.. _section07.appendices:
Appendices
-----------
.. _section-7.appendices.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 identi£ies 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 £or 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).
.. _section-7.appendices.2:
Steam Generator Water-Side Heat Transfer Correlations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. rubric:: Subcooled Water
The Dittus-Boelter correlation :ref:`[7-4] <section-7.references>` is used.
.. math:: N = 0.023 Re^{0.8} Pr^{0.4} = \frac{h_{w} D_{H}}{k}
Bulk liquid properties are used to calculate :math:`Re, Pr` and :math:`k; h_{w}` is the
heat transfer coefficient between the wall surface and the bulk water.
.. rubric:: Nucleate Boiling Water
A correlation developed by Thom, et al. :ref:`[7-5] <section-7.references>` is used.
.. math:: h_{w} = 3.1968 \left( e^{P / 8.65 \times 10^{6}} \right) \frac{1}{0.072} \left( q \right)^{0.5}
:math:`h_{w}` is the heat transfer coefficient between the wall surface and the
bulk water; :math:`q = H_{T} \left( T_{m} - T_{sat} \right)` where :math:`H_{T}` is defined
as in Eq. :ref:`7.3-66 <eq-7.3-66>`; :math:`T_{m}` is the average wall temperature; :math:`P` is the
steam generator pressure.
.. rubric:: Film Boiling Water
A correlation of A. A. Bishop et al. :ref:`[7-6] <section-7.references>` is the following
.. math:: 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 :math:`Re` and :math:`Pr`. All temperature-dependent
properties are calculated with :math:`T_{sat}`. The wall film temperature is
only crudely approximated and the relevant properties are very
insensitive to temperature when above :math:`T_{sat}`. :math:`h_{w}` is the heat
transfer coefficient between the wall surface and the bulk water and :math:`x`
is the local nodal quality. The mass flux used in the Reynold's number
is the local value.
.. rubric:: Superheated Steam
A correlation developed by A. A. Bishop :ref:`[7-7] <section-7.references>` is used.
.. math:: 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 :math:`T_{sat}`. :math:`h_{w}` is the heat transfer coefficient between the wall
surface and the bulk steam.
.. rubric:: Liquid Sodium Heat Transfer
A variation of the Maresca-Dwyer correlation is used :ref:`[7-8] <section-7.references>`. The heat
transfer coefficient between the outside wall and the bulk sodium is the
following.
.. math:: H_{Na} = \frac{N c_{p} \mu}{Pr D_{H}}
where :math:`c_{p}`, :math:`\mu` and :math:`D_{H}` are the specific heat, viscosity and
hydraulic diameter respectively. The Prandtl number is computed
according to the following,
.. math:: 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,
.. math:: Nu = 6.66 + 3.126 POD + 1.184 POD^{2} + 0.0155 \left( Pr Re S \right)^{0.86}
where :math:`POD` is the tube pitch-to-diameter ratio. :math:`S` is calculated
according to the following,
.. math:: S = 1.0 - 1.82\ / \left( Pr E \right)
and :math:`E` is given by,
.. math:: E = 0.000175 Re^{1.32}\ /\ POD^{1.5}
The Nusselt number for molecular conduction is given by the following,
.. math:: Nu = 6.4353 + 3.97 POD + 1.025 POD^{2} - 29494 / (Re + 20363)
.. _section-7.appendices.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 Eqs. :ref:`7.4-8 <eq-7.4-8>` and :ref:`7.4-9 <eq-7.4-9>`. Since the
problem is symmetric with respect to the center point of the cylinder,
as is obvious by looking at :numref:`figure-7.4-3`, only Eq. :ref:`7.4-8 <eq-7.4-8>` is used to
demonstrate the solution scheme.
By defining the angle between the vertical section :math:`L` and the radius
:math:`r_{s}` as :math:`\beta` in :numref:`figure-7.4-3`, Eq. :ref:`7.4-8 <eq-7.4-8>`
can be rewritten as,
(A7.3‑1)
.. _eq-A7-3.1:
.. math:: \alpha_{s} = \left[ \beta - \frac{\text{sin} \left( 2 \beta \right)}{2} \right] / \pi
for :math:`0 \leq \alpha_{s} \leq 1/2` and :math:`0 \leq \beta \leq \pi / 2`. The height of the vapor region
:math:`A_{g}` corresponding to :math:`\alpha_{s}` in :numref:`figure-7.4-3` is :math:`r_{s} - L`,
which can be normalized to the cylinder radius :math:`r_{s}` as
(A7.3‑2)
.. _eq-A7.3-2:
.. math:: \omega \equiv \frac{r_{s} - L}{r_{s}} = 1 - \text{cos} \beta
The aim now is to obtain an expression for :math:`\beta` as a function of :math:`\alpha_{s}`,
in order to avoid using an iterative solution to find :math:`\beta`. The most
straight-forward way would be to generate a polynomial expression in
:math:`\alpha_{s}` for :math:`\beta`. However, a study of the curvature of the :math:`\beta`
vs. :math:`\alpha_{s}` curve indicates that the slope is very steep at :math:`\beta` close to zero,
changes dramatically as :math:`\beta` increases, and levels off as :math:`\beta` approaches
:math:`\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 :math:`\alpha_{s}` on the :math:`\beta` vs. :math:`\alpha_{s}` curve is divided into three
regions based on the slopes along the curve, i.e., :math:`0 \leq \alpha_{s} < \alpha_{1}`,
:math:`\alpha_{1} \leq \alpha_{s} < \alpha_{2}`, and :math:`\alpha_{2} \leq \alpha_{s} \leq 1/2`,
where :math:`\alpha_{1}` and :math:`\alpha_{2}` are chosen
to be at the suitable values 0.015 and 0.225, respectively. Also, the
polynomials of :math:`\beta` for each region are given the forms,
(A7.3‑3)
.. _eq-A7.3-3:
.. math:: \beta = C_{1} \left(1.5 \alpha_{s} \right)^{1 / 3}, 0 \leq \alpha_{s} < \alpha_{1}
(A7.3‑4)
.. _eq-A7.3-4:
.. math:: \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}
(A7.3‑5)
.. _eq-A7.3-5:
.. math:: \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 :math:`\beta_{1}` and :math:`\beta_{2}` are values of :math:`\beta` corresponding to
:math:`\alpha_{1}` and :math:`\alpha_{2}`, respectively, :math:`X_{1}` and
:math:`X_{2}` are defined as
.. math:: X_{1} = \left( \alpha_{2} - \alpha_{1} \right)
and
.. math:: X_{2} = \left( \alpha_{s} - \alpha_{2} \right)
and the coefficients :math:`C_{1}` through :math:`C_{9}` are determined
using a least squares fit separately on each region. The coefficient
values are as follows:
* :math:`C_{1} = 1.00377`
* :math:`C_{2} = 1.41595 \times 10^{1}`
* :math:`C_{3} = 6.98089 \times 10^{1}`
* :math:`C_{4} = 2.76847 \times 10^{1}`
* :math:`C_{5} = 4.47906 \times 10^{1}`
* :math:`C_{6} = 2.37819`
* :math:`C_{7} = 6.38891 \times 10^{-1}`
* :math:`C_{8} = 1.72658`
* :math:`C_{9} = 4.94197 \times 10^{2}`
More significant digits for :math:`C_{1}` through :math:`C_{9}` are used
in the coding. The polynomials :ref:`A7.3-3 <eq-A7.3-3>` through :ref:`A7.3-5 <eq-A7.3-5>`
are chosen such that continuity conditions are satisfied at :math:`\alpha_{s}` equal to
:math:`\alpha_{1}` and :math:`\alpha_{2}`.
Once :math:`\beta` is calculated from one of Eqs. :ref:`A7.3-3 <eq-A7.3-3>`, :ref:`A7.3-4 <eq-A7.3-4>`,
and :ref:`A7.3-5 <eq-A7.3-5>` for a given :math:`\alpha_{s}`, the two-phase interface can be computed as
.. math:: TP = \left(1 - \omega \right) r + CV,
where :math:`\omega` is given in Eq. :ref:`A7.3-2 <eq-A7.3-2>`.
The maximum error, defined as the difference between :math:`\omega`, as calculated
from Eq. :ref:`A7.3-2 <eq-A7.3-2>` with :math:`\beta` obtained by Eqs. :ref:`A7.3-3 <eq-A7.3-3>`,
:ref:`A7.3-4 <eq-A7.3-4>`, or :ref:`A7.3-5 <eq-A7.3-5>`, and
the actual :math:`\omega`, is within :math:`\pm 4.53 \times 10^{-4}`. If more accuracy is
needed, a better value for :math:`\omega` can be obtained by introducing the Newton
iteration method and using the calculated :math:`\omega`
With one iteration, the maximum error could be reduced to :math:`\pm 2.67 \times 10^{-7}`,
and with two iterations, to :math:`\pm 9.17 \times 10^{-13}`.
.. _section-7.appendices.4:
Dictionary of Steam Generator Model Variables
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. rubric:: 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.
.. list-table::
:header-rows: 1
:align: center
:widths: auto
* - Name
- Block
- Units
- Explanation
* - ARM
- SGEN2
- :math:`m^{2}`
- Cross-sectional area of tube wall
* - ARMS
- SGENS2
- :math:`m^{2}`
- Cross-sectional area of tube wall
* - ARNA
- SGEN2
- :math:`m^{2}`
- Flow area of sodium
* - ARNAS
- SGENS2
- :math:`m^{2}`
- Flow area of sodium
* - ARW
- SGEN2
- :math:`m^{2}`
- Flow area of water
* - ARWS
- SGENS2
- :math:`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 :math:`6.66 + \left( 1.184 TBPODS + 3.126 \right) TBPODS`
* - CNAFR1
- SGEN2
- \-
- Constant used in sodium heat transfer coefficient calculation; equal to :math:`6.66 + \left( 1.184 TUBPOD + 3.126 \right) TUBPOD`
* - COILD
- SGEN2
- :math:`m`
- Average diameter of coil in helical coil in the helical coil geometry option for evaporator/steam generator model
* - COILDS
- SGENS2
- :math:`m`
- Average diameter of the coil for the helical coil geometry option in the superheater model
* - DDW
- SGEN2
- :math:`m`
- Hydraulic diameter on water side
* - DDWS
- SGENS2
- :math:`m`
- Hydraulic diameter on water side
* - DELP24
- SGEN2
- \-
- Fraction of total pressure drop in subcooled zone
* - DEWI
- SGEN2
- :math:`m`
- Booster tube outer diameter on water side
* - DEWIS
- SGENS2
- :math:`m`
- Booster tube outer diameter on water side
* - DEWO
- SGEN2
- :math:`m`
- Tube wall inner diameter
* - DEWOS
- SGENS2
- :math:`m`
- Tube wall inner diameter
* - DOUT
- SGEN2
- :math:`m`
- Tube wall outer diameter
* - DOUTS
- SGENS2
- :math:`m`
- Tube wall outer diameter
* - DHNA
- SGEN2
- :math:`m`
- Hydraulic diameter on sodium side
* - DHNAS
- SGENS2
- :math:`m`
- Hydraulic diameter on sodium side
* - DZONE(2)
- SGEN2
- :math:`m / s`
- Velocity of subcooled and superheat zone boundaries
* - FACT1
- SGEN2
- \-
- Not currently used
* - FACT1S
- SGENS2
- \-
- Not currently used
* - FACT2
- SGEN2
- \-
- :math:`1.04 * 10^{4} * TUBPOD^{1.5}`; used in sodium heat transfer coefficient calculation
* - FACT2S
- SGENS2
- \-
- :math:`1.04 * 10^{4} * TBPODS^{1.5}`; used in sodium heat transfer coefficient calculation
* - FACT3
- SGEN2
- :math:`m^{-1}`
- :math:`\pi * DEWO / ARW`
* - FACT3S
- SGENS2
- :math:`m^{-1}`
- :math:`\pi * DEWOS / ARWS`
* - FACT4
- SGEN2
- :math:`m^{-1}`
- :math:`\pi * DOUT / ARNA`
* - FACT4S
- SGENS2
- :math:`m^{-1}`
- :math:`\pi * DOUTS / ARNAS`
* - FACT5
- SGEN2
- \-
- :math:`ARNA / ARM`
* - FACT5S
- SGENS2
- \-
- :math:`ARNAS / ARMS`
* - FACT6
- SGEN2
- \-
- :math:`ARW / ARM`
* - FACT6S
- SGENS2
- \-
- :math:`ARWS / ARMS`
* - FACT7
- SGEN2
- :math:`K`
- Constant used in viscosity function
* - FACT8
- SGEN2
- :math:`kg / m^{3}`
- Constant used in viscosity function
* - FACT9
- SGEN2
- :math:`J / m^{3} - K`
- Density x specific heat for tube wall
* - FACT10
- SGEN2
- \-
- :math:`0.023 * DDW^{-0.2}`; used in subcooled heat transfer coefficient
* - FACT11
- SGEN2
- \-
- :math:`0.0193 * DDW^{-0.2}`; used in film boiling heat transfer coefficient
* - FACT12
- SGEN2
- \-
- :math:`0.0073 * DDW^{-0.114}`; used in superheat zone heat transfer coefficient
* - FACTSP
- SGENS2
- \-
- :math:`0.0073 * DDWS^{-0.114}`; used in superheater heat transfer coefficient
* - FOULR(4)
- SGEN2
- :math:`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
- :math:`m^{2} - K / w`
- Fouling heat resistances on the water side for each heat transfer regime
* - FOULRS
- SGENS2
- :math:`m^{2} - K / w`
- Tube wall heat resistance on the water side plus any fouling heat resistance on water side
* - FOULSI
- SGENS2
- :math:`m^{2} - K / w`
- Fouling heat resistance on the water side
* - FRIC1(4)
- SGEN2
- \-
- Normalizing friction factor in Eqs. :ref:`7.3-56 <eq-7.3-56>` and :ref:`7.3-94 <eq-7.3-94>` for each heat transfer regime
* - FRIC1S
- SGENS2
- \-
- Normalizing fraction factor in superheater
* - GNA
- SGEN2
- :math:`kg / m^{2} - s`
- Sodium side mass flow
* - GNAS
- SGENS2
- :math:`kg / m^{2} - s`
- Sodium side mass flow
* - GWB(100)
- SGEN1
- :math:`kg / m^{2} - s`
- Water side mass flow at each node at beginning of time step
* - GWE(100)
- SGEN1
- :math:`kg / m^{2} - s`
- Water side mass flow at each node at end of time step
* - GWS
- SGENS2
- :math:`kg / m^{2} - s`
- Sodium side mass flow
* - HD
- SGEN2
- :math:`\left(J / kg \right) / \left( BTU / lb \right)`
- Conversion factor for enthalpies since functions are in :math:`BTU / lb`
* - HDNB
- SGEN2
- \-
- Fraction of cell where DNB point lies which is in the nucleate boiling regime
* - HFG
- SGEN2
- :math:`J / kg`
- :math:`h_{fg}`
* - HFSAT
- SGEN2
- :math:`J / kg`
- :math:`h_{f}`
* - HFSATP
- SGEN2
- :math:`BTU / lb`
- :math:`h_{f} / HD`
* - HGSAT
- SGEN2
- :math:`J / kg`
- :math:`h_{g}`
* - HGSATP
- SGEN2
- :math:`BTU / lb`
- :math:`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
- :math:`w / m^{2} - K`
- Heat transfer coefficient between the tube wall surface and the bulk water by cell center
* - HTWS
- SGENS2
- :math:`w / m^{2} - K`
- Heat transfer coefficient between the tube wall surface and the bulk water by cell center in superheater
* - HUNIT
- SGEN2
- :math:`s`
- Current steam generator time step
* - HUNITN
- SGEN2
- :math:`s`
- Newly selected steam generator time step for next step
* - HUNITS
- SGENS2
- :math:`s`
- Current superheater time step
* - HWB(100)
- SGEN1
- :math:`J / kg`
- Enthalpy by node at beginning of step
* - HWBS(100)
- SGENS1
- :math:`J / kg`
- Enthalpy by node at beginning of step
* - HWE(100)
- SGEN1
- :math:`J / kg`
- Enthalpy by node at the end of step
* - HWES(100)
- SGENS1
- :math:`J / kg`
- Enthalpy by node at the end of step
* - H1MIN
- SGENS2
- :math:`w / m^{2} - K`
- Minimum value allowed for :math:`HTW` in subcooled zone
* - H2MIN
- SGEN2
- :math:`w / m^{2} - K`
- Minimum value allowed for :math:`HTW` in nucleate boiling zone
* - H3MIN
- SGEN2
- :math:`w / m^{2} - K`
- Minimum value allowed for :math:`HTW` in film boiling zone
* - H4MIN
- SGEN2
- :math:`w / m^{2} - K`
- Minimum value allowed for :math:`HTW` in superheated zone or for :math:HTWS` in superheater
* - IDNB
- SGEN3
- \-
- Cell number when :math:`DNB` point occurs
* - IDNBL
- SGEN3
- \-
- Value of :math:`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 :math:`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 :math:`NODSC`
* - NODSC1
- SGEN3
- \-
- Node number of subcooled/boiling boundary; :math:`NODSC + 1`
* - NODSC2
- SGEN3
- \-
- :math:`NODSC1 + 1`
* - NODSH
- SGEN3
- \-
- Number of cells within superheated zone in steam generator or evaporator
* - NODSHO
- SGEN3
- \-
- Initial value of :math:`NODSH`
* - NODSHT
- SGENS3
- \-
- Number of cells within superheater
* - NODSH0
- SGEN3
- \-
- Total number of cells in steam generator or evaporator; :math:`NODSC + NODTP + NODSH`
* - NODSH1
- SGEN3
- \-
- :math:`NODSHT + 1`; or total number of nodes in superheater
* - NODT
- SGEN3
- \-
- :math:`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 :math:`NODTP`
* - NODTP0
- SGEN3
- \-
- :math:`NODSC + NODTP`
* - NODTP1
- SGEN3
- \-
- Node number of boiling/superheat boundary; :math:`NODSC + NODTP + 1`
* - NODTP2
- SGEN3
- \-
- :math:`NODTP1 + 1`
* - ON
- SGEN2
- \-
- 1.0
* - PD
- SGEN2
- :math:`Pa / PSI`
- Conversion factor for pressures since functions are in :math:`PSI`
* - PDOT
- SGEN2
- :math:`Pa / s`
- Time derivative of steam generator average pressure
* - PI
- SGEN2
- \-
- :math:`\pi`
* - PICHL
- SGEN2
- :math:`m`
- Longitudinal pitch of the helical tubes in the helical coil geometry option for the evaporator/steam generator model
* - PICHLS
- SGENS2
- :math:`m`
- Longitudinal pitch of the helical tube in the helical coil geometry option for the superheater model
* - PICHT
- SGEN2
- :math:`m`
- Transverse pitch of the helical tubes in the helical coil geometry option for the evaporator/steam generator model
* - PICHTS
- SGENS2
- :math:`m`
- Transverse pitch of the helical tube in the helical coil geometry option for the superheater model
* - PSW
- SGEN2
- \-
- :math:`\rho_{g} / \rho_{f}`
* - PWAVEP
- SGEN2
- :math:`PSI`
- :math:`PWAVES / PD`
* - PWAVES
- SGEN2
- :math:`Pa`
- Steam generator pressure
* - PWAVSP
- SGENS2
- :math:`Pa`
- Pressure in superheater
* - PWVSPP
- SGENS2
- :math:`PSI`
- :math:`PWAVSP / PD`
* - P25
- SGEN2
- \-
- 0.25
* - P5
- SGEN2
- \-
- 0.5
* - QMT(100)
- SGEN2
- :math:`w / m^{3}`
- Volumetric heat source for each cell in the tube wall
* - QMTS(100)
- SGENS2
- :math:`w / m^{3}`
- Volumetric heat source for each cell in the tube wall
* - QST(100)
- SGEN2
- :math:`w / m^{3}`
- Volumetric heat source for each cell in the sodium side
* - QSTS(100)
- SGENS2
- :math:`w / m^{3}`
- Volumetric heat source for each cell on the sodium side
* - QWB(100)
- SGEN1
- :math:`w / m^{3}`
- Volumetric heat source for each cell on the water side at the beginning of step
* - QWBS(100)
- SGENS1
- :math:`w / m^{3}`
- Volumetric heat source for each cell on the water side at the beginning of step
* - QWE(100)
- SGEN1
- :math:`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
- :math:`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
- :math:`w / m^{3}`
- Volumetric heat source for each cell on the water side at the end of step
* - QWTS(100)
- SGENS2
- :math:`w / m^{3}`
- Volumetric heat source for each cell on the water side at the end of step
* - RMDEWO
- SGEN2
- :math:`m^{2} - K / w`
- Tube wall heat resistance on the water side
* - RMDEWS
- SGENS2
- :math:`m^{2} - K / w`
- Tube wall heat resistance on the water side
* - RMDNAA
- SGEN2
- :math:`m^{2} - K / w`
- Tube wall heat resistance on the sodium side
* - RMDNAS
- SGENS2
- :math:`m^{2} - K / w`
- Tube wall heat resistance on the sodium side
* - ROB(100)
- SGEN1
- :math:`kg / m^{3}`
- Water density at each node at beginning of step
* - ROBS(100)
- SGENS1
- :math:`kg / m^{3}`
- Water density at each node at beginning of step
* - ROCPTB
- SGEN2
- \-
- Reserved
* - ROE(100)
- SGEN1
- :math:`kg / m^{3}`
- Water density at each node at end of step
* - ROES(100)
- SGENS1
- :math:`kg / m^{3}`
- Water density at each node at end of step
* - ROFG
- SGEN2
- :math:`kg / m^{3}`
- :math:`\rho_{g} - \rho_{f}`
* - ROFSAT
- SGEN2
- :math:`kg / m^{3}`
- :math:`\rho_{f}`
* - ROGSAT
- SGEN2
- :math:`kg / m^{3}`
- :math:`\rho_{g}`
* - ROHFG
- SGEN2
- :math:`J / m^{3}`
- :math:`\rho_{g} h_{g} - \rho_{f} h_{f}`
* - ROZ1
- SGEN2
- :math:`kg / m^{3}`
- Average water density in subcooled zone
* - TBPODS
- SGENS2
- \-
- Tube pitch-to-diameter ratio
* - TIMCUR
- SGEN2
- :math:`s`
- Time at end of current steam generator time step
* - TIMEIN
- SGEN2
- :math:`s`
- Time at beginning of current primary loop time step
* - TIMENP
- SGEN2
- :math:`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
- :math:`K`
- Temperature at each cell center of tube wall at beginning of step
* - TMBS(100)
- SGENS1
- :math:`K`
- Temperature at each cell center of tube wall at beginning of step
* - TME(100)
- SGEN1
- :math:`K`
- Temperature at each cell center of tube wall at end of step
* - TMES(100)
- SGENS1
- :math:`K`
- Temperature at each cell center of tube wall at end of step
* - TO
- SGEN2
- \-
- 2.0
* - TSB(100)
- SGEN1
- :math:`K`
- Temperature of sodium at each node at beginning of step
* - TSBS(100)
- SGENS1
- :math:`K`
- Temperature of sodium at each node at beginning of step
* - TSC(100)
- SGEN2
- :math:`K`
- Temperature of sodium at each node at beginning of step
* - TSCS(100)
- SGENS2
- :math:`K`
- Temperature of sodium at each cell center at beginning of step
* - TSE(100)
- SGENS2
- :math:`K`
- Temperature of sodium at each node at end of step
* - TSES(100)
- SGENS1
- :math:`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
- :math:`K`
- Temperature of water at each node at beginning of step
* - TWBS(100)
- SGENS1
- :math:`K`
- Temperature of water at each node at beginning of step
* - TWC(100)
- SGENS1
- :math:`K`
- Temperature of water at each cell center at beginning of step
* - TWCS(100)
- SGENSG
- :math:`K`
- Temperature of water at each cell center at beginning of step
* - TWE(100)
- SGEN1
- :math:`K`
- Temperature of water at each node at end of step
* - TWES(100)
- SGENS1
- :math:`K`
- Temperature of water at each node at end of step
* - TWSAT
- SGEN2
- :math:`K`
- Water saturation temperature
* - UWZ1
- SGEN2
- :math:`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
- :math:`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
- :math:`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
- :math:`m`
- Zone length threshold below which the number of nodes is reduced to one
* - ZO
- SGEN2
- \-
- 0.0
* - ZONLB(3)
- SGEN2
- :math:`m`
- Lengths of each zone at beginning of step
* - ZONLE(3)
- SGEN2
- :math:`m`
- Lengths of each zone at end of step
* - ZSG
- SGEN2
- :math:`m`
- Length of steam generator or of evaporator
* - ZSUP
- SGEN2
- :math:`m`
- Length of superheater
.. rubric:: Selected Variables not in COMMON
.. list-table::
:header-rows: 1
:align: center
:widths: auto
* - 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
- :math:`Pa`
- Pressure drop across subcooled zone
* - DELP2
- | SGUNIT
| INIT
- :math:`Pa`
- Pressure drop across nucleate boiling zone
* - DELP3
- | SGUNIT
| INIT
- :math:`Pa`
- Pressure drop across film boiling zone
* - DELP4
- | SGUNIT
| INIT
- :math:`Pa`
- Pressure drop across superheated zone
* - DHF
- SGUNIT
- :math:`J / kg - Pa`
- Derivative of :math:`h_{f}` with respect to pressure
* - DHG
- SGUNIT
- :math:`J / kg - Pa`
- Derivative of :math:`h_{g}` with respect to pressure
* - DRODH
- SGUNIT
- :math:`kg^{2} / m^{3} - J`
- Derivative of respect to enthalpy in superheated zone
* - DRODP
- SGUNIT
- :math:`kg / m^{3} - Pa`
- Derivative of :math:`\rho \left(h, P \right)` with respect to pressure in superheated zone
* - DROF
- SGUNIT
- :math:`kg / m^{3} - Pa`
- Derivative of :math:`\rho_{f}` with respect to pressure
* - DROG
- SGUNIT
- :math:`kg / m^{3} - Pa`
- Derivative of :math:`\rho_{g}` with respect to pressure
* - DTSG(100)
- TSBOP
- :math:`s`
- Array to store time steps over :math:`LMPDOT` steam generator time steps in order to calculate.
* - GDOT
- SGUNIT
- :math:`kg / m^{2} - s`
- in pressure drop calculation
* - GWO
- INIT
- :math:`kg / m^{2} - s`
- Steady state mass flow
* - GWZ
- | SGUNIT
| INIT
- :math:`k / m^{2} - s`
- End of time step regional average mass flow for pressure drop calculation
* - GWZ0
- SGUNIT
- :math:`k / m^{2} - s`
- Beginning of time step regional average mass flow for pressure drop calculation
* - HTAV
- INIT
- :math:`W / m^{2} - K`
- Average heat transfer coefficient at tube wall surface on water side in one-node approximation or region
* - HTAVT
- INIT
- :math:`W / m^{2} - K`
- Total water side average heat transfer coefficient in one-node approximation of region
* - HWAV
- INIT
- :math:`J / kg`
- Average water enthalpy in one-node approximation of region
* - HWIN
- INIT
- :math:`J / kg`
- Steady state inlet water enthalpy
* - HWOUT
- INIT
- :math:`J / kg`
- Steady state outlet water enthalpy
* - IGO
- INIT
- \-
- Indicator which is set when the :math:`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 :math:`\left( = 2 \right)` or calibration factor :math:`\left( = 1 \right)` is to be searched on for subcooled zone
* - IOPT2
- INIT
- \-
- Indicator which shows whether length :math:`\left( = 2 \right)` or calibration factor :math:`\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 :math:`Z_{TP}` in boiling zone when the zone reaches the top of the steam generator or when :math:`Z_{TP}` changes more than a maximum amount allowed
* - ITER
- | SGUNIT
| INIT
- \-
- Iteration counter either on boiling zone length searches in :math:`SGUNIT` or searches in all three zones in :math:`INIT`
* - LMPDOT
- TSBOP
- \-
- Number of time steps and pressures stored in :math:`DTSG` and :math:`PTSG` arrays for calculation
* - PTSG(100)
- TSBOP
- :math:`Pa`
- Array to store pressures over :math:`LMPDOT` steam generator time steps in order to calculate
* - ROAV
- INIT
- :math:`kg / m^{3}`
- Average water density in one-node approximation of region
* - ROEDNB
- | SGUNIT
| INIT
- :math:`kg / m^{3}`
- Water density at the :math:`DNB` point used in pressure drop calculation
* - ROZ2
- | SGUNIT
| INIT
- :math:`kg / m^{3}`
- Average density in nucleate boiling zone for pressure drop calculation
* - ROZ3
- | SGUNIT
| INIT
- :math:`kg / m^{3}`
- Average density in film boiling zone for pressure drop calculation
* - ROZ4
- | SGUNIT
| INIT
- :math:`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
- :math:`s`
- Time difference between beginning of primary loop time step and the end of current steam generator time step
* - TMAV
- INIT
- :math:`K`
- Average tube wall temperature in one-node approximation to region
* - TNAINB
- TSBOP
- :math:`K`
- Inlet sodium temperature at the beginning of primary loop time step
* - TNAINE
- INIT
- :math:`K`
- Inlet sodium temperature at the end of primary loop time step
* - TSAV
- INIT
- :math:`K`
- Average sodium temperature in one-node approximation to region
* - TSHF
- INIT
- :math:`K`
- Sodium temperature at the point of :math:`h_{f}` on the water side at steady state
* - TSHG
- INIT
- :math:`K`
- Sodium temperature at the point of :math:`h_{f}` on the water side at steady state
* - TSIN
- INIT
- :math:`K`
- Steady state sodium inlet temperature
* - TSOUT
- INIT
- :math:`K`
- Steady state sodium outlet temperature
* - TWAV
- | SGUNIT
| INIT
- :math:`K`
- Average water temperature in one-node approximation to region
* - UZ2
- | SGUNIT
| INIT
- :math:`kg / m - s`
- Average viscosity in nucleate boiling zone for pressure drop calculation
* - UZ3
- | SGUNIT
| INIT
- :math:`kg / m - s`
- Average viscosity in film boiling zone for pressure drop calculation
* - UZ4
- TSBOP
- :math:`kg / m - s`
- Average viscosity in superheated zone for pressure drop calculation
* - WNAINB
- TSBOP
- :math:`kg / s`
- Sodium flow rate at beginning of primary loop time step
* - WNAINE
- SGUNIT
- :math:`kg / s`
- Sodium flow rate at end of primary loop time step
* - ZITER
- | SGUNIT
| INIT
- :math:`m`
- Current value of :math:`Z_{TP}` during search on region length in boiling zone calculation
* - ZONLE2
- | SGUNIT
| INIT
- :math:`m`
- Length of nucleate boiling zone used in pressure drop calculation
* - ZONLE3
-
- :math:`m`
- Length of film boiling zone used in pressure drop calculation
.. _section-7.appendices.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 :numref:`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 :ref:`[7-13] <section-7.references>`:
.. math:: \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
.. math:: \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 :math:`\mu` is the viscosity in Pa-s, :math:`\rho` is the density in kg/m\ :sup:`3`, :math:`T` is
the temperature in Kelvins, and the constants :math:`\rho` and :math:`T` are
:math:`\rho^{*} = 317.763` kg/m\ :sup:`3`,
:math:`T^{*} = 647.27` K,
The coefficients in the expression for :math:`\mu_{o}` are:
:math:`a_{0} = 0.018 1583`
:math:`a_{1} = 0.017 7624`
:math:`a_{2} = 0.010 5287`
:math:`a_{3} = -0.003 6744`
and the values for :math:`b_{ij}` are given in :numref:`table-A7.5-1`.
.. _table-A7.5-1:
.. list-table:: Numerical Values of the Coefficients :math:`b_{ij}`
:header-rows: 0
:align: center
:widths: auto
* -
- 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 :ref:`[7-14] <section-7.references>`. The
specific enthalpy of liquid water is given by
.. math::
h_{g} = \left\{ \begin{matrix}
\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{matrix} \right.
and the specific enthalpy of saturated steam is given by
.. math::
h_{g} = \left\{ \begin{matrix}
\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{matrix} \right.
where :math:`h_{f}` and :math:`h_{g}` are the specific enthalpy in units of BTU/lbm,
:math:`P` is the pressure in psia, :math:`P_{CRIT}` is the critical pressure :math:`\left(3208.2 \text{psia} \right)`,
and the constant coefficients are given in :numref:`table-A7.5-2`.
Expressions for the temperatures of subcooled water superheated steam as
functions of pressure and enthalpy are taken from RETRAN-02 :ref:`[7-14] <section-7.references>`:
.. math:: T_{l} = \sum_{i = 0}^{i = i} \sum_{j = 0}^{j = 3} \text{CT1}_{i, j} P^{i, j}
and
.. math:: T_{v} = \sum_{i = 0}^{i = 4} \sum_{j = 0}^{j = 4} \text{CT3}_{i, j} P^{i, j}
where :math:`T_{l}` and :math:`T_{v}` are the subcooled water and superheated steam
temperatures in degrees Fahrenheit, :math:`P` is the pressure in psia, :math:`h` is
the enthalpy in BTU/lbm, and the constant coefficients :math:`\text{CT1}` and
:math:`\text{CT3}` are given in :numref:`table-A7.5-3`. 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 :math:`T_{l}` and
:math:`T_{v}` with respect to enthalpy. The saturation temperature is obtained
from the expression for :math:`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 :ref:`[7-14] <section-7.references>`:
.. math:: v_{l} = \text{exp} \left[ \sum_{i = 0}^{2} \sum_{j = 0}^{4} \text{CN1}_{i, j} P^{i, j} \right]
and
.. math:: v_{v} = \sum_{i = -1}^{2} \sum_{j = 0}^{2} \text{CN2}_{i, j} P^{i, j}
where :math:`v_{l}` and :math:`v_{v}` are the subcooled water and superheated
steam specific volumes in :math:`ft^{3}\ /\ lbm`, :math:`P` is the pressure in
psia, :math:`h` is the enthalpy in BTU/lbm, and the constant coefficients
:math:`\text{CN1}` and :math:`\text{CN2}` are listed in :numref:`table-A7.5-4`. The satruated liquid
water density is computed from the value for :math:`v_{l}` at the ambient
pressure and the saturated steam specific enthalpy at that pressure.
Similarly the saturated steam density is obtained from :math:`v_{v}` at the
ambient pressure and the satruated steam specific enthalpy at that
pressure.
.. _table-A7.5-2:
.. list-table:: Constant Coefficients in Expressions for Saturated Liquid Water and Saturated Steam Enthalpies as Functions of Pressure
:header-rows: 0
:align: center
:widths: auto
* - i
- :math:`\text{CF1}_{i}`
- :math:`\text{CF2}_{i}`
- :math:`\text{CF3}_{i}`
* - 0
- :math:`.6970887859 \times 10^{2}`
- :math:`.8408618802 \times 10^{6}`
- :math:`.9060030436 \times 10^{3}`
* - 1
- :math:`.3337529994 \times 10^{2}`
- :math:`.3637413208 \times 10^{6}`
- :math:`-.1426813520 \times 10^{0}`
* - 2
- :math:`.2318240735 \times 10^{1}`
- :math:`-.4634506669 \times 10^{6}`
- :math:`.1522233257 \times 10^{1}`
* - 3
- :math:`.1840599513 \times 10^{0}`
- :math:`.1130306339 \times 10^{6}`
- :math:`-.6973992961 \times 10^{0}`
* - 4
- :math:`-.5245502284 \times 10^{-2}`
- :math:`-.4350217298 \times 10^{3}`
- :math:`.1743091663 \times 10^{0}`
* - 5
- :math:`.2878007027 \times 10^{-2}`
- :math:`-.3898988188 \times 10^{4}`
- :math:`-.2319717696 \times 10^{-1}`
* - 6
- :math:`.1753652324 \times 10^{-2}`
- :math:`.6697399434 \times 10^{3}`
- :math:`.1694019149 \times 10^{-2}`
* - 7
- :math:`-.4334859629 \times 10^{-3}`
- :math:`-.4730726377 \times 10^{2}`
- :math:`-.6454771710 \times 10^{-4}`
* - 8
- :math:`.3325699282 \times 10^{-4}`
- :math:`.1265125057 \times 10^{1}`
- :math:`.1003003098 \times 10^{-5}`
* -
-
-
-
* - i
- :math:`\text{CG1}_{i}`
- :math:`\text{CG2}_{i}`
- :math:`\text{CG3}_{i}`
* - 0
- :math:`.1105836875 \times 10^{4}`
- :math:`-.2234264997 \times 10^{7}`
- :math:`.9059978254 \times 10^{3}`
* - 1
- :math:`.1436943768 \times 10^{2}`
- :math:`.1231247634 \times 10^{7}`
- :math:`.5561957539 \times 10^{1}`
* - 2
- :math:`.8018288621 \times 10^{0}`
- :math:`-.1978847871 \times 10^{7}`
- :math:`.3434189609 \times 10^{1}`
* - 3
- :math:`.1617232913 \times 10^{-1}`
- :math:`.1859988044 \times 10^{2}`
- :math:`-.6406390628 \times 10^{0}`
* - 4
- :math:`-.1501147505 \times 10^{-2}`
- :math:`-.2765701318 \times 10^{1}`
- :math:`.5918579484 \times 10^{-1}`
* - 5
- :math:`.0000000000 \times 10^{0}`
- :math:`.1036033878 \times 10^{4}`
- :math:`-.2725378570 \times 10^{-2}`
* - 6
- :math:`.0000000000 \times 10^{0}`
- :math:`-.2143423131 \times 10^{3}`
- :math:`.5006336938 \times 10^{-4}`
* - 7
- :math:`.0000000000 \times 10^{0}`
- :math:`.1690507762 \times 10^{2}`
-
* - 8
- :math:`.0000000000 \times 10^{0}`
- :math:`-.4864322134 \times 10^{0}`
-
* - 9
- :math:`-.1237675562 \times 10^{-4}`
-
-
* - 10
- :math:`.3004773304 \times 10^{-5}`
-
-
* - 11
- :math:`-.2062390734 \times 10^{-6}`
-
-
.. _table-A7.5-3:
.. list-table:: Constant Coefficients in Expressions for Temperature as a Function of Pressure and Specific Enthalpy
:header-rows: 0
:align: center
:widths: auto
* -
-
-
- :math:`\text{CT1}_{i, j}`
-
-
* -
- i = 0
- i = 1
- i = 2
- i = 3
-
* - j = 0
- :math:`0.3276275552 \times 10^{2}`
- :math:`0.9763617000 \times 10^{0}`
- :math:`0.1857226027 \times 10^{-3}`
- :math:`-0.4682674330 \times 10^{-6}`
-
* - j = 1
- :math:`0.3360880214 \times 10^{-2}`
- :math:`-0.5595281760 \times 10^{-4}`
- :math:`0.1618595991 \times 10^{-6}`
- :math:`-0.1180204381 \times 10^{-9}`
-
* -
-
-
- :math:`\text{CT3}_{i, j}`
-
-
* -
- i = 0
- i = 1
- i = 2
- i = 3
- i = 4
* - j = 0
- :math:`-0.1179100862 \times 10^{5}`
- :math:`0.2829274345 \times 10^{2}`
- :math:`-0.2678181564 \times 10^{-1}`
- :math:`0.1218742752 \times 10^{-4}`
- :math:`-0.2092033147 \times 10^{-8}`
* - j = 1
- :math:`0.1256160907 \times 10^{3}`
- :math:`-0.3333448495 \times 10^{0}`
- :math:`0.3326901268 \times 10^{-3}`
- :math:`-0.1477890326 \times 10^{-6}`
- :math:`0.2463258371 \times 10^{-10}`
* - j = 2
- :math:`-0.1083713369 \times 10^{0}`
- :math:`0.2928177730 \times 10^{-3}`
- :math:`-0.2972436458 \times 10^{-6}`
- :math:`0.1342639113 \times 10^{-9}`
- :math:`-0.2275585718 \times 10^{-13}`
* - j = 3
- :math:`0.3278071846 \times 10^{-4}`
- :math:`-0.8970959364 \times 10^{-7}`
- :math:`0.9246248312 \times 10^{-10}`
- :math:`-0.4249155515 \times 10^{-13}`
- :math:`0.7338316751 \times 10^{-17}`
* - j = 4
- :math:`-0.3425564927 \times 10^{-8}`
- :math:`0.9527692453 \times 10^{-11}`
- :math:`-0.1001409043 \times 10^{-13}`
- :math:`0.4703914404 \times 10^{-17}`
- :math:`-0.8315044742 \times 10^{-21}`
.. _table-A7.5-4:
.. list-table:: Constant coefficients in Expressions for Specific Volume as a Function of Pressure and Specific Enthalpy
:header-rows: 0
:align: center
:widths: auto
* -
-
-
- :math:`\text{CN2}_{i, j}`
-
-
* -
- i = 0
- i = 1
- i = 2
- i = 3
- i = 4
* - j = 0
- :math:`-0.4117961750 \times 10^{1}`
- :math:`-0.3811294543 \times 10^{-3}`
- :math:`0.4308265942 \times 10^{-5}`
- :math:`-0.9160120130 \times 10^{-8}`
- :math:`0.8017924673 \times 10^{-11}`
* - j = 1
- :math:`-0.4816067020 \times 10^{-5}`
- :math:`0.7744786733 \times 10^{-7}`
- :math:`-0.6988467605 \times 10^{-9}`
- :math:`0.1916720525 \times 10^{-11}`
- :math:`-0.1760288590 \times 10^{-14}`
* - j = 2
- :math:`-0.1820625039 \times 10^{-8}`
- :math:`0.1440785930 \times 10^{-10}`
- :math:`-0.2082170753 \times 10^{-13}`
- :math:`-0.3603625114 \times 10^{-16}`
- :math:`0.7407124321 \times 10^{-19}`
* -
-
-
- :math:`\text{CN2}_{i, j}`
-
-
* -
- i = 0
- i = 1
- i = 2
-
-
* - j = -1
- :math:`-0.1403086182 \times 10^{4}`
- :math:`0.1802594763 \times 10^{1}`
- :math:`-0.2097279215 \times 10^{-3}`
-
-
* - j = 0
- :math:`0.3817195017 \times 10^{0}`
- :math:`-0.5394444747 \times 10^{-3}`
- :math:`0.1855203702 \times 10^{-6}`
-
-
* - j = 1
- :math:`-0.6449501159 \times 10^{-4}`
- :math:`0.8437637660 \times 10^{-7}`
- :math:`-0.2713755001 \times 10^{-10}`
-
-
* - j = 2
- :math:`0.7823817858 \times 10^{-8}`
- :math:`-0.1053834646 \times 10^{-10}`
- :math:`0.3629590764 \times 10^{-14}`
-
-