intermediate version

doc/notebooks/Saturation.ipynb

@@ -1,7 +1,7 @@
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   "name": "",
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+  "signature": "sha256:7e5fb7d44068c577c6639c507162e5bd0d34809bc445e672bb55c26472df5e9a"
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@@ -16,6 +16,11 @@
       "$d$ means a derivative ALONG the saturation line,  \n",
       "$\\partial$ means a partial derivative AT the saturation line (or anywhere in the single phase region).\n",
       "\n",
+      "### References:  \n",
+      "Krafcik and Velasco, DOI 10.1119/1.4858403  \n",
+      "Thorade and Saadat, DOI 10.1007/s12665-013-2394-z\n",
+      "\n",
+      "### Clausius-Clapeyron\n",
       "Clausius-Clapeyron p/T\n",
       "\n",
       "\\begin{equation}\n",
@@ -57,18 +62,14 @@
       "\\frac{dv}{dp} &= \\left(\\frac{\\partial v}{\\partial p}\\right)_T  + \\left(\\frac{\\partial v}{\\partial T}\\right)_p \\frac{dT}{dp}\\\\\n",
       "\\frac{ds}{dp} &= \\left(\\frac{\\partial s}{\\partial p}\\right)_T  + \\left(\\frac{\\partial s}{\\partial T}\\right)_p \\frac{dT}{dp}\n",
       "\\end{split}\n",
-      "\\end{equation}\n",
-      "\n",
-      "### References:  \n",
-      "Krafcik and Velasco, DOI 10.1119/1.4858403  \n",
-      "Thorade and Saadat, DOI 10.1007/s12665-013-2394-z"
+      "\\end{equation}"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The following goes beyond the two cited papers.\n",
+      "### Temporary Names\n",
       "\n",
       "Introduce temporary names for some of the partial derivatives wrt $p$ and $T$:\n",
       "\\begin{equation}\n",
@@ -98,9 +99,14 @@
       "M &= \\frac{d \\rho}{d h} = \\frac{{d \\rho}/{dT}}{{dh}/{dT}} \\\\\n",
       "N &= \\frac{d s}{d h} = \\frac{{ds}/{dT}}{{dh}/{dT}}\n",
       "\\end{split}\n",
-      "\\end{equation}\n",
-      "\n",
-      "Now the rest is just a lot of writing and simple math.\n",
+      "\\end{equation}"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now the rest is not too hard, but with intermediate steps it is quite long.\n",
       "\n",
       "### First example: $d^2 \\rho / dT^2$  \n",
       "The corresponding first derivative can be written in two ways:\n",
@@ -142,12 +148,19 @@
       "\\begin{equation}\n",
       "\\begin{split}\n",
       "\\frac{ds}{dT} \n",
-      "  &= \\left(\\frac{\\partial s}{\\partial T}\\right)_p + \\left(\\frac{\\partial s}{\\partial p}\\right)_T \\frac{dp}{dT} \\\\\n",
+      "  &= \\left(\\frac{\\partial s}{\\partial T}\\right)_p + \\left(\\frac{\\partial s}{\\partial p}\\right)_T \\frac{dp}{dT} \n",
+      "   = C + E\\frac{dp}{dT}\\\\\n",
       "  &= \\left(\\frac{\\partial s}{\\partial T}\\right)_{\\rho} + \\left(\\frac{\\partial s}{\\partial \\rho}\\right)_T \\frac{d \\rho}{dT}\n",
+      "   = Y + Z\\frac{d \\rho}{dT}\n",
       "\\end{split}\n",
       "\\end{equation}\n",
       "Both can be used as starting point for the second derivatives.\n",
-      "\n"
+      "\\begin{split}\n",
+      "\\frac{d^2 s}{dT^2} \n",
+      "  &= \\frac{dC}{dT} + \\frac{dE}{dT}\\frac{dp}{dT} + E\\frac{d^2p}{dT^2}\\\\\n",
+      "  &= \\frac{dY}{dT} + \\frac{dZ}{dT}\\frac{dp}{dT} + Z\\frac{d^2 \\rho}{dT^2}\n",
+      "\\end{split}\n",
+      "Now, which one is nicer to work with? Unusre here"
      ]
     },
     {