making changes to the mass balance page

_toc.yml

@@ -3,6 +3,7 @@ root: index
 options:
   numbered: true
 chapters:
+  - file: conservation/index
   - file: calculus/index
   - file: multivariable-calculus/index
   - file: linear-algebra/index

conservation/index.md

@@ -0,0 +1,20 @@
+# Conservation
+
+**Rate of Change** Accumulation = In – Out + Generation – Consumption
+
+$$
+\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + r_{gen} - r_{cons}
+$$
+
+______________________________________________________________________
+
+**Example 1: Water Draining from a Bucket**
+
+![Water draining from a bucket](../images/cstr1.png)
+
+$$
+\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + r_{gen} - r_{cons}
+$$
+
+```{tableofcontents}
+```

conservation/unsteady_mass_balance.md

@@ -0,0 +1,22 @@
+# **Mass Balance**
+
+$$
+\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out}
+$$
+
+Since density is constant:
+
+$$
+\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0
+$$
+
+Thus:
+
+$$
+\rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0
+$$
+
+______________________________________________________________________
+
+```{tableofcontents}
+```

conservation/unsteady_mole_balance.md

@@ -0,0 +1,41 @@
+# **Mole Balance**
+
+$$
+\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + r_{gen} - r_{cons}
+$$
+
+Expanding:
+
+$$
+\frac{d(C_A V)}{dt} = C_{A0} \dot{V} - C_A \dot{V} - k C_A V
+$$
+
+Rearranging:
+
+$$
+C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V
+$$
+
+Since ( \\frac\{dV}\{dt} = 0 ):
+
+$$
+V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V
+$$
+
+At **steady state**, where ( \\frac\{dC_A}\{dt} = 0 ):
+
+$$
+0 = (C_{A0} - C_A) \dot{V} - k C_A V
+$$
+
+Rearrange:
+
+$$
+(C_{A0} - C_A) \dot{V} = k C_A V
+$$
+
+Solving for ( C_A ):
+
+$$
+C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV}
+$$