mirror of
https://github.com/CoolProp/CoolProp.git
synced 2026-04-23 03:00:17 -04:00
231 lines
8.6 KiB
ReStructuredText
231 lines
8.6 KiB
ReStructuredText
.. _mixtures:
|
|
|
|
********
|
|
Mixtures
|
|
********
|
|
|
|
.. contents:: :depth: 2
|
|
|
|
Theoretical description
|
|
-----------------------
|
|
The mixture modeling used in CoolProp is based on the work of Kunz et al. :cite:`Kunz-BOOK-2007,Kunz-JCED-2012` and Lemmon :cite:`Lemmon-JPCRD-2000,Lemmon-JPCRD-2004,Lemmon-IJT-1999`
|
|
|
|
A mixture is composed of a number of components, and for each pair of components, it is necessary to have information for the excess Helmholtz energy term as well as the reducing function. See below for what binary pairs are included in CoolProp.
|
|
|
|
The numerical methods required for mixtures are far more complicated than those for pure fluids, so the number of flash routines that are currently available are relatively small compared to pure fluids.
|
|
|
|
The only types of inputs that are allowed for mixtures right now are
|
|
|
|
- Pressure/quality
|
|
- Temperature/quality
|
|
- Temperature/pressure
|
|
|
|
.. Used in Python script later on
|
|
.. role:: raw-html(raw)
|
|
:format: html
|
|
|
|
Estimating binary interaction parameters
|
|
----------------------------------------
|
|
|
|
If you have a mixture that you would like to include in your analysis, but for which no interaction parameters are currently available, there are two estimation schemes available. These estimation schemes should be used with extreme caution. If you use these schemes, you should definitely check that you are getting reasonable output values. In general, these estimation schemes should be used for fluids with similar properties.
|
|
|
|
The two schemes available are
|
|
|
|
* ``linear`` - :math:`T_r` and :math:`v_r` are a linear function of molar composition between the two pure fluid values
|
|
* ``Lorentz-Berthelot`` - all interaction parameters are 1.0
|
|
|
|
|
|
Here is a sample of using this in python:
|
|
|
|
.. ipython::
|
|
|
|
In [1]: import CoolProp.CoolProp as CP
|
|
|
|
In [1]: CP.apply_simple_mixing_rule('Helium', 'Xenon', 'linear')
|
|
|
|
In [1]: CP.PropsSI('Dmass','T',300,'P',101325,'Helium[0.5]&Xenon[0.5]')
|
|
|
|
.. warning::
|
|
|
|
Use with caution!! For other mixtures this can give you entirely(!) wrong predictions
|
|
|
|
Using your own interaction parameters
|
|
-------------------------------------
|
|
|
|
If you have your own interaction parameters that you would like to use, you can set them using the ``set_mixture_binary_pair_data`` function. (You can also retrieve them using the ``set_mixture_binary_pair_data`` function). You must do this before you would like to call other functions. Some sample code is below.
|
|
|
|
.. ipython::
|
|
:okexcept:
|
|
|
|
In [1]: import CoolProp.CoolProp as CP
|
|
|
|
# This adds a dummy entry in the library of interaction parameters if the mixture is not already there
|
|
In [1]: CP.apply_simple_mixing_rule('Helium', 'Xenon', 'linear')
|
|
|
|
# If the binary interaction pair is already there, set the configuration flag
|
|
# to allow binary interaction parameters to be over-written
|
|
In [1]: CP.set_config_bool(CP.OVERWRITE_BINARY_INTERACTION, True)
|
|
|
|
# This is before setting the binary interaction parameters
|
|
In [1]: CP.PropsSI('Dmass','T',300,'P',101325,'Helium[0.5]&Xenon[0.5]')
|
|
|
|
In [1]: CP.set_mixture_binary_pair_data(CAS_He, CAS_Xe, 'betaT', 1.0)
|
|
|
|
In [1]: CP.set_mixture_binary_pair_data(CAS_He, CAS_Xe, 'gammaT', 1.5)
|
|
|
|
In [1]: CP.set_mixture_binary_pair_data(CAS_He, CAS_Xe, 'betaV', 1.0)
|
|
|
|
In [1]: CP.set_mixture_binary_pair_data(CAS_He, CAS_Xe, 'gammaV', 1.5)
|
|
|
|
# This is after setting the interaction parameters
|
|
In [1]: CP.PropsSI('Dmass','T',300,'P',101325,'Helium[0.5]&Xenon[0.5]')
|
|
|
|
Once you have constructed an instance of an AbstractState using the low-level interface, you can set the interaction parameters for only that instance by calling the ``set_binary_interaction_double`` and ``get_binary_interaction_double`` functions.
|
|
|
|
.. ipython::
|
|
|
|
In [1]: import CoolProp.CoolProp as CP
|
|
|
|
# This adds a dummy entry in the library of interaction parameters if the mixture is not already there
|
|
In [1]: CP.apply_simple_mixing_rule(CAS_He, CAS_Xe, 'linear')
|
|
|
|
In [1]: AS = CP.AbstractState("HEOS","Helium&Xenon")
|
|
|
|
In [1]: AS.set_binary_interaction_double(0, 1, 'betaT', 0.987)
|
|
|
|
In [1]: AS.get_binary_interaction_double(0, 1, 'betaT')
|
|
|
|
# Here you can see that this call to the high-level interface is untouched (is the same as above)
|
|
In [1]: CP.PropsSI('Dmass','T',300,'P',101325,'Helium[0.5]&Xenon[0.5]')
|
|
|
|
And now, reset the configuration variable
|
|
|
|
.. ipython::
|
|
|
|
In [1]: import CoolProp.CoolProp as CP
|
|
|
|
In [1]: CP.set_config_bool(CP.OVERWRITE_BINARY_INTERACTION, False)
|
|
|
|
Phase Envelope
|
|
--------------
|
|
|
|
You can download the script that generated the following figure here: :download:`(link to script)<methane-ethane.py>`, right-click the link and then save as... or the equivalent in your browser. You can also download this figure :download:`as a PDF<methane-ethane.pdf>`.
|
|
|
|
.. image:: methane-ethane.png
|
|
|
|
Reducing Parameters
|
|
-------------------
|
|
|
|
From Lemmon :cite:`Lemmon-JPCRD-2000` for the properties of Dry Air, and also from Lemmon :cite:`Lemmon-JPCRD-2004` for the properties of R404A, R410A, etc.
|
|
|
|
.. math::
|
|
|
|
\rho_r(\bar x) = \left[ \sum_{i=1}^m\frac{x_i}{\rho_{c_i}}+\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}x_ix_j\zeta_{ij}\right]^{-1}
|
|
|
|
.. math::
|
|
|
|
T_r(\bar x) = \sum_{i=1}^mx_iT_{c_i}+\sum_{i=1}^{m-1}\sum_{j=i+1}^mx_ix_j\xi_{ij}
|
|
|
|
From the GERG 2008 formulation :cite:`Kunz-JCED-2012`
|
|
|
|
.. math::
|
|
|
|
T_r(\bar x) = \sum_{i=1}^{N}x_i^2T_{c,i} + \sum_{i=1}^{N-1}\sum_{j=i+1}^{N}2x_ix_j\beta_{T,ij}\gamma_{T,ij}\frac{x_i+x_j}{\beta_{T,ij}^2x_i+x_j}(T_{c,i}T_{c,j})^{0.5}
|
|
|
|
.. math::
|
|
|
|
\frac{1}{\rho_r(\bar x)}=v_r(\bar x) = \sum_{i=1}^{N}x_i^2\frac{1}{\rho_{c,i}} + \sum_{i=1}^{N-1}\sum_{j=i+1}^N2x_ix_j\beta_{v,ij}\gamma_{v,ij}\frac{x_i+x_j}{\beta^2_{v,ij}x_i+x_j}\frac{1}{8}\left(\frac{1}{\rho_{c,i}^{1/3}}+\frac{1}{\rho_{c,j}^{1/3}}\right)^{3}
|
|
|
|
Excess Helmholtz Energy Terms
|
|
-----------------------------
|
|
From Lemmon :cite:`Lemmon-JPCRD-2004` for the properties of R404A, R410A, etc.
|
|
|
|
.. math::
|
|
|
|
\alpha^E(\delta,\tau,\mathbf{x}) = \sum_{i=1}^{m-1} \sum_{j=i+1}^{m} \left [ x_ix_jF_{ij} \sum_{k}N_k\delta^{d_k}\tau^{t_k}\exp(-\delta^{l_k})\right]
|
|
|
|
where the terms :math:`N_k,d_k,t_k,l_k` correspond to the pair given by the indices :math:`i,j`
|
|
|
|
From Lemmon :cite:`Lemmon-JPCRD-2000` for the properties of Dry Air
|
|
|
|
.. math::
|
|
|
|
\alpha^E(\delta,\tau,\mathbf{x}) = \left \lbrace \sum_{i=1}^{2} \sum_{j=i+1}^{3} x_ix_jF_{ij}\right\rbrace \left[-0.00195245\delta^2\tau^{-1.4}+0.00871334\delta^2\tau^{1.5} \right]
|
|
|
|
|
|
From Kunz and Wagner :cite:`Kunz-JCED-2012` for GERG 2008 formulation
|
|
|
|
.. math::
|
|
|
|
\alpha^E(\delta,\tau,\mathbf{x}) = \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} x_ix_jF_{ij}\alpha_{ij}^r(\delta,\tau)
|
|
|
|
where
|
|
|
|
.. math::
|
|
|
|
\alpha_{ij}^r(\delta,\tau) = \sum_{k=1}^{K_{pol,ij}}\eta_{ij,k}\delta^{d_{ij,k}}\tau^{t_{ij,k}}+\sum_{k=K_{pol,ij}+1}^{K_{pol,ij}+K_{Exp,ij}}\eta_{ij,k}\delta^{d_{ij,k}}\tau^{t_{ij,k}}\exp[-\eta_{ij,k}(\delta-\varepsilon_{ij,k})^2-\beta_{ij,k}(\delta-\gamma_{ij,k})]
|
|
|
|
and is for the particular binary pair given by the indices :math:`i,j`. This term is similar in form to other Helmholtz energy terms for pure fluids though the derivatives are slightly special.
|
|
|
|
Appendix
|
|
--------
|
|
To convert from the form from Lemmon for HFC and Air to that of GERG 2008, the following steps are required:
|
|
|
|
.. math::
|
|
|
|
x_0T_{c0}+(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = x_0^2T_{c0}+(1-x_0)^2T_{c1} + 2x_0(1-x_0)\beta\gamma_T\frac{x_0+(1-x_0)}{\beta x_0 + (1-x_0)}\sqrt{T_{c0}T_{c1}}
|
|
|
|
set :math:`\beta=1`, solve for :math:`\gamma`. Equate the terms
|
|
|
|
.. math::
|
|
|
|
x_0T_{c0}+(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = x_0^2T_{c0}+(1-x_0)^2T_{c1} + 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
|
|
|
|
Move to LHS
|
|
|
|
.. math::
|
|
|
|
[x_0-x_0^2]T_{c0}+[(1-x_0)-(1-x_0)^2]T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
|
|
|
|
Factor
|
|
|
|
.. math::
|
|
|
|
x_0(1-x_0)T_{c0}+(1-x_0)[1-(1-x_0)]T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
|
|
|
|
Expand
|
|
|
|
.. math::
|
|
|
|
x_0(1-x_0)T_{c0}+x_0(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
|
|
|
|
Cancel factors of :math:`x_0(1-x_0)`
|
|
|
|
.. math::
|
|
|
|
T_{c0}+T_{c1}+\xi_{01} = 2\gamma_T\sqrt{T_{c0}T_{c1}}
|
|
|
|
Answer:
|
|
|
|
.. math::
|
|
|
|
\boxed{\gamma_T = \dfrac{T_{c0}+T_{c1}+\xi_{01}}{2\sqrt{T_{c0}T_{c1}}}}
|
|
|
|
Same idea for the volume
|
|
|
|
.. math::
|
|
|
|
\boxed{\gamma_v = \dfrac{v_{c0}+v_{c1}+\zeta_{01}}{\frac{1}{4}\left(\frac{1}{\rho_{c,i}^{1/3}}+\frac{1}{\rho_{c,j}^{1/3}}\right)^{3}}}
|
|
|
|
|
|
Binary pairs
|
|
------------
|
|
|
|
.. note::
|
|
Please hover the mouse pointer over the coefficients to get the full accuracy
|
|
for the listed coefficients. You can also get more information on references
|
|
that are not in bibliography.
|
|
|
|
.. csv-table:: All binary pairs included in CoolProp
|
|
:header-rows: 1
|
|
:file: Mixtures.csv |