Equations Mooney

Viscosity of Systems with Dispersed Phases. A large proportion of coatings are pigmented and, therefore, have dispersed phases. In latex paints, both the pigments and the principal polymer are in dispersed phases. The viscosity of a coating having dispersed phases is a function of the volume concentration of the dispersed phase and can be expressed mathematically by the Mooney equation (96), a convenient form of which is... 

If there is particle—particle interaction, as is the case for flocculated systems, the viscosity is higher than in the absence of flocculation. Furthermore, a flocculated dispersion is shear thinning and possibly thixotropic because the floccules break down to the individual particles when shear stress is appHed. Considered in terms of the Mooney equation, at low shear rates in a flocculated system some continuous phase is trapped between the particles in the floccules. This effectively increases the internal phase volume and hence the viscosity of the system. Under sufficiently high stress, the floccules break up, reducing the effective internal phase volume and the viscosity. If, as is commonly the case, the extent of floccule separation increases with shearing time, the system is thixotropic as well as shear thinning. 

In order to determine the true shear rate at the wall it is necessary to use the Rabinowitsch-Mooney equation ... 

This material is seen to be shear thinning. It is possible that it may exhibit a yield stress but confirmation of this would require measurements at lower shear rates. Note that the Rabinowitsch-Mooney equation is still valid when a non-zero yield stress occurs. 

When data are available in the form of the flow rate-pressure gradient relationship obtained in a small diameter tube, direct scale-up for flow in larger pipes can be done. It is not necessary to determine the r-y curve with the true value of y calculated from the Rabinowitsch-Mooney equation (equation 3.20). 

Equation 3.29 is helpful in showing how the value of the correction factor in the Rabinowitsch-Mooney equation corresponds to different types of flow behaviour. For a Newtonian fluid, n = 1 and therefore the correction factor has the value unity. Shear thinning behaviour corresponds to < 1 and consequently the correction factor has values greater than unity, showing that the wall shear rate yw is of greater magnitude than the value for Newtonian flow. Similarly, for shear thickening behaviour, yw is of a... 

The minus sign has been placed inside the parentheses recognizing the fact that the shear rate y (equal to dvjdr) is negative. yw is the true shear rate at the wall and is related to the flow characteristic (8a/d,) by the Rabino-witsch-Mooney equation ... 

When trying to determine the flow behaviour of a material suspected of exhibiting wall slip, the procedure is first to establish whether slip occurs and how significant it is. The magnitude of slip is then determined and by subtracting the flow due to slip from the measured flow rate, the genuine flow rate can be determined. The standard Rabinowitsch-Mooney equation can then be used with the corrected flow rates to determine the tw-jw curve. Alternatively, the results can be presented as a plot of tw against the corrected flow characteristic, where the latter is calculated from the corrected value of the flow rate. 

This must be done for each of a range of values of the wall shear stress tw. The standard Rabinowitsch-Mooney equation can then be used with the corrected values of uc ... 

Providing that c is less than 0.1, good agreement with the Einstein equation is found when glass spheres are suspended in ethylene glycol. Equation 8.1 has been modified by including a hydrodynamics or crowding factor (/8). The modified Mooney equation (8.2) resembles the Einstein equation when (3 = 0. 

Still other equations for concentrated suspensions depart from the above forms. There are a number of variations on the Mooney equation [393],... 

The Rabinowitch-Mooney equation gives the total volumetric flowrate Q through the pipe as ... 

Accepting the elastic floe model as a reasonable description of the structured suspension, it is possible to calculate a few more parameters from the experimental results. For example C p may be calculatedfrom which, in turn, may be obtained from the plastic viscosity, ripj using the Mooney equation (23),... 

The first term in Eq. (17.17) is the flow rate dne to shearing, and the second one is the flow rate dne to slip. The apparent shear rate in case of wall slip is given by the Mooney equation [5] ... 

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