7.7. Appendices

7.7.1. Listing of Balance-of-Plant Variables

Many of the variables used in the balance‑of‑plant subroutines are listed below in alphabetical order, together with a short definition of each variable. All input variables are included in the list and are preceded by an asterisk. In the case of a variable which has a counterpart in the sodium loop coding (e.g, the segment flow), the counterpart variable name is listed in parentheses after the variable definition.

* ALFANZ(NSEG)

nozzle angle.

* APMWHD(20,NPUMPW)

pump head coefficients and torque coefficients (APMPHD).

* APRMHT(NHTR)

the heat transfer area between the primary and secondary sides of a heater.

* AREAW(NELEW)

cross‑sectional area of an element (AREAEL).

* ARNAS

the superheater sodium flow area per tube.

* AXPMHT(NHTR)

the flow area on the primary side of a heater.

* BENDW(NELEW)

the number of bends in a flow element (BENDNM).

* CBBKCF(NSEG)

nozzle bucket coefficient.

* CHCALB(NCVLVW)

calibration constant for a check valve.

CHDELT(2,NCVLVW)

CHDELT(1) is the closure time for a check valve.

CHDELT(2) is the opening time for a check valve.

* CHEPS1(NCVLVW)

the value of the pressure drop across a check valve or the value of the mass flow rate through the check valve at which the valve begins to close.

* CHEPS2(NCVLVW)

the value of the pressure drop across a check valve or the value of the mass flow rate through the check valve at which the valve begins to open.

* CHPHIW(NCVLVW)

absolute valve characteristic value for a check valve which is fully open.

CHTIME(NCVLVW)

the time since initiation of closure or of opening of a check valve.

* CNNZCF(NSEG)

nozzle velocity coefficient.

* CONSKl(NSEG)

nozzle rotation loss coefficient.

* CONSK2(NSEG)

nozzle moisture loss coefficient.

* CONSK3

turbine exhaust loss coefficient.

* CRRXCF(NSEG)

nozzle reactor coefficient.

* CSAREA(NCVW)

the effective cross‑sectional area of a heater.

* CSARLW(NCVW)

the effective cross‑sectional area of the drain in a heater containing a drain.

* CSARUP(NCVW)

the effective cross‑sectional area of the desuperheating region in a heater containing a desuperheating region.

* CVMLTW(2,NSEGW)

the multiplicity factors at the entrance (1) and exit (2) of a flow segment (CVLMLT).

* CVPHIC(10,NCVLVW)

normalized valve characteristic for a closing check valve as a function of time since the start of valve closure.

* CVPHIO(10,NCVLVW)

normalized valve characteristic for an opening check valve as a function of the time since the start of valve opening.

* CVTIMC(10,NCVLVW)

time table for CVPHIC.

* CVTIMO(10,NCVLVW)

time table for CVPHIO.

* DEWI

the evaporator booster tube outer diameter.

* DEWIS

the superheater booster tube outer diameter.

* DEWOS

the superheater steam tube inner diameter.

* DHLWW(NCVW)

the hydraulic diameter of the drain in a heater containing a drain.

* DHNAS

the superheater sodium hydraulic diameter per tube.

* DHPMHT(NHTR)

the hydraulic diameter of the primary side of a heater.

* DHSHW(NCVW)

the hydraulic diameter of the shell side of a heater.

* DHUPW(NCVW)

the hydraulic diameter of the desuperheating region in a heater containing a desuperheating region.

* DHW(NELEW)

hydraulic diameter of an element (DHELEM).

* DNSW(NCVW)

compressible volume density.

* DOUTS

the superheater steam tube outer diameter.

* DPACC(NRVLVW)

the accumulated pressure drop for a relief valve.

* DPBLD(NRVLVW)

the blowdown pressure drop for a relief valve.

DPELW(NELEW)

pressure drop across an element (DPRSEL).

* DPSET(NRVLVW)

the set pressure drop for a relief valve.

DTSUBO

previous timestep.

* FLOWLS

the relief valve capacity at the accumulated pressure drop.

* FLOWSS(NSEGW)

steady state flow in each segment (FLOSSL).

FLOW4(NSEGW)

segment flow at the end of the current PRIMAR time subinterval. At the beginning of the subinterval, flow is given by FLOW3, and at the end of the PRIMAR timestep, it is given by FLOW2 (FLOSL2 FLOSL3, FLOSL4)

* GAMABL(NSEG)

turbine blade exit angle.

GRAVW(NELEW)

gravity head in an element (GRAVHD).

* G2PW(NELEW)

the orifice coefficient in the momentum equation (G2PRDR).

* HCVI

the initial upstream enthalpy for a relief valve.

HCVW(NCVW)

specific enthalpy of the compressible volumes.

* HEADWR(NPUMPW)

rated pump head (HEADR).

HEADW2(NPUMPW)

pump head at end of PRIMAR timestep (HEADP2).

HEADW3(NPUMPW)

pump head at beginning of timestep (HEADP3).

HEADW4(NPUMPW)

pump head at end of timestep (HEADP4).

HELEW(NELEW)

enthalpy at an element outlet.

* HIGHLW(NCVW)

the height of the drain in a heater containing a drain.

* HIGHUP(NCVW)

the height of the desuperheating region in a heater containing a desuperheating region.

* HTOTO(NHTR)

the initial value of the total heat transfer coefficient between the primary and secondary sides of a heater.

* HTRELV(NCVW)

the elevation of the lowest point of a heater.

* HTRRAD(NCVW)

the radius of a heater.

* IBOPRT

the number of PRIMAR timesteps between full balance‑of‑plant prints.

* ICHVLK(2,NCHVLV)

a flag for the criteria which trigger a check valve to open and to close. ICHVLK(1,IVLV) flags whether an open check valve will start to close based on a pressure criterion (ICHVLK = 1) or on a flow criterion (ICHVLK = 2). ICHVLK(2,IVLV) does the same for the opening criteria of a closed valve.

* ICVLEW(NCVLVW)

the element number of a check valve (initially entered as the user’s number for the element, then changed to the code’s number).

* ICVSGN(M,NSGN)

for M=1, ICVSGN is the user’s number of the compressible volume of a steam generator inlet plenum; for M=2, it is the volume number of the outlet plenum.

* IELPW(NPUMPW)

element number of waterside pump (IELPMP).

* IELVLW(NVLVW)

element number of a valve (ordered by the code‑generated valve number).

* IEMPW(NPUMPW)

type of waterside pump (IEMPMP).

* IFBWCL(NCVW)

flags whether a flow boundary condition is controlled by a table or by the control system, with IFBWCL = 0 if the boundary condition is controlled by a table, = 1 if the boundary condition is controlled by the control system.

* IHTLW(NSEGW)

the user’s number of the volume containing the drain to which the segment is attached.

* IHTUP(NSEGW)

the user’s number of the volume containing the de-superheating section to which the segment is attached.

* IHTSEG(NSEGW)

the user’s number of the heater volume (if any) through which the segment passes.

ILEGW(NLEGS)

the number of compressible volumes in each leg of the water side.

* ILRPW(NPUMPW)

flag for locked rotor (ILRPMP).

* IPMWCL(NPUMPW)

control system flag for waterside pumps (IPMPCL).

* IRVLVW(NRVLVW)

the user’s number for the element assigned to a check valve.

ISEGCV(NCVW,6)

segment numbers of segments attached to each compressible volume (maximum of 6 segments currently allowed).

* ISGIN

entries (1‑10), user number of first segment in leg,

entries (11‑20), code‑generated number of first flow boundary condition pseudo‑segment in leg,

entries (21‑30), code‑generated number of first steam generator inlet pseudo‑segment in leg.

ISGNCV(‑NCVW,6)

for each compressible volume, ISGNCV identifies the flow from each segment attached to the volume as flowing into or out of the volume at steady state. ISGNCV = 1 indicates flow into the volume, while YSGNCV = ‑1 indicates flow out of the volume.

* ISGOUT

entries (1‑10), user number of last segment in leg,

entries (11‑20), code‑generated number of last flow boundary condition pseudo‑segment in leg,

entries (21‑30), code‑generated number of last steam generator inlet pseudo‑segment in leg.

* ITYPW(NELEW)

element type for each element (ITYPEL).

* IVBWCL(NCVW)

flags whether a volume boundary condition is controlled by a table or by the control system, with IVBWCL = 0 if the boundary condition is controlled by a table, = 1 if the boundary condition is controlled by the control system.

IVLELW(NVLVW)

code‑generated element number of a valve (ordered by the code‑generated valve number).

* IVLWCL(NVLVW)

control system flag for waterside valves (IVLVCL).

IVLWCL = 0 if the control system does not control the valve,

IVLWCL = 1 if the control system controls the valve driving function, and

IVLWCL = 2 if the control system controls the valve stem position directly.

* JCVW(M,NSEGW)

compressible volume numbers at each end of a segment (M=1 at the flow inlet, M=2 at the flow outlet) (JCVL).

* JCVlFG(NSEGW)

indicates where a segment attached to a heater volume is attached to the volume, with

JCVlFG = ‑1 if the segment is attached to the bottom of the volume,

= 0 if the segment is attached in between the top and the bottom of the volume,

= 1 if the segment is attached to the top of the volume.

* JFSEW(NSEGW)

first element in a segment (JFSELL).

* JLSEW(NSEGW)

last element in a segment.

* JPRINT (17)

an array of flags through which the user selects which parameters to include in the full balance‑of‑plant print.

LEGBCK(NLEGS)

the translator array from the user’s numbering of the legs on the balance of-plant side to the code’s internal numbering of the legs.

* LEGORD(NLEGS)

lists the order in which the legs into which the balance-of‑plant is divided should be ordered in the output listing.

* LMPDOT

the number of steam generator timesteps averaged to compute the time derivative of pressure in the steam generator.

* NBCCVF(NBCFLO)

the number of the compressible volume to which the flow boundary condition pseudo-segment is attached (input as the user’s c.v. number, then changed to the code’s number).

* NBCCVP(NBCPRS)

the number of the compressible volume which serves as a boundary condition (input as the user’s c.v. number, then changed to the code’s number).

NBCFLO

number of flow boundary condition tables.

* NBCINF(NBCFLO)

table number for the time‑dependent data for the flow boundary conditions.

* NBCINP(NBCPRS)

table number for the time‑dependent data for the compressible volume boundary conditions.

NBCINT

number of interior volumes (volumes which are not boundary condition volumes).

* NBCSEG(NBCFLO)

code‑generated pseudo‑segment number for each flow boundary condition.

NBCPRS

number of volume boundary condition tables.

* NBOREL(M,NELEW)

neighboring element numbers for each element (M=1 for the upstream neighbor, M=2 for the downstream neighbor). NBOREL(1,I) = 0 for the first element in a segment and NBOREL(2,I) = ‑1 for the last element in a segment.

* NCHVST(NCVLVW)

flags the state or each check valve as follows:

= 1, valve is fully open and will begin to close if the pressure drop across the valve is less than the user‑input value CHEPS1.

= 2, valve is fully open and will begin to close if the flow through the valve is less than CHEPSl.

= 3, valve is in the process of closing.

= 4, valve is fully closed but leaking slightly and will begin to open if the pressure drop across the valve becomes greater than the user‑input value CHEPS2.

= 5, valve is fully closed but leaking slightly and will begin to open if the flow through the valve becomes greater than CHEPS2.

= 6, valve is in the process of opening.

* NCVBCW

identifies compressible volumes as boundary condition, steam generator plenum, etc., with NCVBCW

= 0 for a standard interior volume,

= 1 for a volume boundary condition volume,

= 2 for an inlet flow boundary condition volume,

= 3 for an outlet flow boundary condition volume,

= 4 for a steam generator inlet plenum,

= 5 for a steam generator outlet plenum,

= 6 for a heater volume,

= 7 for a turbine.

NCVIN(NLEGS)

user number of first compressible volume in loop.

NCVLBK(NCVLVW)

array which maps the user’s number for a check valve to the code’s number for that check valve.

NCVLTR(NCVLVW)

array which maps the code’s number for a check valve to the user’s number for the same check valve.

NCVLVW

number of check valves in the balance‑of‑plant loop.

NCVOUT(NLEGS)

user number of last compressible volume in loop.

NCVQ(NHTR)

code‑generated compressible volume number of a heater (by code‑generated heater number).

NCVW

number of compressible volumes (NCVT).

NELEW

number‑of elements (NELEMT).

* NELSGW(NELEW)

user’s number of the segment in which an element lies.

* NELSUH

the user’s element number for a superheater.

* NENTRF(NCVW)

flag for the type of floating‑point input data entered for a volume, with NENTRF

= 1 for single‑phase volumes, pressure and temperature entered,

= 2 for single‑phase volumes, pressure and enthalpyentered,

= 3 for two‑phase volumes, pressure and quality entered,

= 4 for two‑phase volumes, temperature and quality entered,

= 5 for two‑phase heater volumes, pressure, two‑phase level and ambient temperature entered,

= 6 for two‑phase heater volumes, temperature, two‑phase level, and ambient temperature entered.

* NFLSEG(NCVW)

flags the type of floating point data entered for an inflow boundary condition, with NFLSEG

= 0 if enthalpy is entered,

= 1 if temperature and pressure are entered for a subcooled liquid boundary condition,

= 2 if temperature and pressure are entered for a superheated steam boundary condition,

= 3 if quality and pressure are entered for a two‑phase boundary condition,

= 4 if quality and temperature are entered for a two‑phase boundary condition.

NHTR

number of heaters in the balance‑of‑plant.

NLEGS

number of legs (a leg is a section of the balance of plant for which all flows and volume pressures are solved simultaneously. For example, the volumes and segments from the inlet to the steam generator might be one leg (a liquid leg), and those from the steam generator to the outlet might be another leg (a vapor leg).

* NLGCVW(NCVW)

the number of the leg of the loop to which a volume belongs.

NLVOL

number of liquid compressible volumes.

* NODMAX(NSEGW)

the maximum number of enthalpy transport nodes into which a segment may be divided.

* NODSC

the number of nodes in the evaporator subcooled zone.

* NODSH

the number of nodes in the evaporator superheated zone.

* NODTP

the number of nodes in the evaporator two‑phase zone.

* NODSHT

the number of nodes in the superheater.

* NOSGW(NSGN)

user’s number for the segment which is at the outlet of the vapor leg which is fed by the steam generator (used for saving plot data only).

NPUMPW

number of pumps in the balance‑of‑plant.

* NPUTRN(NPUMPW)

user’s number of pump.

* NQFLG(NCVW)

user‑assigned heater number for a compressible volume which is a heater.

NSEGCV(NCVW)

number of segments attached to each compressible volume.

NSEGT

the number of flow segments entered by the user (NSEGLT).

NSEGW

total number of segments, including pseudo‑segments generated by flow boundary conditions and steam generator interfaces.

* NSSIN(NSSEG)

the compressible volume number at a supersegment inlet.

* NSSOUT(NSSEG)

the compressible volume number at a supersegment outlet.

* NSUPSG(NCVW)

the number of the supersegment in which a vapor volume is contained.

* NTABVL(NBCPRS)

flags the types of parameters entered in the floating point volume boundary condition table, with NTABVL

= 1 for pressure and enthalpy entered for a liquid volume

= 2 for pressure and temperature entered for a liquid volume,

= 3 for pressure and enthalpy entered for a vapor volume,

= 4 for pressure and temperature entered for a vapor volume,

= 5 for pressure and quality entered for a two‑phase volume,

= 6 for temperature and quality entered for a two-phase volume.

* NTPCVW(NCVW)

compressible volume type, with NTPCVW

= 1 for a subcooled liquid volume,

= 2 for a superheated vapor volume,

= 3 for a two‑phase volume,

= 4 for a pseudo‑volume at the liquid/two‑phase interface in an evaporator.

NTPELW(NELEW)

state of an element, with NTPELW

= 1 for a subcooled liquid element,

= 2 for a superheated vapor element,

= 3 for a two‑phase element.

* NTRNPT

flags whether or not enthalpy transport is used in the vapor leg, with NTRNPT

= 0 if enthalpy transport is used,

= 1 if enthalpy transport is not used.

NVLBCK(NVLVW)

array which takes the number assigned to a valve by the user and gives the number assigned to the valve by the code.

NVLTRN(NVLVW)

array which takes the code‑generated number assigned to a valve and gives the number assigned to the valve by the user.

NVLVW

number of valves in the balance‑of‑plant (NVALVE).

* OMEGAR

turbine rotor angular velocity.

* ORIFLW(NCVW)

the elevation of the drain orifice in a heater which contains a drain.

* ORIFUP(NCVW)

the elevation of the desuperheating region orifice in a heater which contains a desuperheating region.

* PCVI

the initial upstream pressure for a relief valve.

* PCVO

the initial downstream pressure for a relief valve.

* PELEW(NELEW)

pressure at an element outlet.

* PMPFWR(NPUMPW)

rated pump flow (PMPFLR).

PMPHDW

coefficients in centrifugal pump option 2 (PMPHD).

* PMPSWR(NPUMPW)

rated pump speed (PMPSPR).

PMPTQW

torque coefficients in cent. pump option 2 (PMPTQ).

* PMWEFR(NPLPMPW)

pump efficiency (PMPEFR).

* PMWINR(NPUMPW)

moment of inertia, pump and motor (PMPINR).

PMWTQR(NPUMPW)

steady state pump torque (PMPTQR).

* PRESW4(NCVW)

pressure in each compressible volume at the end of the current PRIMAR time subinterval (PRESL4). Pressure at the beginning of the subinterval is PRESW3, and the pressure at the end of the PRYMAR timestep is PRESW2.

PSPDW2(NPUMPW)

pump speed at start of PRIMAR timestep (PSPED2).

PSPDW3(NPUMPW)

pump speed at start of timestep (PSPED3).

PSPDW4(NPUMPW)

pump speed at end of timestep (PSPED4).

* QRATIO(NCVW)

the percentage of incoming energy to a heater lost due to imperfect insulation.

* ROUGHW(NELEW)

the roughness of an element wall (ROUGHL).

* RROTOR(NCVW)

the radius of a turbine rotor.

* RVA(NRVLVW)

the fractional valve area to which a relief valve opens when the set pressure drop is reached.

* RVFRAC(NRVLVW)

the fractional relief valve opening area.

SEGLW(NSEGW)

length of a segment.

* SHHTCC(NCVW)

the shell side condensation coefficient for a heater.

* TABSEG(10,3,NBCFLO)

table for flow boundary condition input data. TABSEG(x,1,y) contains time, TABSEG(x,2,y) contains absolute flows, and TABSEG(x,3,y) contains enthalpies.

* TABVOL(10,4,NBCPRS)

table for compressible volume boundary condition input data. TABVOL(x,1,y) contains time, TABVOL(x,2,y) contains pressures, TABVOL(x,3,y) contains enthalpies, and TABVOL(x,4,y).contains qualities.

* TAMBNT(NCVW)

the ambient temperature for a heater volume.

* TBCP(NELEW)

the specific heat of the tube in an element representing a heater tube bundle.

* TBKPMO(NELEW)

the thermal conductivity of the tube in an element representing a heater tube bundle.

* TBLNLW(NELEW)

the length of the section of the element within the drain for an element representing a tube bundle in a drain cooler or desuperheater/drain cooler.

* TBLNUP(NELEW)

the length of the section of the element within the desuperheating section for an element representing a tube bundle in a desuperheating heater or a desuperheater/drain cooler.

* TBNDLW(NELEW)

the number of nodes for the section of the element within the drain for an element representing a tube bundle in a drain cooler or desuperheater/drain cooler.

* TBNDUP(NELEW)

the number of nodes for the section of the element within the superheating section for an element representing a tube bundle in a desuperheating heater or desuperheater/drain cooler.

* TBNMBR(NELEW)

the total number of tubes in a heater tube bundle.

* TBNODE(NELEW)

the number of nodes for the heat transfer calculation in an element representing a heater tube bundle.

* TBPODS

the superheater bundle pitch‑to‑diameter ratio.

* TBRHO(NELEW)

the tube material density in an element representing a heater tube bundle

* TBTHIK(NELEW)

the tube thickness in an element representing a heater tube bundle.

* TCVW(NCVW)

compressible volume temperature (TLQCV2).

* TEMPLW(NCVW)

the temperature of the drain in a heater containing a drain.

* TEMPUP(NCVW)

the temperature in the desuperheating region in a heater containing a desuperheating region.

* TIMERV(NRVLVW)

the relief valve delay time for opening or closing.

* TPFACE(NCVW)

the two‑phase level in a volume in which liquid and vapor are separated.

TQMBW3(NPUMPW)

motor torque at start of timestep (TQMB3).

TQMBW4(NPUMPW)

motor torque at end of timestep (TQMB4).

TQPBW3(NPUMPW)

pump torque at start of timestep (TQPB3).

TQWSAV(NPUMPW)

torque from PUMPFL (TQBSAV).

* TRGRMI

turbine/generator rotor moment of inertia.

* TRKLSW(NPUMPW)

windage (TRKLSC).

TRQMSW(NPUMPW)

initial steady state speed (TRQMSS).

* TSECHT(NHTR)

the temperature of the secondary fluid in a heater.

* TUBNOS

the number of superheater tubes.

VCALBW(NVLVW)

calibration constant for a standard valve.

* VCONSW(NVLVW)

the proportionality constant between the stem position and the valve characteristic for a standard valve.

* VDAMPW(NVLVW)

damping coefficient for the valve stem position equation.

VDRIVW(NVLVW)

driving function for the valve stem position equation.

* VLVMSW(NVLVW)

valve mass.

* VPHINW(NVLVW)

valve characteristic at the current PRIMAR subinterval.

VPHIW(10,NVLVW)

valve characteristic curve for a standard valve.

* VPOSW(10,NVLVW)

valve stem position for points in VPHIW.

* VSPRGW(NVLVW)

spring constant for the valve stem position equation.

* VSTEMW(NVLVW)

valve stem position.

VSTMWl(NVLVW)

valve stem position from the previous timestep.

* VTABDW(10,NVLVW)

table of driving function vs. time for a standard valve (this array is used to vary driving function with time if the control system is not used to control the valve).

* VTIMW(10,NVLVW)

values of time for VTABDW.

* VOLCVW(NCVW)

volume of each compressible volume (VOLLGC).

* VOLLW(NCVW)

the volume of the drain in a heater containing a drain.

* VOLUP(NCVW)

the volume of the desuperheating region in a heater containin desuperheating region.

* WMOTTK(20,NPUMPW)

motor torque table and times (AMOTTK).

* XCVW(NCVW)

compressible volume quality.

* XKTUBE

the evaporator tube thermal conductivity.

* XLENLW(NCVW)

the length of the drain in a heater containing a drain.

* XLENUP(NCVW)

the length of the desuperheating region in a heater containing a desuperheating region.

* XLENW(NELEW)

length of an element (XLENEL).

* XRXFR(NSEG)

nozzle reaction fraction.

* ZCVW(NCVW)

compressible volume midpoint elevation (ZCVL).

* ZINW(NSEGW)

elevation of the segment inlet (ZINL).

* ZLOWST(NELEW)

the lowest elevation of the element within the heater for an element representing a heater tube bundle.

* ZONLE(3)

the zone lengths in the evaporator. ZONLE(1) is the subcooled zone length, and ZONLE(3) is the superheated zone length; these are both input, with ZONLE(2) (the two‑phase zone length) calculated from EL, ZONLE(1), and ZONLE(3) (ELEV).

* ZOUTLW(NELEW)

elevation of the element outlet (ZOUTEL).

7.7.2. Steam Generator Water-Side Heat Transfer Correlations

Subcooled Water

The Dittus-Boelter correlation [7-4] is used.

\[N = 0.023 Re^{0.8} Pr^{0.4} = \frac{h_{w} D_{H}}{k}\]

Bulk liquid properties are used to calculate \(Re, Pr\) and \(k; h_{w}\) is the heat transfer coefficient between the wall surface and the bulk water.

Nucleate Boiling Water

A correlation developed by Thom, et al. [7-5] is used.

\[h_{w} = 3.1968 \left( e^{P / 8.65 \times 10^{6}} \right) \frac{1}{0.072} \left( q \right)^{0.5}\]

\(h_{w}\) is the heat transfer coefficient between the wall surface and the bulk water; \(q = H_{T} \left( T_{m} - T_{sat} \right)\) where \(H_{T}\) is defined as in Eq. (7.3-66); \(T_{m}\) is the average wall temperature; \(P\) is the steam generator pressure.

Film Boiling Water

A correlation of A. A. Bishop et al. [7-6] is the following

\[N = 0.0193 Re^{0.8} Pr^{1.23} \left[x + \left(1 - x \right) \frac{\rho_{g}}{\rho_{f}} \right]^{0.68} \left( \frac{\rho_{g}}{\rho_{f}} \right)^{0.068} = \frac{h_{w} D_{H}}{k}\]

A modification of the original formulation is used. The original formulation specified that properties appropriate for the wall film temperature be used to calculate \(Re\) and \(Pr\). All temperature-dependent properties are calculated with \(T_{sat}\). The wall film temperature is only crudely approximated and the relevant properties are very insensitive to temperature when above \(T_{sat}\). \(h_{w}\) is the heat transfer coefficient between the wall surface and the bulk water and \(x\) is the local nodal quality. The mass flux used in the Reynold’s number is the local value.

Superheated Steam

A correlation developed by A. A. Bishop [7-7] is used.

\[N = 0.0073 Re^{0.886} Pr^{0.61} = \frac{h_{w} D_{H}}{k}\]

Although the original correlation specified that the film temperature be used to evaluate the relevant properties, the bulk temperature is instead used since these properties are very insensitive to temperature above \(T_{sat}\). \(h_{w}\) is the heat transfer coefficient between the wall surface and the bulk steam.

Liquid Sodium Heat Transfer

A variation of the Maresca-Dwyer correlation is used [7-8]. The heat transfer coefficient between the outside wall and the bulk sodium is the following.

\[H_{Na} = \frac{N c_{p} \mu}{Pr D_{H}}\]

where \(c_{p}\), \(\mu\) and \(D_{H}\) are the specific heat, viscosity and hydraulic diameter respectively. The Prandtl number is computed according to the following,

\[Pr = 0.00212 + 2.329 / \left( 1.8 T - 410.92 \right)\]

Nusselt numbers are computed for both turbulent flow and molecular conduction. The greater of the two is used. For turbulent flow,

\[Nu = 6.66 + 3.126 POD + 1.184 POD^{2} + 0.0155 \left( Pr Re S \right)^{0.86}\]

where \(POD\) is the tube pitch-to-diameter ratio. \(S\) is calculated according to the following,

\[S = 1.0 - 1.82\ / \left( Pr E \right)\]

and \(E\) is given by,

\[E = 0.000175 Re^{1.32}\ /\ POD^{1.5}\]

The Nusselt number for molecular conduction is given by the following,

\[Nu = 6.4353 + 3.97 POD + 1.025 POD^{2} - 29494 / (Re + 20363)\]

7.7.3. Two-Phase Interface Solution Scheme for Heater Cylinders Lying on the Side

The solution scheme described below is used to solve for the two-phase interface, for heater cylinders lying on the side, form the transcendental equations given in Eq. (7.4-8) and Eq. (7.4-9). Since the problem is symmetric with respect to the center point of the cylinder, as is obvious by looking at Figure 7.4.3, only Eq. (7.4-8) is used to demonstrate the solution scheme.

By defining the angle between the vertical section \(L\) and the radius \(r_{s}\) as \(\beta\) in Figure 7.4.3, Eq. (7.4-8) can be rewritten as,

(7.7-1)\[\alpha_{s} = \left[ \beta - \frac{\text{sin} \left( 2 \beta \right)}{2} \right] / \pi\]

for \(0 \leq \alpha_{s} \leq 1/2\) and \(0 \leq \beta \leq \pi / 2\). The height of the vapor region \(A_{g}\) corresponding to \(\alpha_{s}\) in Figure 7.4.3 is \(r_{s} - L\), which can be normalized to the cylinder radius \(r_{s}\) as

(7.7-2)\[\omega \equiv \frac{r_{s} - L}{r_{s}} = 1 - \text{cos} \beta\]

The aim now is to obtain an expression for \(\beta\) as a function of \(\alpha_{s}\), in order to avoid using an iterative solution to find \(\beta\). The most straight-forward way would be to generate a polynomial expression in \(\alpha_{s}\) for \(\beta\). However, a study of the curvature of the \(\beta\) vs. \(\alpha_{s}\) curve indicates that the slope is very steep at \(\beta\) close to zero, changes dramatically as \(\beta\) increases, and levels off as \(\beta\) approaches \(\pi / 2\). A single polynomial of high order to approximate the curve is difficult to obtain without unacceptable errors in some part of the curve, and the computation of the polynomial may be time-consuming. Thus, an alternative method is used and is described as follows.

The range of \(\alpha_{s}\) on the \(\beta\) vs. \(\alpha_{s}\) curve is divided into three regions based on the slopes along the curve, i.e., \(0 \leq \alpha_{s} < \alpha_{1}\), \(\alpha_{1} \leq \alpha_{s} < \alpha_{2}\), and \(\alpha_{2} \leq \alpha_{s} \leq 1/2\), where \(\alpha_{1}\) and \(\alpha_{2}\) are chosen to be at the suitable values 0.015 and 0.225, respectively. Also, the polynomials of \(\beta\) for each region are given the forms,

(7.7-3)\[\beta = C_{1} \left(1.5 \alpha_{s} \right)^{1 / 3}, 0 \leq \alpha_{s} < \alpha_{1}\]
(7.7-4)\[\beta = \frac{\beta_{1} + C_{2} X_{1} + C_{3} X_{1}^{2}}{1 + C_{4} X_{1} + C_{5} X_{1}^{2}}, \alpha_{1} \leq \alpha_{s} < \alpha_{2}\]
(7.7-5)\[\beta = \frac{\beta_{2} + C_{6} X_{2} + C_{7} X_{2}^{2}}{1 + C_{8} X_{2} + C_{9} X_{2}^{2}}, \alpha_{2} \leq \alpha_{2} < 1 / 2\]

where \(\beta_{1}\) and \(\beta_{2}\) are values of \(\beta\) corresponding to \(\alpha_{1}\) and \(\alpha_{2}\), respectively, \(X_{1}\) and \(X_{2}\) are defined as

\[X_{1} = \left( \alpha_{2} - \alpha_{1} \right)\]

and

\[X_{2} = \left( \alpha_{s} - \alpha_{2} \right)\]

and the coefficients \(C_{1}\) through \(C_{9}\) are determined using a least squares fit separately on each region. The coefficient values are as follows:

  • \(C_{1} = 1.00377\)

  • \(C_{2} = 1.41595 \times 10^{1}\)

  • \(C_{3} = 6.98089 \times 10^{1}\)

  • \(C_{4} = 2.76847 \times 10^{1}\)

  • \(C_{5} = 4.47906 \times 10^{1}\)

  • \(C_{6} = 2.37819\)

  • \(C_{7} = 6.38891 \times 10^{-1}\)

  • \(C_{8} = 1.72658\)

  • \(C_{9} = 4.94197 \times 10^{2}\)

More significant digits for \(C_{1}\) through \(C_{9}\) are used in the coding. The polynomials Eq. (7.7-3) through Eq. (7.7-5) are chosen such that continuity conditions are satisfied at \(\alpha_{s}\) equal to \(\alpha_{1}\) and \(\alpha_{2}\).

Once \(\beta\) is calculated from one of Eq. (7.7-3), Eq. (7.7-4), and Eq. (7.7-5) for a given \(\alpha_{s}\), the two-phase interface can be computed as

\[TP = \left(1 - \omega \right) r + CV,\]

where \(\omega\) is given in Eq. (7.7-2).

The maximum error, defined as the difference between \(\omega\), as calculated from Eq. (7.7-2) with \(\beta\) obtained by Eq. (7.7-3), Eq. (7.7-4), or Eq. (7.7-5), and the actual \(\omega\), is within \(\pm 4.53 \times 10^{-4}\). If more accuracy is needed, a better value for \(\omega\) can be obtained by introducing the Newton iteration method and using the calculated \(\omega\) With one iteration, the maximum error could be reduced to \(\pm 2.67 \times 10^{-7}\), and with two iterations, to \(\pm 9.17 \times 10^{-13}\).

7.7.4. Dictionary of Steam Generator Model Variables

Variables in COMMON

Note: Variables in block SGEN1, SGEN2, and SGEN3 apply to the once-through steam generator or to the evaporator in the recirculation type steam generator. Variables in blocks SGENS1, SGENS2, and SGENS3 apply to the superheater only.

Name

Block

Units

Explanation

ARM

SGEN2

\(m^{2}\)

Cross-sectional area of tube wall

ARMS

SGENS2

\(m^{2}\)

Cross-sectional area of tube wall

ARNA

SGEN2

\(m^{2}\)

Flow area of sodium

ARNAS

SGENS2

\(m^{2}\)

Flow area of sodium

ARW

SGEN2

\(m^{2}\)

Flow area of water

ARWS

SGENS2

\(m^{2}\)

Flow area of water

AWB(100)

SGEN1

-

Void fraction at each node in boiling zone at beginning of time step

AWE(100)

SGEN1

-

Void fraction at each node in boiling zone at end of time step

CNAFRS

SGENS2

-

Constant used in sodium heat transfer coefficient calculation; equal to \(6.66 + \left( 1.184 TBPODS + 3.126 \right) TBPODS\)

CNAFR1

SGEN2

-

Constant used in sodium heat transfer coefficient calculation; equal to \(6.66 + \left( 1.184 TUBPOD + 3.126 \right) TUBPOD\)

COILD

SGEN2

\(m\)

Average diameter of coil in helical coil in the helical coil geometry option for evaporator/steam generator model

COILDS

SGENS2

\(m\)

Average diameter of the coil for the helical coil geometry option in the superheater model

DDW

SGEN2

\(m\)

Hydraulic diameter on water side

DDWS

SGENS2

\(m\)

Hydraulic diameter on water side

DELP24

SGEN2

-

Fraction of total pressure drop in subcooled zone

DEWI

SGEN2

\(m\)

Booster tube outer diameter on water side

DEWIS

SGENS2

\(m\)

Booster tube outer diameter on water side

DEWO

SGEN2

\(m\)

Tube wall inner diameter

DEWOS

SGENS2

\(m\)

Tube wall inner diameter

DOUT

SGEN2

\(m\)

Tube wall outer diameter

DOUTS

SGENS2

\(m\)

Tube wall outer diameter

DHNA

SGEN2

\(m\)

Hydraulic diameter on sodium side

DHNAS

SGENS2

\(m\)

Hydraulic diameter on sodium side

DZONE(2)

SGEN2

\(m / s\)

Velocity of subcooled and superheat zone boundaries

FACT1

SGEN2

-

Not currently used

FACT1S

SGENS2

-

Not currently used

FACT2

SGEN2

-

\(1.04 * 10^{4} * TUBPOD^{1.5}\); used in sodium heat transfer coefficient calculation

FACT2S

SGENS2

-

\(1.04 * 10^{4} * TBPODS^{1.5}\); used in sodium heat transfer coefficient calculation

FACT3

SGEN2

\(m^{-1}\)

\(\pi * DEWO / ARW\)

FACT3S

SGENS2

\(m^{-1}\)

\(\pi * DEWOS / ARWS\)

FACT4

SGEN2

\(m^{-1}\)

\(\pi * DOUT / ARNA\)

FACT4S

SGENS2

\(m^{-1}\)

\(\pi * DOUTS / ARNAS\)

FACT5

SGEN2

-

\(ARNA / ARM\)

FACT5S

SGENS2

-

\(ARNAS / ARMS\)

FACT6

SGEN2

-

\(ARW / ARM\)

FACT6S

SGENS2

-

\(ARWS / ARMS\)

FACT7

SGEN2

\(K\)

Constant used in viscosity function

FACT8

SGEN2

\(kg / m^{3}\)

Constant used in viscosity function

FACT9

SGEN2

\(J / m^{3} - K\)

Density x specific heat for tube wall

FACT10

SGEN2

-

\(0.023 * DDW^{-0.2}\); used in subcooled heat transfer coefficient

FACT11

SGEN2

-

\(0.0193 * DDW^{-0.2}\); used in film boiling heat transfer coefficient

FACT12

SGEN2

-

\(0.0073 * DDW^{-0.114}\); used in superheat zone heat transfer coefficient

FACTSP

SGENS2

-

\(0.0073 * DDWS^{-0.114}\); used in superheater heat transfer coefficient

FOULR(4)

SGEN2

\(m^{2} - K / w\)

Tube wall heat resistance on the water side plus any fouling heat resistance on water side for each heat transfer regime

FOULRI(4)

SGEN2

\(m^{2} - K / w\)

Fouling heat resistances on the water side for each heat transfer regime

FOULRS

SGENS2

\(m^{2} - K / w\)

Tube wall heat resistance on the water side plus any fouling heat resistance on water side

FOULSI

SGENS2

\(m^{2} - K / w\)

Fouling heat resistance on the water side

FRIC1(4)

SGEN2

-

Normalizing friction factor in Eq. (7.3-56) and Eq. (7.3-94) for each heat transfer regime

FRIC1S

SGENS2

-

Normalizing fraction factor in superheater

GNA

SGEN2

\(kg / m^{2} - s\)

Sodium side mass flow

GNAS

SGENS2

\(kg / m^{2} - s\)

Sodium side mass flow

GWB(100)

SGEN1

\(kg / m^{2} - s\)

Water side mass flow at each node at beginning of time step

GWE(100)

SGEN1

\(kg / m^{2} - s\)

Water side mass flow at each node at end of time step

GWS

SGENS2

\(kg / m^{2} - s\)

Sodium side mass flow

HD

SGEN2

\(\left(J / kg \right) / \left( BTU / lb \right)\)

Conversion factor for enthalpies since functions are in \(BTU / lb\)

HDNB

SGEN2

-

Fraction of cell where DNB point lies which is in the nucleate boiling regime

HFG

SGEN2

\(J / kg\)

\(h_{fg}\)

HFSAT

SGEN2

\(J / kg\)

\(h_{f}\)

HFSATP

SGEN2

\(BTU / lb\)

\(h_{f} / HD\)

HGSAT

SGEN2

\(J / kg\)

\(h_{g}\)

HGSATP

SGEN2

\(BTU / lb\)

\(h_{g} / HD\)

HN(520)

SGEN2

s

Array which stores all potential time steps from which is selected the minimum

HSTEP

SGEN2

s

Primary loop time step

HTF(4)

SGEN2

-

Calibration factors for heat transfer coefficients for each regime

HTFI(4)

SGEN2

-

Calibration factors for heat transfer coefficients for each regime

HTFS

SGENS2

-

Calibration factor for heat transfer coefficient in superheater

HTW(100)

SGEN2

\(w / m^{2} - K\)

Heat transfer coefficient between the tube wall surface and the bulk water by cell center

HTWS

SGENS2

\(w / m^{2} - K\)

Heat transfer coefficient between the tube wall surface and the bulk water by cell center in superheater

HUNIT

SGEN2

\(s\)

Current steam generator time step

HUNITN

SGEN2

\(s\)

Newly selected steam generator time step for next step

HUNITS

SGENS2

\(s\)

Current superheater time step

HWB(100)

SGEN1

\(J / kg\)

Enthalpy by node at beginning of step

HWBS(100)

SGENS1

\(J / kg\)

Enthalpy by node at beginning of step

HWE(100)

SGEN1

\(J / kg\)

Enthalpy by node at the end of step

HWES(100)

SGENS1

\(J / kg\)

Enthalpy by node at the end of step

H1MIN

SGENS2

\(w / m^{2} - K\)

Minimum value allowed for \(HTW\) in subcooled zone

H2MIN

SGEN2

\(w / m^{2} - K\)

Minimum value allowed for \(HTW\) in nucleate boiling zone

H3MIN

SGEN2

\(w / m^{2} - K\)

Minimum value allowed for \(HTW\) in film boiling zone

H4MIN

SGEN2

\(w / m^{2} - K\)

Minimum value allowed for \(HTW\) in superheated zone or for :math:HTWS` in superheater

IDNB

SGEN3

-

Cell number when \(DNB\) point occurs

IDNBL

SGEN3

-

Value of \(IDNB\) during previous time step

ISTEPW

SGEN3

-

Number of current primary loop time step

LAR

SGEN3

-

Array size limit for nodal arrays

LIM

SGEN3

-

Length of \(SGEN1 COMMON\) block

NCOUNT

SGEN3

-

Number of steam generator time substeps within primary loop step

NODSC

SGEN3

-

Number of cells within subcooled zone

NODSCO

SGEN3

-

Initial value of \(NODSC\)

NODSC1

SGEN3

-

Node number of subcooled/boiling boundary; \(NODSC + 1\)

NODSC2

SGEN3

-

\(NODSC1 + 1\)

NODSH

SGEN3

-

Number of cells within superheated zone in steam generator or evaporator

NODSHO

SGEN3

-

Initial value of \(NODSH\)

NODSHT

SGENS3

-

Number of cells within superheater

NODSH0

SGEN3

-

Total number of cells in steam generator or evaporator; \(NODSC + NODTP + NODSH\)

NODSH1

SGEN3

-

\(NODSHT + 1\); or total number of nodes in superheater

NODT

SGEN3

-

\(NODSH0 + 1\); or total number of nodes in steam generator or evaporator

NODTP

SGEN3

-

Number of cells within boiling zone

NODTPO

SGEN3

-

Initial value of \(NODTP\)

NODTP0

SGEN3

-

\(NODSC + NODTP\)

NODTP1

SGEN3

-

Node number of boiling/superheat boundary; \(NODSC + NODTP + 1\)

NODTP2

SGEN3

-

\(NODTP1 + 1\)

ON

SGEN2

-

1.0

PD

SGEN2

\(Pa / PSI\)

Conversion factor for pressures since functions are in \(PSI\)

PDOT

SGEN2

\(Pa / s\)

Time derivative of steam generator average pressure

PI

SGEN2

-

\(\pi\)

PICHL

SGEN2

\(m\)

Longitudinal pitch of the helical tubes in the helical coil geometry option for the evaporator/steam generator model

PICHLS

SGENS2

\(m\)

Longitudinal pitch of the helical tube in the helical coil geometry option for the superheater model

PICHT

SGEN2

\(m\)

Transverse pitch of the helical tubes in the helical coil geometry option for the evaporator/steam generator model

PICHTS

SGENS2

\(m\)

Transverse pitch of the helical tube in the helical coil geometry option for the superheater model

PSW

SGEN2

-

\(\rho_{g} / \rho_{f}\)

PWAVEP

SGEN2

\(PSI\)

\(PWAVES / PD\)

PWAVES

SGEN2

\(Pa\)

Steam generator pressure

PWAVSP

SGENS2

\(Pa\)

Pressure in superheater

PWVSPP

SGENS2

\(PSI\)

\(PWAVSP / PD\)

P25

SGEN2

-

0.25

P5

SGEN2

-

0.5

QMT(100)

SGEN2

\(w / m^{3}\)

Volumetric heat source for each cell in the tube wall

QMTS(100)

SGENS2

\(w / m^{3}\)

Volumetric heat source for each cell in the tube wall

QST(100)

SGEN2

\(w / m^{3}\)

Volumetric heat source for each cell in the sodium side

QSTS(100)

SGENS2

\(w / m^{3}\)

Volumetric heat source for each cell on the sodium side

QWB(100)

SGEN1

\(w / m^{3}\)

Volumetric heat source for each cell on the water side at the beginning of step

QWBS(100)

SGENS1

\(w / m^{3}\)

Volumetric heat source for each cell on the water side at the beginning of step

QWE(100)

SGEN1

\(w / m^{2} - K\)

Total heat transfer coefficient from tube wall center to bulk water including possible fouling for each cell center

QWES(100)

SGENS1

\(w / m^{2} - K\)

Total heat transfer coefficient from tube wall center to bulk water including possible fouling for each cell center

QWT(100)

SGEN2

\(w / m^{3}\)

Volumetric heat source for each cell on the water side at the end of step

QWTS(100)

SGENS2

\(w / m^{3}\)

Volumetric heat source for each cell on the water side at the end of step

RMDEWO

SGEN2

\(m^{2} - K / w\)

Tube wall heat resistance on the water side

RMDEWS

SGENS2

\(m^{2} - K / w\)

Tube wall heat resistance on the water side

RMDNAA

SGEN2

\(m^{2} - K / w\)

Tube wall heat resistance on the sodium side

RMDNAS

SGENS2

\(m^{2} - K / w\)

Tube wall heat resistance on the sodium side

ROB(100)

SGEN1

\(kg / m^{3}\)

Water density at each node at beginning of step

ROBS(100)

SGENS1

\(kg / m^{3}\)

Water density at each node at beginning of step

ROCPTB

SGEN2

-

Reserved

ROE(100)

SGEN1

\(kg / m^{3}\)

Water density at each node at end of step

ROES(100)

SGENS1

\(kg / m^{3}\)

Water density at each node at end of step

ROFG

SGEN2

\(kg / m^{3}\)

\(\rho_{g} - \rho_{f}\)

ROFSAT

SGEN2

\(kg / m^{3}\)

\(\rho_{f}\)

ROGSAT

SGEN2

\(kg / m^{3}\)

\(\rho_{g}\)

ROHFG

SGEN2

\(J / m^{3}\)

\(\rho_{g} h_{g} - \rho_{f} h_{f}\)

ROZ1

SGEN2

\(kg / m^{3}\)

Average water density in subcooled zone

TBPODS

SGENS2

-

Tube pitch-to-diameter ratio

TIMCUR

SGEN2

\(s\)

Time at end of current steam generator time step

TIMEIN

SGEN2

\(s\)

Time at beginning of current primary loop time step

TIMENP

SGEN2

\(s\)

Time at end of current primary loop time step

TLIM

SGEN2

-

Largest fractional change in selected parameters for new time step selection

TMB(100)

SGEN1

\(K\)

Temperature at each cell center of tube wall at beginning of step

TMBS(100)

SGENS1

\(K\)

Temperature at each cell center of tube wall at beginning of step

TME(100)

SGEN1

\(K\)

Temperature at each cell center of tube wall at end of step

TMES(100)

SGENS1

\(K\)

Temperature at each cell center of tube wall at end of step

TO

SGEN2

-

2.0

TSB(100)

SGEN1

\(K\)

Temperature of sodium at each node at beginning of step

TSBS(100)

SGENS1

\(K\)

Temperature of sodium at each node at beginning of step

TSC(100)

SGEN2

\(K\)

Temperature of sodium at each node at beginning of step

TSCS(100)

SGENS2

\(K\)

Temperature of sodium at each cell center at beginning of step

TSE(100)

SGENS2

\(K\)

Temperature of sodium at each node at end of step

TSES(100)

SGENS1

\(K\)

Temperature of sodium at each node at end of step

TUBNO

SGEN2

-

Number of tubes in steam generator

TUBNOS

SGENS2

-

Number of tubes in superheater

TUBPOD

SGEN2

-

Tube pitch-to-diameter ratio

TWB(100)

SGEN1

\(K\)

Temperature of water at each node at beginning of step

TWBS(100)

SGENS1

\(K\)

Temperature of water at each node at beginning of step

TWC(100)

SGENS1

\(K\)

Temperature of water at each cell center at beginning of step

TWCS(100)

SGENSG

\(K\)

Temperature of water at each cell center at beginning of step

TWE(100)

SGEN1

\(K\)

Temperature of water at each node at end of step

TWES(100)

SGENS1

\(K\)

Temperature of water at each node at end of step

TWSAT

SGEN2

\(K\)

Water saturation temperature

UWZ1

SGEN2

\(kg / m - s\)

Average viscosity in the subcooled zone

VRISE

SGEN2

-

Vertical rise per length of helical tube in the helical coil geometry option for the evaporator/steam generator model

VRISES

SGENS2

-

Vertical rise per length of helical tube in the helical coil geometry option in the superheater model

XKTUBE

SGEN2

\(W / m - K\)

Conductivity of tube wall

XWB(100)

SGEN1

-

Quality at each node in boiling zone at beginning of step

XWE(100)

SGEN1

-

Quality of each node in boiling zone at end of step

ZMAX

SGEN2

\(m\)

Zone length threshold above which the number of nodes is restored to the initial value when the previous number of nodes is one

ZMIN

SGEN2

\(m\)

Zone length threshold below which the number of nodes is reduced to one

ZO

SGEN2

-

0.0

ZONLB(3)

SGEN2

\(m\)

Lengths of each zone at beginning of step

ZONLE(3)

SGEN2

\(m\)

Lengths of each zone at end of step

ZSG

SGEN2

\(m\)

Length of steam generator or of evaporator

ZSUP

SGEN2

\(m\)

Length of superheater

Selected Variables not in COMMON

Name

Routine

Units

Explanation

AWZ2

SGUNIT

INIT

-

Average void fraction in nucleate boiling zone

AWZ3

SGUNIT

INIT

-

Average void fraction in film boiling zone

DELP1

SGUNIT

INIT

\(Pa\)

Pressure drop across subcooled zone

DELP2

SGUNIT

INIT

\(Pa\)

Pressure drop across nucleate boiling zone

DELP3

SGUNIT

INIT

\(Pa\)

Pressure drop across film boiling zone

DELP4

SGUNIT

INIT

\(Pa\)

Pressure drop across superheated zone

DHF

SGUNIT

\(J / kg - Pa\)

Derivative of \(h_{f}\) with respect to pressure

DHG

SGUNIT

\(J / kg - Pa\)

Derivative of \(h_{g}\) with respect to pressure

DRODH

SGUNIT

\(kg^{2} / m^{3} - J\)

Derivative of respect to enthalpy in superheated zone

DRODP

SGUNIT

\(kg / m^{3} - Pa\)

Derivative of \(\rho \left(h, P \right)\) with respect to pressure in superheated zone

DROF

SGUNIT

\(kg / m^{3} - Pa\)

Derivative of \(\rho_{f}\) with respect to pressure

DROG

SGUNIT

\(kg / m^{3} - Pa\)

Derivative of \(\rho_{g}\) with respect to pressure

DTSG(100)

TSBOP

\(s\)

Array to store time steps over \(LMPDOT\) steam generator time steps in order to calculate.

GDOT

SGUNIT

\(kg / m^{2} - s\)

in pressure drop calculation

GWO

INIT

\(kg / m^{2} - s\)

Steady state mass flow

GWZ

SGUNIT

INIT

\(k / m^{2} - s\)

End of time step regional average mass flow for pressure drop calculation

GWZ0

SGUNIT

\(k / m^{2} - s\)

Beginning of time step regional average mass flow for pressure drop calculation

HTAV

INIT

\(W / m^{2} - K\)

Average heat transfer coefficient at tube wall surface on water side in one-node approximation or region

HTAVT

INIT

\(W / m^{2} - K\)

Total water side average heat transfer coefficient in one-node approximation of region

HWAV

INIT

\(J / kg\)

Average water enthalpy in one-node approximation of region

HWIN

INIT

\(J / kg\)

Steady state inlet water enthalpy

HWOUT

INIT

\(J / kg\)

Steady state outlet water enthalpy

IGO

INIT

-

Indicator which is set when the \(DNB\) node is found in the boiling zone so that a switch is made from the nucleate to the film boiling regime

INITER

INIT

-

Counter on the number of iterations in the search on the calibration factor in the film boiling calculation for each iteration on the nucleate boiling regime

IOPT1

INIT

-

Indicator which shows whether length \(\left( = 2 \right)\) or calibration factor \(\left( = 1 \right)\) is to be searched on for subcooled zone

IOPT2

INIT

-

Indicator which shows whether length \(\left( = 2 \right)\) or calibration factor \(\left( = 1 \right)\) is to be searched on for superheated zone

IOPT3

INIT

-

Indicator which shows how many heat transfer regimes there are in the steady state calculation

IPASS

SGUNIT

-

Indicator which stops iterative search on \(Z_{TP}\) in boiling zone when the zone reaches the top of the steam generator or when \(Z_{TP}\) changes more than a maximum amount allowed

ITER

SGUNIT

INIT

-

Iteration counter either on boiling zone length searches in \(SGUNIT\) or searches in all three zones in \(INIT\)

LMPDOT

TSBOP

-

Number of time steps and pressures stored in \(DTSG\) and \(PTSG\) arrays for calculation

PTSG(100)

TSBOP

\(Pa\)

Array to store pressures over \(LMPDOT\) steam generator time steps in order to calculate

ROAV

INIT

\(kg / m^{3}\)

Average water density in one-node approximation of region

ROEDNB

SGUNIT

INIT

\(kg / m^{3}\)

Water density at the \(DNB\) point used in pressure drop calculation

ROZ2

SGUNIT

INIT

\(kg / m^{3}\)

Average density in nucleate boiling zone for pressure drop calculation

ROZ3

SGUNIT

INIT

\(kg / m^{3}\)

Average density in film boiling zone for pressure drop calculation

ROZ4

SGUNIT

INIT

\(kg / m^{3}\)

Average density in superheated zone for pressure drop calculation

R32

SGUNIT

INIT

-

Thom friction factor in nucleate boiling zone

R33

SGUNIT

INIT

-

Thom friction factor in film boiling zone

TIMDIF

TSBOP

\(s\)

Time difference between beginning of primary loop time step and the end of current steam generator time step

TMAV

INIT

\(K\)

Average tube wall temperature in one-node approximation to region

TNAINB

TSBOP

\(K\)

Inlet sodium temperature at the beginning of primary loop time step

TNAINE

INIT

\(K\)

Inlet sodium temperature at the end of primary loop time step

TSAV

INIT

\(K\)

Average sodium temperature in one-node approximation to region

TSHF

INIT

\(K\)

Sodium temperature at the point of \(h_{f}\) on the water side at steady state

TSHG

INIT

\(K\)

Sodium temperature at the point of \(h_{f}\) on the water side at steady state

TSIN

INIT

\(K\)

Steady state sodium inlet temperature

TSOUT

INIT

\(K\)

Steady state sodium outlet temperature

TWAV

SGUNIT

INIT

\(K\)

Average water temperature in one-node approximation to region

UZ2

SGUNIT

INIT

\(kg / m - s\)

Average viscosity in nucleate boiling zone for pressure drop calculation

UZ3

SGUNIT

INIT

\(kg / m - s\)

Average viscosity in film boiling zone for pressure drop calculation

UZ4

TSBOP

\(kg / m - s\)

Average viscosity in superheated zone for pressure drop calculation

WNAINB

TSBOP

\(kg / s\)

Sodium flow rate at beginning of primary loop time step

WNAINE

SGUNIT

\(kg / s\)

Sodium flow rate at end of primary loop time step

ZITER

SGUNIT

INIT

\(m\)

Current value of \(Z_{TP}\) during search on region length in boiling zone calculation

ZONLE2

SGUNIT

INIT

\(m\)

Length of nucleate boiling zone used in pressure drop calculation

ZONLE3

\(m\)

Length of film boiling zone used in pressure drop calculation

7.7.5. Material Properties Data

This Appendix documents material properties correlations employed throughout the balance-of-plant network model, the steam generator model, and the component models for thermal and physical properties data. These data are used in heat transfer and fluid dynamics calculations.

On the sodium side of the steam generator, the correlations used for liquid sodium density, liquid sodium specific heat, and liquid sodium viscosity are documented in Section 12.12.

On the water side of the steam generator and throughout the balance-of-plant models, the dynamic viscosity of steam and water is calculated from [7-13]:

\[\mu = \mu_{o} \text{exp} \left[ \frac{\rho}{\rho^{*}} \sum_{i = 0}^{5} \sum_{j = 0}^{4} b_{ij} \left( \frac{T^{*}}{T} - 1 \right)^{i} \left( \frac{\rho}{\rho^{*}} - 1 \right)^{j} \right]\]

with

\[\mu_{o} = 10^{-6} \left( \frac{T}{T^{*}} \right)^{1 / 2} \left[ \sum_{k = 0}^{3} a_{k} \left( \frac{T^{*}}{T} \right)^{k} \right]^{-1}\]

where \(\mu\) is the viscosity in Pa-s, \(\rho\) is the density in kg/m3, \(T\) is the temperature in Kelvins, and the constants \(\rho\) and \(T\) are

\(\rho^{*} = 317.763\) kg/m3,

\(T^{*} = 647.27\) K,

The coefficients in the expression for \(\mu_{o}\) are:

\(a_{0} = 0.018 1583\)

\(a_{1} = 0.017 7624\)

\(a_{2} = 0.010 5287\)

\(a_{3} = -0.003 6744\)

and the values for \(b_{ij}\) are given in Table 7.7.1.

Table 7.7.1 Numerical Values of the Coefficients \(b_{ij}\)

i = 0

i = 1

i = 2

i = 3

i = 4

i = 5

j = 0

0.5601938

0.162888

-0.130356

0.907919

-0.551119

0.146543

j = 1

0.235622

0.789393

0.673665

1.207552

0.0670665

-0.0843370

j = 2

-0.274637

-0.743539

-0.959456

-0.687343

-0.497089

0.195286

j = 3

0.145831

0.263129

0.346247

0.213486

0.100754

-0.032932

j = 4

-0.0270448

-0.0253093

-0.026776

-0.0822904

0.0602253

-0.0202595

Correlations for the enthalpy of saturated liquid water and saturated steam are taken from the RETRAN-02 code documentation [7-14]. The specific enthalpy of liquid water is given by

\[\begin{split}h_{g} = \begin{cases} \sum_{i = 0}^{8} \text{CF1}_{i} \left[ \text{ln} \left(P \right) \right]^{i} & \text{for}\ 0.1\ \text{psia} \leq P \leq 950\ \text{psia} \\ \sum_{i = 0}^{8} \text{CF2}_{i} \left[ \text{ln} \left(P \right) \right]^{i} & \text{for}\ 950\ \text{psia} \leq P \leq 2550\ \text{psia} \\ \sum_{i = 0}^{8} \text{CF3}_{i} \left[ \left(P_{CRIT} - P \right)^{0.41} \right]^{i} & \text{for}\ 2550\ \text{psia} < P \leq P_{CRIT} \end{cases}\end{split}\]

and the specific enthalpy of saturated steam is given by

\[\begin{split}h_{g} = \begin{cases} \sum_{i = 0}^{11} \text{CG1}_{i} \left[ \text{ln} \left(P \right) \right]^{i} & \text{for} \ 0.1\ \text{psia} \leq P \leq 1500\ \text{psia} \\ \sum_{i = 0}^{8} \text{CG2}_{i} \left[ \text{ln} \left(P \right) \right]^{i} & \text{for}\ 1500\ \text{psia} \leq P \leq 6550\ \text{psia} \\ \sum_{i = 0}^{6} \text{CG3}_{i} \left[ \left(P_{CRIT} - P \right)^{0.41} \right]^{i} & \text{for}\ 2650\ \text{psia} < P \leq P_{CRIT} \end{cases}\end{split}\]

where \(h_{f}\) and \(h_{g}\) are the specific enthalpy in units of BTU/lbm, \(P\) is the pressure in psia, \(P_{CRIT}\) is the critical pressure \(\left(3208.2 \text{psia} \right)\), and the constant coefficients are given in Table 7.7.2. Expressions for the temperatures of subcooled water superheated steam as functions of pressure and enthalpy are taken from RETRAN-02 [7-14]:

\[T_{l} = \sum_{i = 0}^{i = i} \sum_{j = 0}^{j = 3} \text{CT1}_{i, j} P^{i, j}\]

and

\[T_{v} = \sum_{i = 0}^{i = 4} \sum_{j = 0}^{j = 4} \text{CT3}_{i, j} P^{i, j}\]

where \(T_{l}\) and \(T_{v}\) are the subcooled water and superheated steam temperatures in degrees Fahrenheit, \(P\) is the pressure in psia, \(h\) is the enthalpy in BTU/lbm, and the constant coefficients \(\text{CT1}\) and \(\text{CT3}\) are given in Table 7.7.3. The specific heats at constant pressure for subcooled water and superheated steam are calculated as the inverses of the partial derivations of the expressions for \(T_{l}\) and \(T_{v}\) with respect to enthalpy. The saturation temperature is obtained from the expression for \(T_{l}\) evaluated at the ambient pressure and the saturated liquid water specific enthalpy at that pressure.

Correlations for the specific enthalpies of subcooled liquid water and superheated steam as functions of pressure and enthalpy are taken from RETRAN-02 [7-14]:

\[v_{l} = \text{exp} \left[ \sum_{i = 0}^{2} \sum_{j = 0}^{4} \text{CN1}_{i, j} P^{i, j} \right]\]

and

\[v_{v} = \sum_{i = -1}^{2} \sum_{j = 0}^{2} \text{CN2}_{i, j} P^{i, j}\]

where \(v_{l}\) and \(v_{v}\) are the subcooled water and superheated steam specific volumes in \(ft^{3}\ /\ lbm\), \(P\) is the pressure in psia, \(h\) is the enthalpy in BTU/lbm, and the constant coefficients \(\text{CN1}\) and \(\text{CN2}\) are listed in Table 7.7.4. The satruated liquid water density is computed from the value for \(v_{l}\) at the ambient pressure and the saturated steam specific enthalpy at that pressure. Similarly the saturated steam density is obtained from \(v_{v}\) at the ambient pressure and the satruated steam specific enthalpy at that pressure.

Table 7.7.2 Constant Coefficients in Expressions for Saturated Liquid Water and Saturated Steam Enthalpies as Functions of Pressure

i

\(\text{CF1}_{i}\)

\(\text{CF2}_{i}\)

\(\text{CF3}_{i}\)

0

\(.6970887859 \times 10^{2}\)

\(.8408618802 \times 10^{6}\)

\(.9060030436 \times 10^{3}\)

1

\(.3337529994 \times 10^{2}\)

\(.3637413208 \times 10^{6}\)

\(-.1426813520 \times 10^{0}\)

2

\(.2318240735 \times 10^{1}\)

\(-.4634506669 \times 10^{6}\)

\(.1522233257 \times 10^{1}\)

3

\(.1840599513 \times 10^{0}\)

\(.1130306339 \times 10^{6}\)

\(-.6973992961 \times 10^{0}\)

4

\(-.5245502284 \times 10^{-2}\)

\(-.4350217298 \times 10^{3}\)

\(.1743091663 \times 10^{0}\)

5

\(.2878007027 \times 10^{-2}\)

\(-.3898988188 \times 10^{4}\)

\(-.2319717696 \times 10^{-1}\)

6

\(.1753652324 \times 10^{-2}\)

\(.6697399434 \times 10^{3}\)

\(.1694019149 \times 10^{-2}\)

7

\(-.4334859629 \times 10^{-3}\)

\(-.4730726377 \times 10^{2}\)

\(-.6454771710 \times 10^{-4}\)

8

\(.3325699282 \times 10^{-4}\)

\(.1265125057 \times 10^{1}\)

\(.1003003098 \times 10^{-5}\)

i

\(\text{CG1}_{i}\)

\(\text{CG2}_{i}\)

\(\text{CG3}_{i}\)

0

\(.1105836875 \times 10^{4}\)

\(-.2234264997 \times 10^{7}\)

\(.9059978254 \times 10^{3}\)

1

\(.1436943768 \times 10^{2}\)

\(.1231247634 \times 10^{7}\)

\(.5561957539 \times 10^{1}\)

2

\(.8018288621 \times 10^{0}\)

\(-.1978847871 \times 10^{7}\)

\(.3434189609 \times 10^{1}\)

3

\(.1617232913 \times 10^{-1}\)

\(.1859988044 \times 10^{2}\)

\(-.6406390628 \times 10^{0}\)

4

\(-.1501147505 \times 10^{-2}\)

\(-.2765701318 \times 10^{1}\)

\(.5918579484 \times 10^{-1}\)

5

\(.0000000000 \times 10^{0}\)

\(.1036033878 \times 10^{4}\)

\(-.2725378570 \times 10^{-2}\)

6

\(.0000000000 \times 10^{0}\)

\(-.2143423131 \times 10^{3}\)

\(.5006336938 \times 10^{-4}\)

7

\(.0000000000 \times 10^{0}\)

\(.1690507762 \times 10^{2}\)

8

\(.0000000000 \times 10^{0}\)

\(-.4864322134 \times 10^{0}\)

9

\(-.1237675562 \times 10^{-4}\)

10

\(.3004773304 \times 10^{-5}\)

11

\(-.2062390734 \times 10^{-6}\)
Table 7.7.3 Constant Coefficients in Expressions for Temperature as a Function of Pressure and Specific Enthalpy

\(\text{CT1}_{i, j}\)

i = 0

i = 1

i = 2

i = 3

j = 0

\(0.3276275552 \times 10^{2}\)

\(0.9763617000 \times 10^{0}\)

\(0.1857226027 \times 10^{-3}\)

\(-0.4682674330 \times 10^{-6}\)

j = 1

\(0.3360880214 \times 10^{-2}\)

\(-0.5595281760 \times 10^{-4}\)

\(0.1618595991 \times 10^{-6}\)

\(-0.1180204381 \times 10^{-9}\)

\(\text{CT3}_{i, j}\)

i = 0

i = 1

i = 2

i = 3

i = 4

j = 0

\(-0.1179100862 \times 10^{5}\)

\(0.2829274345 \times 10^{2}\)

\(-0.2678181564 \times 10^{-1}\)

\(0.1218742752 \times 10^{-4}\)

\(-0.2092033147 \times 10^{-8}\)

j = 1

\(0.1256160907 \times 10^{3}\)

\(-0.3333448495 \times 10^{0}\)

\(0.3326901268 \times 10^{-3}\)

\(-0.1477890326 \times 10^{-6}\)

\(0.2463258371 \times 10^{-10}\)

j = 2

\(-0.1083713369 \times 10^{0}\)

\(0.2928177730 \times 10^{-3}\)

\(-0.2972436458 \times 10^{-6}\)

\(0.1342639113 \times 10^{-9}\)

\(-0.2275585718 \times 10^{-13}\)

j = 3

\(0.3278071846 \times 10^{-4}\)

\(-0.8970959364 \times 10^{-7}\)

\(0.9246248312 \times 10^{-10}\)

\(-0.4249155515 \times 10^{-13}\)

\(0.7338316751 \times 10^{-17}\)

j = 4

\(-0.3425564927 \times 10^{-8}\)

\(0.9527692453 \times 10^{-11}\)

\(-0.1001409043 \times 10^{-13}\)

\(0.4703914404 \times 10^{-17}\)

\(-0.8315044742 \times 10^{-21}\)
Table 7.7.4 Constant coefficients in Expressions for Specific Volume as a Function of Pressure and Specific Enthalpy

\(\text{CN2}_{i, j}\)

i = 0

i = 1

i = 2

i = 3

i = 4

j = 0

\(-0.4117961750 \times 10^{1}\)

\(-0.3811294543 \times 10^{-3}\)

\(0.4308265942 \times 10^{-5}\)

\(-0.9160120130 \times 10^{-8}\)

\(0.8017924673 \times 10^{-11}\)

j = 1

\(-0.4816067020 \times 10^{-5}\)

\(0.7744786733 \times 10^{-7}\)

\(-0.6988467605 \times 10^{-9}\)

\(0.1916720525 \times 10^{-11}\)

\(-0.1760288590 \times 10^{-14}\)

j = 2

\(-0.1820625039 \times 10^{-8}\)

\(0.1440785930 \times 10^{-10}\)

\(-0.2082170753 \times 10^{-13}\)

\(-0.3603625114 \times 10^{-16}\)

\(0.7407124321 \times 10^{-19}\)

\(\text{CN2}_{i, j}\)

i = 0

i = 1

i = 2

j = -1

\(-0.1403086182 \times 10^{4}\)

\(0.1802594763 \times 10^{1}\)

\(-0.2097279215 \times 10^{-3}\)

j = 0

\(0.3817195017 \times 10^{0}\)

\(-0.5394444747 \times 10^{-3}\)

\(0.1855203702 \times 10^{-6}\)

j = 1

\(-0.6449501159 \times 10^{-4}\)

\(0.8437637660 \times 10^{-7}\)

\(-0.2713755001 \times 10^{-10}\)

j = 2

\(0.7823817858 \times 10^{-8}\)

\(-0.1053834646 \times 10^{-10}\)

\(0.3629590764 \times 10^{-14}\)