ML21294A202
| ML21294A202 | |
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|---|---|
| Site: | Susquehanna  |
| Issue date: | 10/12/2021 |
| From: | Talen Energy, Susquehanna |
| To: | Office of Nuclear Reactor Regulation |
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Text
SSES-FSAR Text Rev. 55 APPENDIX B COMPARTMENT DIFFERENTIAL PRESSURE ANALYSIS DESCRIPTION The computer codes COPDA, FLUD and COTTAP4 were used to evaluate compartment _differential pressure. This appendix describes the computational procedure and the analytical techniques used in FLUD. The analytical basis for COPDA is described in Reference 6B-4 and the analytical basis for COTT AP4 is described in Reference 3.6-10. The set-up of initial conditions, the determination of the thermodynamic state point at subsequent time increments. and computation of energy and mass transport between one time step is discussed in Sections 6B.1, 6B.2 and 6B.3 for FLUD. Selection was made of the control volume and flow path configuraUon that resulted in the best representation of the pressure transients in the compartments along the flow paths from the break.
6B.1 FLUD Calculational Procedure The major differences between FLUD and COPDA {Ref. 6B-4} are the use of steam table curve fits (Section 68.3) instead of table look-ups and the equation of state which is a first-order virial expansion (discussed in 68.1.1 ). - The fluid flow equations (compressible equations! HEM model and integrated momentum equation) used in COPDA have been reproduced in the FLUD Code. lt may be observed from the FLUD flowchart in Fig. 68-_1 that the calculatronal procedures for FLUD and COPDA are very similar.
6B.1.1 Equation of State In this section we describe how FLUD detennines the thermodynamic state for each compartment in a system of interconnected compartments.
Our thermodynamic system {compartment) is assumed to be in equilibrium. The states assumed by the air-steam-water mixture can *be described in terms of thermodynamic coordinates P, V. and T referring to the mixture as a whole. The equation of state is derived from a first order virial expansion as presented in Ref. 6B-1. Using the molecular theory of gasest the following equation of state for an air-steam mixture is obtained assuming negligible air-steam molecular interaction:
(Eq.68-1)
FSAR Rev. 59 68-1
SSES-FSAR Text Rev. 55 where the temperature dependence of the second virial coefficient for steam Bs(T) ls given by2 75 3137 3 2659 2 Bs (T) = 0.0330-
- 10
- 1(r x10-5 + 1.1308) (Eq. 6B- 2)
T .
Eq. 68-1 can be rewritten as the sum of the partial pressure of air Pa and the partial pressure of steam Ps where T
P0
= j\,fV{} ( )
RJ, lbf I ft 2 = 0.37043-, (psia) (Eq.68*.1)
V 11 and (Eq. 6B- 4)
Eq. 6B-4 compares well with the steam tables.2 For example, the relative error in Eq. 6B-4 is less than 1 % for saturated steam at temperatures less than 570°F.
6B.1.2 Compartment Thermodynamic State At any time, the total internal energy E, the air mass Ma, and the vapor mass Mv have know vatues for each compartment. Vapor is defined as a homogeneous mixture of steam and water in unknown proportions.
The internal energy is a function of as many thermodynamic coordinates as are necessary to specify the state of the system. Therefore, for known air and vapor masses and because the compartment volume is originally specified, the compartment internal energy can be expressed as a function of temperature only:
E=E(T) (Eq. 6B-5)
At the saturation temperature T 0, there is a discontinuous change in the slope of E(T) due to a phase change in the compartment atmosphere. Associated with To is the compartment saturation energy E0=E(T0). Equation 6B-5 has two branches: (1) a two-phase branch were E<Eo and T <TO and (2) a superheat branch where E>Eo and T> To*
Along the two-phase branch the vapor portion of the atmosphere has a non-zero water mass component, while along the superheat branch the vapor contains no water.
Having examined the behavior of E(T), we now proceed to solve EO. 6B-5 for the compartment temperature, E be,ng known. Vsat, esat and Vw, ew represent the specific volumes and specific internal energies of saturated steam and water respectively. The dependence of these quantities on temperature is determined empirically from steam FSAR Rev. 59 68-2
SSES-FSAR Text Rev. 55 table curve fits described in Section 6B-3. E0 is calculated to determine on which branch of E(T) the compartment temperature lies. At compartment saturation, the steam mass Ms is identical to Mv and the specific volume of the steam is just Ysat (T0 } .
- Thus, V =My,ur (T.,) (Eq . 6B - 6)
The above equation is easily solved to To by utilizing the inverse of the function Vsat (To),
which is also a steam table curve fit where TO = T sat (V/Mv). The saturation internal energy for the compartment is then given by (Eq. 6B- 7) where Cva = 0.1725 Btu/lbm 0 R is the specific heat at constant volume for air averaged over the temperature range -109.7 to 440.3°F. For the case E<E 0 (the two-phase branch), the explicit dependence of E on Ma, Ms, Mw, and T is (Eq. 6B- 8)
The functions es (Ps,T) and ew(T) are the specific internal energies of steam and water respectively and are also discussed in Section 68. 3. The steam and water masses are functions of temperature only and are given by (Eq . 6B- 9) and where the steam quality x(T) is defined by the following:
(Eq. 6B-10)
For the case E>E 0 (the superheat branch), the explicit dependence of Eis given by (Eq. 68* 11)
The steam mass Ms is not a function of temperature since it is equal to the vapor mass Mv, and of course the water mass is zero.
FSAR Rev. 59 68-3
SSES-FSAR Text Rev. 55 Because Eis a complex function of T as seen by the above, EQ. 6B-5 does not readily lend its~lf to a strictly analytical solution. Instead, FLUD employs a one-pass iterative technique to solve for the temperature .
6B.1 .3 Compartment Initial Conditions The initial thermodynamic state is specified for each compartment by the total compartment pressure P, and the compartment volume V, temperature T, relative humidity <Pr and vapor quality x.
If (!><1.0, the compartment is superheated, the vapor consists entirely of steam, and the steam mass is given by definition as 1'.-1 =i+._V_ (Eq . 68 - 12)
J '+' V (T)
JUI The steam partial pressure is obtained from Eq. 6B-4, and thus the air mass is given by Eq. 6B-3. The internal energy is calculated using Eq. 68-11. If q,=1.0 and x;:;1.0, the compartment is saturated. The steam partial pressure is given by the saturation pressure is given *by the saturation pressure Ps=Psat(T). The saturation pressure of steam Psat is obtained empirically from a curve fit to the steam tabtes. The steam mass is given by Eq. 6B-12 with ¢>=1.0. The vapor mass is identically equal to the steam mass, and the internal energy is computed from Eq. 6B-7. For $=1 .. 0 and x<1 .0, the compartment is two-phase. The vapor and steam masses are given by Eq. 68~9 and the water mass by Eq. 68-9. The steam partial pressure is equal to the saturation pressure P=Psat(T). Therefore, the air mass can be calculated from Eq. 6B-3. However, because the compartment now contains water, the volume accessible to the air and steam V9 is just (Eq. 68-13)
This gas volume V 9 must be used in place of V in Eq. 6B-3 in determining the air mass.
The internal energy is obtained from Eq. 68-8.
6B.1.4 Air and Vapor Component Flow Rates The time-dependent partial pressure of steam is given by Eq. 6B-4 where Vs replace V/Ms. The time-dependent air specific vorume Va is then obtained from Eq. 6B-3. Time-dependent air and steam mass fractions are then calculated as follows:
(Eq. 66*14)
(Eq. 6B .. 14)
Text Rev. 55 The flow rates of the air and vapor components that comprise the gas are calculated from the total flow rate M1 by using the mass fractions of air and vapor in the upstream compartment:
(Eq. 6B-15)
(Eq. 6B-16) 68.2 Energy Transfer Mechanisms There are several mechanisms by which FLUD transfers energy to and from the various compartments and the atmosphere. These mechanisms are:
( 1) Slowdown energy (2) Flow of energy between compartments (3) Compartment heat loads (4) Compartment unit coolers All of these mechanisms add or subtract energy from the system. A continuous accounting of aU energy contributors is kept by FLUD in the form of an overall energy balance to ensure energy conservation. The various energy transfer mechanisms are discussed and the energy balance are discussed below.
6B.2.1 Slowdown Energy Slowdown energy is added to the system of compartments when FLUD is used to analyze a high-energy pipe break problem. The blowdown flow rate M8 specific enthalpy hs, and the split among compartments are assumed to be given as input data.
The rate of energy addition to the system by blowdown H8 is usually a time-varying quantity given by (Eq. 6B
- 17)
This variable energy rate is used to calculate the amount of energy that is placed in one or in the various break compartments during each time step. The total amount of blowdown energy added to the system is the integral of H8 .
(Eq.6B-18)
SSES-FSAR Text Rev. 55 The blowdown energy rate added to the ith compartment is calculated by multiplying the user-supplied split fraction for the ith compartment times the total blowdown energy rate in Eq. 68-17.
6B.2.2 Enthalpy Flow Whenever mass is transferred between compartments or between a compartment and the atmosphere, there is an associated transfer of energy based upon the enthalpy of the upstream compartment. The general relation used to calculate enthalpy flow between compartments is H. = ~ M. h~
I ~ tj /J (Eq.6B-19) where h; represents the total specific enthalpy of the gas in the upstream compartment and Mij rs the flow rate between compartments i and j as discussed in 6B.1.4. The total enthalpy flow rate for the system is (Eq. 6B- 20)
When energy transfer occurs between a compartment and the atmosphere, the relation used to calculate this flow is (Eq.6B-21)
Here Mii represents the total flow from or to the atmosphere from component i and h;;
is the specific enthalpy of the upstream compartment (which may be either compartment i or the atmosphere depending upon the sign of M; 1 ). The total enthalpy flow rate to the atmosphere is
{Eq. 68- 22) and the total amount of energy transferred to the atmosphere is (Eq. 6B- 23) 6B.2.3 Compartment Heat Loads Heat is generated within a compartment in the case where pumps or equipment are operating in that compartment. These heat loads are given with the input data as a FSAR Rev. 59 6B-6
SSES-FSAR Text Rev. 55 const~nt heat rate (Btu/sec) for each compartment 01oad. These heat loads are assumed to be applicable throughout the problem under consideration.
6B.2.4 Unit Coolers Unit coolers or room coolers are present in many situations, especially in compartments that have equipment capable of generating large heat loads. Room coolers can have a variable start temperature which is specified in the input data. The coolers are usually set to begin operating when the compartment temperature exceeds some prescribed limit.
The cooling heat transfer rate is given by (Eq. 6B- 24) where T 000 1 is the cooler cold water inlet temperaturet T is the temperature of the
. compartment, and oc is the cooler constant (Btu/sec-0 R). The cooler constant can be calculated from room cooler specifications and is assumed to be constant throughout the temperature ranges of the room atmosphere and the cooling water temperature.
68.2.5 Energy Balance The energy balance given by the following equations is used to ensure that energy conservation is achieved.
(Eq. 6B-25) where Ei is the total energy in the ith compartment! E1 (0) is the initial compartment energy, and (Eq. 6B* 26)
If an energy balance is achieved, then Eba1 shoutd be zero.
6B.2.6 Blowout Panel Activation Blowout panels are treated as instantaneous one-way switches. Once a blowout panel set pressure is exceeded. the flowpath is open for the duration of the calculation. The actual activation of a blowout panel is made by setting the forward and reverse set pressures equal to zero once the forward set pressure has been exceeded.
SSES-FSAR Text Rev. 55 68.2. 7 Energy and Mass Conservation Energy and mass conservation is then checked by calculating the following quantities:
(Eq. 68- 27) and
{Eq. 6B- 28)
If all mass and energy transfer has been accounted for, then Eba1 and Mbal should be zero (or a very small percentage of the total energy and mass due to computer round-off error).
6B.2.8 Eulerian Integration The time-dependent quantities listed below are integrated according to the following general scheme:
X(T + M) = X(t) + X(t)M (Eq. 6B- 29) where Xis any time dependent variable and X is its time rate of change. The variables intergrated by FLUD are:
Ha blowdown enthalpy flow rate Ma blowaown mass flow rate E energy rate of change Hatm - atmospheric enthalpy flow rate Ma air mass flow rate MV vapor mass flow rate Matm - atmospheric mass flow rate FSAR Rev. 59 68-8
SSES-FSAR Text Rev. 55 6B.3 Thermodynamic Properties of Steam, Water, and Air FLUD uses steam, air, and water properties for various thermodynamic calculations which are performed during each time step. The thermodynamic variables needed in FLUD calculations are:
specific internal energy of air Psat(T) saturation pressure of steam Vsat(T} saturation specific volume of steam es(T,P) specific internal energy of steam Vw(T) specific volume of water ew(T) specific internal energy of water saturation temperature of steam Tsat(V} saturation temperature of steam saturation specific internal energy of steam hsat(T) saturation specific enthalpy of steam ht9(P) enthalpy of vaporization of steam The "known" quantities that can be used to calculate the above nine variables are the macroscopic compartment thermodynamic variables pressure, specific volume , and temperature, P, v, and T respectively.
The air and water properties ea(T)t Vw(T), and ew(T) are calculated by spline 2 3 fitting polynomials to data in the steam and gas tables , . The air property ea(T) was found to be adequately represented by a linear fit. This is no doubt due to the good "ideal gas behavior of air. Thus, (Eq . 6B-30)
The water properties Vw(T) and ew(T) and the steam propertie~ hs81(T) , e0 (T), and es.-,-.(T) are very nearly straight line functions, but small variations were accommodated by using third order spline polynomial fits of the general form:
3 property (T) = ao + a1T + a2T2 + a3T (Eq. 6B-31)
FSAR Rev. 59 68-9
SSES-FSAR Text Rev. 55 For example, for ht9 (P),
2 3 h19 (P) = ao+ a,P + a2P + a3P 3
The accuracy of he curve fits the range between 0.01% and 4% for the various properties.
68.4 References 68-1 Reif, F. J. Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Co., p. 183.
68-2 Kennan, J. H. et al, Steam Tablest John Wiley & Sons, Inc., New York, 1969.
6B-3 Kennan, J. H., and J. Kaye, Gas Tables, John Wiley & Sons, Inc.,
New York, 1948.
68-4 Bechtel Topical Report BN-TOP-4 Rev. 1, October 1977, "Subcompartment Pressure and Temperature Transient Analysis." This report was approved by the NRC in February, 1979.
FSAR Rev. 59 68-10
400 EULERIAN INTEGRATION COMPARTMENT STATE POINT 100 200 300 402 READ IN CHECK INPUT, DETERMINE CALC DATA WRITE OUT INITIAL ATMOSPHERE SET CONSTANTS INPUT DATA CONDITIONS CONDITIONS 448 FLOWS 500 THERMO.
600 BLOWOUT PANELS 603 ENERGY MASS CONSERVATION 605 CHECK TIME STEP 607 CHECK PRINT STEP 700 608 STOP PRINT FSAR REV.65 SUSQUEHANNA STEAM ELECTRIC STATION UNITS 1 & 2 FINAL SAFETY ANALYSIS REPORT BASICE FLUD CALCULATION FLOWCHART FIGURE 6B-1, Rev. 47 AutoCAD: Figure Fsar 6B_1.dwg