S3 Reynolds Number Presentation


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Semester 3
Reynolds Number

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S3 Reynolds Number Presentation

  2. 2. INTRODUCTION  As the water flows from a faucet at a very low velocity, the flow appears to be smooth and steady.  The stream has a fairly uniform diameter and there is little or no evidence of mixing of the various parts of the stream.  This is called laminar flow, a term derived from the word layer, because the fluid appears to be flowing in continuous layers with little or no mixing from one layer to the adjacent layers. Saturday,July27,2013 2 CB306-HYDRAULIC
  3. 3. When the faucet is nearly fully open, the water has a rather high velocity. The elements of fluid appear to be mixing chaotically within the stream. This is a general description of turbulent flow. Saturday,July27,2013 3 CB306-HYDRAULIC
  4. 4. REYNOLDS NUMBER  The behavior of a fluid, particularly with regard to energy losses, it quite dependent on whether the flow is laminar or turbulent, as will be demonstrated later in this chapter.  For this reason we need a means of predicting the type of flow without actually observing it.  Indeed, direct observation is impossible for fluids in opaque pipes. Saturday,July27,2013 4 CB306-HYDRAULIC
  5. 5. It can be shown experimentally and verified analytically that the character of flow in depends on four variables: fluid density, fluids viscosity , Pipe diameter D, and average velocity of flow . Osborne Reynolds was the first to demonstrate that laminar or turbulent flow can be predicted if the magnitude of a dimensionless number, now called the Reynolds number (NR), is known. NR = Saturday,July27,2013 5 CB306-HYDRAULIC
  6. 6. Table 8.1 lists the required units in both the SI metric unit system and the U.S. customary unit system. Converting to these standard units prior to entering data into the calculation for NR is recommended We can demonstrate that the Reynolds number is dimensionless by substituting standard SI units into Eq. (8-1): NR = Saturday,July27,2013 6 CB306-HYDRAULIC
  7. 7.  We can demonstrate that the Reynolds number is dimensionless by substituting standard SI units into Eq. (8-1):  NR =  NR =  Because all units can be cancelled, NR is dimensionless. Saturday,July27,2013 7 CB306-HYDRAULIC
  8. 8.  TABLE 8.1 Standard units for quantities used in the calculation of Reynolds number to ensure that it is dimensionless. Quantity SI Units U.S. Customary Units Velocity m/s Ft/s Diameter m ft Density Kg/m3 or N.s2/m4 Slugs/ft3 or lb.s2/ft4 Dynamic viscosity N.s/m2 or Pa.s or kg/m.s lb.s/ft2 or slugs/ft.s Kinematic viscosity M2/s ft2/s Saturday,July27,2013 8 CB306-HYDRAULIC
  9. 9. CRITICAL REYNOLDS NUMBERS  for practical applications in pipe flow we find that if the Reynolds number for the flow is less than 2000, the flow will be laminar.  If the Reynolds number is greater than 4000, the flow can be assumed to be turbulent.  In the range of Reynolds number between 2000 and 4000, it is impossible to predict which type of flow exist; therefore this range is called the critical region. Saturday,July27,2013 9 CB306-HYDRAULIC
  10. 10. Typical applications involve flow that are well within the laminar flow range or well within the turbulent flow range, so the existence of this region of uncertainty does not cause great difficulty. If the flow in a system is found to be in the critical region, the usual practice is to change the flow rate or pipe diameter to cause the flow to be definitely laminar or turbulent. More precise analysis is then possible. Saturday,July27,2013 10 CB306-HYDRAULIC
  11. 11.  By carefully minimizing external disturbances, it is possible to maintain laminar flow for Reynolds numbers as high as 50000.  However, when NR is greater than about 4000, a minor disturbance of the flow stream will cause the flow to suddenly change from laminar to turbulent.  For this reason, and because we are dealing with practical applications in this book, we assume the following:  If NR < 2000, the flow is laminar.  If NR >4000, the flow is turbulent. Saturday,July27,2013 11 CB306-HYDRAULIC
  12. 12. DARCY’S EQUATION  in the general energy equation  the term hL is defined as the energy loss from the system. one component of the energy loss is due to friction in the flowing fluid. Saturday,July27,2013 12 CB306-HYDRAULIC
  13. 13.  one component of the energy loss is due to friction in the flowing fluid. friction is proportional to the velocity head of the flow and to the ratio of the length to the diameter of the flow stream, for the case of flow in pipes and tubes. this is expressed mathematically as Darcy’s equation:  where  hL = energy loss due to friction (Nm/N,m,lb-ft/lb, or ft)  L = length of flow stream of flow stream (m or ft)  D = pipe diameter (m or ft)  f = friction factor (dimensionless) Saturday,July27,2013 13 CB306-HYDRAULIC
  14. 14. FRICTION LOSS IN LAMINAR FLOW  when laminar flow exists, the fluid seems to flow as several layers, one on another.  because of the viscosity of the fluid, a shear stress is created between the layers of fluid.  energy is lost from the fluid by the action of overcoming the frictional forces produced by the shear stress. Saturday,July27,2013 14 CB306-HYDRAULIC
  15. 15.  because laminar flow is so regular and orderly, we can derive a relationship between the energy loss and the measurable parameters of the flow system. this relationship is known as the hagen poiseuille equation:  the Hagen-Poiseuile equation is valid only for laminar flow (NR <2000).  however we stated earlier that Darcy’s equation, equation (8-3), could also be used to calculate the friction loss for laminar flow. Saturday,July27,2013 15 CB306-HYDRAULIC
  16. 16.  if the two relationships for hL are set equal to each other, we can solve for the value of the friction factor.  Because ρ = γ/g, we get  The Reynolds number is defined as NR = υDρ/µ. Then we have Saturday,July27,2013 16 CB306-HYDRAULIC
  17. 17.  In summary, the energy loss due to friction in laminar flow can be calculated either from the hagen- poiseuille equation,  Or from Darcy’s equation,  Where ƒ= 64/NR. Saturday,July27,2013 17 CB306-HYDRAULIC
  18. 18. FRICTION LOSS IN TURBULENT FLOW  For turbulent flow of fluids in circular pipes it is most convenient to use Darcy’s equation to calculate the energy loss due to friction.  Turbulent flow is rather chaotic and is constantly varying.  For these reasons we must rely on experimental data to determine the value of ƒ.  Test have shown that dimensionless number ƒ is dependent on two other dimensionless numbers, the Reynolds number and relative roughness of the pipe.  The relative roughness is the ratio of the pipe diameter D to the average pipe wall roughness є (Greek letter epsilon). Saturday,July27,2013 18 CB306-HYDRAULIC
  19. 19.  Figure 8.5 illustrates pipe wall roughness (exaggerated) as the height of the peaks of the surface irregularities.  The condition of the pipe surface is very much dependent on the pipe material and the method of manufacture.  Because the roughness is somewhat irregular, averaging techniques are used to measure the overall roughness value. Saturday,July27,2013 19 CB306-HYDRAULIC
  20. 20. Saturday,July27,2013 20 CB306-HYDRAULIC
  21. 21. THE MOODY DIAGRAM  One of the most widely used methods for evaluating the friction factor employs the Moody diagram shown in Fig.8.6.  The diagram shows the friction factor f plotted versus the Reynolds number NR, with a series of parametric curves related to the relative roughness D.  moody diagram.docx  explanation of parts of moody’s diagram..docx Saturday,July27,2013 21 CB306-HYDRAULIC
  22. 22. USE OF THE MOODY DIAGRAM  The Moody Diagram is used to help determine the value of the friction factor f for turbulent flow.  The value of the Reynolds number and the relative roughness must be known.  Therefore, the basic data required are the pipe inside diameter, the pipe material, the flow velocity, and the kind of fluid and its temperature, from which the viscosity can be found.  The following example problems illustrate the procedure for finding f. Saturday,July27,2013 22 CB306-HYDRAULIC
  23. 23. EQUATION FOR THE FRICTION FACTOR  The moody diagram in Fig. 8.6 is a convenient and sufficiently accurate means of determining the value of the friction factor when solving problems by manual calculation.  However, if the calculation are to be automated for solution on a computer or a programmable calculator, we need equation for the friction factor.  In the laminar flow zone. For values below 2000, can be found from Eq. (8-5).  = 64 / R Saturday,July27,2013 23 CB306-HYDRAULIC
  24. 24.  The following equations, which allows the direct calculation of the value of the friction factor for turbulent flows, was developed by P.K. Swamee and A.K. Jain and is reported in Reference 3 :  = Saturday,July27,2013 24 CB306-HYDRAULIC
  25. 25. HAZEN WILLIAMS FORMULA FOR WATER FLOW  The Darcy equation presented in this chapter for calculating energy loss due to friction is applicable for any Newtonian fluid.  An alternate approach is convenient for the special case of the flow of water in pipeline systems.  The Hazen Williams formula is unit-specific. In the U.S. Customary unit system it takes the form. Saturday,July27,2013 25 CB306-HYDRAULIC
  26. 26.  Where :  V= Average velocity of flow (ft/s) Cℎ = Hazen Williams coefficient (dimensionless) R= Hydraulic radius of flow conduit (ft) S= Ratio of ℎL/: energy loss/length of conduit (ft/ft)  The Hazen Williams formula for SI unit is Saturday,July27,2013 26 CB306-HYDRAULIC
  27. 27. NOMOGRAPH FOR SOLVING THE HAZEN-WILLIAMS FORMULA  The nomograph shown in Fig.8.9. allows the solution of the Hazen-Williams formula to be done by simply aligning known quantities with a straight edge and reading the desired unknowns at the intersection of the straight edge with the appropriate vertical axis.  Note that this nomograph is constructed for the value of the Hazen Williams coefficient of If the actual pipe condition warrants the use of a different value of the following formulas can be used to adjust the results. Saturday,July27,2013 27 CB306-HYDRAULIC
  28. 28.  The subscript ”100” refers to the value read from the nomograph for .  The subscript “” refers to the value for the given.  nomograph.docx Saturday,July27,2013 28 CB306-HYDRAULIC
  29. 29. THAT ALL…. THANK YOU… FINAL EXAM:20 DAYS Saturday, July 27, 2013 29 CB306- HYDRAULIC