In the previous article, we derived the Hertz-Knudsen Equation, which describes the mass flux due to evaporation and condensation.
However, the equation contains two unknown parameters: the condensation coefficient σc and the evaporation coefficient σe, which makes it inconvenient for practical use.
Schrage modified the Hertz-Knudsen Equation by adding an average velocity component to the Maxwell-Boltzmann distribution [1].
In this article, we will follow Schrage’s approach to derive the Hertz-Knudsen-Schrage Equation.
Modification of the Hertz-Knudsen Equation
We assume that the velocity distribution follows the Maxwell-Boltzmann distribution, but with an additional average velocity component vz in the z-direction.
The number of particles colliding with the liquid-vapor interface (per unit time and unit area) can be estimated as follows.
The parameters β and Γ are defined as shown below.
The parameter m is the mass of a single molecule, and the molar mass is given by M=mNA, where NA is Avogadro’s number.
R is the molar gas constant, satisfying R=kBNA.
We denote the flux colliding from the vapor side to the liquid-vapor interface as jv, and the flux colliding from the liquid side to the liquid-vapor interface as jl.
For the liquid side, we assume there is no offset in the average velocity, and use the mass flux expression obtained when deriving the Hertz-Knudsen Equation.
We assume that there is a non-zero average velocity only on the vapor side for the following reasons:
Since the number of molecules per unit volume is significantly large on the liquid side, the average velocity on the liquid side is considered to be sufficiently small compared to the vapor side.
Since the number of molecules per unit volume is significantly large on the liquid side, the mean free path is very small, allowing us to ignore non-equilibrium regions.
The ratio of molecules, that actually condense or evaporate upon colliding with the liquid-vapor interface, is denoted as α (assumed to be the same in both directions).
Then, the overall mass flux can be expressed as jLV=α(jl−jv).
Substituting (8) and (9) into this relation, we obtain the Hertz-Knudsen-Schrage Equation as shown below.
jLV=α2πkBm(TlPl−TvΓPv)
Meanwhile, the following relationship should also hold between the mass flux jLV and the average velocity vz.
jLV=ρvvz
Since Γ(vzβ) is a nonlinear function of vz as shown in (7), it is not possible to explicitly determine the mass flux jLV, when the temperature and pressure of the liquid and vapor phases are given.
Moderate Evaporation and Condensation
If the evaporation or condensation is moderate, where the average velocity is sufficiently small, (10) can be further transformed to the explicit form.
If vzβv is sufficiently small, Γ can be approximated as follows.
Γ(vzβv)≃1−vzβvπ
vzβv in this equation can be transformed as follows.
Finally, we obtain the following relation, which is also called the Hertz-Knudsen-Schrage Equation.
jLV=2−α2α2πkBm(TlPl−TvPv)
The Hertz-Knudsen-Schrage equation is cited occasionally in books and papers as a relatively simple tool for modeling evaporation and condensation.
Looking at equation (1), we can see that the Maxwell-Boltzmann distribution is modified using the average velocity component.
It is true that certain amount of mass is moving with an average velocity when evaporation or condensation is occurring.
However, there is no clear physical justification for expressing the velocity distribution of molecules in the form of (1).
It is important to note that this is an assumption to simply handle the velocity distribution of molecules.
References
Robert W. Schrage, “A theoretical study of interphase mass transfer”, Columbia University Press, 1953, doi: 10.7312/schr90162
Hertz-Knudsen Equation (also known as Hertz-Knudsen-Langmuir Equation) is a classical model for representing mass flux due to condensation and evaporation. In this article, we will derive the Hertz-Knudsen Equation starting from the Maxwell-Boltzmann distribution.
U.S. Standard Atmosphere 1976, ISA (International Standard Atmosphere), ICAO Standard Atmosphere are well-known models that describe the variation of atmospheric properties such as temperature and pressure with altitude. This article explains the calculation methods for various atmospheric parameters based on these models.
To include a honeycomb panel in a spacecraft system thermal model, the effective density and thermal conductivity as a panel must be estimated. We will discuss the evaluation methodology for honeycomb core effective properties, and introduce a calculation tool.