Modified Cross Model

Modified Cross model is expressed by the following equation. This model has three parameters: consistency A, exponent n, and reference shear stress t*. Consistency is modified using the temperature dependent function a_T to introduce the effect of temperature on the computed viscosity. There is a very subtle difference between Cross and Modified Cross model, this difference enables (by appropriate choice of parameters) one to easily set up Carreau-Yasuda model to mimic the modified Cross model.

mcross

This model describes the viscosity using a power-law relationship. The factor (A alpha_T/tau*) is like a time constant in the equation. Depending on the value of the exponent used, the nature of the model will change.

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. Refer to the section Power Law Model for a detailed discussion of this concept.

Explanation of Parameters

Parameter	Description	Units	Data Type	Condition	Typical Value
ConstitutiveModel	Describes the model used	None	String	Required	"ModifiedCross"
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 modifies Cross model	Pa s	Constant	Required	1.0e+09
Exponent	Power law index, defines the dependency of viscosity on shear rate.	None	Constant	Required	0.66
ReferenceShearStress	One of the parameters of the modified Cross model.	Pa	Constant	Required	75000
ZeroShearRateLimit	Defines the trunctation limit for shear rate. See Power law model	1/s	Constant	Required	0.01
TemperatureDependence	See Temperature Dependence	None	String	Required	"WLF
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 the 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

Explanation of Parameters

See Also: