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Unformatted text preview: Lecture 16Lecture 27 Control Volume Analysis Review for Test II Nov. 5, 2008 Mass Balances d(m cv )/dt is the rate of change of mass in the control volume. If mass is accumulating in the control volume, d(m cv )/dt is positive. This happens when there is more inflow than outflow. If there is more outflow, than inflow, d(m cv )/dt is negative. The dots on the mass flow imply that they are “rates of flow”. The units are kg/s or lbm/s Multiple inlets and exits: If there are multiple inlets and exits, all of these should be summed up. Hence, e exit i in cv m m dt m d ∑ ∑ = ) ( Mass Balances The product AV is the volumetric flow rate (m 3 /s). It is the product of Area (m 2 ) and Velocity (m/s). In volume flow rates, the mass balance equation is written as: Steady State Form of mass Rate Balance: In steady state, the rate of change of mass in the control volume is zero (d/dt = 0). Hence, However, note that temperature, pressure may be changing with time. In steady state, all quantities must be time invariant. ∑ ∑ = exit e e in i i CV v AV v AV dt dm ∑ = ∑ exit e e in i i v AV v AV Key Equations e i cv m m dt m d  = ) ( e exit i in cv m m dt m d ∑ ∑ = ) ( ∑ ∑ = exit e e in i i CV v AV v AV dt dm ∑ = ∑ exit e e in i i v AV v AV dA V m A n ∫ = ρ Conservation of Energy Time rate of change of the energy contained within the control volume at time t = net rate at which energy is being transferred in by heat transfer at time t  net rate at which energy is being transferred out by work at time t + net rate at which energy is being transferred into the volume accompanying the mass flow Mathematical Form For a oneinlet, oneoutlet control volume, we can write: where E CV = Energy in the control volume at time t Energy transfer by heat into the control volume Energy transfer by work out of the control volume The energy accompanying mass flow is the sum of internal energy, kinetic energy, and the potential energy. The first term denotes inflows while the second term denotes outflows. ) ( ) ( e e e e i i i i CV gz V u m gz V u m W Q dt dE + + + + + = 2 2 2 2 = Q = W Evaluating Work for a Control Volume Because work is done on or by a control volume where matter flows across the control volume, we should separate the work in two parts. The first part is associated with fluid pressure acting on the flow at inlets and outlets. The other part of the work is one that drives a shaft of a turbine, or displacements of the boundary. Consider the flow at the exit. The rate of work done by the flow of fluid is (Force exerted) x (Rate of distance moved) = (pressure x area) x (velocity) = p e A e V e = p e x (Rate of volume flow) = V = Velocity (Please be extremely careful with the double use of symbol V) e e p m v i i i e e e CV v p m v p m W W  + = Flow Work and Form of Control Volume Energy Balance The last two terms are called the flow work at the exit and at the inlet. and at the inlet....
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 Spring '08
 Staff
 Thermodynamics, Energy, Entropy, Steady State, COP

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