.. _section-7.3:
Steam Generator Model
----------------------
.. _section-7.3.1:
Once-Through Steam Generator
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In an once‑through steam generator subcooled feedwater enters the bottom
on the water side and superheated steam comes out the top (see :numref:`figure-7.3-1`). Considering that there is a transition boiling zone, the water
side is naturally divided into three regions. :numref:`figure-7.3-2` shows a
detailed schematic of the once‑through steam generator. The top of the
subcooled zone and the bottom of the boiling zone is defined by the
point of saturated liquid enthalpy. The top of the boiling zone and the
bottom of the superheated vapor zone is defined by the point of
saturated vapor enthalpy. This is the situation during normal
steady‑state operation. Various transient conditions can produce any
situation from a steam generator filled with subcooled water to total
dry‑out on the water side. The current model can calculate this whole
spectrum of conditions with one proviso: there must always be a
subcooled liquid region of some finite length at the inlet of the steam
generator; but this length can be extremely small. Another way of
stating this assumption is that there is no provision for two‑phase
fluid or superheated vapor in the inlet plenum of the steam generator.
Going to the other end of the spectrum of transient cases, the complete
disappearance of both the boiling and superheated vapor zones can be
calculated. If, however, there is a superheated vapor zone, there must,
of course, be a boiling zone. Each of the three zones, therefore, is
treated as a separate calculation with its own node structure and
providing boundary conditions for the adjacent region or regions even as
each zone length changes during the transient.
The steam generator is, of course, one module in the balance‑of‑plant
sequence. As far as the system is concerned, it represents a pressure
drop and a momentum source or sink from a hydrodynamic point of view.
The balance-of‑plant calculation produces pressures at the inlet and
outlet plena of the steam generator as well as the mass flow into the
steam generator. As will be shown later, the steam generator model
itself calculates outlet flows from the steam generator. The steam
generator provides the balance‑of‑plant momentum equation with an
estimate of the pressure drop across the steam generator. This will be
discussed later.
.. _figure-7.3-1:
.. figure:: media/Figure7.3-1.png
:align: center
:figclass: align-center
Once-Through Steam Generator
.. _figure-7.3-2:
.. figure:: media/Figure7.3-2.png
:align: center
:figclass: align-center
Schematic of Once-Through Steam Generator
There are two main points to emphasize here, however. First, the inlet
and outlet pressures provided by the balance‑of‑plant momentum equation
are simply averaged and this average pressure is used at each time step
by the steam generator to calculate properties. Thus no account is taken
of the variation of pressure across the steam generator for the purpose
of calculating properties. For many and probably most transient
calculations of interest, this pressure variation is small and, although
not trivial, can be safely neglected in the total context of the
calculation. There are some transients, however, involving large
pressure reductions downstream of the steam generator, which would
produce a significant pressure variation across the steam generator, the
neglect of which could lead to some level of inaccuracy.
The second point to emphasize is, given the estimate of the pressure
drop across the steam generator, the inlet flow is provided as a
boundary condition for the steam generator. There is no momentum
equation coupled to the mass and energy conservation equations which
characterize the regions in the steam generator to produce velocities.
Instead the mass flows above the subcooled zone result from the mass and
energy equations alone (as shown below), given the inlet flow as a
driving function.
The subcooled liquid and superheated vapor zones each have their own
heat transfer regime. The boiling zone has two heat transfer regimes
separated at the boiling crisis or departure‑from‑nucleate‑boiling (DNB)
point. No smoothing or intermediate regimes are used between these four
regimes. This provides of course calculation convenience. Heat transfer
phenomena in a steam generator are much more complicated. The adequacy
of this heat transfer scheme will be judged by benchmarking against
experimental data. The DNB point is crucial to properly characterizing
the boiling zone. But tracking the point produces calculation
difficulties. This will be explained in some detail later. Suffice it to
say here that the DNB point is assumed to be at the intersection of two
curves, one representing the local heat flux at the tube wall and the
other representing the heat flux required for the boiling crisis to
occur. In this way, the point of maximum boiling heat flux is tracked.
This DNB point is tracked continuously within the node structure of the
boiling zone and the heat flux in the cell where the DNB point is a
prorated average of the two boiling heat transfer regimes since the
volumetric heat flux is always calculated on a cell‑average basis.
Depending on the fineness of the node structure, this produces some
amount of inaccuracy and approximation to the real physical situation.
The following are general forms of the continuity and of the enthalpy
form of the energy conservation equation in one dimension:
.. math:: \frac{\partial\rho}{\partial t} = - \frac{\partial G}{\partial z}
:label: 7.3-1
.. math:: \frac{\partial(\rho h)}{\partial t} = - \frac{\partial\left( Gh \right)}{\partial z} + Q - \left( \tau:\nabla v \right) + \frac{\partial P}{\partial t} + v\frac{\partial P}{\partial z}
:label: 7.3-2
:math:`Q` is a volumetric heat source, :math:`- \left( \tau:\nabla v \right)`
represents viscous dissipation and :math:`v(\partial P/\partial z)` is a
work‑energy conversion term (representing feedwater pump work, for
example). The viscous dissipation term will be dropped for this
application because it is small compared to other terms. The work term
will also be neglected for the same reason although it is possible that
in certain extreme conditions the term could be of some significance.
The general energy equation thus becomes,
.. math:: \frac{\partial(\rho h)}{\partial t} = - \frac{\partial\left( Gh \right)}{\partial z} + Q + \frac{\partial P}{\partial t}
:label: 7.3-3
Incompressible flow is assumed in the subcooled liquid zone. The
balance‑of‑plant momentum equation provides the inlet mass flow, as
stated above and thus the mass flow for the whole zone :math:`\left( \Delta G = 0 \right)`.
Therefore no continuity equation is needed to characterize the region.
The enthalpy level and shape and the subcooled region length are all
that need be solved for with a coupled set of nodal energy conservation
equations. The boundary conditions for flow are the constant mass flow
and for energy, the inlet enthalpy and the saturated liquid enthalpy
unless the zone reaches the top of the steam generator in which case the
region length is known and the outlet enthalpy is calculated. Since is
assumed to be zero, the LHS of the energy equation, :eq:`7.3-3`, can be
simplified. The density changes over the transient as a result of
changes in pressure and enthalpy but these changes are taken into
account by updating the density at the end of each time step in the
transient after new enthalpies and pressures are obtained.
In the boiling region, compressible flow is calculated with sets of
nodal mass and energy conservation equations. The equations are
formulated in terms of the void fraction instead of density and
enthalpy. This is conveniently done since saturation conditions are
always assumed and a no slip condition between phases is assumed; and
also because saturation properties are functions of pressure alone. Thus
simultaneous nodal continuity and energy equations are used to solve for
mass flows and void fractions. The boundary conditions at the bottom of
the zone are the single‑phase liquid flow and the saturated liquid
enthalpy (i.e. void fraction zero). At the top of the zone, there is
either the saturated vapor enthalpy (i.e. void fraction 1.0) or, if the
boiling zone extends all the way to the top of the steam generator, then
the void fraction is a free variable and only lower boundary conditions
are required. When the zone does not extend to the top of the steam
generator, then the upper boundary condition of the saturated vapor
enthalpy is used by requiring that the length of the zone be adjusted
until the upper boundary condition is satisfied.
Compressible flow is also assumed in the superheated vapor region. The
boundary conditions at the bottom of the zone are the saturated vapor
enthalpy and the mass flow calculated at the top of the boiling zone.
Since the length of the zone is known, simultaneous nodal mass and
energy equations are used to solve for enthalpies and mass flows all the
way to the top of the steam generator.
On the sodium side, there is no change of phase and consequently the
liquid can be adequately treated with incompressible flow. There is a
calculation of the sodium flow external to the steam generator
calculation so that, as far as the steam generator is concerned, the
mass flow is given. Therefore no continuity equation is required.
Besides the mass flow, the only boundary condition required is the inlet
sodium temperature at the top of the steam generator. Only the nodal
energy equation is required to solve for the enthalpy shape on the
sodium side. In addition, given the relatively stable and low pressure
conditions on the sodium side, the pressure term in the energy equation
:eq:`7.3-3` can be safely neglected.
The heat capacity of the tube wall separating the water and the sodium
must be taken into account. Therefore its effect on the heat transfer
from the sodium to the water is treated by means of a wall temperature
calculation. With no convective or pressure terms the energy equation
:eq:`7.3-3` becomes much simplified. Also, the density is assumed to be
temperature independent which further facilitates the solution.
.. _section-7.3.2:
Recirculation-Type Steam Generator
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
:numref:`figure-7.3-3` depicts a steam generator which consists of several
evaporators, several superheaters, a steam drum and a recirculation
loop. Subcooled water is pumped into the bottom of the evaporator.
Within the evaporator the water boils and a two‑phase fluid, typically
of very high quality, leaves the top of the evaporator and enters the
steam drum. One exit line from the steam drum transports saturated steam
to the bottom of the superheater where more heat is added so that highly
superheated steam leaves the top available for the turbines. The other
exit line from the steam drum transports saturated liquid to the pump
but before reaching the pump it is mixed with feedwater. This mixture is
substantially subcooled, therefore, and is pumped back to the evaporator
to complete the cycle.
The modeling of the evaporator can be done with the once‑through steam
generator model which is designed to model any situation from a
liquid‑filled steam generator to the normal operating condition for a
once‑through type with superheated vapor exiting the top. Thus the
physical modeling assumptions of the previous section apply to the
evaporator.
The modeling of the superheater is different, however, than the
super-heated vapor zone of the previous section. Without elaborating on
the details, it is sufficient here to say that the momentum equation of
the balance‑of‑plant model is much more tractable if the assumption of
incompressible flow is made for the superheater. Since the superheater
operates at quite high pressures, this incompressibility assumption is
probably adequate in most transient conditions. There may, however, be
certain conditions when the pressure in the superheater is greatly
reduced when inaccuracies may result from this assumption. A study of
this effect would have to be undertaken to decide the issue and it has
not been done so far. The lower boundary condition besides the given
mass flow is the inlet enthalpy (i.e. the saturated vapor enthalpy).
Thus the enthalpy shape is determined given these conditions. The
density is updated each time step during the transient after new
enthalpies and pressures have been determined.
The steam drum is modeled as a zero‑dimensional reservoir in the sense
that perfect mixing of all incoming fluid and the pre‑existing separated
two phases is presumed. There is one proviso here, however: the liquid
level must be tracked so that appropriate action can be taken when
liquid may enter the pipe to the superheater or vapor may enter the pipe
to the recirculation pump.
.. _figure-7.3-3:
.. figure:: media/Figure7.3-3.png
:align: center
:figclass: align-center
Steam Generator with Separate Evaporator and Superheaters
and Recirculation Loop
.. _section-7.3.3:
Numerical Solution Methods
~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-7.3.3.1:
Once‑Through Steam Generator
''''''''''''''''''''''''''''
.. _section-7.3.3.1.1:
General Forms of the Conservation Equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Before the individual solution methods for each water side region and
the sodium side can be considered, general forms of the continuity and
energy equations must be developed. Equations will be given for one node
of the multi‑node system of equations. Integration of the continuity
:eq:`7.3-1` over the length of one cell from :math:`Z_{i}` to :math:`Z_{i + 1}`
gives,
.. math:: \int_{Z_{i}}^{Z_{i + 1}} \frac{\partial}{\partial t} \rho dz = - \int_{Z_{i}}^{Z_{i+1}} \frac{\partial}{\partial z} G dz
:label: 7.3-4
According to Leibnitz's Theorem,
.. math:: \int_{Z_{i}}^{Z_{i+1}} \frac{\partial}{\partial z} \rho dz = \frac{d}{dz} \int_{Z_{i}}^{Z_{i+1}} \rho dz - \rho_{i + 1} \dot{Z}_{i + 1} + \rho _{i} \dot{Z}_{i}
:label: 7.3-5
:eq:`7.3-4` becomes,
.. math:: \frac{d}{dt} \int_{Z_{i}}^{Z_{i + 1}} \rho dz - \rho_{i + 1} \dot{Z}_{i + 1} + \rho_{i} \dot{Z}_{i} = -\int_{Z_{i}}^{Z_{i + 1}} \frac{\partial}{\partial z} G dz
:label: 7.3-6
Therefore,
.. math:: \frac{d}{dt}\left\{ \overline{\rho}_i {\Delta}Z_{i} \right\} - \rho_{i + 1}{\dot{Z}}_{i + 1} + \rho_{i}{\dot{Z}}_{i} = - {\Delta}G_{i}
:label: 7.3-7
Donor‑cell differencing is used to enhance numerical stability. In order
to write the equation in donor‑cell form, let :math:`\rho_{i + 1}` replace
the average value over the interval, :math:`\overline{\rho}_{i}` and simplify,
.. math:: \dot{\rho}_{i + 1} \Delta Z_{i} - \Delta \rho_{i} \dot{Z}_{i} = - \Delta G_{i}
:label: 7.3-8
Integration of the enthalpy form of the energy :eq:`7.3-3` from :math:`Z_{i}` to
:math:`Z_{i + 1}` gives,
.. math:: \int_{Z_{i}}^{Z_{i + 1}} \frac{\partial}{\partial t} \left( \rho h \right) dz = - \int_{Z_{i}}^{Z_{i + 1}} \frac{\partial}{\partial z} \left( G h \right) dz + \int_{Z_{i}}^{Z_{i + 1}} Q dz + \int_{Z_{i}}^{Z_{i + 1}} \frac{\partial}{\partial z} P dz
:label: 7.3-9
According to Leibnitz's theorem,
.. math:: \int_{Z_{i}}^{Z_{i + 1}}{\frac{\partial}{\partial t}(\rho h)dz} = \frac{d}{dt}\int_{Z_{i}}^{Z_{i + 1}}{(\rho h)dz} - {(\rho h)}_{i + 1}{\dot{Z}}_{i + 1} + {(\rho h)}_{i}{\dot{Z}}_{i}
:label: 7.3-10
.. math:: \int_{Z_{i}}^{Z_{i + 1}} \frac{\partial}{\partial t} P dz = \frac{d}{dt} \int_{Z_{i}}^{Z_{i + 1}} P dz - P_{i + 1} \dot{Z}_{i + 1} + P_{i} \dot{Z}_{i}
:label: 7.3-11
:eq:`7.3-9` becomes
.. math::
:label: 7.3-12
\frac{d}{dh} \int_{Z_{i}}^{Z_{i+1}} \left( \rho h \right) dz - \left( \rho h \right)_{i + 1} \dot{Z}_{i + 1} + \left( \rho h \right)_{i} \dot{Z}_{i} = - \int_{Z_{i}}^{Z_{i+1}} \frac{\partial}{\partial z} \left( G h \right) dz + \int_{Z_{i}}^{Z_{i+1}} Q dz \\
+ \frac{d}{dt} \int_{Z_{i}}^{Z_{i+1}} P dz - P_{i + 1} \dot{Z}_{i + 1} + P_{i} \dot{Z}_{i}
Therefore,
.. math::
:label: 7.3-13
\frac{d}{dt} \left\{ \left( \overline{ \rho h } \right)_{i} \Delta Z_{i} \right\} - \left( \rho h \right)_{i + 1} \dot{Z}_{i + 1} + \left( \rho h_{i} \right) \dot{Z}_{i} = - \Delta \left( G h \right)_{i} + Q_{i} \Delta Z_{i} \\
+ \frac{d}{dt} \left\{ \overline{P}_{i} \Delta Z_{i} \right\} - P_{i + 1} \dot{Z}_{i + 1} + P_{i} \dot{Z}_{i}
In order to write :eq:`7.3-13` in donor‑cell form, let
:math:`\left( \rho h \right)_{i + 1}` replace the average value over
the interval :math:`\left[ \rho h \right]_{i}` let :math:`P_{i + 1}` replace :math:`\overline{P}_{i}`, and
simplify,
.. math:: \left( \rho \dot{h} \right)_{i + 1} \Delta Z_{i} - \Delta \left( \rho h \right)_{i} \dot{Z}_{i} = -\Delta \left( G h \right)_{i} + Q_{i} \Delta Z_{i} + \dot{P}_{i + 1} \Delta Z_{i} - \Delta P_{i} \dot{Z}_{i}
:label: 7.3-14
As discussed before, the pressure variation across the steam generator
is neglected, i.e., :math:`\Delta P_{i} = 0`, and :eq:`7.3-14` becomes
.. math:: \left( \rho \dot{h} \right)_{i + 1} \Delta Z_{i} - \Delta \left( \rho h \right)_{i} \dot{Z}_{i} = -\Delta \left( G h \right)_{i} + Q_{i} \Delta Z_{i} + \dot{P} \Delta Z_{i}
:label: 7.3-15
Each of the three regions on the water side is divided into a fixed
number of cells. Since the region lengths vary during the transient, the
cell lengths also vary and are thus a constant fraction of the varying
region length. Thus :math:`\Delta Z_{i} = \frac{1}{n} Z_{x}` where :math:`Z_{x}`
is the current length of region :math:`x` and :math:`n` is the number of cells in region
:math:`x`. Therefore the subscript :math:`i` can be dropped for :math:`\Delta Z_{i}`
in equations for a given region. On the sodium side, the same node structure is used as on the
water side in order to simplify the calculation, although the precise node
structure on the sodium side is not nearly as important as on the water side
since there is no change of phase and properties thus calculated parameters
change gradually and smoothly over the length of the steam generator.
All spatially variable parameters except two are evaluated at the cell
boundary so that there are :math:`n+1` values needed to characterize a region,
where :math:`n` is the number of cells in the region. The two exceptions are the
tube wall enthalpy and the heat flux. The volumetric heat source is most
conveniently calculated on a cell‑average basis so that n cells exactly
encompass the whole of the heat transfer for a given region. If the heat
source were calculated at the cell edge, then half‑cells would have to
be used at the ends of a region where the heat transfer coefficients
change form. In order to calculate the temperature gradients for the
heat flux, linear averages over the cell length are computed from the
cell‑edge values for water and sodium. However, there is no need to
calculate the tube wall temperature at the cell edge and a cell‑centered
value is most convenient for the heat flux calculation. Other parameters
such as enthalpies, void fractions, mass fluxes, etc. are most
conveniently calculated at the cell‑edge since this is where the
boundary conditions are defined.
.. _section-7.3.3.1.2:
Subcooled Liquid Region
^^^^^^^^^^^^^^^^^^^^^^^
In the subcooled region, the mass flow is assumed to be uniform
throughout the zone due to incompressible flow. Each time step during
the transient an updated inlet mass flow is provided to the steam
generator from the explicitly‑coupled momentum equation. Therefore no
continuity equation is required. The coupled set of nodal energy
conservation equations are used to determine the length of the zone and
the nodal enthalpies simultaneously using the inlet enthalpy and the
saturated liquid enthalpy as boundary conditions. Or, alternatively,
when liquid fills the steam generator and the zone length is known, the
outlet enthalpy is instead determined.
Using :eq:`7.3-15` and setting :math:`\dot{ \rho} = 0` because of the incompressible
flow, and recalling that :math:`\Delta Z_{i}` is invarient within a zone, the
following results for node :math:`i`,
.. math:: \dot{h}_{i + 1} \rho_{i + 1} \Delta Z - \Delta \left( \rho h \right)_{i} \dot{Z}_{i} = - \Delta \left( G h \right)_{i} + Q_{i} \Delta Z + \dot{P} \Delta Z
:label: 7.3-16
The following is :eq:`7.3-16` in finite difference form,
.. math::
:label: 7.3-17
h_{i + 1}^{k + 1} \frac{1}{\Delta t} \rho_{i + 1}^{k} \frac{1}{n} Z_{SC}^{k} - h_{i + 1}^{k} \frac{1}{\Delta t} \rho_{i + 1}^{k} \frac{1}{n} Z_{SC}^{k + 1} - \overline{\rho}_{i}^{k} \left(h_{i + 1}^{k} - h_{i}^{k} \right) \frac{i - 1}{n} \\
\times \frac{1}{\Delta t} \left(Z_{SC}^{k + 1} - Z_{SC}^{k} \right) = - G^{k + 1} \left(h_{i + 1}^{k + 1} - h_{i}^{k + 1} \right) + Q_{i}^{k} \frac{1}{n} Z_{SC}^{k + 1} + \dot{P} \frac{1}{n} Z_{SC}^{k + 1}
There are several points to note concerning :eq:`7.3-17`. The
:math:`\Delta Z` in the first term on the LHS is treated semi-implicitly in
time since :math:`h_{i + 1}^{k + 1}` is multiplied by :math:`Z_{SC}^{k}`, the value at the
beginning of the time step and :math:`h_{i + 1}^{k + 1}` is multiplied by :math:`Z_{SC}^{k + 1}`,
the value at the end‑of‑step which needs to be determined. :math:`h_{i + 1}^{k + 1}` cannot be
multiplied by the end‑of‑step value without making the equation set non‑linear.
Making the term semi‑implicit as opposed to fully explicit has been found to
enhance stability in the calculation. The next point is that the density is
entirely explicit in time. It is updated at the end of every time step as
enthalpy and pressure change. This is consistent with the assumption of
incompressibility as density changes slowly and gradually over time. Note also
the use of
:math:`{\overline{\rho}}_{i}^{k}` in the third term on the LHS of
:eq:`7.3-17`. :math:`\Delta \rho h_{i}` becomes :math:`\rho \Delta h_{i}`
in order to preserve the proper sign of the term. Using :math:`\overline{\rho}_{i}^{k}`
which is :math:`0.5 \left( \rho_{i + 1}^{k} + \rho_{i}^{k} \right)`, introduces only
a small error so long as the mesh structure is not too coarse. :math:`\Delta h_{i}`
must be explicit in time because it is multiplied by
:math:`Z_{SC}^{k + 1}` and the equation must be kept linear.
:math:`G^{k + 1}` in the convective term implies that the inlet mass flow is updated
before the steam generator solution begins. The volumetric heat source is
totally explicit since the temperatures used in calculating the :math:`\Delta T` and the
properties contained in the heat transfer correlation (see :numref:`section-7.appendices.2`)
cannot be made implicit without making the equation insoluble.
Rearranging :eq:`7.3-17` according to coefficients there are :math:`n` equations
of the following form,
.. math::
:label: 7.3-18
\left[-G^{k + 1} \right] h_{i}^{k + 1} + \left[ \frac{1}{\Delta t} \rho_{i + 1}^{k} \frac{1}{n} Z_{SC}^{k} + G^{k + 1} \right] h_{i + 1}^{k + 1} \\
+ \left[ -h_{i + 1}^{k} \frac{1}{\Delta t} \rho_{i + 1}^{k} \frac{1}{n} - \overline{\rho}_{i}^{k} \left( h_{i + 1}^{k} - h_{i}^{k} \right) \frac{i - 1}{n} \frac{1}{\Delta t} - Q_{i}^{k} \frac{1}{n} - \dot{P} \frac{1}{n} \right] Z_{SC}^{k + 1} \\
= - \overline{\rho}_{i}^{k} \left(h_{i + 1}^{k} - h_{i}^{k} \right) \frac{i - 1}{n} \frac{1}{\Delta t} Z_{SC}^{k + 1}
:eq:`7.3-18` can be written as,
.. math:: a_{i}h_{i} + b_{i}h_{i + 1} + c_{i} Z_{SC} = d_{i}
:label: 7.3-19
In the case where the liquid region does not reach the top of the steam
generator, :math:`h_{1}` and :math:`h_{n + 1}` are known, since they are
the inlet enthalpy and :math:`h_{f}. h_{2} - h_{n}` and :math:`Z_{SC}` are unknown
(:math:`n` unknowns) and are solved for as follows. (The superscript :math:`k + 1`
is dropped as unnecessary.)
From the first equation of the equation set :eq:`7.3-19`, solve for
:math:`h_{2}`, then for :math:`h_{3}` from the second equation and so
on,
.. math::
:label: 7.3-20
h_{2} &= e_{1} + f_{1} Z_{SC}; & e_{1} &= \frac{d_{1} - a_{1} h_{1}}{b_{1}}, & f_{1} &= -\frac{c_{1}}{b_{1}}
h_{3} &= e_{2} + f_{2} Z_{SC}; & e_{2} &= \frac{d_{2} - a_{2} e_{1}}{b_{2}}, & f_{2} &= -\frac{c_{2} + a_{2} f_{1}}{b_{2}}
h_{i + 1} &= e_{i} + f_{i} Z_{SC}; & e_{i} &= \frac{d_{i} - a_{i} e_{i - 1}}{b_{i}}, & f_{i} &= - \frac{c_{i} + a_{i} f_{i - 1}}{b_{i}}
Finally, in the equation for :math:`h_{n + 1}, Z_{SC}` can be solved
for since :math:`h_{n + 1}` is known. Then each of :math:`h_{2} - h_{n}`
can be solved for since they are all functions of only :math:`Z_{SC}` in
the equation set :eq:`7.3-20`.
If the steam generator is filled with liquid, then the outlet enthalpy
is unknown and there are :math:`n`s equations with :math:`h_{2} - h_{n + 1}`
as the unknowns. :eq:`7.3-16` becomes the following in
finite difference form (the zone length, now constant as the length of
the steam generator, is denoted simply as :math:`Z_{SC}`),
.. math:: \frac{1}{\Delta t} \rho_{i + 1}^{k} \frac{1}{n} Z_{SC} \left(h_{i + 1}^{k + 1} - h_{i + 1}^{k} \right) = - G^{k + 1} \left(h_{i + 1}^{k + 1} - h_{i}^{k + 1} \right) + \frac{1}{n} Z_{SC} \left(Q_{i}^{k} + \dot{P} \right)
:label: 7.3-21
The following equations are simply solved from the bottom to the top of
the steam generator successively,
.. math:: h_{i + 1}^{k + 1} = \frac{ \frac{1}{\Delta t} h_{i + 1}^{k} \rho_{i + 1}^{k} \frac{1}{n} Z_{SC} + G^{k + 1} h_{i}^{k + 1} + Q_{i} \frac{1}{n} Z_{SC} + \dot{P} \frac{1}{n} Z_{SC}}{\frac{1}{\Delta t} \rho_{i + 1}^{k} \frac{1}{n} Z_{SC} + G^{k + 1}}
:label: 7.3-22
.. _section-7.3.3.1.3:
Boiling Region
^^^^^^^^^^^^^^
In the boiling zone, the fluid is treated as compressible. Simultaneous
nodal equations are solved for void fraction, mass flow and region
length. Boundary conditions are the saturated liquid enthalpy and
subcooled region mass flow at the bottom of the boiling zone and either
the saturated vapor enthalpy at the top of the zone or, if there is no
superheated vapor zone, the region length is defined and the outlet
enthalpy is determined. Only the pressure and the volumetric heat source
are treated explicitly in time. The void fraction, the mass flow and the
region length are all treated in fully implicit fashion. An iterative
method is used to solve for the region length. The current value of the
region length is held constant for each pass in the iteration while
nodal void fractions and mass flows are calculated from the mass and
energy equations. When the uppermost void fraction in the boiling zone
is computed at the end of an iteration, its value is compared to 1.0 and
the region length is adjusted appropriately and the iterative process
continues until convergence. When the boiling zone extends to the top of
the steam generator, the same method is used but there is no iteration
since the region length is known.
First, a number of definitions and identities concerning a two‑phase
fluid must be reviewed. :math:`\rho_{f}` and :math:`h_{f}` are the saturated liquid
density and enthalpy respectively. :math:`\rho_{g}` and :math:`h_{g}` are the saturated
vapor density and enthalpy. :math:`\rho_{fg} = \rho_g - \rho_f. h_{fg} = h_g - h_f`
The density of the two‑phase mixture is, with :math:`\alpha` denoting the
void fraction,
.. math:: \rho = \alpha \rho_{fg} + \rho_{f}
:label: 7.3-23
The enthalpy of the two‑phase mixture is, with :math:`\chi` denoting the
quality,
.. math:: h = \chi h_{fg} + h_{f}
:label: 7.3-24
Since homogeneous flow is assumed in the two‑phase zone,
.. math:: \chi = \frac{\alpha\rho_{g}}{\rho} = \frac{\alpha\rho_{g}}{\rho_{f} + \alpha\rho_{fg}}
:label: 7.3-25
Defining :math:`\left(h \rho \right)_{fg}` as :math:`h_{g}\rho_{g} - h_{f}\rho_{f}`,
:eq:`7.3-23`, :eq:`7.3-24`, and :eq:`7.3-25` imply,
.. math:: \left( h \rho \right) = h_{f} \rho_{f} + \alpha \left(h \rho \right)_{fg}
:label: 7.3-26
The general continuity :eq:`7.3-8` is written in terms of the nodal void
fraction, :math:`\alpha`,
.. math:: \left( \dot{\alpha}_{i + 1} \rho_{fg} + \alpha_{i + 1} \dot{\rho}_{fg} + \dot{\rho}_{f} \right) \Delta Z - \left(\alpha_{i + 1} - \alpha_{i} \right) \rho_{fg} \dot{Z}_{i} = -\Delta G_{i}
:label: 7.3-27
The finite difference form of :eq:`7.3-27` is,
.. math::
:label: 7.3-28
\left[ \frac{1}{\Delta t} \left(\alpha_{i + 1}^{k + 1} - \alpha_{i + 1}^{k} \right) \rho_{fg}^{k} + \alpha_{i + 1}^{k + 1} \dot{\rho}_{fg}^{k} + \dot{\rho}_{f}^{k} \right] \frac{1}{n} Z_{TP}^{k + 1} \\
- \left(\alpha_{i + 1}^{k + 1} - \alpha_{i}^{k + 1} \right) \rho_{fg}^{k} \left[ \frac{i - j}{n} \frac{1}{\Delta t} \left( Z_{TP}^{k + 1} - Z_{TP}^{k} \right) + \dot{Z}_{SC} \right] = - \left(G_{i + 1}^{k + 1} - G_{i}^{k + 1} \right)
Rearranging :eq:`7.3-28` according to coefficients of :math:`\alpha_{i + 1}^{k + 1}` and
:math:`G_{i + 1}^{k + 1}` the following results,
.. math::
:label: 7.3-29
\left\{ \frac{1}{\Delta t} \rho_{fg}^{k} \frac{1}{n} Z_{TP}^{k + 1} + \dot{\rho}_{fg}^{k} \frac{1}{n} Z_{TP}^{k + 1} - \rho_{fg}^{k} \left[ \frac{i - j}{n} \frac{1}{\Delta t} \left(Z_{TP}^{k + 1} - Z_{TP}^{k} \right) + \dot{Z}_{SC} \right] \right\} \alpha_{i + 1}^{k + 1} \\
+ G_{i + 1}^{k + 1} + \bigg\{ -\frac{1}{\Delta t} \alpha_{i + 1}^{k} \rho_{fg}^{k} \frac{1}{n} Z_{TP}^{k + 1} + \dot{\rho}_{f}^{k} \frac{1}{n} Z_{TP}^{k + 1} + \alpha_{i}^{k + 1} \rho_{fg}^{k} \bigg[ \frac{i - j}{n} \frac{1}{\Delta t} \\
\times \left(Z_{TP}^{k + 1} - Z_{TP}^{k} \right) + \dot{Z}_{SC} \bigg] - G_{i}^{k + 1} \bigg\} = 0
:eq:`7.3-29` can be written as,
.. math:: a \times \alpha_{i + 1}^{k + 1} + G_{i + 1}^{k + 1} + c = 0
:label: 7.3-30
Writing the general energy :eq:`7.3-15` in terms of :math:`\alpha` results in the
following,
.. math::
:label: 7.3-31
\left[ \dot{\alpha}_{i + 1} \left(h \rho \right)_{fg} + \alpha_{i + 1} \left(h \dot{\rho} \right)_{fg} + \left( h \dot{\rho} \right)_{f} \right] \Delta Z - \left(\alpha_{i + 1} - \alpha_{i} \right) \left(h \rho \right)_{fg} \dot{Z}_{i} \\
= -\Delta \left(G h \right)_{i} + Q_{i} \Delta Z + \dot{P} \Delta Z
The finite difference form of :eq:`7.3-31` is,
.. math::
:label: 7.3-32
\left[ \frac{1}{\Delta t} \left( \alpha_{i + 1}^{k + 1} - \alpha_{i + 1}^{k} \right) \left(h \rho \right)_{fg}^{k} + \alpha_{i + 1}^{k + 1} \left(h \dot{\rho} \right)_{fg}^{k} + \left(h \dot{\rho} \right)_{f}^{k} \right] \frac{1}{n} Z_{TP}^{k + 1} - \left(\alpha_{i + 1}^{k + 1} - \alpha_{i}^{k + 1} \right) \\
\left(h \rho \right)_{fg}^{k} \left[\frac{i - j}{n} \frac{1}{\Delta t} \left(Z_{TP}^{k + 1} - Z_{TP}^{k} \right) + \dot{Z}_{SC} \right] = -\bigg[ G_{i + 1}^{k + 1} \left(h_{f}^{k} + \frac{\alpha_{i + 1}^{k + 1} \rho_{g}^{k} h_{fg}^{k}}{\rho_{f}^{k} + \alpha_{i + 1}^{k + 1} \rho_{fg}^{k}} \right) + \\
G_{i}^{k + 1} \left(h_{f}^{k} + \frac{\alpha_{i}^{k + 1} \rho_{g}^{k} h_{fg}^{k}}{\rho_{f}^{k} + \alpha_{i}^{k + 1} \rho_{fg}^{k}} \right) \bigg] + Q_{i}^{k} \frac{1}{n} Z_{TP}^{k + 1} + \dot{P} \frac{1}{n} Z_{TP}^{k + 1}
Rearranging :eq:`7.3-32` according to coefficients of
:math:`\alpha_{i + 1}^{k + 1}` and :math:`G_{i + 1}^{k + 1}`, the following
results,
.. math::
:label: 7.3-33
\left\{ \frac{1}{\Delta t} \left(h \rho \right)_{fg}^{k} \frac{1}{n} Z_{TP}^{k + 1} + \left( h \dot{\rho} \right)_{fg}^{k} \frac{1}{n} Z_{TP}^{k + 1} - \left(h \rho \right)_{fg}^{k} \left[ \frac{i - j}{n} \frac{1}{\Delta t} \left(Z_{TP}^{k + 1} - Z_{TP}^{k} \right) + \dot{Z}_{SC} \right] \right\} \\
\times \alpha_{i + 1}^{k + 1} + G_{i + 1}^{k + 1} \left(h_{f}^{k} + \frac{\alpha_{i + 1}^{k + 1} \rho_{g}^{k} h_{fg}^{k}}{\rho_{f}^{k} + \alpha_{i + 1}^{k + 1} \rho_{fg}^{k}} \right) + \bigg\{ -\alpha_{i + 1}^{k} \frac{1}{\Delta t} \left(h \rho \right)_{fg}^{k} \frac{1}{n} Z_{TP}^{k + 1} \\
+ \left(h \dot{\rho} \right)_{f}^{k} \frac{1}{n} Z_{TP}^{k + 1} + \alpha_{i}^{k + 1} \left(h \rho \right)_{fg}^{k} \left[ \frac{i - j}{n} \frac{1}{\Delta t} \left(Z_{TP}^{k + 1} - Z_{TP}^{k} \right) + \dot{Z}_{SC} \right] \\
-G_{i}^{k + 1} \left(h_{f}^{k} + \frac{\alpha_{i}^{k + 1} \rho_{f}^{k} h_{fg}^{k}}{\rho_{f}^{k} + \alpha_{i}^{k + 1} \rho_{fg}^{k}} \right) - Q_{i}^{k} \frac{1}{n} Z_{TP}^{k + 1} - \dot{P} \frac{1}{n} Z_{TP}^{k + 1} \bigg\} = 0
If :math:`a'` and :math:`c'` represent the coefficient of :math:`a_{i + 1}^{k + 1}` and the constant
term respectively, :eq:`7.3-33` becomes,
.. math:: a' \alpha_{i + 1}^{k + 1} + G_{i + 1}^{k + 1} \left(h_{f}^{k} + \frac{\alpha_{i + 1}^{k + 1} \rho_{g}^{k} h_{fg}^{k}}{\rho_{f}^{k} + \alpha_{i + 1}^{k + 1} \rho_{fg}^{k}} \right) + c' = 0
:label: 7.3-34
When the mass :eq:`7.3-30` is substituted into :eq:`7.3-34`, a quadratic in
:math:`a_{i + 1}^{k + 1}` results,
.. math::
:label: 7.3-35
\left(\alpha_{i + 1}^{k + 1} \right)^{2} \left[a' \rho_{fg}^{k} - a h_{f}^{k} \rho_{fg}^{k} - a \rho_{g}^{k} h_{fg}^{k} \right] + \alpha_{i + 1}^{k + 1} \left[a' \rho_{f}^{k} - c \rho_{fg}^{k} h_{f}^{k} \right. \\
- c \rho_{g}^{k} h_{fg}^{k} + c' \rho_{fg}^{k} - ah_{f}^{k} \rho_{f}^{k} + \left[ c' \rho_{f}^{k} - c h_{f}^{k} \rho_{f}^{k} \right] = 0
Several points need to be noted here. An inspection of :eq:`7.3-28` and
:eq:`7.3-32` shows that both :math:`\alpha_{i + 1}` and :math:`G_{i + 1}` are
totally implicit in time. It cannot be emphasized enough how much this
feature of the boiling zone numerical solution enhances the stability of
the calculation compared to other numerical schemes which are
semi‑implicit in time which were also tried. In order to preserve a
linearized set of equations, it is necessary that the void fractions and
mass flows be semi‑implicit in time and this has a strong tendency to
produce instabilities. The only quantities which are explicit in time
are the saturation properties which are functions of pressure alone and
the volumetric heat source term. Saturation properties are very well
behaved functions of time since pressure tends to be a relatively stable
function of time in most transients and the saturation properties are
not as sensitive to changes in pressure as other quantities are
sensitive to changes over time.
The last point concerns the :math:`k + 1` superscript on :math:`Z_{TP}`, the length
of the two‑phase zone. What this indicates, as noted above, is that the
current value of the zone length in the iterative process is used in
:eq:`7.3-30` and :eq:`7.3-35`. :eq:`7.3-25`
is solved for :math:`\alpha_{i + 1}^{k + 1}` and then :math:`G_{i + 1}^{k + 1}`
is obtained from the continuity :eq:`7.3-30` for each node. At
first, the value of :math:`Z_{TP}` from the last time step (i.e. :math:`Z_{TP}^{k}`) is
used to solve for :math:`\alpha_{i + 1}^{k + 1}` and :math:`G_{i + 1}^{k + 1}`
over the whole mesh starting at the bottom :math:`\left(i = 1 \right)` where :math:`a_{i}^{k + 1}`
and :math:`a_{i}^{k + 1} \left( = 0 \right)` are the known boundary conditions.
The solution then proceeds upwards until :math:`\alpha_{i + 1}^{k + 1}` is calculated. :math:`\alpha_{i + 1}^{k + 1}`
compared to 1.0 and if it is greater than 1.0, :math:`Z_{TP}` is reduced for the next
iteration and if it is less than 1.0, :math:`Z_{TP}` is increased. The search on :math:`Z_{TP}` continues
until :math:`\alpha_{i + 1}^{k + 1}` is sufficiently close to 1.0. If the boiling zone
extends to the top of the steam generator, the same procedure is used,
but no iteration on :math:`Z_{TP}` is necessary since :math:`Z_{TP}` is a fixed, known
value.
.. _section-7.3.3.1.4:
Superheated Vapor Region
^^^^^^^^^^^^^^^^^^^^^^^^
A compressible treatment of the vapor is used above the boiling zone and
simultaneous nodal mass and energy equations are solved for nodal
enthalpies and mass flows since the region length is known, being the
remainder of the steam generator length after computing new subcooled
and boiling zone lengths. The nodal densities and enthalpies are treated
partially explicitly in time. Boundary conditions are the saturated
vapor enthalpy and the mass flow at the bottom of the zone. The solution
proceeds upwards to the top of the steam generator.
Since there is an expression for :math:`\rho` as a function of enthalpy and
pressure,
.. math:: \dot{\rho} = \frac{\partial \rho}{\partial h} \frac{\partial h}{\partial t} + \frac{\partial \rho}{\partial P} \frac{\partial P}{\partial t}
:label: 7.3-36
By substituting :eq:`7.3-36` into the mass :eq:`7.3-8`, an expression for
:math:`G_{i + 1}` as a function of :math:`h_{i + 1}` results,
.. math:: G_{i + 1} = G_{i} - \left[ \frac{\partial \rho}{\partial h} \dot{h}_{i + 1} + \frac{\partial \rho}{\partial P} \dot{P} \right] \Delta Z + \Delta \rho_{i} \dot{Z}_{i}
:label: 7.3-37
Next, :eq:`7.3-36` is substituted into :eq:`7.3-15`, the following results
.. math::
:label: 7.3-38
\left[ \frac{\partial \rho}{\partial h} \dot{h}_{i + 1} + \frac{\partial \rho}{\partial P} \dot{P} \right] h_{i + 1} \Delta Z + \rho_{i + 1} \dot{h}_{i + 1} \Delta Z - \Delta \left(\rho h \right)_{i} \dot{Z}_{i} \\
= - G_{i + 1} h_{i + 1} + G_{i} h_{i} + Q_{i} \Delta Z + \dot{P} \Delta Z
Now, substituting the expression for :math:`G_{i + 1}` which results
from the continuity :eq:`7.3-37` into the energy :eq:`7.3-38` and
simplifying,
.. math:: \rho_{i + 1} \dot{h}_{i + 1} \Delta Z + \rho_{i} h_{i} \dot{Z}_{i} = - G_{i} h_{i + 1} + \rho_{i} h_{i + 1} \dot{Z}_{i} + G_{i} h_{i} + Q_{i} \Delta Z + \dot{P} \Delta Z
:label: 7.3-39
The finite difference form of :eq:`7.3-39` is,
.. math::
:label: 7.3-40
\frac{1}{\Delta t} \left(h_{i + 1}^{k + 1} - h_{i + 1}^{k} \right) \rho_{i + 1}^{k} \frac{1}{n} Z_{SH}^{k + 1} + h_{i}^{k + 1} \rho_{i}^{k + 1} \frac{i - j}{n} \dot{Z}_{SH}^{k + 1} = -G_{i}^{k + 1} h_{i + 1}^{k + 1} + h_{i + 1}^{k + 1} \rho_{i}^{k + 1} \frac{i - j}{n} \dot{Z}_{SH}^{k + 1} \\
+ G_{i+}^{k + 1} h_{i}^{k + 1} + Q_{i}^{k} \frac{1}{n} Z_{SH}^{k + 1} + \dot{P} \frac{1}{n} Z_{SH}^{k + 1}
:eq:`7.3-40` is entirely implicit in enthalpy, mass flow and zone
length. However, it is necessary to use the beginning‑of‑step value of
:math:`\rho_{i + 1}` since density is a complicated function of enthalpy
and pressure and there is no way to incorporate this function in
:eq:`7.3-20` and preserve linearity. :math:`\rho_{i}`, however, is entirely implicit
since the solution proceeds upwards in the mesh and :math:`\rho_{i}` can be
updated as the solution proceeds along the mesh. By solving for
:math:`h_{i + 1}^{k + 1}`, the following results,
.. math:: h_{i + 1}^{k + 1} = \frac{h_{i}^{k + 1} \left( G_{i}^{k + 1} - \rho_{i}^{k + 1} \frac{i - j}{n} \dot{Z}_{SH}^{k + 1} \right) + \frac{1}{n} Z_{SH}^{k + 1} \left(\frac{1}{\Delta t} h_{i + 1}^{k} \rho_{i + 1}^{k} + Q_{i}^{k} + \dot{P}^{k} \right)}{G_{i}^{k + 1} + \frac{1}{\Delta t} \rho_{i + 1}^{k} \frac{1}{n} Z_{SH}^{k + 1} - \rho_{i}^{k + 1} \frac{i - j}{n} \dot{Z}_{SH}^{k + 1}}
:label: 7.3-41
The finite‑difference form of :eq:`7.3-37` is,
.. math:: G_{i + 1}^{k + 1} = G_{i}^{k + 1} - \left[ \frac{\partial \rho}{\partial h} \frac{\left(h_{i + 1}^{k + 1} - h_{i + 1}^{k} \right)}{\Delta t} + \frac{\partial \rho}{\partial P} \dot{P} \right] \frac{1}{n} Z_{SH}^{k + 1} + \left(\rho_{i + 1}^{k + 1} - \rho_{i}^{k + 1} \right) \dot{Z}_{SH}
:label: 7.3-42
After obtaining :math:`h_{i+1}^{k+1}` from :eq:`7.3-41`,
:math:`\rho_{i+1}^{k+1}` is calculated as a function of :math:`h_{i+1}^{k+1}` and :math:`P^{k+1}`
and these are used to calculate
:math:`G_{i + 1}^{k + 1}` in :eq:`7.3-42`. It should be noted that the partial derivatives
of density with respect to enthalpy and pressure are evaluated with the
:math:`h_{i + 1}^{k + 1}` and :math:`\rho_{i + 1}^{k + 1}` just calculated.
Starting at the bottom of the mesh,
:math:`G_{1}^{k + 1}` and :math:`h_{1}^{k + 1} \left( = h_{g} \right)` are
known and each parameter is solved for at the top of the cell at the :math:`i + 1`
location according to the above procedure up to the top of the steam generator.
.. _section-7.3.3.1.5:
Sodium Side Calculation
^^^^^^^^^^^^^^^^^^^^^^^
Incompressible flow is assumed on the sodium side because it is always
in the liquid phase. Therefore there is no continuity equation. Also,
since the sodium side is at relatively low pressure and has a stable
pressure history, the pressure terms in the energy :eq:`7.3-13` are
neglected. Next, since the sodium flows downward, in order to donor‑cell
the energy equation, :math:`\left( \overline{\rho h} \right)_{i}` is set equal to :math:`\left( \rho h \right)_{i}`. Lastly, it
is convenient to assume that :math:`G` is positive for downward-flowing sodium
which means that :math:`- \Delta \left( G h \right)_{i}` becomes
:math:`-G \left[ h_{i} - h_{i + 1} \right]`. Thus :eq:`7.3-13` becomes,
.. math:: \left(\rho \dot{h} \right)_{i} \Delta Z - \left[\left( \rho h \right)_{i + 1} - \left( \rho h \right)_{i} \right] \dot{Z}_{i + 1} = -G \left(h_{i} - h_{i + 1} \right) + Q_{i} \Delta Z
:label: 7.3-43
Assuming :math:`\dot{\rho} = 0` because of incompressible flow and rewriting
:eq:`7.3-43` in terms of temperature (and neglecting the :math:`\dot{c}_{p}` term as
unimportant), the following results,
.. math:: \overline{\rho}_{i} \overline{c}_{p, i} \dot{T}_{i} \Delta Z - \overline{\rho}_{i} \overline{c}_{p, i} \left(T_{i + 1} - T_{i} \right) \dot{Z}_{i + 1} = -G \overline{c}_{p, i} \left(T_{i} - T_{i + 1} \right) + Q_{i} \Delta Z
:label: 7.3-44
The :math:`\overline{c}_{p, i}` and the :math:`\overline{\rho}_{i}` are computed using the sodium
temperature at the cell‑center. (This is a slight inaccuracy but, given
the fact that liquid sodium properties are so well‑behaved and change so
gradually, it is of small consequence.) The finite difference form of
:eq:`7.3-44` is,
.. math:: \frac{\overline{\rho}_{i}^{k}}{\Delta t} \left(T_{i}^{k + 1} - T_{i}^{k} \right) \Delta Z^{k + 1} - \overline{\rho}_{i}^{k} \left(T_{i + 1}^{k + 1} - T_{i}^{k} \right) \dot{Z}_{i + 1}^{k + 1} = -G^{k + 1} \left(T_{i}^{k + 1} - T_{i + 1}^{k + 1} \right) + \frac{\Delta Z^{k + 1}}{\overline{c}_{p, i}} Q_{i}^{k}
:label: 7.3-45
Note that the :math:`\overline{\rho}_{i}` and the :math:`\overline{c}_{p, i}` are computed with
beginning-of-step temperatures and that the :math:`\Delta Z` and
:math:`\dot{Z}_{i + 1}` are the end-of-step quantities just calculated in the water side
calculation. :math:`Q_{i}` is, of course, explicit in time as usual and the
mass flow provided by the external sodium loop calculation is the new
end-of-time-step value. If :eq:`7.3-45` is now solved for :math:`T_{i}^{k + 1}`,
the following results,
.. math:: T_{i}^{k + 1} = \frac{T_{i + 1}^{k + 1} \left(G^{k + 1} + \overline{\rho}_{i}^{k} \dot{Z}_{i + 1}^{k + 1} \right) + T_{i}^{k} \frac{\Delta Z^{k + 1}}{\Delta t} \overline{\rho}_{i}^{k} + \frac{\Delta Z^{k + 1}}{\overline{c}_{p, i}} Q_{i}^{k}}{ \frac{\Delta Z^{k + 1}}{\Delta t} \overline{\rho}_{i}^{k} + \overline{\rho}_{i}^{k} \dot{Z}_{i + 1}^{k + 1} + G^{k + 1}}
:label: 7.3-46
Starting at the top of the steam generator, with the new inlet sodium
temperatures at the end of the time step, the calculation proceeds
downward to the bottom of the mesh.
.. _section-7.3.3.1.6:
Wall Temperature Calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The heat capacity of the tube wall must be taken into account during the
transient. Since there is no convective term in the energy equation,
central differencing is used. This means that
:math:`\left( \overline{\rho h} \right)_{i} = \frac{1}{2} \left[ \left( \rho h \right)_{i + 1} + \left( \rho h \right)_{i} \right]`
in :eq:`7.3-13` which becomes,
.. math:: \frac{d}{dt} \left\{ \frac{1}{2} \left[\left(\rho h \right)_{i + 1} + \left( \rho h \right)_{i} \right] \Delta Z \right\} - \left(\rho h \right)_{i + 1} \dot{Z}_{i + 1} + \left( \rho h \right)_{i} \dot{Z}_{i} = Q_{i} \Delta Z
:label: 7.3-47
:eq:`7.3-47` is written in terms of temperature and both :math:`\rho`
and :math:`c_{p}` are assumed to be temperature independent. This results in the following,
.. math::
:label: 7.3-48
\frac{d}{dt} \left\{ \frac{1}{2} \left(T_{i + 1} + T_{i} \right) \right\} \Delta Z + \frac{1}{2} \left( T_{i + 1} + T_{i} \right) \left( \dot{Z}_{i + 1} - \dot{Z}_{i} \right) - T_{i + 1} \dot{Z}_{i + 1} \\
+ T_{i} \dot{Z}_{i} = \frac{\Delta Z}{\rho c_{p}} Q_{i}
Since the wall temperature is tracked at the cell center, not the cell
edge, the quantity desired is
:math:`\overline{T_{i}} = \frac{1}{2}(T_{i + 1} + T_{i})`. Thus
:eq:`7.3-48` becomes, after simplification,
.. math:: \dot{\overline{T}} = \frac{1}{\rho c_{p}} Q_{i}^{k} + \frac{1}{2} \frac{1}{\Delta Z^{k + 1}} \left(T_{i + 1}^{k} - T_{i}^{k} \right) \left( \dot{Z}_{i + 1}^{k + 1} + \dot{Z}_{i}^{k + 1} \right)
:label: 7.3-49
The heat source is explicit in time and the :math:`Delta Z` and :math:`\dot{Z}` terms
are the end-of‑step values from the water side calculation. The
difficulty with this equation is calculating the cell‑edge values
:math:`T_{i + 1}` and :math:`T_{i}` since the wall temperatures are tracked at
the cell center. Therefore interpolated or extrapolated estimates of the
cell‑edge values are formed from the cell‑center temperatures.
.. _section-7.3.3.1.7:
Calculation of Boiling Crisis Point
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The point of boiling crisis, or DNB point, in the boiling zone is
computed by tracking the continuously varying intersection of two
functions which is a point within the node structure. The first function
represents the required heat flux for the boiling crisis to occur and
the second is the actual local heat flux at the wall surface.
The DNB heat flux correlation :ref:`[7-3] <section-7.references>` is as follows,
.. math:: F_{D} = 7.84 \times 10^{8} \left[ \chi h_{fg} \rho_{g} / \rho_{f} \sqrt{\frac{G}{1355}} \right]^{-0.667}
:label: 7.3-50
This correlation is evaluated at each cell center in the boiling zone
using the local quality. The inlet mass flux :math:`G` is used instead of the
local mass flux to enhance numerical stability although the original
correlation used the local flow.
An expression for the wall surface heat flux is obtained as follows.
There is a correlation for the heat transfer coefficient at the tube
wall surface but the wall surface temperature is unknown, although the
mid‑wall temperature and the water temperature at saturation are known.
Without going into the details of the correlation here (see :numref:`section-7.appendices.2`), it is known that the heat flux at the
wall surface, :math:`F_{S}`, is equal to :math:`a \left(T_{S} - T_{sat} \right)^{2}`,
where :math:`T_{S}` is the wall surface temperature and :math:`a` is only a function of pressure.
The heat flux between the mid‑wall and the wall surface, :math:`F_{M}`, is
:math:`b \left( T_{M} - T_{S} \right)`, where :math:`b` is the inverse of the
wall heat resistance and :math:`T_{M}` is the mid-wall temperature.
When :math:`F_{S}` is set equal to :math:`F_{M}`, a
quadratic in :math:`\left(T_{S} - T_{sat} \right)` results,
.. math:: a\left( T_{S} - T_{\text{sat}} \right)^{2} + b\left( T_{S} - T_{\text{sat}} \right) - b\left( T_{M} - T_{\text{sat}} \right) = 0
:label: 7.3-51
Thus the wall surface temperature is obtained and then the heat flux at
the wall surface, :math:`F_{S}`, which is computed at each cell center over the
length of the boiling zone. There are thus two functions, :math:`F_{S}` and
:math:`F_{D}`, with values at each cell. In order to obtain the intersection of
these two functions and thus the point of boiling crisis, a linear
approximation is made to each function proceeding two cells at a time
along the length of the steam generator until an intersection of the two
lines is reached. The intersection point is tracked exactly and the
nucleate boiling and film boiling heat transfer coefficients are
prorated in the cell where the intersection occurs. This method gives a
smoothly varying, stable calculation of the DNB point.
.. _section-7.3.3.1.8:
Disappearing and Appearing Regions
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
When the length of a zone is reduced below a certain value, the number
of nodes within the zone is reduced from whatever initial number there
were in the zone to only one node. However, no matter how long the zone
is, no more than the initial number of nodes will be used in the zone.
Reducing the number of nodes for small zone lengths greatly enhances
numerical stability while reducing computer time and, so long as the
criterion for the node reduction is not too large, very little accuracy
in the calculation is sacrificed. When the node structure is collapsed
to one node, the cell‑edge quantities at the inlet and outlet to the
zone remain unchanged while the intermediate values are eliminated. The
tube wall temperature in the center of the new l‑node region is formed
as an average of two wall temperatures nearest the center of the old
multi‑node region. When the zone length increases beyond a certain value
and there is only one node in the region then the number of nodes is
reset at the original value and the values of parameters at intermediate
nodes must be initialized. This is done with simple linear fits (close
enough considering the short lengths involved) between the end point
values for sodium temperatures, for enthalpies in the subcooled and
superheated zones and for mass flows in the boiling and superheated
regions. When the boiling zone has only one node and when it is
reinitialized at the original number of nodes, the whole region is
assumed to be in the nucleate boiling regime.
When a second length threshold is reached as a region's length is
reduced, then the region is eliminated entirely. This only applies to
the boiling and superheated zones and they must disappear in order. That
is if there is a superheated zone, there must be a boiling zone no
matter how small. When the superheated zone disappears, then the boiling
zone outlet enthalpy (i.e. void fraction or quality) becomes a free
variable rather than fixed at :math:`h_{g}` as it is when a superheated zone
exists. The criterion for elimination of the superheated zone is a
length criterion but the criterion for recreating the superheated zone
is that the outlet enthalpy of the boiling zone be somewhat higher than
:math:`h_{g}`. The mass flow at the outlet of the newly created superheated
zone is assumed the same as the outlet flow from the boiling zone. The
outlet enthalpy is set to the value of the criterion and the boiling
zone outlet enthalpy is set to :math:`h_{g}`. The length of the new superheated
zone is computed according to the following formula,
.. math:: Z_{SH} = Z_{TP} \frac{h_{out} - h_{g}}{h_{out} - h_{f}}
:label: 7.3-52
where :math:`Z_{SH}` is the new superheated length, :math:`Z_{TP}` is the old boiling
zone length before being reduced by :math:`Z_{SH}` and :math:`h_{out}` is the enthalpy
calculated at the outlet of the steam generator. Values for the sodium
and tube wall temperatures are interpolated according to the new node
and zone structure.
The criterion for elimination of the boiling zone is again a length
criterion but the criterion for recreating the boiling zone is that the
outlet enthalpy of the subcooled zone be somewhat higher than :math:`h_{f}`.
The mass flow at the outlet of the newly created boiling zone is assumed
the same as the outlet flow from the subcooled zone. The outlet enthalpy
is set to the value of the criterion and the subcooled zone outlet
enthalpy is set to :math:`h_{f}`. The length of the new boiling zone is
calculated as follows,
.. math:: Z_{TP} = Z_{SC} \frac{h_{out} - h_{f}}{h_{out} - h_{in}}
:label: 7.3-53
where :math:`Z_{TP}` is the new boiling length, :math:`Z_{SC}` is the old subcooled
zone length before being reduced by :math:`Z_{TP}`, :math:`h_{out}` is the enthalpy
calculated at the outlet of the steam generator, and :math:`h_{in}` is the
inlet enthalpy. Values for the sodium and tube wall temperatures are
interpolated as for the superheated zone creation above according to the
new node and zone structure.
.. _section-7.3.3.1.9:
Miscellaneous Numerical Issues
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Nothing has been said so far about the calculation of the time
derivative of pressure which appears in the energy :eq:`7.3-15`. At each
call to the steam generator routine, the balance‑of‑plant calculation
provides the new steam generator pressure at the end of the time step.
It would seem natural to form :math:`\dot{P}` with the current :math:`\Delta t` and
:math:`P^{k}` and :math:`P^{k + 1}`. This, however, can lead to significant
numerical instabilities caused by large temporary spikes in :math:`P` even
when the time‑averaged value of :math:`\dot{P}` is well-behaved and much more
gradual than would be predicted by using the stepwise values of :math:`P`.
Therefore, a moving average of :math:`\dot{P}` is computed over the last 2 :math:`m`
timesteps so that,
.. math:: \dot{P} = \frac{ \sum_{i = m + 1}^{2 m} P_{i} \Delta t_{i} - \sum_{i = 1}^{m} P_{i} \Delta t_{i}}{ \frac{1}{2} \sum_{i = 1}^{2 m} \Delta t_{i}}
:label: 7.3-54
There is a time step selector currently in the code which chooses a new
timestep size for the next step based on information from the current
step. However, this selector is preliminary and merely chooses the new
step according to fractional changes in a number of parameters over the
step. That is, the new time step is computed as follows,
.. math:: \Delta t' = \text{Min} \left(X \left| \frac{\Delta t Y_{i}^{k + 1}}{Y_{i}^{k + 1} - Y_{i}^{k}} \right| \right)
:label: 7.3-55
where :math:`X` is an input fractional change in the quantity :math:`Y_{i}, \Delta t`
is the current time step, :math:`\Delta t'` is the new time step and Min indicates
that the minimum over all the :math:`Y_{i}` values is computed. The various
quantities indicated by :math:`Y` include the sodium side flow, the zone
boundaries, the steam generator pressure, the nodal water mass flows,
the void fractions in the boiling zone, the enthalpies in the subcooled
and superheated zones, the sodium temperatures and the tube wall
temperatures. There is also a minimum value criterion for the new step
set by the steam generator. There is in effect a maximum value for the
time step which is set for the primary loop calculation. However, there
is rarely any need for the steam generator to have a larger time step
than the primary loop calculation and, almost always, it is the
hydrodynamics on the water side of the steam generator which determines
the time step.
There are two artificial limitations that are superimposed on the
boiling zone calculation that need to be pointed out. The first is that
the void fraction solved for in each successive node as the solution
proceeds up the zone must be larger than the previous value at the last
node. The code simply requires that the new void fraction be at least
0.001 larger than the last. This may seem like a major limitation in the
solution method but, in practice, it is rarely used. When it is used, it
has very little consequence for the calculation. The only time this fix
is used is when there is an extremely flat void fraction profile (which
can be caused by a number of things, very low water flow, for example).
In the absence of this fix, there is occasionally a tendency for
numerical instabilities to form when there is an extremely flat void
profile. The other artificial limitation is that the boiling zone length
is not allowed to change more than 1% in a time step. This can have some
significant implications for the course of a transient. Without this
limitation, there can be a tendency for the boiling zone length to
adjust too rapidly to changing conditions, which can cause significant
numerical instabilities since a large change in :math:`Z_{TP}` over a short
time produces a large :math:`\dot{Z}_{TP}` and a large perturbation in the equation
set. Since the effect is largely artificial, this limitation within
certain bounds is consistent with the physical conditions of the case.
However, limiting the change in :math:`Z_{TP}` means that the void fraction at
the top of the zone may not be equal to 1.0 and, if it is too different,
then basic assumptions of the model are, of course, being violated. It
must be emphasized, however, that this limitation, is not used
frequently and, when it is used, the outlet void fraction nearly always
remains close to 1.0 while, over a number of time steps, the region
length changes to accommodate the changing conditions but avoiding large
temporary values of :math:`\dot{Z}_{TP}` which could drive the calculation
unstable.
.. _section-7.3.3.1.10:
Calculation of Pressure Drop for Momentum Equation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Although the balance‑of‑plant calculation does the actual computation
which produces the inlet flow for the steam generator and the pressure
boundary conditions at the inlet and outlet plena of the steam
generator, the steam generator must provide the pressure drops to the
balance-of-plant for this calculation. For the details of the momentum
equation solution, it is necessary to refer to the balance‑of‑plant
description (:numref:`section-7.2`). All that will be done here is to describe the
pressure drop calculation itself. For each of the four zones
corresponding to each of the four heat transfer regimes :math:`i`, the
following is the pressure drop :math:`\Delta P` across the zone,
.. math:: \Delta P_{i} = - \overline{G}_{i}^{2} \left(\frac{1}{\rho_{t}} - \frac{1}{\rho_{b}} \right) - Z_{i} \bigg[ \overline{\rho}_{i} 9.8 + \dot{\overline{G}}_{i} + \overline{G}_{i}^{2} 0.31 FR_{i} \left( \frac{\overline{G}_{i} D_{H}}{\overline{\mu}_{i}} \right)^{-0.25} \frac{1}{2} \frac{1}{D_{H} \overline{\rho}_{i}} R_{i} \bigg]
:label: 7.3-56
where :math:`\overline{G}_{i}` is an average of the mass flow both spatially over the
zone length and temporally over the time step; :math:`\rho_{t}` is the
density at the top of the zone and :math:`\rho_{b}` at the bottom; :math:`Z_{i}` is the
zone length; :math:`\overline{\rho}_{i}` the average density over the zone; :math:`\dot{\overline{G}}_{i}` is
the average mass flow over the zone length at the end of the time step
minus the average mass flow over the zone at the beginning of the time
step divided by the time step; :math:`FR_{i}` is a calibration factor computed
in steady state to provide the proper steady state pressure drop; :math:`D_{H}`
is the water side hydraulic diameter; and :math:`\overline{\mu}_{i}` is the average
viscosity for the regime. :eq:`7.3-56` is used to compute
:math:`\Delta P_{i}` for the subcooled, the nucleate boiling, the film boiling and
superheated zones. The factor :math:`R_{i}` is 1.0 for the subcooled and the
superheated zones. :math:`R_{i}` is computed according to the following formula
in the two boiling zones,
.. math:: R_{i} = 0.95819 - \left( 1167.2 R_{i}' - 505. \right) R_{i}'
:label: 7.3-57
.. math:: R_{i}' = \frac{\overline{\chi}_{i}}{\left( \frac{P}{1.724 \times 10^{6}} \right)^{2.448} + 16.217}
:label: 7.3-58
where :math:`P` is the steam generator pressure and :math:`\overline{\chi}_{i}` is the average
quality over each boiling zone. This formula is the Thom correlation
:ref:`[7-2] <section-7.references>` with constants appropriate for the units used in the code.
Once the :math:`\Delta P_{i}`\ s have been calculated, then the :math:`\Delta P_{i}`\ s for
the boiling zones and the superheated vapor zone are summed and divided
by the sum of all four :math:`\Delta P_{i}`\ s. This gives the current fraction of
the total pressure drop across the steam generator that is above the
subcooled zone. The balance‑of‑plant model uses its current pressures at
the inlet and the outlet plena of the steam generator to provide the
total pressure drop and estimates the pressure at the top of the
subcooled zone with the fractional pressure drop referred to above. The
pressure at the top of the subcooled zone provides the balance‑of‑plant
momentum equation with a pressure boundary condition and it can merely
include the subcooled liquid zone of the steam generator as the last in
a series of incompressible liquid segments bounded by plena which
stretches from the feed water inlet to the lower end of the compressible
zones in the steam generator (i.e. the subcooled/boiling boundary). Thus
the inlet flow (constant throughout the subcooled zone) is determined as
part of the solution matrix in the total balance‑of‑plant momentum
equation. Of course, the steam generator must provide the
balance‑of‑plant model with information about the subcooled zone. It
needs the average density, the length of the zone, the friction
normalization factor :math:`FR_{1}`, the average viscosity and the
hydraulic diameter. It is clear that, in the case when the subcooled
zone extends to the top of the steam generator, the whole steam
generator becomes merely one incompressible segment in the
balance‑of‑plant matrix from the feedwater inlet to the turbines.
.. _section-7.3.3.2:
Recirculation‑Type Steam Generator
''''''''''''''''''''''''''''''''''
As noted before, the modeling of the evaporator in this node of the
steam generator model is done with the same coding as is used for the
once-through type steam generator. The difference is that, when it is
used as an evaporator, the outlet enthalpy will be less than or equal to
:math:`h_{g}`. Since this is within the envelope of cases for which the
once‑through modeling was designed, this has already been described
above. The modeling of the steam drum is described elsewhere in detail
(See :numref:`section-7.4`).
There remains only the discussion of the modeling of the superheater. As
noted above, it is assumed that incompressible flow is adequate to
describe the superheater. The mass flow through the superheater is
determined by the balance‑of‑plant momentum equation. The lower enthalpy
boundary condition is :math:`h_{g}`. There are no region boundaries to
calculate since there is only single phase vapor flow in the
superheater. Since :math:`\dot{\rho} = 0` is assumed, the energy :eq:`7.3-15`
becomes, for the constant superheater length,
.. math:: \rho_{i + 1} \dot{h}_{i + 1} \Delta Z = - G\left(h_{i + 1} - h_{i} \right) + Q_{i} \Delta Z + \dot{P} \Delta Z
:label: 7.3-59
The finite difference form of :eq:`7.3-59` is,
.. math:: \rho_{i + 1}^{k} \Delta Z \frac{h_{i + 1}^{k + 1} - h_{i + 1}^{k}}{\Delta t} = - G^{k + 1} \left( h_{i + 1}^{k + 1} - h_{i}^{k + 1} \right) + Q_{i}^{k} \Delta Z + \dot{P} \Delta Z
:label: 7.3-60
:eq:`7.3-60` is then solved for :math:`h_{i + 1}^{k + 1}` resulting in,
.. math:: h_{i + 1}^{k + 1} = \frac{\rho_{i + 1}^{k} \frac{\Delta Z}{\Delta t} h_{i + 1}^{k} + G^{k + 1}h_{i}^{k + 1} + Q_{i}^{k} \Delta Z + \dot{P} \Delta Z}{\rho_{i + 1}^{k} \frac{\Delta Z}{\Delta t} + G^{k + 1}}
:label: 7.3-61
:eq:`7.3-61` is solved for each successive :math:`h_{i + 1}^{k + 1}` up to the top
of the superheater. :math:`\Delta Z` is, of course, a constant. The lower boundary
conditions are :math:`h_{g}` and :math:`\rho_{g}`. :math:`G^{k + 1}` is provided by the
balance‑of‑plant momentum equation.
The sodium side temperatures are solved according to the same method as
used in the once‑through steam generator. All the same assumptions
concerning incompressible flow and donor‑cell differencing are made. The
only difference is that the :math:`\dot{Z}` terms are eliminated and :math:`\Delta Z`
becomes a constant because of the constant superheater length. Thus
:eq:`7.3-46` becomes,
.. math:: T_{i}^{k + 1} = \frac{T_{i + 1}^{k + 1} G^{k + 1} + T_{i}^{k} \frac{\Delta Z}{\Delta t} \overline{\rho}_{i}^{k} + \frac{\Delta Z}{\overline{c}_{p,i}} Q_{i}^{k}}{\frac{\Delta Z}{\Delta t} \overline{\rho}_{i}^{k} + G^{k + 1}}
:label: 7.3-62
As before, starting at the top of the steam generator, with the new
inlet sodium temperature at the end of the time step, the calculation
proceeds downward to the bottom of the mesh.
The same considerations apply to the tube wall temperature calculation
in the superheater. The solution method is precisely the same as for the
once-through steam generator except that the :math:`\dot{Z}` terms are now
eliminated because of the constant superheater length. Thus :eq:`7.3-49`
becomes simply,
.. math:: \dot{\overline{T}}_{i} = \frac{1}{\rho c_{p}} Q_{i}^{k}
:label: 7.3-63
Thus there is no difficulty in calculating the :math:`\Delta T` term as there was
in :eq:`7.3-49` because it is eliminated.
.. _section-7.3.3.3:
Steady State Solution
'''''''''''''''''''''
.. _section-7.3.3.3.1:
Once‑Through Type Steam Generator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. _section-7.3.3.3.1.1:
Superheated Vapor Region
++++++++++++++++++++++++
For a given reactor power, the product of flow rate and enthalpy change
across the steam generator must be the same on both the water and sodium
sides. In other words, the following holds,
.. math:: RP = G_{w} A_{w} \left(h_{w, out} - h_{w, in} \right) = G_{s} A_{w} \overline{c}_{p,s} \left(T_{s, in} - T_{s, out} \right)
:label: 7.3-64
where :math:`RP` is the reactor power (or a fraction thereof for multiple
steam generators); :math:`G_{w}` and :math:`G_{s}` are the water and sodium flow
rates; :math:`A_{w}` and :math:`A_{s}` are the water and sodium flow areas; :math:`h_{w, out}`
and :math:`h_{w, in}` are the outlet steam and inlet water enthalpies; :math:`T_{s, in}`
and :math:`T_{s, out}` are the inlet and outlet sodium temperatures; and
:math:`\overline{c}_{p, s}` is the average specific heat for sodium over the length of
the steam generator. This relationship must determine the above flow
rates, water enthalpies, geometry and sodium temperatures. Constraints
on any of these parameters must translate into constraints on the other
parameters according to the above relationship. For the present purpose,
however, geometry, flow rates and inlet and outlet enthalpies are
assumed to have been determined elsewhere.
For the steady state, then, :math:`h_{w, out}` is presumed. If :math:`h_{w, out} > h_{g}`,
then there is a superheated vapor zone. (Of course, in any true
steady‑state operation for a once‑through steam generator, there will be
a superheated vapor zone. However, this "steady‑state" calculation
described here produces starting conditions for a transient calculation
of any nature which may be very different than the normal operational
conditions.) For the superheated vapor zone, then, the following
relation holds,
.. math:: G_{w}A_{w} \left(h_{w, out} - h_{g} \right) = G_{s} A_{s} \overline{c}_{p, s} \left( T_{s, in} - T_{s, hg} \right)
:label: 7.3-65
where :math:`T_{s,hg}` is the sodium temperature at the point of saturated
vapor enthalpy on the water side and :math:`\overline{c}_{p, s}` is the average value of
the specific heat over the superheated vapor zone. If :math:`\overline{c}_{p, s}` is
known, then :eq:`7.3-65` can be used to determine :math:`T_{s,hg}`. As a
practical matter, :math:`c_{p,s}` varies only 2-3% and quite smoothly over the
length of the steam generator. If :eq:`7.3-65` is solved for :math:`T_{s,hg}`
using :math:`c_{p,s}` calculated with :math:`T_{s, in}`, then the average of :math:`T_{s, in}`
and :math:`T_{s, hg}` are used to calculate :math:`c_{p, s}` and if this process is
repeated several times, then there is a negligible error in
:math:`\overline{c}_{p, s}` and thus in :math:`T_{s, hg}`.
In order to fully characterize the superheated zone, either the length
of the zone must be specified or some calibrating factor on the water
side heat transfer coefficient must be specified. This will become clear
as the solution description continues. It is also possible to calibrate
the sodium side heat transfer coefficient but it is assumed that the
water side coefficient involves much more uncertainty. First, the form
of the water side heat transfer coefficient is as follows,
.. math:: H_{T} = \frac{1}{\frac{1}{H_{w} CF} + WR + FL}
:label: 7.3-66
where :math:`H_{T}` is the total water coefficient; :math:`H_{w}` is the heat transfer
coefficient between the tube wall surface and the bulk fluid; :math:`CF` is
some calibration factor to be determined; :math:`WR` is the heat resistance of
the tube wall; and :math:`FL` is some additional resistance to take into
account fouling at the wall surface. :math:`H_{w}` is calculated for each heat
transfer regime according to correlations given in the :numref:`section-7.references`. The
:math:`H_{w}`\ s represent values for experimental conditions and therefore
may need to be calibrated for full scale cases. :math:`WR` is calculated from
the tube wall geometry and properties. :math:`FL` is specified by the code
user in the input.
Before a full nodewise solution is obtained, a rough guess to initialize
the iterative process of the final solution is required. The case of
specifying the zone length :math:`Z_{SH}` and computing the calibration factor
is considered first. The following equates the average heat transfer in
the zone from the bulk sodium to the tube wall with the portion of
reactor power known to be derived from the superheated zone,
.. math:: G_{s} \frac{1}{Z_{SH}} \overline{c}_{p, s} \left(T_{s, in} - T_{s, hg} \right) = H_{s} \left( \overline{T}_{s} - \overline{T}_{m} \right) \frac{2 \pi r_{s} Z_{SH}}{A_{s} Z_{SH}}
:label: 7.3-67
:math:`T_{s, hg}` is obtained from :eq:`7.3-64`.
:math:`\overline{T}_{s} = \frac{1}{2} \left(T_{s, in} + T_{s, hg} \right)`. :math:`r_{s}` is the outer
tube wall radius. :math:`\overline{T}_{m}` averaged over the height of the zone.
:math:`H_{s}` is the total heat transfer on the sodium side including the wall
resistance,
.. math:: H_{s} = \frac{1}{\frac{1}{H_{Na}} + WRNA}
:label: 7.3-68
where :math:`H_{Na}` is the heat transfer coefficient from the tube surface to
the bulk sodium and WRNA is the wall resistance on the sodium side of
the tube. The correlation for :math:`H_{Na}` is given in :numref:`section-7.appendices.2`.
:math:`H_{Na}` is a function of :math:`G_{s}`, geometry and temperature. The temperature used
is :math:`\overline{T}_{s}`. :eq:`7.3-67` is solved for :math:`\overline{T}_{m}` which is inserted
into the equation analogous to :eq:`7.3-67` on the water side,
.. math:: G_{s} \frac{1}{Z_{SH}} \left(h_{w, out} - h_{g} \right) = \frac{1}{\frac{1}{H_{w}CF} + WR + FL} \left(\overline{T}_{m} - \overline{T}_{w} \right) \frac{2 \pi r_{w} Z_{SH}}{A_{w} Z_{SH}}
:label: 7.3-69
where :math:`\overline{T}_{W}` is the temperature derived from
:math:`\overline{H}_{w} = \frac{1}{2} \left(h_{w, out} + h_{g} \right)`, :math:`r_{w}` is
the inner tube wall radius and :math:`H_{w}` is calculated with properties
derived from :math:`\overline{h}_{w}`. Thus :eq:`7.3-69` can be solved for :math:`CF`, the
initial guess for the calibration factor on the superheated zone heat
transfer coefficient when the zone length is specified. Alternatively,
if the calibration factor :math:`CF` is specified, and the zone length is
unknown, then :eq:`7.3-67` and :eq:`7.3-69` form a set with two unknowns,
:math:`\overline{T}_{m}` and :math:`Z_{SH}`, which can be solved for easily. If there is only
one node in the superheated zone, then the steady state solution is
finished at this point.
In order to solve the nodal equations, a similar method is used for each
node as in the 1-node approximation above. First, there is the nodal
energy balance for cell :math:`i` from node :math:`i` to node :math:`i + 1` (the solution
proceeds from the bottom to the top of the mesh),
.. math:: G_{w}A_{w}\left( h_{w,i + 1} - h_{w,i} \right) = G_{s}A_{s}\ {\overline{c}}_{p,s,i}(T_{s,i + 1} - T_{s,i})
:label: 7.3-70
where :math:`\overline{c}_{p, s, i}` is the specific heat corresponding to :math:`\frac{1}{2} \left(T_{s, i + 1} + T_{s, i} \right) = \overline{T}_{s, i}` This
equation can be solved for :math:`T_{s, i + 1}` which results in
.. math:: T_{s, i + 1} = \left[ \frac{G_{w} A_{w}}{G_{s} A_{s} \overline{c}_{p, s, i}} \right] h_{w, i + 1} + \left[T_{s, i} - \frac{G_{w} A_{w}}{G_{s} A_{s} \overline{c}_{p, s, i}} h_{w, i} \right]
:label: 7.3-71
which can be written as,
.. math:: T_{s, i + 1} = a h_{w, i + 1} + b
:label: 7.3-72
Next, the sodium side heat transfer is described by the following,
.. math:: G_{s} \frac{1}{\Delta z} \overline{c}_{p, s, i} \left(T_{s, i + 1} - T_{s, i} \right) = H_{s, i} \left[\frac{1}{2} \left(T_{s, i + 1} + T_{s, i} \right) - T_{m, i} \right] \frac{2 \pi r_{s} \Delta Z}{A_{s} \Delta Z}
:label: 7.3-73
where :math:`H_{s, i}` is calculated according to :eq:`7.3-68` with
:math:`\overline{T}_{s, i}` and :math:`T_{m, i}` is the cell-center value and
:math:`\Delta Z` is of course :math:`\frac{1}{n} Z_{SH}` with :math:`n` the number
of cells in the zone. :eq:`7.3-73` is solved, for
:math:`T_{m, i}` in the following,
.. math:: T_{m, i} = \left[ \frac{1}{2} - \frac{G_{s} \overline{c}_{p, s, i} A_{s}}{H_{s, i} 2 \pi r_{s} \Delta Z} \right] T_{s, i + 1} + \left[ \frac{1}{2} + \frac{G_{s} \overline{c}_{p, s, i} A_{s}}{H_{s, i} 2 \pi r_{s} \Delta Z} \right] T_{s, i}
:label: 7.3-74
which can be written as,
.. math:: T_{m, i} = c T_{s, i + 1} + d
:label: 7.3-75
and if :eq:`7.3-72` is substituted into :eq:`7.3-75`,
.. math:: T_{m, i} = a c h_{w, i + 1} + \left(bc + d \right) = eh_{w, i + 1} + f
:label: 7.3-76
Similarly, the water side heat transfer is described by the following,
.. math:: G_{w} = \frac{1}{\Delta Z} \left(h_{w, i + 1} - h_{w, i} \right) = H_{T, i} \left(T_{m, i} - \overline{T}_{w, i} \right) \frac{2 \pi r_{w} \Delta Z}{A_{w} \Delta Z}
:label: 7.3-77
where :math:`H_{T, i}` is calculated according to :eq:`7.3-66`. Next
:math:`\overline{T}_{w, i}` is replaced by :math:`\frac{1}{2} \left[T_{w, i+1} + T_{w, i} \right]`
Then, use is made of the relation,
.. math:: h_{w, i + 1} - h_{w, i} = \overline{c}_{p, w, i} \left(T_{w, i + 1} - T_{w, i} \right)
:label: 7.3-78
:eq:`7.3-78` is solved for :math:`T_{w, i + 1}` and then,
.. math:: \overline{T}_{w, i} = T_{w, i} + \frac{1}{2 \overline{c}_{p, w, i}} \left(h_{w, i + 1} - h_{w, i} \right)
:label: 7.3-79
:eq:`7.3-79` and :eq:`7.3-76` are substituted into :eq:`7.3-77`, which
results in the following,
.. math:: h_{w, i + 1} = \frac{G_{w} h_{w, i} - H_{T, i} \frac{2 \pi r_{w} \Delta Z}{A_{w}} \left[f - T_{w, i} + \frac{h_{w, i}}{2 \overline{c}_{p, w, i}} \right]}{G_{w} + \frac{H_{T, i} 2 \pi r_{w} \Delta Z}{A_{w}} \left[\frac{1}{2 \overline{c}_{p, w, i}} - e \right]}
:label: 7.3-80
When :math:`h_{w, i + 1}` is computed, :math:`T_{m, i}` and :math:`T_{s, i + 1}`
can also be found from :eq:`7.3-76` and :eq:`7.3-72`.
Now that the basic method of solution of the nodal equation has been
described, several points need to be emphasized. First, in the three
equation set, :eq:`7.3-70`, :eq:`7.3-73`, and :eq:`7.3-77`,
there are the three unknowns :math:`T_{s, i + 1}`, :math:`T_{m, i}` and :math:`h_{w, i + 1}` which are solved
for. However, :eq:`7.3-70` and :eq:`7.3-73` presume that
:math:`\overline{c}_{p, s, i}` is known.
:eq:`7.3-73` presumes that :math:`H_{s, i}` is known and this means that an
average sodium temperature for the cell must be known. :eq:`7.3-77`
presumes that :math:`H_{T, i}` and :math:`\overline{c}_{p, w, i}` are known and various physical
properties are required to compute :math:`H_{T, i}`. All of these quantities are
functions of :math:`\overline{T}_{s, i}` and :math:`\overline{T}_{w, i}` (or :math:`\overline{h}_{w, i}`)
which require :math:`T_{s, i + 1}` and :math:`T_{w, i + 1}` (or :math:`h_{w, i + 1}`).
Therefore, there is an iterative process required to produce better and
better values for the various parameters after starting the whole
process with some initial estimate of the :math:`T_{s, i}`\ s and the
:math:`h_{w, i}`\ s. On the very first pass, the nodal values of :math:`T_{s, i}` and
:math:`h_{w, i}` are simply linear interpolations between the end values but,
since there are a number of iterations before the final result is
obtained, the effect of this initial assumption is negligible.
Secondly, the main goal of the iterative procedure is to produce either
a calibration factor for a specified length or a zone length for a
specified calibration factor. The length or calibration factor is simply
assumed in the above solution of the nodal equations and when the
solution of all the nodal values is complete, then a new length or
calibration factor is chosen if the result is not yet acceptable. The
nodal solution begins at the point of saturated vapor enthalpy where the
:math:`h_{w, i}` is simply :math:`h_{g}` and the :math:`T_{s, i}` is :math:`T_{s, hg}`, produced
from :eq:`7.3-65`. It should be mentioned here that :eq:`7.3-65`
produces a better and better value of :math:`T_{s, hg}` on each iteration since the value
of :math:`\overline{c}_{p, s}` used is refined by a recomputation after each iteration
by using the new nodal :math:`T_{s, i}`\ s calculated in the iteration.
The nodal solution proceeds upwards from :math:`h_{g}` and :math:`T_{s, hg}`
to the top of the steam generator where values of :math:`h_{w, out}` the
:math:`T_{s, in}` are calculated. At this point, the criterion for convergence
needs to be determined. The new values of :math:`h_{w, out}` and :math:`T_{s, in}` can
both be compared to the externally calculated values. It was decided to
use :math:`T_{s, in}` as the criterion and thus the length or calibration factor
is adjusted until the :math:`T_{s, i + 1}` at the top of the steam
generator is as close to :math:`T_{s, in}` as necessary according to an input
criterion. There is only one last point to note. This is that for each
outer iteration which uses a new length or calibration factor, there are
ten inner iterations for each node in the nodal solution which are
necessary to converge on values of the heat transfer coefficients and
:math:`c_{p}`\ s for an assumed length or calibration factor. If this inner
iteration is not done, the outer iteration will not converge.
.. _section-7.3.3.3.1.2:
Subcooled Liquid Region
+++++++++++++++++++++++
Because of the difficulty of obtaining a solution for the boiling zone
unless the length of the boiling zone is known, the subcooled zone
solution, which may not have its length specified, is completed before
the boiling zone. Once the lengths for both the superheated vapor zone
and the subcooled liquid zone are determined, the length of the boiling
zone is, of course, merely the remainder of the steam generator length.
There is a boiling zone, however, only when :math:`h_{w, out} > h_{f}`.
.. math:: G_{w}A_{w}\left( h_{w,t} - h_{f} \right) = G_{s}A_{s}{\overline{c}}_{p,s}(T_{s,t} - T_{s,hf})
:label: 7.3-81
where :math:`h_{w, t}` is :math:`h_{g}` when there is a superheated vapor zone and
:math:`h_{w, out}` when there is not; :math:`T_{s, t}` is :math:`T_{s, hg}` when there is a
superheated vapor zone and :math:`T_{s, in}` when there is not; :math:`\overline{c}_{p, s}` is the
average value of the specific heat over the boiling zone. As is similar
to :eq:`7.3-65`, :math:`\overline{c}_{p, s}` is first calculated with
:math:`T_{s, t}` and then with an average of :math:`T_{s, t}` and :math:`T_{s, hf}`
and this process is repeated several times thus producing better and better values of
:math:`T_{s, hf}`.
As was the case with the superheated zone, either the length of the
subcooled zone or the calibration factor on its heat transfer
coefficient must be specified. The total water side heat transfer
coefficient is calculated as in :eq:`7.3-66` with a different correlation
for :math:`H_{w}` and perhaps a different value for :math:`FL`.
Again, as with the superheated zone, a one-node initial guess for the
subcooled zone length :math:`Z_{SC}` or calibration factor is obtained starting
with an equation analogous to :eq:`7.3-67`,
.. math:: G_{s} \frac{1}{Z_{SC}} \overline{c}_{p, s} \left(T_{s, u} - T_{s, out} \right) = H_{s} \left( \overline{T}_{s} - \overline{T}_{m} \right) \frac{2 \pi r_{s} Z_{SC}}{A_{s} Z_{SC}}
:label: 7.3-82
where :math:`T_{s, u}` is either :math:`T_{s, in}` when there is no boiling zone or
:math:`T_{s, hf}` which is obtained from :eq:`7.3-81`, :math:`\overline{T}_s = \frac{1}{2}\left(T_{s,u} - T_{s,out} \right)`, :math:`\overline{T}_{m}`
is the midwall tube temperature averaged over the height of the zone and :math:`H_{s}` is
defined in :eq:`7.3-68`. Again :math:`\overline{T}_{m}` is obtained
from :eq:`7.3-82` and inserted into the analogous equation to :eq:`7.3-69`,
.. math:: G_{s} \frac{1}{Z_{SC}} \left(h_{w, u} - h_{w, in} \right) = \frac{1}{\frac{1}{H_{w} CF} + WR + FL } \left( \overline{T}_{m} - \overline{T}_{w} \right) \frac{2 \pi r_{w} Z_{SC}}{A_{w} Z_{SC}}
:label: 7.3-83
where :math:`h_{w, u}` is :math:`h_{w, out}` when there is no boiling zone and
:math:`h_{f}` when there is a boiling zone and :math:`\overline{T}_{w}` is the
temperature derived from :math:`\frac{1}{2} \left(h_{w, u} + h_{w, in} \right)`.
:eq:`7.3-83` is solved for the
initial guess for :math:`CF` when the zone length is specified. Alternatively,
if the calibration factor :math:`CF` is specified, then the :eq:`7.3-82` and
:eq:`7.3-83` form a set with two unknowns, :math:`\overline{T}_{m}` and :math:`Z_{SC}`
which are then solved for. If there is only one node in the subcooled zone, the
steady state solution is finished here.
In order to solve the nodal equations, the same method is used as was
used for the superheated vapor zone except the nodewise solution
proceeds from the top of the subcooled zone to the bottom of the steam
generator. The nodal energy balance for a cell is exactly the same as
:eq:`7.3-70`. However, since the direction of solution is from top to
bottom, :math:`T_{s, i}` is solved for,
.. math:: T_{s, i} = \left[ \frac{G_{w} A_{w}}{G_{s} A_{s} \overline{c}_{p, s, i}} \right] h_{w, i} + \left[ T_{s, i + 1} - \frac{G_{w} A_{w}}{G_{s} A_{s} \overline{c}_{p, s, i}} h_{w, i + 1} \right]
:label: 7.3-84
which is written as,
.. math:: T_{s, i} = a h_{w, i} + b
:label: 7.3-85
As before, the sodium side heat transfer is described by :eq:`7.3-73`. As
before, :eq:`7.3-73` is solved for :math:`T_{m, i}` which results in :eq:`7.3-34`.
However, since :math:`T_{s, i}` is the unknown now and not :math:`T_{s, i + 1}`,
:eq:`7.3-74` is written as,
.. math:: T_{m, i} = c t_{s, i} + d
:label: 7.3-86
and :eq:`7.3-85` is substituted into :eq:`7.3-86`,
.. math:: T_{m, i} = ac h_{w, i} + \left( bc + d \right) = e h_{w, i} + f
:label: 7.3-87
The water side heat transfer is described by :eq:`7.3-77` just as for the
superheated vapor zone, but :eq:`7.3-78` is solved for :math:`T_{w, i}` and,
.. math:: \overline{T}_{w, i} = T_{w, i + 1} - \frac{1}{2 \overline{c}_{p, w, i}} \left(h_{w, i + 1} - h_{w, i} \right)
:label: 7.3-88
:eq:`7.3-88` and :eq:`7.3-87` are substituted
into :eq:`7.3-77` which results in the following,
.. math:: h_{w, i} = \frac{G_{w} h_{w, i + 1} - H_{T, i} \frac{2 \pi r_{w} \Delta Z}{A_{w}} \left[f - T_{w, i + 1} + \frac{h_{w, i + 1}}{2 \overline{c}_{p, w, i}} \right]}{G_{w} + \frac{H_{T, i} 2 \pi r_{w} \Delta Z}{A_{w}} \left[e - \frac{1}{2 \overline{c}_{p, w, i}} \right]}
:label: 7.3-89
As before, when :math:`h_{w, i}` is computed, :math:`T_{m, i}` and :math:`T_{s, i}` can also be
found from :eq:`7.3-87` and :eq:`7.3-85`. The same general considerations apply
to the solution of the subcooled zone equation set as applied to the
superheated vapor zone equation set. The criterion for convergence is
now that :math:`T_{s, i}` at the bottom of the mesh be arbitrarily close to
:math:`T_{s, out}`. The subcooled zone length or the calibration factor :math:`CF` is
adjusted until this is achieved.
.. _section-7.3.3.3.1.3:
Boiling Zone
++++++++++++
If :math:`h_{w, out} > h_{f}`, then there is a boiling zone. It has already
been shown how the sodium temperatures were determined at each end of
the zone, :math:`T_{s, in}` or :math:`T_{s, hg}` (depending on whether or not there is a
superheated zone) and :math:`T_{s, hf}`. The water enthalpies at the ends of the
zone are either :math:`h_{w, out}` or :math:`h_{g}` and :math:`h_{f}`. Since the lengths of
the superheated vapor and subcooled liquid zones have been determined
either by input specification or by the steady state calculation before
the boiling zone calculation begins, the length of the boiling zone is
also determined and it remains, therefore, to adjust the heat transfer
in the boiling zone so that with the known zone length the enthalpy
condition at the top of the zone is obtained. That is, either :math:`T_{s, in}`
or :math:`T_{s, hg}` is obtained on the sodium side and :math:`h_{w, out}` or :math:`h_{g}` on
the water side, although the actual criterion used, as in the other
zones, is the sodium temperature. There is a complicating factor when
trying to adjust the heat transfer in the boiling zone which does not
exist in the other two zones, however. This is the fact that there are
two heat transfer regimes in the boiling zone separated at the DNB
point. Both the calibrating factors on both the heat transfer
coefficients cannot be adjusted independently or the solution won't
converge. Therefore, the calibrating factor is fixed arbitrarily at a
constant value in the nucleate boiling zone while the calibrating factor
in the film boiling zone is adjusted to obtain the proper outlet
conditions. A related problem is to determine where the DNB point is in
the zone. This is done by using the method outlined in the transient
section as the solution proceeds up the mesh and when the intersection
of the two functions is determined by extrapolation of the two
functions, the DNB point is predicted there.
The method of solution, then, is to proceed upwards in the mesh
node‑by-node in the nucleate boiling zone using the known zone length
and the assumed heat transfer coefficient calibration factor and
extrapolating ahead to determine the DNB point. When the DNB point is
found in a particular cell, then the calculation for that cell is
repeated in order to use the properly prorated heat transfer coefficient
(apportioned between the nucleate and the film boiling coefficients) for
that cell. Then the solution switches to the film boiling zone and then
proceeds to the top of the boiling zone and compares the sodium
temperature obtained at the top to either :math:`T_{s, hg}` or :math:`T_{s, in}`. Before
the calculation proceeds node‑by‑node through the film boiling zone on
the first iteration, however, an initial guess is made of the
calibration factor in the zone is made on a one‑node basis just as was
done with :eq:`7.3-67` and :eq:`7.3-69`
in the superheated vapor zone. The only difference is that the sodium
and water :math:`\Delta T` and :math:`\Delta h` are now
appropriate for the boiling zone and :math:`\overline{T}_{w}` becomes :math:`T_{sat}`. The
calibration factor applied to the film boiling heat transfer coefficient
is adjusted on each iteration until the sodium temperature at the top of
the zone satisfies the criterion. If, as the calculation proceeds
upwards in the mesh, no DNB point is predicted before the top of the
boiling zone is reached, then it is assumed there is no film boiling
zone and the calibration factor in the nucleate boiling zone, otherwise
constant, is searched upon until the criterion at the top of the zone is
satisfied.
In order to solve the nodal equations, a method similar to that used in
the superheated vapor zone is used. First a nodal energy balance
equation exactly the same as :eq:`7.3-70` is solved for
:math:`T_{s, i + 1}` as in :eq:`7.3-71` and rewritten as in
:eq:`7.3-72`. Again, :math:`T_{m, i}` is solved for from the sodium
side heat transfer :eq:`7.3-73` resulting in :eq:`7.3-74`,
rewritten as :eq:`7.3-75` and then :eq:`7.3-76`. The water side heat transfer
equation is slightly simpler than in the superheated zone because the water temperature
is known at each node to be :math:`T_{sat}`. :eq:`7.3-77` becomes,
.. math:: G_{w} = \frac{1}{\Delta Z} \left(h_{w, i + 1} - h_{w, i} \right) = H_{T, i} \left( T_{m, i} - T_{sat} \right) \frac{2 \pi r_{w} \Delta Z}{A_{w} \Delta Z}
:label: 7.3-90
:eq:`7.3-90` is solved for :math:`h_{w, i + 1}` after :math:`T_{m, i}` is
replaced with :math:`e h_{w, i + 1} + f` from :eq:`7.3-76`. The result
is the following,
.. math:: h_{w,i + 1} = \frac{G_{w} h_{w, i} + H_{T, i} \frac{2 \pi r_{w} \Delta Z}{A_{w}} \left(f - T_{sat} \right)}{G_{w} - h_{w, i} \frac{2 \pi r_{w} \Delta Z}{A_{w}} e}
:label: 7.3-91
As before, when :math:`h_{w, i + 1}` is computed, :math:`T_{m, i}` is found from
:eq:`7.3-76` and :math:`T_{s, i + 1}` from :eq:`7.3-72`. The same
considerations concerning the calculation of properties and heat
transfer coefficients during the iterative process and concerning the
inner and outer iterations apply to the boiling zone as apply to the
superheated zone.
.. _section-7.3.3.3.1.4:
Friction Factors for Momentum Equation
++++++++++++++++++++++++++++++++++++++
As mentioned before in the section concerning the calculation of the
pressure drops in the steam generator, the factor :math:`FR_{i}` for each heat
transfer regime in :eq:`7.3-56` needs to be defined for steady state
conditions. If the "steady state" calculation does not include all heat
transfer regimes, then the :math:`FR_{i}` for the missing zones is arbitrarily
set to 1.0. First an unnormalized pressure drop :math:`\Delta P_{i}^{u}` is
calculated for heat transfer regime :math:`i`,
.. math:: \Delta P_{i}^{u} = -G_{w}^{2} \left( \frac{1}{\rho_{t}} - \frac{1}{\rho_{b}} \right) - Z_{i} \left[ \overline{\rho}_{i} 9.8 + G_{w}^{2} 0.31 \left( \frac{G_{w} D_{h}}{ \overline{\mu}_{i}} \right)^{-0.25} \frac{1}{2} \frac{1}{D_{\mu} \overline{\rho}_{i}} R_{i} \right]
:label: 7.3-92
where all parameters except :math:`G_{w}` are defined as in :eq:`7.3-56`. Next, a
normalized pressure drop :math:`\Delta P_{i}^{n}` is calculated,
.. math:: \Delta P_{i}^{n} = \Delta P_{i}^{u} \frac{\Delta P_{SG}}{\sum_{i} \Delta P_{i}^{u}}
:label: 7.3-93
where :math:`\Delta P_{SG}` is the specified pressure drop across the entire steam
generator length. The :math:`FR_{i}`\ s are calculated as follows,
.. math:: FR_{i} = \frac{\frac{\Delta P_{i}^{n}}{Z_{i}} + 9.8 \overline{\rho}_{i} + \frac{G_{w}^{2}}{Z_{i}} \left(\frac{1}{\rho_{t}} - \frac{1}{\rho_{b}} \right) 2 D_{H} \overline{\rho}_{i}}{0.31 G_{w}^{2} \left( \frac{G_{w} D_{H}}{\overline{\mu}_{i}} \right)^{-0.25} R_{i}}
:label: 7.3-94
.. _section-7.3.3.3.2:
Recirculation‑Type Steam Generator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Only the superheater steady‑state condition needs to be described here
since, as was explained above, the evaporator steady‑state is described
in the once‑through steam generator section which can account for the
situation with :math:`h_{w, out} \leq h_{g}`. The steady state solution for the
superheater is identical with the steady‑state solution for the
superheated vapor zone covered in the once‑through steam generator
section. Obviously the option of the fixed length with the search on the
calibration factor is the relevant option for the superheater. The water
side outlet enthalpy is specified and the inlet water enthalpy is
:math:`h_{g}`. The sodium inlet temperature, as before, is externally
determined and the sodium temperature at the outlet of the superheater
must be determined just as it is in :eq:`7.3-65`. The solution proceeds
upwards to the top of the superheater and the sodium temperature
calculated at the top is compared to :math:`T_{s, in}` and the calibration
factor is adjusted until they are sufficiently close.