Fluid Flow in Pipes Pipes are closed conduits through which fluids or gages flows. Conduits may flow full or partially full. Pipes are referred to as conduits (usually circular) which flow full. Conduits flowing partially full are called open channels, which will be discussed in next Series. Fluid flow in pipes may be steady or unsteady. In steady flow, there are two types of flow that exist; they are called laminar flow and turbulent flow. INTRODUCTION

Liquid or gas flow through pipes or ducts is commonly used in heating and cooling applications and fluid distribution networks. The fluid in such applications is usually forced to flow by a fan or pump through a flow section. We pay particular attention to friction, which is directly related to the pressure drop and head loss during flow through pipes and ducts. The pressure drop is then used to determine the pumping power requirement. A typical piping system fittings or elbows to route the fluid, valves to control the flow rate, and pumps to pressurized the fluid. Although the theory of fluid flow is reasonably well understood, theoretical solutions are obtained only for a few simple cases such as fully developed laminar flow in a circular pipe. Therefore, we must rely on experimental r esults and empirical relations for most fluid flow problems rather than closed-form analytical solutions. Nothing that the experimental results are obtained under carefully controlled laboratory conditions and that no two systems are exactly alike, we must not be so naive as to view the results obtained as “exact.” An error of 10 percent (or more) in friction factors calculated using the relations in this chapter is the “norm” rather than the “exception.”

Circular pipes can withstand large pressure differences between the inside and the outside without undergoing any significant distortion, but noncircular pipes cannot.

Average velocity Vavg is defined as the average speed through a cross section. For fully developed laminar pipe flow, Vavg is half of maximum velocity.

Laminar Flow

The flow is said to be laminar when the path of individual fluid particles does not cross or intersect. The flow is always laminar when the Reynolds number Re is less than 2,000. Turbulent Flow

The flow is said to be turbulent when the path of individual particles is irregular and continuously cross each other. Turbulent flow normally occurs when the Reynolds number exceed 2,000.

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FLUID FLOW IN PIPES Laminar flow in circular pipes can be maintained up to values of Re as high as 50,000. However, in such cases this type of flow is inherently unstable, and the least disturbance will transform it instantly into turbulent flow. On the other hand, it is practically impossible for turbulent flow in a str aight pipe to persist at values of Re much below 2,000, because any turbulence that is set up will be damped out by viscous friction. REYNOLDS NUMBER

The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow ve locity, 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

=

= = =

= average flow velocity (m/s),

in m), and

= characteristic length of the geometry (diameter in this case,

= kinematic viscosity of the fluid (m2/s). Note that the Reynolds number is a dimensionless

quantity. Also, kinematic viscosity has the unit m 2/s, and can be viewed as viscous diffusivity for momentum. At large Reynolds numbers, the inertial forces, which are proportional to the fluid density and the square of the fluid velocity, are large relative to the viscous forces, and thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid. At small or moderate Reynolds numbers, however, the viscous forces are large enough to suppress these fluctuations and to keep the fluid “in line.” Thus the flow is turbulent in the first case and laminar in the second.

=

The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number,

.

The value of the critical Reynolds number is different for different geometries and flow conditions. For internal flow in a circular pipe, the generally accepted value of the critical Reynolds number is

2300

.

For flow through noncircular pipes, the Reynolds number is based on the hydraulic diameter Dh defined as

= 4

where is the cross-sectional area of the pipe and is the wetted perimeter. The hydraulic diameter is defined such that it reduces to ordinary diameter D for the circular pipes.

; = = 4 2

FLUID FLOW IN PIPES THE ENTRACE REGION

Consider a fluid entering a circular pipe at a uniform velocity. Because of the no-slip condition, the fluid particles in the layer in contact with the surface of the pipe come to a complete stop. This layer also causes the fluid particles in the adjacent layers to slow down gradually as a result of friction. To make up for this velocity reduction, the velocity of the fluid at the midsection of the pipe has to increase to keep the mass flow rate through the pipe constant. As a result, a velocity gradient develops along the pipe. The region of the flow in which the effects of the viscous s hearing forces caused by fluid viscosity are felt is called the velocity boundary layer or just the boundary layer. The hypothetical boundary surface divides the flow in a pipe into two regions: the boundary layer region, in which the viscous effects and the velocity changes are significant, and the irrotational (core) flow region, in which the frictional effects are negligible and the velocity remains essentially constant in the radial direction.

The development of the flow velocity boundary layer in a pipe. (the developed average velocity profile is parabolic in laminar flow, as shown, but somewhat flatter or fuller in turbulent flow)

HEAD LOSSES IN PIPE FLOW

Head losses in pipes may be classified into two; the major head loss, which is caused by pipe friction along straight sections of pipe of uniform diameter and uniform roughness, and minor head loss, which are caused by changes in velocity or directions of flow, and are commonly expressed in terms of kinetic energy. Major Head Loss A. Darcy-Weisbach Formula (pipe-friction equation)

ℎ = 2

= friction factor

= length of pipe in meters or feet = pipe diameter in meter or feet

= mean or average v elocity of flow in m/s or ft/s

For non-circular pipes, use D = 4R, where R is the hydraulic radius. 3

FLUID FLOW IN PIPES

Value of

For Laminar Flow:

= 64 For Turbulent Flow: For smooth pipes,

between 3,000 and 10,000 10,00 0 (Blasius)

= 0. 3164

For smooth and ro ugh pipes, turbulent (Colebrook equation)

1 = −2log + 2.51 −2log 3.7

This equation was plotted in 1944 by Moody into what is now called the Moody chart for pipe friction. B. Manning’s Formula

The Manning Formula is one of the best-known open-channel formulas and is commonly used in pipes. The formula is given by:

= 1

= roughness coefficient = hydraulic radius

= slope of the energy grade line =

C. Hazen-Williams Formula

The Hazen-Williams formula is the widely used in waterworks industry. This formula is applicable only to the flow of water in pipes larger than 50mm (2in) and velocity les s than 3m/s. This formula was designed for flow in both pipes and open channels but is more commonly used in pipes.

= 0.849 . . = Hazen-Williams coefficient

= hydraulic radius

= slope of the energy grade line =

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FLUID FLOW IN PIPES Illustrative Examples:

= 1.1.3 × 10 /

1. Water having a kinematic viscosity velocity of 4.5 m/s. Is the flow laminar or turbulent?

flows in a 100 mm diameter pipe at a

Solution:

The kinetic viscosity of water, and average velocity are given. The Reynolds number is to be determined, to v erify if the flow is laminar or turbulent.

Assumption:

1. The flow is steady and incompressible. 2 The entrance effects are negligible, and thus the flow is fully developed. 3 The pipe involves no components such as bends, valves, and connectors. 4 The piping section involves no work devices such as a pump or a turbine.

Properties:

The kinematic viscosity, diameter of pipe and velocity are given to be , , and , respectively.

Analysis:

10 / = 100

= 4.5 /

= 1.1.3 ×

First we need to determine the value of Reynolds number. The Reynolds number is

5 /(0. 1) = = 346,154 = = 4.1.3×10 / which is greater than 2,300. Therefore, the flow is turbulent. 2. Oil of specific gravity 0.80 flows in a 200 m m diameter pipe. Find the critical velocity. Use .

8.14×10 ∗∗

=

Solution:

The specific gravity of oil, dynamic viscosity and diameter of pipe are given. The average velocity is to be determined

Assumption:

1. The flow is steady and incompressible. 2 The entrance effects are negligible, and thus the flow is fully developed. 3 The pipe involves no components such as bends, valves, and connectors. 4 The piping section involves no work devices such as a pump or a turbine.

Properties:

The specific gravity of oil, dynamic viscosity and diameter of pipe are given to be , , and , respectively.

Analysis:

= 0.80 = 8.1 8.144 × 10 ∗∗ = 200 = 2,000 = = ∗ ( 2, 0 00( 00 8.14×10 ∗ ) = 1.0175 / = 0.20 ((0.80× 80 × 1000 1000 / / ) 1.0175 / The critical velocity in pipe occurred when velocity can evaluate as

Therefore, the average velocity of oil in t he pipe is

. The value of average

.

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FLUID FLOW IN PIPES Assignment (Laminar and Turbulent Flow) 1. Why are liquids usually transported in circular pipes? 2. Consider a person walking first in air and then in water at the same speed. For which will the Reynolds number be higher? 3.

Consider the flow of air and water in pipes of the same diameter, at the same temperature, and at the same mean velocity. Which flow is more likely to be turbulent? Why?

4. What is hydraulic diameter? How is it defined? What is it e qual to for a circular pipe of diameter D? 5. For laminar flow condition, what size of pipe will deliver 6 liters per second of oil having kinematic viscosity of ?

6.1×10 / 0.001 / / = 1.0 1 .08 8 × 10 = 1.51× 51 × 10 /) = 4.0 4.066 × 10 / = 1. 02× 02 × 10 /) = 1.15× 15 × 10 /) = 1.18× 18 × 10 /

6. A fluid at through a 100-mm-diameter pipe. Determine whether the flow is laminar or turbulent if the fluid is (a) hydrogen ( ), (b) air ( , (c) gasoline (

mercury (

), (d) water (

, or glycerin (

, (e)

)

7. Which fluid at room temperature re quires a larger pump to flow at a specific velocity in a given pipe: water, gasoline, or glycerin? Why?

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FLUID FLOW IN PIPES Illustrative Example 3. What is the hydraulic radius of a re ctangle air duct 200 mm by 350 mm? 4. What commercial size of new iron pipe shall be use d to carry 4,490 gpm with a loss of head of 10.56 feet per mile? Assume f = 0.019. 5. Glycerin (sg = 1.26 and μ = 1.49 Pa-s) flows through a rectangular conduit 300 mm by 450 mm at the rate of 160 lit/sec. a. Is the flow laminar or turbulent? b. Determine the head lost per kilometer length of pipe. 6. Oil with sg = 0.95 at 200 lit/sec through a 500 m of 200 mm diameter pipe (f = 0.0225). Determine (a) the head loss and (b) the pressure dro p if the pipe slopes down at 10˚ in the direction of flow.

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FLUID FLOW IN PIPES Minor Head Loss

The fluid in a typical piping system passes through various fittings, valves, bends, elbows, tees, inlets, exits, e xits, enlargements, and contractions in addition to the pipes. These components interrupt the smooth flow of the fluid and cause additional losses because of the flow separation and mixing they induce. In a typical system with long pipes, these losses are minor compared to the total head loss in the pipes (the major losses) and are called minor losses. Although this is generally true, in some cases the minor losses may be greater than the major losses. This is the case, for example, in systems with several turns and valves in a short distance. The head loss introduced by a completely open valve, for example, may be negligible. But a partially closed valve may cause the largest head loss in the system, as e videnced by the drop in the flow rate. Flow through valves and fittings is very complex, and a theoretical analysis is generally not plausible. Therefore, minor losses are determined experimentally, usually by the manufacturers o f the components.

ℎ = 2

8

FLUID FLOW IN PIPES

9

FLUID FLOW IN PIPES Piping Networks

Most piping systems encountered in practice such as the water distribution systems in cities or commercial or residential establishments involve numerous parallel and series connections as well as several sources (supply of fluid into the system) and loads (discharges of fluid from the system). A piping project may involve the design of a new system or the expansion of an existing system. The engineering objective in such projects is to design a piping system that will deliver the specified flow rates at specified pressures reliably at minimum total (initial plus operating and maintenance) cost. Once the layout of the system is prepared, the determination of the pipe diameters and the pressures throughout the system, while remaining within the budget constraints, typically requires solving the system repeatedly until the optimal solution is reached. Computer modeling and analysis of such systems make this tedious task a simple chore. Pipes Connected in Series

For pipes of different diameters c onnected in series as shown in the figure below, the discharge in all pipes are all equal and the head lost is e qual to the sum of the individual head losses.

= = = = ℎ + ℎ + ℎ + ℎ Pipes Connected in Parallel

= + + = ℎ = ℎ = ℎ

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FLUID FLOW IN PIPES Illustrative Problem 7. Two pipes, each 300 m long, are connected in series. The flow of water through the pipes is 150 lit/sec with a total loss of 15 m. If one pipe has a diameter of 300 mm, what is the diameter of the other pipe? Neglect minor losses and assume f = 0.02 for both pipes. 255mm 8. Two pipes 1 and 2 are in series. If the roughness coefficients n2 = 2n 2 n1 and the diameter D1 = 500 mm, find the diameter D 2 if the slope of their e nergy grade lines are to be the same. 648 mm 9. Two pipes 1 and 2 having the same length and diameter are in parallel. If the flow in pipe 1 is 750 lit/sec, what is the flow in pipe 2 if the friction factor f of the second pipe is twice that of the first pipe? 530 lit/sec 10. A pipe network consists of pipe 1 from A to B, then at B it is connected to pipelines 2 and 3, where it merges again at Joint C to form a single pipeline 4 up to point D. Pipe line 1, 2, and 4 are in series connection pipelines 2 and 3 are in parallel to each other. If the rate of flow from A to B is 10 lit/sec and assuming f = 0.02 for all the pipes. Determine the flow in each pipe and the total he ad lost from A to D. HL = 1.683 m Pipelines 1 2 3 4

Length (m) 3,000 2,200 3,200 2,800

Diameter (mm) 200 300 200 400

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FLUID FLOW IN PIPES RESERVOIR PROBLEMS Types of Reservoir Problems Type 1: Given the discharge in one of the

pipes, or the pressure at the junction P, and the required is the elevation one of the reservoirs or the diameter or length of the one of the pipes, and Type 2: Given all the pipe properties and

elevation of all reservoirs, find the flow in each pipe, which can be solved by trial and error. In any of these types, the main objective is to locate the position (elevation) of the energy at the junction P. this position represents the water surface of an imaginary reservoir at P. the difference in elevation between this surface and the surface of another reservoir is the head lost in the pipe leading to that reservoir. Illustrative Problem 11. A 1,200-mm-diameter concrete pipe 1,800-m-long carries 1.35 m3/s from reservoir A, whose water surface elevation 50 m, and discharges into two concrete pipes, each 1,350 m long and 750 mm in diameter. One of the 750-mm-diameter pipe discharges into reservoir B in which the water surface is at elevation 44 m. Determine the elevation of the water surface of reservoir C into which the other 750-mm-diameter pipe is flowing. Assume f = 0.02 for all pipes. 12. Three reservoirs A, B, and C are connected respectively with pipes 1, 2, and 3 joining at a common junction P whose elevation is 366 m. Reservoir A is at elevation 933 m and reservoir B is at elevation 844 m. The properties of each pipe are as follows: L1 = 1500 m, D1 = 600 mm, f1 = 0.02; L2 = 1000m, D2 = 450 mm, f2 = 0.025; L3 = 900m, D3 = 500 mm, f3 = 0.018. A pressure gage at junction P reads 4950 kPa. What is the flow in pipe 3 in m3/s and the elevation of reservoir C . 13. Determine the flow in each pipe in t he figure shown and the elevation of reservoir C if the inflow to reservoir A is 515 Lit/sec.

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FLUID FLOW IN PIPES 14. Three reservoirs A, B, and C are connected respectively with pipes 1,2, and 3. Determine the flow in pipe in the three re servoirs. Reservoirs

Elevation (m)

Pipe

Length (m)

A B C

80 50 10

1 2 3

1800 2000 4000

Diameter (mm) 400 500 800

f 0.02 0.025 0.03

Pipe Networks The following conditions must be satisfied in any pipe network: 1. The algebraic sum of the pressure drop ( head loss) around any closed loop must be zero and, 2. The flow entering a junction must be equal to the flow leaving it. Pipe network problems are usually solved by numerical methods using computer since any analytical solution requires the use of many simultaneous equations, some of which are nonlinear. Hardy Cross Method The procedure suggested by Hardy Cross requires that the flow in each pipe be assumed so that the principle of continuity is satisfied at each junction. A correction to the assumed flow is computed successively for each pipe loop in the network until the correction is reduced to an acceptable value Let

Qa = assumed flow Q

= true flow

α

= correctin

Q = Qa + α

Using Darcy-Weisbach formula:

ℎ = . Let say

= . ℎ = From condition 1

ℎ + ℎ + ℎ + ⋯ = 0 13

FLUID FLOW IN PIPES

+ + + ⋯ = 0 ∑ = 0 ∑ ( + ) = 0 ∑ + 2 ∑ +∑ + ∑ = 0 If α is small, the te rm containing α2 may be neglected. Hence;

∑ + 2 ∑ = 0 = − ∑∑ Illustrative Problem 15. The pipe network shown in the figure represents a spray rinse system. Find the flow in each pipe. Assume C1 = 120 for all pipes. 0.3 m3/s 600m – 300 mm

600m – 300 mm

A

600m – 300 mm

B

600m – 300 mm H

600m – 300 mm

600m – 300 mm G

0.1 m3/s

D

C

F

0.1 m3/s

E

0.1 m3/s

0.1 m3/s

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