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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= 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 V = W +V CV

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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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