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Deformation of block of wood when one surface is heated

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See the attached diagram.

A block of wood that is 3 inches wide x 6 inches long x .380 inches thick is fixed/anchored to a structure.

The top surface of the wood block is heated to 300 degrees C. The bottom surface is cooled to an undetermined temperature.

The differential in temperature causes the top surface to expand/elongate while the bottom wood surface contracts. The difference in the expansion and contraction causes the end of wood block to deflect/deform by 1.25 inches.

The wood properties as well as a thermal conductivity are given.

1) Determine the cold temperature on the bottom surface of the wood that would cause enough temp differential between the hot & cold to cause the end of the wood to deflect as shown.

2) Based on the deflection of 1.25 inches, we are also asked to determine the vertical force, horizontal force, and actual force that caused the deformation.

If this were a traditional structural mechanics problem, I could reasonably determine that the vertical load (P) that could cause a deflection of 1.25 inches is approx. 144 lbs.

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Solution Summary

The deformation of blocks of wood when one surface is heated is determined.

See Also This Related BrainMass Solution

Head-box issues in Pulp Mill : Theoretical Modelling Question

The process in question uses a centrifugal fan pump to convey a 1000 gallon per minute flow containing 2% finely ground wood pulp to a head box...which in turn is supposed to ocnvert turbulent flow to laminar flow and then deposit the water & pulp slurry on a high speed moving wire in order to form a wet fiber mat.

However, the head box that we have unfortunately does not create laminar flow --- it instead creates flow vectors that direct a high percentage of the slurry flow from the center of the moving wire to the edges of the moving wire leaving us with a dimensionally imperfect fiber mat that deposits more fiber on the edges of the moving wire form than in the center ---- this causes numerous downstream quality and physical sheet strength properties.

Rather than replace or completely re-design the head box, which would be extremely expensive and time consuming, we crafted what can only be called "flow tubes" that when welded together as a unit and placed directly in front of the turbulent flow vectors from the head box, seem to actually straighten out the flow patter and ultimately produces laminar flow.....which is what we want!!!

Please see the attached diagrams that I have put together which show the "plan view" of the existing head box and flow pattern and then the plan view and cross-sections of the flow tube device that I mentioned above.

Here is the question....is there a way in which the flow patterns can be empirically modeled or derived mathematically?
In otherwords, can math and/or science (e.g. fluid dynamics) prove that this "flow tube" device can actually work?

Second, is there a flow equation that can be used to calculate the theoretical flow in each flow tube given the 1000 gallon per minute flow at a 2% consistency???

Any thoughts, direction, or help in deriving a more empirical understanding of how this device should work...or otherwise straightening out the turbulent flow in my head box would be very greatly appreciated.

Please see attached file.

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