fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...

, a shell balance can be used to determine how fluid

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

changes across a flow. A shell is a

differential element In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathema ...

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 In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

is created within the shell through fluid entering and leaving the shell and by

shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...

. In addition, there are

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

and

gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

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 in many situations. For example, flow in a pipe, the 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.

Requirements for Shell Balance Calculations

The fluid must exhibit: *

Laminar flow In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...

*No bends or curves *

Steady State In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...

*Two boundary conditions Boundary Conditions are used to find constants of integration. *

Fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...

Solid Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...

Boundary:

No-slip condition In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary. The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary. C ...

, the velocity of a liquid at a solid is equal to the velocity of the solid. *

Liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...

Gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...

Boundary:

Shear Stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...

= 0. *Liquid - Liquid Boundary: Equal

and

on both liquids.

Performing shell balances

A fluid is flowing between and in contact with two horizontal surfaces of contact area A. A differential shell of height Δy is utilized (see diagram below). Shell Balance Diagram

The top surface is moving at velocity U and the bottom surface is stationary.
*

Density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

of fluid = ρ *

Viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...

of fluid = μ *Velocity in x direction =

V_x

, shown by the diagonal line above. This is what a shell balance is solving for.

Conservation of Momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

is the Key of a Shell Balance
*(Rate of

momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

in) - (rate of momentum out) + (sum of all forces) = 0
To perform a shell balance, follow the following basic steps: #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'' + Δ''y''. Shear stress = ''τ''_''yx'', area = ''A'', momentum = ''τ''_''yx''''A''. #Find momentum from the 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. #Find

gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...

force on the shell. #Find

forces. #Plug into conservation of momentum and solve for ''τ''_''yx''. #Apply Newton's law of viscosity for a

Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...

''τ''_''yx'' = -''μ''(''dV''_''x''/''dy''). #Integrate to find the equation for velocity and use Boundary Conditions to find constants of integration. Boundary 1: Top Surface: y = 0 and V_x = U Boundary 2: Bottom Surface: y = D and V_x = 0 For examples of performing shell balances, visit the resources listed below.

Resources

* *{{cite book , last = Harriott , first = Peter , author2=W. McCabe , author3=J. Smith , title = Unit Operations of Chemical Engineering: Seventh Edition , publisher = McGraw-Hill Professional , year=2004 , pages = 68–132 , url = https://books.google.com/books?id=u3SvHtIOwj8C&dq=unit+operations+of+chemical+engineering&pg=PP1, isbn = 9780072848236 Fluid mechanics