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Started to work on the new incompressible enthalpy formulation

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Jorrit Wronski
2014-12-17 10:42:55 +01:00
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Web/fluid_properties/.gitignore vendored
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@@ -1,3 +1,4 @@
/report/
*.pdf
*.csv
*.bib.bak

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Web/fluid_properties/Incompressibles.bib
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@@ -2,6 +2,19 @@
% Encoding: ASCII
@Book{Cengel2007,
Title = {Thermodynamics, An Engineering Approach},
Author = {Yunus A. \c{C}engel and Michael A. Boles},
Publisher = {McGraw-Hill},
Year = {2007},
Address = {New York},
Edition = {6th edition},
Owner = {jowr},
Timestamp = {2014.12.16}
}
@Book{ASHRAE2001,
Title = {{2001 ASHRAE Handbook: Fundamentals}},
Author = {{American Society of Heating, Refrigerating and Air-Conditioning Engineers}},
@@ -34,6 +47,20 @@
Timestamp = {2014.09.22}
}
@Article{Kostic2006,
Title = {Analysis of Enthalpy Approximation for Compressed Liquid Water},
Author = {Milivoje M. Kostic},
Journal = {Journal of Heat Transfer},
Year = {2006},
Number = {5},
Pages = {421--426},
Volume = {128},
Doi = {0.1115/1.2175090},
Owner = {jowr},
Timestamp = {2014.12.16}
}
@Book{Melinder2010,
Title = {{Properties of Secondary Working Fluids for Indirect Systems}},
Author = {{\AA}ke Melinder},

145
Web/fluid_properties/Incompressibles.rst
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@@ -118,7 +118,7 @@ corresponds to the new mixture syntax in CoolProp v5.
#Density of a lithium bromide solution at 300 K and 1 atm.
In [1]: PropsSI('D','T',300,'P',101325,'INCOMP::LiBr[0.23]')
#Density of a lithium bromide solution at 300 K and 1 atm.
In [1]: PropsSI('D','T',300,'P',101325,'INCOMP::LiBr-23%')
@@ -176,6 +176,103 @@ that typically lies in the middle of the allowed range. Dashed red lines indicat
the limits in terms of concentration as well as the freezing temperature.
.. _IncompThermo:
Thermodynamics of Incompressible Fluids
---------------------------------------
For an incompressible fluid, the specific at constant volume and at constant
pressure are the same allowing us to drop the subscripts, :math:`c_p=c_v=c`. Using
temperature :math:`T` and pressure :math:`p` as state variables, we can simplify
the normal thermodynamic relation as described below. working with brines and
mixtures, the concentration :math:`x` has to be considered as well. Following
the same approach as for the compressible fluids, we regard mixtures with different
compositions as independent fluids. This should be kept in mind when comparing
properties for different compositions. Setting the reference state for one
composition will always affect all fluids consisting of the same components.
The approach described in textbooks like Cengel and Boles :cite:`Cengel2007`
is that the internal energy :math:`u` only depends on temperature and does not
change with pressure.
.. Alternatively, use cancel package with \cancelto{0}{x-d} command
.. math::
du &= \overbrace{ \left( \frac{\partial u}{\partial T} \right)_p}^{=c_p=c_v=c} dT &+ \overbrace{\left( \frac{\partial u}{\partial p} \right)_T}^{\stackrel{!}{=}0} dp \\
By using the fourth Maxwell relation, we can extend the simplifications to the
entropy formulation
.. math::
ds &= \left( \frac{\partial s}{\partial T} \right)_p dT &+ \left( \frac{\partial s}{\partial p} \right)_T dp \\
&= \underbrace{ \left( \frac{\partial h}{\partial T} \right)_p}_{=c_p=c_v=c} T^{-1} dT &-\underbrace{\left( \frac{\partial v}{\partial T} \right)_p}_{\stackrel{!}{=} 0} dp \\
As indicated by the braces above, the fluids implemented in CoolProp do also follow
the second common assumption of a constant specific volume :math:`v` that does
change neither with temperature nor with pressure. It should be highlighted, that
this simplification violates the integrity of the implemented equations since there
are changes in density as a function of temperature for all incompressible fluids.
Employing :math:`h=u+pv`, we can derive the impact on enthalpy as well by
rewriting the equation in terms of our state variables :math:`p` and :math:`T`.
.. math::
dh &= \overbrace{ \left( \frac{\partial h}{\partial T} \right)_p}^{=c_p=c_v=c} dT + \left( \frac{\partial h}{\partial p} \right)_T dp \\
&= du + \underbrace{p dv}_{\stackrel{!}{=} 0} + v dp \\
Using only polynomials for the heat capacity functions, we can derive internal
energy and entropy by integrating the specific heat capacity in temperature, only
enthalpy requires an integration in both :math:`T` and :math:`p`
.. _BaseValue:
.. math::
c &= \sum_{i=0}^n x^i \cdot \sum_{j=0}^m C_{c}[i,j] \cdot T^j \text{ yielding } \\
u &= \int_{0}^{1} c\left( x,T \right) dT
= \sum_{i=0}^n x^i \cdot \sum_{j=0}^m \frac{1}{j+1} \cdot C_{c}[i,j]
\cdot \left( T_{1}^{j+1} - T_{0}^{j+1} \right) \text{ and } \\
s &= \int_{0}^{1} \frac{c\left( x,T \right)}{T} dT
= \sum_{i=0}^n x^i \cdot \left(
C_{c}[i,0] \cdot \ln\left(\frac{T_{1}}{T_{0}}\right)
+ \sum_{j=0}^{m-1} \frac{1}{j+1} \cdot C_{c}[i,j+1] \cdot \left( T_{1}^{j+1} - T_{0}^{j+1} \right)
\right) \\
h &= u + v \cdot \left( p_{1} - p_{0} \right)
According to Melinder :cite:`Melinder2010` and Skovrup :cite:`Skovrup2013`,
using a centred approach for the independent variables enhances the fit quality.
Therefore, all solutions have a base temperature and concentration in the original
works as well as in CoolProp: :math:`x_\text{in} = x - x_\text{base}`
and :math:`T_\text{in} = T - T_\text{base}`, this technique does not affect the calculation
of the derived quantity internal energy since the formula contains temperature differences.
However, integrating :math:`c(x_\text{in},T_\text{in})T_\text{in}^{-1}dT_\text{in}` for the entropy requires some changes due to
the logarithm.
.. warning::
You must **not** use the base temperature :math:`T_\text{base}`
as reference temperature for your thermodynamic states. This will lead to an
error caused by a division by zero during the integration carried out to
obtain the entropy.
To structure the problem, we introduce a variable :math:`f(j,T)`,
which will be expressed by a third sum. As a first step for simplification, one
has to expand the the binomial :math:`(T-T_{base})^n` to a series. Only
containing :math:`j` and :math:`T`, :math:`f` is independent from :math:`x_\text{in}` and
can be computed outside the loop for enhanced computational efficiency. An
integration of the expanded binomial then yields the final factor :math:`F` to
be multiplied with the other coefficients and the concentration.
.. math::
s &= \int_{0}^{1} \frac{c\left( x_\text{in},T_\text{in} \right)}{T_\text{in}} dT_\text{in} = \sum_{i=0}^n x_\text{in}^i \cdot \sum_{j=0}^m C_{c}[i,j] \cdot F(j,T_\text{in,0},T_\text{in,1}) \\
F &= (-1)^j \cdot \ln \left( \frac{T_\text{in,1}}{T_\text{in,0}} \right) \cdot T_{base}^j + \sum_{k=0}^{j-1} \binom{j}{k} \cdot \frac{(-1)^k}{j-k} \cdot \left( T_\text{in,1}^{j-k} - T_\text{in,0}^{j-k} \right) \cdot T_{base}^k
.. _Equations:
Equations
@@ -221,52 +318,6 @@ Python module `CPIncomp <https://github.com/CoolProp/CoolProp/tree/master/dev/in
contains the source code for the fits used in CoolProp as well as the code to
generate the fitting reports. Feel free to browse the code.
Using only polynomials for the heat capacity functions, we can derive internal
energy and entropy by integrating the specific heat capacity
.. _BaseValue:
.. math::
c &= \sum_{i=0}^n x^i \cdot \sum_{j=0}^m C_{c}[i,j] \cdot T^j \text{ yielding } \\
u &= \int_{0}^{1} c\left( x,T \right) dT
= \sum_{i=0}^n x^i \cdot \sum_{j=0}^m \frac{1}{j+1} \cdot C_{c}[i,j]
\cdot \left( T_{1}^{j+1} - T_{0}^{j+1} \right) \text{ and } \\
s &= \int_{0}^{1} \frac{c\left( x,T \right)}{T} dT
= \sum_{i=0}^n x^i \cdot \left(
C_{c}[i,0] \cdot \ln\left(\frac{T_{1}}{T_{0}}\right)
+ \sum_{j=0}^{m-1} \frac{1}{j+1} \cdot C_{c}[i,j+1] \cdot \left( T_{1}^{j+1} - T_{0}^{j+1} \right)
\right) \\
According to Melinder :cite:`Melinder2010` and Skovrup :cite:`Skovrup2013`,
using a centred approach for the independent variables enhances the fit quality.
Therefore, all solutions have a base temperature and concentration in the original
works as well as in CoolProp: :math:`x_\text{in} = x - x_\text{base}`
and :math:`T_\text{in} = T - T_\text{base}`, this technique does not affect the calculation
of the derived quantity internal energy since the formula contains temperature differences.
However, integrating :math:`c(x_\text{in},T_\text{in})T_\text{in}^{-1}dT_\text{in}` for the entropy requires some changes due to
the logarithm.
.. warning::
You must **not** use the base temperature :math:`T_\text{base}`
as reference temperature for your thermodynamic states. This will lead to an
error caused by a division by zero during the integration carried out to
obtain the entropy.
To structure the problem, we introduce a variable :math:`f(j,T)`,
which will be expressed by a third sum. As a first step for simplification, one
has to expand the the binomial :math:`(T-T_{base})^n` to a series. Only
containing :math:`j` and :math:`T`, :math:`f` is independent from :math:`x_\text{in}` and
can be computed outside the loop for enhanced computational efficiency. An
integration of the expanded binomial then yields the final factor :math:`F` to
be multiplied with the other coefficients and the concentration.
.. math::
s &= \int_{0}^{1} \frac{c\left( x_\text{in},T_\text{in} \right)}{T_\text{in}} dT_\text{in} = \sum_{i=0}^n x_\text{in}^i \cdot \sum_{j=0}^m C_{c}[i,j] \cdot F(j,T_\text{in,0},T_\text{in,1}) \\
F &= (-1)^j \cdot \ln \left( \frac{T_\text{in,1}}{T_\text{in,0}} \right) \cdot T_{base}^j + \sum_{k=0}^{j-1} \binom{j}{k} \cdot \frac{(-1)^k}{j-k} \cdot \left( T_\text{in,1}^{j-k} - T_\text{in,0}^{j-k} \right) \cdot T_{base}^k
The Different Fluids
--------------------