point g" only the following pulse magnitude will remain:
= 20psi Page 16
FAI/03-07 Page C-1 of C-16 Rev. 1 Date: 03/06/03 APPENDIX C Point Beach CFC 2B EPRI TBR Waterhammer Calculations Using MathCad 2000
POINT BEACH CFC 2B VOIDED t
a, VOIDED
_d]
EL2 a
Figure 1 Open Loop Configuration Pressure & Temperature Note, pressures listed as "psi" are absolute (psia) or differential (psid) unless otherwise stated Patm := 14.7-psi T~oid := 217.1 -F Tpipe minia := 75-F Pipe Geometrv EL, := 33.2-ft EL,:= 72.0-ft L~b:= 36.8-ft Lk:= 139.4-ft Led =83.6-ft Lde 129.2-ft Lef := 4.8-ft Lfg:= 118.6-ft Llsik := 400.ft rDabf:
13.124-in Dbcd 7.98 1-in IDag = 22.624-in ODbcd := 8 625-in Pressure above reservoir and above heat sink (absolute)
Temperature in the void when the pumps restart (i.e. surface temperature of piping) [Ref. FAI/97-60 Rev. 2] (Assumed average T in void at 26 sec)
Temperature of the fluid and piping when the transient starts [Ref. FAI/97-60 Rev. 2]
Elevation of node "1" [Ref. FAI/97-60 Rev. 2]
Elevation of node "2" [Ref. FA1197-60 Rev. 2]
Length from node "a" to node b [Ref. FAI/97-60 Rev. 2]
Length from node "b" to node "c" [Ref. FAI/97-60 Rev. 2]
Length from node "c" to node "d" [Ref. FAI/97-60 Rev. 2]
Length from node ad" to node "e" [Ref. FAI/97-60 Rev. 2]
Length from node "e" to node 1f" [Ref. FAI/97-60 Rev. 2]
Length from node "fT to node g"9 [Ref. FAI/97-60 Rev. 2]
Length from node "g" to the ultimate heat sink [Ref. N/A -not used]
I.D. of piping along path from "a' to "b" to "f' [Ref. FAI/97-60 Rev. 2]
I.D. of piping along path from "b" to "c" to "d" [Ref. FAI/97-60 Rev. 2]
I.D. of piping along remaining path from 'a" to "g" [Ref. FA1197-60 Rev. 2]
O.D. of piping along path from "b" to 'c" to 'd" [Ref. FA1197-60 Rev. 2]
Page 2
FAI/03-07, Page C-3 of C-16 Rev. 2, Date: 03106/03 Flows Qabf := 800 gal Flow along path from "a" to "b" to "f" during steady state condition without min voiding [Ref. FAI/97-60 Rev. 2]
Qbcd 886-ga Flow along path from 'b" to "c" to 'd' during steady state condition without min voiding [Ref. FAI197-60 Rev. 21 Qg := 5200.- ga Flow along path from 'a' to "g during steady state condition without voiding min
[Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
FCU Characteristics Nube := 240 Number of tubes in cooler [Ref. FAI/97-60 Rev. 2]
IDwbe := 0.527-in Internal diameter of tubes [Ref. FAI/97-60 Rev. 2]
Itub, := 22-ft Length of tubes [Ref. FAI/97-60 Rev. 2]
Pump Characteristics q := 240.8-ft Pump shutoff head [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
Al := 0.2547--
1st order pump curve coefficient [Ref. Chuck Richardson Emails dated ft2 1/27/03 & 1/28/03]
2 A2
-0.5783 -
2nd order pump curve coefficient [Ref. Chuck Richardson Emails dated ft5 1/27/03 & 1/28/03]
Hpump(Qp):= A2-Qp2 + Al.Qp + H5 Pump curve equation Other Inputs KVIV := 165.447 Vwtr ~f
- = 0.04ft Vwtr 2phase = 6.ft3 lb Pwtr:= 62- --
TdeS :
95*F ft2 Rgs,:= 1717.-
sec -R Valve frictional flow coefficient for throttled globe valve [Ref. FAI/97-60 Rev. 2]
Volume of water that is left in the FCU when the pump restarts [Ref.
FAI/97-60 Rev. 2]
Volume of water that flows into the cooler after voiding has started and before the pumps restart. This volume of water is exposed to two phase flow conditions. [Ref. N/A -not used]
Water density Design temp of the system Gas constant Page 3
FAI/03-07, Page C-4 ot C-1 6 Rev. 2, Date: 03/06/03 Pump Flow Rate Equation Qtotn..a := Qg + Qbcd + Qabf Haorm := Hpump(Qtot..rmaj)
Qtotn0 ax = 6.886 x 103 gal Hno0M = 109 ft min The total system flow rate is solved at any pump operating point using:
-Al -.A12 A2-(H, - Hd)
Qpump(Hd).-2A 2*A2 Qpump(Honn) = 6.886x I gal m30 00
= 100
-100 I
0 2000 4000 6000 8000 I 104 GPM Pump Curve 00 Operating Point PUMP CURVE & OPERATING POINT Figure 2 SW Pump Curve Page 4
FAII03-07, Page C-5 of C-16 Rev. 2, Date: 03/06/03 7.4.1 Initial Velocity & FLOW COEFFICIENT PREDICTION The water at the front of the void (point Ed") is assumed to not move or simplification of this problem. More detailed hydraulic modeling may be performed to determine the reverse or forward flow at point "d". In many cases this flow is less than 10% of the incoming flow.
After combining parallel paths the system is then simplified to:
Figure 3 Simplified Open Loop Model In terms of the initial flow diagram (Figure 1), the flow area for each path is calculated:
A
-bf
- =
E ])abf 4
Aabf =0.939 fCt Atcd := TID 2d 4
Abcd = 0.347 ft-A Sg:t IDa2 Aag = 2.792 f The velocity for each path is calculated:
Qabf Vabf A=a-Aabf Qbcd VIDd
- Abcd Qag ag=
A.g Vag - 4.2-f Vabf = 1.9-S ft Vb~d =57 S
Calculate equivalent velocity for all other loads:
V~q-Qabf + Qag AVbf + Aag Veq = 3.583 It S
Page 5
FAI/03-07, Page C-6 of C-1 6 Rev. 2, Date: 03/06/03 The flow coefficient for each path is calculated.
The flow resistance from point 'a' to point Vb and from point "f" to point "g" are assumed to have a negligible effect on the flow split to the different paths. In an actual plant system, the engineer may choose to use values from a previously qualified system hydraulic model to determine a more accurate initial velocity.
V2 hf = K--
=>
2-g 2-g hf V2 2-g-H...
K Vbf a= 2 Vabf 2-g-H...n Kbcd :=
X Vbcd 2-g-Hnor Kag:=
2 Vag Kab, = 1.941 x 103 Kbrd = 216 Kag = 406 An equivalent flow coefficient for the "other loads" path (Figure 1) is calculated from:
[
A bf Kohe Aabf A29 N,
)R1f)K Kether = 3 5 Aethr = Aabf IDother = IDabf An equivalent flow coefficient from all other loads is calculated from:
2-g H1nor Kother =
Veq Aother = Aabf + Aag Kother = 544.321 oh 4 -Aother 5
)~other :=
IDother = 2.18 ft The flow coefficient for the path to the void is calculated by subtracting the flow coefficient downstream of the void along this path. To simplify this sample problem only the valve resistance downstream of the void is considered:
Koid := Kbcd - Kviv Kvoid = 51 The pressure in the void is assumed to correspond to the saturation pressure for the void temperature.
Pvjid = 16.3.psi Absolute pressure based on saturation pressure of Tvoid shown above.
Comes from TREMOLO output. [Ref. FAI/97-60, Rev. 2]
Page 6
The pump total developed head (TDH) is written by using Bernoulli's equation:
Helm + EL, + TDH = H,,,d + EL, + Hf where the following terms are defined in terms of feet H20 HItm
= atmospheric pressure head EL,
= elevation of node "1" TDH = total developed head from pump EL,
= elevation of node "2" Hf
= frictional losses from point "1" to "2" The frictional losses are written using Darcy's formula with an appropriate units conversion factor:
Hf = 0.00259*K 10,, Q-where ID4 KOSS= loss coefficient o
= flow rate in gpm ID
= pipe diameter in inches Two equations for the total developed head (TDH) by the pump are written with a corresponding flow balance and initial guesses for the simultaneous solution of these equations:
Qvoid := 1 QOth: =.5 TDH := 300 Given Qother -
TDH = 0.00259-Koth,-
4 (in) frictional losses along 'other" path equal the total developed head Qvoid Patm Pvoid )ft-1 TDH = 0.0025K 4
P
-g L, -
Bernoulli's along the "void" path Qotber + Qvoid = Qpump(tDH ft) C-g nuni pump curve The solution to the simultaneous equations is solved and defined as "Results".
Results := Find(TDHvQother1Qvoid)
TDH:= Resultsn-ft TDH = 100.189 ft Qother := Results-gal Qthr = 5.767 x 103 gal mmn min Qvod := Results2. gal T min The initial velocity is then:
Qvoid = 1.331 x I gal min The total resistance for this path is:
Votiaid:=
V-ot
= 8.5-Kvold = 51 Abcd Check: is the velocity within the RBM bounds?
Vinitial
< 20 ft/sec
===> yes, velocity is within bounds of RBM runs Page 7
7.4.2 VOID & WATER COLUMN LENGTHS The volume of piping that is voided is calculated:
Vpipe-voided := Lcd'-
IDbcd 4
The void of the fan cooler unit is calculated:
Vfu := Nube'Lbe IDrube 4
The equivalent void length is then:
Vpipe voided = 29 ft3 Vfc" = 8 ft3 Vpipe voided + Vfcu Lao:=
Atcd Lao = 107 ft The initial water column length is assumed to be the distance from point "a" to point "c". The discussion that follows explains why point 'a' was chosen.
Ignoring the FCU, the flow area changes from the closure point to node 'a" are the same as the area changes from the closure point to node "g" on the return side. The transmission coefficients calculated for the return side demonstrate that less than 10% of the pressure pulse propagates to the header. Because of the similar flow area changes, less than 10% of any pressure would propagate into the supply header upstream of point "a". In general, this indicates that the header acts like a large pressurized reservoir during the void closure process and water in the supply header does not add to the inertia of the decelerating water column.
Note: if desired, a plant could select a length all the way back to the pumps. However, this is considered excessively conservative.
The length if the accelerating water column is then:
Lwo := Lab + Lk Lwo = 176.2 ft Check: are the lengths within the bounds of the RBM runs?
Lao < 100 ft Lwo < 400 ft
===>> yes lengths are within bounds of RBM runs Page 8
FAI/03-07, Page C-9 of C-16 Rev. 2, Date: 03/06/03 7.4.3 GAS RELEASE AND MASS OF AIR CONCENTRATED IN VOID The mass of air concentrated in the void during the void phase of the transient is calculated by assuming that the water that has experienced boiling and subsequent condensation releases its air as described in Section 5 of the User's Manual.
For this problem, the tube volume only will be credited, assuming a draining of the FCU in which the headers do not remain full. This mass of water will release 50% of its non-condensable gas.
Vr,, = 7.998 ft0 or Vf,,, = 226 liter from 7.4.2 This represents the mass of water in the tubes which will lose 50% of its non-condensable gas.
The concentration of gas is obtained from Figure 5-3.
Tdes = 95 F Tdes - 32F
= 35 degO 1.81F CONar:= 18.5 mg liter From Figure 5-3 marn := 0.5 CON,,-Vfcu mair = 2095 mg Check: is the mass of air within bounds of UM?
for void closure in 8' piping there should be at least 900 mg of air per Table 5.2
===> yes, mass if air is within RBM run bounds.
Page 9
7.4.4 Cushioned VELOCITY The graphs presented in Appendix A for the velocity ratios are solutions to the simultaneous differential equations that capture the acceleration of the advancing column and pressurization of the void.
In order to determine the cushioned velocity the following terms that are needed are repeated:
Vatial -= 8.538 -
S Kyoid = 51 Lao = 106.622 ft Lwo = 176.2 ft nar =2.095 x 103 mg T,,,d = 217.1 F Check: is the temperature within the bounds of the RBM?
Tvoid > 200 F===> yes, the temperature is within the RBM run bounds 7.4.4.1 Air Cushioning If only credit for air cushioning is considered then Figure A-1 3 from Appendix A is selected. This figure corresponds to 10" piping while the sample problem has 8" piping. 10' piping bounds the 8" piping since the inertia modeled in the 10. piping runs is greater than that in the 8" piping runs and the velocity has reached a steady state until the final void closure occurs. This is apparent by comparison of the 4" and 10" RBM run results for the same gas mass; the velocity is reduced more in the smaller pipe case. If the pipe size at a given plant is not shown then the Velocity Ratio chart for the next larger size pipe will always be bounding.
Figure A-13 corresponds to an initial velocity of 10 fps. The initial velocity calculated in this sample problem is less. The higher velocity chart is selected because the higher momentum associated with the higher velocity bounds the lower velocity. If the Initial velocity at a plant Is not shown then the Velocity Ratio chart for the next larger velocity will always be bounding.
For a K of 51 as calculated in the sample problem, from Figure A-1 0 the ratio of the second to initial velocity is:
lvcusfon =88%
only air cushion credited lVLnill l
Therefore, the final closure velocity will be reduced by 12% just considering air in this sample problem. Pressure "clipping" is not included here and is calculated later.
Page 10
7.4.4.2 Air and Steam Cushioning The velocity that results by considering steam cushioning is found using Figure A-40 from Appendix A. Note that the condensing surface temperature was verified being within the bounds of the RBM run limitations so steam condensation cushioning may be credited. The steam and air cushioning result in a ratio of cushioned to initial velocity of:
Vcusuon 83%
lVinittall air and steam cushioning The cushioned velocity is then:
Vcushon := 0.83'VitilI V.,td.. = 7 S 7.4.5 SONIC VELOCITY The sonic velocity is calculated from Equation 5-1 and 5-2 in the main body of the User's Manual.
Pvoid = 16.3 ps where B=bulk modulus of water E=Young's modulus for steel OD=outside diameter of pipe t=wall thickness B := 319000psi E:= 28.10 6psi C
B C *-
Pwtr B ODbCd E. (ODbcd - ID J)
C = 4274 f-s Page 1 1
7.4.6 PEAK PRESSURE PULSE WITH NO "CLIPPING" The peak waterhammer pressure is calculated using the Joukowski equation with a coefficient of 1/2 for a water on water closure:
I APno-cLipping := 2 Pwtr(C Vcustuon APno-cipping = 203 psi 7.4.7 RISE TIME The rise time is calculated by using equation 5-4 from the UM.
ms := 0.OOls TR := 0.5sec{. vc 0 TR = 39ms sec Page 12
7.4.8 TRANSMISSION COEFFICIENTS The pressure pulse may be affected by rarefaction waves as it is developing and the peak may be "clipped". In addition, the pressure may be attenuated as it propagates through the system as a result of area changes. In order to calculate each of these effects, the transmission coefficients at junctions is required. The transmission coefficients are calculated consistent with section 5.3 of the UM.
At points "fT and "g" the transmission coefficients are calculated using Equation 5-8 from the UM; for simplification here the sonic velocity is assumed to be constant up and downstream of the junction:
2 -Amacident Aincident + E Aj J
X =
2 Ab_ d Tf = 0.312
=> this fraction of the incident pulse Abcd + A.bf + AWir continues past point "fT and the remainder of the incident pulse returns towards the initiation point.
2-Aabf g = 0.288
=> this fraction pulse that is incident upon Aabf + Ang + Ang point "g" continues past point "g" and the remainder of the incident pulse returns towards the initiation point.
Tiotal3 =Ef-rg Ttotal = 0.09 When the pressure pulse travels past point "g" only 10% of the pulse will continue on. 69% of the incident pulse was reflected as a negative pulse at point "fi and then 71% of the pulse that was incident upon point "g" was reflected back as a negative pulse. the net reflection effect is:
1Pzf =Prc (-69%) + (31%P,,,n)
.(-71%) =Pc(-69% - 47%-71%) =91%
This reflection travels back to the initiation point. The pulse at the initiation point is 9% of its original value when this reflection arrives. For simplicity, the compounding effect of the "f" node transmission coefficient on the reflected wave from node 'g" is ignored.
The transmission coefficient evaluation needs to consider the control valve. The transmission coefficient at the control valve is calculated by assuming the valve acts like an orifice as the pressure pulse propagates through it. Equation 5-14 provides a simple relationship for an orifice flow coefficient in terms of its diameter ratio (f3). This equation is used to back calculate an equivalent 3 ratio for the control valve knowing its coefficient and assuming Co=0.6.
0.5 Initial guess for the iteration below r
j3=roo
-1I
- Kvlv
=0.347 0.6-02 Page 13
For this i ratio and for the approximate waterhammer pressure already solved, the control valve will have a slight effect on the pressure pulse propagation by inspection of Figure 5-15. The reflection from this interaction will add approximately 10% to the incident pulse.
In general what this means is that 10% of the pulse magnitude is reflected in a positive sense back towards the initiation point. To account for this effect, the peak pressure pulse is conservatively increased by 10%.
7.4.9 DURATION The pressure pulse is reduced to approximately 10% of its peak value as a result of the reflections from the area changes at points 'f" and "g". As a result, the time that it takes the pressure pulse to travel to point 'g* and back may be used to calculate the pressure pulse duration.
(Lde + Lef + Lfg)-2 C
TMeg = 118.2ms Time for pulse to travel to and from point "g". Note that reflections from 'a" and Vb" are not credited.
The total duration is conservatively increased by adding the rise time.
TD:= TDeg + TR TD = 157 ms 7.4.10 PRESSURE CLIPPING The peak pressure is checked for 'clippingn using Table 5-3.
Le:= Lde + Lef + Lfg Le = 252.6 ft TR 2 = 84 ft 2
Utot
=.09 This corresponds to the conditions in row two of the table referenced and no pressure clipping is expected.
IAP := I.1-APno chppingl 1.1 is from the control valve AP = 223 psi Page 14
7.4.11 PRESSURE PULSE SHAPE The pulse shape is then characterized by four points.
Psys 9PS-I this is the steady state system pressure [Ref. FAI/97-60 Rev. 2]
Using an index, i=0,1,2,3 i := O.. 3 time, :=
I.la2 pressure :=
Psys AP + Psys AP +PsysI PsysI This provides the following values, which are plotted below.r O 1
( 19 i 0.039me u
242 i
time = 018s pressure = 22psi
%0..157/
19 Pressure Pulse 300 200 time (Ms) trace 1 Calculate the area underneath the curve to get the pressure impulse:
integral := 'R-AP + AP- (TDeg - TR) integral = 1.817 x 105 kg ms impulse:= integral-Abcd impulse = 1.318x 10 3bf-s Page 15
7.4.12 FLOW AREA ATTENUATION To simplify the analysis of the SW structures, the approach suggested here is to take the initiating pressure pulse and propagate the pulse through the system. For this example problem, the duration of the pulse is assumed to remain unchanged as it travels. In reality, the duration of the pulse is shortened as it approaches negative reflection sites. Maintaining the duration conservatively increases the impulse.
As the pressure pulse propagates through the system it will be atenuated/amplified by flow area changes. For this example, only the downstream propagation is considered. The pulse will be attenuated by the increase in area at f" and "gi.
The transmission coefficients were previously calculated.
incident pulse transmitted pulse transmission pulse AP =223 psi APf := Tf-AP APf = 70psi APf 70 psi APg:= Tg.tPf APg = 20psi Downstream of point 'g" only the following pulse magnitude will remain:
APg =20psi Page 16
FAI/03-07 Page D-1 of D-16 Rev. I Date: 03/06/03 APPENDIX D Point Beach CFC 2D EPRI TBR Waterhammer Calculations Using MathCad 2000
FAI/03-07, Page D-2 of D-1 6 Rev. 1, Date: 03/06103 POINT BEACH CFC 2D EL 19~~
Figure 1 Open Loop Configuration Pressure & Temperature Note, pressures listed as "psi' are absolute (psia) or differential (psid) unless otherwise stated P,,,
- = 14.7-psi T,,d := 204.4-F Tpipe-iniial := 75 F Pipe Geometry EL, := 33.2-ft EL,7:= 30.3-ft Lb := 36.8-ft Lb,:= 161.8-ft Lcd := 465-ft LdC:= 161.2-ft Lef:= 6.4*ft lg := 86.8-ft
- LLg_,
- = 400-ft IDabf := 13.124-in IDbcd := 7.981-in rDag:= 22.624-in ODWd := 8.625-in Pressure above reservoir and above heat sink (absolute)
Temperature in the void when the pumps restart (i.e. surface temperature of piping) [Ref. FAI/97-60 Rev. 2] (Assumed average T in void at 26 sec)
Temperature of the fluid and piping when the transient starts [Ref. FAI/97-60 Rev. 2]
Elevation of node "1" [Ref. FAI/97-60 Rev. 2]
Elevation of node "2" [Ref. FA1197-60 Rev. 2]
Length from node "a" to node "b" [Ref. FAI/97-60 Rev. 2]
Length from node 'b" to node "c" [Ref. FAI/97-60 Rev. 2]
Length from node "c" to node "d" [Ref. FAI/97-60 Rev. 2]
Length from node "d" to node 'e" [Ref. FAI/97-60 Rev. 2]
Length from node "e" to node If" [Ref. FAI/97-60 Rev. 2]
Length from node "I" to node 'g" [Ref. FAI/97-60 Rev. 2]
Length from node "g" to the ultimate heat sink [Ref. N/A -not used]
I.D. of piping along path from "a" to "b" to I " [Ref. FAI/97-60 Rev. 2]
I.D. of piping along path from "b" to "c" to "d" [Ref. FAI/97-60 Rev. 2]
I.D. of piping along remaining path from "a" to "g" [Ref. FAI/97-60 Rev. 2]
O.D. of piping along path from "b" to "c" to "d" [Ref. FAI/97-60 Rev. 2]
Page 2
FAI103-07, Page D-3 of D-16 Rev. 1, Date: 03/06/03 Flows Qbf := 800 gal min Qbcd := 941m 8.
min Qag := 5100- ga mnu Flow along path from "a' to "b* to Tf" during steady state condition without voiding [Ref. FAI/97-60 Rev. 2]
Flow along path from "b" to "c' to "d' during steady state condition without voiding [Ref. FAI/97-60 Rev. 2]
Flow along path from VaW to "g during steady state condition without voiding
[Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
FCU Characteristics Nbr := 240 ED1 bbe = 0.527-in
.,b,,b := 22-ft Number of tubes in cooler [Ref. FAI/97-60 Rev. 2]
Internal diameter of tubes [Ref. FAI/97-60 Rev. 2]
Length of tubes [Ref. FAI/97-60 Rev. 2]
Pump Characteristics Ei := 240.8-ft Al := 0.2547--
ft2 2
A2 := -0.5783--
ft5 Pump shutoff head [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
1st order pump curve coefficient [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
2nd order pump curve coefficient [Ref. Chuck Richardson Emails dated 1/27/03 & 1/28/03]
Hpump(Qp):= A2-Qp2 + Al.Qp + H, Pump curve equation Other Inputs KV1. := 139.326 Vwtr
,:= 0.0-ft3 Vwtr 2phase = 6ft3 p,
1tr:= 62.- b ft3 Tdes:= 95-F ft2 Ras.= 1717.-
sec2*R Valve frictional flow coefficient for throttled globe valve [Ref. FAI/97-60 Rev. 2]
Volume of water that is left in the FCU when the pump restarts [Ref.
FAI/97-60 Rev. 2]
Volume of water that flows into the cooler after voiding has started and before the pumps restart. This volume of water is exposed to two phase flow conditions. [Ref. N/A -not used]
Water density Design temp of the system Gas constant Page 3
FAI/03-07, Page D-4 of D-16 Rev. 1, Date. 03/06/03 Pump Flow Rate Equation QtotnonnS = Qg + Qcd + Qbf HAno
- = Hpump(Qtotnorma)
QtotnS,,m
6.841 x 103 gal Horm
0ft min The total system flow rate is solved at any pump operating point using:
-Al -
Al2 - 4.A2.(H5 - Hd)
Qpump(Hd):=2A 2.A2 Qpump(H.Onn) = 6.841 x 10 gal min 0
2000 4000 6000 8000 1-104 GPM Pump Curve 00 Operating Point PUMP CURVE & OPERATING POINT Figure 2 SW Pump Curve Page 4
7.4.1 Initial Velocity & FLOW COEFFICIENT PREDICTION The water at the front of the void (point "d") is assumed to not move or simplification of this problem. More detailed hydraulic modeling may be performed to determine the reverse or forward flow at point d". In many cases this flow is less than 10% of the incoming flow.
After combining parallel paths the system is then simplified to:
Figure 3 Simplified Open Loop Model In terms of the initial flow diagram (Figure 1), the flow area for each path is calculated:
Aabf :=
lDabf 4
Aabr = 0.939 ft-AbCd := -.IDkd 4
Abcd = 0.347 ft2 A4g = 7:.IDag2 4
Aag = 2.792 ft2 The velocity for each path is calculated:
Vab
=-
Vb:d =
Abcd Vag :=-
Aag Vag = 4.1-Vabf = 1.9-S Vcd = 6-ft S
Calculate equivalent velocity for all other loads:
V Qabf + Qg Aabr + Aag Veq = 3.523-S Page 5
The flow coefficient for each path is calculated.
The flow resistance from point 'a" to point "b" and from point If" to point "g" are assumed to have a negligible effect on the flow split to the different paths. In an actualplant system, the engineermay choose to use values from a previously qualified system hydraulic model to determine a more accurate initial velocity.
V2 hf =K--
=>
2-g K = LiVfV2 Kabf 2=
2gH.H VabW 2 g*gH 00rr Kbcd =
X Vbcd Kag :=
g Vag2 Kabf = 1.972x 103 Kbcd = 195 Kag = 429 An equivalent flow coefficient for the 'other loads' path (Figure 1) is calculated from:
K other := r A b I Aabf Aag X
_91 Kother = 36 Aether := Aaof IDother := IDabf An equivalent flow coefficient from all other loads is calculated from:
Kother 2=2g*H..
V. 2 eq Aother := Aabf + Aag Kother = 571.989 IDother :=(4Aher IDother = 2.18 ft The flow coefficient for the path to the void is calculated by subtracting the flow coefficient downstream of the void along this path. To simplify this sample problem only the valve resistance downstream of the void is considered:
Kvold := Kcd - Kvlv Kvold = 56 The pressure in the void is assumed to correspond to the saturation pressure for the void temperature.
PVod := 12.7-psi Absolute pressure based on saturation pressure of Tvoid shown above.
Comes from TREMOLO output. [Ref. FAI/97-60, Rev. 2]
Page 6
The pump total developed head (TDH) is written by using Bernoulli's equation:
Hltm + EL, + TDH = Hvold + EL, + Hf where the following terms are defined in terms of feet H20 Haitm
= atmospheric pressure head EL,
= elevation of node "1" TDH = total developed head from pump EL,
= elevation of node "2" Hf
= frictional losses from point "1 to "2" The frictional losses are written using Darcy's formula with an appropriate units conversion factor.
Q2 Hf = 0.00259-Kjo,,- Q where ID4 Kloss = loss coefficient 0
= flow rate in gpm ID
= pipe diameter in inches Two equations for the total developed head (TDH) by the pump are written with a corresponding flow balance and initial guesses for the simultaneous solution of these equations:
Qv.id = *1 Qthe: = *5 TDH := 300 Given TDH = 0.00259-Kotr-4 in frictional losses along 'other" path equal the total developed head Q1d2 Painn Pvoid tI TDH = 0.00259*Kvld-K
+ (EL 2 -ELI_-+
- t
( IDW )
Pwtr g Pwtr g I.in)
Bernoulli's along the "void" path Qother + Qvoid = Qpump(TDH-lft)- rrun pump curve The solution to the simultaneous equations is solved and defined as 'Results".
Results := Find(TDHQothe,,Qv oid)
TDH := Resultsn-ft Qother := Results, g TDH = 95.792ft Qother = 5.501 x 10 gal min Qvo.d := Results-ga Qvoid = 1.706x 13 Ea min min The initial velocity is then:
The total resistance for this path is:
V
- =-
Vinnial = 10.9-K 1old = 56 Abcd S
Check: is the velocity within the RBM bounds?
Vini,,a,
< 20 ft/sec
===> yes, velocity is within bounds of RBM runs Page 7
FAI/03-07, Page D-8 of D-16 Rev. 1, Date: 03/06/03 7.4.2 VOID & WATER COLUMN LENGTHS The volume of piping that is voided is calculated:
Vpipe votded = Lcd- [IDbcd Vpipe-voided = 1611 4
The void of the fan cooler unit is calculated:
Vf. := Nmbe Ltube IDtube Vfr, = 8 ft 4
The equivalent void length is then:
Vpipe-voided + Vfcu Lao :=
Lao = 70ft Abcd The initial water column length is assumed to be the distance from point 'a' to point "c". The discussion that follows explains why point 'a" was chosen.
Ignoring the FCU, the flow area changes from the closure point to node 'a" are the same as the area changes from the closure point to node Ng" on the return side. The transmission coefficients calculated for the return side demonstrate that less than 10% of the pressure pulse propagates to the header. Because of the similar flow area changes, less than 10% of any pressure would propagate into the supply header upstream of point 'a". In general, this indicates that the header acts like a large pressurized reservoir during the void closure process and water in the supply header does not add to the inertia of the decelerating water column.
Note: if desired, a plant could select a length all the way back to the pumps. However, this is considered excessively conservative.
The length if the accelerating water column is then:
Lwo := Lab + Lt, Lwo = 198.6ft Check: are the lengths within the bounds of the RBM runs?
Lao < 100 ft Lwo < 400 ft
===>> yes lengths are within bounds of RBM runs Page 8
FAI/03-07, Page D-9 of D-1 6 Rev. 1, Date: 03/06103 7.4.3 GAS RELEASE AND MASS OF AIR CONCENTRATED IN VOID The mass of air concentrated in the void during the void phase of the transient is calculated by assuming that the water that has experienced boiling and subsequent condensation releases its air as described in Section 5 of the User's Manual.
For this problem, the tube volume only will be credited, assuming a draining of the FCU in which the headers do not remain full. This mass of water will release 50% of its non-condensable gas.
VfrU = 7.998 ft3 or Vf,, = 226 liter from 7.4.2 This represents the mass of water in the tubes which will lose 50% of its non-condensable gas.
The concentration of gas is obtained from Figure 5-3.
Tdes = 95 F Tdes - 32F =35 degC 1.8F CON,,,:= 18.5 mg liter From Figure 5-3 mr
-:= 0.5-CONr,-Vfg, Mar = 2095 mg Check: is the mass of air within bounds of UM?
for void closure in 8" piping there should be at least 900 mg of air per Table 5.2
===> yes, mass if air is within RBM run bounds.
Page 9
FA1/3-07, Page D-10 of D-16 Rev. 1, Date: 03/06/03 7.4.4 Cushioned VELOCITY The graphs presented in Appendix A for the velocity ratios are solutions to the simultaneous differential equations that capture the acceleration of the advancing column and pressurization of the void.
In order to determine the cushioned velocity the following terms that are needed are repeated:
Vim,,:w = 10.94 1-S Kvoid = 56 Lao = 69.522 ft Lwo = 198.6ft Mar =2.095 X 103 mg Tvoid = 204.4F Check: is the temperature within the bounds of the RBM?
Tvoid > 200 F===> yes, the temperature is within the RBM run bounds 7.4.4.1 Air Cushioning If only credit for air cushioning is considered then Figure A-13 from Appendix A is selected. This figure corresponds to 1o" piping while the sample problem has 8" piping. 10" piping bounds the 8" piping since the inertia modeled in the 10" piping runs is greater than that in the 8" piping runs and the velocity has reached a steady state until the final void closure occurs. This is apparent by comparison of the 4" and 10" 'RBM run results for the same gas mass; the velocity is reduced more in the smaller pipe case. If the pipe size at a given plant is not shown then the Velocity Ratio chart for the next larger size pipe will always be bounding Figure A-1 3 corresponds to an initial velocity of 10 fps. The initial velocity calculated in this sample problem is less. The higher velocity chart is selected because the higher momentum associated with the higher velocity bounds the lower velocity. l the initial velocity at a plant is not shown then the Velocity Ratio chart for the next larger velocity will always be bounding.
For a K of 56 as calculated in the sample problem, from Figure A-13 the ratio of the second to initial velocity is:
l
=i7%
only air cushion credited Therefore, the final closure velocity will be reduced by 13% just considering air in this sample problem. Pressure "clipping' is not included here and is calculated later.
Page 10
7.4.4.2 Air and Steam Cushioning The velocity that results by considering steam cushioning is found using Figure A-40 from Appendix A. Note that the condensing surface temperature was verified being within the bounds of the RBM run limitations so steam condensation cushioning may be credited. The steam and air cushioning result in a ratio of cushioned to initial velocity of:
l Vcuson l
lVnmltal l
air and steam cushioning The cushioned velocity is then:
VchO
= 9.1 s
7.4.5 SONIC VELOCITY The sonic velocity is calculated from Equation 5-1 and 5-2 in the main body of the User's Manual.
Pvod = 12.7 psi where B=bulk modulus of water E=Young's modulus for steel OD=outside diameter of pipe t=wall thickness B := 319000psi E:= 28-10 psi B
C *-
Pwtr
.- 1+
B.ODbd E.(bcdbcd)
[
2 C = 4274 f-s Page 1 1
7.4.6 PEAK PRESSURE PULSE WITH NO "CLIPPING" The peak waterhammer pressure is calculated using the Joukowski equation with a coefficient of 1/2 for a water on water closure:
Apnocilpping I2Pwtr C Vcusbun 2
APncOciippjns = 260 psi 7.4.7 RISE TIME The rise time is calculated by using equation 5-4 from the UM.
ms := O.OOls TR := 0.5sec.
i
)
TR = 28 ms secf 3 Page 12
7.4.8 TRANSMISSION COEFFICIENTS The pressure pulse may be affected by rarefaction waves as it is developing and the peak may be "clipped". In addition, the pressure may be attenuated as it propagates through the system as a result of area changes. In order to calculate each of these effects, the transmission coefficients at junctions is required. The transmission coefficients are calculated consistent with section 5.3 of the UM.
At points "f" and 'g" the transmission coefficients are calculated using Equation 5-8 from the UM; for simplification here the sonic velocity is assumed to be constant up and downstream of the junction:
2'Aincident Aincident +
E Aj 2fAfd
= 0.312
=> this fraction of the incident pulse Abcd + Aabf + if continues past point f" and the remainder of the incident pulse returns towards the initiation point.
2-Aabf fig = 0.288
=> this fraction pulse that is incident upon Aabf a + +Ag point UgS continues past point "`" and the remainder of the incident pulse returns towards the initiation point.
'rtotl =
f-rg
=rwal 0.09 When the pressure pulse travels past point gu only 10% of the pulse will continue on. 69% of the incident pulse was reflected as a negative pulse at point `ff and then 71% of the pulse that was incident upon point g"9 was reflected back as a negative pulse. the net reflection effect is:
lref =Pic&69%) +
(3-%6P 1
(71%) =Px~(-9% -47%-71%) = 91%
This reflection travels back to the initiation point. The pulse at the initiation point is 9% of its original value when this reflection arrives. For simplicity, the compounding effect of the "f" node transmission coefficient on the reflected wave from node "g" is ignored.
The transmission coefficient evaluation needs to consider the control valve. The transmission coefficient at the control valve is calculated by assuming the valve acts like an orifice as the pressure pulse propagates through it. Equation 5-14 provides a simple relationship for an orifice flow coefficient in terms of its diameter ratio (1). This equation is used to back calculate an equivalent 13 ratio for the control valve knowing its coefficient and assuming Co=0.6.
0.5 Initial guess for the iteration below 2
1 1:=roof Ij
_ K~Iv,1 10.361 Page 13
FA1103-07, Page D-14 of D-16 Rev. 1, Date: 03106/03 For this 1 ratio and for the approximate waterhammer pressure already solved, the control valve will have a slight effect on the pressure pulse propagation by inspection of Figure 5-15. The reflection from this interaction will add approximately 10% to the incident pulse.
In general what this means is that 10% of the pulse magnitude is reflected in a positive sense back towards the initiation point. To account for this effect, the peak pressure pulse is conservatively increased by 10%.
7.4.9 DURATION The pressure pulse is reduced to approximately 10% of its peak value as a result of the reflections from the area changes at points "'f and "9". As a result, the time that it takes the pressure pulse to travel to point 'go and back may be used to calculate the pressure pulse duration.
-Meg :=
(Lde + Lef + Lrg) 2 C
TDeg= 119.1 Ms Time for pulse to travel to and from point "gm. Note that reflections from 'a" and "b" are not credited.
The total duration is conservatively increased by adding the rise time.
TD:= TD.9 + TR TD = 147 ms 7.4.10 PRESSURE CLIPPING The peak pressure is checked for 'clipping' using Table 5-3.
L,:= Ld, + Lef + Lfg Le = 254.4 ft TR-C =61ft 2
Ttotaw = 0.09 This corresponds to the conditions in row two of the table referenced and no pressure clipping is expected.
1AP := l l APno-cippingI 1.1 is from the control valve AP = 286 psi Page 14
7.4.11 PRESSURE PULSE SHAPE The pulse shape is then characterized by four points.
lPsys := 19psIl this is the steady state system pressure [Ref. FAI/97-60 Rev. 2]
Using an index, i=0,1,2,3 i = O.. 3 time, :=
Oms TD - TR TD pressure :=
Psy[
AP + Psys AP + Psys Psys This provides the following values, which are plotted below.r 0 I 9 "
t 0.028)
[305) time = 0.119 s pressure = 305 psi 0.147 19 )
Pressure Pulse 400 300 a"
,; 200 10 150 time (Mns) trace I Calculate the area underneath the curve to get the pressure impulse:
integral := TR-AP + AP-(TDeg - TR) integral = 2.345 x 105 kg ms impulse := integral Abd impulse = 1.701 x 103 Ibf -s Page 15
FAI/03-07, Page D-16 of D-16 Rev. 1, Date: 03/06/03 7.4.12 FLOW AREA ATTENUATION To simplify the analysis of the SW structures, the approach suggested here is to take the initiating pressure pulse and propagate the pulse through the system. For this example problem, the duration of the pulse is assumed to remain unchanged as it travels. In reality, the duration of the pulse is shortened as it approaches negative reflection sites. Maintaining the duration conservatively increases the impulse.
As the pressure pulse propagates through the system it will be atenuated/amplified by flow area changes. For this example, only the downstream propagation is considered. The pulse will be attenuated by the increase in area at "f" and "g'. The transmission coefficients were previously calculated.
incident pulse transmitted pulse transmission pulse AP= 286 psi APf := r-AP APf = 89 psi APf 89 psi APg:= tgApf APg = 26 psi Downstream of point ugf, only the following pulse magnitude will remain:
APg = 26 psi Page 16
FAI/03-07 Page El of E2 Rev. 1 Date: 03/06/03 APPENDIX E SERVICE WATER PUMP CURVE CALCULATIONS USING MICROSOFT EXCEL 97
FAI/03-07 Page E2 of E2 Rev. 1 Date: 03/06103 Below is the pump-head curve for the Unit 1 & 2 Service Water Pump Curves for the Point Beach Waterhammer EPRI TBR Analysis. This curve was generated from the Unit 1 & 2 WATER data sent to FAI from Chuck Richardson (WEPCo) on January 27, 2002 (Unit 1) and January 28, 2002 (Unit 2). Therefore, from the data below using the EXCEL, the pump curve coefficients can be calculated using a polynomonial (A2'Q 2 + AI'Q + H) curve-fit features within EXCEL. As shown below, the coefficients are A2 = -0.5783 and A, = 0.2547.
Flow (gpm) 1814.05 2824.87 3823.52 4500 4739.93 5795.15 6735.92 7532.61 Head (ft) 237.95 214.81 196.22 182.09 177.08 155.69 125.45 71.99 Flow (cu fW/s) 4.042 6.294 8.519 10.027 10.561 12.913 15.009 16.784 Point Beach SW Pump Curve 250 200
- 150
-Pump curve data I10 Poly. (Pump curve data) 50 0
0.000 5.000 10.000 15.000 20.000 Flow (cu !t/ sec)
FAI/03-07 Page F-1 of F-5 Rev. 1 Date: 03/06/03 APPENDIX F EPRI TBR MAX FORCE/IMPULSE CALCULATIONS FOR POINT BEACH CFC 1A USING MICROSOFT EXCEL 97
FAI/03-07 Page F-2 of F-5 Rev. 1 Date: 03/06/03 POINT BEACH CFC1 A INPUTS:
Wave Speed (C) =
Rise Time =
Duration =
Peak Pressure =
Area (8-line)
Area (6' line)
Area (2.5" line)
Trans. Coeff. (elbows) =
Trans. Coelf. (8" x 6") =
Trans. Coeff. (6" x 2.5k) =
4274 0.048 0.127 191 50 28 89 4.79 1
0.7759 0 9234 ft/s sec sec psia sq In sq in sq in Rate =
3979.166667 psVsec Direction - Downstream Towards Throttle Valve Flow Element Pipe Area (sq In) Length (ft) 35 50 3.16 36 50 18 37 50 3
38 50 7
39 50 12 40 50 22.5 41 50 2
42 50 2
43 S0 13 P1 -time 0
0 0
0 0
0 0
0 0
P2 -time P3 -time P4 - time P5 -time 0.00074 0 04874 0 07974 0.12774 0.00495 0.05295 0 08395 0.13195 0 00565 0.05365 0.08465 0.13265 0 00729 0 05529 0 08629 0.13429 0.01010 0.05810 0.08910 0.13710 0 01536 0.06336 0.09436 0.14236 0 01583 0.06383 0.09483 0.14283 0 01630 0.06430 0.09530 0.14330 0.01934 0 06734 0.09834 0.14634 Direction - Upstream Towards Fan Cooler Flow Element Pipe Area (sq In) Length (if) 34(-)
50 14 83 33 50 6.5 32 50 4
31 50 6.4167 30 28.89 4.1 29 28.89 1.5 28 28.89 7.667 27 28.89 2.25 26 28.89 2.25 P1 -time 0
0 0
0 0
0 0
0 0
P2-time P3 -time P4-time PS -time 0.00347 0 05147 0.08247 0.13047 0.00499 0.05299 0.08399 0.13199 0.00593 0 05393 0 08493 0.13293 0.00743 0.05543 0 08643 0.13443 0.00839 0.05639 0.08739 0.13539 0.00874 0.05674 0.08774 0.13574 0 01053 0 05853 0.08953 0.13753 0 01106 0 05906 0.09006 0.13806 001158 0.05958 0.09058 0.13858 Direction - Downstream Towards Throttle Valve Flow Element Delta-Time (s)
Force (Ibt) Impulse (Ibf-s) 35 0.00074 147.100686 7.060832943 36(')
0.00421 592.495592 28.43978842 37(-)
0.00070 98.7492653 4.739964736 38 0.00164 325.855951 15 64108563
FAI/03-07 Page F-3 of F-5 Rev. 1 Date: 03/06/03 39 0.00281 558 610201 26.81328966 40(g) 0.00526 907.069922 43 53935626 41 0.00047 93.1017002 4.46888161 42( )
000047 65.8328436 3.159976491 43 0 00304 605.161051 29.04773046 Direction - Upstream Towards Fan Cooler Flow Elem Delta-T (s)
Force (lbf) Impulse (ibf-s) 34(')
0 00347 488.150535 23.43122568 33 0 00152 302.580526 14.52386523 32 0.00094 186.2034 8.937763219 31(l) 000150 231.750205 11.12400986 30 0.00096 85.5598052 4.106870648 29 0.00035 31.3023677 1.502513652 28(l) 0 00179 147.748396 7.091923028 27(l) 0.00053 43.3590573 2.081234748 26(l) 0.00053 43.3590573 2.081234748 Notes:
()
denotes the flow element as a 45-degree elbow
(#) denotes the flow element as a 30-degree elbow (1) -denotes the flow element as a reducing tee
FAI/03-07 Page F-4 of F-5 Rev. 1 Date: 03/06/03 POINT BEACH CFCtA INPUTS:
Wave Speed (C) =
Rise Time =
Durahon =
Peak Pressure =
Area (8-line)
Area (8-nne)
Area (2 5-line)
Trans Coeff. (elbows) -
Trans. Coeff (8-x 6e) =
Trans Coeff. (6" x 2 5")"
4274 0048 0127 191 50 28 89 479 Ws sec sec psia sq In sq In sq In Rate.
eC8/C4 psVsec
=2'C7/(C7+C7+C8)
=2'CB/(CB+C8+C9)
DirectIon
- Downstream Towards Thrt Flow Element 35 35 37 38 39 40 41 42 43 Pipe Area (sq In) 50 50 50 50 50 so 50 50 50 Length (tt)
Pt -time 316 18 3
7 12 22 5 2
2 13 0
0 0
0 0
0 0
0 0
P2 -time
=C17/C3
=(C17+CIB)/C3
=(C17+CI8+Cl9)/C3
=(CI 7+C18+C19+C20)1C3
=(C17+Cl8.Clg+C20+C21)1C3
=(C17+C18+CI9+C20eC21.C22YC3
=(C17+Cl8+C19+C20.C21+C22+C23)1C3
=(017+C18+Clg+C20+C21+C22.C23.C24YC3
=(CI 7+CI B+CI 9+C20.C21 +C22+C23+C24+C25)/C3 P3 -time
=E17+C4
=E1 8+C4
=E19+C4
=E20+C4
=E21+C4
=E22+C4
=E23+C4
=E24+C4
=E25+C4 P4 - time
=E17+(C5-C4)
=EI8+(C5-C4)
=EI9+(C5-C4)
=E20+(C5.C4)
.E21.(C5-C4)
=E22+(C5-C4)
=E23+(C5-C4)
=E24.(CS-C4)
=E25+(CS-C4)
PS -time E1 7+C5
=E18S+C5
=E19+C5
=E20+CS
=E21 +C5
=E22+C5
=E23+C5
-E24*C5
-E25+C5 DIrection - Upstream Towards Fan Co Flow Element 34(*)
33 32 31 30 29 28 27 25 Pipe Area (sq In) 50 50 50
=C8
=C8
=C8
=C8
=C8 Length (ft) 1483 65 4
8 4187 41 1 5 7 667 225 2 25 0
0 0
0 0
0 0
0 0
P1 -tIme P2 -time
=C30IC3
=(C30+C31 )C3
=(C30+C31 +C32)YC3
=(C30+C31.C32*C33)1C3
=(C00C311+C32+C33+34YOC
=(C30+C31 +C32+C33+C34+C35)/C3
=(C30+C31.C32+C33+C34+C35+C36)/C3
=(C30+C31.C32+C33+C34+C35+C38+C37)/C3
=(C30.C31 *C32+C33+C34+C35+C36+C37.C38y/C3 P3 -tIme
=E30.C4
=E31+C4
=E32+C4
.E33+C4
=E34+C4
=E35+C4
=E36+C4
=E37+C4
=E38+C4 P4 -time
=E30+(C5-C4)
=E31.(C5-C4)
=E32+(C5-C4)
=E33+(C5-C4)
.E34+(C5-C4)
=E35+(C5-C4)
=E36+(C5-C4)
=E37+(C5-C4)
=E38+(CS-C4)
PS -time
=E30+C5
=E31+C5
=E32+C5
=E33+C5
-E34+C5
- E35+C5
=E36+C5
=E37+C5
=E38+C5 DirectIon - Downstream Towards Thrc Flow Element Delta-Time (s) 35 36(")
37(.)
38
=EI 7
=E1l8-E17
=1E19-1E18
=E20-E19 Force (Ibl')
=B43'F68S17
=B44TF6'1$'COS(4S'PIY1 80)
=B4SF6819'COS(45'PiY1 80)
=548'F6820 Impulse (Ibf-s)
=C43-C4
=C44-C4
=C45-C4
=C468C4
FAI/03-07 Page F-5 of F-5 Rev. 1 Date: 03/06/03 39 40(1) 41 42(-)
43
=E21-E20
=E22-E21
=E23-E22
=E24-E23 sE25-E24
=B4r'F6'B21
=B48'F6822'COS(30'Pl(Y 80)
=849'F66B23
-S50'F6624'COS(4S'P1(Yl 80)
.55VF8'B25 Direction - Upstream Towards Fan Co Flow Elem 34(-)
33 32 31(l) 30 29 28(I) 27(I) 26()
Defta-T (a)
-E30
.E31-E30 mE32-E31 mE33-E32
.E34-E33
.E3S-E34 zE36-E35
=E37-E36
=038-137 Force (IbI)
=855IF6B301COS(45'PIYI 80)
.856'FS'B31
-857'FBB32
=F6'B58833'C1
.B59'F6834'Cll
=B60'F8B35Cl i
.B6P'F6'B36'C1IPC12
=B62'F6B37rC11PC12 BS3'F6'B38BC11PCi2
=C47rC4
=C48rC4
=C49'C4
=CSO'C4
-C51 C4 Impulse (lbf-&)
.CSS'C4
=CSO'C4
=C57'C4
.CS8'C4
.C59'C4
=C80'C4
=C6SPC4
=C62'C4
=C63'C4 Notes:
(-) - denotes the flow element as a 45-de (U) - denotes the flow element as a 30-do (I) - denotes the flow element as a reduc
FAI/03-07 Page G-1 of G-5 Rev. 1 Date: 03/06/03 APPENDIX G EPRI TBR MAX FORCE/IMPULSE CALCULATIONS FOR POINT BEACH CFC IC USING MICROSFOT EXCEL 97
FAI/03-07 Page G-2 of G-5 Rev. 1 Date: 03/06/03 POINT BEACH CFC1 C INPUTS:
Wave Speed (C) =
Rise Time =
Duration =
Peak Pressure =
Area (8" line)
Area (6" line)
Area (2.5" line)
Trans. Coeff. (elbows) =
Trans. Coeff. (8" x 6") =
Trans. Coeff. (6" x 2.5") =
4274 0 039 0.118 222 50 28.89 4.79 1
0.7759 0.9234 fts sec sec psia sq in sq in sq in Rate =
5692 307692 psi/sec Direction - Downstream Towards Throttle Valve Flow Element Pipe Area (sq in) Length (ft) 42 50 0 65 43 50 8.035 44 50 5.3 45 50 11 46 50 11 47 50 22 48 50 11 49 50 15 50 50 4
51 50 3
P1 -time 0
0 0
0 0
0 0
0 0
0 P2 -time 0.00015 0.00203 0.00327 0.00585 0.00842 0.01357 0.01614 0.01965 0.02059 0.02129 P3 -time 0.03915 0.04103 0.04227 0.04485 0.04742 0.05257 0.05514 0.05865 0.05959 0.06029 P4 - time 0.07915 0.08103 0.08227 0.08485 0.08742 0.09257 0.09514 0.09865 0.09959 0.10029 P5 -time 0.11815 0.12003 0.12127 0.12385 0.12642 0.13157 0.13414 0.13765 0.13859 0.13929 Direction - Upstream Towards Fan Cooler Flow Element Pipe Area (sq in) Length (ft) 41 40 39 38 37 36 35 34 33 32 31 30 50 50 50 50 28.89 28.89 28.89 28.89 28.89 28.89 28.89 28.89 8.711 1.654 1.689 0.583 1.612 3.73 0.583 1.579 5.521 4.625 2.25 2.25 P1 -time 0
0 0
0 0
0 0
0 0
0 0
0 P2 -time 0.00204 0.00243 0.00282 0.00296 0.00333 0.00421 0.00434 0.00471 0.00600 0.00709 0.00761 0.00814 P3 -time 0.04104 0.04143 0.04182 0.04196 0.04233 0.04321 0.04334 0.04371 0.04500 0.04609 0.04661 0.04714 P4 -time 0.08104 0.08143 0.08182 0.08196 0.08233 0.08321 0.08334 0.08371 0.08500 0.08609 0.08661 0.08714 P5 -time 0.12004 0.12043 0.12082 0.12096 0.12133 0.12221 0.12234 0.12271 0.12400 0.12509 0.12561 0.12614 Direction - Downstream Towards Throttle Valve Flow Element 42 43 44 45 46(A) 47(@)
48 49(-)
50(*)
51 Delta-Time (s) 0.00015 0.00188 0.00124 0.00257 0.00257 0.00515 0.00257 0.00351 0.00094 0.00070 Force (Ibf) 43.2849789 535.068932 352.939059 732.515028 732.515028 1200.08237 732.515028 706.317742 188.351398 199.776826 Impulse (lbf-s) 1.688114179 20.86768835 13.7646233 28.5680861 28.5680861 46.80321226 28.5680861 27.54639192 7.345704512 7.79129621
FAI/03-07 Page G-3 of G-5 Rev. 1 Date 03/06/03 Direction - Upstream Towards Fan Cooler Flow Elem Delta-T (s)
Force (Ibf) 41 0.00204 580.08531 40 0 00039 110.143623 39(!)
0.00040 112.474353 38(&)
0.00014 38.8232965 37 0.00038 62.0249509 36($)
0.00087 143.519272 35(-)
0.00014 15.8618905 34(-)
0.00037 42.9604205 33 0.00129 212.431609 32(!)
0.00108 164.332897 31 (!)
0.00053 79.9457336 30(!)
0.00053 79.9457336 Impulse (Ibf-s) 22.62332709 4.29560131 4.386499766 1.514108563 2.418973084 5.597251614 0.618613731 1.675456399 8.284832752 6.408982977 3.117883611 3.117883611 Notes: All other flow elements are assumed to be 90-degree elbows
(*) - denotes the flow element as a 45-degree elbow
(#) - denotes the flow element as a 30-degree elbow
(@) - denotes the flow element as a 55-degree elbow (I) - denotes the flow element as a reducing tee (A) - denotes the flow element as flow orifice
($) - denotes the flow element as a flow control valve
(&) - denotes the flow element as a 8" x 6" reducer
FAI/03-07 Page G-4 of G-5 Rev. 1 Date: 03/06/03 POINT BEACH CFC1C INPUTS; Wave Speed (C) =
Rise Time=
Duration =
Peak Pressure -
Area (8 line)
Area (6 line)
Area (2 5 line)
Trans Coeff (elbows) a Trans. Coeff (t x 8j)
=
Trans Coeff (6 x 2 5')
Direction Downstream Towards 4274 0 039 222 50 28 89 4.79 1
-2 C7/(C7+C7+C8)
=2-C8/(C8+C8+Cg)
It/s sec sec psta sq in sq in sq In Rate -
=C6/C4 psVsec Flow Element 42 43 44 45 46 47 48 49 50 51 Pipe Area (sq In) 50 50 50 50 50 50 50 50 50 50 Length (ht)
P1 -time 0 65 8 035 53 11 11 22 11 15 4
3 0
0 0
0 0
0 0
0 0
a P2 -time
.C171C3
=(C17eCI8YC3
=(C17+CIB+C19g/C3
=(C17+C1 8+Cl 9+C20)/C3
=(C 7.CI 8+CIO+C20+C21 )C3
=(C1 7+CI 8+Clg+C20+C21 +C22)/C3
=(C17+C18+C19+C20+C21+C22eC23YC3
=(C17+C1B+C19+C20+C21eC22eC23+C24Y/C3
=(C17+C18.C19.C20+C21+C22+C23+C24.C25)/C3
.(Cl7eCI B.CI +C20+C21.C22eC23.C24+C25+C26)/C3 P3 -time P4 - time
.E17+C4
=E17+(C5-C4)
=Et8+C4
=E18+(C5-C4)
=E19+C4
=E19+(C5-C4)
- E20+C4
- E20+(C5-C4)
- E21+C4
=E21+(C5-C4)
- E22+C4
=E22+(C5-C4)
- E23+C4
=E23+(C5-C4)
- E24+C4
=E24+(C5-C4)
=E25+C4
=E25+(C5-C4)
=E26+C4
.E26+(C5-C4)
P5 -time
.E1 7+C5
=EIB+C5
=E19.C5
=E20+C5
=E21 +C5
=E222CS
=E23+C5
=E24+C5
=E25+CS
=E26+C5 Direction -Upstream Towards Fs Flow Element 41 40 39 38 37 36 35 34 33 32 31 30 Pipe Area (sq In) 50 50 50 50
-C8
=C8
=C8
- C8
=C8 2889 2889 2889 Length (It)
P1 -time 8711 1 654 1 689 0 583 1 612 3 73 0 583 1.579 5521 4 625 2 25 225 0
0 0
0 0
0 0
0 0
0 0
0 P2 -time
=C3I/C3
=(C31.C32)/3
=(C31 eC32+C33)1C3
.(C31 +C32+C33+C34)1C3
=(C3i +C32+C33+C34+C35YC3
.(C31 +C32+C33+C34+C35+C38)/C3
.(C31 +C32.C33+C34+C35.C38.C37YC3
=(C31.C32+C33+C34+C354C36+C37.C38VC3
.(C31 +C32+C33+C34+C35+C38+C37.C38.C39)1C3
.(C31 eC32.C33+C34.C35+C36*C37eC38+C39+C40yIC3
.(C31 +C32+C33+C34+C354C38.C37+C38+C39+C40+C41 )/C3
.(C31+C32+C33+C34+C35+C36.C37+C38+C39+C40+C41
+C42YC3 P3 -time P4 -time
=E31+C4
=E31+(C5-C4)
=E32+C4
=E32+(C5-C4)
=E33+C4
=E33+(C5-C4)
- E34+C4
.E34+(C5-C4)
=E35+C4
=E35+(C5-C4)
=E36+C4
=E36+(C5-C4)
=E37+C4
=E37+(C5-C4)
=E38+C4
=E38+(C5-C4)
=E39+C4
=E39+(CS-C4)
=E40+C4
=E40+(C5-C4)
=E41+C4
=E41+(C5-C4)
=E42+C4
=E42+(C5-C4)
P5 time
=E31+C5
=E32+C5
=E33+C5
=E34+C5
=E35+C5
=E36+Cs
=E37+C5
=E38+C5
=E39+C5
=E40+C5
=E41+C5
=E42+C5 Direction - Downstream Towards Flow Ebment Delta-Tlme (s)
Force (Ibf)
=B45 F68517 Impulse (Ibt-s)
=C45 C4 42
=E17
FAI/03-07 Page G-5 of G-5 Rev. 1 Date: 03/06/03 43 44 45 46(A) 47(0) 48 49(')
50(.)
51
=El8-E17
=E19-ElB
=E20-E19
-E21-E20
-E22-E21
=E23-E22
-=-24-E23
=E25-E24
=E26-E25
=B47'F6Bl8
=B8'F6'B19
=B49'F&B20
=850'F6a82i
=135 lF6'22'COS(35'PI(Yl 80)
=BS2'F68B23
=B53'F6824'COS(45'Pl(yl 80)
=S54'F6B25'COS(45'PI(/1 80)
=BSS'F6826
=C47rC4
-C48'C4
.C49'C4
-CSO'C4
-CS1'C4
.-CS2'C4
=C53'C4
=C54'C4
=CSS'C4 Direction - Upstream Towards Fs Flow Elem Delta-T (a) 41
=E31 40
=E32-E31 39(1)
-E33-E32 38(&)
- E34-E33 37
=E35-E34 36(S)
- E36-E35 35(t)
- E37-E36 34(*)
=E38-E37 33
=E39-E38 32(1)
-E40-E39 31(l)
-E41-E40 30(l)
=E42-E41 Force (Ibf)
=B59¶r6B31
=B60'F6532
=B6'F6B'33
=B62'FB'34
=863'FB835
=B64'F68B36
=865'F6B37'COS(45'Pi(YI80)
-B686FGB38'COS(45'PI(Y1 80)
-B67'F6'39
=B68'F6'B4GC12
=B69'F8B4r'C12
=670'F6B42'C12 Impulse (1bZ-rn)
=CS93C4
-C60'C4
-C61rC4
-C62'C4
-C63'C4
-C64'C4
=C6S5C4
=C66'C4
=CG7TC4
=C68'C4
=C69'C4
-C70'C4 Notes: All other flow elements ar
(-) - denotes the flow element as a
(#). denotes the flow element as a (0) denotes the flow element as t (I)
- denotes the flow element as a (A) - denotes the flow element as bh (S)
- denotes the flow element as a
(&)- denotes the flow element as a
FAI/03-07 Page H-1 of H-5 Rev. 1 Date: 03106/03 APPENDIX H EPRI TBR MAX FORCE/IMPULSE CALCULATIONS FOR POINT BEACH CFC 2B USING MICROSFOT EXCEL 97
FAI/03-07 Page H-2 of H-5 Rev. 1 Date: 03/06/03 POINT BEACH CFC2B INPUTS:
Wave Speed (C) =
Rise Time =
Duration =
Peak Pressure =
Area (8" line)
Area (6" line)
Area (2.5" line)
Trans. Coeff. (elbows) =
Trans. Coeff. (8" x 6") =
Trans. Coeff. (6" x 2.5") =
4274 0.039 0.157 223 50 28.89 4.79 1
0.7759 0.9234 fts sec sec psia sq in sq in sq in Rate =
5717.948718 psi/sec Direction - Downstream Towards Throttle Valve Flow Element Pipe Area (sq in) Length (ft) 41 50 10.33 42 50 13 43 50 6
44 50 6.2 45 50 36 46 50 7
47 50 5
48 50 17.1 49 50 7
50 50 24 51 50 2
52 50 1
53 50 14.1 P1 -time 0
0 0
0 0
0 0
0 0
0 0
0 0
P2 -time 0.00242 0.00546 0.00686 0.00831 0.01674 0.01837 0.01954 0.02354 0.02518 0.03080 0.03127 0.03150 0.03480 P3 -time 0.04142 0.04446 0.04586 0.04731 0.05574 0.05737 0.05854 0.06254 0.06418 0.06980 0.07027 0.07050 0.07380 P4 - time 0.12042 0.12346 0.12486 0.12631 0.13474 0.13637 0.13754 0.14154 0.14318 0.14880 0.14927 0.14950 0.15280 P5 -time 0.15942 0.16246 0.16386 0.16531 0.17374 0.17537 0.17654 0.18054 0.18218 0.18780 0.18827 0.18850 0.19180 Direction - Upstream Towards Fan Cooler Flow Element Pipe Area (sq in) 40 50 39 50 38 28.89 37 28.89 36 28.89 35 28.89 Length (ft) 1.16 13.063 2.167 6.5 2.25 2.25 P1 -time 0
0 0
0 0
0 P2 -time 0.00027 0.00333 0.00383 0.00536 0.00588 0.00641 P3 -time 0.03927 0.04233 0.04283 0.04436 0.04488 0.04541 P4 -time 0.11827 0.12133 0.12183 0.12336 0.12388 0.12441 P5 -time 0.15727 0.16033 0.16083 0.16236 0.16288 0.16341 Direction - Downstream Towards Flow Element Delta-Time (s) 41 0.00215 42 0.00304 43 0.00140 44(#)
0.00145 45 0.00842 46 0.00164 47 0.00117 48 0.00400 49 0.00164 50(@)
0.00562 51 0.00047 52(^)
0.00023 Throttle Valve Force (lbf) Impulse (Ibf-s) 613.401845 23.92267197 869.599127 33.91436593 401.353443 15.65278428 359.168353 14.00756578 2408.12066 93.91670566 468.245684 18.26158166 334.461203 13.0439869 1143.85731 44.61043519 468.245684 18.26158166 1298.80702 50.65347396 133.784481 5.217594759 47.2999569 1.844698318
FAI/03-07 Page H-3 of H-5 Rev. 1 Date: 03/06/03 53 0.00330 943.180591 36.78404305 Direction - Upstream Towards Fan Cooler Flow Elem Delta-T (s)
Force (Ibf) 40 0.00027 77.594999 39(!)
0.00306 677.95278 38 0.00051 83.7552793 37(!)
0.00152 194.915965 36(!)
0.00053 62.3057254 35(!)
0.00053 62.3057254 Impulse (lbf-s) 3.02620496 26.44015841 3.266455894 7.601722646 2.429923292 2.429923292 Notes: All other flow elements are assumed to be 90-degree elbows
(*) - denotes the flow element as a 45-degree elbow
(#) - denotes the flow element as a 30-degree elbow
(@) - denotes the flow element as a 36-degree elbow
(!) - denotes the flow element as a reducing tee
FAI/03-07 Page H 4 of H-5 Rev t Date 03106/03 POINT BEACH CFC2B lI Wave Speed (C) =
Rise Time -
Duration -
Peak Pressure Area (8 line)
Area (6 ltne)
Area (2 5 line)
Trans CoeHf. (elbows)=
Trans. CoeHf. (8 x6) =
Trans Coeff. (6 x 2 5) =
Direction - Downstream 4274 0 039 0157 223 50 28 89 4 79 t
=2C7/(C7+C7+C8)
=2 C8t(C8+C8+C9) ft/s sec sec psia sq In sq in sq In Rate.
=CG/C4 psi/sec Flow Element 41 42 43 44 45 46 47 48 49 50 51 52 53 Pipe Area (sq in) 50 50 50 50 50 50 50 50 50 50 50 50 50 Length (tt)
P1 -time 1033 13 6
62 36 7
5 17.1 7
24 2
1 14.1 0
0 0
0 0
0 0
0 0
0 0
0 0
P2 -time
-C17/C3
=(C17+C18)/C3
=(C17+C18+C19)/C3
=(C17+C18+CI9+C20YC3
=(C17+CI8+C19+C20+C21)/C3
.(C17+C18+C19+C20+C21 +C22)/C3
=(C17+C18+C19+C20.C21+C22+C23)/C3
=(C17+C18+C19+C20+C21 +C22+C23+C24)/C3
=(C17+CI8+Cl9+C20+C21.C22+C23+C24+C25)/C3
=(C17+C18+C19+C20+C21 +C22+C23+C24+C25+C26)/C3
=(C17+C18+C19+C20+C21 +C22+C23+C24+C25+C26+C27)/C3
=(C17+C18+CI9+C20+C21 +C22+C23+C24+C25+C26+C27+C28)1C3
=(C17+C18+C19+C20+C21+C22+C23+C24+C25*C26+C27+C28+C29)/C3 P3 -time
=E1 7+C4
.E18+C4
=E19+C4
=E20+C4
=E21+C4
=E22+C4
=E23+C4
=E24+C4
=E25+C4
=E26+C4
=E27+C4
=E28+C4
- E29+C4 P4 - time PS -time
=E17+(C5-C4)
=EI7+C5
=E18+(C5-C4)
=E18+C5
=E19+(C5-C4)
=E19+C5
=E20+(C5-C4)
E20+C5
=E21+(C5-C4)
=E21+C5
=E22+(C5-C4)
=E22+C5
-E23+(C5-C4) =E234C5
=E24+(C5-C4)
=E244C5
=E25+(CS-C4)
=E25+C5
=E264(C5-C4)
=E26+C5
=E27+(C5-C4)
=E27+C5
=E28+(C5-C4)
=E28+CS
=E29+(C5-C4)
=E29+C5 Direction - Upstream To Flow Element 40 39 38 37 36 35 Pipe Area (sq In) 50 50 28 89 28 89
=C8
=C8 Length (ft)
P1 -time 1.16 13 063 2167 65 225 225 0
0 0
0 0
0 P2 -time
-C34/C3
=(C34+C35)/C3
=(C34+C35+C36)/C3
=(C34+C35+C36+C37YC3
=(C34+C35+C36+C37+C38)1C3
.-(C34+C35+C36+C37+C38+C39)/C3 P3 -time
=E34+C4
-E35+C4
=E36+C4
=E37+C4
-E38+C4
=E39+C4 P4 -time P5 -time
=E34+(C5-C4)
E34+CS
=E35+(C5-C4)
=E35+C5
=E36+(C5-C4)
=E36+C5
=E37+(C5-C4)
=E37+C5
=E38+(C5-C4)
=E38+C5
=E39+(C5-C4)
=E39+C5 Direction - Downstream Flow Element Delta-Time (s) 41
=E17-E34 42
-E18-E17 43
=E19-E18 44(#)
=E20-E19 45
=E21-E20 46
-E22-E21 47
-E23-E22 48
-E24-E23 49
=E25-E24 50(@)
=E26-E25 Force (Ibf)
Impulse (Ibf-s)
B44 F6'817
=C44'C4
=B45'F6'B1 8 C45'C4
.B46'F6'B19
=C46'C4
=I47'F6'B20'C0S(30'P1Y/1 80) =C47'C4
.B48'F6'B21
=C48'C4
=B49sF68B22
.C49sC4
=B50'F6'B23
=C50'C4
=B51'F6'B24
-CS1'C4
.B52'F6'825
-C52'C4
=B53'F6'B26'C0S(36'PI(/180) =C53'C4
FAI/03-07 Page H-5 of H-5 Rev. 1 Date 03/06/03 51 52(')
53
-E27-E26
=E28-E27
=E29-E28
=B54'F6'B27
-C54-C4
=B55F6-B28'COS(45'Pt)/180) -C5S'C4
=B56'F6'829
-C56'C4 Direction - Upstream To' Flow Elem Delta-T (s) 40
-E34 39(1)
-E35-E34 38
=E36-E35 37(1)
=E37-E36 36(1)
=E38-E37 35(l) sE39-E38 Force (Ibt)
=B60-F68B34
=B61'F6-835-C1 t
-B62-F6-B36
=F6-B63-B37-C1 t
=B64-F6-B38C11 C1 2
=B65-F6-B39-C l C12 Impulse (IbI.s)
=C60-C4
-C61PC4
=C62-C4
-C63-C4
=C64-C4
-C650C4 Notes: All other flow ele
(-) - denotes the flow elerr
() - denotes the flow elen
() - denotes the flow ele (I) - denotes the flow elerr
FAI/03-07 Page I-1 of 1-5 Rev. I Date: 03/06/03 APPENDIX I EPRI TBR MAX FORCE/IMPULSE CALCULATIONS FOR POINT BEACH CFC 2D USING MICROSOFT EXCEL 97
FAI/03-07 Page 1-2 of 1-5 Rev 1 Date: 03/06/03 POINT BEACH CFC2D INPUTS:
Wave Speed (C) =
Rise Time =
Duration =
Peak Pressure =
Area (8' line)
Area (6" line)
Area (2.5" line)
Trans. Coeff. (elbows) =
Trans. Coeff. (8 x 6-) =
Trans. Coeff. (6" x 2.5") =
4274 0.028 0.147 286 50 28 89 4.79 1
0.7759 0.9234 ft/s sec sec psia sq in sq in sq in Rate =
10214.3 psi/sec Direction - Downstream Towards Throttle Valve Flow Element Pipe Area (sq in)
Length (ft) 48 50 6.85 49 50 9
50 50 7
51 50 6
52 50 8
53 50 4.3 54 50 7
55 50 40 56 50 9
57 50 6 9 58 50 4
59 50 37.2 60 50 3
61 50 2.9 62 50 12.9 P1 -time 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
P2 -time 0.00160 0.00371 0.00535 0.00675 0 00862 0.00963 0.01127 0.02062 0.02273 0.02434 0 02528 0.03398 0 03469 0.03536 0.03838 P3 -time 0.02960 0.03171 0.03335 0.03475 0.03662 0.03763 0.03927 0.04862 0.05073 0.05234 0.05328 0.06198 0.06269 0.06336 0.06638 P4 - time P5 -time 0.12060 0.14860 0.12271 0.15071 0.12435 0.15235 0.12575 0.15375 0.12762 0.15562 0.12863 0.15663 0.13027 0.15827 0.13962 0.16762 0.14173 0.16973 0.14334 0.17134 0.14428 0.17228 0.15298 0.18098 0.15369 0.18169 0.15436 0.18236 0.15738 0.18538 Direction - Upstream Towards Fan Cooler Flow Element Pipe Area (sq in)
Length (ft) 47 50 2.75 46 50 3.667 45 28.89 6.438 44 28.89 3.667 43 28.89 3.667 42 28.89 5.625 41 28.89 2.25 40 28.89 2.25 P1 -time 0
0 0
0 0
0 0
0 P2 -time 0.00064 0.00150 0 00301 0.00387 0.00472 0.00604 0.00657 0.00709 P3 -time 0.02864 0.02950 0.03101 0.03187 0.03272 0.03404 0.03457 0.03509 P4 -time 0.11964 0.12050 0.12201 0.12287 0.12372 0.12504 0.12557 0.12609 P5 -time 0.14764 0.14850 0.15001 0.15087 0.15172 0.15304 0.15357 0.15409 Direction - Downstream Towards Throttle Valve Flow Element 48 49 50 51 52(-)
53(")
54 55 56 57(.)
58 59(#)
Delta-Time (s) 0.00160 0.00211 0.00164 0.00140 0.00187 0.00101 0.00164 0.00936 0.00211 0.00161 0 00094 0.00870 Force (Ibf) 818.5289792 1075.439535 836.4529715 716.9596898 675.956078 363.3263919 836.4529715 4779.731265 1075.439535 583.0121173 477.9731265 3596.201955 Impulse (lbf-s) 22.91881142 30.11230697 23.4206832 20.07487131 18.92677018 10.17313897 23.4206832 133.8324754 30.11230697 16.32433928 13.38324754 100.6936547
FAI/03-07 Page 1-3 of 1-5 Rev. 1 Date: 03/06/03 60 61(-)
62 0.00070 0 00068 0.00302 3stream Towards Fan Cooler Flow Elem Delta-T (s) 47 0.00064 46 0.00086 45(!)
0.00151 44(S) 0.00086 43 0.00086 42(l) 0.00132 41 (!)
0.00053 40(1) 0.00053 358 4798449 245 0340783 1541.463333 Force (Ibf) 328 6065245 438.1818638 344 8679016 196.4322142 196.4322142 278.250345 111.300138 111.300138 10.03743566 6 860954192 43.16097333 Impulse (Ibf-s) 9.200982686 12.26909219 9.656301244 5.500101998 5.500101998 7.791009659 3.116403864 3.116403864 Notes:
(*) - denotes the flow element as a 45-degree elbow
(#) - denotes the flow element as a 36-degree elbow (I) - denotes the flow element as a reducing tee (S) - denotes the flow element as a flow control valve
FAI/03-07 Page 1.4 ol 1.5 Rev. 1 Date 03/06/03 POINT BEACH CFC2D INPUTS Wave Speed (C) =
Rise Time n Duration.
Peak Pressure.
Area (8 line)
Area (6 line)
Area (2.5 line)
Trans Coeff (elbows)
Trans Coeff. (8 x 6')
Trans Coeif (6 x 2 S) -
Directlon - Downstream Towat 4274 0 028 0 147 286 50 28 89 4 79 1
=2 C7/(C7+C7+C8)
2'C81(C8+C8+C9) ftts sec sec psia sq In sq In sq In Rate
=C61C4 psi/sec Flow Element 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 Pipe Area (sq In) 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 Length (ft)
P1 -time 6 85 9
7 6
8 43 7
40 9
6 9 4
37 2 3
29 129 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
P2 -time
.C17/C3
=(C17.CIB)/C3
=(C17.CIB.Cl9)Y03
=(C17.C18.C19.C20)/03
-(C1 7.CI 8+CI19.C20.C21)I03
-(Cl 7.C18+Cl9.C20.C21.C22)/03
=(C17.01 8+019.C20i+C21.C22.C23yIC3
-(Cl 7+018.01l9.C20.C21.C22.C23.C24)/C3
=(C17+C1l8+Cl9.C20.C21.C22.C23.C24+C25)103
=(C17.01l.09.C20.C21 +C22+C23.C244C25.C26yoC3
=(01 7401 8I-09.C20.C21.C22.C23.C24.C25.C26,C27),C3
.(C17.Cl8.C19.C20.C21.aC22.C23.C24.C25.C26.C27,C2 8 ),C3
=(C1 7+C18.C19.C20.021.C22.C23.C24.C25,C26.C27.C28+o29YC3
=(C1 7+C18.01l9.C20.C21 +C22.C23.C24.C25.C26.027,C28.C29.C30),C3
=(Cl7.CIl8+Cl9+C20+C21.C22.C23.C24+C25.C26.C27.C28.C29.C30.C31)1C3 P3 -time
=E17.C4
=El8.C4
=Et9.C4
=E20.C4
.1E21
.04
=E22.C4
=E23.C4
=E24.C4
=E25.C4
=E26+C4
=E27.C4
=E28.C4
=E29.C4
=E30.C4
=E31+C4 P4 -time
=E17.(C5-C4)
-EIB.(C5-C4)
=EI9.(C5 04I)
-E20.(CS-C4)
=E214(C5-C4)
-E22+(C5-C4)
=E23.(C5 04I)
=E24+(C5-C4)
=E25.(C5-C4)
.E26+(C5-C4)
=E27.(CS-C4)
=E28+(C5-C4)
=E29.(C5-C4)
=E30.(C5-C4)
=E31,(C5-C4)
P5 -time
=E17.C5
=EIB.C5 sE20.C5
=E21+C5
=E22+C5
=E23.C5
=E24+C5
-E25.C5
=E26+C5
=E27+C5
=E28+C5
=E29.C5
=E30.C5
=E31 +C5 Direction - Upstream Towards Flow Element 47 46 45 44 43 42 41 40 Direction. Downstream Towai Flow Element 48 49 50 51 52( )
53(*)
54 55 56 57(*)
58 Pipe Area (sq In) 50 50 28 89 28 89
-C8
=C8
=C8
-C8 Delta-Time (s)
-E1 7
- E128-E127
- E19.E18
=E20-E19
=E21-E20
.E22-E21
=E23-E22
.E24.E23
=E25-E24
=E26-E25
=E27-E26 Length (It)
P1 -time 2.75 3 667 6 438 3 667 3 667 5 625 225 2 25 0
0 0
0 0
0 0
0 P2 -time C36/C3
- (C36+C37)/C3
=(C36+C37+C38)/C3
-(C36+C37+C38+C39)/C3
=(C36.C37.C38+C39.C40yC3
=(C36+C37+C38+C39+C40+C41)/C3
=(C36+C37+C38+C39+C40+C41+C42)1C3
=(C36+C37+C38+C39+C40+C41 +C42+C43YC3 P3 -time
=E36+C4
=E37+C4
-E38+C4
=E39+C4
- E40+C4
- E41sC4
.E42+C4
=E43+C4 P4 -time
=E36.(CS-C4)
=E37+(C5-C4)
=E38+(C5-C4)
.E39+(CS-C4)
=E40+(CS-C4)
=E41.(CS-C4)
=E42+(C5-C4)
=E43+(C5-C4)
PS -time
=E36+CS
=E37+CS
=E38+C5
-E39.C5
=E40+C5
=E414C5
=E424C5
=E43+CS Force (bi)
Impulse (ibt-s)
=B47F6'817
=C47-C4
.B48-F6-818
=C480C4
=B49'F6-819
- C49-C4
- 850F6-B20
=C500C4
=B51 F6'821 0OS(45 PIy1 80) =C51-C4
=B52 F6 B22'COS(45 Piy1 80) =CS20C4
-853-F6-B23
=C530C4
- BS4-F6-B24 CS4-C4
=B55-F6'825
=CSS-C4
=B56'F6*826'C0S(45 PIy1 80) =C56-C4
=B57-F6-B27
- C57-C4
FAI103-07 Page 1-5 ot 1.5 Rev I Date: 03/06103 59(m) 60 61(-)
62
=E28-E27
=B58'F66B28'OS(36-PiQ/leO) =C58-C4
-E29-E28
- 259'F65B29
.C59'C4
=E30-E29
=B60'F61B30'COS(4S-PiQI1ao)
-C60-C4
..E31-E30
=B6P'F6'B31
=C61PC4 Direction - Upstream Towards Flow Elem 47 46 45(1) 44(S) 43 42(1) 41(i) 40(i)
Delta-T (a)
Force (Ib/)
-E36
- B65-F6-B36
=E37-E36
=B66-F6-B37
.E38-E37
-B67-F6-B38-C 1
=E39-E38
=F6-B68-B39C1 1
.E40-E39
-B69-F68B40-C11
=E41-E40
=B70-F6-B41tC11-C12
=E42-E41
-B71-F6-B42-C11-C12
.E43-E42
=B72-F6-B43-C11-C12 Impulse (lbf-s)
-C65-C4
=C66-C4
-C67-C4
=C68-C4
=C69-C4
-C70-C4
-C71 C4
=C72-C4 Notes:
() - denotes the flow element as
(#) - denotes the flow element a:
(I) - denotes the flow element as (S) - denotes the flow element as