Thermodynamics with ASCEND - Revision history

Jpye: /* Example 5 */

2012-04-10T00:33:54Z

Example 5

← Older revision		Revision as of 00:33, 10 April 2012
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	! <math>y_w^L</math>		! <math>y_w^L</math>
	! <math>y_w^V</math>		! <math>y_w^V</math>
	! <~~span class="texhtml"~~>''T'' / [''K'']~~</span>~~		! <math>T</math> / [K]
	\|-		\|-
	\| 0.99		\| 0.99

Jpye: /* Calculating properties for a single component */

2012-04-10T00:32:10Z

Calculating properties for a single component

← Older revision		Revision as of 00:32, 10 April 2012
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	The ASCEND thermodynamics library has two models for vapour components ('''<tt>ideal_vapour_component</tt>''' and '''<tt>Pitzer_vapor_component</tt>''') and one model for liquid components ('''<tt>Rackett_liquid_component</tt>'''). All these models are similar in concept; only the EOS changes. We'll use the ideal vapour model to illustrate the main concepts.		The ASCEND thermodynamics library has two models for vapour components ('''<tt>ideal_vapour_component</tt>''' and '''<tt>Pitzer_vapor_component</tt>''') and one model for liquid components ('''<tt>Rackett_liquid_component</tt>'''). All these models are similar in concept; only the EOS changes. We'll use the ideal vapour model to illustrate the main concepts.

	The model defines the equations needed to calculate the molar volume <math>{\bar v}</math>, enthalpy <math>{\bar h}</math> and Gibbs free energy <math>{\bar g}</math>, based on the temperature <~~span class="texhtml"~~>''T''</~~span~~>, pressure <~~span class="texhtml"~~>''p''</~~span~~> and component properties.		The model defines the equations needed to calculate the molar volume <math>{\bar v}</math>, enthalpy <math>{\bar h}</math> and Gibbs free energy <math>{\bar g}</math>, based on the temperature <math>T</math>, pressure <math>p</math> and component properties.

	The first equation needed by the ideal gas model is the well known equation of state,		The first equation needed by the ideal gas model is the well known equation of state,
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	{{eq\|:\|<math>\bar g = {\bar g}(p,T)</math>\|6}}		{{eq\|:\|<math>\bar g = {\bar g}(p,T)</math>\|6}}

	As a summary, this model has three equations ({{eqref\|1}}, {{eqref\|4}}, {{eqref\|6}}), and five variables (<~~span class="texhtml"~~>''p''</~~span~~>, ''T'', <math>\bar g</math>, <math>\bar h</math>, <math>\bar v</math>).		As a summary, this model has three equations ({{eqref\|1}}, {{eqref\|4}}, {{eqref\|6}}), and five variables (<math>p</math>, <math>T</math>, <math>\bar g</math>, <math>\bar h</math>, <math>\bar v</math>).

	=== ASCEND implementation ===		=== ASCEND implementation ===

Jpye: /* Example 4 */

2012-04-09T18:21:13Z

Example 4

← Older revision		Revision as of 18:21, 9 April 2012
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	END howto_thermo_ex8;</source>		END howto_thermo_ex8;</source>

	Use of the thermodynamics model is slightly more complex. First, you should define the components data, then you choose the models for the phases with the <tt>phases_data</tt> model. This model's first parameter is a phase definition (it can be '<tt>V</tt>', '<tt>L</tt>' or '<tt>VL</tt>'), then the name for the vapour model (or <tt>none</tt> if there's not vapour phase), and then the liquid model name. The last parameter is for a second liquid phase model when <tt>LL</tt> equilibrium is implemented; for now, you should just use <tt>none</tt>.		Use of the thermodynamics model is slightly more complex. First, you should define the components data, then you choose the models for the phases with the <tt>phases_data</tt> model. This model's first parameter is a phase definition (it can be <tt>'V'</tt>, <tt>'L'</tt> or <tt>'VL'</tt>), then the name for the vapour model (or <tt>none</tt> if there's not vapour phase), and then the liquid model name. The last parameter is for a second liquid phase model when <tt>'LL'</tt> equilibrium is implemented; for now, you should just use <tt>none</tt>.

	The <tt>select_mixture_type</tt> model builds the chosen models so that they can be sent to the main thermodynamic model as a parameter. This may seem complicated at first, but as you probable will never need to change that piece of code, you don't need to worry about it too much. Just peek at the library code when you feel brave enough.		The <tt>select_mixture_type</tt> model builds the chosen models so that they can be sent to the main thermodynamic model as a parameter. This may seem complicated at first, but as you probable will never need to change that piece of code, you don't need to worry about it too much. Just peek at the library code when you feel brave enough.

Jpye: /* Example 4 */

2012-04-09T18:20:33Z

Example 4

← Older revision		Revision as of 18:20, 9 April 2012
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	END howto_thermo_ex8;</source>		END howto_thermo_ex8;</source>

	Use of the thermodynamics model is slightly more complex. First, you should define the components data, then you choose the models for the phases with the <tt>phases_data</tt> model. This model's first parameter is a phase definition (it can be <tt>V</tt>, <tt>L</tt>~~<~~ or <tt>VL</tt>), then the name for the vapour model (or <tt>none</tt> if there's not vapour phase), and then the liquid model name. The last parameter is for a second liquid phase model when <tt>LL</tt> equilibrium is implemented; for now, you should just use <tt>none</tt>.		Use of the thermodynamics model is slightly more complex. First, you should define the components data, then you choose the models for the phases with the <tt>phases_data</tt> model. This model's first parameter is a phase definition (it can be '<tt>V</tt>', '<tt>L</tt>' or '<tt>VL</tt>'), then the name for the vapour model (or <tt>none</tt> if there's not vapour phase), and then the liquid model name. The last parameter is for a second liquid phase model when <tt>LL</tt> equilibrium is implemented; for now, you should just use <tt>none</tt>.

	The <tt>select_mixture_type</tt> model builds the chosen models so that they can be sent to the main thermodynamic model as a parameter. This may seem complicated at first, but as you probable will never need to change that piece of code, you don't need to worry about it too much. Just peek at the library code when you feel brave enough.		The <tt>select_mixture_type</tt> model builds the chosen models so that they can be sent to the main thermodynamic model as a parameter. This may seem complicated at first, but as you probable will never need to change that piece of code, you don't need to worry about it too much. Just peek at the library code when you feel brave enough.

Jpye: /* Phase Equilibria */

2012-04-09T18:18:38Z

Phase Equilibria

Jpye: /* Example 5 */

2012-04-09T18:01:54Z

Example 5

← Older revision		Revision as of 18:01, 9 April 2012
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	'''Build the VL equilibrium chart for water-ethanol at atmospheric pressure.'''		'''Build the VL equilibrium chart for water-ethanol at atmospheric pressure.'''

	The model presented in the last example specifies <~~span class="texhtml"~~>''p''</~~span~~> and <~~span class="texhtml"~~>''T''</~~span~~>. To build an equilibrium chart, it's better to specify <~~span class="texhtml"~~>''p''</~~span~~> and a mole fraction, and to free <~~span class="texhtml"~~>''T''</~~span~~>. That way, we can be sure to have an equilibrium. The following method, added to the last example model, uses that trick. We'll also use the Pitzer vapour model, in the hope of improving accuracy.		The model presented in the last example specifies <math>p</math> and <math>T</math>. To build an equilibrium chart, it's better to specify <math>p</math> and a mole fraction, and to free <math>T</math>. That way, we can be sure to have an equilibrium. The following method, added to the last example model, uses that trick. We'll also use the Pitzer vapour model, in the hope of improving accuracy.

	<source lang="a4c">MODEL howto_thermo_ex9 REFINES cmumodel;		<source lang="a4c">MODEL howto_thermo_ex9 REFINES cmumodel;

Jpye: /* Relative volatility model */

2012-04-09T18:00:00Z

Relative volatility model

← Older revision		Revision as of 18:00, 9 April 2012
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	This model has extra equations for multi-phase systems in equilibrium. The first group just equates pressures and temperature on all phases to the ones in the reference phase.		This model has extra equations for multi-phase systems in equilibrium. The first group just equates pressures and temperature on all phases to the ones in the reference phase.

	{{eq\|:\|<~~span class="texhtml"~~>''p~~''<sup>''~~j~~''</sup>~~ = ''p~~''<sup>~~ref</~~sup></span~~>\|20}}		{{eq\|:\|<math>p^j = p^{ref}</math>\|20}}

	{{eq\|:\|<~~span class="texhtml"~~>''T~~''<sup>''~~j~~''</sup>~~ = ''T~~''<sup>~~ref</~~sup></span~~>\|21}}		{{eq\|:\|<math>T^j = T^{ref}</math>\|21}}

	This accounts for <span class="texhtml">''m'' − 1</span> equations where <span class="texhtml">''m''</span> is the number of phases (the reference phase is excluded).		This accounts for <span class="texhtml">''m'' − 1</span> equations where <span class="texhtml">''m''</span> is the number of phases (the reference phase is excluded).
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	Now, recall the variable <span class="texhtml">''s''</span>, a.k.a. <tt>slack_PhaseDisappearance</tt>. There is one of these for each mixture. One problem with multiphase models is that models are written thinking of the existence of a phase, and if by some thermodynamics change the phase ceases to exist the equations become invalid. By using the following equation for each phase		Now, recall the variable <span class="texhtml">''s''</span>, a.k.a. <tt>slack_PhaseDisappearance</tt>. There is one of these for each mixture. One problem with multiphase models is that models are written thinking of the existence of a phase, and if by some thermodynamics change the phase ceases to exist the equations become invalid. By using the following equation for each phase

	{{eq\|:\|<~~span class="texhtml"~~>''s~~''<sup>''~~j~~''</sup>&~~gamma~~;<sup>''~~j~~''</sup> = 0~~</~~span~~>\|22}}		{{eq\|:\|<math>s^j = \gamma^j</math>\|22}}

	It's possible for a phase to ''disappear''. If <span class="texhtml">γ<sup>''j''</sup></span> is not null, which means the phase exists, then <span class="texhtml">''s''<sup>''j''</sup></span> must be zero and by the already-presented relation <~~span class="texhtml">~~		It's possible for a phase to ''disappear''. If <span class="texhtml">γ<sup>''j''</sup></span> is not null, which means the phase exists, then <span class="texhtml">''s''<sup>''j''</sup></span> must be zero and by the already-presented relation <math>\sum_{i}{y_i + s} = 1</math> the mixture properties must be properly calculated. On the other hand, if <span class="texhtml">γ<sup>''j''</sup></span> becomes zero then <span class="texhtml">''s''<sup>''j''</sup></span> can be relaxed and the model continues to bee solved without regard for that mixture.
	~~</span~~>

	{~~\| class="wikitable"~~
	\|-
	~~\| <span style="font-size: x-large; font-family: serif;">∑</span>\| ''y''<sub>''~~i~~''</sub>~~ + ''s'' = 1\|-
	~~\| ''i''\| \|}~~

	the mixture properties must be properly calculated. On the other hand, if <span class="texhtml">γ<sup>''j''</sup></span> becomes zero then <span class="texhtml">''s''<sup>''j''</sup></span> can be relaxed and the model continues to bee solved without regard for that mixture.

	The library can be used to perform just <span class="texhtml">''T''</span> and <span class="texhtml">''p''</span> equilibrium and fractions are computed by specification of relative volatilities. This kind of equilibrium is easier to compute and can be used as a first guess to solve the full equilibrium.		The library can be used to perform just <span class="texhtml">''T''</span> and <span class="texhtml">''p''</span> equilibrium and fractions are computed by specification of relative volatilities. This kind of equilibrium is easier to compute and can be used as a first guess to solve the full equilibrium.

Jpye: /* Phase Equilibria */

2012-04-09T17:55:12Z

Phase Equilibria

Jpye: /* Mixtures */

2012-04-09T17:53:09Z

Mixtures

Jpye: /* Calculating properties for a single component */

2012-04-09T17:51:38Z

Calculating properties for a single component

← Older revision		Revision as of 18:18, 9 April 2012
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	{{eq\|:\|<math>s^j = \gamma^j</math>\|22}}		{{eq\|:\|<math>s^j = \gamma^j</math>\|22}}

	It's possible for a phase to ''disappear''. If <~~span class="texhtml"~~>&gamma~~;<sup>''~~j''</~~sup></span~~> is not null, which means the phase exists, then <span class="texhtml">''s''<sup>''j''</sup></span> must be zero and by the already-presented relation <math>\sum_{i}{y_i + s} = 1</math> the mixture properties must be properly calculated. On the other hand, if <~~span class="texhtml"~~>&gamma~~;<sup>''~~j''</~~sup></span~~> becomes zero then <~~span class="texhtml"~~>''s~~''<sup>''~~j''</~~sup></span~~> can be relaxed and the model continues to bee solved without regard for that mixture.		It's possible for a phase to ''disappear''. If <math>\gamma^j</math> is not null, which means the phase exists, then <span class="texhtml">''s''<sup>''j''</sup></span> must be zero and by the already-presented relation <math>\sum_{i}{y_i + s} = 1</math> the mixture properties must be properly calculated. On the other hand, if <math>\gamma^j</math> becomes zero then <math>s^j</math> can be relaxed and the model continues to bee solved without regard for that mixture.

	The library can be used to perform just <~~span class="texhtml"~~>''T''</~~span~~> and <~~span class="texhtml"~~>''p''</~~span~~> equilibrium and fractions are computed by specification of relative volatilities. This kind of equilibrium is easier to compute and can be used as a first guess to solve the full equilibrium.		The library can be used to perform just <math>T</math> and <math>p</math> equilibrium and fractions are computed by specification of relative volatilities. This kind of equilibrium is easier to compute and can be used as a first guess to solve the full equilibrium.

	The following set of equations is a special representation of relative volatilities		The following set of equations is a special representation of relative volatilities
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	{{eq\|:\|<math>{\bar \alpha}^j y_i^j = {\bar \alpha}_i^j y_i^{\mathrm{ref}}</math>\|23}}		{{eq\|:\|<math>{\bar \alpha}^j y_i^j = {\bar \alpha}_i^j y_i^{\mathrm{ref}}</math>\|23}}

	where <~~span class="texhtml"~~>''j'' = 1,..~~''n''<sub>~~phases~~</sub> −~~ 1</~~span~~>, and <~~span class="texhtml"~~>''i'' = 1,..~~''n''<sub>~~components</~~sub></span~~>. To make it easier to understand the goal of this representation let's see what these equations represent for a two phase system (V,L) of the two components (A,B).		where <math>j=1,..n_{\mathrm{phases}} - 1</math>, and <math>i = 1,..n_{\mathrm{components}}</math>. To make it easier to understand the goal of this representation let's see what these equations represent for a two phase system (V,L) of the two components (A,B).

	{{eq\|:\|<math>{\bar \alpha} y_A^V = \alpha_A^V y_A^L</math>\|24}}		{{eq\|:\|<math>{\bar \alpha} y_A^V = \alpha_A^V y_A^L</math>\|24}}
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	{{eq\|:\|<math>{\bar \alpha} y_B^V = \alpha_B^V y_B^L</math>\|25}}		{{eq\|:\|<math>{\bar \alpha} y_B^V = \alpha_B^V y_B^L</math>\|25}}

	in the relative volatility model you fix the <~~span class="texhtml"~~>α</~~span~~> values and compute the ~~<~~math{\bar \alpha~~}<~~/math~~>~~. Note that <math>\alpha_A^V / {\bar \alpha}</math> will be a K-value. Before seeing the model in action let's do some accounting. To simplify, we suppose a two-phase system, since ASCEND is not yet able to compute liquid-liquid models.		in the relative volatility model you fix the <math>\alpha</math> values and compute the <math>\bar \alpha</math>. Note that <math>\alpha_A^V / {\bar \alpha}</math> will be a K-value. Before seeing the model in action let's do some accounting. To simplify, we suppose a two-phase system, since ASCEND is not yet able to compute liquid-liquid models.

	'''FIXME: add the equation-counting table here.'''		'''FIXME: add the equation-counting table here.'''
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	for each partial component and each phase except the reference one.		for each partial component and each phase except the reference one.

	When using this model <math>\bar \alpha</math> is fixed and equal to one and the <~~span class="texhtml"~~>α</~~span~~>'s are released and computed by equation 23. This equation becomes a computation of K-values. Now let's see the infamous accounting table for this model.		When using this model <math>\bar \alpha</math> is fixed and equal to one and the <math>\alpha</math>'s are released and computed by equation 23. This equation becomes a computation of K-values. Now let's see the infamous accounting table for this model.

	FIXME equation-counting for VL-equilibrium G-model.		FIXME equation-counting for VL-equilibrium G-model.
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	When solving multi-phase models you should always check <tt>state.slack_PhaseDisappearance</tt>. Only if it's zero the phase exists. Sometimes you can get strange results if you're looking at properties of a non-existent phase.		When solving multi-phase models you should always check <tt>state.slack_PhaseDisappearance</tt>. Only if it's zero the phase exists. Sometimes you can get strange results if you're looking at properties of a non-existent phase.

	If you run the model, you'll get only a liquid phase and <~~span class="texhtml"~~>~~''H''~~</~~span~~> = -282.351 kJ/mol. With <~~span class="texhtml"~~>''T''</~~span~~> = 400 K, you'll get <~~span class="texhtml"~~>~~''H''~~</~~span~~> = -233.022 kJ/mol, and so <~~span class="texhtml"~~>&Delta~~;''H''~~</~~span~~> = 49.329 kJ/mol.		If you run the model, you'll get only a liquid phase and <math>\bar h</math> = -282.351 kJ/mol. With <math>T</math> = 400 K, you'll get <math>\bar h</math> = -233.022 kJ/mol, and so <math>\Delta \bar h</math> = 49.329 kJ/mol.

	=== Example 5 ===		=== Example 5 ===

← Older revision		Revision as of 17:55, 9 April 2012
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	As mixtures are models that group simple components, phase equilibria groups the same mixture at different phases. By defining <span class="texhtml">γ<sup>''j''</sup></span> to be the molar fraction of phase <span class="texhtml">''j''</span> we have the usual fraction relation:		As mixtures are models that group simple components, phase equilibria groups the same mixture at different phases. By defining <span class="texhtml">γ<sup>''j''</sup></span> to be the molar fraction of phase <span class="texhtml">''j''</span> we have the usual fraction relation:

	:<math>\sum_{i=1}^n{\gamma^j} = 1</math>		{{eq\|:\|<math>\sum_{i=1}^n{\gamma^j} = 1</math>\|15}}

	Now you must have another three equations to account for the phases:		Now you must have another three equations to account for the phases:

	:<math>{\bar v}^p = \sum_i{\gamma^i {\bar v}^i}</math>		{{eq\|:\|<math>{\bar v}^p = \sum_i{\gamma^i {\bar v}^i}</math>\|16}}

	:<math>{\bar h}^p = \sum_i{\gamma^i {\bar h}^i}</math>		{{eq\|:\|<math>{\bar h}^p = \sum_i{\gamma^i {\bar h}^i}</math>\|17}}

	:<math>{\bar g}^p = \sum_i{\gamma^i {\bar g}^i}</math>		{{eq\|:\|<math>{\bar g}^p = \sum_i{\gamma^i {\bar g}^i}</math>\|18}}

	where <math>{\bar v}^p</math>, <math>{\bar h}^p</math>, <math>{\bar g}^p</math> are the mixture properties defined by equations 8, 11 and 13 for each phase.		where <math>{\bar v}^p</math>, <math>{\bar h}^p</math>, <math>{\bar g}^p</math> are the mixture properties defined by equations 8, 11 and 13 for each phase.
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	Another group of equations (one for each component) are needed to define a global fraction for a component.		Another group of equations (one for each component) are needed to define a global fraction for a component.

	:<math>y_i = \sum_p {\gamma^p y_i^p}</math>		{{eq\|:\|<math>y_i = \sum_p {\gamma^p y_i^p}</math>\|19}}

	=== Relative volatility model ===		=== Relative volatility model ===
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	This model has extra equations for multi-phase systems in equilibrium. The first group just equates pressures and temperature on all phases to the ones in the reference phase.		This model has extra equations for multi-phase systems in equilibrium. The first group just equates pressures and temperature on all phases to the ones in the reference phase.

	:<span class="texhtml">''p''<sup>''j''</sup> = ''p''<sup>ref</sup></span>		{{eq\|:\|<span class="texhtml">''p''<sup>''j''</sup> = ''p''<sup>ref</sup></span>\|20}}

	:<span class="texhtml">''T''<sup>''j''</sup> = ''T''<sup>ref</sup></span>		{{eq\|:\|<span class="texhtml">''T''<sup>''j''</sup> = ''T''<sup>ref</sup></span>\|21}}

	This accounts for <span class="texhtml">''m'' − 1</span> equations where <span class="texhtml">''m''</span> is the number of phases (the reference phase is excluded).		This accounts for <span class="texhtml">''m'' − 1</span> equations where <span class="texhtml">''m''</span> is the number of phases (the reference phase is excluded).
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	Now, recall the variable <span class="texhtml">''s''</span>, a.k.a. <tt>slack_PhaseDisappearance</tt>. There is one of these for each mixture. One problem with multiphase models is that models are written thinking of the existence of a phase, and if by some thermodynamics change the phase ceases to exist the equations become invalid. By using the following equation for each phase		Now, recall the variable <span class="texhtml">''s''</span>, a.k.a. <tt>slack_PhaseDisappearance</tt>. There is one of these for each mixture. One problem with multiphase models is that models are written thinking of the existence of a phase, and if by some thermodynamics change the phase ceases to exist the equations become invalid. By using the following equation for each phase

	:<span class="texhtml">''s''<sup>''j''</sup>γ<sup>''j''</sup> = 0</span>		{{eq\|:\|<span class="texhtml">''s''<sup>''j''</sup>γ<sup>''j''</sup> = 0</span>\|22}}

	It's possible for a phase to ''disappear''. If <span class="texhtml">γ<sup>''j''</sup></span> is not null, which means the phase exists, then <span class="texhtml">''s''<sup>''j''</sup></span> must be zero and by the already-presented relation <span class="texhtml">		It's possible for a phase to ''disappear''. If <span class="texhtml">γ<sup>''j''</sup></span> is not null, which means the phase exists, then <span class="texhtml">''s''<sup>''j''</sup></span> must be zero and by the already-presented relation <span class="texhtml">
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	The following set of equations is a special representation of relative volatilities		The following set of equations is a special representation of relative volatilities

	:<math>{\bar \alpha}^j y_i^j = {\bar \alpha}_i^j y_i^{\mathrm{ref}}</math>		{{eq\|:\|<math>{\bar \alpha}^j y_i^j = {\bar \alpha}_i^j y_i^{\mathrm{ref}}</math>\|23}}

	where <span class="texhtml">''j'' = 1,..''n''<sub>phases</sub> − 1</span>, and <span class="texhtml">''i'' = 1,..''n''<sub>components</sub></span>. To make it easier to understand the goal of this representation let's see what these equations represent for a two phase system (V,L) of the two components (A,B).		where <span class="texhtml">''j'' = 1,..''n''<sub>phases</sub> − 1</span>, and <span class="texhtml">''i'' = 1,..''n''<sub>components</sub></span>. To make it easier to understand the goal of this representation let's see what these equations represent for a two phase system (V,L) of the two components (A,B).

	:<math>{\bar \alpha} y_A^V = \alpha_A^V y_A^L</math>		{{eq\|:\|<math>{\bar \alpha} y_A^V = \alpha_A^V y_A^L</math>\|24}}

	:<math>{\bar \alpha} y_B^V = \alpha_B^V y_B^L</math>		{{eq\|:\|<math>{\bar \alpha} y_B^V = \alpha_B^V y_B^L</math>\|25}}

	in the relative volatility model you fix the <span class="texhtml">α</span> values and compute the <math{\bar \alpha}</math>. Note that <math>\alpha_A^V / {\bar \alpha}</math> will be a K-value. Before seeing the model in action let's do some accounting. To simplify, we suppose a two-phase system, since ASCEND is not yet able to compute liquid-liquid models.		in the relative volatility model you fix the <span class="texhtml">α</span> values and compute the <math{\bar \alpha}</math>. Note that <math>\alpha_A^V / {\bar \alpha}</math> will be a K-value. Before seeing the model in action let's do some accounting. To simplify, we suppose a two-phase system, since ASCEND is not yet able to compute liquid-liquid models.
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	In the full equilibrium model, the Gibbs free energy for each phase is equated. To choose full equilibrium, you should set the variable '''<tt>equilibrated</tt>''' to <tt>TRUE</tt>.		In the full equilibrium model, the Gibbs free energy for each phase is equated. To choose full equilibrium, you should set the variable '''<tt>equilibrated</tt>''' to <tt>TRUE</tt>.

	:<math>{\bar g}_i^j = {\bar g}_i^{\mathrm{ref}}</math>		{{eq\|:\|<math>{\bar g}_i^j = {\bar g}_i^{\mathrm{ref}}</math>\|26}}

	for each partial component and each phase except the reference one.		for each partial component and each phase except the reference one.

← Older revision		Revision as of 17:53, 9 April 2012
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	For example, for an ideal gas mixture, we would have the following:		For example, for an ideal gas mixture, we would have the following:

	:<math>p {\bar v}_i^p = {\bar R} T</math>		{{eq\|:\|<math>p {\bar v}_i^p = {\bar R} T</math>\|7}}

	:<math>{\bar v}_i = {\bar v}_i^p</math>		{{eq\|:\|<math>{\bar v}_i = {\bar v}_i^p</math>\|8}}

	:<math>{\bar v} = \sum_i {y_i {\bar v}_i}</math>		{{eq\|:\|<math>{\bar v} = \sum_i {y_i {\bar v}_i}</math>\|9}}

	The first equation comes from the <tt>ideal_vapor_component</tt> model (and there is one such equation for each component in the mixture). The second equation comes from the <tt>ideal_vapor_mixture</tt> model (again, one for each component). The last one is generic for all mixture models (and is part of the <tt>phase_partials</tt> model from which all mixture models are inherited).		The first equation comes from the <tt>ideal_vapor_component</tt> model (and there is one such equation for each component in the mixture). The second equation comes from the <tt>ideal_vapor_mixture</tt> model (again, one for each component). The last one is generic for all mixture models (and is part of the <tt>phase_partials</tt> model from which all mixture models are inherited).
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	The same concepts apply to enthalpy and Gibbs free energy. The equations for ideal vapour mixing and mixture calculation are the following:		The same concepts apply to enthalpy and Gibbs free energy. The equations for ideal vapour mixing and mixture calculation are the following:

	:<math>p {\bar h}_i^p = {\bar h}_i^p</math>		{{eq\|:\|<math>p {\bar h}_i^p = {\bar h}_i^p</math>\|10}}

	:<math>{\bar h} = \sum_i {y_i {\bar h}_i}</math>		{{eq\|:\|<math>{\bar h} = \sum_i {y_i {\bar h}_i}</math>\|11}}

	:<math>{\bar g}_i = {\bar g}_i^p + {\bar R} T \ln(y_i)</math>		{{eq\|:\|<math>{\bar g}_i = {\bar g}_i^p + {\bar R} T \ln(y_i)</math>\|12}}

	:<math>{\bar g} = \sum_i {y_i {\bar g}_i}</math>		{{eq\|:\|<math>{\bar g} = \sum_i {y_i {\bar g}_i}</math>\|13}}

	This model has one extra special equation whose meaning will be easier to understand after talking about phase equilibria:		This model has one extra special equation whose meaning will be easier to understand after talking about phase equilibria:

	:<math>\sum_{i} {y_i + s = 1}</math>		{{eq\|:\|<math>\sum_{i} {y_i + s = 1}</math>\|14}}

	This equation introduces a new variable <span class="texhtml">''s''</span> called <tt>slack_PhaseDisappearance</tt>. With this formulation of the molar fraction sum, when <span class="texhtml">''s''</span> is fixed at 0, normal calculations are performed; when <span class="texhtml">''s''</span> is free, if the phase ceases to exist, <span class="texhtml">''s''</span> will be different from zero but simulation can continue. It's a clever and important trick for multi-phase equilibria. We will return to this issue later.		This equation introduces a new variable <span class="texhtml">''s''</span> called <tt>slack_PhaseDisappearance</tt>. With this formulation of the molar fraction sum, when <span class="texhtml">''s''</span> is fixed at 0, normal calculations are performed; when <span class="texhtml">''s''</span> is free, if the phase ceases to exist, <span class="texhtml">''s''</span> will be different from zero but simulation can continue. It's a clever and important trick for multi-phase equilibria. We will return to this issue later.

← Older revision		Revision as of 17:51, 9 April 2012
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	The first equation needed by the ideal gas model is the well known equation of state,		The first equation needed by the ideal gas model is the well known equation of state,

	{{~~eqn~~\|:\|<math>p{\bar v}={\bar R}T</math>\|1}}		{{eq\|:\|<math>p{\bar v}={\bar R}T</math>\|1}}

	where <math>{\bar R}</math> is the ideal gas constant. To define <math>{\bar h}</math>, one can use the constant-pressure molar heat capacity, <math>{\bar c}_p</math>. The heat capacity is define implicitly as a polynomial function of temperature:		where <math>{\bar R}</math> is the ideal gas constant. To define <math>{\bar h}</math>, one can use the constant-pressure molar heat capacity, <math>{\bar c}_p</math>. The heat capacity is define implicitly as a polynomial function of temperature:

	:<math>{\bar c}_p = a + b T + c T^2 + d T^3</math>		{{eq\|:\|<math>{\bar c}_p = a + b T + c T^2 + d T^3</math>\|2}}

	where ''a'', ''b'', ''c'', and ''d'' are defined (as <tt>cpvap''a..d''</tt>) in the components data file. Remembering that, for an ideal gas, <math>{\bar h}</math> is defined as		where ''a'', ''b'', ''c'', and ''d'' are defined (as <tt>cpvap''a..d''</tt>) in the components data file. Remembering that, for an ideal gas, <math>{\bar h}</math> is defined as

	:<math>{\bar h} - {\bar h}_0 = \int_{T_0}^{T} {\bar c}_p dT</math>		{{eq\|:\|<math>{\bar h} - {\bar h}_0 = \int_{T_0}^{T} {\bar c}_p dT</math>\|3}}

	results in the following explicit equation for <math>{\bar h}</math>:		results in the following explicit equation for <math>{\bar h}</math>:

	:<math>{\bar h} = {\bar h}_0 + a (T-T_0) + \frac{1}{2}{b (T^2 - T_0^2)} + \frac{1}{3}c (T^3 - T_0^3) + \frac{1}{4}d (T^4 - T_0^4)</math>		{{eq\|:\|<math>{\bar h} = {\bar h}_0 + a (T-T_0) + \frac{1}{2}{b (T^2 - T_0^2)} + \frac{1}{3}c (T^3 - T_0^3) + \frac{1}{4}d (T^4 - T_0^4)</math>\|4}}

	Note that, if it were ''not'' an ideal gas, <math>{\bar h}</math> would also depend on <span class="texhtml">''p''</span>. Looking back again to your old thermodynamics textbook, it can then be proven that		Note that, if it were ''not'' an ideal gas, <math>{\bar h}</math> would also depend on <span class="texhtml">''p''</span>. Looking back again to your old thermodynamics textbook, it can then be proven that

	:<math>d ( \frac{\bar g}{{\bar R}T} ) = \frac{\bar v}{{\bar R}T} dp - \frac{\bar h}{{\bar R}T^2}dT</math>		{{eq\|:\|<math>d ( \frac{\bar g}{{\bar R}T} ) = \frac{\bar v}{{\bar R}T} dp - \frac{\bar h}{{\bar R}T^2}dT</math>\|5}}

	Substitution of equations 1 and 4 on equation 5 followed by equation yields an equation for <math>{\bar g}</math> with the form		Substitution of equations 1 and 4 on equation 5 followed by equation yields an equation for <math>{\bar g}</math> with the form

	:<math>\bar g = {\bar g}(p,T)</math>		{{eq\|:\|<math>\bar g = {\bar g}(p,T)</math>\|6}}

	As a summary, this model has three equations (~~eqs~~ 1, 4, 6), and five variables (<span class="texhtml">''p''</span>, ''T'', <math>\bar g</math>, <math>\bar h</math>, <math>\bar v</math>).		As a summary, this model has three equations ({{eqref\|1}}, {{eqref\|4}}, {{eqref\|6}}), and five variables (<span class="texhtml">''p''</span>, ''T'', <math>\bar g</math>, <math>\bar h</math>, <math>\bar v</math>).

	=== ASCEND implementation ===		=== ASCEND implementation ===