Derivation of the Hertz-Knudsen-Schrage Equation

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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\sigma_c and the evaporation coefficient σe\sigma_e, 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\overline{v}_z 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.

νz=0NV(m2πkBT)32exp[m(vx2+vy2+(vzvz)2)2kBT]vz dvxdvydvz=NVm2πkBT0exp[m(vzvz)22kBT]vz dvz\begin{align} % \label{eq:velocity_distribution} \nu_z &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{0} \frac{N}{V} \left( \frac{m}{2\pi k_\mathrm{B} T} \right)^{\frac{3}{2}} \exp \left[ -\frac{m(v_x^2 + v_y^2 + (v_z - \overline{v}_z)^2)}{2k_\mathrm{B}T} \right] |v_z|~dv_x dv_y dv_z \notag \\ &= - \frac{N}{V} \sqrt{\frac{m}{2\pi k_\mathrm{B} T}} \int_{-\infty}^{0} \exp \left[ -\frac{m (v_z - \overline{v}_z)^2}{2k_\mathrm{B}T} \right] v_z~dv_z \end{align}

The integral with respect to w=vzvzw = v_z - \overline{v}_z can be evaluated as follows.

vz(w+vz)exp[mw22kBT]dw=vzwexp[mw22kBT]dw+vzvzexp[mw22kBT]dw=kBTm[exp[mw22kBT]]vz+vz0exp[mw22kBT]dw+vz0vzexp[mw22kBT]dw=kBTmexp[mvz22kBT]+vz22πkBTmvz0vzexp[mw22kBT]dw=kBTmexp[mvz22kBT]+vz22πkBTm[1erf(vzm2kBT)]\begin{align} &\int_{-\infty}^{-\overline{v}_z} (w + \overline{v}_z) \exp \left[ -\frac{m w^2}{2k_\mathrm{B}T} \right]dw \notag \\ &= \int_{-\infty}^{-\overline{v}_z} w \exp \left[ -\frac{m w^2}{2k_\mathrm{B}T} \right]dw + \overline{v}_z \int_{-\infty}^{-\overline{v}_z} \exp \left[ -\frac{m w^2}{2k_\mathrm{B}T} \right]dw \notag \\ &= - \frac{k_\mathrm{B}T}{m} \left[ \exp \left[ -\frac{m w^2}{2k_\mathrm{B}T} \right] \right]_{-\infty}^{-\overline{v}_z} + \overline{v}_z \int_{-\infty}^{0} \exp \left[ -\frac{m w^2}{2k_\mathrm{B}T} \right]dw + \overline{v}_z \int^{-\overline{v}_z}_{0} \exp \left[ -\frac{m w^2}{2k_\mathrm{B}T} \right]dw \notag \\ &= - \frac{k_\mathrm{B}T}{m} \exp \left[- \frac{m \overline{v}_z^2}{2k_\mathrm{B}T} \right] + \frac{\overline{v}_z}{2} \sqrt{\frac{2\pi k_\mathrm{B}T}{m}} - \overline{v}_z \int^{\overline{v}_z}_{0} \exp \left[ -\frac{m w^2}{2k_\mathrm{B}T} \right]dw \notag \\ &= - \frac{k_\mathrm{B}T}{m} \exp \left[ -\frac{m \overline{v}_z^2}{2k_\mathrm{B}T} \right] + \frac{\overline{v}_z}{2} \sqrt{\frac{2\pi k_\mathrm{B}T}{m}} \left[1 - \mathrm{erf} \left( \overline{v}_z \sqrt{\frac{m}{2k_\mathrm{B}T}} \right) \right] \end{align}

erf(x) is the error function, defined as shown in (3), and the shape of the function is shown in Figure 1.

erf(x)=2π0xexp[t2]dt\begin{equation} % \label{eq:erf} \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 \exp \left[ -t^2 \right] dt \end{equation}

hertz-knudsen-schrage-1 Figure 1: Error Function.

Substituting t=att = a t', we can rewrite it in a form that includes an additional parameter aa.

erf(x)=2aπ0xaexp[(at)2]dt\begin{equation} % \label{eq:erfa} \mathrm{erf}(x) = \frac{2 a}{\sqrt{\pi}} \int^{\frac{x}{a}}_0 \exp \left[ -(a t')^2 \right] dt' \end{equation}

As a result, the mass flux impinging on the liquid-vapor interface is expressed as follows.

j=mνz=mNVm2πkBT{kBTmexp[mvz22kBT]vz22πkBTm[1erf(vzm2kBT)]}=NkBTVm2πkBTexp[mvz22kBT]vzmN2V[1erf(vzm2kBT)]=Pβπexp[vz2β2]Pvzβ2[1erf(vzβ)] =βPπ{exp[vz2β2]vzβπ[1erf(vzβ)]}=βPπ Γ(vzβ)\begin{align} &j = m \nu_z \notag \\ &= - \frac{mN}{V} \sqrt{\frac{m}{2\pi k_\mathrm{B} T}} \left\{ - \frac{k_\mathrm{B}T}{m} \exp \left[ -\frac{m \overline{v}_z^2}{2k_\mathrm{B}T} \right] - \frac{\overline{v}_z}{2} \sqrt{\frac{2\pi k_\mathrm{B}T}{m}} \left[1 - \mathrm{erf} \left( \overline{v}_z \sqrt{\frac{m}{2k_\mathrm{B}T}} \right) \right] \right\} \notag \\ &= \frac{N k_\mathrm{B}T}{V} \sqrt{\frac{m}{2\pi k_\mathrm{B} T}} \exp \left[ -\frac{m \overline{v}_z^2}{2k_\mathrm{B}T} \right] - \frac{\overline{v}_z mN}{2V} \left[1 - \mathrm{erf} \left( \overline{v}_z \sqrt{\frac{m}{2k_\mathrm{B}T}} \right) \right] \notag \\ &= P \frac{\beta}{\sqrt{\pi}} \exp \left[- \overline{v}_z^2 \beta^2 \right] - P \overline{v}_z \beta^2 \left[ 1 - \mathrm{erf}(\overline{v}_z \beta) \right] \notag \\ &= \frac{\beta P}{\sqrt{\pi}} \left\{ \exp \left[- \overline{v}_z^2 \beta^2 \right] - \overline{v}_z \beta \sqrt{\pi} \left[ 1 - \mathrm{erf}(\overline{v}_z \beta) \right] \right\} \notag \\ &= \frac{\beta P}{\sqrt{\pi}} ~ \Gamma \left( \overline{v}_z \beta \right) % = \frac{\rho}{2\sqrt{\pi} \beta} ~ \Gamma \left( \overline{v}_z \beta \right) \end{align}

The parameters β\beta and Γ\Gamma are defined as shown below. The parameter mm is the mass of a single molecule, and the molar mass is given by M=mNAM = m N_\mathrm{A}, where NAN_\mathrm{A} is Avogadro’s number. RR is the molar gas constant, satisfying R=kBNAR = k_\mathrm{B} N_\mathrm{A}.

β=m2kBT=M2RT\begin{equation} \beta = \sqrt{\frac{m}{2k_\mathrm{B}T}} = \sqrt{\frac{M}{2RT}} \end{equation} Γ(vzβ)=exp[vz2β2]vzβπ[1erf(vzβ)]\begin{equation} \Gamma \left( \overline{v}_z \beta \right) = \exp \left[ - \overline{v}_z^2 \beta^2 \right] - \overline{v}_z \beta \sqrt{\pi} \left[ 1 - \mathrm{erf} \left( \overline{v}_z \beta \right) \right] \end{equation}

We denote the flux colliding from the vapor side to the liquid-vapor interface as jvj_v, and the flux colliding from the liquid side to the liquid-vapor interface as jlj_l. 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.

jv=βvPπΓ=m2πkBΓPvTv\begin{equation} j_v = \frac{\beta_v P}{\sqrt{\pi}} \Gamma = \sqrt{\frac{m}{2\pi k_\mathrm{B}}} \frac{\Gamma P_v}{\sqrt{T_v}} \end{equation} jl=m2πkBPlTl\begin{equation} j_l = \sqrt{\frac{m}{2\pi k_\mathrm{B}}} \frac{P_l}{\sqrt{T_l}} \end{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\alpha (assumed to be the same in both directions). Then, the overall mass flux can be expressed as jLV=α(jljv)j^{LV} = \alpha (j_l - j_v). Substituting (8) and (9) into this relation, we obtain the Hertz-Knudsen-Schrage Equation as shown below.

jLV=αm2πkB(PlTlΓPvTv)\begin{equation} % \label{eq:schrage} j^{LV} = \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left( \frac{P_l}{\sqrt{T_l}} - \frac{\Gamma P_v}{\sqrt{T_v}} \right) \end{equation}

Meanwhile, the following relationship should also hold between the mass flux jLVj^{LV} and the average velocity vz\overline{v}_z.

jLV=ρvvz\begin{equation} % \label{eq:Schrage1953_3.1-14} j^{LV} = \rho_v \overline{v}_z \end{equation}

Since Γ(vzβ)\Gamma(\overline{v}_z \beta) is a nonlinear function of vz\overline{v}_z as shown in (7), it is not possible to explicitly determine the mass flux jLVj^{LV}, 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\overline{v}_z \beta_v is sufficiently small, Γ\Gamma can be approximated as follows.

Γ(vzβv)1vzβvπ\begin{equation} % \label{eq:gamma} \Gamma ( \overline{v}_z \beta_v ) \simeq 1 - \overline{v}_z \beta_v \sqrt{\pi} \end{equation}

vzβv\overline{v}_z \beta_v in this equation can be transformed as follows.

vzβv=12πjLVjlTvTlPlPv\begin{equation} % \label{eq:Schrage1953_3.1-15} \overline{v}_z \beta_v = \frac{1}{2 \sqrt{\pi}} \frac{j^{LV}}{j_l} \frac{\sqrt{T_v}}{\sqrt{T_l}} \frac{P_l}{P_v} \end{equation}

The relation of (13) can be confirmed as follows.

12πjLVjlTvTlPlPv=12πρvvzm2πkBPlTlTvTlPlPv=12πmPvkBTvvzm2πkBPlTlTvTlPlPv=vzm2kBTv=vzβv\begin{align} &\frac{1}{2 \sqrt{\pi}} \frac{j^{LV}}{j_l} \frac{\sqrt{T_v}}{\sqrt{T_l}} \frac{P_l}{P_v} = \frac{1}{2 \sqrt{\pi}} \frac{\rho_v \overline{v}_z}{\sqrt{\frac{m}{2\pi k_\mathrm{B}}} \frac{P_l}{\sqrt{T_l}}} \frac{\sqrt{T_v}}{\sqrt{T_l}} \frac{P_l}{P_v} \notag \\ &= \frac{1}{2 \sqrt{\pi}} \frac{\frac{m P_v}{k_\mathrm{B}T_v} \overline{v}_z}{\sqrt{\frac{m}{2\pi k_\mathrm{B}}} \frac{P_l}{\sqrt{T_l}}} \frac{\sqrt{T_v}}{\sqrt{T_l}} \frac{P_l}{P_v} = \overline{v}_z \sqrt{\frac{m}{2 k_\mathrm{B} T_v}} = \overline{v}_z \beta_v \end{align}

Substituting (12) and (13) into (10), we can rewrite jLVj^{LV} as follows.

jLV=αm2πkB{PlTlPvTv(1vzβvπ)}=αm2πkB{PlTlPvTv(112jLVjlTvTlPlPv )}=αm2πkB(PlTlPvTv)+αm2πkBPvTv×12jLVjlTvTlPlPv=αm2πkB(PlTlPvTv)+m2πkBPvTvm2πkBPlTl×α2jLVTvTlPlPv=αm2πkB(PlTlPvTv)+α2jLV\begin{align} j^{LV} &= \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left\{ \frac{P_l}{\sqrt{T_l}} - \frac{P_v}{\sqrt{T_v}} \left( 1 - \overline{v}_z \beta_v \sqrt{\pi} \right) \right\} \notag \\ &= \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left\{ \frac{P_l}{\sqrt{T_l}} - \frac{P_v}{\sqrt{T_v}} \left( 1 - \frac{1}{2} \frac{j^{LV}}{j_l} \frac{\sqrt{T_v}}{\sqrt{T_l}} \frac{P_l}{P_v}  \right) \right\} \notag \\ &= \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left( \frac{P_l}{\sqrt{T_l}} - \frac{P_v}{\sqrt{T_v}} \right) + \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}} \frac{P_v}{\sqrt{T_v}} \times \frac{1}{2} \frac{j^{LV}}{j_l} \frac{\sqrt{T_v}}{\sqrt{T_l}} \frac{P_l}{P_v} \notag \\ &= \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left( \frac{P_l}{\sqrt{T_l}} - \frac{P_v}{\sqrt{T_v}} \right) + \frac{\sqrt{\frac{m}{2\pi k_\mathrm{B}}} \frac{P_v}{\sqrt{T_v}}}{\sqrt{\frac{m}{2\pi k_\mathrm{B}}} \frac{P_l}{\sqrt{T_l}}} \times \frac{\alpha}{2} j^{LV} \frac{\sqrt{T_v}}{\sqrt{T_l}} \frac{P_l}{P_v} \notag \\ &= \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left( \frac{P_l}{\sqrt{T_l}} - \frac{P_v}{\sqrt{T_v}} \right) + \frac{\alpha}{2} j^{LV} \end{align} (1α2)jLV=αm2πkB(PlTlPvTv)\begin{equation} \left( 1- \frac{\alpha}{2} \right) j^{LV} = \alpha \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left( \frac{P_l}{\sqrt{T_l}} - \frac{P_v}{\sqrt{T_v}} \right) \end{equation}

Finally, we obtain the following relation, which is also called the Hertz-Knudsen-Schrage Equation.

jLV=2α2αm2πkB(PlTlPvTv)\begin{gather} % \label{eq:schrage2} j^{LV} = \frac{2 \alpha}{2 - \alpha} \sqrt{\frac{m}{2\pi k_\mathrm{B}}}\left( \frac{P_l}{\sqrt{T_l}} - \frac{P_v}{\sqrt{T_v}} \right) \end{gather}

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

  1. Robert W. Schrage, “A theoretical study of interphase mass transfer”, Columbia University Press, 1953, doi: 10.7312/schr90162

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