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Anonymous Poster #1

### Boundary Layer Thickness Problem ?

09/07/2012 5:36 PM

I am generally inexperienced in how to conduct "boundary layer" calculations and am unsure how to approach the problem I will now describe:

I have a steel strip (x, ft. wide) traveling in a closed looped fashion at a strip speed (speed, fpm) of ~1500 ft./min. In part of the loop, the strip is submerged in a quiescent water bath and then exits vertically from the bath into ambient air and travels vertically. I want to estimate the volume of water still on the strip a vertical distance (say L, ft.) from the water surface. I expect this is a "boundary layer thickness" calculation, but I have no knowledge as to what assumptions can be made in the calculation. Please provide any equations and assumptions made. FYI, this is a real life situation, not a "Homework" problem.

Thanks

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Join Date: Sep 2012
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#1

### Re: Boundary Layer Thickness Problem ?

09/07/2012 6:51 PM

￼￼ assistance.... Boundary layer is a layer adjacent to a surface where viscous effects are important. The fluid particles at the flat plate surface have zero velocity and they act as a retardant to reduce velocity of adjacent particles in the vertical direction. Similar actions continue by other particles until at the edge of the boundary layer where the particles' velocity is 99% of the free stream velocity. Bound- ary layers can also be measured by more significant parameters. The main boundary layer parameters are as follows: The displacements thickness, δ∗ is defined as the distance by which the external streamlines are shifted due to the presence of the boundary layer: ∗��u δ= (1−u)dy (1) ∞ The momentum thickness represents the height of the free-stream flow which would be needed to make up the deficiency in momentum flux within the boundary layer due to the shear force at the surface. The momentum thick- ness for an in-compressible boundary layer is given by: ��uu θ= u (1−u )dy (2) ∞∞ The skin-friction coefficient is defined as: Cf = τ0 dy (3) 1 ρu2 2∞ τ0 = (∂u)y=0 (4) ∂y 2 ￼￼￼￼￼￼ The Reynolds number is a measure of the ratio of inertia forces to viscous forces. It can be used to characterize flow characteristics aver a flat plate. Values under 500,000 are classified as Laminar flow where values from 500,000 to 1,000,000 are deemed Turbulent flow. Is it important to distinguish be- tween turbulent and non turbulent flow since the boundary layer thickness varies. Visit the following webpage... I found same extremely informative http://kai.gemba.org/pdf/MAE440/MAE440Exp03.pdf

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#2

### Re: Boundary Layer Thickness Problem ?

09/09/2012 1:31 AM

You want to know how much water is lifted out of the bath. What's the surface of the steel look like?

What is this and why does it matter? Won't the water just "rain" back down into the bath?

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#3

### Re: Boundary Layer Thickness Problem ?

09/11/2012 1:58 PM

I can only make a suggestion about the layer thickness.

If you consider an element dy * dz * l at the distance y from the steel band surface which moves with Wo upwards it is possible to write that this element has on one side a shear force τ*dz*l, on the other side (τ+dτ)*dz*l and is under the gravity force g*dm= g*dy*dz*l. The force balance leads to the relationship: dτ=g*dy→dτ/dy=g. We know that τ=η* dv/dy and that η=ρ*ν so that we obtain: dτ/dy=η*(d2v/dy2) and finally: d2v/dy2=g/ν. Integrating the equation we obtain the evolution of the fluid in the film perpendicular to the movement: v= Wo-g/(2*ν)*y^2. At a distance

y*= (2*ν*Wo*ν/g )^0.5

the velocity is zero (ν=kinematic viscosity and g= earth acceleration, Wo is the band transport velocity). For y>y* the velocities are <0 which means that the transported fluid flows back down. As one sees the problem is not the same as for the boundary layer where the fluid is "stopped" by the friction to the wall, in this case the friction is moving the fluid!

If I did not make any error the value for y* at 20°C will be about 1.25 mm. It decreases with temperature increase.

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