# Shell 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.

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
#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

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 = ρ

viscosity 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 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 at y and leaves the system at y+Δy

Shear Stress = Τyx momentum = Tyx * (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 Shell

4. Find Pressure Forces

5. Plug into Conservation of Momentum and Solve for Tyx

6. Apply Newtons law of viscosity for a Newtonian Fluid

Tyx = -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

For 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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Shell model — In nuclear physics, the nuclear shell model is a model of the atomic nucleus which uses the Pauli principle to describe the structure of the nucleus in terms of energy levels. The model was developed in 1949 following independent work by several… …   Wikipedia

• Balance wheel — Wheel Wheel (hw[=e]l), n. [OE. wheel, hweol, AS. hwe[ o]l, hweogul, hweowol; akin to D. wiel, Icel. hv[=e]l, Gr. ky klos, Skr. cakra; cf. Icel. hj[=o]l, Dan. hiul, Sw. hjul. [root]218. Cf. {Cycle}, {Cyclopedia}.] [1913 Webster] 1. A circular… …   The Collaborative International Dictionary of English

• Racing shell — Toronto rowers in a 4+ oar shell In watercraft, a racing shell (also referred to as just a fine boat (UK) or just shell) is an extremely narrow, and often disproportionately long, rowing boat specifically designed for racing or exercise. It is… …   Wikipedia

• World of Ghost in the Shell — The anime and manga series Ghost in the Shell takes place in a cyberpunk version of Earth in the near future. The series focuses on Japan but several other nations figure prominently in some stories. The world of Ghost in the Shell features… …   Wikipedia

• Artillery — For other uses of the term, see Artillery (disambiguation). Warfare Military history Eras Prehistoric Ancient …   Wikipedia

• Economic Affairs — ▪ 2006 Introduction In 2005 rising U.S. deficits, tight monetary policies, and higher oil prices triggered by hurricane damage in the Gulf of Mexico were moderating influences on the world economy and on U.S. stock markets, but some other… …   Universalium

• Business and Industry Review — ▪ 1999 Introduction Overview        Annual Average Rates of Growth of Manufacturing Output, 1980 97, Table Pattern of Output, 1994 97, Table Index Numbers of Production, Employment, and Productivity in Manufacturing Industries, Table (For Annual… …   Universalium

• M60 (танк) — У этого термина существуют и другие значения, см. M60. 105 mm Gun Full …   Википедия

• Life Sciences — ▪ 2009 Introduction Zoology       In 2008 several zoological studies provided new insights into how species life history traits (such as the timing of reproduction or the length of life of adult individuals) are derived in part as responses to… …   Universalium

• Glossary of rowing terms — In competitive rowing, the following specialized terms are important in the corresponding aspects of the sport: Boat Classes In competitive rowing events, abbreviations are used for different boat classes.;Weight: *L or Lt:Lightweight rowing *If… …   Wikipedia