CHAPTER 1: Water Flow in Pipes University of Palestine Engineering Hydraulics 2nd semester 2010-2011 CHAPTER 1: Water Flow in Pipes

Description of A Pipe Flow Water pipes in our homes and the distribution system Pipes carry hydraulic fluid to various components of vehicles and machines Natural systems of “pipes” that carry blood throughout our body and air into and out of our lungs.

Description of A Pipe Flow Pipe Flow: refers to a full water flow in a closed conduits or circular cross section under a certain pressure gradient. The pipe flow at any cross section can be described by: cross section (A), elevation (h), measured with respect to a horizontal reference datum. pressure (P), varies from one point to another, for a given cross section variation is neglected The flow velocity (v), v = Q/A.

Difference between open-channel flow and the pipe flow The pipe is completely filled with the fluid being transported. The main driving force is likely to be a pressure gradient along the pipe. Open-channel flow Water flows without completely filling the pipe. Gravity alone is the driving force, the water flows down a hill.

Types of Flow Steady and Unsteady flow For a steady flow The flow parameters such as velocity (v), pressure (P) and density (r) of a fluid flow are independent of time in a steady flow. In unsteady flow they are independent. For a steady flow For an unsteady flow If the variations in any fluid’s parameters are small, the average is constant, then the fluid is considered to be steady

Types of Flow Uniform and non-uniform flow A flow is uniform if the flow characteristics at any given instant remain the same at different points in the direction of flow, otherwise it is termed as non-uniform flow. For a uniform flow For a non-uniform flow

Types of Flow Examples: The flow through a long uniform pipe diameter at a constant rate is steady uniform flow. The flow through a long uniform pipe diameter at a varying rate is unsteady uniform flow. The flow through a diverging pipe diameter at a constant rate is a steady non-uniform flow. The flow through a diverging pipe diameter at a varying rate is an unsteady non-uniform flow.

Types of Flow Laminar and turbulent flow Laminar flow: The fluid particles move along smooth well defined path or streamlines that are parallel, thus particles move in laminas or layers, smoothly gliding over each other. Turbulent flow: The fluid particles do not move in orderly manner and they occupy different relative positions in successive cross-sections. There is a small fluctuation in magnitude and direction of the velocity of the fluid particles Transitional flow The flow occurs between laminar and turbulent flow

Types of Flow Reynolds Experiment Reynolds performed a very carefully prepared pipe flow experiment.

Increasing flow velocity

Types of Flow Reynolds Experiment Reynold found that transition from laminar to turbulent flow in a pipe depends not only on the velocity, but only on the pipe diameter and the viscosity of the fluid. This relationship between these variables is commonly known as Reynolds number (NR) It can be shown that the Reynolds number is a measure of the ratio of the inertial forces to the viscous forces in the flow

Types of Flow Reynolds number where V: mean velocity in the pipe [L/T] D: pipe diameter [L] : density of flowing fluid [M/L3] : dynamic viscosity [M/LT] : kinematic viscosity [L2/T]

Types of Flow

Types of Flow It has been found by many experiments that for flows in circular pipes, the critical Reynolds number is about 2000 Flow laminar when NR < Critical NR Flow turbulent when NR > Critical NR The transition from laminar to turbulent flow does not always happened at NR = 2000 but varies due to experiments conditions….….this known as transitional range

Types of Flow Laminar Vs. Turbulent flows low velocities Laminar flows characterized by: low velocities small length scales high kinematic viscosities NR < Critical NR Viscous forces are dominant. Turbulent flows characterized by high velocities large length scales low kinematic viscosities NR > Critical NR Inertial forces are dominant

Types of Flow Example 1 40 mm diameter circular pipe carries water at 20oC. Calculate the largest flow rate (Q) which laminar flow can be expected.

Energy Head in Pipe Flow Water flow in pipes may contain energy in three basic forms: 1- Kinetic energy. 2- potential energy. 3- pressure energy. Bernoulli Equation Energy per unit weight of water OR: Energy Head

Energy Head in Pipe Flow Energy head and Head loss in pipe flow

Energy Head in Pipe Flow Kinetic head Pressure head Elevation head = + + Notice that: In reality, certain amount of energy loss (hL) occurs when the water mass flow from one section to another. The energy relationship between two sections can be written as:

Energy Head in Pipe Flow Example

Energy Head in Pipe Flow Example In the figure shown: Where the discharge through the system is 0.05 m3/s, the total losses through the pipe is 10 v2/2g where v is the velocity of water in 0.15 m diameter pipe, the water in the final outlet exposed to atmosphere.

Energy Head in Pipe Flow Calculate the required height (h =?) below the tank

Energy Head in Pipe Flow Without calculation sketch the (E.G.L) and (H.G.L)

Basic components of a typical pipe system

Calculation of Head (Energy) Losses In General: When a fluid is flowing through a pipe, the fluid experiences some resistance due to which some of energy (head) of fluid is lost. Energy Losses (Head losses) Major Losses Minor losses Loss due to the change of the velocity of the flowing fluid in the magnitude or in direction as it moves through fitting like Valves, Tees, Bends and Reducers. loss of head due to pipe friction and to viscous dissipation in flowing water

Head Losses in Pipelines Calculation of Head (Energy) Losses Head Losses in Pipelines Part A:Major Losses

Losses of Head due to Friction Energy loss through friction in the length of pipeline is commonly termed the major loss hf This is the loss of head due to pipe friction and to the viscous dissipation in flowing water. Several studies have been found the resistance to flow in a pipe is: - Independent of pressure under which the water flows - Linearly proportional to the pipe length, L - Inversely proportional to some water power of the pipe diameter D - Proportional to some power of the mean velocity, V - Related to the roughness of the pipe, if the flow is turbulent

Energy Head & Head loss in pipe flow Losses of Head due to Friction Energy Head & Head loss in pipe flow

Major losses formulas Several formulas have been developed in the past. Some of these formulas have faithfully been used in various hydraulic engineering practices. Darcy-Weisbach ( f ) Hazen-William (CHW) Manning (n) The Chezy Formula The Strickler Formula

The resistance to flow in a pipe is a function of: Major losses formulas The resistance to flow in a pipe is a function of: The pipe length, L The pipe diameter, D The mean velocity, V The properties of the fluid . The roughness of the pipe, (the flow is turbulent).

Darcy-Weisbach Equation Major losses formulas Darcy-Weisbach Equation Where: f is the friction factor L is pipe length D is pipe diameter Q is the flow rate hL is the loss due to friction It is conveniently expressed in terms of velocity (kinetic) head in the pipe The friction factor is function of different terms: Renold number Relative roughness

Major losses formulas Friction Factor: (f ) For Laminar flow: (NR < 2000) [depends only on Reynolds’ number and not on the surface roughness] For turbulent flow in smooth pipes (e/D = 0) with 4000 < NR < 105 is

Major losses formulas Colebrook-White Equation for f For turbulent flow ( NR > 4000 ) with e/D > 0.0, the friction factor can be founded from: Th.von Karman formulas: Colebrook-White Equation for f

Major losses formulas There is some difficulty in solving this equation So, Miller improve an initial value for f , (fo) The value of fo can be use directly as f if: Pandtle - Colebrook Equation There are other Equation such as Karman Equation see Text book 2

Friction Factor f The thickness of the laminar sublayer d decrease with an increase in NR laminar flow NR < 2000 pipe wall e f independent of relative roughness e/D Smooth f varies with NR and e/D e pipe wall transitionally rough Colebrook formula turbulent flow e pipe wall f independent of NR NR > 4000 rough

Moody diagram A convenient chart was prepared by Lewis F. Moody and commonly called the Moody diagram of friction factors for pipe flow, There are 4 zones of pipe flow in the chart: A laminar flow zone where f is simple linear function of Re A critical zone (shaded) where values are uncertain because the flow might be neither laminar nor truly turbulent A transition zone where f is a function of both Re and relative roughness A zone of fully developed turbulence where the value of f depends solely on the relative roughness and independent of the Reynolds Number

Moody diagram

Moody diagram Laminar Marks Reynolds Number independence Transition critical

Moody diagram Notes: Colebrook formula is valid for the entire nonlaminar range (4000 < Re < 108) of the Moody chart In fact , the Moody chart is a graphical representation of this equation

Moody diagram Bonus: Find the theoretical formulation for friction factor for laminar flow.

Typical values of the absolute roughness (e) are given

Absolute roughness Materials Roughness

Problems (head loss) Three types of problems for uniform flow in a single pipe: Type 1: Given the kind and size of pipe and the flow rate head loss ? Type 2: Given the kind and size of pipe and the head loss flow rate ? Type 3: Given the kind of pipe, the head loss and flow rate size of pipe ?

Problems type I (head loss)

Example 2 The water flow in Asphalted cast Iron pipe (e = 0.12mm) has a diameter 20cm at 20oC. Is 0.05 m3/s. determine the losses due to friction per 1 km Type 1: Given the kind and size of pipe and the flow rate head loss ? Moody f = 0.018

Example 3 The water flow in commercial steel pipe (e = 0.045mm) has a diameter 0.5m at 20oC. Q=0.4 m3/s. determine the losses due to friction per 1 km Type 1: Given the kind and size of pipe and the flow rate head loss ?

Example 3-cont. Use other methods to solve f 1- Cole brook

Example 3-cont. 2- Pandtle - Colebrook Equation

Problems type II (head loss)

Method for solution of Type 2 problems

Relative roughness e/D

Example 4:

Example 4:

Example 5: Cast iron pipe (e = 0.26), length = 2 km, diameter = 0.3m. Determine the max. flow rate Q , If the allowable maximum head loss = 4.6m. T=10oC Type 2: Given the kind and size of pipe and the head loss flow rate ?

Example 5:cont. V= 0.82 m/s , Q = V*A = 0.058 m3/s Another solution? Trial 1 Trial 2 Another solution? V= 0.82 m/s , Q = V*A = 0.058 m3/s

Example 6: Compute the discharge capacity of a 3-m diameter, wood stave pipe in its best condition carrying water at 10oC. It is allowed to have a head loss of 2m/km of pipe length. Type 2: Given the kind and size of pipe and the head loss flow rate ? Solution 1: Table 3.1 : wood stave pipe: e = 0.18 – 0.9 mm, take e = 0.3 mm At T= 10oC, n = 1.31x10-6 m2/sec

Solve by trial and error: Iteration 1: Assume f = 0.02 From moody Diagram: Iteration 2: update f = 0.0122 From moody Diagram: Iteration f V NR 0 0.02 2.45 5.6106 Solution: 1 0.0122 3.14 7.2106 2 0.0121 Convergence Another solution?

Example 6:Another solution? Compute the discharge capacity of a 3-m diameter, wood stave pipe in its best condition carrying water at 10oC. It is allowed to have a head loss of 2m/km of pipe length. Type 2: Given the kind and size of pipe and the head loss flow rate ? Solution 2: At T= 10oC, n = 1.31x10-6 m2/sec Table 3.1 : wood pipe: e = 0.18 – 0.9 mm, take e = 0.3 mm From moody Diagram:

f = 0.0121

Problems type III (head loss)

Example 7:

Example 7:cont.

Example 8: Estimate the size of a uniform, horizontal welded-steel pipe installed to carry 14 ft3/sec of water of 70oF (20oC). The allowable pressure loss is 17 ft/mi of pipe length. Solution 2: From Table : Steel pipe: ks = 0.046 mm Darcy-Weisbach: Let D = 2.5 ft, then V = Q/A = 2.85 ft/sec Now by knowing the relative roughness and the Reynolds number: We get f =0.021

Example 8:cont. A better estimate of D can be obtained by substituting the latter values into equation a, which gives A new iteration provide V = 4.46 ft/sec NR = 8.3 x 105 e/D = 0.0015 f = 0.022, and D = 2.0 ft. More iterations will produce the same results.

Example 9:

Empirical Formulas 1 Hazen-Williams Simplified

Empirical Formulas 1

Empirical Formulas 1

Empirical Formulas 1

Empirical Formulas 1 Where: CH = corrected value CHo = value from table Vo = velocity at CHo V = actual velocity

Empirical Formulas 2 Manning This formula has extensively been used for open channel designs. It is also quite commonly used for pipe flows Manning Simplified

Empirical Formulas 2 n = Manning coefficient of roughness (See Table) Rh and S are as defined for Hazen-William formula.

Empirical Formulas 2

Empirical Formulas 2

Empirical Formulas 3 The Chezy Formula where C = Chezy coefficient

Empirical Formulas 3 It can be shown that this formula, for circular pipes, is equivalent to Darcy’s formula with the value for [f is Darcy Weisbeich coefficient] The following formula has been proposed for the value of C: [n is the Manning coefficient]

The Strickler Formula: Empirical Formulas 4 The Strickler Formula: where kstr is known as the Strickler coefficient. Comparing Manning formula and Strickler formula, we can see that

Empirical Formulas Relations between the coefficients in Chezy, Manning , Darcy , and Strickler formulas.

Example 10 New Cast Iron (CHW = 130, n = 0.011) has length = 6 km and diameter = 30cm. Q= 0.32 m3/s, T=30o. Calculate the head loss due to friction using: Hazen-William Method b) Manning Method

Head Losses in Pipelines Calculation of Head (Energy) Losses Head Losses in Pipelines Part B:Minor Losses

Minor Losses Additional losses due to entries and exits, fittings and valves are traditionally referred to as minor losses

Flow pattern through a valve Minor Losses It is due to the change of the velocity of the flowing fluid in the magnitude or in direction [turbulence within bulk flow as it moves through and fitting] Flow pattern through a valve

Minor Losses The minor losses occurs du to : It has the common form Valves Tees Bends Reducers And other appurtenances It has the common form “minor” compared to friction losses in long pipelines but, can be the dominant cause of head loss in shorter pipelines

Losses due to contraction Minor Losses Losses due to contraction A sudden contraction in a pipe usually causes a marked drop in pressure in the pipe due to both the increase in velocity and the loss of energy to turbulence. Along centerline Along wall

Minor Losses Value of the coefficient Kc for sudden contraction V2

Head Loss Due to a Sudden Contraction Minor Losses Head Loss Due to a Sudden Contraction 96

Minor Losses Head losses due to pipe contraction may be greatly reduced by introducing a gradual pipe transition known as a confusor

Head Loss Due to Gradual Contraction (reducer or nozzle) Minor Losses Head Loss Due to Gradual Contraction (reducer or nozzle) a 100 200 300 400 KL 0.2 0.28 0.32 0.35 A different set of data is : 98

Losses due to Enlargement Minor Losses Losses due to Enlargement A sudden Enlargement in a pipe

Minor Losses Head losses due to pipe enlargement may be greatly reduced by introducing a gradual pipe transition known as a diffusor

smaller head loss than in the case of an abrupt expansion Minor Losses Note that the drop in the energy line is much larger than in the case of a contraction abrupt expansion gradual expansion smaller head loss than in the case of an abrupt expansion

Head Loss Due to a Sudden Enlargement Minor Losses Head Loss Due to a Sudden Enlargement or :

Head Loss Due to Gradual Enlargement (conical diffuser) Minor Losses Head Loss Due to Gradual Enlargement (conical diffuser) a 100 200 300 400 KL 0.39 0.80 1.00 1.06

Minor Losses Gibson tests

Loss due to pipe entrance Minor Losses Loss due to pipe entrance General formula for head loss at the entrance of a pipe is also expressed in term of velocity head of the pipe

increasing loss coefficient Minor Losses Different pipe inlets increasing loss coefficient

Head Loss at the Entrance of a Pipe (flow leaving a tank) Minor Losses Head Loss at the Entrance of a Pipe (flow leaving a tank) Reentrant (embeded) KL = 0.8 Sharp edge KL = 0.5 Slightly rounded KL = 0.2 Well rounded KL = 0.04

Another Typical values for various amount of rounding of the lip Minor Losses Another Typical values for various amount of rounding of the lip

Loss at pipe exit (discharge head loss) Minor Losses Loss at pipe exit (discharge head loss) In this case the entire velocity head of the pipe flow is dissipated and that the discharge loss is

Head Loss at the Exit of a Pipe (flow entering a tank) Minor Losses Head Loss at the Exit of a Pipe (flow entering a tank) KL = 1.0 the entire kinetic energy of the exiting fluid (velocity V1) is dissipated through viscous effects as the stream of fluid mixes with the fluid in the tank and eventually comes to rest (V2 = 0).

Loss of head in pipe bends Minor Losses Loss of head in pipe bends

Minor Losses Miter bends For situations in which space is limited,

Loss of head through valves Minor Losses Loss of head through valves

Minor Losses

The loss coefficient for elbows, bends, and tees Minor Losses The loss coefficient for elbows, bends, and tees

Loss coefficients for pipe components (Table) Minor Losses Loss coefficients for pipe components (Table)

Minor loss coefficients (Table)

Minor loss calculation using equivalent pipe length Minor Losses Minor loss calculation using equivalent pipe length

Minor Losses Note that the above values are average typical values, actual values will depend on the make (manufacturer) of the components. See: Catalogs Hydraulic handbooks !!

Energy and hydraulic grade lines Minor Losses Energy and hydraulic grade lines Unless local effects are of particular interests the changes in the EGL and HGL are often shown as abrupt changes (even though the loss occurs over some distance)

Minor Losses Example 11 In the figure shown two new cast iron pipes in series, D1 =0.6m , D2 =0.4m length of the two pipes is 300m, level at A =80m , Q = 0.5m3/s (T=10oC).there are a sudden contraction between Pipe 1 and 2, and Sharp entrance at pipe 1. Fine the water level at B e = 0.26mm v = 1.31×10-6 Q = 0.5 m3/s

Minor Losses Solution

Minor Losses ZB = 80 – 13.36 = 66.64 m

Minor Losses Example 12 A pipe enlarge suddenly from D1=240mm to D2=480mm. the H.G.L rises by 10 cm calculate the flow in the pipe

Solution

Minor Losses Solution

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