Fluid Flow Through an Expanding Channel - Assignment Example

FLUID FLOW THROUGH AN EXPANDING CHANNEL Student University FLUID FLOW THROUGH AN EXPANDING PIPE Introduction Fluids flow in pipes that may have a uniform or non-uniform cross-section. Depending on the shape of the pipe, the fluid will flow so that it can fill the pipe cross-section at all times. When the fluid flows through the tunnels, the pressure developed will depend on whether the tunnel is open all closed. When the tunnel is open, the pressure developed is always equivalent to the atmospheric pressure while the pressure in a closed tunnel may be greater, less than or equal to the atmospheric pressure depending on the radius of the tunnel as well as the experienced physical conditions. It is common for a fluid to develop a shear stress when it flows over any solid boundary; this will tend to slow the velocity of the fluid since its velocity is opposed by the developed frictional force. Fluid properties The fluid is incompressible i.e. the density of the fluid remains constant in the tunnel. The flow is laminar The fluid fills the tunnel The mass of the fluid is conserved throughout the length of the tunnel The fluid flow is continuous The surface roughness of the tunnel is constant throughout its length . In the flow through the expanding tunnel, key factors on fluids are considered; For the flow to be steady; Conservation of mass equation Therefore; In cylindrical coordinate system, a fluid point can be located by the parameters as shown in figure 1 Figure 1:a system showing the cylindrical coordinate From the cylindrical coordinate system as shown on figure 1; Change with the space position. is fixed The independent variable therefore is while the dependent variables are in this system Therefore, to determine a position in the X direction in the system, the expression below will be used. Therefore the velocity vector in this system will be as given below Where; Are the main parameters of velocity in the cylindrical system Therefore equation (1) becomes; Where; When the liquid is incompressible, the density is constant. Therefore, equation (4) becomes; Figure 2: tunnel section Figure 1 shows a tunnel with varying areas, the tunnel expands as the distance from the inlet increases. To analyze the fluid flow through the tunnel, the whole system is divided into three tunnel sections. The areas and the velocities corresponding to those tunnels are also considered. The quantity of the fluid (Q) entering the tunnel must be equal to the quantity of the fluid leaving the tunnel. In this case, therefore; But, Then equation (7) becomes; Continuity equation This is the same as equation (6). This shows that the mass of the fluid is conserved for the flow to be continuous. To determine the velocity at any point in the tunnel, the formula below can be used. This implies that, the velocity is indirectly proportional to the cross-sectional area. To determine the pressure loss in the tunnel, the velocity and the viscosity of the fluid are very important parameters to be considered. The figure below will be used to derive and determine the pressure loss. Figure 3: tunnel dimensions The shear force experienced on the walls of the tunnel will be obtained by; Where; For the incompressible fluid; Equation (10) then becomes; To determine the force due to the difference in the pressure between the inlet and the outlet of the tunnel, the equation 14 is applied. Where; For the flow to be laminar and steady; Therefore; Then; Simplifying; The velocity is zero when and maximum when, therefore, to obtain the velocity at any point on the radius, integration based on the limits is necessary. In that; Integrating equation 19, the velocity, v, can be obtained as; Equation 20 can be written as; To obtain the maximum velocity achieved at any section of the tunnel, then put This will result to a velocity formula of; Figure 4: variation of the velocity with respect to the distance from the tunnel wall Therefore the average velocity obtained at any cross-section along the tunnel is half the maximum velocity. Therefore, dividing equation 22 by 2, then equation 23 will be obtained as; Rearranging equation 23, the pressure drop in the tunnel can then be determined as; Therefore; This is then the amount of fluid at any specific section along the tunnel length. When the fluid moves from a smaller cross-sectional area to a larger section, the velocity of the fluid reduces since the same amount of fluid passes through the section. This concept remains the same as already derived until the fluid reaches a point (in section 3 from figure 1); the nose cone and then exits at a contraction point. These factors change the fluid velocities developed earlier due to the changing cross-sectional areas and the inclusion of a constant velocity at the nose cone. At the nose cone therefore, there are two velocities involved; Therefore the resultant velocity (V) achieved at that point; To determine the size of the cone, the continuity equation will be used; Therefore; The cone area therefore will be; At the contraction section, the velocity can also be determined as follows; Figure 5: contraction section To derive the equation for determining the properties of the contraction section, figure 5 will be used; Figure 6: flow through the contraction section Assumptions; Velocity at point 3 is equal to zero Velocity at point 4 is equal to V4 The pressure at point 3 is equal to the pressure at point 4 Where; From figure 5; Energy equation Then; Therefore, the velocity at the contraction section is; The velocities at the different sections of the tunnel can therefore be determined and the sizes of the sections can also be determined when the quantity of the fluid and the velocity at a particular point are known. Schematic diagram Conclusion The conservation of mass and the continuity of fluid flow are the key concepts used in the determining the sizes of the cross-sections as well as the fluid velocity and the pressure distributions. At points in the cross-sections where the cross-section is very small, the velocity experienced at such points is very high. This is because; the flow in the tunnels has to be kept constant. This means that, the amount of fluid at any given section should be the same. This can be then achieved by the high velocity so that the amount of fluid passing at a given section is high making the flow constant. The formulas can be used to determine all the required properties of the fluid and the cross-sections of the tunnel. The design of the different tunnels will be based on the derived equations for safe flow to be achieved. References Kundu, P. K., Cohen, I. M., Dowling, D. R., & Tryggvason, G. (2016). Fluid mechanics. Massey, B. S., & Ward-Smith, J. (2012). Mechanics of fluids. London, Spon Press. Schlichting, H. (1949). Lecture series "boundary layer theory". Washington, DC, National Advisory Committee for Aeronautics. Vandyke, M. (2012). ˜An album of fluid motion. Stanford, Calif, Parabolic Press. White, F. M. (1986). Fluid mechanics. New York, McGraw-Hill. White, F. M. (2007). Viscous fluid flow. Boston [u.a.], McGraw-Hill. Read More