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Closed Conduit HydraulicsPowerPoint Presentation

Closed Conduit Hydraulics

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Hydraulics of Closed Conduit Flow

- Synonyms- closed conduit flow- pipe flow- pressurized flow
- Objectives – to introduce- basic concepts of closed conduit flow, - its hydraulics, and - design method

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Concepts – Reynolds Number

- Reynolds Number (ratio of inertia force to viscous force)V = velocity (ft/sec)D = pipe diameter (ft) = density of fluid (lbm/ft3) = dynamic viscosity of fluid (lbm/ftsec or lbfsec/ft2) = kinematic viscosity (ft2/sec)

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Concepts – Froude Number

- Froud Number (ratio of inertia force to gravitational force)
- V = velocityg = gravitational accelerationh = depth of water

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Concepts – turbulent flow

- Turbulent flow - Critical Re (laminar to turbulent) in the order of 1000

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Concepts – uniform & steady flow

- Uniform flow – constant characteristics with respect to space
- Steady flow – constant characteristics with respect to time. Often adopted when establishing pipe system design parameters (pressure & flow at certain locations). Consider unsteady (transient) phenomena to refine design (pipe pressure class and thickness)

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Conservation of Mass

- For steady & incompressible flow, dS/dt = 0I = OV1A1 = V2A2 ViAi = VoAo

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Conservation of Momentum

- Newton’s 2nd law – the resultant of all external forces on a system is equal to the time rate of change of momentum of this system

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Conservation of Momentum

- Consider this control volume (CV) of fluid in a pipe elbow

x1=v1t

1

1’

2

2’

x2=v2t

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Conservation of Momentum

- In a time t the fluid originally at Section 1 moves to 1’, and that at Section 2 moves to 2’
- The control volume lost momentum equal to that of the fluid contained between 1 and 1’(A1x1)V1 = A1V12t = (QV1)tAt the same time it gained momentum (QV2)t

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Conservation of Momentum

- The time rate of change of momentum is (QV2)- (QV1)
- Hence, the 2nd Law becomes
- This is the momentum equation for steady flow. Use this convention:
- QVx1 Fx = QVx2
- QVy1 Fy = QVy2
- Where depends on the direction of the force w.r.t. the coordinate system

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Application of Momentum Eq.

- Forces on a pipe elbow:Taking momentum balance in the x direction,QV1 + (PA)1 – Fx = Q(0)Fx = (PA)1 + QV1

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Application of Momentum Eq.

- Taking momentum balance in the y direction,External y force = (PA)2 - FyRate of change of momentum = QV2 (where V2 is in the negative direction) (PA)2 - Fy = QV2 Fy = (PA)2 - QV2 = (PA)2 + QV2

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Conservation of Energy

- In pipeline design, most often consider steady state – flow not varying with time - first
- Steady state (SS) Bernoulli Equation along a streamline:

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Conservation of Energy

- Pressure head p/
- Elevation head z
- Velocity head V2/2g
- Piezometric head p/ + z
(hydraulic grade line)

- Total head p/ + z + V2/2g
(energy grade line)

- Head Loss h

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Example 3-1

- A plane jet of unit discharge q0 strikes a boundary at an angle of 45, what will be the ratio of q1/q2 for the divided flow?

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Head Losses

- Include mostly 2 types of losses:
- Friction Loss- resulting from friction between the fluid and pipe wall
- Minor Loss- resulting from pipe entrance, transition, exit, valve and other in-line structures

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Friction Loss

- Most useful head loss equation for closed-conduit flow – Darcy-Weisbach equation

Pipe length

Friction head loss

Pipe velocity

Dimensionless Friction coefficient

Gravitational acceleration

Pipe diameter

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Friction Loss

- Darcy-Weisbach equation- derived from basic relationships of physics - dimensionless, app. to all unit systems- determined from experimental data
- Other friction loss relationships – Hazen-Wiliams, Manning, Chezy, etc. – are also used in the industry, but are less accurate and will not be discussed here

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Darcy-Weisbach (cont’d)

- Turbulent flow in rough pipesTransition between turbulent smooth and rough pipes

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Friction Loss

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Darcy-Weisbach

- Most recent development of Darcy Weisbach coefficient - Explicit equation [Swamee and Jain, 1976] applicable to entire turbulent flow regime (smooth, transition and rough pipes):

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Minor Loss

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Minor Loss

- American Water Works Association – Steel Pipe, A guide for design and installation, Manual of Water Supply Practices, M11, 4th Edition, 2004

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Minor Loss

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Minor Loss

- Valve manufacturer has loss curves typically presented in terms of Cv vs. valve opening degrees. Cv is defined as the flow rate in gallons per minute of 60 water that flows through the valve under 1 psi of head loss.

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Globe Valve

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Angle Valve

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Example 3-2

- p. 2.24 of Mays’ Hydraulic Design Handbook – Calculate f and e/D from given discharge

V2/2g=1.21 m

Atmospheric Pressure

P=3MPa

L=2500 m

El. 200 m

D=27 in

El. 100 m

Q=1.8 cms

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Example 3-3

- Same problem but now we have an 20” in-line ball valve with a 20” bore opened at 70 from closed position, a contraction and expansion section each connected to the valve, and 2 90 elbows with r/D=2. What is the f now?

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