mirror of
https://github.com/CoolProp/CoolProp.git
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Added derivation for Plank-Einstein generalized form - will work nicely, simplifying PlanckEinstein, PlankEinstein2 and Aly-Lee all into one form. To be implemented tomorrow.
Signed-off-by: Ian Bell <ian.h.bell@gmail.com>
This commit is contained in:
Ian Bell
175
doc/notebooks/CP0.ipynb
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175
doc/notebooks/CP0.ipynb
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@@ -0,0 +1,175 @@
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{
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"metadata": {
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"name": ""
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},
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"nbformat": 3,
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"nbformat_minor": 0,
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"worksheets": [
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{
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"cells": [
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{
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"cell_type": "heading",
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"level": 1,
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"metadata": {},
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"source": [
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"Conversion of Aly-Lee form to Planck-Einstein"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"from __future__ import division\n",
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"from sympy import *\n",
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"init_session(quiet=True)\n",
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"init_printing()"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"IPython console for SymPy 0.7.5 (Python 2.7.2-32-bit) (ground types: python)\n"
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]
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}
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],
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"prompt_number": 2
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"alpha0 = a_k*log(c_k + d_k*exp(b_k*tau/T_c))\n",
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"display(diff(alpha0, tau, 2)*(-tau**2).together().ratsimp())"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"latex": [
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"$$- \\frac{a_{k} b_{k}^{2} d_{k} \\tau^{2} e^{\\frac{b_{k} \\tau}{T_{c}}}}{T_{c}^{2} \\left(c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right)} \\left(- \\frac{d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}}{c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}} + 1\\right)$$"
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],
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"metadata": {},
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"output_type": "display_data",
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"text": [
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" \u239b b_k\u22c5\u03c4 \u239e \n",
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" \u239c \u2500\u2500\u2500\u2500\u2500 \u239f b_k\u22c5\u03c4 \n",
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" \u239c T_c \u239f \u2500\u2500\u2500\u2500\u2500 \n",
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" 2 2 \u239c d_k\u22c5\u212f \u239f T_c \n",
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"-a_k\u22c5b_k \u22c5d_k\u22c5\u03c4 \u22c5\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 1\u239f\u22c5\u212f \n",
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" \u239c b_k\u22c5\u03c4 \u239f \n",
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" \u239c \u2500\u2500\u2500\u2500\u2500 \u239f \n",
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" \u239c T_c \u239f \n",
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" \u239d c_k + d_k\u22c5\u212f \u23a0 \n",
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"\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
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" \u239b b_k\u22c5\u03c4\u239e \n",
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" \u239c \u2500\u2500\u2500\u2500\u2500\u239f \n",
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" 2 \u239c T_c \u239f \n",
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" T_c \u22c5\u239dc_k + d_k\u22c5\u212f \u23a0 "
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]
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}
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],
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"prompt_number": 4
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"And then with some manual simplification:\n",
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"$$\\frac{c_p^0}{R} = - \\frac{a_{k} (b_{k}\\tau/T_c)^{2} c_k d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}}{ \\left(c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right)^2} $$\n",
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"\n",
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"Two important identities:\n",
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"$$ \\left(\\frac{1}{\\cosh x}\\right) = \\frac{2e^{-x}}{1+e^{-2x}}$$\n",
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"$$ \\left(\\frac{1}{\\sinh x}\\right) = \\frac{2e^{-x}}{1-e^{-2x}}$$\n",
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"which yield\n",
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"$$ \\left(\\frac{1}{\\cosh x}\\right)^2 = \\frac{4e^{-2x}}{(1+e^{-2x})^2}$$\n",
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"$$ \\left(\\frac{1}{\\sinh x}\\right)^2 = \\frac{4e^{-2x}}{(1-e^{-2x})^2}$$"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"B, D, T = symbols('B, D, T', real = True, positive = True)\n",
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"# From Aly-Lee:\n",
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"term1 = ((D/T)/sinh(D/T))**2\n",
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"# Planck Einstein\n",
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"term2 = (B/T)**2*exp(B/T)/(exp(B/T)-1)**2\n",
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"diff1 = term2 - term1\n",
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"display(diff1.rewrite(exp).subs(B, 2*D).simplify())\n",
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"term2 = (2*D/T)**2*exp(2*D/T)/(exp(2*D/T)-1)**2\n",
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"display(term2)\n",
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"print 'which is the same as the Aly-Lee sinh term with b_k = 2*D'\n",
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"\n",
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"B, F, T = symbols('B, F, T', real = True, positive = True)\n",
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"# From Aly-Lee:\n",
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"term1 = ((F/T)/cosh(F/T))**2\n",
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"# Planck Einstein\n",
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"term2 = (B/T)**2*exp(-B/T)/(exp(-B/T)+1)**2\n",
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"diff2 = term2 - term1\n",
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"display(diff2.rewrite(exp).subs(B, 2*F).simplify())\n",
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"# So Aly-Lee cosh term can be directly converted to Plank-Einstein2 term with b_k = 2*F, and c=1"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"latex": [
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"$$0$$"
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],
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"metadata": {},
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB5wID+0cG/gsMfN8Z5BUY+L4wzDdg\nYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxylQEA9FISlFfRJtkAAAAASUVORK5C\nYII=\n",
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"text": [
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"0"
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]
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},
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{
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"latex": [
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"$$\\frac{4 D^{2} e^{\\frac{2 D}{T}}}{T^{2} \\left(e^{\\frac{2 D}{T}} - 1\\right)^{2}}$$"
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],
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"metadata": {},
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAABLBAMAAACsFlnLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nimrffr4AAAAASUVORK5CYII=\n",
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"text": [
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" 2\u22c5D \n",
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" \u2500\u2500\u2500 \n",
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" 2 T \n",
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" 4\u22c5D \u22c5\u212f \n",
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"\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
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" 2\n",
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" \u239b 2\u22c5D \u239e \n",
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" \u239c \u2500\u2500\u2500 \u239f \n",
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" 2 \u239c T \u239f \n",
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"T \u22c5\u239d\u212f - 1\u23a0 "
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]
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},
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"which is the same as the Aly-Lee sinh term with b_k = 2*D\n"
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]
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},
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{
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"latex": [
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"$$0$$"
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],
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"metadata": {},
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB5wID+0cG/gsMfN8Z5BUY+L4wzDdg\nYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxylQEA9FISlFfRJtkAAAAASUVORK5C\nYII=\n",
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"text": [
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"0"
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]
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}
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],
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"prompt_number": 10
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}
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],
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"metadata": {}
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}
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]
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}
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Reference in New Issue
Block a user