Relationship To Mach and Reynolds Numbers in Gases
The Knudsen number can be related to the Mach number and the Reynolds number:
Noting the following:
Dynamic viscosity,
Average molecule speed (from Maxwell-Boltzmann distribution),
thus the mean free path,
dividing through by L (some characteristic length) the Knudsen number is obtained:
where
- is the average molecular speed from the Maxwell–Boltzmann distribution,
- T is the thermodynamic temperature,
- μ is the dynamic viscosity,
- m is the molecular mass,
- kB is the Boltzmann constant,
- ρ is the density, .
The dimensionless Mach number can be written:
where the speed of sound is given by
where
- U∞ is the freestream speed,
- R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1),
- M is the molar mass,
- is the ratio of specific heats, and is dimensionless.
The dimensionless Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by ,
yields the Knudsen number.
The Mach, Reynolds and Knudsen numbers are therefore related by:
Read more about this topic: Knudsen Number
Famous quotes containing the words relationship to, relationship, mach, reynolds, numbers and/or gases:
“Artists have a double relationship towards nature: they are her master and her slave at the same time. They are her slave in so far as they must work with means of this world so as to be understood; her master in so far as they subject these means to their higher goals and make them subservient to them.”
—Johann Wolfgang Von Goethe (1749–1832)
“I began to expand my personal service in the church, and to search more diligently for a closer relationship with God among my different business, professional and political interests.”
—Jimmy Carter (James Earl Carter, Jr.)
“Physics is experience, arranged in economical order.”
—Ernst Mach (1838–1916)
“If you have great talents, industry will improve them: if you have but moderate abilities, industry will supply their deficiency.”
—Sir Joshua Reynolds (1723–1792)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)
“The bird is not in its ounces and inches, but in its relations to Nature; and the skin or skeleton you show me, is no more a heron, than a heap of ashes or a bottle of gases into which his body has been reduced, is Dante or Washington.”
—Ralph Waldo Emerson (1803–1882)