Mooney correction

A complete determination with the capillary rheometer includes the plotting of apparent flow curves with several dies having different lengths and diameters. This enables several corrections to be carried out, e.g. the Bag-ley correction, to separate the input pressure loss from the flow resistance inside the die the Weihenberg-Rabinowitsch correction to determine the true shear rate, and the Mooney correction to determine wall slippage speed [19, 20], and therefore the differentiated determination of the material s flow properties and the formulation of the material law. 

At this point, it is important to remark that there are needed corrections for end effects and wall slip. A discussion on Bagley and Mooney correction techniques is given in reference [2]. 

OtherRota.tiona.1 Viscometers. Some rotational viscometers employ a disk as the inner member or bob, eg, the Brookfield and Mooney viscometers others use paddles (a geometry of the Stormer). These nonstandard geometries are difficult to analy2e, particularly for an infinite bath, as is usually employed with the Brookfield and the Stormer. The Brookfield disk has been analy2ed for Newtonian and non-Newtonian fluids and shear rate corrections have been developed (22). Other nonstandard geometries are best handled by determining iastmment constants by caUbration with standard fluids. 

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... 

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 ... 

Some fluids, as discussed in Section VI, may appear to slip at the wall. This factor may be accounted for quantitatively by methods given by Mooney (M15), but corrections for this factor are necessary before the proposed correlating methods may be applied. 

Some further remarks concerning the Mooney-Rivlin equation are in place (14, 112). In dry rubber networks Ca in extension is often of the same order of magnitude as Cx, so that we are by no means confronted with a minor correction. In unilateral compression C2 is almost zero, and perhaps slightly negative. The constant Cx increases with the crosslinking density and with the temperature the ratio C2/C( in extension seems... 

Choosing the correct flow period is a very important aspect in the forming and shaping process. Mooney cure curves give good data on the flow period. The flow period also depends on the flow distances in the cavities of the mould and the viscosity of the rubber stock and the repeat pressure application cycles, called "bumping" cycles on the shop floor. 

Nakajima and Hamel50 have derived expressions for calculating shear stresses from the Mooney torque values to give viscosities in agreement with those obtained from other instruments, and also an expression to correct for the edge effects.51 Bristow52 derived non-standard Mooney parameters for natural rubbers to improve the distinction between different grades. 

Mooney-Rivlin correction. In the 1940 s, Mooney and Rivlin showed that, generally, the basic force-elongation relationship must be corrected by a term proportional to the reciprocal extension ratio ... 

Mooney M (1931) Trans Soc Rheol 2 210. It was evident even at the time of Navier that the meaningful way to quantify any level of wall slip is to express it in terms of the extrapolation length b [24]. Equation (4a) clearly shows that any correction would enter as the ratio of b to a characteristic dimension of the flow apparatus, e.g., the diameter D of capillary dies. It is unfortunate that Mooney abandoned the notion of the extra polation length b in favor of the slip velocity vs... 

The velocity difference across the test material is velocity to the velocity difference without slip [(v(,p, — v yp)/vbob] is the factor used to correct the shear rate. It must be noted that Mooney (1931) also suggested that one may use only two concentric cylinder systems and this suggestion was interpreted in a quantitative manner by Yoshimura and Prud homme (1988a) who derived the necessary equations for such systems. 

Qiu and Rao (Qiu, C. G. and Rao, M. A. J. Texture Stud., submitted) determined slip coefficients and slip velocities for apple sauce in a concentric cylinder viscometer as well as the effect of insoluble solids content on them. Three concentric cylinder units specified in the theory of Mooney (42.) were employed. Rotational speeds were determined with the different concentric cylinder systems at the same magnitude of torque. Figure 2 shows, for one sample of apple sauce, the shear rates uncorrected and corrected for slip plotted against the shear stress. The magnitudes of the flow behavior index of the power law model (Equation 2) did not change significantly due to correction for wall slip however, the magnitudes of the consistency index increased due to wall slip corrections. 

Wall effects in capillary flow of suspensions such as apple sauce occur as a result of velocity gradient near the wall that in turn causes the suspended particles to move away from the wall region. The net result is slip of the fluid at the wall (23., 24.) The correct shear rate can be calculated from flow rates corrected for slip. The procedure, due to Mooney (42.), requires the use of several capillaries of different different length to diameter ratios has been applied to food suspensions by Higgs (38) and Kokini and Plutchok (22.) to show that slip effects are significant. These results also suggest caution in using... 

Computer) The Mooney-Rivlin plots often show poor linearity at low strains. Considering that a likely systematic error is an inaccurate value of L0> the unstretched length of the sample, the observed A should be corrected by a multiplicative constant close to 1.0 to get the true stretch ratio a A. Using the form of the Mooney-Rivlin equation... 

The difference between single and twin-bore capillary rheometers (Fig. 9) cannot be dealt with in this contribution. Basically this is a matter of being able to make the so-called Bagley correction with less effort and fewer errors. Likewise, no reference can be made here as to how dilatant respectively structural viscose behaviour (Rabinowitsch correction) and the strain properties of ceramic bodies or wall slipping effects (according to the Mooney method) can be determined with the capillary rheometer. 

The occurrence of this phenomenon may be tested by comparing the viscosity ftmctions obtained using capillaries of similar length-to-radius ratios, L/R, but of different radii. Any apparent wall slip may then be corrected for and the true viscosity of the fluid determined by extrpolating the results obtained to infinite pipe diameter. In the relation developed by Mooney [1931], apparent wall shear rates obtained for constant length-to-radius ratio are plotted against L/K). 

Solution cross-linked elastomers also exhibit stress-strain isotherms in elongation that are closer in form to those expected from the simplest molecular theories of rubberlike elasticity. Specifically, there are large decreases in the Mooney-Rivlin 2C correction constant described in... 

Mark, J. E., The Volume Dependence of the C2 Correction in the Mooney-Rivlin Elastic Equation of State. J. Polym. Sci., Part C Polym. Symp. 1970,31,97-106. 

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