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\title{Models of Turbulent Transport and Transfer of Disperse Phase Mass in Liquids}
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             \author[1]{?????? ???????? ???????????  }

             \affil[1]{  Kazan State Power Engineering University}

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\date{\small \em Received: 15 December 2013 Accepted: 2 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


Using the theory of turbulent migration of particles and boundary layer models, expressions for calculation of particle transport coefficients and of mass transfer within the continuous phase are obtained. Various cases of turbulent direct flow of particles through the carrier turbulent medium are considered: smooth and rough channel; channel filled with fine random packing and apparatuses with mechanical mixing. Examples of calculation of mass transfer from droplets and solid particles are included and their agreement with experimental data of other authors is demonstrated.

\end{abstract}


\keywords{turbulence, direct flow, momentum transfer, mass transfer, mixing, extraction, dissolution.}

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\let\tabcellsep& 	 	 		 \par
???????????? ?????. ??? ????? ???? ?????????? ????? ??? ????? ??????????? ????????? ???????? ??? ??????? ? ???????????? ????????. ?????????? ???????? ? ????????? ?????????. ??????????? ??????? ? ???????????? ????????. ?????????? ???????? ? ????????? ?????????. ??????????? ????????????? ????? ????????? ???????? ?????? ??? ????????? ??????????????? ???????? ?????????? ???? ? ?????? ????????.\par
??? ????????????? ??????? ???? ???? ???????? ?????? ? ???????????? ?????? ?????? ??????????? ????????? ????????????? \hyperref[b13]{[14]}.\par
?.Ð?"?????? ?????? d ? ??? ?? ????????? ? ????????? ??????? ?? ????????????? ?????? ? ????????? â??": d ? << â??".\par
??? ????? ????????????? ?????? ??????? ????????? ????????, ????????? ? ???????? ???????? ?????.\par
??????????? ??????? ????????????? ??????? ????? ???????????? ?.?. ???????????????? (d ? <1 ???); ??????????????? (1< d ? <20 ???) ? ??????????????? (20< d ? <200 ???).\par
?. ??????? ????? ????? ??????? ? ???????????, ? ? ?????? ???????? ?????????? ?? ????? ???????? ??????????? ?????. ???????????????? ?????? ??????????????? ????????????.   
\section[{?. ????? ????? ???????}]{?. ????? ????? ???????}? = + y v v (6) , 6 . 11 , ò ? ? R y y v v ? < = + +\textbf{(7)}\par
????? ?????????????? (3) ???????? ????????? ( ), / ln 1 1 1 * R R R u ? ? ? + = (8) ??? R 1 = * u ? 1 /? -???????????? ??????? ??????? ???????, ? ?????? ???????? R 1 = 11.6; ? = 0.4 - ????????? ??????????????.\par
???????? ???????? ????????? (5) ? \hyperref[b7]{(8)}. Ð?"?? ????? ?? ?????? ????????????????? ???????? ??????? ????? ????? ?????????????? ???????? ???????? ? ??????????? \hyperref[b6]{[7,}\hyperref[b9]{10]} (9) ??? ?? ???????-????????? n = 0,66. ??????, ??? ????????, ??? ???????? ? ????? ???????????????? ??????????? n = 0,6-0,57.\par
?? ????????? (5), \hyperref[b7]{(8)}   ? * u ???????????? ???????? ? ?????? ? ????????, ñ / ì \hyperref[b12]{[13]}.\par
Ð?"?? ??????? ???????? ????? C f?? = ?/4, ??? ???????????? ?????? ? ??????? ?????????? ??????; ?=0.316/Re ? 0.25 ; Re ? =U ?? d ? /? (??? ?????????????? ?????? d=d ? ); ý d -????????????? ??????? ???????, ?; ? = ñâ v ñâ ; / 4 ? ? à d ý ???????? ????????? ?????, 3 3 ì / ì ; v à -???????? ??????????? ???????,= ? ?? /? ? . ?????? ?=0.25d ? ? ???????? R ? . 25 . 0 * v u d R ý = ?\textbf{(14)}\par
??????????? ???? ????????? ??? ????????????? ???????? ???????? ??????????? ??? ???????????? ?????????? ???? ??? ???????????? ????, ????? ????????? ??? ??????????? ?????????? (????????, ??? ??????????? ????????). 
\section[{III. ??????????? ???????? ??????}]{III. ??????????? ???????? ??????}\par
Ð?"???????????????? ?????????????? ????????? ?????? ? ???????????? ????? ?????????? ??????? ??????? ?????????? ? ??????????????, ??? ? ??????????. ??????? ??????????, ??? ?????? ?????????? ??????? ??????? ? ????????? (????? 3-5 ??, ? ??????????? ?? ????????????????? ??????? ????? ? ?????????? ????????????????? ???) ?? ????????? ? ???????????? ?????????? ?????. Ð?"?? ????? ?????? ?????? ??? ????????????? ????????????????? ????????? ? ??????????? ???????? ???????????? ????????????? ???????? ?????? ?????????? ?????????.\par
? ???????????? ? ??????? ???????????? ???????? ????? ???????????????? ??????? ?? ??????? ?? ????????? ??????? ????????????? ? ? ? ? , ??? ? ???????? ??????? ???????????? ?????????????? ????????? ??????????? ??????, ? -1 , ? ?????? ??????????, ?. ?? ????????????????? ?????? ??? ???????? ? ? ? ? < 0.01, ??????? ????????? ?????? ????????????? ??????????? ????? ????????? 100\% (µ 2 p =1), µ 2 p = (1-? E ? p ) -1 \hyperref[b13]{[14]}. ??????? ??? ??????? ????????? ??? ?????? ??????????? ????????? ?????????? ????????????? ??????????? ????? ????? ???????? ???????: . 01 . 0 9 ae 2 < ? = µ ? ? ? ? f d ÷ p E (15) ?????? ???????? [1] ????????? ??? ?????? ???????? ?????? ? ??????, ??? ?? ????????? ? E ? u * /(0.1R) ? ? ? = ?|?Î?"??|?d ? 2 /(18µ ? ), ?|?Î?"??|? ? 0: , 134 . 0 3 . 0 * ae ae u R f d ÷ ? µ ? ? µ ? = ? < (16) ??? ? ? = 2?f -??????? ??????? ??????????? ?????????, ? -1 ; f -??????? ???????????? ????????? ?????, ? -1 ; R -?????? ??????, ? (??? ??????? R=d ? /2,u R d u R ÷ ? µ ? µ ? < < ?\par
) ????? ?????????? ?? ????????? \hyperref[b13]{[14]} .1 ò p ÷ Å v D ? ? + = (19)\par
??????????? ???????? ???????? ?????? ? ? (???????????? ????????) ?????? ????? ?????????????? ????????? (3) ? ???????? ? ? = ? / (1+ ? E ? p ).\par
??????????? ???????? ?????? ?? ???????????, ?.?. ?? ??????? ?? ??????? ? ???????????? ?????? ????? ?????????????. ??????? ( ) ( ). / ln 1 1 1 1 * ? ? ? ? ? ? + + = R R R u p E ÷ ? ? ? ? ? (20)\par
? ?????????????? ????????? ??????????? ??? ?????????? ?????????? ??????, ??????? ????????????? ?????????????? ??????, ????????????? ????????????? ???????? ?????????? ?????? ? ?????? ?????? ??? ? ??????????? ????????? ??????? ????? ???. ????????????? ???????? ????? ?????????? ? ?????? ??????? ?? ????? ???????????? ? ???? \hyperref[b5]{[6,}\hyperref[b6]{7]}.  ???????? R 1 ??? ??????? (23) ???????? ? ???? \hyperref[b5]{[6]}.ae 1 exp , ÷ F V ? ? ? ? = ? ? ? ? ? ?\textbf{(}\par
( )2 1 1 * 1 ? ? ? ? ? ? ? = ? R arctg u u R ,\textbf{(25)}\par
? ??? ??????? (24) ( )2 / 3 1 2 1 * 1 3 6 ? ? ? ? ? ? + + ? = ? ? k k u u R , (\textbf{26})\par
??? ? 1 = 3 ln ( )  ???????? ? ? ??????, ????????? ??????? ???????? ???????? ?????????? ???????, ?????????? ????? ??????? ???????? ? ?????? \hyperref[b1]{[2,}\hyperref[b5]{6,}\hyperref[b10]{11]}.1 ) 1 ( 1 2 + ? + Â Â Â ; ? 2 = 6 arctg 3 ) 2 ( Â Â ? ; ? = R 1 -0,3332 / 1 ae ae 2 / 1 ae ae ò 4 4 ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? = H PW ñð ? ? ? ? ? ? ? ? , (\textbf{30})\par
??? ???????? ???????? ?????????? ???????, ??/? 3 ; Î?"P??????? ????????, ??;W ????????? ???????? ???????? ? ??????, ?/?; H????? ??????, ?.\par
????????? Î?" P ?? ????????? Ð?"????-???????? ?? (30) ???????.  2 / 1 3 ae ò 2 4 ? ? ? ? ? ? ? ? = d W ñð ? ? ? ? , (\textbf{31}2 0 ? u u u k + = , (\textbf{32})\par
??? u k???????? ??????????????? ????????, ?/?; u ???????? ????????, ????????? ????????????? ??????????? ?????, ?/?.\par
???????? u ? ??????????????? ??? ???????? ????????? ??????? ????????????? ??????????? ??????? ???????? ???????, ?????????? ????? ???????? ?????????? ??????? ? ??????.\par
( ) ???????????? ????????? ????????????? ??????????? (? 2 ????) ???????????, ???? ? ????? ????????? ?????? ???????????? ???????, ???????? «???????» \hyperref[b3]{[4]}.3 / 1 3 3 / 1 2 ? ? ? ? ? ? ? ? = ? d W d d u ñð k k ? ? ?\textbf{(}\par
??????? ??????????? ?????????? ?? ??????????? ???????????? ?????????? ????????? ????????? ?? ????????? (Re ? >50) \hyperref[b10]{[11]}.( ) ( ) 5 , 0 5 , 1 2 ae ò 2 / Re / 4 ? ? ? ? ý ý d = , (\textbf{34})\par
??? Re ? = W ?? d ? / ?????? ?????????? ??? ???????; d ? = 4? ?? /? ?????????????? ??????? ???????, ?; ? ?????????? ????????? ????? ???????, ? 3 /? 3 ; ? ????????? ???????????, ? 2 /? 3 ; ???????????? ??????????????? ????????????? ??????????? ???? \hyperref[b3]{[4]}.\par
??? ??????? ???????????? ??????????? ?? ?????? ? ???????????? ????????? ? ?????? ? ????????? ???????? ??????????????, ??? ?????????? ???? ?? ????????? ??????????? ???????. ???? ?? ?????????? ???? ?????? ????????? ??????????? ????????? ???????, ?? ?????????? ????????? ?????????????? ??????????? ?? ?????? ???????? ? ?????????, ??? ??????? ????????????? ??????????. 
\section[{V. ??????????? ??? ?????????????}]{V. ??????????? ??? ?????????????}\par
Ð?"???? ?????????? ?????????? ???? ?????????????? ????????? ??? ??????? ?????????? ?? ?????? ? ??????? ?????? ? ???????? ? ???????? ??? ???????????? ??????.\par
??????????? ??????? ???????? ?????????????? ???????? ?????????? ?????? ? ?????????????? ????? ???????? ??????? ???????, ??????? ???? ?? ????? ??????? ?????????????? ???????. ? ?????? \hyperref[b14]{[15]} ???????????? ??????? ???????? ???????? ?????? ???????????? ??? ?????????????? ????????? (?????????? (32)):2 îáò 2 2 0 u u u u k + + = ? ,\textbf{(35)}\par
??? u ??????????? ???????????????:ì ê ì d d u u ae ê îáò 6 ? ? ? ? ? = , (\textbf{36})??? u ? -???????? ???????? ????? ??????? ???????, ?/?: u ? = ? n d ? , n -??????? ????????, ? -1 ; d ? - ??????? ???????, ?; ? -??????????? ????????????? ??? ???????? ?????, ? k = f(Re ? ).\par
??????? ???????????? ???????? ?? ????????? ??????????? ????? ?????? ?? ????????? \hyperref[b5]{[6,}\hyperref[b6]{7]}  ?????? ?????? ??????? ??????????? ? ???????? ???? ??? ??????????? ??????? ??????.\par
?? ??????? 2 ???????? ?????????? ????????? ? ????????????????? ?????? \hyperref[b14]{[15]} ??? ??????????? ?????????????? ????? ? ???????? ??????? ???????. Ð?"?????? ?????? ?????????????? ????? -30, 8?10 -6 ; 593?10 -6 , ?.    \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-2.png}
\caption{\label{fig_0}3 ?}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}}\end{figure}
         \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.632051282051282\textwidth}P{0.002564102564102564\textwidth}P{0.002564102564102564\textwidth}P{0.001282051282051282\textwidth}P{0.001282051282051282\textwidth}P{0.011538461538461539\textwidth}P{0.011538461538461539\textwidth}P{0.001282051282051282\textwidth}P{0.021794871794871794\textwidth}P{0.001282051282051282\textwidth}P{0.002564102564102564\textwidth}P{0.01282051282051282\textwidth}P{0.001282051282051282\textwidth}P{0.05769230769230769\textwidth}P{0.07692307692307691\textwidth}P{0.00641025641025641\textwidth}P{0.005128205128205128\textwidth}}
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\multicolumn{14}{l}{?????????? ? ???????????? ??????, ??? ?????? ??????? ?? ??????????? ?????, ??? ?????? ?? ?????? ? ???????? ?????????? ???, ??? ?????? ???????? ????? ? ???? ??????? ?? ?????????. ??????????? ???????????? ???????? ?????? * * (??? ??????? ae ae 4 . 13 134 . 0}\tabcellsep ( ) C Volume XIV Issue II Version\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Global Journal of Researches in Engineering\end{longtable} \par
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\caption{\label{tab_3}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.5941747572815534\textwidth}P{0.013754045307443364\textwidth}P{0.04951456310679612\textwidth}P{0.1127831715210356\textwidth}P{0.07427184466019418\textwidth}P{0.005501618122977346\textwidth}}
\tabcellsep \tabcellsep \tabcellsep \multicolumn{3}{l}{?????? ?????????????? ?}\\
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\caption{\label{tab_6}}\end{figure}
 			\footnote{© 2014 Global Journals Inc. (US)} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}
