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 steadystate balance 4. The mass balance equation 5. The steadystate flow
a) Onedimensional flow in a pipe
b) Onedimensional flow – Average velocity
c) Onedimensional 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 steadystate 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 steadystate flow
a) Onedimensional 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 crosssection:
ρ1V1A1 = ρ2V2A2 = constant
University of Ottawa, CHG 2312, P. Mehrani 6 4. The steadystate 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 steadystate mass balance
c) Onedimensional 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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 Spring '11
 STADNIK
 Fluid Dynamics, Kinetic Energy, Mass, Mass Balance, Velocity

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