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@@ -191,67 +191,71 @@ compositions as independent fluids. This should be kept in mind when comparing
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properties for different compositions. Setting the reference state for one
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composition will always affect all fluids consisting of the same components.
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The approach described in textbooks like Cengel and Boles :cite:`Cengel2007`
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is that the internal energy :math:`u` only depends on temperature and does not
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change with pressure.
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.. Alternatively, use cancel package with \cancelto{0}{x-d} command
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.. math::
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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 \\
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By using the fourth Maxwell relation, we can extend the simplifications to the
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entropy formulation
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.. math::
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ds &= \left( \frac{\partial s}{\partial T} \right)_p dT &+ \left( \frac{\partial s}{\partial p} \right)_T dp \\
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&= \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 \\
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As indicated by the braces above, the fluids implemented in CoolProp do also follow
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the second common assumption of a constant specific volume :math:`v` that does
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change neither with temperature nor with pressure. It should be highlighted, that
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this simplification violates the integrity of the implemented equations since there
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are changes in density as a function of temperature for all incompressible fluids.
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Employing :math:`h=u+pv`, we can derive the impact on enthalpy as well by
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rewriting the equation in terms of our state variables :math:`p` and :math:`T`
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as shown by Skovrup :cite:`Skovrup1999`.
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.. 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 \\
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.. math::
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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 \\
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&= \left( \frac{\partial u}{\partial T} \right)_v dT + \left( v - T \left( \frac{\partial v}{\partial T} \right)_p \right) dp \\
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&= du + \underbrace{p dv}_{\stackrel{!}{=} 0} + v dp \quad \text{ with $v\stackrel{!}{=}v_0=$ const } \\
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The two assumptions used above :math:`\left( \partial v / \partial T \right)_p \stackrel{!}{=} 0`
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and :math:`\left( \partial u / \partial T \right)_p \stackrel{!}{=} \left( \partial u / \partial T \right)_v`
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imply that :math:`v` is constant under all circumstances. Hence, we have to use
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the specific volume at reference conditions to calculate enthalpy from the
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integration in :math:`T` and :math:`p`. Future work could provide a more accurate
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formulation of entropy and enthalpy by implementing the term
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:math:`\left( \partial v / \partial T \right)_p \neq 0`.
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Using only polynomials for the heat capacity functions, we can derive internal
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energy and entropy by integrating the specific heat capacity in temperature.
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.. The approach described in textbooks like Cengel and Boles :cite:`Cengel2007`
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.. is that the internal energy :math:`u` only depends on temperature and does not
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.. change with pressure.
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..
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.. .. Alternatively, use cancel package with \cancelto{0}{x-d} command
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..
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.. .. math::
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..
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.. 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 \\
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..
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.. By using the fourth Maxwell relation, we can extend the simplifications to the
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.. entropy formulation
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..
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.. .. math::
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..
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.. ds &= \left( \frac{\partial s}{\partial T} \right)_p dT &+ \left( \frac{\partial s}{\partial p} \right)_T dp \\
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.. &= \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 \\
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..
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.. As indicated by the braces above, the fluids implemented in CoolProp do also follow
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.. the second common assumption of a constant specific volume :math:`v` that does
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|
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.. change neither with temperature nor with pressure. It should be highlighted, that
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.. this simplification violates the integrity of the implemented equations since there
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.. are changes in density as a function of temperature for all incompressible fluids.
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..
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.. Employing :math:`h=u+pv`, we can derive the impact on enthalpy as well by
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.. rewriting the equation in terms of our state variables :math:`p` and :math:`T`
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.. as shown by Skovrup :cite:`Skovrup1999`.
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..
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.. .. 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 \\
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..
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.. .. math::
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.. 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 \\
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.. &= \left( \frac{\partial u}{\partial T} \right)_v dT + \left( v - T \left( \frac{\partial v}{\partial T} \right)_p \right) dp \\
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.. &= du + \underbrace{p dv}_{\stackrel{!}{=} 0} + v dp \quad \text{ with $v\stackrel{!}{=}v_0=$ const } \\
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..
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.. The two assumptions used above :math:`\left( \partial v / \partial T \right)_p \stackrel{!}{=} 0`
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.. and :math:`\left( \partial u / \partial T \right)_p \stackrel{!}{=} \left( \partial u / \partial T \right)_v`
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.. imply that :math:`v` is constant under all circumstances. Hence, we have to use
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.. the specific volume at reference conditions to calculate enthalpy from the
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.. integration in :math:`T` and :math:`p`. Future work could provide a more accurate
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.. formulation of entropy and enthalpy by implementing the term
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.. :math:`\left( \partial v / \partial T \right)_p \neq 0`.
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..
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.. Using only polynomials for the heat capacity functions, we can derive internal
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.. energy and entropy by integrating the specific heat capacity in temperature.
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.. _BaseValue:
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.. math::
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c &= \sum_{i=0}^n x^i \cdot \sum_{j=0}^m C_{c}[i,j] \cdot T^j \text{ yielding } \\
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u &= \int_{0}^{1} c\left( x,T \right) dT
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= \sum_{i=0}^n x^i \cdot \sum_{j=0}^m \frac{1}{j+1} \cdot C_{c}[i,j]
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\cdot \left( T_{1}^{j+1} - T_{0}^{j+1} \right) \text{ and } \\
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s &= \int_{0}^{1} \frac{c\left( x,T \right)}{T} dT
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= \sum_{i=0}^n x^i \cdot \left(
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C_{c}[i,0] \cdot \ln\left(\frac{T_{1}}{T_{0}}\right)
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+ \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)
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\right) \\
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h &= u + v_{0} \cdot \left( p_{1} - p_{0} \right)
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.. note::
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The internal routines for the incompressibles were updated 2015-02-10, the documentation is not fully updated.
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We are going to add the new equation as soon as possible, probably mid-March 2015. Please be patient.
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.. .. math::
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..
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.. c &= \sum_{i=0}^n x^i \cdot \sum_{j=0}^m C_{c}[i,j] \cdot T^j \text{ yielding } \\
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.. u &= \int_{0}^{1} c\left( x,T \right) dT
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.. = \sum_{i=0}^n x^i \cdot \sum_{j=0}^m \frac{1}{j+1} \cdot C_{c}[i,j]
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.. \cdot \left( T_{1}^{j+1} - T_{0}^{j+1} \right) \text{ and } \\
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.. s &= \int_{0}^{1} \frac{c\left( x,T \right)}{T} dT
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.. = \sum_{i=0}^n x^i \cdot \left(
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.. C_{c}[i,0] \cdot \ln\left(\frac{T_{1}}{T_{0}}\right)
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.. + \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)
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.. \right) \\
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.. h &= u + v_{0} \cdot \left( p_{1} - p_{0} \right)
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..
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According to Melinder :cite:`Melinder2010` and Skovrup :cite:`Skovrup2013`,
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using a centred approach for the independent variables enhances the fit quality.
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@@ -278,10 +282,9 @@ be multiplied with the other coefficients and the concentration.
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.. math::
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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}) \\
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\int_{0}^{1} \left( \frac{\partial s}{\partial T} \right)_p dT &= \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}) \\
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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
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.. _Equations:
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Equations
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Ian Bell