The Power-Law Model

From the previous equations, the fraction released q (t) /as a function of time t is obtained  

This equation describes the entire fractional release curve for Case II drug transport with axial and radial release from a cylinder. Again, (4.10) indicates that the smaller dimension of the cylinder p or L) determines the total duration of the phenomenon. When p L, (4.10) can be approximated by 

Peppas and coworkers [64,68] introduced a semiempirical equation (the so-called power law) to describe drug release from polymeric devices in a generalized way  

From the values of A listed in Table 4.1, only the two extreme values 0.5 and 1.0 for thin films (or slabs) have a physical meaning. When A = 0.5, pure Fickian diffusion operates and results in diffusion-controlled drug release. It should be recalled here that the derivation of the relevant (4.3) relies on short-time approximations and therefore the Fickian release is not maintained throughout the release process. When A = 1.0, zero-order kinetics (Case II transport) are justified in accord with (4.4). Finally, the intermediate values of A (cf. the inequalities in Table 4.1) indicate a combination of Fickian diffusion and Case II transport, which is usually called anomalous transport. 

It is interesting to note that even the more realistic model adhering to the Case II radial and axial drug release from a cylinder, (4.10), can be described by the power-law equation. In this case, pure Case II drug transport and release is approximated (Table 4.1) by the following equation  

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A common choice of functional relationship between shear viscosity and shear rate, that u.sually gives a good prediction for the shear thinning region in pseudoplastic fluids, is the power law model proposed by de Waele (1923) and Ostwald (1925). This model is written as the following equation... 

Step 3 - using the calculated velocity field, find the shear rate and update viscosity using the power law model. 

VISCA Calculates shear dependent viscosity using the power law model. 

TREF == REFERENCE TEMPERATURE C AAA tRCO = COEFFICIENT b IN THE POWER LAW MODEL... 

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... 

Rules. Eliminate temperature terms in the denominator. (Terms with negative exponents in the power law model are considered to belong to the denominator, in the hyperbolic model. Author.)... 

A useful two-parameter model is the power-law model, or Ostwald-de Waele law to identify its first proponents. The relation between shear stress and shear rare is given by ... 

For a shear-thickening fluid the same arguments can be applied, with the apparent viscosity rising from zero at zero shear rate to infinity at infinite shear rate, on application of the power law model. However, shear-thickening is generally observed over very much narrower ranges of shear rate and it is difficult to generalise on the type of curve which will be obtained in practice. 

In order to overcome the shortcomings of the power-law model, several alternative forms of equation between shear rate and shear stress have been proposed. These are all more complex involving three or more parameters. Reference should be made to specialist works on non-Newtonian flow 14-171 for details of these Constitutive Equations. 

If it is known that a particular form of relation, such as the power-law model, is applicable, it is not necessary to maintain a constant shear rate. Thus, for instance, a capillary tube viscometer can be used for determination of the values of the two parameters in the model. In this case it is usually possible to allow for the effects of wall-slip by making measurements with tubes covering a range of bores and extrapolating the results to a tube of infinite diameter. Details of the method are given by Farooqi and Richardson. 21 ... 

The rheological properties of a particular suspension may be approximated reasonably well by either a power-law or a Bingham-plastic model over the shear rate range of 10 to 50 s. If the consistency coefficient k is 10 N s, /m-2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham-plastic model ... 

What will be the pressure drop, when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power-law model Calculate the centre-line velocity for this pressure drop for the Bingham-plastic model. 

A Newtonian liquid of viscosity 0.1 N s/m2 is flowing through a pipe of 25 mm diameter and 20 m in lenglh, and the pressure drop is 105 N/m2. As a result of a process change a small quantity of polymer is added to the liquid and this causes the liquid to exhibit non-Newtonian characteristics its rheology is described adequately by the power-law model and the flow index is 0.33. The apparent viscosity of the modified fluid is equal to ihc viscosity of the original liquid at a shear rate of 1000 s L... 

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

A liquid w hose rheology can be represented by the power law model is flowing under streamline conditions through a pipe of 5 mm diameter. If the mean velocity of flow in I nt/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law index n ... 

The Arrhenius plots for both sets of kinetic parameters together with experimental points are shown in Fig. 5.4 -32. Experimental points scatter uniformly on both sides of the straight lines indicating that the power-law model with the evaluated rate constant can be satisfactorily used to describe the kinetic experiments under consideration. 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

The power of this technique is two-fold. Firstly, the viscosity can be measured over a wide range of shear rates. At the tube center, symmetry considerations require that the velocity gradient be zero and hence the shear rate. The shear rate increases as r increases until a maximum is reached at the tube wall. On a theoretical basis alone, the viscosity variation with shear rate can be determined from very low shear rates, theoretically zero, to a maximum shear rate at the wall, yw. The corresponding variation in the viscosity was described above for the power-law model, where it was shown that over the tube radius, the viscosity can vary by several orders of magnitude. The wall shear rate can be found using the Weissen-berg-Rabinowitsch equation ... 

The plot of viscosity versus shear rate is shown in Fig. 3-3b, in which the line represents Eq. (3-24), with n = 0.77 and m = 1.01 dyn s /cm2 (or poise ). In this case the power law model represents the data quite well over the entire range of shear rate, so that n = n is the same for each data point. If this were not the case, the local slope of log T versus log rpm would determine a different value of n for each data point, and the power law model would not give the best fit over the entire range of shear rate. The shear rate and viscosity would still be determined as above (using the local value of n for each data point), but the viscosity curve could probably be best fit by some other model, depending upon the trend of the data (see Section III). 

If the data (either shear stress or viscosity) exhibit a straight line on a log-log plot, the fluid is said to follow the power law model, which can be represented as... 

The two viscous rheological properties are m, the consistency coefficient, and n, the flow index. The apparent viscosity function for the power law model in terms of shear rate is... 

You would like to determine the pressure drop in a slurry pipeline. To do this, you need to know the rheological properties of the slurry. To evaluate these properties, you test the slurry by pumping it through a 1 in. ID tube that is 10 ft long. You find that it takes a 5 psi pressure drop to produce a flow rate of 100 cm3/s in the tube and that a pressure drop of 10 psi results in a flow rate of 300cm3/s. What can you deduce about the rheological characteristics of the slurry from these data If it is assumed that the slurry can be adequately described by the power law model, what would be the values of the appropriate fluid properties (i.e., the flow index and consistency parameter) for the slurry ... 

A film of paint, 3 mm thick, is applied to a flat surface that is inclined to the horizontal by an angle 9. If the paint is a Bingham plastic, with a yield stress of 150 dyn/cm2, a limiting viscosity of 65 cP, and an SG of 1.3, how large would the angle 9 have to be before the paint would start to run At this angle, what would the shear rate be if the paint follows the power law model instead, with a flow index of 0.6 and a consistency coefficient of 215 (in cgs units) ... 

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) models—the power law and Bingham plastic—can be evaluated in a similar manner. The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results. 

Because the shear stress and shear rate are negative in pipe flow, the appropriate form of the power law model for laminar pipe flow is... 

A coal slurry is to be transported by pipeline. It has been determined that the slurry may be described by the power law model, with a flow index of 0.4, an apparent viscosity of 50 cP at a shear rate of 100 s-1, and a density of 90 lbm/ft3. What horsepower would be required to pump the slurry at a rate of 900 gpm through an 8 in. sch 40 pipe that is 50 mi long ... 

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