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06/21/05 23:12:50 (3 years ago)
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powell
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Major update: added reaction rate coefficients to this handout.

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  • trunk/matml/transport/resources/evaprat/evaprat.tex

    r174 r181  
    3434\end{equation} 
    3535This condition, called the Clapeyron equation, applies across liquid-solid, 
    36 liquid-gas and solid-gas transition in pressure-temperature space.  For 
     36liquid-gas and solid-gas transitions in pressure-temperature space.  For 
    3737example, in most substances, the volume change during melting $\Delta V_m$ is 
    3838positive, and since melting increases entropy, $\Delta S_m/\Delta V_m$ is 
    … …  
    4545  $\ $\pdfximage{SLVmet.png}\pdfrefximage\pdflastximage$\ $ 
    4646  $\ $\pdfximage{SLVwater.png}\pdfrefximage\pdflastximage$\ $ 
    47   \caption{Schematic phase diagrams for a single component: typical metal 
     47  \caption{Schematic phase diagrams for an element/compound: typical metal 
    4848    (left), water or silicon (right).} 
    4949  \label{fig:PTphases} 
    50 \end{figure}In evaporation, the volume of gas is so much larger than of the liquid than 
    51 $\Delta V_v$ is approximately the gas volume, which on a molar basis is $RT/P$ 
    52 according to the ideal gas law; thus: 
     50\end{figure}In evaporation, the volume of the gas is so much larger than that 
     51of the liquid that $\Delta V_v$ is approximately the gas volume, which on a 
     52molar basis is $RT/P$ according to the ideal gas law; thus: 
    5353\begin{equation} 
    5454  \label{eq:almostclausclap} 
    … …  
    7070where $B$ is the integration constant.  If $\Delta H_v$ depends linearly on 
    7171temperature, a third term $C\ln T$ is added, and in some cases a fourth term 
    72 $DT$ is added as well.  In tables, this is summarized: 
     72$DT$ is also added to improve fit to data.  In tables\footnote{For example, 
     73  {\em Smithells Metals Handbook} (E. Brandes ed., Butterworth, 1983) has these 
     74  constants for numerous metals.}, this is summarized: 
    7375\begin{equation} 
    7476  \label{eq:finalclausclap} 
    … …  
    7779 
    7880For a multi-component liquid or solid, the vapor pressure of each component is 
    79 activity times the vapor pressure of the pure substance. 
     81its activity $a$ times the vapor pressure of the pure substance.  From this 
     82point on, $\bar{P}_v$ will denote the vapor pressure of the pure substance and 
     83$P_v=a\bar{P}_v$ its partial pressure in solution. 
    8084 
    8185\paragraph{Evaporation Into a Vacuum: the Langmuir Equation} 
    8286 
    83 For a gas with energy distribution given by the Boltzmann equation, the number 
    84 of atoms (moles) striking a surface per unit area per unit time is given by: 
     87For a gas with energies following a Boltzmann distribution, the number of atoms 
     88(moles) striking a surface per unit area per unit time is given by: 
    8589\begin{equation} 
    8690  \label{eq:langmuir} 
    … …  
    97101  temperature, much stronger than convective heat transfer coefficients or 
    98102  evaporation rates. 
    99 \item Vapor pressure is inversely correlated with melting point. 
    100 \item $P_v$ and $J$ curves do not change dramatically across the melting point. 
     103\item Vapor pressure decreases with increasing melting point. 
     104\item $\bar{P}_v$ and $J$ curves do not change their slopes or shapes 
     105  dramatically across melting points. 
    101106\end{itemize} 
    102107\begin{figure}[htbp] 
    103108  %% Redo these figures with my own data... 
    104   \centering \pdfimageresolution 220 
     109  \centering \pdfimageresolution 230 
    105110  $\ $\pdfximage{elem-pv.png}\pdfrefximage\pdflastximage$\ $ 
    106111  $\ $\pdfximage{elem-J.png}\pdfrefximage\pdflastximage$\ $ 
    107   \caption{Vapor pressures (left) and evaporation rates (right) of pure 
    108     elements.  Small circles represent melting points.  From Schiller, Heisig 
    109     and Panzer, {\em Electron Beam Technology}, Wiley, 1982.} 
     112  \caption{Vapor pressures $\bar{P}_v$ (left) and evaporation rates $J$ (right) 
     113    of pure elements.  Small circles represent melting points.  From Schiller, 
     114    Heisig and Panzer, {\em Electron Beam Technology}, Wiley, 1982.} 
    110115  \label{fig:metalvap} 
    111116\end{figure} 
     117 
     118\paragraph{Reaction Rate Coefficients} 
     119 
     120For an element or compound preferentially evaporating from a solution into a 
     121vacuum, a first order heterogeneous rate coefficient $k''$ can describe its 
     122removal from the surface: 
     123\begin{equation} 
     124  \label{eq:heteratecoeff} 
     125  J_B = k''_B C_B, 
     126\end{equation} 
     127where $J_B$ is the molar flux of species $B$ from the surface (the ``reaction 
     128rate'') and $C_B$ is its molar concentration at the surface of the solution. 
     129The units of $k''$ are length/time, and this can be thought of as a diffusive 
     130average velocity of the solute atoms through the solution near the surface. 
     131 
     132Note that if the surface concentration is not known, one can often estimate a 
     133mass transfer coefficient $h_D$ relating flux to difference between the surface 
     134and bulk concentrations.  The sum of the two resistances due to diffusion 
     135through the surface boundary layer $1/h_D$ and evaporation $1/k''$ then 
     136describes the total resistance to transport from the bulk to the gas phase. 
     137 
     138Because a solute's partial vapor pressure $P_{vB}$ is its activity $a_B$ 
     139times its pure vapor pressure $\bar{P}_{vB}$, its evaporation flux $J_B$ is the 
     140product of activity and evaporation rate of the pure substance: 
     141\begin{equation} 
     142  \label{eq:partialvap} 
     143  J_B = \frac{P_{vB}}{\sqrt{2\pi M_BRT}} = 
     144  \frac{a_B\bar{P}_{vB}}{\sqrt{2\pi M_BRT}}. 
     145\end{equation} 
     146For a dilute solution of solute $B$ in solvent $A$ in the Henry's Law 
     147r\'{e}gime, the activity is proportional to mole fraction $X_B$, and the 
     148proportionality constant defines the activity coefficient $\gamma_B$: 
     149$a_B=\gamma_BX_B$.  The mole fraction, in turn, is the quotient of solute 
     150concentration $C_B$ and the sum of all concentrations, which in dilute solution 
     151is approximately the solvent molar density: 
     152\begin{equation} 
     153  \label{eq:molefrac} 
     154  X_B = \frac{C_B}{\sum C_i} \simeq \frac{C_B}{C_A} = \frac{C_B M_A}{\rho}, 
     155\end{equation} 
     156where $\rho$ is the solution density.  Substituting equation \ref{eq:molefrac} 
     157and the definition of the activity coefficient into equation 
     158\ref{eq:partialvap} yields an expression of solute flux $J_B$ which is 
     159proportional to concentration $C_B$, and an expression for reaction rate 
     160coefficient $k''_B$: 
     161\begin{equation} 
     162  \label{eq:finalrate} 
     163  J_B = \frac{\gamma_BC_BM_A}{\rho}\frac{\bar{P}_{vB}}{\sqrt{2\pi M_BRT}} = 
     164  k''_B C_B, 
     165\end{equation} 
     166\begin{equation} 
     167  \label{eq:finalratecoeff} 
     168  k''_B = \frac{\gamma_BM_A}{\rho}\frac{\bar{P}_{vB}}{\sqrt{2\pi M_BRT}}. 
     169\end{equation} 
     170This rate coefficient has many uses, {\em e.g.} in models of batch and 
     171continuous flow reactors, this helps to assess the effectiveness of vacuum 
     172distillation processes for solute removal. 
    112173 
    113174\paragraph{Evaporation Ratio} 
    … …  
    116177``Analysis of Multicomponent Evaporation in Electron Beam Melting and Refining 
    117178of Titanium Alloys,'' {\em Metall. Mater. Trans.} {\bf 38B}, 1227-1239 (1997).] 
    118 For a dilute solution of solute $B$ in solvent $A$ evaporating into a vacuum, 
    119 the Evaporation Ratio is defined as: 
     179For a dilute solution (liquid or solid) of solute $B$ in solvent $A$ 
     180evaporating into a vacuum, the Evaporation Ratio $ER_B$ is defined as: 
    120181\begin{equation} 
    121182  \label{eq:evratdef} 
    122183  ER_{\rm B} = \frac{\rm wt\%B_{vapor}/wt\%A_{vapor}} 
    123   {\rm wt\%B_{liq/sol}/wt\%A_{liq/sol}}. 
     184  {\rm wt\%B_{solution}/wt\%A_{solution}}. 
    124185\end{equation} 
    125186This ratio will be equal to the equivalent ratio of mole fractions, and the