02 closed_conduit_02_momentum_stud

# 02 closed_conduit_02_momentum_stud - Conservation Laws...

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Unformatted text preview: Conservation Laws Conservation of Momentum The linear momentum equation for a control volume requires Total time rate of change of linear momentum External forces acting on control volume D ( m v ) Dt = Σ F where Da i Dt = da i dt + v da i d x We can define the forces in the direction of flow, resulting in Net R.O.C. within C.V. + Net R.O.C. leaving C.V. = Pressure forces on C.V. + Gravitational force within C.V. + Shear force on C.V. Mathematically, this is : For steady flow, the time rate of change is zero. CE 3200 1 of 14 Closed Conduit Momentum Net Rate of Change Leaving the Control Volume Considering only the surfaces with flow normal to the control surfaces (i.e. the inflow and outflow), we can re-write the equation as the sum of forces: In order to assume a uniform velocity, we will define the momentum correction factor, β , defined with With this definition, we can reduce the net rate of change leaving the control volume to We will assume that β = 1 , reducing the above to For a control volume of uniform cross-section ( A is constant everywhere), External Forces: Force Due to Pressure at Boundaries We can then define the average pressure at a boundary as p , yielding a pressure force of Note that these two pressure forces act opposite of each other. CE 3200 2 of 14 Closed Conduit Momentum We will also assume an average area for the control volume, reducing the above to a total pressure force to External Forces: Force Due to Gravity We can define the angle of the pipeline with sinθ = (Δ z ) / ( L ) and then write the x-component of the total weight of the control volume as External Forces: Force Due to Shear The forces acing on the control volume due to shear can be written as The forces due to shear act on all of the surfaces of the pipeline ( A s ), Defining the perimeter of the pipeline as P , we can re-write the above integral to Our new momentum equation is (assuming constant cross-section): CE 3200 3 of 14 Closed Conduit Momentum Substituting our definition for sin θ into the previous equation yields We define the total head, or energy per unit weight, at sections 1 and 2 as Then the head loss , Δ h , between sections 1 and 2 is defined as Note, at this point, all of the head loss is due to friction from the pipe walls, and...
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02 closed_conduit_02_momentum_stud - Conservation Laws...

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