Shell Momentum Balance / Methods of Shell Momentum Balance (Chemical Engineering-Basic Chemical Engineering)

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Chemical Engineering

 

Shell Momentum Balance

Methods of Shell Momentum Balance

Shell Momentum Balance

•      In fluid mechanics  it may be necessary to determine how a fluid velocity changes across the flow. This can be done with a shell balance.

•      A shell is a differential element of the flow. By looking at the momentum and forces on one small portion, it is possible to integrate over the flow to see the larger picture of the flow as a whole.

•      The balance is determining what goes into and out of the shell. Momentum enters and leaves the shell through fluid entering and leaving the shell and through shear stress. In addition, there are pressure and gravity forces on the shell.

•      The goal of a shell balance is to determine the velocity profile of the flow. The velocity profile is an equation to calculate the velocity based on a specific location in the flow. From this, it is possible to find a velocity for any point across the flow.

•      When fluid flow occurs in a single direction everywhere in a system, shell balances are useful devices for applying the principle of conservation of momentum.

•      An example is incompressible laminar flow of fluid in a straight circular pipe. Other examples include flow between two wide parallel plates or flow of a liquid film down an inclined plane.

Shell Momentum Balance Equation

·         Rate of entry of momentum into shell - Rate of efflux of momentum from the shell + Sum of all the forces acting on the fluid in the shell = 0.

  • In order for a shell balance to work, the flow must:
  • Be laminar flow
  • Be without bends or curves in the flow
  • Steady State
  • Have two boundary conditions
  • Boundary Conditions are used to find constants of integration.
  • Fluid - Solid Boundary: No-slip condition, the velocity of a liquid at a solid is equal to the velocity of the solid
  • Liquid - Gas Boundary: Shear Stress = 0
  • Liquid - Liquid Boundary: Equal velocity and shear stress on both liquids

 

•      The following is an outline of how to perform a basic shell balance.

•      If fluid is flowing between two horizontal surfaces, each with area A touching the fluid, a differential shell of height Δy can be drawn between them as shown in the diagram below.

Diagram of the Shell Balance in fluid mechanics



•      Conservation of Momentum is the Key of a Shell Balance

•      In above Figure, top surface velocity U and Bottom surface is stationary, density of fluid and viscosity considered and velocity in x direction.

•      Conservation of Momentum is the Key of a Shell Balance

•      rate of momentum in - rate of momentum out + sum of all forces = 0

To perform a shell balance, follow the following basic steps

•      1. Find momentum from shear stress

•      (Momentum from Shear Stress Into System) - (Momentum from Shear Stress Out of System)

•      Momentum from Shear Stress goes into the shell at y and leaves the system at y + Δy.

•      Shear stress = τyx, area = A, momentum = τyxA

•      2. Find momentum from flow

•      Momentum flows into the system at x = 0 and out at x = L

•      The flow is steady state. Therefore, the momentum flow at x = 0 is equal to the moment of flow at x = L. Therefore, these cancel out

•      3. Find gravity force on the shell

•      4. Find pressure forces

•      5. Plug into conservation of momentum and solve for τyx

•      6. Apply Newton's law of viscosity for a Newtonian fluid

                τyx = -μ(dVx/dy)

•      7. Integrate to find equation for velocity and use Boundary Conditions to find constants of integration

•      Boundary 1: Top Surface: y = 0 and Vx = U

•      Boundary 2: Bottom Surface: y = D and Vx = 0

 


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