Reference ID: MET-4A3F | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The viscosity of polymeric and glass‑forming materials varies strongly with temperature.
Predicting this variation is essential for process engineers who must size pumps, design
extrusion lines, and control cooling cycles. The Williams‑Landel‑Ferry (WLF) equation
provides a widely accepted empirical relationship that links the logarithm of the
viscosity ratio to the temperature distance from the glass‑transition temperature.
It is routinely used in polymer processing, coating operations, and any application
where a material transitions from a glassy to a rubbery state.
Methodology & Formulas
The calculation follows the logical steps implemented in the reference Python script,
expressed here in purely algebraic form.
Convert temperatures from Celsius to Kelvin. T_K = T_C + ΔT_K Tg_K = Tg_C + ΔT_K ΔT_K denotes the constant offset required for the Celsius‑to‑Kelvin conversion.
Determine the temperature difference relative to the glass‑transition point. ΔT = T_K - Tg_K
This difference will be negative when the target temperature is below the glass‑transition temperature.
Apply the base‑10 logarithmic form of the WLF equation. rhs = - (C1 · ΔT) / (C2 + ΔT) C1 and C2 are the WLF constants; C2 is expressed in Kelvin (its numerical value matches the Celsius representation).
Compute the viscosity ratio. ratio = 10^{rhs}
This yields the factor by which the reference viscosity changes at the target temperature.
Obtain the absolute viscosity at the target temperature. μ = μ_g · ratio μ_g is the reference viscosity measured at the glass‑transition temperature.
The resulting expression for the target viscosity can be written compactly as:
This formula enables engineers to predict how a material’s flow resistance will evolve
with temperature, supporting design decisions and process optimization without the need
for extensive experimental testing.
The Williams‑Landel‑Ferry (WLF) equation relates the shift factor a_T to temperature relative to a reference temperature (usually the glass‑transition temperature, T_g). It is expressed as
log a_T = –C₁ (T – T_ref) / [C₂ + (T – T_ref)]
where C₁ and C₂ are material‑specific constants. By converting viscosity data measured at one temperature to any other temperature, the WLF equation enables process engineers to design and control operations (extrusion, molding, coating) without extensive experimental trials.
Perform rheological measurements at several temperatures around the polymer’s T_g (typically –20 °C to +30 °C relative to T_g). Plot log a_T versus (T – T_ref) and fit the data to the WLF form; the slope and intercept give C₁ and C₂. Commercial software often provides a built‑in WLF fitting routine, or you can use linear regression on the transformed equation:
log a_T = –C₁ ΔT / (C₂ + ΔT).
If literature values are available for a similar grade, they can be used as a starting point.
The WLF equation is most accurate within about ±30 °C of T_g, where the polymer exhibits viscoelastic behavior governed by free‑volume changes. At temperatures well above T_g (the rubbery or melt region), the Arrhenius relationship often provides a better fit:
log a_T = –E_a / (2.303 R) (1/T – 1/T_ref).
For high‑temperature processing, it is common to use a combined approach—WLF near T_g and Arrhenius at higher temperatures—to maintain continuity in the shift factor.
1. **Calculate a_T** for the operating temperature using the WLF equation with the determined C₁, C₂, and T_ref.
2. **Scale the reference viscosity** (or relaxation time) by a_T: η(T) = η_ref · a_T.
3. **Update material property tables** in the simulation (e.g., viscosity vs. shear rate) with the temperature‑adjusted values.
4. **Validate** the model by comparing simulated torque or pressure profiles against pilot‑scale runs.
Most simulation packages (e.g., ANSYS Polyflow, COMSOL) allow user‑defined functions or lookup tables where you can input the WLF‑adjusted viscosity curve directly.
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Worked Example: Predicting Viscosity at a Lower Temperature Using the WLF Equation
Scenario
A process engineer at a polymer‑processing plant needs to estimate the melt viscosity of a polymer at 30 °C. The polymer’s glass‑transition temperature (Tg) is 45 °C, and the viscosity at Tg (μg) is known to be 1.0 × 10⁶ Pa·s. The engineer will use the Williams‑Landel‑Ferry (WLF) equation with the standard constants C₁ = 17.0 and C₂ = 51.0.
Knowns (Input Parameters)
Tg = 45.0 °C
T = 30.0 °C
μg = 1.0 × 10⁶ Pa·s
C₁ = 17.0 (dimensionless)
C₂ = 51.0 (°C)
Step‑by‑Step Calculation
Calculate the temperature difference relative to Tg:
\[
\Delta T = T - T_g = 30.0 - 45.0 = -15.0\ \text{°C}
\]
Insert the values into the WLF equation (logarithmic form):
\[
\log_{10}\!\left(\frac{\mu}{\mu_g}\right) = -\frac{C_1\,\Delta T}{C_2 + \Delta T}
\]
Evaluate the right‑hand side (RHS):
\[
\text{RHS} = -\frac{17.0 \times (-15.0)}{51.0 + (-15.0)}
= \frac{255.0}{36.0}
= 7.083\ (\text{rounded to 3 d.p.})
\]
Solve for the viscosity ratio:
\[
\frac{\mu}{\mu_g} = 10^{\text{RHS}} = 10^{7.083} \approx 1.212 \times 10^{7}
\]
