Fluid Mechanics for Mechanical Engineering Viscous Flow in Ducts

SEQUENCE OF CHAPTER 8 Introduction Objectives 8.1 Flows Characteristics in Pipe 8.2 Fully Developed Laminar Flow 8.3 Fully Developed Turbulent Flow 8.4 Friction Loss 8.5 Minor Loss Summary

Introduction Now, we cover fluid with internal viscous friction attributed by the viscosity properties and friction between the flows and any adjacent walls. We will look into how to analyse the laminar and turbulent pipe flows, and to calculate friction losses due to pipe walls as well as pressure losses due to fitting components such as valves, junctions, faucets and flow measurement apparatus.

Objectives At the end of this chapter, you should be able to : understand and differentiate between laminar and turbulent pipe flows in terms of velocity profile and pressure distribution, use relevant formulae and charts to calculate friction and other minor losses in pipes, calculate flow-rate from pressure difference in obstacle based measurement apparatus.

Flow Characteristics in Pipes This type of flow is also known as internal flow where the pipe is assumed to be completely filled with the fluid. The fluid motion is generated by pressure difference between two points and is constrained by the pipe walls. The direction of the flow is always from a point of high pressure to a point of low pressure. If the fluid does not completely fill the pipe, such as in a concrete sewer, the existence of any gas phase generates an almost constant pressure along the flow path. If the sewer is open to atmosphere, the flow is known as open-channel flow and is out of the scope of this chapter .Hence this chapter only concentrates on internal pipe flows only.

Flow Characteristics in Pipes Flow in pipes can be divided into two different regimes, i.e. laminar and turbulence. The experiment to differentiate between both regimes was introduced in 1883 by Osborne Reynolds (1842–1912), an English physicist who is famous in fluid experiments in early days.

Flow Characteristics in Pipes The Reynolds’ experiment is depicted in Fig. 1. Figure 1 Experiment for Differentiating Flow Regime

Flow Characteristics in Pipes From Fig. 1, the dye is used to mark the flow path of the fluid. In order to demonstrate the transition between laminar and turbulent regime, the - Q is varied. For a constant diameter pipe, the cross sectional area is also constant. Thus, by virtue of mass conservation, the velocity V is directly proportional to Q. For laminar regime, the flow velocity is kept small, thus the generated flow is very smooth which is shown as a straight tiny line formed by the dye. When the flow velocity is increased, the flow becomes slightly unstable such that it contains some temporary velocity fluctuation of fluid molecules and this mark the transition regime between both regimes. Then, the velocity can be increased further so that the fluid flow is completely unstable and the dye is totally mixed with the surrounding fluid. This phenomenon is known as turbulence.

Flow Characteristics in Pipes This graph clearly shows a smooth velocity of laminar flow and a fluctuated velocity for turbulent flow. Clearly, one of the main critical parameters that determines the flow regimes is the velocity. This parameter, together with fluid properties, namely density  and dynamic viscosity , as well as pipe diameter D, forms the dimensionless Reynolds number, that is From Reynolds’ experiment, he suggested that Re < 2000 for laminar flows and Re > 4000 for turbulent flows. The range of Re between 2000 and 4000 represents transitional flows.

Example 1 Consider a water flow in a pipe having a diameter of D = 20 mm which is intended to fill a 0.35 liter container. Calculate: (a) the minimum time required if the flow is laminar, (b) the maximum time required if the flow is turbulent. Use density  = 998 kg/m3 and dynamic viscosity  = 1.1210–3 kg/ms. Solution: (a) For laminar flow, use Re =VD/ = 2100: Hence, the minimum time t is (b) For turbulent flow, use Re = VD/ = 4000:

Flow Characteristics in Pipes In many cases of pipe flows, it may begin from a tank as shown in Fig. 2. The common velocity profile for laminar pipe flow is parabolic. However, at a position in the pipe where the fluid just exits from the reservoir, the velocity profile is almost uniform. This uniform flow can also be seen as a representation of inviscid flow since the fluid molecules has no relative motion from one to another. The transition from the initially uniform flow and a fully developed parabolic occurred in the entrance region. In this region, the flow is formed by a mixture between the following two regions: 1. Inviscid core, where the velocity profile is uniform and the viscous effect is negligible, 2. Boundary layer, where it allows velocity variation from pipe walls with no-slip condition to the core and the viscous effect is dominant.

Flow Characteristics in Pipes Figure 2 Velocity Profiles at the Entrance and Fully Developed Regions Figure 3 Pressure Distribution in a Horizontal Pipe

Flow Characteristics in Pipes The entrance region can be represented by entrance length e, which can be empirically determined by the following formulae for both regimes: Laminar: Turbulent: Due to different boundary layer thickness in the inviscid core, the pressure distribution behaves non-linearly in this region and the pressure slope is not constant as shown in Fig. 3. However, after the flow is fully developed, the slope becomes constant and the pressure drop p is directly caused only by viscous effect. By projecting the graph back towards the tank, we can estimate the pressure drop due to entrance flow. Hence, by using the Bernoulli equation with losses, the pressure value at all position along the same pipe can be calculated.

Flow Characteristics in Pipes From Fig. 3, we can also deduce that there are two types of pressure loss; the first is known as friction or major loss and is caused by friction which reduces the fluid pressure linearly with gradient -p/ , and the second is known as minor loss and is generated by sudden change in flow direction as in the entrance flow. The friction loss is proportional to the pipe length, while minor losses can be emulated by sudden pressure drop. In this case, we can summarise that minor losses represent pressure losses in developing flow which is experiencing disturbances and changes in internal pipe geometry. Now, we can apply the modified Bernoulli equation with head loss hL between two points along a horizontal pipe of length  with constant diameter D. The modified Bernoulli equation can be written as For constant diameter and horizontal pipe, V1 = V2 and z1= z2. Then, the head loss can be formulated as

Fully Developed Laminar Flow Head loss due to friction hf, (11) f is known as the Darcy friction factor. For laminar flow, it is defined as (12)

Fully Developed Turbulent Flow In turbulent pipe flow, we can anticipate that the velocity profile and the shear stress will be somewhat similar to the second graph of Fig. 7. At any point in vicinity of the wall the velocity is still small and the flow is locally laminar. However, towards the centreline, the flow becomes turbulent but stable. If we relate this pattern with the velocity gradient , we will have a non-linear distribution of shear stress which is maximum at the wall but reduces significantly towards the centreline. Figure 7 Turbulent Velocity Profile and Shear Stress Distribution

Fully Developed Turbulent Flow From Fig. 7, we can see that the velocity gradient u/r near the wall is greater compared to the gradient for laminar flow, thus the wall shear stress w is very high but still finite. However, near the pipe centreline, the gradient becomes smaller which forms a nearly inviscid profile. Hence, we can deduce that a highly turbulent flow can be approximated by an inviscid flow with a finite friction factor. Therefore, we can divide a general turbulent flow into three regions: 1. Viscous sub-layer, the flow is locally laminar and the laminar shear stress is dominant, 2. Outer layer, the flow is locally turbulent and the turbulent shear stress is dominant, 3. Overlap layer, transition between the above two layers..

Friction Loss For laminar flow in rough pipes, the friction factor f is dominantly caused by viscous friction due to molecular interaction. Hence, we can use Eq. (12) for all occasions involving laminar flow. However, for turbulent flow, the profile at the core of the pipe is close to inviscid profile and the friction factor f is much due to the existence of viscous sublayer near the wall. Thus, if the wall surface is rougher, the resulting viscous sublayer is thicker. The roughness of a pipe is measured in length which is defined as equivalent roughness . The values of  for typical pipes are listed in Table 1. Table 1 Equivalent Roughness for Typical New Pipes

Friction Loss For turbulent flow, the friction factor f can be obtained by using the graphical representation of the Colebrook formula which is the Moody chart as shown in Figure 11 Table 11: The moody chart

Friction Loss In some texts, the same experimental data are refitted to a simpler form of correlation which can be solved directly with 2% error. This correlation is known as the Haaland formula which takes the following form: After knowing the friction factor f for the pipe, we can calculate the major head loss due to friction for a fluid flowing in the pipe. If fluid properties,  and , pipe length  and relative roughness of the pipe wall  are all known, provided that other variables are also known, the problem can be one of the following types: 1. Determine pressure loss p or friction head loss hf, 2. Determine volumetric flow-rate Q or average velocity V, 3. Determine pipe diameter D. After knowing f, then hf can be calculated via Eq. (11),

Minor Losses Apart from major loss due to friction, there are also other forms of losses which are caused by changes in internal pipe geometries and by fitted components. These types of losses are referred to as minor losses. There are four types of minor losses: 1. Sudden or gradual flow expansion and flow contraction, 2. Entrance and exit flows to and from reservoirs or tanks, 3. Bends, elbows, junctions and other fittings, 4. Valves, including those completely opened or partially closed. Minor loss is denoted by hm and is expressed as proportional to the velocity head, i.e. where K is the loss coefficient for each case. This coefficient K can either be derived analytically or taken from experimental or commercial data.

Minor Losses If we have a number of fittings along a pipe, the total head loss is be the summation of friction head loss, or major loss, with all minor losses, i.e.

Example 5 Water flows from the ground floor to the second level in a three-storey building through a 20 mm diameter pipe (drawn-tubing,  = 0.0015 mm) at a rate of 0.75 liter/s. The layout of the whole system is illustrated in Figure below. The water flows out from the system through a faucet with an opening of diameter 12.5 mm. Calculate the pressure at point (1).

Example 5 Solution: From the modified Bernoulli equation, we can write In this problem, p2 = 0, z1 = 0. Thus, The velocities in the pipe and out from the faucet are respectively The Reynolds number of the flow is

Example 5 Solution: The roughness d = 0.0015/20 = 0.000075. From the Moody chart,   0.022 (or,0.02191 via the Colebrook formula). The total length of the pipe is Hence, the friction head loss is The total minor loss is Therefore, the pressure at (1) is

Summary This chapter has summarized on the aspect below: You should be able to understand the concept of viscous flow in pipes and to be able to differentiate a laminar flow against a turbulent flow. In addition, you should also be able to calculate major and minor losses for both types of flow in order to calculate the overall pressure or head loss in a pipe system.

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