fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...

, a shell balance can be used to determine the velocity profile of a moving fluid, i.e,. how fluid

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

changes with position across a flow cross section. A "shell" is a differential element of the flow. By looking at the

momentum In Newtonian mechanics, momentum (: momenta or momentums; 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. ...

and

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

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

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

. 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 eve ...

and gravitational forces on the shell. 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 *No bends or curves * Steady state *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 may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...

Solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...

Boundary:

No-slip condition In fluid dynamics, the no-slip condition is a Boundary conditions in fluid dynamics, boundary condition which enforces that at a solid boundary, a viscous fluid attains zero bulk velocity. This boundary condition was first proposed by Osborne Reyno ...

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

Liquid Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...

- Gas Boundary:

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

= 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 ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

of fluid = ρ *

Viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

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

*(Rate of

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 (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

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 cha ...

''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

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