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 CarreauYasuda model to mimic the modified Cross model.
This model describes the viscosity using a powerlaw 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.
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 
1.0e05 
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 
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