Model Sisko

If the liquid flowing in a pipe/tube can be described by the Sisko model (q =kj n-i +r ), then the pressure drop is given (to a very good approximation) by 

Where V is the average velocity in the pipe (= Q/na ) of radius a and length L This equation reduces to the power-law version (see relevant equation above) when poo is zero. It is very good for describing the flow of most emulsions and dispersions through pipes. 

The Casson fluid is essentially the same as its Bingham cousin, but with all the components raised to the half power, so 

Notice that this equation reduces to the Newtonian version at very high wall shear stresses, when the wall shear stress Oiy is much greater than the yield stress Oo. 

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The power law model can be extended by including the yield value r — Tq = / 7 , which is called the Herschel-BulMey model, or by adding the Newtonian limiting viscosity,. The latter is done in the Sisko model, 77 +. These two models, along with the Newtonian, Bingham, and Casson... 

If r/0 (i], 1] ) and (ky)2 >S> 1, the Carreau model reduces to the equivalent of a power law model with a high shear limiting viscosity, called the Sisko model ... 

If the value of p is set equal to 1/2 in the Sisko model, the result is equivalent to the Bingham plastic model ... 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

O = [i ocK Sisko model for high-shear rate data containing noo (Brodkey, 1967)... 

The power law does not describe the regions of the viscosity curve near y = 0 and y —> oo. To this end, the Ellis model at low shear rates and the Sisko model at high shear rates can be used (2). The models are given by... 

Other practical applications for the Sisko model have been reported for yogurt and polymer liquid crystals. ... 

If the linear plot shows curvature, the data should be plotted on a logarithmic basis. From this plot we can see if we have a reasonable straight line, or a straight-line and some curvature. If a fair straight line is seen, then the data can be submitted for power-law regression and the k and n extracted and used for prediction. Last, if there is considerable curvature, then the Sisko model, or if necessary the complete Cross equation is indicated. However, here we have a problem, because the equation is nonlinear and simple regression analysis is inadequate. However, most modem viscometers have suitable software for this purpose, otherwise other specialised commercial mathematical software. 

The Bingham model is the most non-Newtonian example of the Sisko model, which is itself a simplification of the Cross (or Carreau model) under the appropriate conditions [3]. 

Qoss model Sisko model Sisko model Cross or Sisko model Cross model... 

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