======Menu Regression======

===== Evaluation of the Measurements =====

The measurement data entered can be evaluated using various predefined regression functions. The data determined in this way is then automatically transferred to the data record for the corresponding material. Measurement series can only be evaluated if they have been assigned a specific measurement type (rheology, density, specific heat capacity, etc.). No evaluation is possible for //Free measurement types//.

The regression can be performed either in the main menu via Regression or in the Edit Material window, which opens when you click the Edit button. There, you can click directly on the Regression button under the various tabs Thermodynamics, Density, Rheology, Technology, Tribology, Molecular Weight and Fibre Degradation. 

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===== Rheology =====
When evaluating rheological data, a non-linear regression is performed on the measurement series depending on the shear rate and temperature. First, a test temperature for a measurement series must be defined as the reference temperature. The following mathematical approaches are available:

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The flow properties of polymers depend on the shear rate and temperature. 

Description of shear rate dependence:
* Carreau approach

$$η =  \frac{A \cdot α_{T}} {(1+B \cdot α_{T} \cdot \dot \gamma)^C}$$

  * Yasuda approach

$$η (\dot \gamma, T) = η_∞ \cdot a_T + \frac{A \cdot a_T - η_∞ \cdot a_T} {(1 + (B \cdot a_T \cdot \dot \gamma)^κ)^\frac{1-C}{κ}}$$

  * Polynomial approach

$$η = ln (α_0 + α_1 \cdot ln(\dot \gamma) + α_2 \cdot ln(\dot \gamma)^2 + α_2 \cdot T + α_{22} \cdot T^2 + α_{12} \cdot T \cdot ln(\dot \gamma))$$

  * Power approach

$$η = K \cdot α_T \cdot \dot \gamma^{n-1}$$

Description of temperature dependence:
* WLF (Tb, Ts)

$$log(a_T) = \frac {C_1\cdot(T_B-T_S)} {C_2+(T_B-T_S)} - \frac {C_1\cdot(T-T_S)} {C_2+(T-T_S)}$$

  * WLF (C1, C2)

$$ln(a_T) = - \frac {C_1 \cdot (T-T_B)} {C_2+(T-T_B)}$$

  * Arrhenius

$$ln(a_T)=\frac{E}{R} (\frac{1}{T+273.15}-\frac{1}{T_B+273.15}) $$

Description of pressure dependence:
* Beta

$$β=ln[\frac{η(p2)}{η(p1)}] \cdot \frac{1}{(p2-p1)}$$

In some cases, the regression does not produce a meaningful result. This problem can often be solved by selecting a different reference temperature and/or a different calculation approach. Nevertheless, it should be noted that regression always produces an approximate result, which often cannot provide the desired accuracy for mathematical reasons.

===== Thermodynamical Data =====
When evaluating the thermodynamic data and densities, a distinction must be made between whether the molecular structure of the polymer is amorphous or semi-crystalline. Different curves result depending on the molecular structure. 

==== Specific Heat Capacity ====
  * amorphous polymers

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  * partially crystalline polymers

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Where there are significant changes in the curve, the characteristic temperature of the material is $Tg$ for amorphous materials and $Tk$ for semi-crystalline materials. To evaluate the material data, first determine the characteristic temperature and then perform a linear regression in the range $T$ >>$Tg$ or $Tk$ using the following equation.

$$c_p(T) = c_{p,0} + c_{p,m}\cdot T$$

==== Enthalpies ====
The following enthalpy curves result for amorphous or semi-crystalline materials:
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For semi-crystalline materials, a tangent must first be calculated at low temperatures. The enthalpy at this point is then divided into enthalpy components below the tangent, as shown in the following figure, on the one hand into the solid enthalpy $∆h_F$ and on the other hand into the melting enthalpy $∆h_A$.

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For amorphous materials, the enthalpy value at $T_g$ must be calculated as solid enthalpy $∆h_F$ (in case of doubt, linear interpolation of the enthalpy values of the neighbouring measured values).

==== Thermal conductivity ====
For amorphous or semi-crystalline materials, the following thermal conductivity curves over temperature result:

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These curves also show significant changes in the characteristic temperatures $Tg$ and $Tk$. To evaluate the material data, the characteristic temperature must again be determined and a linear regression must be performed in the range $T$ >> $Tg$ and $Tk$ using the following equation: 

$$λ(T) = λ_0 + λ_m \cdot T$$

===== Density =====
Density measurements are generally performed at constant temperature and varying pressures. However, to evaluate the densities, it is necessary to plot them at constant pressures and varying temperatures. Here, too, the curves for amorphous and semi-crystalline polymers differ.

  * partially crystalline polymers

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  * amorphous polymers

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Depending on whether density or specific volume was measured, a linear regression must be performed using one of the functions listed below: 

Specific volume:

$$v = v_0 + v_m \cdot T$$

Density:

$$ρ = ρ_0 - ρ_m \cdot T$$

where: 

$$ρ = \frac{1}{v}$$

For the evaluation, a pressure must be selected from the list of available pressures.

Description of pressure dependence:

Kappa:

$$ \kappa = - \frac{1}{V} \frac{dV}{dp} $$

==== Technological data ====
Minimum, maximum and average particle diameters as well as shear stress are data that must all be entered manually. No calculation is performed.

==== Tribological data ====
For further calculations with the REX / PSI / SIGMA programmes, various friction coefficients of the materials must be specified.