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Fig. 11. Clear annulus model: normalized core radius vs. Reynolds number. Slip velocities determined in the manner described in the preceding section are plotted against the slurry con- sistency in Figure 9, Also shown on this figure is the curve predicted by the Kozeny-Carman equation, Equation (21), with Equation (33) being used to predict the Kozeny factor. The data encompass all the runs in the pipe for

Figure 11 Clear annulus model: normalized core radius vs. Reynolds number. Slip velocities determined in the manner described in the preceding section are plotted against the slurry con- sistency in Figure 9, Also shown on this figure is the curve predicted by the Kozeny-Carman equation, Equation (21), with Equation (33) being used to predict the Kozeny factor. The data encompass all the runs in the pipe for

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The clear annulus model is described in part by the equations for the liquid phase shear stress: the relationship between the mean and maximum velocities (and hence between the mean residence time and the breakthrough time)
The clear annulus model is described in part by the equations for the liquid phase shear stress: the relationship between the mean and maximum velocities (and hence between the mean residence time and the breakthrough time)
Slurry Flow System
Slurry Flow System
Fig. 2. Flow circuit.
Fig. 2. Flow circuit.
Fig. 3. Settling velocities: Carroll’s correlation for Kozeny factor. g y was one twentieth of its zero frequency value. The Fourier transform of the system residence time distribution (the system transfer function) was calculated as the quotient of the output and input peak transforms, and the distribu- tion itself was obtained by inverting the transform, again using the fast Fourier transform algorithm. in water, and the manganese tracer solution was mixed with an approximately equal volume of inactive solution which served as a carrier. Both tracers were injected by means of syringes with 13 gauge needles 9 cm long, so that the tip could be positioned at the flow axis, The dura- tion of the injection was less than 1 s.
Fig. 3. Settling velocities: Carroll’s correlation for Kozeny factor. g y was one twentieth of its zero frequency value. The Fourier transform of the system residence time distribution (the system transfer function) was calculated as the quotient of the output and input peak transforms, and the distribu- tion itself was obtained by inverting the transform, again using the fast Fourier transform algorithm. in water, and the manganese tracer solution was mixed with an approximately equal volume of inactive solution which served as a carrier. Both tracers were injected by means of syringes with 13 gauge needles 9 cm long, so that the tip could be positioned at the flow axis, The dura- tion of the injection was less than 1 s.
Fig. 4. Settling velocities: Labrecque correlation for Kozeny factor.
Fig. 4. Settling velocities: Labrecque correlation for Kozeny factor.
Fig. 5. Settling velocities: Modified Labrecque correlation for Kozeny factor. Model parameters were initially determined by applying nonlinear regression in the time domain using the decon- voluted impulse responses (Harrison, Felder, and Rous- seau, 1974). The results offered little improvement over simpler methods based on moment differences and peak maximum lags, however, and so the latter methods were used throughout most of the study.
Fig. 5. Settling velocities: Modified Labrecque correlation for Kozeny factor. Model parameters were initially determined by applying nonlinear regression in the time domain using the decon- voluted impulse responses (Harrison, Felder, and Rous- seau, 1974). The results offered little improvement over simpler methods based on moment differences and peak maximum lags, however, and so the latter methods were used throughout most of the study.
EXPERIMENTAL RESULTS
EXPERIMENTAL RESULTS
Fig, 7. Deconvoluted liquid phase residence time distribution. The improvement of the fit when Labrecque’s correla- tion is used is clear from visual inspection of Figure 4 and is confirmed by the calculated correlation coefficient R = 0.965. This agreement between theoretical predic- tion and experimental data suggests the appropriateness of the model for the pulp slurry used in this study and validates two assumptions in particular: the representation of interphase drag by Darcy’s law and the applicability of the Kozeny-Carman equation for the viscous flow resist- ance,
Fig, 7. Deconvoluted liquid phase residence time distribution. The improvement of the fit when Labrecque’s correla- tion is used is clear from visual inspection of Figure 4 and is confirmed by the calculated correlation coefficient R = 0.965. This agreement between theoretical predic- tion and experimental data suggests the appropriateness of the model for the pulp slurry used in this study and validates two assumptions in particular: the representation of interphase drag by Darcy’s law and the applicability of the Kozeny-Carman equation for the viscous flow resist- ance,
Typical scaled responses are shown in Figure 8. The mean residence times of each phase were determined from the peak response times (the symmetry of the peaks made this technique preferable to the more cumbersome method of moments), and the slip velocity was calculated from the difference in the residence times.
Typical scaled responses are shown in Figure 8. The mean residence times of each phase were determined from the peak response times (the symmetry of the peaks made this technique preferable to the more cumbersome method of moments), and the slip velocity was calculated from the difference in the residence times.
Fig. 13. Residence time distributions for four slurry consistencies. "The greatest deviations between experiment and theory appearing in Figure 9 are at the lower pulp consistencies, It is apparent that both laminar flow models provide a far more accurate representation of the behavior of the system than does the dispersion model. It is difficult to choose between the clear annulus and power law models on the basis of goodness of fit; both models qualitatively simulate the experimental response, but the data points show a degree of axial dispersion about the mean which cannot be represented by either model. The spread of the experimental distribution may be attributable in part to imperfect collimation of the response peaks, but it is more likely a consequence of two contraventions of several as- sumptions inherent in the models, specifically, a flat core
Fig. 13. Residence time distributions for four slurry consistencies. "The greatest deviations between experiment and theory appearing in Figure 9 are at the lower pulp consistencies, It is apparent that both laminar flow models provide a far more accurate representation of the behavior of the system than does the dispersion model. It is difficult to choose between the clear annulus and power law models on the basis of goodness of fit; both models qualitatively simulate the experimental response, but the data points show a degree of axial dispersion about the mean which cannot be represented by either model. The spread of the experimental distribution may be attributable in part to imperfect collimation of the response peaks, but it is more likely a consequence of two contraventions of several as- sumptions inherent in the models, specifically, a flat core
which the Reynolds number was less than 1 100, so that flow instabilities and pulse overlapping were not compli- cating factors. The scatter in the data is substantial, but no more could be expected in view of the low magnitudes of the slip velocities and approximations in the data anal- ysis required to determine them. The trend predicted by the model can be clearly observed; however, the model appears to consistently underpredict the slip velocity, a point to which we will subsequently return.
which the Reynolds number was less than 1 100, so that flow instabilities and pulse overlapping were not compli- cating factors. The scatter in the data is substantial, but no more could be expected in view of the low magnitudes of the slip velocities and approximations in the data anal- ysis required to determine them. The trend predicted by the model can be clearly observed; however, the model appears to consistently underpredict the slip velocity, a point to which we will subsequently return.
Fig. 14. Predicted ys. measured solid phase mean residence time.
Fig. 14. Predicted ys. measured solid phase mean residence time.
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