{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Cristallisation d’un produit actif pharmaceutique\n", "\n", "On effectue une cristallisation d’un produit actif pharmaceutique dans un cristallisoir agité avec une vitesse relative entre les cristaux et la solution de 0,1 m/s. La vitesse de la cinétique d'intégration à la surface (réaction hétérogène) est d'ordre 2 et s'écrit :\n", "\n", "$r=2.10^{-5}(c-c^*)^2\\quad[kg/(m^2.s)]$\n", "\n", "Le coefficient de transfert de matière autour des cristaux est déterminé à partir de la corrélation suivante :\n", "\n", "$Sh=2+0.95Re^{0.5}Sc^{0.33}$ pour 2Pour déterminer la vitesse de réaction d'intégration et l'efficacité de procédé, il faut connaître la concentration en soluté à l'interface (i.e. où la cristallisation a lieu). Pour cela, il faut déterminer le profil de concentration en soluté dans la couche limite autour du cristal (i.e. la zone où la concentration en soluté est plus faible que dans la solution à cause de la consommation à la surface par la cristallisation).\n", ">\n", ">Il faut donc établir un bilan sur la matière différentiel (i.e. sur un élément de volume dans la couche limite). On établira ce bilan différentiel en coordonnées rectilignes (en supposant que l'épaisseur de la couche limite est petite par rapport à la taille du cristal) et en régime permanent. Dans ces conditions, on peut démontrer pour une cinétique de réaction hétérogène suivant la loi $r=k_r(c-c^*)^{or}$ que l'efficacité du procédé suit l'expression suivante :\n", ">$$\\eta=(1-\\eta Da)^{or}$$\n", ">avec un nombre de Damkholer (rapport entre la réaction et le transport de matière dans la couche limite $k=D/\\delta$) qui s'écrit :\n", ">$$Da=\\frac{k_r(c_0-c^*)^{or-1}}{k}$$\n", ">Le code ci dessous permet de calculer :\n", ">1. l'épaisseur de la couche limite (et donc le coefficient de transfert de matière)\n", ">2. le nombre de Damkholer\n", ">3. le facteur d'efficacité\n", ">4. le flux de matière à l'interface du cristal qui est aussi la vitesse de croissance des cristaux\n", ">\n", ">Notez que la détermination du facteur d'efficacité, $\\eta$, est plus ou moins difficile selon l'ordre de réaction, $or$ : pour des ordres supérieurs à 2 la solution est déterminée numériquement par la méthode de Newton." ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Sh= 6.76e+01\n", "Da= 0.148\n", "Efficiency= 0.782\n", "Concentration à l interface 100.88437949142293\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Taux de croissance ou flux de matière s integrant à l interface 1.56e-05 en kg/(m2.s)\n" ] } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from scipy.optimize import newton\n", "#DATA\n", "#constant for the second order reaction\n", "kr=2e-5\n", "o_r=2\n", "#hydro\n", "u=0.1 #m/s\n", "#solution\n", "c0=101. #g/L\n", "ceq=100 #g/L\n", "ro=1000 #kg/m3\n", "D=1.e-9 #m2/s\n", "mu=0.001 #Pa s\n", "#cristal\n", "d0=5.e-4 #m\n", "roc=2000 #kg/m3\n", "\n", "#COEFFICIENT DE MATIERE\n", "#Reynolds\n", "Re=ro*u*d0/mu\n", "#Schmidt\n", "Sc=mu/(ro*D)\n", "#Sherwood\n", "Sh=2+(0.95*Re**0.5*Sc**0.33)\n", "print ('Sh=',\"%.2e\"%Sh)\n", "k=Sh*D/d0\n", "delta=D/k\n", "\n", "#DAMKHOLER\n", "Da=(kr*(c0-ceq)**(o_r-1))/k\n", "print ('Da=', round(Da,3))\n", "\n", "#EFFICIENCY \n", "if o_r==1 :\n", " def ef(Da):\n", " eff=1/(1+Da)\n", " return eff\n", "if o_r==2 :\n", " def ef(Da):\n", " eff=1/Da + 1/(2*Da**2)-np.sqrt(1+4*Da)/(2*Da**2)\n", " return eff\n", "if o_r>2 :\n", " def ef(Da):\n", " def f(eff):\n", " return eff-(1-eff*Da)**o_r\n", " eff=newton(f,x0=0.5)\n", " return eff\n", " \n", "ci=ceq+(ef(Da)*(c0-ceq)**o_r)**(1/o_r)\n", "\n", "print ('Efficiency=',round(ef(Da),3)) \n", "print ('Concentration à l interface', ci)\n", "x=np.logspace(-2,2,100)\n", "plt.semilogx(x,ef(x))\n", "plt.annotate('Eff', xy=(Da, ef(Da)), xytext=(10,ef(Da)),arrowprops=dict(facecolor='black', shrink=0.05),)\n", "plt.annotate('Da', xy=(Da, ef(Da)), xytext=(Da, 0),arrowprops=dict(facecolor='black', shrink=0.05),)\n", "plt.title('Evolution du facteur d efficacite en fonction du Damkholer')\n", "plt.xlabel('Da')\n", "plt.ylabel('Efficacite')\n", "plt.show()\n", "\n", "#taux de croissance\n", "N=ef(Da)*kr*(c0-ceq)**o_r\n", "print ('Taux de croissance ou flux de matière s integrant à l interface', \"%.2e\"%N, 'en kg/(m2.s)')\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**2) Estimer d'après ces résultats, le temps nécessaire à une augmentation de 10% du diamètre des cristaux. Vous effectuerez un bilan matière sur un cristal en négligeant l'effet de l'augmentation du diamètre sur le taux de croissance des cristaux.**\n", ">Au fur et à mesure de l'intrégration des molécules solubilisées sur le cristal, la taille et le volume du cristal va croitre. Le volume du cristal croît car il est alimenté par le flux de matière déterminé dans le 1). Pour décrire cette croissance, il faut effectuer un bilan sur le volume du cristal. L'enveloppe pour le bilan est alors la surface du cristal. L'accumulation dans le cristal est alors égale au flux de matière entrant : \n", ">$$ Accumulation = Entrée - Sortie +- terme source$$\n", ">$$cdV=S.N.dt$$\n", ">$$\\rho_c\\frac{4}{3}\\pi\\frac{dr_p^3}{dt}=4\\pi r_p^2N$$\n", ">$$\\frac{dr_p}{dt}=\\frac{N}{\\rho_c}$$\n", ">$$r_p=r_{p0}+\\frac{N}{\\rho_c}t$$\n", ">Le temps pour que le diamètre augmente de 10% est alors :\n", ">$$t=0.1\\frac{r_{p0}\\rho_c}{N}$$\n" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "temps nécessaire à la croissance de 10% du cristal 3.20e+03 s\n" ] } ], "source": [ "t=0.1*(d0/2)*roc/N\n", "print ('temps nécessaire à la croissance de 10% du cristal', \"%.2e\"%t, 's')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**3) Commenter l'effet des conditions opératoires sur la croissance des cristaux :\n", "a) effet de la vitesse relative solution/cristaux\n", "b) effet de la concentration de sursaturation**" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#Données pour les cas experimentaux A, B et C\n", "u=[0.1, 0.05, 0.1]\n", "c0=[101.,101.,110.]\n", "\n", "#Création de tableau pour stocker les valeurs correspondant aux 3 cas\n", "Re=np.zeros(3)\n", "Sh=np.zeros(3)\n", "k=np.zeros(3)\n", "Da=np.zeros(3)\n", "eff=np.zeros(3)\n", "ci=np.zeros(3)\n", "N=np.zeros(3)\n", "t=np.zeros(3)\n", "\n", "#Calcul du transfert de matière, du damkholer, de l'efficacité, du flux de matière et de l'efficacité pour les trois cas\n", "\n", "for i in range(len(u)):\n", " Re[i]=ro*u[i]*d0/mu\n", " Sh[i]=2+(0.95*Re[i]**0.5*Sc**0.33)\n", " k[i]=Sh[i]*D/d0\n", " Da[i]=(kr*(c0[i]-ceq)**(o_r-1))/k[i]\n", " eff[i]=1/Da[i] + 1/(2*Da[i]**2)-np.sqrt(1+4*Da[i])/(2*Da[i]**2)\n", " ci[i]=ceq+(eff[i]*(c0[i]-ceq)**o_r)**(1/o_r)\n", " N[i]=eff[i]*kr*(c0[i]-ceq)**o_r\n", " t[i]=0.1*(d0/2)*roc/N[i]\n", " \n", " \n", "print (' A B C ')\n", "print (' --------------------------------------------------------------')\n", "print ('u(m/s) | ', round(u[0],3), ' ', round(u[1],3),' ', round(u[2],3))\n", "print ('Dc(kg/m3) | ', round(c0[0]), ' ', round(c0[1]),' ', round(c0[2]))\n", "print ('Re | ', round(Re[0]), ' ', round(Re[1]),' ', round(Re[2]))\n", "print ('Sh | ', round(Sh[0],3), ' ', round(Sh[1],3),' ', round(Sh[2],3))\n", "print ('Da | ', round(Da[0],3), ' ', round(Da[1],3),' ', round(Da[2],3))\n", "print ('ci(kg/m3) | ', round(ci[0],2), ' ', round(ci[1],2),' ', round(ci[2],2))\n", "print ('Efficacite | ', round(eff[0],3), ' ', round(eff[1],3),' ', round(eff[2],3))\n", "print ('N(mol.m-2.s-1) | ', \"%.2e\"%N[0], ' ', \"%.2e\"%N[1],' ', \"%.2e\"%N[2])\n", "print ('t(s) | ', round(t[0]), ' ', round(t[1]),' ', round(t[2]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">a) Lorsque la vitesse relative entre cristaux et solution augmente (comparaison entre B et A) la couche limite de diffusion autour du cristal est réduite et le coefficient de transfert de matière, Sh, augmente. Le nombre de Damkhöler est alors réduit : la cristallisation est moins limité par la diffusion. La concentration à l'interface est plus élevée : la diffusion permet de mieux transporter le réactif vers l'interface. Le facteur d'efficacité augmente et la croissance des cristaux est plus rapide.\n", ">\n", ">b) Une sursaturation multipliée par 10 (comparaison A et C) devrait induire une augmentation de 100 fois de la vitesse de la réaction d'intégration à la surface (la réaction est d'ordre 2). Mais; lorsque la vitesse de réaction augmente; le nombre de Damkholer augmente : le phénomène est donc de plus en plus limité par la diffusion. Le facteur d'efficacité est donc réduit de 0,723 à 0,304. Une augmentation de sursaturation d'un facteur 100 n'engendre donc pas une croissance 100 fois plus rapide mais seulement 42 fois plus rapide à cause de cette perte d'efficacité.\n" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "41.97492437583958\n", "0.4197492437583958\n" ] } ], "source": [ "print (N[2]/N[1])\n", "print (eff[2]/eff[1])\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }