FPROPS/Thermal conductivity - Revision history

Jpye: /* Critical part */

2014-03-27T08:12:02Z

Critical part

← Older revision		Revision as of 08:12, 27 March 2014
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	== Critical part ==		== Critical part ==

	The approach attributed to Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data, although the equations presented in each of these references appear somewhat different (Specifically, in Lemmon and Jacobsen, <math>q_D</math> has replaced <math>{\tilde q}_D^{-1}</math> from Vesovic et al, and <math>\tilde\chi</math> has been divided by <math>T/T_c</math>). Using the formulae of Lemmon and Jacobsen, the following parameters are required as correlation data:		The approach attributed to Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data, although the equations presented in each of these references appear somewhat different (Specifically, in Lemmon and Jacobsen, <math>q_D</math> has replaced <math>{\tilde q}_D^{-1}</math> from Vesovic et al, and the function <math>\tilde\chi \left(\rho,T\right)</math> has been divided by <math>T/T_c</math>). Using the formulae of Lemmon and Jacobsen, the following parameters are required as correlation data:

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Jpye: /* Critical part */

2014-03-27T08:10:10Z

Critical part

← Older revision		Revision as of 08:10, 27 March 2014
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	== Critical part ==		== Critical part ==

	The approach attributed to Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data, although the equations presented in each of these references appear somewhat different. Using the formulae of Lemmon and Jacobsen, the following parameters are required as correlation data:		The approach attributed to Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data, although the equations presented in each of these references appear somewhat different (Specifically, in Lemmon and Jacobsen, <math>q_D</math> has replaced <math>{\tilde q}_D^{-1}</math> from Vesovic et al, and <math>\tilde\chi</math> has been divided by <math>T/T_c</math>). Using the formulae of Lemmon and Jacobsen, the following parameters are required as correlation data:

	{\| border="1" cellpadding="3" cellspacing="0"		{\| border="1" cellpadding="3" cellspacing="0"

Jpye: /* Critical part */

2014-03-27T08:01:29Z

Critical part

@@ Line 60: / Line 60: @@
 == Critical part ==
-This part is not yet implemented, but it is noted that a common form attributed to Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data.
+The approach attributed to Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data, although the equations presented in each of these references appear somewhat different. Using the formulae of Lemmon and Jacobsen, the following parameters are required as correlation data:
 == References ==
 <references/>

Jpye: /* Ideal part */

2014-03-25T07:52:01Z

Ideal part

← Older revision		Revision as of 07:52, 25 March 2014
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	with <math>\tau = T^\ast / T</math>, and <math>T^\ast</math> is a scaling temperature provided as part of the correlation data. The specific correlation above would be writen with <math>T^\ast = \epsilon / k</math> hence <math>\tau = 1 / T^\star</math>.		with <math>\tau = T^\ast / T</math>, and <math>T^\ast</math> is a scaling temperature provided as part of the correlation data. The specific correlation above would be writen with <math>T^\ast = \epsilon / k</math> hence <math>\tau = 1 / T^\star</math>.

	(Note that the reduced effective cross section is dimensionless, and is reduced from the 'real' effective cross-section area using ~~<math>~~<math>\mathcal{C}_\lambda^\ast = \frac{\mathcal{C}_\lambda}{\pi \sigma^2}</math>.		(Note that the reduced effective cross section is dimensionless, and is reduced from the 'real' effective cross-section area using <math>\mathcal{C}_\lambda^\ast = \frac{\mathcal{C}_\lambda}{\pi \sigma^2}</math>.)

	Note that one publication<ref name="Lemmon2004"/> provides the function <math>\lambda_0</math> in the form		Note that one publication<ref name="Lemmon2004"/> provides the function <math>\lambda_0</math> in the form

Jpye: /* Ideal part */

2014-03-25T07:51:41Z

Ideal part

← Older revision		Revision as of 07:51, 25 March 2014
Line 30:		Line 30:

	with <math>\tau = T^\ast / T</math>, and <math>T^\ast</math> is a scaling temperature provided as part of the correlation data. The specific correlation above would be writen with <math>T^\ast = \epsilon / k</math> hence <math>\tau = 1 / T^\star</math>.		with <math>\tau = T^\ast / T</math>, and <math>T^\ast</math> is a scaling temperature provided as part of the correlation data. The specific correlation above would be writen with <math>T^\ast = \epsilon / k</math> hence <math>\tau = 1 / T^\star</math>.

			(Note that the reduced effective cross section is dimensionless, and is reduced from the 'real' effective cross-section area using <math><math>\mathcal{C}_\lambda^\ast = \frac{\mathcal{C}_\lambda}{\pi \sigma^2}</math>.

	Note that one publication<ref name="Lemmon2004"/> provides the function <math>\lambda_0</math> in the form		Note that one publication<ref name="Lemmon2004"/> provides the function <math>\lambda_0</math> in the form

Jpye: /* Critical part */

2014-03-25T07:43:45Z

Critical part

← Older revision		Revision as of 07:43, 25 March 2014
Line 58:		Line 58:
	== Critical part ==		== Critical part ==

	This part is not yet implemented, but it is noted that a common form attributed by Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data.		This part is not yet implemented, but it is noted that a common form attributed to Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data.


	== References ==		== References ==
	<references/>		<references/>

Jpye at 07:43, 25 March 2014

2014-03-25T07:43:32Z

← Older revision		Revision as of 07:43, 25 March 2014
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	This form can be accommodated by the form of Lemmon and Jacobsen by setting <math>N_i = \mathrm{d}_i</math>, <math>d_i = i</math>, and <math>t_i</math> and <math>l_i</math> to zero, and <math>\rho^\ast</math> and <math>T^\ast</math> to one.		This form can be accommodated by the form of Lemmon and Jacobsen by setting <math>N_i = \mathrm{d}_i</math>, <math>d_i = i</math>, and <math>t_i</math> and <math>l_i</math> to zero, and <math>\rho^\ast</math> and <math>T^\ast</math> to one.

			== Critical part ==

			This part is not yet implemented, but it is noted that a common form attributed by Olchowy and Sengers is used in both the carbon dioxide data<ref name="Vesovic1990"/> and nitrogen<ref name=Lemmon2004/> data.


	== References ==		== References ==
	<references/>		<references/>

Jpye: /* Ideal part */

2014-03-25T07:40:14Z

Ideal part

← Older revision		Revision as of 07:40, 25 March 2014
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	Note that one publication<ref name="Lemmon2004"/> provides the function <math>\lambda_0</math> in the form		Note that one publication<ref name="Lemmon2004"/> provides the function <math>\lambda_0</math> in the form

	:<math>\lambda_0 = N_1 \frac{\mu_0\left(T\right)}{1 \, \mathrm{\mu Pa \cdot s}} + N_2 \tau^{t_2} + N_3 \tau^{t_3}</math>		:<math>\frac{\lambda_0}{1\,\mathrm{\mu Pa \cdot s}} = N_1 \frac{\mu_0\left(T\right)}{1\,\mathrm{\mu Pa \cdot s}} + N_2 \tau^{t_2} + N_3 \tau^{t_3}</math>

	and this form does not comply with the above ratio of power series, since zero-density viscosity is, in turn, defined as a ratio of power series (see [[FPROPS/Viscosity]]). This form could be accommodated by 'flagging' one of the terms in the numerator power series in a special way, and providing a pointer to the appropriate <math>\mu_0</math> data/function, perhaps.		and this form does not comply with the above ratio of power series, since zero-density viscosity is, in turn, defined as a ratio of power series (see [[FPROPS/Viscosity]]). This form could be accommodated by 'flagging' one of the terms in the numerator power series in a special way, and providing a pointer to the appropriate <math>\mu_0</math> data/function, perhaps.

Jpye: /* Ideal part */

2014-03-25T07:39:30Z

Ideal part

← Older revision		Revision as of 07:39, 25 March 2014
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	The ideal (zero-density limit) part has been expressed in several places<ref name="Vesovic1990">V Vesovic, W A Wakeham, G A Olchowy, J V Sengers, J T R Watson and J Millat, 1990. [http://www.nist.gov/data/PDFfiles/jpcrd391.pdf The Transport Properties of Carbon Dioxide], J Phys Chem Ref Data '''19''', 763. {{doi\|10.1063/1.555875}}.</ref><ref name="Lemmon2004">E W Lemmon and R T Jacobsen, 2004. [http://www.boulder.nist.gov/div838/theory/refprop/NAO.PDF Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air] Int J Thermophys '''25'''. {{doi\|10.1023/B:IJOT.0000022327.04529.f3}}.</ref><ref name="Vesovic1994">V Vesovic, 1994. "On Correlating the Transport Properties of Supercritical Fluids", in ''Supercritical Fluids: Fundamentals for Application'', Springer, pp 273-283. {{doi\|10.1007/978-94-015-8295-7_10}}. ''(Note the error in equation (6) in this reference.)''</ref> as:		The ideal (zero-density limit) part has been expressed in several places<ref name="Vesovic1990">V Vesovic, W A Wakeham, G A Olchowy, J V Sengers, J T R Watson and J Millat, 1990. [http://www.nist.gov/data/PDFfiles/jpcrd391.pdf The Transport Properties of Carbon Dioxide], J Phys Chem Ref Data '''19''', 763. {{doi\|10.1063/1.555875}}.</ref><ref name="Lemmon2004">E W Lemmon and R T Jacobsen, 2004. [http://www.boulder.nist.gov/div838/theory/refprop/NAO.PDF Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air] Int J Thermophys '''25'''. {{doi\|10.1023/B:IJOT.0000022327.04529.f3}}.</ref><ref name="Vesovic1994">V Vesovic, 1994. "On Correlating the Transport Properties of Supercritical Fluids", in ''Supercritical Fluids: Fundamentals for Application'', Springer, pp 273-283. {{doi\|10.1007/978-94-015-8295-7_10}}. ''(Note the error in equation (6) in this reference.)''</ref> as:

	:<math>\frac{\lambda_0\left(T\right)}{1\,\mathrm{\~~muPa~~\cdot s}} = \frac{0.177568 \sqrt{T} \left(c_p^0/R\right)}{\sigma^2 \sqrt{M} \; \mathcal{C}_\lambda^\ast\left(T^\star\right)}</math>.		:<math>\frac{\lambda_0\left(T\right)}{1\,\mathrm{\mu Pa\cdot s}} = \frac{0.177568 \sqrt{T} \left(c_p^0/R\right)}{\sigma^2 \sqrt{M} \; \mathcal{C}_\lambda^\ast\left(T^\star\right)}</math>.

	Here, <math>c_p^0/R</math> is the isobaric heat capacity divided by the gas constant (either both specific or both molar), <math>\sigma</math> is length scaling parameter (in units of nm, which is messy), a constant particular to the fluid in question, and <math>M</math> is relative molecular mass of the fluid.		Here, <math>c_p^0/R</math> is the isobaric heat capacity divided by the gas constant (either both specific or both molar), <math>\sigma</math> is length scaling parameter (in units of nm, which is messy), a constant particular to the fluid in question, and <math>M</math> is relative molecular mass of the fluid.

Jpye: /* Ideal part */

2014-03-25T07:39:19Z

Ideal part

← Older revision		Revision as of 07:39, 25 March 2014
Line 9:		Line 9:
	The ideal (zero-density limit) part has been expressed in several places<ref name="Vesovic1990">V Vesovic, W A Wakeham, G A Olchowy, J V Sengers, J T R Watson and J Millat, 1990. [http://www.nist.gov/data/PDFfiles/jpcrd391.pdf The Transport Properties of Carbon Dioxide], J Phys Chem Ref Data '''19''', 763. {{doi\|10.1063/1.555875}}.</ref><ref name="Lemmon2004">E W Lemmon and R T Jacobsen, 2004. [http://www.boulder.nist.gov/div838/theory/refprop/NAO.PDF Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air] Int J Thermophys '''25'''. {{doi\|10.1023/B:IJOT.0000022327.04529.f3}}.</ref><ref name="Vesovic1994">V Vesovic, 1994. "On Correlating the Transport Properties of Supercritical Fluids", in ''Supercritical Fluids: Fundamentals for Application'', Springer, pp 273-283. {{doi\|10.1007/978-94-015-8295-7_10}}. ''(Note the error in equation (6) in this reference.)''</ref> as:		The ideal (zero-density limit) part has been expressed in several places<ref name="Vesovic1990">V Vesovic, W A Wakeham, G A Olchowy, J V Sengers, J T R Watson and J Millat, 1990. [http://www.nist.gov/data/PDFfiles/jpcrd391.pdf The Transport Properties of Carbon Dioxide], J Phys Chem Ref Data '''19''', 763. {{doi\|10.1063/1.555875}}.</ref><ref name="Lemmon2004">E W Lemmon and R T Jacobsen, 2004. [http://www.boulder.nist.gov/div838/theory/refprop/NAO.PDF Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air] Int J Thermophys '''25'''. {{doi\|10.1023/B:IJOT.0000022327.04529.f3}}.</ref><ref name="Vesovic1994">V Vesovic, 1994. "On Correlating the Transport Properties of Supercritical Fluids", in ''Supercritical Fluids: Fundamentals for Application'', Springer, pp 273-283. {{doi\|10.1007/978-94-015-8295-7_10}}. ''(Note the error in equation (6) in this reference.)''</ref> as:

	:<math>\lambda_0\left(T\right) = \frac{0.177568 \sqrt{T} \left(c_p^0/R\right)}{\sigma^2 \sqrt{M} \; \mathcal{C}_\lambda^\ast\left(T^\star\right)}</math>.		:<math>\frac{\lambda_0\left(T\right)}{1\,\mathrm{\muPa\cdot s}} = \frac{0.177568 \sqrt{T} \left(c_p^0/R\right)}{\sigma^2 \sqrt{M} \; \mathcal{C}_\lambda^\ast\left(T^\star\right)}</math>.

	Here, <math>c_p^0/R</math> is the isobaric heat capacity divided by the gas constant (either both specific or both molar), <math>\sigma</math> is length scaling parameter (in units of nm, which is messy), a constant particular to the fluid in question, and <math>M</math> is relative molecular mass of the fluid.		Here, <math>c_p^0/R</math> is the isobaric heat capacity divided by the gas constant (either both specific or both molar), <math>\sigma</math> is length scaling parameter (in units of nm, which is messy), a constant particular to the fluid in question, and <math>M</math> is relative molecular mass of the fluid.