3-4%20Mass%20Balance-Thermodynamic1

3-4%20Mass%20Balance-Thermodynamic1 - CHG 2312 CHG Fluid...

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Unformatted text preview: CHG 2312 CHG Fluid Flow Review of the Balance Equation and the Mass anc Balance Outline 1. The balance equation 2. The mass balance equation 3. The steady-state balance 4. The mass balance equation 5. The steady-state flow a) One-dimensional flow in a pipe b) One-dimensional flow – Average velocity c) One-dimensional flow – Velocity distribution 6. The energy balance 1. The balance equation Accumulation = creation - destruction + flow in - flow out University of Ottawa, CHG 2312, P. Mehrani 3 2. The mass balance equation Mass Balance: o o Also known as the principle of conservation of mass, continuity equation or material balance. Creation and destruction terms are zero. Accumulation of mass within the chosen boundaries = flow of mass in - flow of mass out Rate of increase of mass within the chosen boundaries = flow rate of mass in - flow rate of mass out • dmcv • = min − m out dt University of Ottawa, CHG 2312, P. Mehrani 4 3. The steady-state balance Implies that nothing is changing with respect to time. 0 = mass flow rate in - mass flow rate out 5 University of Ottawa, CHG 2312, P. Mehrani 4. The steady-state flow a) One-dimensional flow in a pipe: System boundary Flow out Flow in 2 1 Mass flow rate in at point 1 = Mass flow rate out at point 2 ∫ ρV dA =∫ area 1 area 2 ρV dA If the fluid velocity and density are constant across the cross-section: ρ1V1A1 = ρ2V2A2 = constant University of Ottawa, CHG 2312, P. Mehrani 6 4. The steady-state flow b) One dimensional flow - Average Velocity: o W hen the density is uniform across the cross section of a pipe or channel: • mass flow rate m Volumetric Flow Rate = Q = = density ρ Average Velocity = Vaverage = Q A A1 V1 = A2 V2 University of Ottawa, CHG 2312, P. Mehrani 7 5. The steady-state mass balance c) One-dimensional flow – Velocity Distribution Inviscid: Laminar: Turbulent: University of Ottawa, CHG 2312, P. Mehrani 8 6. The energy balance Consider only: o Internal energy (u) o Kinetic energy (ke) o Potential energ y (pe) Energy can not be created or destroyed. Accumulation = Energy flow in - Energy flow out University of Ottawa, CHG 2312, P. Mehrani 9 6. The energy balance Accumulation = Energy flow in - Energy flow out d[m(u + pe + ke)]sys = (u + pe + ke)in dmin − (u + pe + ke)out dmout + dQ + dW University of Ottawa, CHG 2312, P. Mehrani 10 6. The energy balance d[m(u + pe + ke )]sys = (u + pe + ke )in dmin − (u + pe + ke )out dmout + dQ + dW Potential Energy: o System : 1kg steel ball lifted a distance dz. o Insulated: dQ = 0. o No m atter flows in or out: dmin = dmout = 0. m d (u + pe + ke o o o )sys = dW No friction heating: dusys = 0. Initial and final velocity are zero: d(ke)sys = 0. W ork done on the system: dW = F dz = msys g dz pe = gz md( pe )sys = (mgdz ) 11 University of Ottawa, CHG 2312, P. Mehrani 6. The energy balance d[m(u + pe + ke )]sys = (u + pe + ke )in dmin − (u + pe + ke )out dmout + dQ + dW Kinetic Energy: o System : 1kg steel ball is thrown horizontally. o Insulated: dQ = 0. o No m atter flows in or out: dmin = dmout = 0. o No friction heating: dusys = 0. o No change in elevation: d(pe)sys = 0. o W ork done on the system : dW = F dx = msys asys dx md (u + pe + ke )sys = dW m d(ke )sys = m a dx University of Ottawa, CHG 2312, P. Mehrani V2 ke = 2 12 6. The energy balance d[m(u + pe + ke )]sys = (u + pe + ke )in dmin − (u + pe + ke )out dmout + dQ + dW dW = dWinj + dWn. f . Injection dW rk( Pv) in dmin − ( Pv) out dmout + dWn. f . Wo= : o The work necessary to inject dm across the system boundary, i.e. work done by a normal stress (pressure) at the control surface. dWinj = Fdx = PAdx = − PdV dWinj = + ( Pv) in dmin − ( Pv) out dmout o dW en system For an op= dWinj + dWn . f . dW = (Pv)in dmin − (Pv)out dmout + dWn.f. University of Ottawa, CHG 2312, P. Mehrani 13 6. Energy balance equation d[m(u + pe + ke)]sys = (u + pe + ke)in dmin − (u + pe + ke)out dmout + dQ + dW dW = ( Pv) in dmin − ( Pv) out dmout + dWn. f . ⎡⎛ ⎛ ⎛ V 2 ⎞⎤ V2 ⎞ V2 ⎞ ⎟ dmin − ⎜ u + Pv + gz + ⎟⎥ = ⎜ u + Pv + gz + ⎟ dmout + dQ + dWn. f . d ⎢m⎜ u + gz + ⎜ ⎜ ⎜ ⎟ ⎟ 2 ⎠⎦ sys ⎝ 2 ⎠ in 2 ⎟ out ⎝ ⎠ ⎣⎝ u + Pv = h University of Ottawa, CHG 2312, P. Mehrani 14 ...
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