transport phenomena lec16 21b

D E P A R T M E N T O F C I V I L E N G I N E E R I N G

B I T S P I L A N I , R A J A S T H A N

B Y

D R . S H I B A N I K H A N R A J H A

A U G U S T 2 0 1 5

Flow Analysis using Control Volumes

Lecture 16-21

Course: CE F212 Transport Phenomena 3 0 3

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Topics to be considered

Conservation of Mass: The Continuity Equation Derivation of Continuity Equation Fixed, Non-deforming Control Volume Moving, Non-deforming Control Volume Deforming Control Volume

Newton’s second law: The linear momentum and Moment-of-momentum equations Derivation of the Linear Momentum Equation Application of the linear Momentum Equation Derivation of the Moment of Momentum Equation Application of the Moment of Momentum Equation

First law of thermodynamics: The energy equation Derivation of the Energy Equation Application of the Energy Equation Comparison of the Energy Equation with the Bernoulli Equation Application of the Energy Equation to Non-uniform Flows Combination of the Energy Equation and the Moment of Momentum Equation


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Flow analysis using control volumes

Control volume approach can be used to solve many fluid mechanics problems

The control volume formulas are derived fromthe equations representing basic laws appliedto a collection of mass (a system)

The control volume or Eulerian view isgenerally less complicated and, therefore,more convenient to use than the system orLagrangian view.


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Learning Objects


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Select an appropriate finite CV to solve a fluid mechanics problem.

Apply basic laws to the contents of a finite CV to get important answers

How to apply these basic laws?

How to express these basic laws based on CV method?

Review of Reynolds Transport Theorem


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This is the fundamental relation between the rate of change of any arbitrary extensive property, B, of a system and the variations of this property associated with a control volume.

Newton’s second law:Derivation of the Linear Momentum equation

Newton’s second law deals with system momentumand forces

Mathematically, we can write

When a CV is coincident with a system at an instant of time,the forces acting on the system and the forces acting on thecontents of the coincident CV are instantaneously identical,that is

Time rate of change of the linear momentum of

the system

Sum of external forces= acting on the system

sys

sys

dDt

DFV

FFsyscontents of the coincident

control volume


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Newton’s second law:Derivation of the Linear Momentum equation

The Reynolds transport theorem, with b set equal to the velocity(momentum per unit mass) and B sys being the system momentum,allows us to write the following mathematical equation

As particles of mass move into or out of a control volume through thecontrol surface, they carry linear momentum in or out

For fixed and non-deforming control volume, the mathematicalstatement of Newton’s second law is

Above equation is known as linear momentum equation

cscvsys

dAdt

dDt

DnV.VVV ˆ

cscv

dAdt

FnV.VV ˆcontents of thecontrol volume


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Newton’s second law:Application of the Linear Momentum equation

The linear momentum equation for an inertial CV is a vector equation

Several important generalities about the application of the aboveequation can be listed as follows:

cscv

dAdt

FnV.VV ˆcontents of thecontrol volume


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The Linear Momentum Equations


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For a fixed and nondeforming control volume, the control volume formulation of Newton’s second law

Vector Form of Momentum Equation


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The sum of all forces (surface and body forces) acting on a Non-accelerating control volume is equal to the sum of the rate of change of momentum inside the control volume and the net rate of flux of momentum out through the control surface.

Example 10 Linear Momentum – Change in Flow Direction


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As shown in Figure, a horizontal jet of water exits a nozzle with a uniform speed of V1=10 ft/s, strike a vane, and is turned through an angle θ. Determine the anchoring force needed to hold the vane stationary. Neglect gravity and viscous effects.

Example 10 Solution


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The x and z direction components of linear momentum equation

Example 11 Linear Momentum – Weight, pressure, and Change in Speed


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Determine the anchoring force required to hold in place a conical nozzle attached to the end of a laboratory sink faucet when the water flowrate is 0.6 liter/s. The nozzle mass is 0.1kg. The nozzle inlet and exit diameters are 16mm and 5mm, respectively. The nozzle axis is vertical and the axial distance between section (1) and (2) is 30mm. The pressure at section (1) is 464 kPa. to hold the vane stationary. Neglect gravity and viscous effects.

Example 11 Solution


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Example 11 Solution


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Example 11 Solution


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Water flows through a horizontal, 180° pipe bend. The flow cross section area is constant at a value of 0.1ft2 through the bend. The magnitude of the flow velocity everywhere in the bend is axial and 50 ft/s. The absolute pressure at the entrance and exit of the bend are 30 psia and 24 psia, respectively. Calculate the horizontal (x and y) components of the anchoring force required to hold the bend in place.



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Example 12 Solution


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Example 12 Solution


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Example 13 Linear Momentum –Pressure, Change in Speed, and Friction


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Air flows steadily between two cross sections in a long, straight portion of 4-in. inside diameter pipe as indicated in Figure, where the uniformly distributed temperature and pressure at each cross section are given, If the average air velocity at section (2) is 1000 ft/s, we found in that the average air velocity at section (1) must be 219 ft/s. Assuming uniform velocity distributions at sections (1) and (2), determine the frictional force exerted by the pipe wall on the air flow between sections (1) and (2).

Example 13


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Example 13 Solution


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Example 13 Solution


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transport phenomena lec16 21b

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