# STEADY FLOW THROUGH PIPES DARCY WEISBACH EQUATION FOR FLOW IN PIPES. HAZEN WILLIAM S FORMULA, LOSSES IN PIPELINES, HYDRAULIC GRADE LINES AND ENERGY

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1 STEADY FLOW THROUGH PIPES DARCY WEISBACH EQUATION FOR FLOW IN PIPES. HAZEN WILLIAM S FORMULA, LOSSES IN PIPELINES, HYDRAULIC GRADE LINES AND ENERGY LINES 1

2 SIGNIFICANCE OF CONDUITS In considering the convenience and necessities in every day life, it is truly amazing to note the role played by conduits in transporting fluid. For example, the water in our homes is normally conveyed through pressure pipelines, from the distribution system, so that it will be available when and where we want it. Moreover, virtually all of this water leaves our homes as dilute wastes through sewers, another type of conduits. Oil is often transferred from their source by pressure pipelines to refineries while gas is conveyed by pipelines into a distribution network for supply. Thus, it can be seen that the fluid flow in conduits is of immense practical significance in civil/environmental engineering. 2

3 PIPE FLOW SYSTEM A pipe is a closed conduit, generally of circular cross-section, used to carry water or any other fluid. When the pipe is running full, the flow is under pressure. But if the pipe is not running full (as in case of sewer pipes, culverts etc), the flow is not under pressure. In such case the atmospheric pressure exists inside the pipe. We will discuss the flow of pipes under pressure only. 3

4 HYDRAULIC RADIUS Hydraulic Radius = R h = A/P Where, A is cross-sectional Area P is the Wetted Perimeter (length of boundary in contact with water) 4

5 HYDRAULIC RADIUS For Pipe Flow: A R 2 & P 2R R h 2 R 2R R 2 D 4 5

6 HEAD (REF: Energy head topic) 2 2 p1 V p V z h 2 2 L z g g g g In above equation each term has the dimensions of length. Thus p/g, called the pressure head, represents the energy per unit weight stored in the fluid by virtue of the pressure under which the fluid exists. Z called the elevation head or potential head, represents the potential energy per pound of fluid; V 2/ 2g, called the velocity head, represents the kinetic energy per pound of fluid. We call the sum of these three terms the total head, usually denoted by H, so that H 2 p V z g 2g 6

7 GRAPHICAL REPRESENTATION OF PRESSURE HEAD AND VELOCITY HEAD If pressure head of liquid flowing in a pipe be plotted as vertical ordinates on the centre line of the pipe, then the line joining the tops of such ordinates is known as Hydraulic Grade Line (HGL). If the sum of pressure heads and velocity heads of a liquid flowing in a pipe be plotted as vertical ordinates on the center line of the pipe then the line joining the tops of such ordinates is known as Energy Grade Line (EGL) or Total Energy Line (TEL). In other words EGL lies over the HGL by an amount equal to the velocity heads as shown in the figure. 7

9 LOSS OF HEAD IN PIPES When the water is flowing in a pipe, it experiences some resistance to its motion, whose effect is to reduce the velocity and ultimately the available head of water. Though there are many type of losses, yet the major loss is due to frictional resistance of the pipe only. The frictional resistance depends upon the roughness of the inside surface of pipe. It has been experimentally found that more the roughness of inside surface of pipe, greater will be the resistance. This friction is known as fluid friction and the resistance is known as frictional resistance. 9

10 LOSSES Fluids have losses due to friction in the pipe and minor losses associated with tees, elbow, valves etc. Bernoulli s Equation becomes, Where, h f friction head loss. h m minor head loss h f h m g V z p g V z p g g 10

11 1. FRICTIONAL LOSSES IN PIPE FLOW In fluid flow, the friction head loss can be calculated by considering the pressure losses along the pipelines. In a horizontal pipe of diameter D carrying a steady flow there will be a pressure drop in a length L of the pipe. Equating the frictional resistance to the difference in pressure forces, and manipulating resulted into the following expression: This equation is known as Darcy-Weisbach (D-W) equation, in which f is the friction factor. It should be noted that f is dimensionless, and the value is not constant. 11

12 1. FRICTIONAL LOSSES IN PIPE FLOW 12

13 REYNOLDS NUMBER The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid, among other things. After exhaustive experiments in the 1880s, Osborne Reynolds discovered that the flow regime depends mainly on the ratio of inertial forces to viscous forces in the fluid. This ratio is called the Reynolds number and is expressed for internal flow in a circular pipe as Where, V is flow velocity (m/s), d is diameter in this case, in m). 13

14 2. MINOR LOSSES In addition to head loss due to friction, there are always other head losses due to pipe expansions and contractions, bends, valves, and other pipe fittings. These losses are usually known as minor losses (h m ). In case of a long pipeline, the minor losses maybe negligible compared to the friction losses, however, in the case of short pipelines, their contribution may be significant. These are: Losses due to pipe fittings Sudden Enlargement Sudden Contraction Bends etc. 14

15 DARCY-WEISBACH (D-W) EQUATION 15

16 PROBLEM NO.01 16

17 PROBLEM NO.02 17

18 PROBLEM NO.03 18

19 PROBLEM NO.04 19

20 HAZEN-WILLIAMS EQUATION 20

21 HAZEN-WILLIAMS FORMULA For calculating energy loss to friction the special case of the flow of water (Newtonian fluid) in pipeline systems. Limited to the flow of water in pipe larger than 2.0 in and smaller than 6.0 ft in diameter. Velocity of flow should not exceed 10.0 ft/s Developed for water at

22 v = Average velocity of flow (ft/s) C h = Hazen-Williams coefficient (dimensionless) R = Hydraulic radius of flow conduit (ft) s v C R s h 0.54 = Ratio of h L /L: energy loss/length of conduit (ft/ft) 22

23 v ChR s v = Average velocity of flow (m/s) C h = Hazen-Williams coefficient (dimensionless) R = Hydraulic radius of flow conduit (m) s = Ratio of h L /L: energy loss/length of conduit (m/m) 23

24 24

25 PROBLEM NO.05 For what velocity of flow of water in a new, clean, 6-in diameter pipe with an energy loss of 6.1 m of head occur over a length of m? Compute the volume flow rate at that velocity, using the design value of Ch for NEW cast iron pipe. SOLUTION: Then, S= h L /L = (6.1 m)/(304.8 m)= 0.02 R= D/4 = (0.154 m)/4 = m C h = 130 = 0.85C h R 0.63 s 0.54 = 0.85(130)(0.0385) 0.63 (0.02) 0.54 = m/s Q = A = (0.019 m m/s) = m 3 /s 25

26 OTHER FORMS OF THE HAZEN-WILLIAMS FORMULA 26

27 PIPE FLOW ANALYSIS 27

28 PIPES IN SERIES When two or more pipes of different diameters or roughness are connected in such a way that the fluid follows a single flow path throughout the system, the system represents a series pipeline. In a series pipeline the total energy loss is the sum of the individual minor losses and all pipe friction losses. Discharge = Q = Q 1 = Q 2 = Q 3 = --- Head losses = h L = h L1 +h L2 +h L

29 PIPES IN PARALLEL A combination of two or more pipes connected between two points so that the discharge divides at the first junction and re-joins at the next is known as pipes in parallel. Here the head loss between the two junctions is the same for all pipes. Discharge = Q = Q 1 + Q 2 + Q Head losses = h L = h L1 =h L2 =h L3 =

30 PIPE NETWORKS In municipal distribution systems, pipes are frequently interconnected so that the flow to a given outlet may come by several different paths, as in Fig. 30

31 PIPE NETWORKS As a result, we often cannot tell by inspection which way the flow travels, as in pipe BE. Nevertheless, the flow in any network, however complicated, must satisfy the basic relations of continuity and energy as follows: 1. The flow into any junction must equal the flow out of it. 2. The flow in each pipe must satisfy the pipe-friction laws for flow in a single pipe. 3. The algebraic sum of the head losses around any closed loop must be zero. 31

32 PIPE NETWORK ANALYSIS Pipe network analysis involves the determination of the pipe flow rates and pressure heads at the outflows points of the network. The flow rate and pressure heads must satisfy the continuity and energy equations. The earliest systematic method of network analysis (Hardy-Cross Method) is known as the head balance or closed loop method. This method is applicable to system in which pipes form closed loops. The outflows from the system are generally assumed to occur at the nodes junction. For a given pipe system with known outflows, the Hardy-Cross method is an iterative procedure based on initially iterated flows in the pipes. At each junction these flows must satisfy the continuity criterion, i.e. the algebraic sum of the flow rates in the pipe meeting at a junction, together with any external flows is zero. 32

33 PROCEDURE: Step-1: By careful inspection assume the most reasonable distribution of flows that satisfies condition 1. Step-2: Write condition 2 for each pipe in the form: h where K and n are constants for each pipe. L KQ n 33

34 PROCEDURE: Step-3: To investigate condition 3, compute the algebraic sum of the head losses around each elementary loop, h L =.KQ n. Consider losses from clockwise flows as positive, counterclockwise negative. Only by good luck will these add up to zero on the first trial. Step-4: Adjust the flow in each loop by a correction Q to balance the head in that loop and give KQ n = 0. The heart of this method lies in the following determination of Q. For any pipe, we may write Q = Q 0 + Q 34

35 PROCEDURE: where Q is the correct discharge and Q o is the assumed discharge. Then, for each pipe, h L KQ n n n1 K( Q0 Q) K( Q0 Q...) If Q is small compared with Q o, we may neglect the terms of the binomial series after the second one, so that h L KQ n 0 QKnQ For a loop, h L = KQ n = 0, so because Q is the same for all pipes in that loop, n1 0 n n1 KQ0 Q KnQ0 0 35

36 PROCEDURE: As we must sum the corrections of head loss in all pipes arithmetically (treating all terms as positive), we may solve this equation for Q, Q n KQ 0 KQ Q n1 0 n1 0 since, h L /Q = KQ n-l. We emphasize again that we must sum the numerator of algebraically, with due account of each sign, while we must sum the denominator arithmetically. Note that the in the numerator gives this quantity the same sign as the n1 head loss. Q Q 0 0 n hl h L / Q 0 36

37 PROCEDURE: The negative sign in Eq. indicates that when there is an excess of head loss around a loop in the clockwise direction, we must subtract the Q from clockwise Q o values and add it to counterclockwise ones. The reverse is true if there is a deficiency of head loss around a loop in the clockwise direction. Step-5: After we have given each loop a first correction, the losses will still not balance, because of the interaction of one loop upon another (pipes which are common to two loops receive two independent corrections, one for each loop). So we repeat the procedure, arriving at a second correction, and so on, until the corrections become negligible. 37

38 PROCEDURE: As values of K appear in both the numerator and denominator of the first form, we can use values proportional to the actual K to find the distribution. The second form is more convenient for use with pipe-friction diagrams for water pipes. An attractive feature of this approximation method is, that errors in computation have the same effect as errors in judgment and the process eventually corrects them. 38

39 PROBLEM NO.06 If the flow into and out of a two-loop pipe system are as shown in Fig., determine the flow in each pipe using only a basic scientific calculator. The K values for each pipe were calculated from the pipe and minor loss characteristics and from an assumed value of f, and n = 2, 39

40 SOLUTION: As a first step, assume a flow in each pipe such that continuity holds at all junctions. Take clockwise flows as positive. Calculate Q for each loop, make corrections to the assumed Qs, and repeat several times until the Qs are quite small. 40

41 41

42 PROBLEM NO.07 Find the magnitude and direction of the flow in network lines ab and bc (Fig. P8.117) after making two sets of corrections. The numbers on the figure are the K values of each line; take n = 2.0. Start by assuming initial flows as follows: 9 cfs in lines ab and cd, 6 cfs in lines ac and bd, and 3 cfs in line bc. 42

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