Template:T-H Model

Introduction
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:


 * $$L(V,U)=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}$$

where: •	 $$\phi $$ is one of the three parameters to be determined.

•	 $$b$$ is the second of the three parameters to be determined (also known as the activation energy for humidity).

•	 $$A$$ is a constant and the third of the three parameters to be determined.

•	 $$U$$ is the relative humidity  (decimal or percentage).

•	 $$V$$ is temperature (in absolute units).

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 the T-H relationship, or:


 * $$ln(L(V,U))=ln(A)+\frac{\phi }{V}+\frac{b}{U}$$

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 where the term for the stress which is kept at a fixed value becomes another constant (in addition to the $$\ln (A)$$  constant). In the next two figures, data obtained from a temperature and humidity test were analyzed and plotted on Arrhenius paper. In the first figure, life is plotted versus temperature with relative humidity held at a fixed value. In the second figure, life is plotted versus relative humidity with temperature held at a fixed value.

Note that the Life vs. Stress plots 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, $$\eta )$$. For example, the points shown in the first figure represent $$\eta $$  at each of the test temperature levels (two temperature levels were considered in this test).

A look at the Parameters $$\phi $$ and  $$b$$
Depending on which stress type is kept constant, it can be seen from the linearized T-H relationship that either the parameter $$\phi $$  or the parameter  $$b$$  is the slope of the resulting line. If, for example, the humidity is kept constant then $$\phi $$  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, $$\phi $$  is a measure of the effect that temperature has on the life, and  $$b$$  is a measure of the effect that relative humidity has on the life. The larger the value of $$\phi ,$$  the higher the dependency of the life on the temperature. Similarly, the larger the value of $$b,$$  the higher the dependency of the life on the humidity.

T-H Data
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.

Acceleration Factor
The acceleration factor for the T-H relationship is given by:


 * $${{A}_{F}}=\frac=\frac{A{{e}^{\tfrac{\phi }+\tfrac{b}}}}{A{{e}^{\tfrac{\phi }+\tfrac{b}}}}={{e}^{\phi \left( \tfrac{1}-\tfrac{1} \right)+b\left( \tfrac{1}-\tfrac{1} \right)}}$$

where: •	 $${{L}_{USE}}$$ is the life at use stress level.

•	 $${{L}_{Accelerated}}$$ is the life at the accelerated stress level.

•	 $${{V}_{u}}$$ is the use temperature level.

•	 $${{V}_{A}}$$ is the accelerated temperature level.

•	 $${{U}_{A}}$$ is the accelerated humidity level.

•	 $${{U}_{u}}$$ is the use humidity level.

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 as shown in the next two figures.