Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ CHAPTER FIVE VISCOUS FLOW IN PIPE 5.1 General characteristics of pipe flow Before we apply the governing equation to pipe flow we will discuss some of the basic concepts of pipe flow, we will assume that the conduit is round . For all flows involved in this chapter we assume that the pipe is completely filled with the fluid being transported. 5.1.1 Laminar or Turbulent Flow The flow of fluid in pipe may be laminar flow or it may be turbulent flow. if water runs through a pipe of diameter (D) with average velocity (V), the following characteristics are observed by injecting naturally buoyant dye as shown. for small enough flow rate the dye streak will remain as well defined line as it flows along (laminar flow), for somewhat large the dye streak fluctuates in time and space (transitional flow). For large enough flow rate the dye streak almost immediately becomes blurred and spread across the entire pipe in random fashion (Turbulent flow). ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 1 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ The next curves represent the x-component of velocity as a function of time at a point (A) in the flow. For pipe flow the most important dimensionless parameter is Reynolds number. Reynolds Number (Re): is the ratio of the inertia effect to viscous effect in the flow. 𝑅𝑒 = 𝜌𝑉𝐷 𝜇 V : average velocity in the pipe. : density of fluid : viscosity of fluid In general the flow in round pipe Re < 2100 Laminar for Transitional for 2100 < Re < 4000 Turbulent for Re > 4000 ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 2 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Example: Water at a temperature of 20 oC flows through pipe of diameter D = 2 cm . Determine: (a) the minimum time taken to fill a glass of volume 0.5 l , if the flow in pipe is laminar. (b) the maximum time taken to fill the glass if the flow is turbulent. Solution: (a) if the flow in pipe is to remain laminar the minimum time to fill the glass will occur if the Re is the maximum allowed , Re = 2100 From the table B.1 for water at 20 oC 𝜌 = 998.2 𝑘𝑔⁄𝑚3 𝜇 = 1.002 × 10−3 𝑁. 𝑠⁄𝑚2 𝜌𝑉𝐷 𝑅𝑒 𝜇 2100 × 1.002 × 10−3 𝑅𝑒 = = 2100 ⇒ 𝑉 = = 𝜇 𝜌𝐷 998.2 × 0.02 V = 0.1 m/s 𝑄 = 𝐴𝑉 = 𝜋 2 𝜋 𝐷 𝑉 = (0.02)2 (0.1) = 3.14 × 10−5 𝑚3 ⁄𝑠 4 4 the time ∀ 0.5 × 10−3 𝑡= = = 15.9 𝑠𝑒𝑐 𝑄 3.14 × 10−5 (b) if the flow in pipe is turbulent the maximum time occur at Re = 4000 ⇒ ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 3 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ 𝑅𝑒 𝜇 4000 × 1.002 × 10−3 𝑉= = = 0.2 𝑚/𝑠 𝜌𝐷 998.2 × 0.02 𝑄 = 𝐴𝑉 = 𝜋 4 𝜋 𝐷2 𝑉 = (0.02)2 (0.2) = 6.28 × 10−5 𝑚3 ⁄𝑠 4 ∀ 0.5 × 10−3 𝑡= = = 7.96 𝑠𝑒𝑐 𝑄 6.28 × 10−5 5.1.2 Entrance region and fully developed Flow Any fluid in pipe has to enter the pipe at some location the region of flow near where the fluid enters the pipe is termed the entrance region as in figure. The fluid typically enters the pipe with nearly uniform velocity profile at section (1) , as the fluid moves through the pipe , viscous effects cause it to stick to the pipe wall (the no slip condition). thus a boundary layer is produced along the fluid reaches the end of the entrance length at section (2), beyond which the velocity profile dose not vary with x. ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 4 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ 5.1.3 Energy Equation For one-dimensional, incompressible, steady flow with friction and shaft work. The energy equation is, 𝑝 𝑉2 𝑝 𝑉2 + 𝑔𝑍) = ( + + 𝑔𝑍) + 𝑊𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 + 𝑙𝑜𝑠𝑠𝑒𝑠 ( + 𝜌 2 𝜌 2 1 2 where 𝑊𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 𝑊̇𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 = 𝑚̇ (𝑁. 𝑚⁄𝑘𝑔 = 𝑚2 ⁄𝑠 2 ) Dividing by gravity (g) 𝑝 𝑉2 𝑝 𝑉2 + 𝑍) = ( + + 𝑍) + ℎ𝑠 + ℎ𝐿 ( + 𝛾 2𝑔 𝛾 2𝑔 1 2 ℎ𝑠 = ℎ𝐿 = 𝑊𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 𝑊̇𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 𝑊̇𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 = = 𝑔 𝑚̇ 𝑔 𝛾𝑄 𝑙𝑜𝑠𝑠𝑒𝑠 = 𝑒𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 𝑔 (𝑚) ( 𝑁. 𝑚⁄𝑁 ) = (𝑚) 5.2 Dimensional analysis of pipe flow: It is necessary to determine the head loss ( hL ) - (pressure drop) that occurs in a pipe flow. The overall head loss for the pipe system consists of the head loss due to viscous effects in the straight pipes, termed the Major loss, and denoted ( hL major ) and the head loss in the various pipe components (valves, elbows, … etc.), termed the, Minor loss and denoted ( hL minor ). hL = hL major + hL minor ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 5 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ The head loss designations of “major” and “minor” do not necessarily reflect the relative importance of each type of loss. For a pipe system that contains many components and a relatively short length of pipe, the minor loss may actually be larger than the major loss. 5.2.1 Major losses For any fully developed, steady, incompressible pipe flow - whether the pipe is horizontal or on a hill, the Darcy-Weisbach is used to determine the major head loss ( hL major ), 𝑙 𝑉2 ℎ𝐿 𝑚𝑎𝑗𝑜𝑟 = 𝑓 𝐷 2𝑔 where V is average velocity l is pipe length D is pipe diameter f is friction factor , The friction factor f is dependent on the behavior of the flow. for Laminar flow, 𝑓 = 64⁄𝑅𝑒 for Turbulent flow , f is determined from the Colebrook formula, 𝜀 ⁄𝐷 2.51 = −2.0 log ( + ) 3.7 𝑅𝑒 √𝑓 √𝑓 1 𝜀 ⁄𝐷 ∶ 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑅𝑜𝑢𝑔ℎ𝑛𝑒𝑠𝑠 ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 6 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ The Colebrook equation is represented in chart , which is called Moody chart. ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Page 7 of 11 Chapter (5) - Viscous Flow in Pipes Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Example: Air under standard conditions flows through a 4.0-mm diameter drawn tubing with an average velocity of V = 50 m/s. For such conditions the flow would normally be turbulent. However, if precautions are taken to eliminate disturbances to the flow (the entrance to the tube is very smooth, the air is dust free, the tube does not vibrate, etc.), it may be possible to maintain laminar flow. (a) Determine the head loss in a 0.1-m section of the tube if the flow is laminar. (b) Repeat the calculations if the flow is turbulent. (c) Calculations the pressure drop in the tube Solution: for standard temperature and pressure conditions the density and viscosity of air are 𝜌 = 1.23 𝑘𝑔⁄𝑚3 𝜇 = 1.79 × 10−5 𝑁. 𝑠⁄𝑚2 𝑅𝑒 = 𝜌 𝑉 𝐷 1.23 × 50 × 0.004 = = 13,700 𝜇 1.79 × 10−5 which would normally indicate turbulent flow. (a) If the flow were laminar, then 𝑓 = 64⁄𝑅𝑒 = 64⁄13700 = 0.00467 and the pressure drop in a 0.1-m-long horizontal section of the pipe would be (50)2 𝑙 𝑉2 0.1 ℎ𝐿 = 𝑓 = (0.00467) × × = 14.8 𝑚 𝐷 2𝑔 0.004 2 × 9.81 (b) If the flow is turbulent from table (8.1) 𝜀 = 0.0015 𝑚𝑚, so that 𝜀 0.0015 = = 0.000375 𝐷 4 ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 8 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ From Moody chart at 𝑅𝑒 = 1.37 × 104 𝑎𝑛𝑑 f = 0.028 𝜀 𝐷 = 0.000375 (50)2 𝑙 𝑉2 0.1 (0.028) ℎ𝐿 = 𝑓 = × × = 89.1 𝑚 𝐷 2𝑔 0.004 2 × 9.81 (c) pressure drop in the tube from energy equation 𝑝 𝑉2 𝑝 𝑉2 + 𝑍) = ( + + 𝑍) + ℎ𝑠 + ℎ𝐿 ( + 𝛾 2𝑔 𝛾 2𝑔 1 2 𝑍1 = 𝑍2 , 𝑉1 = 𝑉2 𝑝1− 𝑝2 ∆𝑝 = = ℎ𝐿 ⇒ 𝛾 𝛾 , hs = 0 ∆𝑝 = 𝜌 𝑔 ℎ𝐿 for Laminar flow ∆𝑝 = 1.23 × 9.81 × 14.8 = 0.179 𝑘𝑃𝑎 for Turbulent flow ∆𝑝 = 1.23 × 9.81 × 89.1 = 1.076 𝑘𝑃𝑎 5.2.2 Minor losses this losses are due to the components (valves, elbows, bends, tees … etc.) added to the pipe system. The most common method used to determine these head losses or pressure drops is to specify the loss coefficient, KL which is defined as ℎ𝐿 𝑚𝑖𝑛𝑜𝑟 ∆𝑝 𝑉2 𝐾𝐿 = 2 = ⇒ ℎ𝐿 𝑚𝑖𝑛𝑜𝑟 = 𝐾𝐿 1 2 (𝑉 /2𝑔) 2𝑔 (2 𝜌𝑉 ) The actual value of KL is strongly dependent on the geometry of the component considered. It may also be dependent on the fluid properties. ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 9 of 11 Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Typical values of KL for such components are given in Table 8.2. ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Page 10 of 11 Chapter (5) - Viscous Flow in Pipes Course Notes in Fluid Mechanics University of Benghazi Lecturer: Dr. Nabil Elsharif Faculty of Engineering ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ • The energy equation for incompressible, steady flow between two locations is given by 5.2.3 Noncircular Conduits For noncircular ducts the previous correlation can be used by introducing the hydraulic diameter ( Dh ) where 𝐷ℎ = 4𝐴 𝑃 A : cross sectional area of the duct P : the wetted perimeter of the duct ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Chapter (5) - Viscous Flow in Pipes Page 11 of 11