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=Temperature-Humidity Relationship=
#REDIRECT [[Temperature-Humidity_Relationship]]
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==Introduction==
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The Temperature-Humidity (T-H) relationship, a variation of the Eyring  relationship, has been proposed for predicting the life at use conditions when temperature and humidity are the accelerated stresses in a test. This combination model is given by:
 
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::<math>L(V,U)=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>
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where:
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• <math>\phi </math>  is one of the three parameters to be determined.
 
• <math>b</math>  is the second of the three parameters to be determined (also known as the activation energy for humidity).
 
• <math>A</math>  is a constant and the third of the three parameters to be determined.
 
• <math>U</math>  is the relative humidity  (decimal or percentage).
 
• <math>V</math>  is temperature (in absolute units ).
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The T-H relationship can be linearized and plotted on a Life vs. Stress plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (Temp-Hum), or:
 
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::<math>ln(L(V,U))=ln(A)+\frac{\phi }{V}+\frac{b}{U}</math>
 
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Since life is now a function of two stresses, a Life vs. Stress plot can only be obtained by keeping one of the two stresses constant and varying the other one. Doing so will yield a straight line as described by Eqn. (ln Temp-Hum), where the term for the stress which is kept at a fixed value becomes another constant (in addition to the  <math>\ln (A)</math>  constant). In Figs. 1 and 2, data obtained from a temperature and humidity test were analyzed and plotted on Arrhenius paper. In Fig. 1, life is plotted versus temperature with relative humidity held at a fixed value. In Fig. 2, life is plotted versus relative humidity with temperature held at a fixed value.
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[[Image:ALTA9.1.png|thumb|center|300px|Life vs. Temperature plot at a fixed relative humidity.]]
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[[Image:ALTA9.2.png|thumb|center|300px|Life vs. Relative Humidity plot at a fixed temperature.]]
 
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Note that the Life vs. Stress plots in both Figs. 1 and 2 are plotted on a log-reciprocal scale. Also note that the points shown in these plots represent the life characteristics at the test stress levels (the data set was fitted to a Weibull distribution, thus the points represent the scale parameter,  <math>\eta )</math> . For example, the points shown in Fig. 1 represent  <math>\eta </math>  at each of the test temperature levels (two temperature levels were considered in this test).
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==A look at the Parameters  <math>\phi </math>  and  <math>b</math>==
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Depending on which stress type is kept constant, it can be seen from Eqn. (ln Temp-Hum) that either the parameter  <math>\phi </math>  or the parameter  <math>b</math>  is the slope of the resulting line. If, for example, the humidity is kept constant (Fig. 1) then  <math>\phi </math>  is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of product life to the temperature. In other words,  <math>\phi </math>  is a measure of the effect that temperature has on the life, and  <math>b</math>  is a measure of the effect that relative humidity has on the life. The larger the value of  <math>\phi ,</math>  the higher the dependency of the life on the temperature. Similarly, the larger the value of  <math>b,</math>  the higher the dependency of the life on the humidity. For example, it can be seen by comparing Figs. 1 and 2 that, for this data set, temperature has a greater effect on the life than humidity.
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==T-H Data==
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When using the T-H relationship, the effect of both temperature and humidity on life is sought. For this reason, the test must be performed in a combination manner between the different stress levels of the two stress types. For example, assume that an accelerated test is to be performed at two temperature and two humidity levels. The two temperature levels were chosen to be 300K and 343K. The two humidity levels were chosen to be 0.6 and 0.8. It would be wrong to perform the test at (300K, 0.6) and (343K, 0.8). Doing so would not provide information about the temperature-humidity effects on life. This is because both stresses are increased at the same time and therefore it is unknown which stress is causing the acceleration on life. A possible combination that would provide information about temperature-humidity effects on life would be (300K, 0.6), (300K, 0.8) and (343K, 0.8). It is clear that by testing at (300K, 0.6) and (300K, 0.8) the effect of humidity on life can be determined (since temperature remained constant). Similarly the effects of temperature on life can be determined by testing at (300K, 0.8) and (343K, 0.8) since humidity remained constant.
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==Acceleration Factor==
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The acceleration factor for the T-H relationship is given by:
 
 
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::<math>{{A}_{F}}=\frac{{{L}_{USE}}}{{{L}_{Accelerated}}}=\frac{A{{e}^{\tfrac{\phi }{{{V}_{u}}}+\tfrac{b}{{{U}_{u}}}}}}{A{{e}^{\tfrac{\phi }{{{V}_{A}}}+\tfrac{b}{{{U}_{A}}}}}}={{e}^{\phi \left( \tfrac{1}{{{V}_{u}}}-\tfrac{1}{{{V}_{A}}} \right)+b\left( \tfrac{1}{{{U}_{u}}}-\tfrac{1}{{{U}_{A}}} \right)}}</math>
 
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where:
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• <math>{{L}_{USE}}</math>  is the life at use stress level.
 
• <math>{{L}_{Accelerated}}</math>  is the life at the accelerated stress level.
 
• <math>{{V}_{u}}</math>  is the use temperature level.
 
• <math>{{V}_{A}}</math>  is the accelerated temperature level.
 
• <math>{{U}_{A}}</math>  is the accelerated humidity level.
 
• <math>{{U}_{u}}</math>  is the use humidity level.
 
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The acceleration Factor is plotted versus stress in the same manner used to create the Life vs. Stress plots. That is, one stress type is kept constant and the other is varied (see Figs. 3 and 4).
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[[Image:ALTA9.3.gif|thumb|center|300px|Acceleration Factor vs. Temperature at a fixed relative humidity.]]
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[[Image:ALTA9.4.gif|thumb|center|300px|Acceleration Factor vs. Humidity at a fixed temperature.]]
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==T-H Exponential==
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By setting  <math>m=L(U,V)</math>  in Eqn. (Temp-Hum) the exponential  <math>pdf</math>  becomes:
 
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::<math>f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}\cdot {{e}^{-\tfrac{t}{A}\cdot {{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
 
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===T-H Exponential Statistical Properties Summary===
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====Mean or MTTF====
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The mean,  <math>\overline{T},</math>  or Mean Time To Failure (MTTF) for the T-H exponential model is given by:
 
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::<math>\overline{T}=\mathop{}_{0}^{\infty }t\cdot f(t,V,U)dt</math>
 
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Substituting Eqn. (t-h exp pdf) yields:
 
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::<math>\begin{align}
  & \overline{T}= & \mathop{}_{0}^{\infty }t\cdot \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dt \\
& = & A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}} 
\end{align}</math>
 
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====Median====
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The median,  <math>\breve{T},</math> for the T-H exponential model is given by:
 
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::<math>\breve{T}=0.693\cdot A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>
 
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====Mode====
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The mode,  <math>\tilde{T},</math>  for the T-H exponential model is given by:
 
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::<math>\tilde{T}=0</math>
 
====Standard Deviation====
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The standard deviation,  <math>{{\sigma }_{T}}</math> , for the T-H exponential model is given by:
 
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::<math>{{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>
 
====T-H Exponential Reliability Function====
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The T-H exponential reliability function is given by:
 
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::<math>R(T,V,U)={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
 
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This function is the complement of the T-H exponential cumulative distribution function or:
 
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::<math>R(T,V,U)=1-Q(T,V,U)=1-\mathop{}_{0}^{T}f(T)dT</math>
 
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and:
 
 
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::<math>R(T,V,U)=1-\mathop{}_{0}^{T}\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dT={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
 
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====Conditional Reliability====
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The conditional reliability function for the T-H exponential model is given by:
 
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::<math>R(T,t,V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
 
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====Reliable Life====
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For the T-H exponential model, the reliable life, or the mission duration for a desired reliability goal,  <math>{{t}_{R}},</math>  is given by:
 
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::<math>R({{t}_{R}},V,U)={{e}^{-\tfrac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
 
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::<math>\ln [R({{t}_{R}},V,U)]=-\frac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}</math>
 
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or:
 
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::<math>{{t}_{R}}=-A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\ln [R({{t}_{R}},V,U)]</math>
 
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===Parameter Estimation===
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====Maximum Likelihood Estimation Method====
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Substituting the T-H model into the exponential log-likelihood equation yields:
 
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::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}\cdot {{e}^{-\tfrac{{{T}_{i}}}{A}\cdot {{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}}} \right] \\
&  & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}+\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>
 
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where:
 
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::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}</math>
 
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::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}</math>
 
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and:
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• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
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• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
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• <math>A</math>  is the T-H parameter (unknown, the first of three parameters to be estimated).
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• <math>\phi </math>  is the second T-H parameter (unknown, the second of three parameters to be estimated).
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• <math>b</math>  is the third T-H parameter (unknown, the third of three parameters to be estimated).
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• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{U}_{i}}</math>  is the relative humidity level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
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• <math>S</math>  is the number of groups of suspension data points.
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• ..  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
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• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
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• <math>FI</math>  is the number of interval data groups.
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• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the  <math>{{i}^{th}}</math>  group of data intervals.
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• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  <math>{{i}^{th}}</math>  interval.
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• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the  <math>{{i}^{th}}</math>  interval.
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The solution (parameter estimates) will be found by solving for the parameters  <math>A,</math>  <math>\phi </math>  and  <math>b</math>  so that  <math>\tfrac{\partial \Lambda }{\partial A}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial \phi }=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial b}=0</math> .
 
{{t-h weibull}}
 
{{t-h lognormal}}

Latest revision as of 05:29, 15 August 2012

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