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Herschel Bulkley Model

Herschel Bulkley Model

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Herschel Bulkley Model

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

hers_bulk_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.

HB2

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

Polymer_name

{

 

ConstitutiveModel =

"HerschelBulkley"

   

Density =

ρ

   

SpecificHeat =

Cp(T)

   

Conductivity =

K(T)

   

CoeffOfThermalExpansion =

βΤ

   

VolumetricHeatSource =

Qvol

   

Consistency =

A

   

Exponent =

n

 

YieldStress =

τ0

   

StressExponentGrowth =

m

 

TemperatureDependence =

"None"}

 

Explanation of Parameters

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 electrical heating

W/m^3

Constant

Required

0.0

Consistency

One of the parameters of the Herschel—Bulkley 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 Power law model

1/s

Constant

Required

0.01

YieldStress

One of 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

"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.

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

 

 

See Also:

Polymer Material Properties