Mathematical Equations used in PET Preform Design {Part II}

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Designing the wall thickness of a PET preform 

involves considering various factors such as required bottle strength, size, material properties, and processing conditions. I can give you an overview of the mathematical equations for this.


1-     Taper Thickness Equations 
          1-1           Linear Taper Equation:
                                                Tt = Tb - (Tb - Te) * (L - Lt) / L
Where:
  • Tt = taper thickness
  • Tb = bottom/maximum thickness
  • Te = top/minimum thickness
  • L = total length
  • Lt = length of taper section
          1-2           Exponential Taper Equation:
                                                       Tt = Tb * e^(-k(L - Lt))
Where:
  • k = taper coefficient determined experimentally (typical 0.05-0.1)
           1-3           PET Technology Taper Equation:
                                                       Tt = Tb - x * (Tb - Te)
Where:
  • x = fractional distance from bottom (0 at bottom, 1 at top)
This equation assumes a linear thickness reduction along the taper length.
            1-4           Economic Taper Equation:
Minimizes material volume while maintaining strength. Calculations are more complex.
The linear and exponential equations are commonly used in preform design as they provide a smooth thickness transition between bottom and top. The coefficient or constants need to be determined through testing. Proper taper is important for even stress distribution during blow molding.

2-    Body Thickness Equations 
              2-1           Design Failure Equation:
                                               T = σy / P
Where:
  • T = wall thickness
  • σy = material yield strength (around 5000 psi for PET)
  • P = internal pressure during blow molding
                2-2           PET Technology Design Equation:
                                                        T = 0.90 * σy / P
  •  Uses a safety factor of 0.9.
                2-3           Milhofer Creep Equation:
                                                 T = K * P^(1/3)
Where 
  • K is 0.14-0.16 for PET. Accounts for both pressure and creep.
                  2-4           Owens-Illinois Creep Equation:
                                                 T = 0.14 * (P/150)^(1/3)
Where 
  • P is in psi.
                     2-5           SIFY Blow Molding Creep Equation:
                                                          T = 0.135 * (P/150)^(1/3)
More conservative than Owens-Illinois.
All these equations relate wall thickness (T) to the internal pressure (P) during blow molding. The Design Failure equation provides a minimum thickness based on yield strength, while the creep equations incorporate time-dependent wall thinning during molding.


3- End Cap thickness Equations 
                        3-1           Milhofer Equation:
                                            T = K * P^(1/3)
Where:
  • T = end cap thickness (inches)
  • P = inside container pressure (psi)
  • K = constant based on material properties, typically 0.16-0.18 for PET
                        3-2           PET Technology Equation:
                                     T = 0.175 * P^(1/3)
Slightly more conservative than the Milhofer equation.
                         3-3           Owens-Illinois Equation:
                                                  T = 0.175 * (P/150)^(1/3) * W
Where:
  • W = wall thickness factor (typically 1.0-1.3)
                         3-4   Container Research Institute Equation:
                                                                  T = 0.185 * P^(1/3)
Slightly more conservative than PET Technology equation.
The equations all follow a similar form based on applying elastic buckling theory to the end cap geometry. The constant K or coefficient varies slightly between equations depending on the safety factor used. P is typically the final blow molding pressure. Wall thickness factor W can be adjusted based on preform/container design.
 
4- Gat Wall Thickness Equations 
                                     4-1 Milhofer Equation:
                                                     T = K * Q^(1/3)
Where:
  • T = gate/sprue wall thickness (inches)
  • Q = flow rate (cm^3/sec)
  • K = constant, typically 0.07-0.09 for PET
                                    4-2    Owens-Illinois Equation:
                                                          T = 0.09 * (Q/30)^(1/3)
Where:
  • Q is in cm^3/sec
                                   4-3 PET Technology Equation:
                                                                    T = 0.08 * Q^(1/3)
                                      4-4 Husky Injection Molding Systems Equation:
                                                                  T = 0.08 * (Q/30)^(1/3)
Where:
  • Q is in cm^3/sec
These equations relate gate thickness to the flow rate of molten PET during injection molding. A higher flow rate requires a thicker gate section to avoid velocity stagnation/shear issues.
The constant K or coefficient varies slightly between equations. Typical Q values for preforms range from 20-80 cm^3/sec depending on size/weight. Wall thickness is usually specified in inches. These equations provide a starting point which may need to be optimized based on particular mold/material/process conditions.



5- Calculating the L/T and wall thickness of a PET preform
 involves using mathematical equations based on the preform's dimensions and material properties. Here are the general equations commonly used for these calculations:
                      5-1   Hoop Stress Equation:
                                              σh = PD/2t
Where:
σh = Hoop stress
  • P = Internal pressure
  • D = Inner diameter of the preform
  • t = Wall thickness
·        This calculates the stress in the hoop direction of the preform bottom.
As mentioned previously, this calculates hoop stress in the walls. A higher L/t ratio will result in higher hoop stresses.
 
                    5-2.   Axial Stress Equation:
                                           σa = PD/4t
Where:
  • σa = Axial stress
  • P, D, t are the same as above
·        This calculates the stress in the axial direction of the preform bottom.
Similarly, a higher L/t ratio increases axial stresses.
                     5-3.   von Mises Stress Equation:
                                          σvm = √(σh2 + σa2 - σhσa)
·        This determines the effective stress by combining the hoop and axial stresses.
                    5-4.   Maximum Shear Stress Equation:
                                            τmax = (εh - εa)/2
This calculates the maximum shear stress.
The wall thickness t is adjusted until the max stresses are within the allowable limits for the PET material at the expected pressures. A safety factor is also commonly applied.
Other considerations include geometric stability, molding process factors, and base thickness requirements to support the preform in blow molding..
 
                      5-5.   Critical Buckling Load Equation:
                                            Pcr = (π2EI)/(KL)2
Where:
  • Pcr = Critical buckling load
  • E = Elastic modulus of PET
  • I = Area moment of inertia
  • K = Column effective length factor
  • L = Length of the preform
A higher L/t ratio will reduce the critical buckling load of the preform, increasing susceptibility to buckling.
A higher L/t ratio allows for a higher stretch ratio, which improves material distribution and properties in the final blown PET container.
·        The optimum L/t ratio balances these factors to avoid excessive stresses and instability while enabling sufficient stretch ratios for the target container.

This was prepared and written by
     Eng/ Mohamed Bayoumi 
   Mobile +201550289138

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