diff --git a/_toc.yml b/_toc.yml index b77ecb462..fddbe2dbb 100644 --- a/_toc.yml +++ b/_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 diff --git a/calculus/index.md b/calculus/index.md deleted file mode 100644 index 133ab03eb..000000000 --- a/calculus/index.md +++ /dev/null @@ -1,6 +0,0 @@ -# Calculus - -This module covers topics in differential and integral calculus of one variable. - -```{tableofcontents} -``` diff --git a/conservation/index.md b/conservation/index.md new file mode 100644 index 000000000..8912d7a6a --- /dev/null +++ b/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} +``` diff --git a/conservation/unsteady_mass_balance.md b/conservation/unsteady_mass_balance.md new file mode 100644 index 000000000..c0a4f60d8 --- /dev/null +++ b/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} +``` diff --git a/conservation/unsteady_mole_balance.md b/conservation/unsteady_mole_balance.md new file mode 100644 index 000000000..196f01b51 --- /dev/null +++ b/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 ): + +$$ +C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} +$$ diff --git a/images/CSTR2.png b/images/CSTR2.png new file mode 100644 index 000000000..e48ef35ff Binary files /dev/null and b/images/CSTR2.png differ diff --git a/images/cstr1.png b/images/cstr1.png new file mode 100644 index 000000000..8b15fa2c2 Binary files /dev/null and b/images/cstr1.png differ