# 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:

1. Be laminar flow
2. Be without bends or curves in the flow
4. Have two boundary conditions

Boundary Conditions are used to find constants of integration.

1. Fluid - Solid Boundary: No-slip condition, the velocity of a liquid at a solid is equal to the velocity of the solid
2. Liquid - Gas Boundary: Shear Stress = 0
3. 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 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 direction = ${\displaystyle V_{x}}$, 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 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

For examples of performing shell balances, visit the resources listed below.