Fluid Mechanics (Analysis of Fluid Flow motion) Essay Example

Mathematical modeling of viscous incompressible flow through a diffuser

a) Design flow

In this design we consider oil of viscosity 0.048 kg/m-s flowing at a mean velocity of 0.5m/s before the diffuser assuming that it flows in pipe of 0.1.5 m diameter before the diffuser. We assume the oil density is 900 kg/m3. The entire pipe is to be taken horizontal. The conical nose base diameter is 0.1m.the conical nose rotates at 1200rpm and the external tunnel is fixed

Calculating the Reynolds number we have.

Fluid Mechanics (Analysis of Fluid Flow motion)

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b) Governing equation

  1. Continuity equation

We use cylindrical coordinate to as shown below;

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Using the mass conservation in the control volume;

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  1. Momentum equation

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The summation of force includes the surface forces plus the body forces;

  1. The body forces includes the gravitational forces that is

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  1. The surface force is due to shearing stresses on all the surfaces of the control volume.

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Therefore the momentum equation simplifies to;

The momentum equation in r direction

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The momentum equation in
Fluid Mechanics (Analysis of Fluid Flow motion) 35 direction

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The momentum equation in z direction

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Fluid Mechanics (Analysis of Fluid Flow motion) 38The Lapsian operator
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C) Assumptions

  1. The tunnel before the diffuser is infinitely long such that the flow is fully developed,

  2. The flow is steady that is the flow properties do not change with time therefore all the time derivatives becomes zero,

  3. The fluid is incompressible and Newtonian with constant properties such as density and viscosity,

  4. A constant pressure drop is applied along the axial direction therefore the pressure gradient is constant all over,

  5. The flow is entirely parallel and there is no movement along the radial direction,

  6. The flow before the conical the nose do not tend to swirl and therefore all derivative with respect to
    Fluid Mechanics (Analysis of Fluid Flow motion) 40 are equal to zero,

  7. In this analysis we can assume the gravitational forces.

Fluid Mechanics (Analysis of Fluid Flow motion) 41Fluid Mechanics (Analysis of Fluid Flow motion) 42Therefore the Continuity differential equation reduce to

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Which means that
Fluid Mechanics (Analysis of Fluid Flow motion) 44 is not a function of z. again considering assumption ii the flow is steady therefore
Fluid Mechanics (Analysis of Fluid Flow motion) 45is not a function of time t henceFluid Mechanics (Analysis of Fluid Flow motion) 46 can only be a function of r. simplifying the momentum equation in the z direction (axial direction) we have

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Fluid Mechanics (Analysis of Fluid Flow motion) 50Fluid Mechanics (Analysis of Fluid Flow motion) 51Fluid Mechanics (Analysis of Fluid Flow motion) 52Fluid Mechanics (Analysis of Fluid Flow motion) 53Fluid Mechanics (Analysis of Fluid Flow motion) 54

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In the diffuser region the continuity would require that

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Considering our
Fluid Mechanics (Analysis of Fluid Flow motion) 66therefore

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Pressure actually increases

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When the fluid particle slides over the nose it will have two velocity components that is the tangential component and the axial component. To determine the axial component we must assume the nose does not rotate.

And therefore we apply the boundary condition for the equation,

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At r=D/2 the axial velocity is equal zero and therefore

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At r=R the axial velocity is equal to zero

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At r=D/2+(R-D/2)/2 velocity is maximum and therefore

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Evaluating for
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The axial velocity will be equal to

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To find the average velocity

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In our design flow

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To determine the tangential component we assume there is no end effect and therefore axial velocity is equal to zero, since the nose rotates at constant velocity therefore there is no variation of angular velocity with the change in the angle. The continuity equation reduce to,

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Fluid Mechanics (Analysis of Fluid Flow motion) 90Fluid Mechanics (Analysis of Fluid Flow motion) 91Fluid Mechanics (Analysis of Fluid Flow motion) 92Fluid Mechanics (Analysis of Fluid Flow motion) 93Fluid Mechanics (Analysis of Fluid Flow motion) 94Fluid Mechanics (Analysis of Fluid Flow motion) 95We also assume there is no motion along the r and therefore radial velocity is equal to zero and

the particles do not accelerate tangentially due to constant angular velocity. The pressure does

not also vary tangentially. Therefore the momentum equation reduce to

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Therefore

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Integrating

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Applying the boundary conditions

At r=D/2
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At r=R
Fluid Mechanics (Analysis of Fluid Flow motion) 103 0=Fluid Mechanics (Analysis of Fluid Flow motion) 104 solving this simultaneously the velocity profile is defined as

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The motion of fluid components moves over the nose with this two velocity components.

References

WHITE, F. M. (2001). Fluid mechanics. Boston, Mass, WCB/McGraw-Hill.

ÇENGEL, Y. A., & CIMBALA, J. M. (2006). Fluid mechanics: fundamentals and applications. Boston, McGraw-HillHigher Education.

SHAUGHNESSY, E. J., KATZ, I. M., & SCHAFFER, J. P. (2005). Introduction to fluid mechanics. New York, Oxford University Press. 

WEAST, R. C., & SELBY, S. M. (1995). Handbook of tables for mathematics. Cleveland, CRC Press.