Arrhenius Relationship - Revision history

Cruz Daniel: changed undefined phi term in log likelihood equation under maximum likelihood section to use reliability term instead - suggestion from Sayyed Vazirizade at University of Arizona

2021-10-29T16:32:06Z

changed undefined phi term in log likelihood equation under maximum likelihood section to use reliability term instead - suggestion from Sayyed Vazirizade at University of Arizona

← Older revision		Revision as of 16:32, 29 October 2021
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	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \
	& \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [~~\Phi (z_~~{Ri}^{\prime \prime })-~~\Phi (z_~~{Li}^{\prime \prime })]		& \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	\end{align}\,\!</math>		\end{align}\,\!</math>

Richard House: /* Formulation */

2019-02-07T16:42:19Z

Formulation

← Older revision		Revision as of 16:42, 7 February 2019
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	*<math>{{E}_{a}}\,\!</math> is the activation energy <math>(\text{eV})\,\!</math>.		*<math>{{E}_{a}}\,\!</math> is the activation energy <math>(\text{eV})\,\!</math>.

	*<math>k\,\!</math> is the Boltzmann's constant <math>(8.~~617385~~\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}})\,\!</math>.		*<math>k\,\!</math> is the Boltzmann's constant <math>(8.6173303\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}})\,\!</math>.

	*<math>T\,\!</math> is the absolute temperature <math>(\text{K})\,\!</math>.		*<math>T\,\!</math> is the absolute temperature <math>(\text{K})\,\!</math>.

Sharon Honecker: /* Conditional Reliability Function */ changed R(t,T,V) to R((t|T),V)

2017-02-08T21:03:48Z

Conditional Reliability Function: changed R(t,T,V) to R((t|T),V)

← Older revision		Revision as of 21:03, 8 February 2017
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	The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:		The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:

	::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>		::<math>R((t\|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>

	or:		or:

	::<math>R(T,t,V)={{e}^{-\left[ {{\left( \tfrac{T+t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }} \right]}}\,\!</math>		::<math>R((t\|T),V)={{e}^{-\left[ {{\left( \tfrac{T+t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }} \right]}}\,\!</math>

	====Reliable Life====		====Reliable Life====

Sharon Honecker: /* Conditional Reliability */ changed R(t,T,V) to R((t|T),V)

2017-02-08T21:01:54Z

Conditional Reliability: changed R(t,T,V) to R((t|T),V)

← Older revision		Revision as of 21:01, 8 February 2017
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	The Arrhenius-exponential conditional reliability function is given by:		The Arrhenius-exponential conditional reliability function is given by:

	::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>		::<math>R((t\|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

	====Reliable Life====		====Reliable Life====

Sharon Honecker: /* Bounds on the Parameters */

2017-02-01T21:49:00Z

Bounds on the Parameters

← Older revision		Revision as of 21:49, 1 February 2017
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	Since the standard deviation, <math>{{\widehat{\sigma }}_{{{T}'}}}\,\!</math>, and the parameter <math>C\,\!</math> are positive parameters, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})\,\!</math> and <math>\ln (C)\,\!</math> are treated as normally distributed. The bounds are estimated from:		Since the standard deviation, <math>{{\widehat{\sigma }}_{{{T}'}}}\,\!</math>, and the parameter <math>\widehat{C}\,\!</math> are positive parameters, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})\,\!</math> and <math>\ln (\widehat{C})\,\!</math> are treated as normally distributed. The bounds are estimated from:

	::<math>\begin{align}		::<math>\begin{align}

Sharon Honecker: /* Activation Energy and the Parameter B */ Replace E_A with E_a and K with k.

2017-02-01T21:43:23Z

Activation Energy and the Parameter B: Replace E_A with E_a and K with k.

← Older revision		Revision as of 21:43, 1 February 2017
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	Depending on the application (and where the stress is exclusively thermal), the parameter <math>B\,\!</math> can be replaced by:		Depending on the application (and where the stress is exclusively thermal), the parameter <math>B\,\!</math> can be replaced by:

	::<math>B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzman}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}}\,\!</math>		::<math>B=\frac{{{E}_{a}}}{k}=\frac{\text{activation energy}}{\text{Boltzman}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}}\,\!</math>

	Note that in this formulation, the activation energy <math>{{E}_{A}}\,\!</math> must be known a priori. If the activation energy is known then there is only one model parameter remaining, <math>C.\,\!</math> Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat <math>B\,\!</math> as one of the model parameters. <math>B\,\!</math> has the same properties as the activation energy. In other words, <math>B\,\!</math> is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the specific stress. Parameter <math>B\,\!</math> may also take negative values. In that case, life is increasing with increasing stress. An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.		Note that in this formulation, the activation energy <math>{{E}_{a}}\,\!</math> must be known a priori. If the activation energy is known then there is only one model parameter remaining, <math>C.\,\!</math> Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat <math>B\,\!</math> as one of the model parameters. <math>B\,\!</math> has the same properties as the activation energy. In other words, <math>B\,\!</math> is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the specific stress. Parameter <math>B\,\!</math> may also take negative values. In that case, life is increasing with increasing stress. An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

	[[Image:ALTA6.5.png\|center\|400px\|Behavior of the parameter ''B''.]]		[[Image:ALTA6.5.png\|center\|400px\|Behavior of the parameter ''B''.]]

Sharon Honecker: /* Formulation */

2017-02-01T21:41:05Z

Formulation

← Older revision		Revision as of 21:41, 1 February 2017
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	*<math>V\,\!</math> represents the stress level (formulated for temperature and '''temperature values in absolute units, degrees Kelvin or degrees Rankine''').		*<math>V\,\!</math> represents the stress level (formulated for temperature and '''temperature values in absolute units, degrees Kelvin or degrees Rankine''').

	*<math>C\,\!</math> is one of the model parameters to be determined, ( <math>C>0\,\!</math>).		*<math>C\,\!</math> is one of the model parameters to be determined, <math>(C>0)\,\!</math>.

	*<math>B\,\!</math> is another model parameter to be determined.		*<math>B\,\!</math> is another model parameter to be determined.

Sharon Honecker: /* Formulation */

2017-02-01T21:40:21Z

Formulation

← Older revision		Revision as of 21:40, 1 February 2017
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	*<math>L\,\!</math> represents a quantifiable life measure, such as mean life, characteristic life, median life, or <math>B(x)\,\!</math> life, etc.		*<math>L\,\!</math> represents a quantifiable life measure, such as mean life, characteristic life, median life, or <math>B(x)\,\!</math> life, etc.

	*<math>V\,\!</math> represents the stress level (formulated for temperature and '''temperature values in absolute units, degrees Kelvin or degrees Rankine''' ).		*<math>V\,\!</math> represents the stress level (formulated for temperature and '''temperature values in absolute units, degrees Kelvin or degrees Rankine''').

	*<math>C\,\!</math> is one of the model parameters to be determined, ( <math>C>0\,\!</math>).		*<math>C\,\!</math> is one of the model parameters to be determined, ( <math>C>0\,\!</math>).

Sharon Honecker: /* Formulation */ Change E_A to E_a and K to k

2017-02-01T21:38:44Z

Formulation: Change E_A to E_a and K to k

← Older revision		Revision as of 21:38, 1 February 2017
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	The Arrhenius reaction rate equation is given by:		The Arrhenius reaction rate equation is given by:

	::<math>R(T)=A{{e}^{-\tfrac{{{E}_{A}}}{K\cdot T}}}\,\!</math>>		::<math>R(T)=A{{e}^{-\tfrac{{{E}_{a}}}{k\cdot T}}}\,\!</math>

	where:		where:
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	*<math>A\,\!</math> is an unknown nonthermal constant.		*<math>A\,\!</math> is an unknown nonthermal constant.

	*<math>{{E}_{A}}\,\!</math> is the activation energy ( <math>eV\,\!</math> ).		*<math>{{E}_{a}}\,\!</math> is the activation energy <math>(\text{eV})\,\!</math>.

	*<math>K\,\!</math> is the Boltzmann's constant (<math>8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}})\,\!</math>		*<math>k\,\!</math> is the Boltzmann's constant <math>(8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}})\,\!</math>.

	*<math>T\,\!</math> is the absolute temperature (~~Kelvin~~).		*<math>T\,\!</math> is the absolute temperature <math>(\text{K})\,\!</math>.

	The activation energy is the energy that a molecule must have to participate in the reaction. In other words, the activation energy is a measure of the effect that temperature has on the reaction.		The activation energy is the energy that a molecule must have to participate in the reaction. In other words, the activation energy is a measure of the effect that temperature has on the reaction.

Sharon Honecker: /* Activation Energy and the Parameter B */ Remove minus sign in B = Ea/k.

2017-02-01T02:32:51Z

Activation Energy and the Parameter B: Remove minus sign in B = Ea/k.

← Older revision		Revision as of 02:32, 1 February 2017
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	Depending on the application (and where the stress is exclusively thermal), the parameter <math>B\,\!</math> can be replaced by:		Depending on the application (and where the stress is exclusively thermal), the parameter <math>B\,\!</math> can be replaced by:

	::<math>B=-\frac{{{E}_{A}}}{K}=-\frac{\text{activation energy}}{\text{Boltzman}{{\text{n}}^{\prime }}\text{s constant}}=-\frac{\text{activation energy}}{8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}}\,\!</math>		::<math>B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzman}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}}\,\!</math>

	Note that in this formulation, the activation energy <math>{{E}_{A}}\,\!</math> must be known a priori. If the activation energy is known then there is only one model parameter remaining, <math>C.\,\!</math> Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat <math>B\,\!</math> as one of the model parameters. <math>B\,\!</math> has the same properties as the activation energy. In other words, <math>B\,\!</math> is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the specific stress. Parameter <math>B\,\!</math> may also take negative values. In that case, life is increasing with increasing stress. An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.		Note that in this formulation, the activation energy <math>{{E}_{A}}\,\!</math> must be known a priori. If the activation energy is known then there is only one model parameter remaining, <math>C.\,\!</math> Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat <math>B\,\!</math> as one of the model parameters. <math>B\,\!</math> has the same properties as the activation energy. In other words, <math>B\,\!</math> is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the specific stress. Parameter <math>B\,\!</math> may also take negative values. In that case, life is increasing with increasing stress. An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.