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If the volumetric expansion coefficient does change appreciably with temperature, or the increase in volume is significant, then the above equation will have to be integrated: (+) = = (()) where () is the volumetric expansion coefficient as a function of temperature T, and and are the initial and final temperatures respectively.
where γ is the heat capacity ratio, α is the volumetric coefficient of thermal expansion, ρ = N/V is the particle density, and = (/) is the thermal pressure coefficient. In an extensive thermodynamic system, the application of statistical mechanics shows that the isothermal compressibility is also related to the relative size of fluctuations ...
β is the coefficient of volume expansion (equal to approximately 1/T for ideal gases) T s is the surface temperature; T ∞ is the bulk temperature; L is the vertical length; D is the diameter; ν is the kinematic viscosity. The L and D subscripts indicate the length scale basis for the Grashof number.
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure (C P) to heat capacity at constant volume (C V).
Some formulations for the Grüneisen parameter include: = = = = = ( ) where V is volume, and are the principal (i.e. per-mass) heat capacities at constant pressure and volume, E is energy, S is entropy, α is the volume thermal expansion coefficient, and are the adiabatic and isothermal bulk moduli, is the speed of sound in the medium ...
The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): = = Here is the thermal expansion coefficient: = is the isothermal compressibility (the inverse of the bulk modulus):
is pressure, temperature, volume, entropy, coefficient of thermal expansion, compressibility, heat capacity at constant volume, heat capacity at constant pressure. Maxwell's relations are a set of equations in thermodynamics which are derivable from the symmetry of second derivatives and from the definitions of the thermodynamic potentials .
β is the thermal expansion coefficient (equals to 1/T, for ideal gases, where T is absolute temperature). is the kinematic viscosity; α is the thermal diffusivity; T s is the surface temperature; T ∞ is the quiescent temperature (fluid temperature far from the surface of the object) Gr x is the Grashof number for characteristic length x