Chapter06

# Chapter06 - Control-Volume Method Global view as opposed to...

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Control Control - - Volume Method Volume Method z Global view as opposed to a detailed view Determine relations between what flows in and out of a control volume Indirect computation of forces No information about details of flow within the CV z Basic equations we will use… Mass Conservation Momentum Conservation Bernoulli’s Equation (if needed) z The first thing we must do is select an appropriate control volume

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Control Control - - Volume Selection Volume Selection Two primary guidelines… z Choose boundaries that make u rel •n easy to evaluate z Choose boundaries that make pressure integral easy to evaluate…suppose we focus only on x momentum z Can’t always satisfy both…trigonometric functions appear in u rel •n for the 2 nd example above Å No trigonometric functions appear in u rel •n Trigonometric functions appear in u rel •n Æ Å Unknown p in the injection slot is irrelevant Unknown p in the injection slot is part of solution Æ
Solution Strategy Solution Strategy Preferred Sequence of Operations… 1. Apply mass conservation…establishes u rel •n for momentum and energy conservation 2. Apply momentum conservation…use as many components as are relevant 3. Count equations ( N e ) and unknowns ( N u ) 4. If N e < N u , apply energy conservation…Bernoulli’s Equation will suffice for now 5. Check the physics, dimensional consistency and limiting cases if any are obvious

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Useful Control Useful Control - - Volume Theorem Volume Theorem - - 1 1 z In evaluating the net pressure force on a control volume, we can replace pressure, p , by ( p-p a ) where p a is a constant reference pressure z We have shown that, for an infinitesimal volume z Can build up an arbitrary, finite volume as the sum of many V …pressures cancel at interfaces, i.e., p A n i + p A n i+1 = 0 z So, for a finite control volume
Useful Control Useful Control - - Volume Theorem Volume Theorem - - 2 2 z Since integration is a linear operation… z For the integral of p a , we have z Since p a is constant, p a = 0 , so that z Reflects the fact that constant pressure exerts no net force on an object

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Fluid Statics Revisited Fluid Statics Revisited - - 1 1 z To test any new theory, it is good engineering practice to test it on a problem for which the solution is known z We know the solution for the following fluid- statics problem What we know about the solution… F = F x i + F z k F x = force on the projection of arc AB on a vertical plane (viz., surface OA) F z = weight of the column of fluid above arc AB
Fluid Statics Revisited Fluid Statics Revisited - - 2 2 z In the absence of motion, mass conservation simplifies to 0 = 0 z Letting f = g = g k , momentum conservation simplifies to z Also, Bernoulli’s Equation simplifies to the hydrostatic relation… z We select our control volume to be bounded by arc OABCD and end planes whose unit normals are normal to the page ( + j and j )

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Fluid Statics Revisited Fluid Statics Revisited - - 3 3 z The closed-surface integral is the sum of integrals
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Chapter06 - Control-Volume Method Global view as opposed to...

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