Dimensionless numbers refer to physical parameters that have no units of measurement. These numbers often appear in calculations used by process engineers. As long as consistent units are used, dimensionless numbers remain the same whether metric or other units are used in the equations. Here are some dimensionless numbers often used in chemical engineering fluid dynamics calculations:
Reynolds number (Re). Reynolds numbers express the ratio of inertial forces to viscous forces in a flowing fluid, and represent a way to quantify the importance of these two types of forces under a given set of flow conditions. When calculating pressure, heat transfer or head loss in pipes, it is important to know whether a fluid is exhibiting laminar flow, turbulent flow, or a mixture of the two. Re is typically used as a criterion for determining whether pipe flow is laminar or turbulent. High Reynolds numbers are associated with turbulent flow, where inertial forces dominate and flow is chaotic and characterized by eddies and vortices. Low Reynolds numbers are associated with laminar flow, where flow paths are smooth and viscous forces dominate as defined by Equation (1). The term is named for U.K. physicist Osborne Reynolds.
Re = (ρ v L) / μ (1)
ρ is fluid density
v is fluid velocity
L is characteristic linear dimension (traveled length of the fluid)
μ is fluid’s dynamic viscosity
Prandtl Number (Pr). Prandtl numbers represent the ratio between kinematic viscosity and thermal diffusivity of a fluid. It is used in calculations that involve heat transfer in flowing fluids because it provides a measure of the relative thickness of the thermal and momentum boundary layers. A fluid’s Prandtl number is based on its physical properties alone. For many gases (with the notable exception of hydrogen), Pr lies in the range of 0.6 to 0.8 over a wide range of conditions. Named after German physicist Ludwig Prandtl, Pr can be calculated using the following equation:
Pr = (CPμ) / k (2)
CP is fluid specific heat capacity
μ is dynamic viscosity
k is thermal conductivity
Nusselt number (Nu). In heat transfer at the boundary or surface of a flowing fluid, the Nusselt number is the ratio of convective to conductive heat transfer across the boundary over a given length. When Nu is close to one, convection and conduction are of similar magnitude, which is characteristic of laminar flow. Larger Nusselt numbers are associated with higher convection and turbulent flow. Named for German engineer Wilhelm Nusselt, Nu can be calculated with the following equation:
Nu = (hl) / k (3)
h is heat transfer coefficient
l is characteristic length (for heat transfer in pipes, l is equal to the pipe diameter)
k is thermal conductivity
Sherwood Number (Sh). Somewhat analogous to the Nusselt number, but for mass transfer, rather than heat transfer, the Sherwood number is a ratio of convective and diffusive mass transfer in a fluid. Named for American engineer Thomas Kilgore Sherwood, Sh can be calculated using the following equation:
Sh = (hD l) / D (4)
hD is mass transfer coefficient
l is characteristic length
D is molecular diffusivity
Froude number (Fr). As the ratio between inertial and gravitational forces, the Froude number can be used to determine the resistance of an object moving through a fluid. Named for English engineer William Froude, Fr can be calculated with the following equation:
Fr = v / (g l)1/2 (5)
v is velocity
l is characteristic length
g is acceleration due to gravity
Grashof Number (Gr). The Grashof number expresses the ratio of buoyancy to viscous force in a fluid. It can serve to correlate heat and mass transfer due to thermally induced natural convection at a solid surface immersed in a fluid. Named after German engineer Franz Grashof, Gr is shown in the following equation:
Gr = (L3 βgΔT) / v2 (6)
L is characteristic length
β is volumetric thermal expansion coefficient
ΔT is the difference between surface temperature and bulk temperature of the fluid
v is kinematic viscosity
g is acceleration due to gravity
Mach number (Ma). Mach number is the ratio of fluid velocity to the velocity of sound in that medium. In chemical engineering, Ma is commonly used in calculations involving high-velocity gas flow. The Mach number is named for Austrian physicist Ernst Mach. It can be calculated with the following equation:
Ma = u/v (7)
u is velocity of the fluid
v is the velocity of sound in that medium
Schmidt number (Sc). The Schmidt numbers is the ratio of kinematic viscosity to diffusivity in a fluid, and characterizes fluid flow where there are molecular momentum and mass-diffusion convection processes occurring simultaneously. Named for German engineer Heinrich Schmidt, the number can be calculated using the following equation:
Sc = μ / ρD (8)
μ is dynamic viscosity
ρ is fluid density
D is diffusivity ■