Power Law Model

This is a two parameter model and it is described by the following equation.

powerlaw1

This model describes the viscosity using a power-law relationship. The exponent n determines the nature of the relationship.

n < 1 Shear thinning or Pseudoplastic

n > 1 Shear thickening or dilalatant

n = 1 Newtonian

For most polymers, Exponent n is less than 1. When the exponent is less than 1, viscosity of the polymer decreases with increase in the shear rate. This behavior is called shear thinning. On the other hand, when the exponent is greater than 1, viscosity increases with shear rate and this behavior is called shear thickening. The following figure shows viscosity as a function of exponent (consistency = 1.05e+04)

powerlaw2

One of the disadvantages of the power-law model is, as the shear rate tends to zero (when the exponent is less than 1) the viscosity tends to infinity. This setback is overcome in HyperXtrude by defining a ZeroShearRateLimit. Typically, this value is chosen based on the experimental data used to determine the model coefficients. For instance, if the ZeroShearRateLimit is set to 10.0, then HyperXtrude will use this value to compute the viscosity whenever the effective shear rate of the flow field is less than the ZeroShearRateLimit. The following figure shows this effect for a power-law model with consistency 1.05e+04 and exponent 0.34.

powerlaw3

In addition to this, a few additional parameters arise due the temperature dependence of material data; these variables are discussed in Temperature Dependence.

Syntax

Syntax of the data packet Polymer is as follows:

Polymer	PolymerName	{
	ConstitutuveModel =	"PowerLaw"
	Density =	ρ
	SpecificHeat =	Cp(T)
	Conductivity =	K(T)
	CoeffOfThermalExpansion =	βT
	VolumetricHeatSource =	Qvol
	Consistency =	A
	Exponent =	n
	ZeroShearRate =	γ0
	TemperatureDependence =	"None" }

Explanation of Parameters

Parameter	Description	Units	Data Type	Condition	Typical Value
ConstitutiveModel	Describes the model used	None	String	Required	"PowerLaw"
Density	Density of the polymer	kg/m^3	Constant	Required	995.0
SpecificHeat	Specific heat at constant pressure	J/kg/K	Constant / F(T)	Required	2000.0
Conductivity	Thermal conductivity	W/m/K	Constant / F(T)	Required	0.167
CoeffOfThermalExpansion	Indicates the change in volume with change in temperature	1/K	Constant	Required	1.0e-05
VolumetricHeatSource	Heat generated/ removed in the volume by methods like electrical heating	W/m^3	Constant	Required	0.0
Consistency	One of the parameters of the power law model. When n=1 it is same as viscosity.	Pa s^n	Constant	Required	1.0e+04
Exponent	Power law index, defines the dependency of viscosity on shear rate.	None	Constant	Required	0.66
ZeroShearRateLimit	See above	1/s	Constant	Required	0.01
TemperatureDependence	See Temperature Dependence	None	String	Required	"Exp(-Beta(DeltaT))"
ReferenceTemperature	Temperature at which data is calculated for the initialization step.	K	Constant	Required only if TD is not "None"	533
FreezeTemperature	This is the no flow temperature. Below this temperature, material ceases to flow.	K	Constant	Required only if TD is not "None"	350
ActivationEnergy	A parameter required by Arrhenius model.	J/mol	Constant	Required only if TD is Exp(Q/RT)	16628
UniversalGasConstant	A parameter from state equation PV = nRT, R is universal Gas constant.	J/mol/K	Constant	Required only if TD is Exp(Q/RT)	8.314
TemperatureSensitivity	A derived parameter which has the same physical meaning as Q/R.	K	Constant	Required only if TD is Exp(Tb/T)	2000 K
WLFConstant1	Constant C1 of WLF model	None	Constant	Required only if TD is WLF	17.44
WLFConstant2	Constant C2 of WLF model. This is like DeltaT, hence the value is same in K and Celsius.	K	Constant	Required only if TD is WLF	51.6
GlassTransitionTemperature	Temperature below with polymer molecules ceases to move (frozen). There are few definitions of this term.	K	Constant	Required only if TD is WLF	320
Beta	Parameter in the relationship Exp(-Beta(DeltaT))	None	Constant	Required only if TD is Exp(-Beta(DeltaT))	0.005

F(T) - Function of Temperature. Can be specified as a TABLE1 or TCL function.

TD - TemperatureDependence

Power Law Model

Power Law Model

Power Law Model

Explanation of Parameters

Parameter

Description

Units

Data Type

Condition

Typical Value

See Also: