# Shell Balance

In

fluid mechanics , it may be necessary to determine how a fluidvelocity 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 throughshear stress . In addition, there arepressure andgravity 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.**Applications**Shell Balances can be used for many situations. For example, flow in a pipe, flow of multiple fluids around each other, or flow due to pressure difference. Although terms in the shell balance and boundary conditions will change, the basic set up and process is the same. This system is useful to analyze any fluid flow that holds true for the requirements listed below.

**Requirements**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 conditionsBoundary 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: Equalvelocity andshear stress on both liquids**Performing shell balances**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 the them as shown in the diagram below.In this example,the top surface is moving at velocity U and the bottom surface is stationary

density of fluid = ρviscosity of fluid = μ

velocity in x direnction = $V\_x$ , shown by the diagonal line above. This is what a shell balance is solving for.Conservation of Momentum is the Key of a Shell Balancerate of

momentum in - rate of momentum out + sum of all forces = 0To 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 at y and leaves the system at y+Δy

Shear Stress = Τ

_{yx}momentum = T_{yx}* (Area)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 Shell4. Find

Pressure Forces5. Plug into Conservation of Momentum and Solve for T

_{yx}6. Apply Newtons law of viscosity for a

Newtonian Fluid T

_{yx}= -μ(dV_{x}/dy)7. Integrate to find equation for velocity and use Boundary Conditions to find constants of integration

Boundary 1: Top Surface: y = 0 and V

_{x}= UBoundary 2: Bottom Surface: y = D and V

_{x}= 0For examples of performing shell balances, visit the resourses listed below.

**Resources***cite web

title = Problem Solutions in Transport Phenomena : Fluid Mechanics Problems

url= http://www.syvum.com/eng/fluid/

accessdate = 2007-10-06*cite book

last = Harriott

first = Peter

coauthors = W. McCabe , J. Smith

title = Unit Operations of Chemical Engineering: Seventh Edition

publisher = McGraw-Hill Professional

date=2004

pages = 68–132

url = http://books.google.com/books?id=u3SvHtIOwj8C&pg=PP1&dq=unit+operations+of+chemical+engineering&sig=-CXkpW8Zq3Mi9p-tCwvF0RUniYs#PPP1,M1

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2010.*

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