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Conservation laws • Laws of conservation of mass, energy, and momentum. • Conservation laws are first applied to a fixed quantity of matter called a closed system or just a system, and then extended to regions in space called control volumes. • The conservation relations are also called balance equations since any conserved quantity must balance during a process.

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**Continuity Equation Conservation of mass for a system**

Time rate of change of the system mass =0 For a system and a fixed nondeforming control volume that are coincident at an instant of time ,with B - mass and b=1 we see at Figure 5.1 (p. 194) System and control volume at three different instances of time. (a) System and control volume at time t – δt. (b) System and control volume at time t, coincident condition. (c) System and control volume at time t + δt.

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• Time rate of change of the mass of the system = time rate of change in the mass of the control volume + net rate of flow of mass through control surface. • Time rate of change in the mass of the control volume = • Net rate of flow of mass through control surface = • For steady flow, • Net mass flow rate through the control surface = where m mass flow rate( slug/s;kg /s)

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Continuity Equation • Conservation of mass: for a fixed, non deforming control volume • More commonly used: • Average velocity:

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Example 1 • Seawater flows steadily through a simple conical-shaped nozzle. If the nozzle exit velocity =20 m/s, determine the minimum pumping capacity in m3/s.

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Example 2 • Airflows steadily between two sections in a long straight portion of 4-in diameter pipe. The uniformly distributed temperature and pressure at each section are given. If the average air velocity at section (2) is 100 ft/s, calculate the average air velocity at section (1).

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Example 3 Moist air enters a dehumidifier at the rate of 22 slugs/hr. Determine the mass flow rate of the dry air and the water vapor leaving the dehumidifier.

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Example 4 A bathtub is being filled with water from a faucet. The rate of flow from the faucet is steady at 9 gal/min. The tub volume is approximated by a rectangular space.

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Figure E5.5b (p. 199)

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**Moving, Non deforming Control Volume**

V = W +VCV V= absolute velocity, V W= relative velocity of fluid seen by an observer moving with the control volume velocity, Vcv Continuity equation for a moving non-deforming control volume

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**Example 5. An airplane moves forward at a speed of 971 km/hr**

Example 5. An airplane moves forward at a speed of 971 km/hr. The frontal intake area of the jet engine is 0.80 m2, and the entering air density is kg/m3. A stationary observer determines that relative to the earth, the jet engine exhaust gases moves away from the engine with a speed of 1050 km/hr. The engine exhaust area is m2, and the exhaust gas density is kg/m3. Estimate the mass flow arte of fuel into the engine in kg/hr.

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MAE 3130: Fluid Mechanics Lecture 5: Fluid Kinematics Spring 2003

MAE 3130: Fluid Mechanics Lecture 5: Fluid Kinematics Spring 2003

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