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CoolProp 8.0.0
An open-source fluid property and humid air property database
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Definition at line 8 of file TransportRoutines.h.
#include <TransportRoutines.h>
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Definition at line 972 of file TransportRoutines.cpp.
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Definition at line 784 of file TransportRoutines.cpp.
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Definition at line 779 of file TransportRoutines.cpp.
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The simplified critical conductivity term of Olchowy and Sengers.
Olchowy, G. A. & Sengers, J. V. (1989), "A Simplified Representation for the Thermal Conductivity of Fluids in the Critical Region", International Journal of Thermophysics, 10, (2), 417-426
\[ \lambda^{(c)} = \frac{\rho c_p R_DkT}{6\pi\eta\zeta}(\Omega-\Omega_0) \]
\[ \Omega = \frac{2}{\pi}\left[ \left( \frac{c_p-c_v}{c_p}\right)\arctan(q_d\zeta)+\frac{c_v}{c_p}q_d\zeta \right] \]
\[ \Omega_0 = \frac{2}{\pi}\left[1-\exp\left(-\frac{1}{(q_d\zeta)^{-1}+(q_d\zeta\rho_c/\rho)^2/3} \right) \right] \]
\[ \zeta = \zeta_0\left(\frac{p_c\rho}{\Gamma\rho_c^2}\right)^{\nu/\gamma}\left[\left.\frac{\partial \rho(T,\rho)}{\partial p} \right|_{T}- \frac{T_R}{T}\left.\frac{\partial \rho(T_R,\rho)}{\partial p} \right|_{T} \right]^{\nu/\gamma}, \]
where \(\lambda^{(c)}\) is in W \(\cdot\)m \(^{-1}\) \(\cdot\)K \(^{-1}\), \(\zeta\) is in m, \(c_p\) and \(c_v\) are in J \(\cdot\)kg \(^{-1}\cdot\)K \(^{-1}\), \(p\) and \(p_c\) are in Pa, \(\rho\) and \(\rho_c\) are in mol \(\cdot\)m \(^{-3}\), \(\eta\) is the viscosity in Pa \(\cdot\)s, and the remaining parameters are defined in the following tables.
It should be noted that some authors use slightly different values for the "universal" constants
Coefficients for use in the simplified Olchowy-Sengers critical term
| Parameter | Variable | Value |
|---|---|---|
| Boltzmann constant | \(k\) | \(1.3806488\times 10^{-23}\) J \(\cdot\)K \(^{-1}\) |
| Universal amplitude | \(R_D\) | 1.03 |
| Critical exponent | \(\nu\) | 0.63 |
| Critical exponent | \(\gamma\) | 1.239 |
| Reference temperature | \(T_R\) | 1.5 \(T_c\) |
Recommended default constants (see Huber (I&ECR, 2003))
| Parameter | Variable | Value |
|---|---|---|
| Amplitude | \(\Gamma\) | 0.0496 |
| Amplitude | \(\zeta_0\) | 1.94 \(\times\) 10 \(^{-10}\) m |
| Effective cutoff | \(q_d\) | 2 \(\times\) 10 \(^{9}\) m |
Definition at line 727 of file TransportRoutines.cpp.
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Definition at line 850 of file TransportRoutines.cpp.
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Definition at line 798 of file TransportRoutines.cpp.
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Definition at line 830 of file TransportRoutines.cpp.
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Definition at line 840 of file TransportRoutines.cpp.
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The general dilute gas conductivity term formed of a ratio of polynomial like terms.
\[ \lambda^0 = \frac{A_i\displaystyle\sum_iT_r^{n_i}}{B_i\displaystyle\sum_iT_r^{m_i}} \]
with \(\lambda^0\) in W/m/K, T_r is the reduced temperature \(T_{r} = T/T_{red}\)
Definition at line 677 of file TransportRoutines.cpp.
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Definition at line 1294 of file TransportRoutines.cpp.
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Definition at line 865 of file TransportRoutines.cpp.
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dh/dx derived using sympy:
E1,x,x0,E2,beta,gamma = symbols('E1,x,x0,E2,beta,gamma')
h = E1*(1 + x/x0)*pow(1 + E2*pow(1 + x/x0, 2/beta), (gamma-1)/(2*beta))
ccode(simplify(diff(h,x)))
Definition at line 1005 of file TransportRoutines.cpp.
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Definition at line 1063 of file TransportRoutines.cpp.
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Definition at line 945 of file TransportRoutines.cpp.
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Definition at line 882 of file TransportRoutines.cpp.
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This term is given by
\[ \Delta\lambda(\rho,T) = \displaystyle\sum_iA_i\tau^{t,i}\delta^{d_i} \]
As used by Assael, Perkins, Huber, etc., the residual term is given by
\[ \Delta\lambda(\rho,T) = \displaystyle\sum_i(B_{1,i}+B_{2,i}(T/T_c))(\rho/\rho_c)^i \]
which can be easily converted by noting that \(\tau=Tc/T\) and \(\delta=\rho/\rho_c\)
Definition at line 696 of file TransportRoutines.cpp.
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Definition at line 711 of file TransportRoutines.cpp.
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Definition at line 1142 of file TransportRoutines.cpp.
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Definition at line 317 of file TransportRoutines.cpp.
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Implement the method of:
Chung, Ting Horng, et al. "Generalized multiparameter correlation for nonpolar and polar fluid transport properties." Industrial & engineering chemistry research 27(4) (1988): 671-679.
Definition at line 630 of file TransportRoutines.cpp.
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Definition at line 347 of file TransportRoutines.cpp.
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Definition at line 600 of file TransportRoutines.cpp.
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The dilute gas viscosity term that is based on collision integral or effective cross section.
\[ \eta^0 = \displaystyle\frac{A\sqrt{MT}}{\sigma^2\mathfrak{S}(T^*)} \]
\[ \mathfrak{S}(T^*)=\exp\left(\sum_ia_i[\ln T^*]^{t_i}\right) \]
with \(T^* = \frac{T}{\varepsilon/k}\) and \(\sigma\) in nm, M is in kg/kmol. Yields viscosity in Pa-s.
Both the collision integral \(\mathfrak{S}^*\) and effective cross section \(\Omega^{(2,2)}\) have the same form, in general we don't care which is used. The are related through \(\Omega^{(2,2)} = (5/4)\mathfrak{S}^*\) see Vesovic(JPCRD, 1990) for CO \(_2\) for further information
Definition at line 28 of file TransportRoutines.cpp.
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Definition at line 87 of file TransportRoutines.cpp.
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Definition at line 593 of file TransportRoutines.cpp.
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Definition at line 581 of file TransportRoutines.cpp.
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The general dilute gas viscosity from used for ECS.
\[ \eta^0 = \displaystyle\frac{26.692\times 10^{-9}\sqrt{MT}}{\sigma^2\Omega^{(2,2)}(T^*)} \]
\[ \Omega^{(2,2)}(T^*)=1.16145(T^*)^{-0.14874}+0.52487\exp(-0.77320T^*)+2.16178\exp(-2.43787T^*) \]
with \(T^* = \frac{T}{\varepsilon/k}\) and \(\sigma\) in nm, M is in kg/kmol. Yields viscosity in Pa-s.
Definition at line 9 of file TransportRoutines.cpp.
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A dilute gas viscosity term formed of summation of power terms.
\[ \eta^0 = \displaystyle\sum_ia_iT^{t_i} \]
with T in K, \(\eta^0\) in Pa-s
Definition at line 57 of file TransportRoutines.cpp.
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A dilute gas viscosity term formed of summation of power terms of the reduced temperature.
\[ \eta^0 = \displaystyle\sum_ia_i(T/T_c)^{t_i} \]
with T in K, \(\eta^0\) in Pa-s
Definition at line 72 of file TransportRoutines.cpp.
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Calculate the viscosity using the extended corresponding states method.
This method is covered in depth in
Bell, I. H.; Wronski, J.; Quoilin, S. & Lemort, V. (2014), Pure and Pseudo-pure Fluid Thermophysical Property Evaluation and the Open-Source Thermophysical Property Library CoolProp, Industrial & Engineering Chemistry Research, 53, (6), 2498-2508
which is originally based on the methods presented in
Huber, M. L., Laesecke, A. and Perkins, R. A., (2003), Model for the Viscosity and Thermal Conductivity of Refrigerants, Including a New Correlation for the Viscosity of R134a, Industrial & Engineering Chemistry Research, v. 42, pp. 3163-3178
and
McLinden, M. O.; Klein, S. A. & Perkins, R. A. (2000), An extended corresponding states model for the thermal conductivity of refrigerants and refrigerant mixtures, Int. J. Refrig., 23, 43-63
Definition at line 1203 of file TransportRoutines.cpp.
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Definition at line 614 of file TransportRoutines.cpp.
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Definition at line 235 of file TransportRoutines.cpp.
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Definition at line 403 of file TransportRoutines.cpp.
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From Michailidou-JPCRD-2014-Heptane
Definition at line 336 of file TransportRoutines.cpp.
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Definition at line 324 of file TransportRoutines.cpp.
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Higher-order viscosity term from friction theory of Sergio Quinones-Cisneros.
Several functional forms have been proposed and this function attempts to handle all of them
\[ \eta_{HO} = \kappa_ap_a + \kappa_r\Delta p_r + \kappa_i p_{id}+\kappa_{aa}p_a^2 + \kappa_{drdr}\Delta p_r^2 + \kappa_{rr}p_{r}^2 + \kappa_{ii}p_{id}^2 +\kappa_{rrr}p_r^3 + \kappa_{aaa}p_a^3 \]
Watch out that sometimes it is \(\Delta p_r\) and other times it is \(p_r\)!
1e5 for conversion from Pa -> bar
\[ p_r = T \frac{\partial p}{\partial T}\right|_{\rho}/1e5 \]
\[ p_a = p - p_r \]
\[ p_{id} = \rho R T / 1e5 \]
\[ \Delta p_r = p_r - p_{id} \]
\[ \psi_1 = \exp(\tau)-c_1 \]
\[ \psi_2 = \exp(\tau^2)-c_2 \]
\[ \kappa_i = (A_{i,0} + A_{i,1}\psi_1 + A_{i,2}\psi_2)\tau \]
\[ \kappa_a = (A_{a,0} + A_{a,1}\psi_1 + A_{a,2}\psi_2)\tau^{N_a} \]
\[ \kappa_{aa} = (A_{aa,0} + A_{aa,1}\psi_1 + A_{aa,2}\psi_2)\tau^{N_{aa}} \]
\[ \kappa_r = (A_{r,0} + A_{r,1}\psi_1 + A_{r,2}\psi_2)\tau^{N_r} \]
\[ \kappa_{rr} = (A_{rr,0} + A_{rr,1}\psi_1 + A_{rr,2}\psi_2)\tau^{N_{rr}} \]
\[ \kappa_{drdr} = (A_{drdr,0} + A_{drdr,1}\psi_1 + A_{drdr,2}\psi_2)\tau^{N_{drdr}} \]
\[ \kappa_{aa} = (A_{aa,0} + F_{Aaa,1}\psi_1 + F.Aaa[2]\psi_2)\tau^{N_{aa}} \]
\[ \kappa_{rrr} = (A_{rrr,0} + A_{rrr,1}\psi_1 + A_{rrr,2}\psi_2)\tau^{N_{rrr}} \]
\[ \kappa_{aaa} = (A_{aaa,0} + A_{aaa,1}\psi_1 + A_{aaa,2}\psi_2)\tau^{N_{aaa}} \]
| HEOS | The instance to use |
Definition at line 358 of file TransportRoutines.cpp.
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The modified Batschinski-Hildebrand contribution to the viscosity.
\[ \Delta\eta = \displaystyle\sum_{i}a_{i}\delta^{d1_i}\tau^{t1_j}\exp(\gamma_i\delta^{l_i})+\left(\displaystyle\sum_{i}f_i\delta^{d2_i}\tau^{t2_i}\right)\left(\frac{1}{\delta_0(\tau)-\delta}-\frac{1}{\delta_0(\tau)}\right) \]
where \(\tau = T_c/T\) and \(\delta = \rho/\rho_c\)
\[ \delta_0(\tau) = \displaystyle\frac{\displaystyle\sum_{i}g_i\tau^{h_i}}{\displaystyle\sum_{i}p_i\tau^{q_i}} \]
The more general form of \(\delta_0(\tau)\) is selected in order to be able to handle all the forms in the literature
Definition at line 103 of file TransportRoutines.cpp.
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Definition at line 312 of file TransportRoutines.cpp.
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An empirical form for the initial density dependence.
Given by the polynomial-like form
\[ \eta^1 = \sum_i n_i\delta^{d_i}\tau^{t_i} \]
where the output is in Pa-s
Definition at line 162 of file TransportRoutines.cpp.
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The initial density dependence term \(B_{\eta}\) from Rainwater-Friend theory.
The total contribution from this term is given by
\[ \eta_{RF} = \eta_0B_{\eta}\rho \]
where \(\eta_0\) is the dilute gas viscosity in Pa-s and \(\rho\) is the molar density in mol/m \(^3\) and \(B_{\eta}\) is in m^3/mol.
\[ B_{\eta}(T) = B_{\eta}^*(T^*)N_A\sigma_{\eta}^3 \]
where \(N_A\) is Avogadros number \(6.022\times 10^{23}\) mol \(^{-1}\) and \(\sigma_{\eta}\) is in m.
\[ B_{\eta}^*(T^*) = \displaystyle\sum_ib_i(T^*)^{t_i} \]
IMPORTANT: This function returns \(B_{\eta}\), not \(\eta_{RF}\)
Definition at line 140 of file TransportRoutines.cpp.
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Definition at line 541 of file TransportRoutines.cpp.
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Viscosity hardcoded for Methanol.
From Xiang et al., A New Reference Correlation for the Viscosity of Methanol, J. Phys. Chem. Ref. Data, Vol. 35, No. 4, 2006
Definition at line 440 of file TransportRoutines.cpp.
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Definition at line 520 of file TransportRoutines.cpp.
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Definition at line 562 of file TransportRoutines.cpp.
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Definition at line 498 of file TransportRoutines.cpp.
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Definition at line 1266 of file TransportRoutines.cpp.
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Definition at line 305 of file TransportRoutines.cpp.
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Definition at line 252 of file TransportRoutines.cpp.
1.9.4