Herschel Bulkley Model

Regularized Herschel-Bulkley model is a four-parameter model and it is given by the following equation.

This model describes the viscosity using a power-law relationship and a yield criterion. The exponent n determines the nature of the power law relationship and the yield criterion is implemented using the regularized model with yield stress t0 and regularization parameter m. Higher the value of m, closer the approximation to the actual yield stress behavior of Herschel-Bulkley model. For practical purposes, value of m=10000 should be suffice. This regularization parameter controls the exponential growth of stress. The original Herschel-Bulkley model, which is given by the following equation, suffers from the disadvantage of stress growing very large when the shear rate tends to zero. This will lead to numerical instability and also increase the difficulty of obtaining a converged solution. Hence, this model is regularized using the parameter m.

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

Syntax

Syntax of the data packet Polymer is as follows:


Polymer	PolymerName	{
	ConstitutiveModel =	"Cross"
	Density =	ρ
	SpecificHeat =	Cp (T)
	Conductivity =	K (T)
	CoeffOfThermalExpansion	βT
	Volumetric Heat Source =	Qvol
	TimeConstant =	Α
	Exponent =	n
	YieldStress =	τ0
	StressExponentGrowth=	m
	TemperatureDependence =	"None" }


Parameter	Description	Units	Data Type	Condition	Typical Value
ConstitutiveModel	Describes the model used	None	String	Required	"HerschelBulkely"
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 elctrical heating	W/m63	Constant	Required	0.0
Consistency	One of the parameters of the Herschel—Bulkley model. When n=1 it is same as viscosity	Pa	Constant	Required	0.01
Exponent	Power law index, defines the dependency of viscosity on shear rate.	None	Constant	Required	0.66
ZeroShearRateLimit	. See Power law model	1/s	Constant	Required	0.01
YieldStress	Oneo f the parameters of the Herschel-Bulkley model.	Pa	Constant	Required	1.0E+05
StressExponentGrowth	One of the parameters of the Herschel—Bulkley model. It is also known as the regularization parameter. It controls the exponential growth of stress	s	Constant	Required
TemperatureDependence	See temperature Dependence	None	string	Required	"WLF"
ReferenceTemeprature	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 onlyh 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(Delta T)	0.005

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

TD - TemperatureDependence