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Unformatted text preview: Conservation Laws Conservation of Energy The derivation for the DarcyWeisbach equation assumed a form for the head loss that is based on the conservation of energy. The Energy equation for a control volume requires that the Total time R.O.C. of internal energy within a C.V. = Heat added to C.V. Work done by C.V. Mathematically, from the first law of thermodynamics dE dt = dH dt dW dt Schematically, our system looks like By applying the Reynolds transport theorem, we can also define where CV is the control volume, CS is the control surface and e is the internal energy per unit mass, given by where z is the elevation of the fluid mass having velocity, v , and internal energy, u . CE 3200 1 of 12 Closed Conduit Energy Combining, this yields Net Rate of Change in the Control Volume Given our assumptions of steady state ( d/dt = 0 ) and incompressibility Net Rate of Change Entering/Leaving the Control Volume Z CS ρe v · n dA Substituting our definition of e into the above yields We can simplify the above by noting that the control surface in impermeable except at the inlet and outlet. This means that v · n is zero everywhere except the inlet and outlet areas, noted as A 1 and A 2 . Also note, if we expanded the above expression fully we would have something of the form Z A . . . + v 3 + . . . dA As we defined an average velocity in the derivation of the conservation of mass, we will again define an average velocity, V . However, since this quantity is cubed, then integrated, we must define an energy correction factor, α , where CE 3200 2 of 12 Closed Conduit Energy With this definition, we can define the internal energy per unit mass in the control volume at the inlet as We have also assumed that the energy at either the inlet or outlet is essentially the same no matter the location in the cross section. This makes the term inde pendent of the integration variable, and we can remove it from the integration, yielding Recall from our derivation of the conservation of mass (note, we are still in vector form) = Z...
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 Fall '08
 Smith
 Fluid Dynamics, Thermodynamics, Energy, Conduit Energy

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