Merge branch 'main' into pysmo_custom_sampling

docs/reference_guides/model_libraries/generic/unit_models/solid_liquid/thickener0d.rst

@@ -1,89 +1,120 @@
 Thickener (0D)
 ==============
 
-.. warning::
-    The Thickener model is currently in beta status and will likely change in the next release as a more predictive version is developed.
+The ``Thickener0D`` unit model is a predictive model for clarifiers and thickeners based on the following references:
 
-The ``Thickener0D`` unit model is an extension of the :ref:`SLSeparator <reference_guides/model_libraries/generic/unit_models/solid_liquid/sl_separator:Generic Solid-Liquid Separator>` model which adds constraints to estimate the area and height of a vessel required to achieve the desired separation of solid and liquid based on experimental measurements of the settling velocity. This model is based on correlations described in:
+[1] R. Burger, F. Concha, K.H. Karlsen, A. Narvaez, Numerical simulation of clarifier-thickener units treating ideal suspensions with a flux density function having two inflection points, Mathematical and Computer Modelling 44 (2006) 255–275, doi:10.1016/j.mcm.2005.11.008
 
-[1] Coulson & Richardson's Chemical Engineering, Volume 2 Particle Technology & Separation Processes (4th Ed.), Butterworth-Heinemann (2001)
+[2] N.G. Barton, C.-H. Li, S.J. Spencer, Control of a surface of discontinuity in continuous thickeners, Journal of the Australian Mathematical Society Series B 33 (1992) 269–289
 
-Sizing Thickeners and Pinch Point
----------------------------------
-
-The approach for sizing the thickener vessel used in this model relies on identifying a pinch point in the thickener which is the limiting condition for the settling velocity of the suspension as described in [1]. The pinch point is described by the condition:
-
-.. math:: max \left( \frac{(Y - Y_{under})}{u(Y)} \right)
-
-where :math:`Y` is the mass based liquid-to-solid ratio, :math:`u(Y)` is the settling velocity of the suspension as a function of :math:`Y` and :math:`Y_{under}` is the liquid-solid ratio at the thickener underflow. Correlations exist which can predict :math:`u(Y)` however in many cases it is necessary to measure this experimentally. Additionally, whilst there are techniques for embedding the maximization operation within an equation-oriented model, these approaches tend to be highly non-linear and require the user to provide good values for scaling parameters and thus have not been implemented yet.
-
-In the current implementation of the model, :math:`Y_{pinch}` and :math:`u_{pinch}` are created as user-defined input variables which should generally be fixed at appropriate values. Users may choose to add a correlation for :math:`u(Y)` if they choose.
 
 Degrees of Freedom
 ------------------
 
-The ``Thickener0D`` model has 5 degrees of freedom, which are generally chosen from:
+The ``Thickener0D`` model has 6 degrees of freedom. Four of these are used to define the solids settling velocity and must be provided by the user.
+
+* the solid particle size ``particle_size``,
+* the maximum achievable solid fraction ``solid_fraction_max``,
+* two empirical coefficients for calculating the settling velocity ``v1``, ``C``.
 
-* the liquid recovery fraction or underflow liquid-to-solid ratio,
-* the liquid-to-solid ratio and settling velocity at the pinch (critical) point,
-* the cross-sectional area of the thickener,
-* the depth of the clarification zone,
-* the depth of the sedimentation zone or the required settling time (experimental).
+Additionally, the user must provide 2 additional degrees of freedom regarding the design and/or operation of the thickener such as:
+
+* the cross-sectional area of the settler ``area``,
+* the total volumetric flowrate or volume fraction of solids at the underflow, or
+* the total volumetric flowrate or volume fraction of solids at the overflow.
 
 Model Structure
 ---------------
 
-The ``Thickener0D`` model has the same structure as the :ref:`SLSeparator <reference_guides/model_libraries/generic/unit_models/solid_liquid/sl_separator:Model Structure>`.
+The ``Thickener0D`` model contains two separators, ``solid_split`` and ``liquid_spit`` to separate the solid and liquid flows. The separations are assumed to be based on total flow with no change in composition, temperature or pressure. The ``Thickener0D`` model has 6 Ports (2 inlet and 4 outlet):
+
+* solid_inlet,
+* liquid_inlet,
+* solid_overflow,
+* liquid_overflow,
+* solid_underflow,
+* liquid_underflow.
 
 Additional Constraints
 ----------------------
 
-The ``Thickener0D`` model adds the following additional constraints beyond those written by the :ref:`SLSeparator <reference_guides/model_libraries/generic/unit_models/solid_liquid/sl_separator:Additional Constraints>`.
+The ``Thickener0D`` model adds the following constraints to calculate the split fractions of the solid and liquid stream.
+
+Total volumetric flowrates at the feed  overflow and underflow are calculated as the sum of the volumetric flowrates of the solid and liquid streams at each location.
+
+.. math:: Q_{x,t} = Q_{solid,x,t} + Q_{liquid,x,t}
+
+where :math:`Q` represents volumetric flowrate and :math:`x` represents the feed, overflow or underflow point.
 
-The cross-sectional area of the thickener is calculated using:
+The flux density at the overflow and underflow are calculated using the following constraint:
 
-.. math:: A = \frac{S_t}{\rho_{liq, t}} \times \frac{(Y_{pinch, t}-Y_{under, t})}{u_{pinch, t}}
+If :math:`0 \leqslant \epsilon_{x,t} \leqslant \epsilon_{max}`:
 
-where :math:`A` is the cross-section area of the thickener, :math:`S_t` is the mass flowrate of solids entering the thickener, :math:`\rho_{liq}` is the mass density of the liquid phase, :math:`Y_{pinch}` and :math:`Y_{under}` are the mass-based liquid-to-solid ratios at the pinch point and underflow respectively and :math:`u_{pinch}` is the sedimentation velocity of the suspension at the pinch point (Eqn 5.54, pg. 198 in [1]).
+.. math:: F_{x,t} = v_0 \times \epsilon_{x, t} \times(1- \frac{\epsilon_{x,t}}{\epsilon_{max}})^C + v_1 \times \epsilon_{x,t}^2 \times(\epsilon_{max}-\epsilon_{x,t})
 
-The liquid-solid ratio at the underflow can be calculated using:
+otherwise:
 
-.. math:: Y_{under, t} = \frac{L_{under, t}}{S_t}
+.. math:: F_{x,t} = 0
 
-where :math:`L_{under}` is the liquid mass flowrate at the underflow.
+where :math:`F` is the flux density, :math:`v_0` is the Stokes velocity of a single particle, :math:`\epsilon` is the solid volume fraction, :math:`\epsilon_{max}` is the maximum attainable solid volume fraction, and :math:`v_1` and :math:`C` are empirical constants [2].
 
-The total height of the thickener is calculated using:
+The solids fraction at the overflow and underflow is described using the following equation (derived from [1]).
 
-.. math:: H = \frac{S_t \tau_{t}}{A\rho_{sol, t}} \times \left( 1+\frac{\rho_{sol}}{\rho_{liq}} \times Y_{avg} \right) + H_{clarified}
+.. math:: Q_{feed, t} \times \epsilon_{feed,t} = A \times (F_{overflow,t} + F_{underflow,t}) - Q_{overflow,t} \times (\epsilon_{overflow,t} - \epsilon_{feed,t}) + Q_{underflow,t} \times (\epsilon_{underflow,t} - \epsilon_{feed,t})
 
-where :math:`H` is the total height of the thickener, :math:`H_{clarified}` is the height of the clarification zone, :math:`\tau` is the empirically measured settling time required to achieve the desired underflow conditions and :math:`Y_avg` is the average liquid-solid ratio in the thickener (assumed to be linear) (Eqn 5.55, pg. 198 in [1]).
+where :math:`A` is the cross-sectional area of the thickener (assumed constant in space and time).
+
+Conservation of solids is enforced using the following constraint:
+
+.. math:: Q_{feed,t} \times \epsilon_{feed,t} = Q_{overflow,t} \times \epsilon_{overflow,t} + Q_{underflow,t} \times \epsilon_{underflow,t}
+
+Two constraints are written to define the solid volume fraction at the feed and underflow point (no constraint is written for the overflow point).
+
+.. math:: Q_{solid,x,t} = \epsilon_{x,t} \times (Q_{solid,x,t} + Q_{liquid,x,t})
+
+The solids volume fraction at the overflow and underflow are bounded by the following inequality constraints:
+
+.. math:: \epsilon_{t,x} \leqslant \epsilon_{max}
+
+Finally, the Stokes velocity is calculated using Stokes Law:
+
+.. math:: 18 \times v0_t \times \mu_{liquid,t} = (\rho_{solid,t} - \rho_{liquid,t}) \times g \times d_p^2
+
+where :math:`\mu_{liquid}` is the viscosity of the liquid phase, :math:`\rho_{liquid}` and :math:`\rho_{solid}` are the densities of the liquid and solid phases, :math:`g` is the acceleration due to gravity and :math:`d_p` is the solid particle diameter.
 
 Variables
 ---------
 
-The ``Thickener0D`` adds the following variables in addition to those in the :ref:`SLSeparator <reference_guides/model_libraries/generic/unit_models/solid_liquid/sl_separator:Variables>`.
-
-===================== ======================== ====== =====================================================================
-Variable              Name                     Index  Notes
-===================== ======================== ====== =====================================================================
-:math:`A`             area                     None   Cross-sectional area of thickener
-:math:`H`             height                   None   Total height of thickener
-:math:`H_{clarified}` height_clarified         None   Height of clarification zone in thickener (height above feed point)
-:math:`u_{pinch}`     settling_velocity_pinch  time   Settling velocity of suspension at pinch point
-:math:`Y_{pinch}`     liquid_solid_pinch       time   Liquid-solid ratio at pinch point
-:math:`Y_{under}`     liquid_solid_underflow   time   Liquid-solid ratio at underflow
-:math:`\tau`          settling_time            time   Settling time in thickener
-===================== ======================== ====== =====================================================================
+The ``Thickener0D`` adds the following variables to those contained in the Separators.
+
+============================= ========================= ====== ========================================================
+Variable                      Name                      Index  Notes
+============================= ========================= ====== ========================================================
+:math:`A`                     area                      None   Cross-sectional area of thickener
+:math:`Q_{feed}`              flow_vol_feed             time   Total volumetric flowrate at feed point
+:math:`Q_{overflow}`          flow_vol_overflow         time   Total volumetric flowrate at overflow point
+:math:`Q_{underflow}`         flow_vol_underflow        time   Total volumetric flowrate at underflow point
+:math:`\epsilon_{feed}`       solid_fraction_feed       time   Volumetric solids fraction at feed point
+:math:`\epsilon_{overflow}`   solid_fraction_overflow   time   Volumetric solids fraction at overflow point
+:math:`\epsilon_{underflow}`  solid_fraction_underflow  time   Volumetric solids fraction at underflow point
+:math:`F_{overflow}`          flux_density_overflow     time   Solids flux density at overflow point
+:math:`F_{underflow}`         flux_density_underflow    time   Solids flux density at underflow point
+:math:`d_p`                   particle size             time   Solid particle size
+:math:`v0`                    v0                        time   Stokes velocity of isolated particle
+:math:`v1`                    v1                        None   Empirical parameter in settling velocity correlation
+:math:`C`                     C                         None   Empirical parameter in settling velocity correlation
+math:`\epsilon_{max}`         solid_fraction_max        None   Maximum achievable volumetric solids fraction
+============================= ========================= ====== ========================================================
 
 .. module:: idaes.models.unit_models.solid_liquid.thickener
 
-SLSeparator Class
+Thickener0D Class
 -----------------
 
 .. autoclass:: Thickener0D
   :members:
 
-SLSeparatorData Class
+Thickener0DData Class
 ---------------------
 
 .. autoclass:: Thickener0DData