The evaluation committee hereby recommends the Department of Mechanical Engineering, BUET, Dhaka, to accept the thesis,"NUMERICAL SIMULATION OF NON-NEWTONIAN FLUIDFLOWTHROUGHCONCENTRICANNULIWITHCENTERBODYROTATION", submitted by Md. Mamunur Rashid, in partial fulfillment of the master's degree. of Science in Mechanical Engineering. This is to confirm that the work presented in this thesis is the result of the studies that the candidate has carried out under the guidance of Dr.
Showkat J ahan Chowdhury, Professor, Department of Mechanical Engineering at Bangladesh University of Engineering and Technology (BUET) for his kind consent to supervise the thesis. The author is also very grateful to Dr. Sadrul Islam, Professor, Department of Mechanical Engineering in Bangladesh University of Engineering and Technology (BUET), for providing assistance at various stages of the work. Measurement of the axial and tangential velocity components is presented in nondimensional form for two fluids, one Newtonian and the other a shear-thinning non-Newtonian fluid.
The numerical predictions were confirmed by comparing them with the experimentally derived axial and tangential velocity profiles obtained for a Newtonian fluid and a non-Newtonian shear thinning polymer.
CHAPTER-l
INTRODUCTION
- BACKGROUND
- MOTIVATION BEHIND THE SELECTION OF THE STUDY
- IMPORTANCE OF NUMERICAL INVESTIGATION
- THE PROBLEMS AND OBJECTIVES
A detailed computational investigation will be carried out on non-Newtonian fluid flow through concentric annulus with body center rotation with Glucose and CMC as working fluid. The phenomenon of non-Newtonian fluid flow is very important in all pharmaceutical and many chemical industries. A computer program capable of predicting the flow of non-Newtonian fluids will be of great importance to the aforementioned fields.
The experimental investigation of non-Newtonian fluid flow is not only expensive and labor intensive, it is in many cases even impossible. The numerical simulation of concentric annular flow with rotation of the central body of a non-Newtonian fluid is not found in the literature. In the present study, a detailed computational investigation will be carried out on the non-Newtonian fluid flow through concentric annuli with central body rotation using glucose and CMC as working fluid.
The geometry and dimensions of the non-Newtonian fluid flow are based on the experimental studies of Escudier et al.
The Present Objective
THESIS OUTLINE
Finally, Chapter 6 presents the conclusions of the work and ends with some suggestions for future research.
CHAPTER-2
LITERATURE REVIEW
INTRODUCTION
PREVIOUS WORK
Experimental validation of extruder characteristics and typical temperature profiles in the screw channel for the flow of a highly viscous fluid shows good agreement and supports the model. 1991) numerically investigated, by solving the modified Navier-Stokes equations, the hydrodynamic evolution of non-Newtonian fluid flow in the entrance region of a pipe with a porous wall. The locations where the shear stress was zero and the velocity gradient was zero were displaced by amounts that, like the secondary flows measured in the 0.5 eccentric ring, were almost within measurement precision. In all cases, the corresponding fluctuations in the cross-flow direction were suppressed threefold due to the molecular stretching in a manner consistent with previous observations in the pipe flow.
This was less apparent in the narrower gap of the eccentric arrangements due to the low velocities, and corresponding deviations from the logarithmic wall law were observed. 1992) studied numerically the initiation of axisymmetric (toroidal) vortices in the flow between concentric cylinders of infinite length whose interior was rotating. The critical aspect ratio was found to exceed a maximum in the case of a wide ring like non-Newtonian character. They noted that large speed increases (for constant rotational speed) produce a progressive. decrease in tangential velocity level that is similar for Glucose and CMC. fluids, except for anomalous behavior for CMC at low Reynolds number.
In another article, Escudier et al. 1995a) demonstrated the applicability of turbulent annular flow in the absence of rotation of the central body using scaling criteria proposed by Hoyt (1991) for drag-reducing fluids in pipe flow.
CHAPTER-3
DEVELOPMENT AND MODELLING OF THE NUMERICAL SCHEME
INTRODUCTION
LAMINAR FLOW OF VISCOUS INCOMPRESSIBLE FLUIDS .1 Flow Through a Pipe- The HagenPoiseuille Flow
The term p(DviDt) in Eq. 3.6) vanishes because Vz =vz(r) and va =Vr =0, since both the radial and tangential velocity components are zero. Equation (3.8) is the balance of the shear force and compression force so that no acceleration or deceleration occurs. Since we have shown in the present case that Vz =vz(r) and p=p(z), Eq. 3.8) cannot be valid unless the pressure is a linear function of z.
With this condition, the constant B is obtained from Eq. Substituting the values of the constants A and B into Eq. 3.10) gives the axial velocity distribution of the Hagen-Poiseuille flow through a pipe. Equation (3.14) can also be obtained from the equilibrium condition in the z direction, which requires that the shear force acting on the circumferential surface of the cylinder is equal to the compressive force acting on the cross-sectional area of the cylinder. The velocity distribution given in Eq. 3.14) can only occur when the flow reaches a fully developed state and remains laminar.
In other words, the flow is initially, immediately after a pipe entrance, nearly uniform across the diameter of the pipe, and the velocity distribution does not completely transform from a nearly uniform profile to a parabolic profile until the flow has traveled about 100 pipe diameters of length from the entrance. From the solution of the velocity distribution in Eq. 3.14), becomes the shear stress at the wall after differentiation. The friction coefficient Cr given in Eq. 3.26) has been verified experimentally by G. Since Hagen Poiseuille's parabolic velocity distribution is an exact solution of the Navier-Stokes equations, it can be said that the Navier-Stokes equations have been verified to hold at least in the case of parallel flow.
The velocity distribution of a Newtonian viscous fluid flowing uniformly parallel to the axis in the annular space between two coaxial cylinders of radii fJ and r2 is given by Eq. 3.34). From the above equations, it can be seen that the shear stresses on both walls are positive, but the velocity gradient on the wall of the outer cylinder is negative. Assume that the flow is marginal, so that we have only the tangential component of the velocity Ye. Let COl be the uniform angular velocity of the inner cylinder.
The Navier-Stokes equation [Eq. Since, Ye is a function of r only, it follows from axial symmetry that the pressure in equation [3.45] must be either a function of r or a constant. Similarly, the shear stress in the inner cylinder wall can be obtained from Eqn.
CHAPTER-4
GOVERNING DIFFERENTIAL EQUATIONS AND SOLUTION PROCEDURE
- THE FORM OF GOVERNING DIFFERENTIAL EQUATIONS ADOPTED FOR USE IN THIS STUDY
- DISCRETIZED GOVERNING DIFFERENTIAL EQUATIONS
- DIFFERENCING SCHEMES USED IN TIllS STUDY
- SOLUTION PROCEDURE
- Grid and Variable Arrangement
- The Momentum Equations
- The Pressure and Velocity Corrections
- BOUNDARY CONDITIONS
- SOLUTION ALGORITHM
A review of the discretization method for the numerical solution of fluid flow problems is given by Patankar [1980]. The source terms are evaluated by integrating the volumetric source S
The name Hybrid denotes the combination of Central Difference Scheme (CDS) and Upwind Difference Scheme (UDS). This section discusses the differential schemes used to evaluate the nominal value of the dependent variable convection cell in terms of the surrounding nodal values. The value of
UDS also meets the property of transportivity and thus the boundedness of the solution is guaranteed. This layout realizes one of the main advantages of the staggered grid: the difference Pp-PE can be used to calculate the compressive force acting on the control volume for the speed u. Calculation of the diffusion coefficient and the mass flow at the surfaces of the u-control volume (Figure 4.2) would require appropriate interpolation.
The neighboring coefficients anb stand for the combined convection-diffusion effect at the control volume surfaces. The procedure developed for calculating the flow field has been named SIMPLE, which stands for Semi-Implicit Method for Pressure-Linked Equations. Numerical solution of the governing equation for the transport of momentum is achieved by using the SIMPLE algorithm.
For concentric rings of flow with body center rotation in the calculation domain can be done by inserting an 'internal' boundary condition. In this study, the convergence criterion is that the sum of the normalized absolute residual at all computational nodes is defined as .
CHAPTER-5
RESULTS AND DISCUSSIONS
- INTRODUCTION
- SOLUTION DOMAIN, COMPUTATIONAL GRID AND BOUNDARY CONDITIONS
- GRID INDEPENDENCE TEST
- RESULTS FOR NON-NEWTONIAN FLUIDS
- RESULTS FOR NEWTONIAN FLUIDS
- VALIDATION OF THE NUMERICAL MODEL
- CLOSURE
In Figure 5.3 at the last station the maximum speed for experimental result is 1.4 and for numerical solution it is 1.65. The tangential velocity levels within the annular gap increase progressively close to the center body. The tangential velocity increases rapidly over a thin layer to match the peripheral velocity of the center body.
It can be seen that the tangential velocity gradually transforms to developed profiles and the length to hydraulic diameter ratio curve XlDh = 104 of all figures is compared with experimental results of Escudier et al. Experimental tangential velocity curves of Escudier et al. 1995) are not in accordance with our numerical prediction. The tangential velocity gradients at the inner and outer surfaces are related by the following equation: [Escudier et al.
So that the tangential velocity gradient in the inner layer must be significantly higher than in the outer layer. From Figure 5.6 we see that the tangential velocity profile is 1995) showed that the tangential velocity reveals three distinct regions across the radial for all Reynolds number. The last curve (in length to hydraulic diameter ratio, XlDh = 104) of Figure 5.9 shows the developed velocity profile compared to experimental result of Escudier et al.
For both Figures 5.7 and 5.9, the maximum velocity occurs near the center of the annuli for Newtonian fluid. The gradual change of tangential velocity profile is shown in concentric annuli with midbody rotation. 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.
1995) showed for a Newtonian fluid that the tangential velocity gradients at the inner and outer surfaces are related to the following. So for the Newtonian fluid flow, the tangential velocity gradient in the inner layer must be significantly higher than in the outer layer.
CHAPTER-6
CONCLUSIONS AND RECOMMENDATIONS
GENERAL
SUMMARY OF MAIN FINDINGS AND ACHIEVEMENTS
SUGGESTION FOR FUTURE WORK
APPENDIX-A
Structurc of tilc Matilcmatical foundation
APPENDIX-B
