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6 . Property Balance Math + Mass balances

6 . Property Balance Math + Mass balances. CH EN 374: Fluid Mechanics. Review: Material Derivative. H ow f changes for a fluid particle. H ow f changes in each direction. H ow a fluid particle moves through the field. H ow the field itself changes with time. Example from the Homework….

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6 . Property Balance Math + Mass balances

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  1. 6. Property Balance Math + Mass balances CH EN 374: Fluid Mechanics

  2. Review: Material Derivative How f changes for a fluid particle How f changes in each direction How a fluid particle moves through the field How the field itself changes with time

  3. Example from the Homework…

  4. Overall Change for Control Volume • Take an extensive property B • Examples: mass, momentum, energy • Overall change inside control volume: • Affected by: • Change inside CV • Flow in and out Control Volume

  5. Reynold’s Transport Theory (RTT) • How is this similar to the material derivative? • Gives a macroscopic or integral material balance Total change of property Change within the control volume. Flow in and out across the control surface. Control Volume Control Surface

  6. Change of B inside control volume. • What will make this term change with time? • Density changing • Control volume changing shape or size

  7. Flow in and out of control volume. • is a vector normal to the control surface at any point • is the velocity component normal to surface—that’s what goes in or out

  8. vavg A • For each inlet and outlet: • For the total, add together inlets and outlets

  9. Simplification • In this class we will only use RTT for: • Incompressible flows • Fixed control volumes • How does this simplify the equation? Control Volume Control Surface

  10. What did I want you to get out of all that? • Be able to describe what each term in the RTT means • Know the assumptions that go into simple material balances • Now we turn this into balances for different properties. • Today: Mass • Monday: Momentum • Wednesday: Mechanical Energy

  11. Conservation of Mass • Mass: • What’s B? • What’s b? • What’s dB/dt? • What’s ?

  12. Conservation of Mass

  13. Problem • A desktop computer is to be cooled by a fan whose flow rate is 0.40 m3/min. Determine the mass flow rate of air through the fan at an elevation of 3400 m where the air density is 0.7 kg/m3.

  14. Problem • A cylindrical water tank of radius whose top is open to the atmosphere is initially filled with water to a height of . A discharge plug near the bottom of the tank is pulled out and water begins to drain through a circular hole of diameter . • If we assume that energy losses through the hole at the bottom of the tank are negligible, the average velocity of the exiting water is given by , where is gravitational acceleration and is the height of water in the tank (this will be a function of time. This is called Toricelli’s Law and we will derive it ourselves next week. • Determine • If is 3 ft, is 4 ft, and is 0.5 in, how long will it take for the water level in the tank to drop to 2 ft?

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