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Unformatted text preview: Fundamental Laws of Hydraulics for a Control Volume (Brief Review) By M.S. Ghidaoui, Spring 2002 An Important Note: This chapter is essentially a review of the material covered in CIVL 151 under the heading Reynolds Transport Theorem (RTT). Therefore, for more details you should consult your CIVL151 notes. Indeed, this course assumes that you know all the concepts learned in CIVL151. Questions in the midterm and final exams may require concepts from that course! Therefore, I highly recommend that you embark on a thorough review of the fluid mechanics notes. General Balance Equation: Let P define a physical quantity (property) of the flow. For example, P may be mass ) ( m ; linear momentum ) ( V m ; angular momentum ) ( r V m × ; energy ) ( E or Entropy ) ( S . Take the following general control volume: The time rate of change of P within the control volume ) . ( V C = Time Rate of production/ destruction of P within the control volume + Time rate of input of P into control volume  Time rate of output of P from control volume ⇒ out in V C P P dt n destructio oduction d dt dP  + = P of / Pr . Or in out V C P P dt dP dt n destructio oduction d  + = . P of / Pr This is the general form of a balance (conservation) equation for a property. Mass Balance (Continuity): In this case, the property is mass. That is m P = Mass conservation principle states that the mass of a system cannot be created or destroyed. Of course, species can chemically react with one another to produce different species or species can change phase (e.g., from liquid to gas) but the mass of each specie and the total mass of all species remain constant with time. Mathematically, one can write the following: ⇒ ( 29 m of / Pr = n destructio oduction dt d ∫ = V C V C V d P . . ρ ∫ ⋅ = Ain in dA n V P ) ( ρ ) ( ≤ ⋅ n V ∫ ⋅ = Aout out dA n V P ) ( ρ ) ( ≥ ⋅ n V ⇒ ∫ ∫ ∫ ⋅ + ⋅ + = Aout Ain V C dA n V dA n V V d dt d ) ( ) ( . ρ ρ ρ Or ∫ ∫ ∫ ⋅ ⋅ = Aout Ain V C dA n V dA n V V d dt d ) ( ) ( . ρ ρ ρ This is the most general continuity (mass balance) equation for a control volume containing a single phase and single component fluid. Example: Spring runoff in a region is accumulated in a reservoir of trapezoidal section, length l , bottom width B , and side slope z . If the daily inflow of water is ) ( t Q and losses are negligible, how fast will the water level in the reservoir rise. Modify your previous answer to include evaporation losses at a rate of E , per unit area of water surface and infiltration at a rate I per unit area of wetted base B (that is, assume the slanted sides of the trapezoid are impermeable and allow no infiltration). Show that the time for the level to change from 1 H to 2 H , during a time period in which the inflow ) ( t Q is zero, is as follows: dh BI E zh B zh B T H H ∫ + + + = 2 1 ) 2 ( 2 h 1 z Solution: (a)...
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This note was uploaded on 03/19/2012 for the course CIVL 000 taught by Professor Kk during the Spring '10 term at HKUST.
 Spring '10
 KK

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