conservation laws

1. CHE 160A - Unit Operations 1
Conservation Laws
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2. Conservation Laws
3 fundamental laws that govern fluid flow:
1 Conservation of mass
2 Newton’s 2nd law - conservation of momentum
3 1st law of thermodynamics - conservation of energy
Fluid fields: quantity defined as a function of position and time
throughout a given region.
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3. Conservation Laws
Conservation of Mass
The conservation of mass equation:
∂
∂t
Z
CV
ρdV +
Z
CS
ρ(v · n)dA = 0
[Rate of mass efflux from a CV] - [Rate of mass flow into a CV] +
[Rate of accumulation of mass within a CV] = 0
Specific cases:
1 Steady flow:
ZZ
C.S.
ρ(v · n)dA = 0
2 Constant density / steady flow:
ZZ
C.S.
(v · n)dA = 0
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4. Conservation Laws
Worksheet - Activity 1
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5. Conservation Laws
Conservation of Momentum
Integral Expression
∂
∂t
ZZZ
CV
ρvdV +
ZZ
CS
ρv(v · n)dA =
X
F
X
F =
X
Fbody +
X
Fsurf
This is a vector equation.
X
Fx =
ZZ
CS
ρvx(vnx + vny + vnz)dA +
∂
∂t
ZZZ
CV
ρvxdV
X
Fy =
ZZ
CS
ρvy(vnx + vny + vnz)dA +
∂
∂t
ZZZ
CV
ρvydV
X
Fz =
ZZ
CS
ρvz(vnx + vny + vnz)dA +
∂
∂t
ZZZ
CV
ρvzdV
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6. Conservation Laws
worksheet - Activity 2
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7. Conservation Laws
Navier Stokes
Navier Stokes Incompressible Flow:
ρ
Dvx
Dt
= −
∂P
∂x
+ µ
∂2vx
∂x2
+
∂2vx
∂y2
+
∂2vx
∂z2
+ ρgx
ρ
Dvy
Dt
= −
∂P
∂y
+ µ
∂2vy
∂x2
+
∂2vy
∂y2
+
∂2vy
∂z2
+ ρgy
ρ
Dvz
Dt
= −
∂P
∂z
+ µ
∂2vz
∂x2
+
∂2vz
∂y2
+
∂2vz
∂z2
+ ρgz
ρ
Dv
Dt
= −∇P + µ∇2
v + ρg
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8. Conservation Laws
Navier Stokes
Common assumptions / conditions:
Steady state: ρ
∂v
∂t
= 0
Rectilinear flow: ρv · ∇v = 0
Inviscid flow: µ = 0
Reduction of dimension
Neglect body force: ρg = 0
No pressure gradient: ∇P = 0
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9. Conservation Laws
Navier Stokes - General solution procedure
Choose a coordinate system Ask yourself:
1 What direction is the flow?
2 In what direction does the velocity change?
Determine the driving force for flow (pressure, shear, gravity)
Write the boundary conditions
Guess the form of the solution →what should it look like?
Simplify the conservation equations
Solve the resulting differential equations
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10. Conservation Laws
Worksheet - Activity 3
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