.. _section-5.4.8:

Stratified Volume Model
~~~~~~~~~~~~~~~~~~~~~~~

In addition to the uniform mixing compressible volume model described in
:numref:`section-5.4.4`, PRIMAR‑4 contains a stratified temperature model for the
liquid in a compressible volume. This stratified model can be used for
an outlet plenum and/or for a pool. This model borrows from the
PLENUM‑2A model [5‑7] of Howard and Lorenz, but the PRIMAR‑4 model has
been extended beyond the capabilities of the PLENUM‑2A model. Borrowed
from PLENUM‑2A is the concept of a small number of distinct temperature
regions in the coolant, separated by horizontal interfaces. Also,
borrowed are the concept of distinct stages in the calculation, a plume
height correlation, and a correlation for interface rise due to
entrainment of a hot layer into a cooler plume rising from the core
outlet. One extension of the PRIMAR‑4 model is the provision for
handling up transients as well as down transients: PLENUM‑2A will only
handle transients in which the core outlet temperature is cooler than
the plenum temperature, whereas the PRIMAR‑4 model will also handle
transients in which the core outlet temperature is hotter than the
plenum temperature. Another extension is the option to handle a
horizontal discharge from an IHX into a cold pool: PLENUM‑2A will only
handle a vertical discharge from the core into an outlet plenum. The
code handles up to three regions and five stages, whereas PLENUM‑2A
considers only two regions and three stages. Also, the PRIMAR‑4 model
treats thermal conduction across the interface between regions, and this
model includes detailed multi‑node wall temperature calculations.

:numref:`figure-strat1` and :numref:`figure-strat2` shows the various stages and cases considered in this
stratified model. At the start of a transient in which the core outlet
temperature is dropping, the plume in the outlet plenum goes to the top
of the plenum; and the outlet plenum is fully mixed, giving stage 1. As
the core outlet temperature and velocity drop, the plume no longer
reaches the top of the plenum. This leads to the start of stage 2 in
which the outlet coolant goes to layer 1. In stage 2 the layer boundary
is at the elevation of the core outlet. After enough cool liquid has
entered layer 1 to fill one quarter of its volume, stage 3, case 3.1
begins. In this case, the plume coolant still goes to layer 1, but the
interface between layers rises as liquid is added to layer 1. In this
case, the plume also entrains hot liquid from the interface into layer
1. If the core outlet temperature at the start of the transient becomes
hotter than the outlet plenum temperature, then stage 3, case 3.2 is
entered. In this case, the core outlet coolant goes to a top hot layer,
entraining cool outlet plenum liquid as it passes through. The three
layer cases of stages 4 and 5 can occur in the later stages of a
transient if the core outlet temperature starts out rising and later
falls, or if the core outlet temperature starts out falling and later
rises. If the coolant inlet into the volume is horizontal, as in the
discharge of an IHX into a cold pool, then only stages 1, 3, and 5 are
used.

.. _figure-strat1:

..  figure:: media/image10.png
	:align: center
	:figclass: align-center

	Stratified Volume Stages.

.. _figure-strat2:
..  figure:: media/image11.png
	:align: center
	:figclass: align-center

	Stratified Volume Stages (Cont'd).

The jet height or plume height is calculated from an equation given by
Yang [5‑8]:

.. math::
   :label: eq-5.4-181

	h_{\text{jet}} = 1.0484 \text{Fr}^{0.785}

where

:math:`h_{\text{jet}}` = height of the jet or the plane

.. math::
   :label: eq-5.4-182

	\text{Fr} = \text{ Froude number } = \frac{v_{\text{o}}^{2} \rho_{\text{plume}}}{g~ r_{o}^{2} \left( \rho_{\text{plume}} - \rho_{\text{plenum}} \right)}

:math:`v_{\text{o}}` = core exit velocity

:math:`\rho_{\text{plume}}` = density of the plume

:math:`\rho_{\text{plenum}}` = density of the plenum

:math:`r_{\text{o}}` = core effective radius

:math:`g` = acceleration of gravity

For entrainment at an interface, Howard and Lorenz give

.. math::
   :label: eq-5.4-183

	w_{\text{ent}} = 0.2 \pi \rho_{\text{plume}} V_{\text{j}} d_{\text{j}}^2 F_{\text{f}}^{- 1.1}

where

:math:`v_{\text{j}}` = plume average velocity at the interface and

:math:`d_{\text{j}}` = plume effective diameter at the interface

:math:`w_{\text{ent}}` = entrainment rate (kg/s).

The values of :math:`v_{\text{j}}` and :math:`d_{\text{j}}` depend on elevation and on
whether the interface occurs within the zone of flow establishment or in
the zone of established flow. The elevation change, :math:`z_{\text{o}}`, from
the core outlet to the top of the zone of flow establishment is

.. math::
   :label: eq-5.4-184

	z_{o} = \frac{r_{o}}{.111}

For :math:`z < z_{\text{o}}`, or the zone of flow establishment:

.. math::
   :label: eq-5.4-185

	\frac{v_{\text{j}}}{v_{\text{o}}} = \frac{.25 + .02095 \left( \frac{z}{d}_{\text{o}} \right) + .003969 \left( \frac{z}{d}_{\text{o}} \right)^{2}}{\left\lbrack \frac{1}{2} + .1052\left( \frac{z}{d}_{\text{o}} \right)  \right\rbrack^{2}}

and

.. math::
   :label: eq-5.4-186

	\frac{d_{\text{j}}}{d_{\text{o}}} = 1 + .2104 \left( \frac{z}{d}_{\text{o}} \right)

For :math:`z > z_{\text{o}}`, or the zone of established flow,

.. math::
   :label: eq-5.4-187

	\frac{v_{\text{j}}}{v_{\text{o}}} = \frac{2.018}{\frac{z}{d}_{\text{o}}}

and

.. math::
   :label: eq-5.4-188

	\frac{d_{\text{j}}}{d_{\text{o}}} = .8649 \left( \frac{z}{d}_{\text{o}} \right)

For the wall temperatures, multi‑node treatments are used. The vertical
wall around the outside of the outlet plenum or pool is treated with a
number of vertical nodes. Each vertical node contains a number of
lateral nodes, with coolant in contact with the first node. There is
also an option to have another coolant compressible volume in contact
with the last lateral node to account for heat transfer from a hot
outlet plenum to a cold pool. The model has an option for a horizontal
wall at the top or bottom of the plenum. This wall is handled with a 1‑D
multinode treatment. Again, the first node is in contact with the plenum
liquid, and the last node can be in contact with the coolant in another
compressible volume.

The input for this model is as follows:

.. _table-5.4-4:

.. list-table:: Input for the Stratified Volume Model
    :header-rows: 1
    :align: center
    :widths: auto

    * - Block
      - Location
      - Name
      - Meaning
    * - 3
      - 1313
      - NSTRCV
      - Number of stratified compressible volumes
    * - 3
      - 1314‑1316
      - ICVSTR(ICVST)
      - 1 for vertical coolant inlet, as in an outlet plenum
        2 for a horizontal coolant inlet
    * - 3
      - 1320‑1322
      - NUMWAL(ICVST)
      - Number of wall sections
    * - 3
      - 1323‑1325
      - IFSTWL(ICVST)
      - Wall number of the first wall section
    * - 3
      - 1326‑1334
      - IWLHRZ(IW)
      - 0 for a vertical wall
        1 for a horizontal wall at the top of a CV
        2 for a horizontal wall at the bottom of a CV
    * - 3
      - 1335‑1343
      - NVNDWL(IW)
      - Number of vertical nodes in a vertical wall.
        NVNDWL = 1 for a horizontal wall
    * - 3
      - 1344‑1352
      - NLNDWL(IW)
      - Number of lateral nodes in a wall section,
        Max. = 8
    * -
      - Note: sum(NVNDWL\*NLNDWL) <= 300.
      -
      -
    * - 3
      - 1353‑1361
      - ICV2WL(IW)
      - Number of the CV in contact with the outer side of the wall section.
        = 0 for an adiabatic outer boundary. If ICV2WL > 38, ICV2WL = the temperature of a constant temperature heat sink.
    * - 3
      - 1362
      - IDBSTR
      - Debug flag for stratified temperature model
        = 0, no debug prints
        = 1, final results only
        = 5, everything
    * - 3
      - 1363
      - ISTDBS
      - PRIMAR time step when stratified debug starts.
    * - 3
      - 1364
      - ISTSTP
      - Stop the run at PRIMAR step ISTSTP. Not used if ISTSTP = 0 or NSTRCV = 0.
    * - 3
      - 1365
      - IFT16
      - Write out stratified CV output to STRATCV.dat every IFT16 PRIMAR steps. No output if IFT16 = 0.
    * - 18
      - 5008
      - RCORE
      - Core radius for use in Froude number.
    * - 18
      - 5009‑5017
      - HCSTWL(IW)
      - Coolant heat transfer coefficient at the inner surface of the wall section.
    * - 18
      - 5027‑5035
      - ASTWL(IW)
      - Area of the wall section
    * - 18
      - 5036‑5107
      - HINVWL(I,IW)
      - Thickness/thermal conductivity of node I in the wall section I = 1 - 8
    * - 18
      - 5108‑5179
      - XMCSTW(I,IW)
      - Mass x heat capacity of node I in the wall.
    * - 18
      - 5180‑5182
      - ZINST(ICVST)
      - z of inlet, used only for a vertical inlet. Otherwise ZOUTEL(IELL) is used.
    * - 18
      - 5183‑5185
      - VOLBLI(ICVST)
      - Volume below the inlet, not used if there is a cover gas in the CV. In this case, the information is obtained from other input for the CV.
    * - 18
      - 5186‑5188
      - EPSTST(ICVST)
      - Minimum temperature difference for switching stages.
    * - 18
      - 5189‑5191
      - XLENTR(ICVST)
      - Entrainment length. A hot plume with a flow rate Wh, rising through a cool layer of thickness dz, will entrain cool liquid at a rate (dz/XLENTR) x Wh.