Post on 15-Jul-2016
description
transcript
1
• Flat plat boundary layer analysis cannot be applied to curved objects i.e. cylinder and sphere– Flow reversal occurs– The boundary layer detaches from the wall surface
Boundary Layer over Curvature
2
• For a flow over a curved body, the x-coordinate is measured along the curved surface
• The y-coordinate is measured normal to the surface
4
• Turbulent eddies formed due to separation and cannot convert their rotational energy back into pressure head
• To prevent, streamlining reduces adverse pressure gradient beyond the max thickness and delays separation
5
• There is also an effect on heat transfer coefficient i.e. flow across a cylinder inside heat exchanger
Nu = hd/k
High h ( turbulent)
Low h (thick boundary layer)
6
• Several dimensionless groups in boundary layer analysis which explain the overall behaviour of the flow
• Re = inertial forces/viscous forces (V∞L/)– High inertial force = transition from laminar to turbulent
• Pr = momentum diffusivity/thermal diffusivity (/)– Relative thickness of the hydrodynamic and thermal boundary
layers
• Sc = momentum diffusivity/mass diffusivity (/Dab)– Relative thickness of the hydrodynamic and concentration
layers• Nu = heat transfer by convection/conduction (hΔT/kΔTL)• St = convective/diffusive mass transport (KL/D)
7
Similar relationship can also be observed between concentration (Sc) and hydrodynamic boundary layers
hydrodynamic
Thermal
h 1/T
8
Boundary Layer Analysis• We use example of convective transport on a flat plate • By making the following assumptions:• Steady state: ( )/t = 0• Incompressible flow: ρ = constant• Constant properties: k, Dab, Cp, ρ, μ• Newtonian fluid: xy = -Vy/x, yx = -Vxy• Two dimensional flow: ( )/z = 0, Vz = 0• No energy generation: q’ = 0• No species generation: R’ = 0
9
• Continuity:
• Momentum:
• Energy:
• Mass:
0
yV
xV yx
2
2
2
21yV
xV
xP
yVV
xVV xxx
yx
x
2
2
2
21yV
xV
yP
yV
VxV
V yyyy
yx
2
2
2
2
yT
xT
yTV
xTV yx
2
2
2
2
yC
xCD
yCV
xCV aa
aba
ya
x
10
• Next, we use some dimensionless equations as defined below:
• x* = x/L, y* = y/L (L = plate length)
• U* = Vx/V∞, V* = Vy/V∞ (V∞ = free stream velocity)
• T* = (T-Ts)/(T∞-Ts) (Ts = wall surface; T = free steam T)
• C* = (Ca-Cas)/(Ca∞-Cas) (Cas = wall surface; Ca = free stream conc.)
• P* = P/ρv∞2
• We also need to derive the partial derivative terms• For example:• x*/x = 1/L, y*/y = 1/L• U*/x = 1/V(Vx/x), V*/y = 1/V(Vy/y)• U*/y = 1/V(Vx/y), V*/y = 1/V(Vy/y)
11
• The equations are transformed into dimensionless forms as below:
• Continuity:
• Momentum:
• Energy:
• Mass:
0*
*
*
*
yV
xU
2*
*2
2*
*2
*
*
*
**
*
**
Re1
yU
xU
xP
yUV
xUU
2*
*2
*2
*2
*
**
*
**
PrRe1
yT
xT
yTV
xTU
2*
*2
2*
*2
*
**
*
**
Re1
yC
xC
ScyCV
xCU
2*
*2
2*
*2
*
*
*
**
*
**
Re1
yV
xV
yP
yVV
xVU