Fluid Mechanics and Applications MECN 3110

1. Fluid Mechanics and Applications MECN 3110 Inter American University of Puerto Rico Professor: Dr. Omar E. Meza Castillo

2. Chapter 6 Viscous Flow in Ducts

3. Course Objectives • To describe the appearance of laminar flow and turbulent flow. • State the relationship used to compute the Reynolds number. • Identify the limiting values of the Reynolds number by which you can predict whether flow is laminar or turbulent. • Compute the Reynolds number for the flow of fluids in round pipes and tubes. • State Darcy’s equation for computing the energy loss due to friction for either laminar and turbulent flow. • Define the friction factor as used in Darcy’s equation • Determine the friction factor using Moody’s diagram for specific values of Reynolds number and the relative roughness of the pipe. • Major and Minor losses in Pipe Systems. Thermal Systems Design Universidad del Turabo

4. Introduction • This chapter is completely devoted to an important practical fluid engineering problem: flow in ducts with various velocities, various fluids, and various duct shapes. Piping Systems are encountered in almost very engineering design and thus have been studied extensively. • The basic piping problem is this: Given the pipe geometry and its added components (such as fitting, valves, bends, and diffusers) plus the desired flow rate and fluid properties, what pressure drop is needed to drive the flow? Of course, it may be stated in alternative form: Given the pressure drop available from a pump, what flow rate will ensue? The correlations discussed in this chapter are adequate to solve most such piping problems

5. Reynolds Number Regimes • 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. • High-viscosity, low-Reynolds-number, laminar flow

6. Reynolds Number Regimes • 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. • Low-viscosity. High-Reynolds-number, turbulent flow

7. Reynolds Number Regimes

8. Reynolds Number Regimes • The changeover is called transition to turbulent. Transition depends on many effects, such as wall roughness or fluctuations in the inlet stream, but the primary parameter is the Reynolds number. • Studies present the following approximate ranges that commonly occur: • 0 < Re < 1: highly viscous laminar “creeping” motion • 1 <Re<100: laminar, strong Reynolds number dependence • 100 <Re <103: laminar, boundary layer theory useful • 103 <Re <104: transition to turbulence • 104<Re<106: turbulent, moderate Reynolds number dependence • 106<Re<∞: turbulent, slight Reynolds number dependence

9. Reynolds Number Regimes • In 1883 Osborne Reynolds, British engineering professor 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 is known. • The following equation shows the basic definition of the Reynolds number, Re: • The value of 2300 is for transition in pipes. Other geometries, such as plates, airfoils, cylinders, and spheres, have completely different transition Reynolds numbers.

10. Critical Reynolds Number • 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. Re < 2000 : Laminar flow • If the Reynolds number is greater than 4000, the flow can be assumed to be turbulent. Re>4000 : Turbulent flow • In the range of Reynolds numbers between 2000 and 2000, it is impossible to predict which type of flow exists; therefore this range is called the critical region.

11. Application Problems

12. Problem • Statement: Determine whether the flow is laminar or turbulent if glycerin at 25oC flows in a pipe with a 150-mm inside diameter. The average velocity of low is 3.6 m/s. • Solution: • Because Re=708, which is less than 2000, the flow is laminar

13. Problem • Statement: Determine whether the flow is laminar or turbulent if water at 70oC flows in a 1-in Type K copper tube with a flow rate of 285 L/min. • Solution: For a 1-in Type K copper tube, D=0.02527m and A=5.017 x 10-4 m2. Then we have • Because Reynolds number is greater than 4000, the flow is turbulent.

14. Head Loss – The Friction Factor • In the general energy equation • Julius Weisbach in 1850 established that hf is proportional to (L/D), and G.H.L Hagen shown that for turbulent flow, hf is proportional to V2. The proposed correlation, still as effective today as in 1850, is • This expression is called Darcy’s Equation. The dimensionless parameter f is called the Darcy Friction factor.

15. Friction Loss in Laminar Flow • Because laminar flow is so regular and orderly, we can derive a relation between the energy loss and the measurable parameters of the flow system. • This relationship is known as the Hagen-Pouseuille equation: • The Hagen-Pouseuille equation is valid only for laminar flow (Re<2000). • If the two previous relationships for hf are set equal to each other, we can solve for the value of the friction factor:

16. Friction Loss in Laminar Flow • In summary, the energy loss due to friction in laminar flow can be calculated either from Hagen-Pouseuille equation or Darcy’s equation. The pipe friction factor decrease inversely with Reynolds number.

17. Application Problems

18. Problem • Statement: Determine the energy loss if glycerin at 25oC flows 30 m through a 150-mm-diamter pipe with an average velocity of 4.0 m/s. • Solution: First, we must determine whether the flow is laminar or turbulent by evaluating the Reynolds number: • Because Re=768, which is less than 2000, the flow is laminar

19. Problem • Using Darcy’s Equation • This means that 13.2 NM of energy is lost by each newton of the glicerin as it flow along the 30 m of pipe.

20. 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 f. • The following figure illustrate pipe wall roughness (exaggerated) as the height of the peaks of the surface irregularities. • Because the roughness is somewhat irregular, averaging techniques are used to measure the overall roughness value

21. Friction Loss in Turbulent Flow • For commercially available pipe and tubing, the design value of the average wall roughness has been determined as shown in the following table

22. Relative Roughness of Pipe Material

23. Moody Diagram It is the graphical representation of the function f(ReD, ε/D)

24. Moody Diagram • Several important observations can be made from these curves: • For a given Reynolds number flow, as the relative roughness is increased, the friction factor f decreases. • For a given relative roughness, the friction factor f decreases with increasing Reynolds number until the zone of complete turbulent is reached. • Within the zone of complete turbulence, the Reynolds number has no effect on the friction factor. • As the relative roughness increases, the value of Reynolds number at which the zone of complete turbulence begins alto increases.

25. Application Problems

26. Problem • Check your ability to read the Moody Diagram correctly by verifying the following values for friction factors for the given values of Reynolds number and relative roughness:

27. Problem • Statement: Determine the friction factor f if water at 70oC is flowing at 9.14 m/s in an uncoated ductile iron pipe having an inside diameter of 25 mm. • Solution: The Reynolds number must first be evaluated to determine whether the flow is laminar or turbulent:

28. Problem • Thus, the flow is turbulent. Now the relative roughness must be evaluated. From previous table we find ε=2.4x10-4 m. Then , the relative roughness is • The final steps in the procedure are as follows: • Locate the Reynolds number on the abscissa of the Moody Diagram. • Project vertically until the curve for ε/D =0.00961538 is reached. • Project horizontally to the left, and read f=0.038

29. Problem • Statement: In chemical processing plant, benzene at 50oC (SG=0.86) must be delivered to point B with a pressure of 550 kPa. A pump is located at point A 21m below point B, and the two points are connected by 240 m of plastic pipe having an inside diameter of 50 mm. If the volume flow rate is 110 L/min, calculate the required pressure at the outlet of the pump.

30. Problem • Solution: Using the energy equation we get the following relation: • Mass Balance • Energy Balance

31. Problem • The evaluation of the Reynolds number is the first step. The type of flow, laminar or turbulent, must be determined. • For a 50-mm pipe, D=.050 m and A=1.963 x 10-3 m2. Then, we have

32. Problem • For turbulent flow, Darcy’s equation should be used: • With the Reynolds number and the relative roughness we obtain the friction factor from the Moody’s Diagram f=0.018

33. Problem • You should have the pressure as follows:

34. Fundamental Equation of Fluid Mechanics • In order to apply previous equation to a piping system, we must extend the Bernoulli equation to account for losses which result from pipe fittings, valves, and direct losses (friction) within the pipes themselves. The extended Bernoulli equation may be written as: • Additionally, at various points along the piping system we may need to add energy to provide an adequate flow. This is generally achieved through the use of some sort of prime mover, such as a pump, fan, or compressor.

35. Fundamental Equation of Fluid Mechanics • For a system containing a pump or pumps, we must include an additional term to account for the energy supplied to the flowing stream. This yields the following form of the energy equation: • Finally, if somewhere in the piping system a component extracts energy from the fluid stream, such as a turbine, the energy equation takes the form:

36. Losses in Piping System • Friction Factor: The total head loss hf in a piping system are typically categorized as major and minor losses. • Majorlosses are associated with the pipe-wall skin friction over the length of the pipe, and • Minor losses in piping systems are generally characterized as any losses which are due to pipe inlets and outlets, fittings and bends, valves, expansions and contractions, filters and screens, etc. • Minor losses are not necessarily smaller than major losses.

37. Major Losses • Major losses of head in a piping system are the direct result of fluid friction in pipes and ducting. The resulting head losses are usually computed through the use of friction factors. Friction factors for ducts have been compiled for both laminar and turbulent flows. Two widely adopted definitions of the friction factor are the Darcy and Fanning friction factors. • The head loss due to flow of a fluid at an average velocity V through a length L of pipe with a diameter D is (Darcy-Weisbach) or

38. Major Losses (Fanning) Where: fD-W: Darcy-Weisbach friction factor fF: Fanning friction factor L: Length of considered pipe D: Pipe diameter V2/2gc: Velocity head

39. Roughness Height (e or ε) for Certain Common Pipes ε

40. Friction Factor f • The Moody diagram is sufficient to determine the friction factor and, hence, the head loss for given flow conditions. If we should desire to generate a computer-based solution, the translation of the Moody diagram into tabular form to use in interpolation is awkward. To say the least. What is needed is a simple algebraic expression in the form f(ReD, ε/D). Historically, the implicit expression of Colebrook has been accepted as the most accurate in the turbulent zone.

41. Friction Factor f • Benedict suggests the expression proposed by Swamee and Jain, i.e., • While for ε/D>10-4Haaland recommends

42. Friction Factor f • For situations where ε/D is very small, as in natural-gas pipelines, Haaland proposes Where n ~ 3 • The use of the Swamee-Jain or Haaland provide an explicit formula of the friction factor in turbulent flow, and is thus the preferred technique.

43. Friction Factor f • For laminar flow (Re<2000) the usual Darcy-Weisbach friction factor representation is • For turbulent flow in smooth pipes (ε/D=0) with 4000<Re<105 is • For turbulent flow (Re>4000) the friction factor can be founded from the Moody diagram

44. Friction Factor f • Churchill devised a single expression that represents the friction factor for laminar, transition and turbulent flow regimes. This expression, which is explicit for the friction factor given the Reynolds number and relative roughness, is • where • and

45. Friction Factor f • In our discussion so far we have been concerned only with circular pipes, but for a variety of reason conduit cross sections often deviate from circular. The appropriate characteristic length to use in evaluating the Reynolds number for noncircular cross-sectional areas is the hydraulic diameter. The hydraulic diameter is defined as • To use the hydraulic diameter concept, the Reynolds number is defined as μ (m2/s) T  (oC)

46. Application Problems

47. Example 2 • Find the head loss due to friction in galvanized-iron pipe 30 cm diameter and 50 m long through which water is flowing at a velocity of 3 m/s assume that water flowing at 20oC. ε

48. Minor Losses • Minor losses are due to the change of the velocity of the flowing fluid in magnitude or direction. They are most often calculated using the concept of a loss coefficient or equivalent friction length method. In the loss coefficient method, a constant or variable factor K is defined as: • The associated head loss is related to the loss coefficient through

49. Minor Losses • The Minor Losses occurs at: • Valves • Tees • Bends • Reducers • Valves • And other appurtenances

50. Minor Losses