RESULTS AND DISCUSSIONS

5.0 INTRODUCTION

In

this Chapter, the results of numerical simulation of non-Newtonian fluid flow through concentric annuli with center body rotation are presented and compared with the experiments of Escudier et al. (1995). The results are obtained by the numerical method described in Chapter 4.

5.1 SOLUTION DOMAIN, COMPUTATIONAL GRID AND BOUNDARY CONDITIONS

The solution domain is shown in Figure 1.0. The domain was bounded by the inlet plane, exit plane, outside solid wall, inside wall with constant rotational speed and the axis symmetry. The entire investigation domain is divided into 2000 x 32 grids. The distribution of these grids were nonuniform. A fine grid spacing was used near the solid walls and a relative course grid was used in the flow region.

For the present study the following values of parameters are chosen.

For CMC:

Power law index n 0.75

Consistency index K ^{0.04 N-s/m2}

Density p ^lOOOkg/m3

For

Glucose:

Power law index n 1.00

Consistency index K ^{0.01 N-s/m2}

Density

P

^{1134 kg/m3}

Both

Cases:

Outer Radius Ro 0.0502 m

Inner Radius Ri 0.0254 m

Length X 5.775 m

Rotational speed of inner pipe N 126 rpm

5.2

GRID

INDEPENDENCE TEST

It is necessary to test whether the predicted results are independent of grid. At a constant Reynolds number, Re = 800 with 32x22, 42x32 and 52x32 grids. The 52x32 grid gave reasonably grid predictions when compared to the theoretical predicted of Yuan et al. [1969] as shown in Figure 5. I I.

5.3 RESULTS FOR NON-NEWTONIAN FLUIDS

Axial Velocity Profiles:

Figures 5. I, 5.3 and 5.5 represent the developing velocity profiles Le. dimensionless velocity as a function of dimensionless radius. The profiles at different non- dimensional axial distances are shown in such a way so that the gradual change in profiles from flat to developed parabolic type can be easily inspected. The last curve (at length to hydraulic diameter ratio, XlDh = 104 )in each case shows the developed velocity profile and compared with experimental results of Escudier et al. (1995). The prediction is for the iso-viscous laminar flow of fluids. From Figure 5. I the last curve of the graph shows the developed velocity profile and points of experimental result from Escudier et. al. (1995). The maximum velocity for experimental result is 1.7 and for numerical solutions it is 1.5. Hence a percentage deviation of 12% in maximum velocity is observed at the last station. In the Figure 5.3 at the last station the maximum velocity for experimental result is 1.4 and for numerical solution it is 1.65. Hence a percentage deviations of 17% in maximum velocity observed. For both figures 5. I and 5.3, it is seen for laminar flow (low Reynolds number respectively I IO and 350) the maximum velocity occurs near the inner wall. Higher rotation of inner pipe gives rise to higher shear stress adjacent to the inner wall resulting lower viscosity, consequently maximum velocity near the inner wall occurs due to lower shear stress. The flow becomes turbulent at Re=4400, the present prediction is carried out under the laminar flow assumption. This is because the study of turbulent flow is beyond the scope of present study. The results obtained for Re=4400 is presented in Figure 5.5. This Figure shows that the maximum velocity for experimental result is 1.22 and for numerical solution it is 1.423. Hence a percentage deviation of 16% in maximum velocity observed at the last station. Due to existence of turbulence the experimental velocity profile appears to be flatter than that by the present numerical prediction.

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Tangential Velocity Profiles:

Figures 5.2, 5.4 and 5.6 represent the tangential velocity profiles. The tangential velocity levels within the annular gap are progressively increased close to the center body. The same qualitative behavior was found by Escudier et al. (1995). The tangential velocity increases rapidly across a thin layer to match the peripheral speed of the center body. It is seen that the tangential velocity gradually transforms to developed profiles and the curve for the length to hydraulic diameter ratio XlDh = 104 of all the figures are compared with experimental results of Escudier et aI. (1995). All figures refer to data for the rotation speed of 126 rpm. In Figure 5.2, it is observed that the curves for length to hydraulic diameter ratio of 50 and 104 superimpose. Experimental tangential velocity curves of Escudier et al. (1995) are not in agreement with our numerical prediction. The reason is that Escudier et aI. (1995) mentioned that in their showed experiments turbulent diffusion was present. Due to this turbulent diffusion the fluid particles move from higher velocity region to lower velocity region and hence uniform velocity occurs at the central region of the annuli. But in our numerical scheme turbulent diffusion was not considered. In Figure 5.8 Newtonian laminar tangential flow for Glucose at Reynold number 800 is compared with the analytical data for Equation (3.50) and it is found to be in excellent agreement. This shows the validity of present numerical predictions. In the experiments of Escudier et aI. (1995), the center body is slightly distorted, and so it is impossible to achieve a concentric geometry over the entire length of the test section. In our numerical prediction this eccentricity was not considered. In contrast to the negligible influence of friction factor, rotation has a strong influence on the tangential mean velocities which generally reveal a triple layer structure of speed, Nouri and Law (1994) also reported very similar observation for CMC. As the Reynolds number is increased the tangential.velocity levels within the annular gap are progressively reduced except for Re 110, which is shown in Figure 5.12. This has also been observed in the experiment of Escudier et aI. (1995). The tangential velocity gradients at the inner and outer surfaces are related by the following equation: [Escudier et aI. (1995)]

( J

²

dw R

^o

dw

- -0)+ - -

dr , R, dr

⁰

So that the tangential velocity gradient in the inner layer must be substantially higher than in the outer layer. From Figure 5.6 we see that the tangential velocity profile is

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gradually developed. Escudier et al. (1995) has shown that the tangential velocity reveal three distinct regions across the radial for all Reynolds number. Flow regions has been categorized as region adjacent to inner pipe, central region of almost constant velocity and region adjacent to outer pipe.Present prediction reveals a pattern of exponential decay of tangential velocity towards the outer wall. This is due to laminar assumption in the present study.

5.4 RESULTS FOR NEWTONIAN FLUIDS

Axial Velocity Profiles:

Figures 5.7 and 5.9 represent the developing velocity profiles. The profiles at different non-dimensional axial distance is shown in such a way so that the gradual changes in profiles from flat to developed parabolic type can be easily inspected. The last curve (at length to hydraulic diameter ratio, XlDh = 104 ) of Figure 5.9 shows the developed velocity profile compared with experimental result of Escudier et al. (1995). The last curve (at length to hydraulic diameter ratio, XlDh = 104) of Figure 5.7 is compared with laminar Newtonian profile. From Figure 5.9, the maximum velocity for experimental result is 1.22 and numerical solution 1.38. Hence percentage of deviation of 13% in maximum velocity observed at the last station. Hence again the difference may have occurred due to developed of turbulence by the inner rotating pipe. From Figure 5.7, the maximum velocity for laminar Newtonian profile is 1.484 and numerical solution 1.49. Hence percentage of deviation 0.5% in maximum velocity profile observed. This indicate the validity of the present methodology. For both Figures 5.7 and 5.9 the maximum velocity occurs near center of the annuli for newtonian fluid.

Tangential Velocity Profiles:

Figures 5.8 and 5.10 represents the tangential velocity profiles for Newtonian fluids.

The gradual change of tangential velocity profile are shown in concentric annuli with center body rotation. The last curve (at length to hydraulic diameter ratio, XlDh =104 ) of Figure 5.8 is in excellent agreement with the theoretical data. The last curve (at length to hydraulic diameter ratio, XlDh = 104 ) of Figure 5.10 is compared with the experimental data of Escudier et al. (1995). Also in the case of Newtonian fluid it is shown that as the Reynolds number is increased, the tangential velocity levels within the annular gap are progressively reduced. The same qualitative behavior was found by

Escudier et al. (1995) and Nouri and Law (1994). Due to the turbulent diffusion of fluid at Reynold number=1200 this predicted results shown in Figure 4.10 are not in good agreement with our numerical prediction. This has already been explained for the case of non-Newtonian flow cases. Escudier et al. (1995) showed for a Newtonian fluid, the tangential velocity gradients at the inner and outer surfaces are related by the following

dw (R

^o

)2 ^dW

dr

ⁱ =⁽⁰

+ R

ⁱ

dr

°

So that for the Newtonian fluid flow the tangential velocity gradient in the inner layer must be substantially higher than in the outer layer. This expression has a consequence of the torque being constant within the annular gap and the assumption of laminar sublayers at each surface. The situation for a non-Newtonian fluid is more complex, although qualitatively, the same trend evidently exist. The present prediction failed to reproduce this behavior.

5.5 VALIDATION OF THE NUMERICAL MODEL

To validate the present methodology primarily a series of predictions are performed for Reynold numbers, 110, 350, 800, 1200 and 4400. The numerical predictions are compared with the experimental results of Escudier et al. (1995) at length to hydraulic diameter ratio 104. It appears that the predictions of the present methodology are in good agreement with experimental data and hence it may be concluded that present numerical model has the capability of predicting laminar axial and tangential flows through concentric annuli with center body rotation with reasonable accuracy.

5.6 CLOSURE

The present numerical methodology is used to simulate the laminar axial and tangential flow through concentric annuli with center body rotation. Numerical predictions are compared with experimental data and found to have reasonably good matching. The flow was investigated for different Reynolds number. Useful non-dimensionalized results are provided with discussions.

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