Chapter 2 (Shell Momentum Balances and Velocity Distributions in Laminar Flow) - Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar

Chapter 2 (Shell Momentum Balances and Velocity Distributions in Laminar Flow)

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Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow ChE 771: Advanced Transport Phenomena Dr Rami Jumah Department of Chemical Engineering Jordan University of Science and Technology
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Outline Molecular and Convective Momentum Transport Shell Momentum Balance Boundary Conditions Flow of a Falling Film Flow Through a Circular Tube Flow Through an Annulus Flow of Two Adjacent Immiscible Fluids 2
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Molecular and Convective Momentum Transport Recall: The total momentum flux is the sum of the convective and molecular momentum fluxes: 3
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Shell Momentum Balance The problems discussed in this chapter are approached by setting up momentum balances for appropriately selected volumes of fluid within the flows. These volumes, which are fixed in space, are open systems through which the fluid is flowing. We call these volumes “shells” because they are thin in one dimension, specifically in the direction in which the fluid velocity varies . For steady flow, the momentum balance over a shell is In this chapter we apply this statement only to one component of the momentum, namely the component in the direction of flow. The momentum balance is applied only to flows in which there is just one velocity component, which depends on just one spatial variable. 4
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Procedure for Setting up and Solving Viscous Flow Problems Draw a sketch of the flow geometry being studied, including your best guess as to that the velocity distribution will look like. Identify the nonvanishing velocity component and the spatial variable on which it depends. Draw the “shell” - a volume that is thin in the direction in which the velocity varies. Write a momentum balance for the thin shell. Let the thickness of the shell approach zero and make use of the definition of the first derivative to obtain a differential equation for the momentum flux. 5
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Procedure for Setting up and Solving Viscous Flow Problems Identify all boundary conditions (solid-liquid, liquid-liquid, liquid-free surface, momentum flux values at boundaries, symmetry for zero flux). Integrate the differential equation for the momentum flux to get the momentum-flux distribution. Insert the expression for the total momentum flux ( ), including Newton’s law of viscosity, and obtain a differential equation for the velocity. Integrate this equation to get the velocity distribution. Use the velocity distribution to get other quantities, such as the maximum velocity, average velocity, or forces on solid surfaces. 6
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Assumptions The flow is always assumed to be at steady-state. The flow is laminar. The flow is assumed to be fully-developed (neglect entrance and exit effects) velocity f (flow direction).
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