From 2e9799d8acad00d52d63068a32a8e3836c97e754 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Wed, 12 Mar 2025 14:50:53 -0500 Subject: [PATCH 01/33] making changes --- calculus/index.md | 78 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 78 insertions(+) diff --git a/calculus/index.md b/calculus/index.md index 133ab03eb..5ce846dcf 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,6 +1,84 @@ # Calculus This module covers topics in differential and integral calculus of one variable. +# Conservation Principles and Mass Balance + +![Conservation Notes](./image.png) + +## 1. Conservation (Rate of Change) +The general conservation equation: + +$$ +\text{Accumulation} = \text{In} - \text{Out} + \text{Generation} - \text{Consumption} +$$ + +For a general quantity \( A \): + +$$ +\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +$$ + +![Water draining from a bucket](./image.png) + +$$ +\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +$$ + +Since no generation or consumption occurs: + +$$ +\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} +$$ + +![Continuous Stirred Tank Reactor (CSTR)](./image.png) + +$$ +A \rightarrow B +$$ + +$$ +r_A = -k C_A +$$ + +$$ +\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} +$$ + +Since **mass density is constant**, we assume: + +$$ +\frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{d\rho}{dt} = 0 +$$ + +$$ +\frac{d(C_A V)}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +$$ + +$$ +C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V +$$ + +Since \( \frac{dV}{dt} = 0 \), we simplify: + +$$ +V \frac{dC_A}{dt} = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V +$$ + +At **steady state**, where \( \frac{dC_A}{dt} = 0 \): + +$$ +0 = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V +$$ + +$$ +C_A \left( \dot{m}_{\text{out}} + kV \right) = \dot{m}_{\text{in}} C_{A0} +$$ + +$$ +C_A = \frac{\dot{m}_{\text{in}} C_{A0}}{\dot{m}_{\text{out}} + kV} +$$ + +![Conservation and Mass Balance Notes](./image.png) ```{tableofcontents} ``` From eb373066f62bbb40c85c4404e5aee8382e38a774 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 12 Mar 2025 20:15:52 +0000 Subject: [PATCH 02/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 5ce846dcf..cfc3c2a24 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,18 +1,20 @@ # Calculus This module covers topics in differential and integral calculus of one variable. + # Conservation Principles and Mass Balance ![Conservation Notes](./image.png) ## 1. Conservation (Rate of Change) + The general conservation equation: $$ \text{Accumulation} = \text{In} - \text{Out} + \text{Generation} - \text{Consumption} $$ -For a general quantity \( A \): +For a general quantity ( A ): $$ \frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} @@ -58,13 +60,13 @@ $$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V $$ -Since \( \frac{dV}{dt} = 0 \), we simplify: +Since ( \\frac\{dV}\{dt} = 0 ), we simplify: $$ V \frac{dC_A}{dt} = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V $$ -At **steady state**, where \( \frac{dC_A}{dt} = 0 \): +At **steady state**, where ( \\frac\{dC_A}\{dt} = 0 ): $$ 0 = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V From 82a32abb41ef2e6f45fb55d2db5efee9e22cfea5 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 14:33:36 -0500 Subject: [PATCH 03/33] working --- calculus/index.md | 76 ++++++++++++++++++++++++++++++----------------- 1 file changed, 48 insertions(+), 28 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index cfc3c2a24..c25837a21 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,86 +1,106 @@ # Calculus +# Conservation -This module covers topics in differential and integral calculus of one variable. +**(Rate of change)** +Accumulation = In – Out + Generation – Consumption -# Conservation Principles and Mass Balance +$$ +\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +$$ -![Conservation Notes](./image.png) +--- -## 1. Conservation (Rate of Change) +## **Example 1: Water Draining from a Bucket** -The general conservation equation: +![Water draining from a bucket](./image.png) $$ -\text{Accumulation} = \text{In} - \text{Out} + \text{Generation} - \text{Consumption} +\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ -For a general quantity ( A ): +For **mass balance**: $$ -\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +\frac{dm}{dt} = \dot{m}_{\text{in}} - 10 $$ -![Water draining from a bucket](./image.png) +--- + +## **Example 2: Continuous Stirred Tank Reactor (CSTR)** + +![CSTR Diagram](./image.png) + +Reaction: $$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +r_A = -kC_A V $$ -Since no generation or consumption occurs: +**Assumption:** +Mass density is constant. + +--- +### **Mass Balance** $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} $$ -![Continuous Stirred Tank Reactor (CSTR)](./image.png) +Since density \( \rho \) is constant: $$ -A \rightarrow B +\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 $$ -$$ -r_A = -k C_A -$$ +Thus: $$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} +\rho \frac{dV}{dt} = 0 \Rightarrow \frac{dV}{dt} = 0 $$ -Since **mass density is constant**, we assume: +--- +### **Mole Balance** $$ -\frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{d\rho}{dt} = 0 +\frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ +Expanding: + $$ -\frac{d(C_A V)}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +\frac{d(C_A V)}{dt} = C_{A0} \dot{V} - C_A \dot{V} - k C_A V $$ +Rearranging: + $$ -C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V +C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -Since ( \\frac\{dV}\{dt} = 0 ), we simplify: +Since \( \frac{dV}{dt} = 0 \): $$ -V \frac{dC_A}{dt} = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V +V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -At **steady state**, where ( \\frac\{dC_A}\{dt} = 0 ): +At **steady state** where \( \frac{dC_A}{dt} = 0 \): $$ -0 = \dot{m}_{\text{in}} C_{A0} - \dot{m}_{\text{out}} C_A - k C_A V +0 = (C_{A0} - C_A) \dot{V} - k C_A V $$ +Rearrange: + $$ -C_A \left( \dot{m}_{\text{out}} + kV \right) = \dot{m}_{\text{in}} C_{A0} +C_A ( \dot{V} + kV ) = C_{A0} \dot{V} $$ +Solving for \( C_A \): + $$ -C_A = \frac{\dot{m}_{\text{in}} C_{A0}}{\dot{m}_{\text{out}} + kV} +C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} $$ -![Conservation and Mass Balance Notes](./image.png) ```{tableofcontents} ``` From 7f08f88f0fb97b22042b3294df1f673046ce85ed Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 19:34:52 +0000 Subject: [PATCH 04/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 26 +++++++++++++------------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index c25837a21..0cbe71ca1 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,14 +1,14 @@ # Calculus + # Conservation -**(Rate of change)** -Accumulation = In – Out + Generation – Consumption +**(Rate of change)** Accumulation = In – Out + Generation – Consumption $$ \frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ ---- +______________________________________________________________________ ## **Example 1: Water Draining from a Bucket** @@ -24,7 +24,7 @@ $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - 10 $$ ---- +______________________________________________________________________ ## **Example 2: Continuous Stirred Tank Reactor (CSTR)** @@ -36,17 +36,17 @@ $$ r_A = -kC_A V $$ -**Assumption:** -Mass density is constant. +**Assumption:** Mass density is constant. ---- +______________________________________________________________________ ### **Mass Balance** + $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} $$ -Since density \( \rho \) is constant: +Since density ( \\rho ) is constant: $$ \frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 @@ -58,9 +58,10 @@ $$ \rho \frac{dV}{dt} = 0 \Rightarrow \frac{dV}{dt} = 0 $$ ---- +______________________________________________________________________ ### **Mole Balance** + $$ \frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ @@ -77,13 +78,13 @@ $$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -Since \( \frac{dV}{dt} = 0 \): +Since ( \\frac\{dV}\{dt} = 0 ): $$ V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -At **steady state** where \( \frac{dC_A}{dt} = 0 \): +At **steady state** where ( \\frac\{dC_A}\{dt} = 0 ): $$ 0 = (C_{A0} - C_A) \dot{V} - k C_A V @@ -95,12 +96,11 @@ $$ C_A ( \dot{V} + kV ) = C_{A0} \dot{V} $$ -Solving for \( C_A \): +Solving for ( C_A ): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} $$ - ```{tableofcontents} ``` From 1b9bf441debc3d43b5f436c8e911bbcac435ace4 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 14:46:22 -0500 Subject: [PATCH 05/33] dd --- calculus/index.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 0cbe71ca1..7ecf6188f 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,3 +1,4 @@ + # Calculus # Conservation @@ -5,14 +6,14 @@ **(Rate of change)** Accumulation = In – Out + Generation – Consumption $$ -\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + r_{\text{gen}} - r_{\text{cons}} $$ ______________________________________________________________________ ## **Example 1: Water Draining from a Bucket** -![Water draining from a bucket](./image.png) + $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} @@ -28,7 +29,7 @@ ______________________________________________________________________ ## **Example 2: Continuous Stirred Tank Reactor (CSTR)** -![CSTR Diagram](./image.png) + Reaction: From 9041dfeca8839eff21a0310d246d6bd95d80273c Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 19:46:29 +0000 Subject: [PATCH 06/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/calculus/index.md b/calculus/index.md index 7ecf6188f..c3970ba77 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,4 +1,5 @@ + # Calculus # Conservation From 6be55b88c5a9480cb1154000996fb5d5fe054915 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:01:29 -0500 Subject: [PATCH 07/33] ss --- calculus/index.md | 123 ++++++++++++++++++++++++---------------------- 1 file changed, 65 insertions(+), 58 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index c3970ba77..dcd6bcf12 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -4,105 +4,112 @@ # Conservation -**(Rate of change)** Accumulation = In – Out + Generation – Consumption +### (Rate of Change) +Accumulation = In – Out + Generation – Consumption -$$ -\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + r_{\text{gen}} - r_{\text{cons}} -$$ +``` +dA/dt = ṁ_in - ṁ_out + g_gen - g_cons +``` -______________________________________________________________________ +--- ## **Example 1: Water Draining from a Bucket** - +![Water draining from a bucket](./image.png) -$$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} -$$ +``` +dm/dt = ṁ_in - ṁ_out + g_gen - g_cons +``` For **mass balance**: -$$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - 10 -$$ +``` +dm/dt = ṁ_in - 10 +``` -______________________________________________________________________ +--- ## **Example 2: Continuous Stirred Tank Reactor (CSTR)** - - -Reaction: +![CSTR Diagram](./image.png) -$$ +**Reaction:** +``` r_A = -kC_A V -$$ +``` -**Assumption:** Mass density is constant. +**Assume mass density is constant.** -______________________________________________________________________ +--- ### **Mass Balance** +``` +dm/dt = ṁ_in - ṁ_out +``` -$$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} -$$ - -Since density ( \\rho ) is constant: +Since density (ρ) is constant: -$$ -\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 -$$ +``` +d(ρV)/dt = ρ dV/dt = 0 +``` Thus: -$$ -\rho \frac{dV}{dt} = 0 \Rightarrow \frac{dV}{dt} = 0 -$$ +``` +ρ dV/dt = 0 → dV/dt = 0 +``` -______________________________________________________________________ +--- ### **Mole Balance** - -$$ -\frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} -$$ +``` +d(n_A)/dt = ṁ_A,in - ṁ_A,out + g_gen - g_cons +``` Expanding: -$$ -\frac{d(C_A V)}{dt} = C_{A0} \dot{V} - C_A \dot{V} - k C_A V -$$ +``` +d(C_A V)/dt = C_A0 ṁ_V - C_A ṁ_V - k C_A V +``` Rearranging: -$$ -C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V -$$ +``` +C_A dV/dt + V dC_A/dt = (C_A0 - C_A) ṁ_V - k C_A V +``` -Since ( \\frac\{dV}\{dt} = 0 ): +Since `dV/dt = 0`: -$$ -V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V -$$ +``` +V dC_A/dt = (C_A0 - C_A) ṁ_V - k C_A V +``` -At **steady state** where ( \\frac\{dC_A}\{dt} = 0 ): +At **steady state**, where `dC_A/dt = 0`: -$$ -0 = (C_{A0} - C_A) \dot{V} - k C_A V -$$ +``` +0 = (C_A0 - C_A) ṁ_V - k C_A V +``` Rearrange: -$$ -C_A ( \dot{V} + kV ) = C_{A0} \dot{V} -$$ +``` +(C_A0 - C_A) ṁ_V = k C_A V +``` + +Solving for `C_A`: -Solving for ( C_A ): +``` +dC_A/dt = ((C_A0 - C_A) ṁ_V) / V - k C_A +``` -$$ -C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} -$$ +At **steady state**, this simplifies further: -```{tableofcontents} +``` +C_A (ṁ_V + kV) = C_A0 ṁ_V +``` + +Final expression: + +``` +C_A = (C_A0 ṁ_V) / (ṁ_V + kV) ``` From 622a0c7a7cc68550ba6a3d09a34d259e566f7650 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:01:41 +0000 Subject: [PATCH 08/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 14 +++++++++----- 1 file changed, 9 insertions(+), 5 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index dcd6bcf12..38a8e77fe 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -5,13 +5,14 @@ # Conservation ### (Rate of Change) -Accumulation = In – Out + Generation – Consumption + +Accumulation = In – Out + Generation – Consumption ``` dA/dt = ṁ_in - ṁ_out + g_gen - g_cons ``` ---- +______________________________________________________________________ ## **Example 1: Water Draining from a Bucket** @@ -27,22 +28,24 @@ For **mass balance**: dm/dt = ṁ_in - 10 ``` ---- +______________________________________________________________________ ## **Example 2: Continuous Stirred Tank Reactor (CSTR)** ![CSTR Diagram](./image.png) **Reaction:** + ``` r_A = -kC_A V ``` **Assume mass density is constant.** ---- +______________________________________________________________________ ### **Mass Balance** + ``` dm/dt = ṁ_in - ṁ_out ``` @@ -59,9 +62,10 @@ Thus: ρ dV/dt = 0 → dV/dt = 0 ``` ---- +______________________________________________________________________ ### **Mole Balance** + ``` d(n_A)/dt = ṁ_A,in - ṁ_A,out + g_gen - g_cons ``` From c642e25f65d0135dbb4e3946a3b5af0823c9c4ae Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:07:18 -0500 Subject: [PATCH 09/33] ds --- calculus/index.md | 120 +++++++++++++++++++--------------------------- 1 file changed, 50 insertions(+), 70 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 38a8e77fe..464b247bd 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,119 +1,99 @@ - - -# Calculus - # Conservation ### (Rate of Change) +Accumulation = In – Out + Generation – Consumption -Accumulation = In – Out + Generation – Consumption - -``` -dA/dt = ṁ_in - ṁ_out + g_gen - g_cons -``` +$$ +\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +$$ -______________________________________________________________________ +--- ## **Example 1: Water Draining from a Bucket** ![Water draining from a bucket](./image.png) -``` -dm/dt = ṁ_in - ṁ_out + g_gen - g_cons -``` +$$ +\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +$$ For **mass balance**: -``` -dm/dt = ṁ_in - 10 -``` +$$ +\frac{dm}{dt} = \dot{m}_{\text{in}} - 10 +$$ -______________________________________________________________________ +--- ## **Example 2: Continuous Stirred Tank Reactor (CSTR)** ![CSTR Diagram](./image.png) **Reaction:** - -``` +$$ r_A = -kC_A V -``` +$$ **Assume mass density is constant.** -______________________________________________________________________ +--- ### **Mass Balance** +$$ +\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} +$$ -``` -dm/dt = ṁ_in - ṁ_out -``` - -Since density (ρ) is constant: +Since density \( \rho \) is constant: -``` -d(ρV)/dt = ρ dV/dt = 0 -``` +$$ +\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 +$$ Thus: -``` -ρ dV/dt = 0 → dV/dt = 0 -``` +$$ +\rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 +$$ -______________________________________________________________________ +--- ### **Mole Balance** - -``` -d(n_A)/dt = ṁ_A,in - ṁ_A,out + g_gen - g_cons -``` +$$ +\frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} +$$ Expanding: -``` -d(C_A V)/dt = C_A0 ṁ_V - C_A ṁ_V - k C_A V -``` +$$ +\frac{d(C_A V)}{dt} = C_{A0} \dot{V} - C_A \dot{V} - k C_A V +$$ Rearranging: -``` -C_A dV/dt + V dC_A/dt = (C_A0 - C_A) ṁ_V - k C_A V -``` +$$ +C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V +$$ -Since `dV/dt = 0`: +Since \( \frac{dV}{dt} = 0 \): -``` -V dC_A/dt = (C_A0 - C_A) ṁ_V - k C_A V -``` +$$ +V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V +$$ -At **steady state**, where `dC_A/dt = 0`: +At **steady state**, where \( \frac{dC_A}{dt} = 0 \): -``` -0 = (C_A0 - C_A) ṁ_V - k C_A V -``` +$$ +0 = (C_{A0} - C_A) \dot{V} - k C_A V +$$ Rearrange: -``` -(C_A0 - C_A) ṁ_V = k C_A V -``` - -Solving for `C_A`: - -``` -dC_A/dt = ((C_A0 - C_A) ṁ_V) / V - k C_A -``` - -At **steady state**, this simplifies further: - -``` -C_A (ṁ_V + kV) = C_A0 ṁ_V -``` +$$ +(C_{A0} - C_A) \dot{V} = k C_A V +$$ -Final expression: +Solving for \( C_A \): -``` -C_A = (C_A0 ṁ_V) / (ṁ_V + kV) -``` +$$ +C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} +$$ From 51c9f740bd16c710b434a332ed2ddd22be9c27ad Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:07:57 +0000 Subject: [PATCH 10/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 26 ++++++++++++++------------ 1 file changed, 14 insertions(+), 12 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 464b247bd..a745275b2 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,13 +1,14 @@ # Conservation ### (Rate of Change) -Accumulation = In – Out + Generation – Consumption + +Accumulation = In – Out + Generation – Consumption $$ \frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ ---- +______________________________________________________________________ ## **Example 1: Water Draining from a Bucket** @@ -23,27 +24,27 @@ $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - 10 $$ ---- +______________________________________________________________________ ## **Example 2: Continuous Stirred Tank Reactor (CSTR)** ![CSTR Diagram](./image.png) -**Reaction:** -$$ +**Reaction:** \$$ r_A = -kC_A V -$$ +$\$ **Assume mass density is constant.** ---- +______________________________________________________________________ ### **Mass Balance** + $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} $$ -Since density \( \rho \) is constant: +Since density ( \\rho ) is constant: $$ \frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 @@ -55,9 +56,10 @@ $$ \rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 $$ ---- +______________________________________________________________________ ### **Mole Balance** + $$ \frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ @@ -74,13 +76,13 @@ $$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -Since \( \frac{dV}{dt} = 0 \): +Since ( \\frac\{dV}\{dt} = 0 ): $$ V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -At **steady state**, where \( \frac{dC_A}{dt} = 0 \): +At **steady state**, where ( \\frac\{dC_A}\{dt} = 0 ): $$ 0 = (C_{A0} - C_A) \dot{V} - k C_A V @@ -92,7 +94,7 @@ $$ (C_{A0} - C_A) \dot{V} = k C_A V $$ -Solving for \( C_A \): +Solving for ( C_A ): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} From 8ee28c9418394f112e6b7db0154420f8bbc285df Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:16:36 -0500 Subject: [PATCH 11/33] s --- calculus/index.md | 13 ++++++++++--- 1 file changed, 10 insertions(+), 3 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index a745275b2..56b1f6845 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,5 +1,7 @@ # Conservation +**(Rate of Change)** +Accumulation = In – Out + Generation – Consumption ### (Rate of Change) Accumulation = In – Out + Generation – Consumption @@ -10,7 +12,7 @@ $$ ______________________________________________________________________ -## **Example 1: Water Draining from a Bucket** +## **Example: Water Draining from a Bucket** ![Water draining from a bucket](./image.png) @@ -26,11 +28,16 @@ $$ ______________________________________________________________________ -## **Example 2: Continuous Stirred Tank Reactor (CSTR)** +## **Example: Continuous Stirred Tank Reactor (CSTR)** ![CSTR Diagram](./image.png) +<<<<<<< HEAD +**Reaction:** +$$ +======= **Reaction:** \$$ +>>>>>>> 51c9f740bd16c710b434a332ed2ddd22be9c27ad r_A = -kC_A V $\$ @@ -71,7 +78,7 @@ $$ $$ Rearranging: - +s $$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ From 0ae275b7a9fb1938fd8b612d372c2d4379b60162 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:17:02 +0000 Subject: [PATCH 12/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 23 +++++++++-------------- 1 file changed, 9 insertions(+), 14 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 56b1f6845..1fe07b420 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,7 +1,7 @@ # Conservation -**(Rate of Change)** -Accumulation = In – Out + Generation – Consumption +**(Rate of Change)** Accumulation = In – Out + Generation – Consumption + ### (Rate of Change) Accumulation = In – Out + Generation – Consumption @@ -32,14 +32,11 @@ ______________________________________________________________________ ![CSTR Diagram](./image.png) -<<<<<<< HEAD -**Reaction:** -$$ -======= -**Reaction:** \$$ ->>>>>>> 51c9f740bd16c710b434a332ed2ddd22be9c27ad -r_A = -kC_A V -$\$ +# \<<\<<\<<< HEAD **Reaction:** \$\$ + +**Reaction:** \$\$ + +> > > > > > > 51c9f740bd16c710b434a332ed2ddd22be9c27ad r_A = -kC_A V \$\$ **Assume mass density is constant.** @@ -77,11 +74,9 @@ $$ \frac{d(C_A V)}{dt} = C_{A0} \dot{V} - C_A \dot{V} - k C_A V $$ -Rearranging: -s -$$ +Rearranging: s \$$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V -$$ +$\$ Since ( \\frac\{dV}\{dt} = 0 ): From d68f82409922ad6bb8a4c5f36f66e6e9d230357b Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:18:46 -0500 Subject: [PATCH 13/33] ww --- calculus/index.md | 20 ++++++++------------ 1 file changed, 8 insertions(+), 12 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 1fe07b420..aeb9fb11f 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -4,13 +4,11 @@ ### (Rate of Change) -Accumulation = In – Out + Generation – Consumption - $$ \frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ -______________________________________________________________________ +--- ## **Example: Water Draining from a Bucket** @@ -26,7 +24,7 @@ $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - 10 $$ -______________________________________________________________________ +--- ## **Example: Continuous Stirred Tank Reactor (CSTR)** @@ -40,15 +38,14 @@ ______________________________________________________________________ **Assume mass density is constant.** -______________________________________________________________________ +--- ### **Mass Balance** - $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} $$ -Since density ( \\rho ) is constant: +Since density \( \rho \) is constant: $$ \frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 @@ -60,10 +57,9 @@ $$ \rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 $$ -______________________________________________________________________ +--- ### **Mole Balance** - $$ \frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ @@ -78,13 +74,13 @@ Rearranging: s \$$ 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 ): +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 ): +At **steady state**, where \( \frac{dC_A}{dt} = 0 \): $$ 0 = (C_{A0} - C_A) \dot{V} - k C_A V @@ -96,7 +92,7 @@ $$ (C_{A0} - C_A) \dot{V} = k C_A V $$ -Solving for ( C_A ): +Solving for \( C_A \): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} From f725c9f1da39a1e860b184f13548728a27cf26ae Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:19:10 +0000 Subject: [PATCH 14/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 18 ++++++++++-------- 1 file changed, 10 insertions(+), 8 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index aeb9fb11f..b4ab98d6d 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -8,7 +8,7 @@ $$ \frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ ---- +______________________________________________________________________ ## **Example: Water Draining from a Bucket** @@ -24,7 +24,7 @@ $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - 10 $$ ---- +______________________________________________________________________ ## **Example: Continuous Stirred Tank Reactor (CSTR)** @@ -38,14 +38,15 @@ $$ **Assume mass density is constant.** ---- +______________________________________________________________________ ### **Mass Balance** + $$ \frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} $$ -Since density \( \rho \) is constant: +Since density ( \\rho ) is constant: $$ \frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 @@ -57,9 +58,10 @@ $$ \rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 $$ ---- +______________________________________________________________________ ### **Mole Balance** + $$ \frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} $$ @@ -74,13 +76,13 @@ Rearranging: s \$$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $\$ -Since \( \frac{dV}{dt} = 0 \): +Since ( \\frac\{dV}\{dt} = 0 ): $$ V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -At **steady state**, where \( \frac{dC_A}{dt} = 0 \): +At **steady state**, where ( \\frac\{dC_A}\{dt} = 0 ): $$ 0 = (C_{A0} - C_A) \dot{V} - k C_A V @@ -92,7 +94,7 @@ $$ (C_{A0} - C_A) \dot{V} = k C_A V $$ -Solving for \( C_A \): +Solving for ( C_A ): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} From 2f7a4d9e579bb96c1c8138344da34e1f584c095d Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:28:24 -0500 Subject: [PATCH 15/33] G --- calculus/index.md | 30 +++++++++++++++--------------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index b4ab98d6d..94337f5c8 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,11 +1,10 @@ # Conservation -**(Rate of Change)** Accumulation = In – Out + Generation – Consumption - -### (Rate of Change) +**Rate of Change** +Accumulation = In – Out + Generation – Consumption $$ -\frac{dA}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} $$ ______________________________________________________________________ @@ -15,13 +14,13 @@ ______________________________________________________________________ ![Water draining from a bucket](./image.png) $$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} + g_{\text{gen}} - g_{\text{cons}} +\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} $$ For **mass balance**: $$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - 10 +\frac{dm}{dt} = \dot{m}_{in} - 10 $$ ______________________________________________________________________ @@ -30,11 +29,10 @@ ______________________________________________________________________ ![CSTR Diagram](./image.png) -# \<<\<<\<<< HEAD **Reaction:** \$\$ - -**Reaction:** \$\$ - -> > > > > > > 51c9f740bd16c710b434a332ed2ddd22be9c27ad r_A = -kC_A V \$\$ +**Reaction:** +$$ +r_A = -k C_A V +$$ **Assume mass density is constant.** @@ -43,7 +41,7 @@ ______________________________________________________________________ ### **Mass Balance** $$ -\frac{dm}{dt} = \dot{m}_{\text{in}} - \dot{m}_{\text{out}} +\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} $$ Since density ( \\rho ) is constant: @@ -63,7 +61,7 @@ ______________________________________________________________________ ### **Mole Balance** $$ -\frac{d(n_A)}{dt} = \dot{n}_{A,\text{in}} - \dot{n}_{A,\text{out}} + g_{\text{gen}} - g_{\text{cons}} +\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + g_{gen} - g_{cons} $$ Expanding: @@ -72,9 +70,11 @@ $$ \frac{d(C_A V)}{dt} = C_{A0} \dot{V} - C_A \dot{V} - k C_A V $$ -Rearranging: s \$$ +Rearranging: + +$$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V -$\$ +$$ Since ( \\frac\{dV}\{dt} = 0 ): From ec471af7d13ec34b688c5dcb87914312f7f44559 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:28:29 +0000 Subject: [PATCH 16/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 8 +++----- 1 file changed, 3 insertions(+), 5 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 94337f5c8..43dfff2da 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,7 +1,6 @@ # Conservation -**Rate of Change** -Accumulation = In – Out + Generation – Consumption +**Rate of Change** Accumulation = In – Out + Generation – Consumption $$ \frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} @@ -29,10 +28,9 @@ ______________________________________________________________________ ![CSTR Diagram](./image.png) -**Reaction:** -$$ +**Reaction:** \$$ r_A = -k C_A V -$$ +$\$ **Assume mass density is constant.** From b2f896ac4469b079ba5a53ebcf766b54d8a8bde3 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:29:54 -0500 Subject: [PATCH 17/33] S --- calculus/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/calculus/index.md b/calculus/index.md index 43dfff2da..c1feb61eb 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -20,7 +20,7 @@ For **mass balance**: $$ \frac{dm}{dt} = \dot{m}_{in} - 10 -$$ +$$ --> ______________________________________________________________________ From 1cc40b1b6f86011bc19e413b2f307016ea9b6344 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:30:06 +0000 Subject: [PATCH 18/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index c1feb61eb..cf6095b15 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -29,8 +29,8 @@ ______________________________________________________________________ ![CSTR Diagram](./image.png) **Reaction:** \$$ -r_A = -k C_A V -$\$ + +r_A = -k C_A V \$\$ **Assume mass density is constant.** From 1ac62c03c29234389e1aafe4d09c4f0d220cf205 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:34:11 -0500 Subject: [PATCH 19/33] nothing --- calculus/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/calculus/index.md b/calculus/index.md index cf6095b15..d9f9e8886 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,4 +1,3 @@ -# Conservation **Rate of Change** Accumulation = In – Out + Generation – Consumption @@ -97,3 +96,4 @@ Solving for ( C_A ): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} $$ + From b3e7fd31c63d32efb0acfa747abbb0f7134a3bef Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:34:17 +0000 Subject: [PATCH 20/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index d9f9e8886..26b8414e3 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,4 +1,3 @@ - **Rate of Change** Accumulation = In – Out + Generation – Consumption $$ @@ -96,4 +95,3 @@ Solving for ( C_A ): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} $$ - From 1c9b774f904e705592a6e7b5e52facc740ce53b9 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:34:45 -0500 Subject: [PATCH 21/33] nothing --- calculus/index.md | 97 ----------------------------------------------- 1 file changed, 97 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 26b8414e3..e69de29bb 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,97 +0,0 @@ -**Rate of Change** Accumulation = In – Out + Generation – Consumption - -$$ -\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} -$$ - -______________________________________________________________________ - -## **Example: Water Draining from a Bucket** - -![Water draining from a bucket](./image.png) - -$$ -\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} -$$ - -For **mass balance**: - -$$ -\frac{dm}{dt} = \dot{m}_{in} - 10 -$$ --> - -______________________________________________________________________ - -## **Example: Continuous Stirred Tank Reactor (CSTR)** - -![CSTR Diagram](./image.png) - -**Reaction:** \$$ - -r_A = -k C_A V \$\$ - -**Assume mass density is constant.** - -______________________________________________________________________ - -### **Mass Balance** - -$$ -\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} -$$ - -Since density ( \\rho ) is constant: - -$$ -\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 -$$ - -Thus: - -$$ -\rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 -$$ - -______________________________________________________________________ - -### **Mole Balance** - -$$ -\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + g_{gen} - g_{cons} -$$ - -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 ): - -$$ -C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} -$$ From fa830fdd081926549a204d4b39653f8d97d3d711 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Fri, 14 Mar 2025 15:42:42 -0500 Subject: [PATCH 22/33] sa --- calculus/index.md | 99 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 99 insertions(+) diff --git a/calculus/index.md b/calculus/index.md index e69de29bb..48d46b14f 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -0,0 +1,99 @@ +# Conservation + +**Rate of Change** +Accumulation = In – Out + Generation – Consumption + +$$ +\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} +$$ + +--- + +## **Example: Water Draining from a Bucket** + +![Water draining from a bucket](./image.png) + +$$ +\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} +$$ + +For **mass balance**: + +$$ +\frac{dm}{dt} = \dot{m}_{in} - 10 +$$ + +--- + +## **Example: Continuous Stirred Tank Reactor (CSTR)** + +![CSTR Diagram](./image.png) + +**Reaction:** +$$ +r_A = -k C_A V +$$ + +**Assume mass density is constant.** + +--- + +### **Mass Balance** +$$ +\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} +$$ + +Since density \( \rho \) is constant: + +$$ +\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 +$$ + +Thus: + +$$ +\rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 +$$ + +--- + +### **Mole Balance** +$$ +\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + g_{gen} - g_{cons} +$$ + +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 \( C_A \): + +$$ +C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} +$$ From 742d76ac2c7bb7be5dc27fb1c568fc9401121d36 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 14 Mar 2025 20:44:16 +0000 Subject: [PATCH 23/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 198 +++++++++++++++++++++++----------------------- 1 file changed, 99 insertions(+), 99 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 48d46b14f..43dfff2da 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -1,99 +1,99 @@ -# Conservation - -**Rate of Change** -Accumulation = In – Out + Generation – Consumption - -$$ -\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} -$$ - ---- - -## **Example: Water Draining from a Bucket** - -![Water draining from a bucket](./image.png) - -$$ -\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} -$$ - -For **mass balance**: - -$$ -\frac{dm}{dt} = \dot{m}_{in} - 10 -$$ - ---- - -## **Example: Continuous Stirred Tank Reactor (CSTR)** - -![CSTR Diagram](./image.png) - -**Reaction:** -$$ -r_A = -k C_A V -$$ - -**Assume mass density is constant.** - ---- - -### **Mass Balance** -$$ -\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} -$$ - -Since density \( \rho \) is constant: - -$$ -\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 -$$ - -Thus: - -$$ -\rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 -$$ - ---- - -### **Mole Balance** -$$ -\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + g_{gen} - g_{cons} -$$ - -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 \( C_A \): - -$$ -C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} -$$ +# Conservation + +**Rate of Change** Accumulation = In – Out + Generation – Consumption + +$$ +\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} +$$ + +______________________________________________________________________ + +## **Example: Water Draining from a Bucket** + +![Water draining from a bucket](./image.png) + +$$ +\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} +$$ + +For **mass balance**: + +$$ +\frac{dm}{dt} = \dot{m}_{in} - 10 +$$ + +______________________________________________________________________ + +## **Example: Continuous Stirred Tank Reactor (CSTR)** + +![CSTR Diagram](./image.png) + +**Reaction:** \$$ +r_A = -k C_A V +$\$ + +**Assume mass density is constant.** + +______________________________________________________________________ + +### **Mass Balance** + +$$ +\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} +$$ + +Since density ( \\rho ) is constant: + +$$ +\frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 +$$ + +Thus: + +$$ +\rho \frac{dV}{dt} = 0 \quad \Rightarrow \quad \frac{dV}{dt} = 0 +$$ + +______________________________________________________________________ + +### **Mole Balance** + +$$ +\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + g_{gen} - g_{cons} +$$ + +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 ): + +$$ +C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} +$$ From 726d1da2ec8bf1e0f9ff151b43477a41c8ce5327 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Tue, 18 Mar 2025 05:03:01 -0500 Subject: [PATCH 24/33] images --- calculus/index.md | 4 ++-- images/CSTR2.png | Bin 0 -> 8368 bytes images/cstr1.png | Bin 0 -> 6293 bytes 3 files changed, 2 insertions(+), 2 deletions(-) create mode 100644 images/CSTR2.png create mode 100644 images/cstr1.png diff --git a/calculus/index.md b/calculus/index.md index 43dfff2da..c5e32cce3 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -10,7 +10,7 @@ ______________________________________________________________________ ## **Example: Water Draining from a Bucket** -![Water draining from a bucket](./image.png) +![Water draining from a bucket](../images/cstr1.png) $$ \frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} @@ -26,7 +26,7 @@ ______________________________________________________________________ ## **Example: Continuous Stirred Tank Reactor (CSTR)** -![CSTR Diagram](./image.png) +![CSTR Diagram](../images/CSTR2.png) 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z2ToO{L#Xw^tID2BpQ-x;x+q?0+uu>~;}Bq6uf~mFyz*#o21(cC)iqHhuF=gDJ3c>ZRrgH;0b${cU;0>!j+*L@4|nwpHJE9yVG@3#k%zzc z=#hDpmZ;^W(t*Fe z=Y?jxP$&__1ViRTl-<6}j|3xM9v5ypBkGC>4c(MGc*>&+ y5m_lFf`QTS-kv|)sHt?&qTuBJc*JxNHWl#4=}Psq8SqyEguHH|^IGdp#J>Q-(u>Ui literal 0 HcmV?d00001 From eda1fdb56c89c659e1325ed5972ca5407dd794f9 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Tue, 18 Mar 2025 05:26:21 -0500 Subject: [PATCH 25/33] perfect --- calculus/index.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index c5e32cce3..b4ff00f61 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -8,7 +8,7 @@ $$ ______________________________________________________________________ -## **Example: Water Draining from a Bucket** +**Example: Water Draining from a Bucket** ![Water draining from a bucket](../images/cstr1.png) @@ -24,7 +24,7 @@ $$ ______________________________________________________________________ -## **Example: Continuous Stirred Tank Reactor (CSTR)** +**Example: Continuous Stirred Tank Reactor (CSTR)** ![CSTR Diagram](../images/CSTR2.png) @@ -42,7 +42,7 @@ $$ \frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} $$ -Since density ( \\rho ) is constant: +Since density is constant: $$ \frac{d(\rho V)}{dt} = \rho \frac{dV}{dt} = 0 @@ -59,7 +59,7 @@ ______________________________________________________________________ ### **Mole Balance** $$ -\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + g_{gen} - g_{cons} +\frac{d(n_A)}{dt} = \dot{n}_{A,in} - \dot{n}_{A,out} + r_{gen} - r_{cons} $$ Expanding: @@ -74,7 +74,7 @@ $$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -Since ( \\frac\{dV}\{dt} = 0 ): +Since \( \frac{dV}{dt} = 0 \): $$ V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V @@ -92,7 +92,7 @@ $$ (C_{A0} - C_A) \dot{V} = k C_A V $$ -Solving for ( C_A ): +Solving for \( C_A \): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} From 4b1e143d617224c86447097ec776dc9cafd37dc5 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Tue, 18 Mar 2025 10:27:20 +0000 Subject: [PATCH 26/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- calculus/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index b4ff00f61..10aff9341 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -74,7 +74,7 @@ $$ C_A \frac{dV}{dt} + V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V $$ -Since \( \frac{dV}{dt} = 0 \): +Since ( \\frac\{dV}\{dt} = 0 ): $$ V \frac{dC_A}{dt} = (C_{A0} - C_A) \dot{V} - k C_A V @@ -92,7 +92,7 @@ $$ (C_{A0} - C_A) \dot{V} = k C_A V $$ -Solving for \( C_A \): +Solving for ( C_A ): $$ C_A = \frac{C_{A0} \dot{V}}{\dot{V} + kV} From fd26f297acb46c75844e99facf01b6c0b44bf84d Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Tue, 18 Mar 2025 05:34:36 -0500 Subject: [PATCH 27/33] perfect working --- calculus/index.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/calculus/index.md b/calculus/index.md index 10aff9341..889169912 100644 --- a/calculus/index.md +++ b/calculus/index.md @@ -3,28 +3,28 @@ **Rate of Change** Accumulation = In – Out + Generation – Consumption $$ -\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} +\frac{dA}{dt} = \dot{m}_{in} - \dot{m}_{out} + r_{gen} - r_{cons} $$ ______________________________________________________________________ -**Example: Water Draining from a Bucket** +**Example 1: Water Draining from a Bucket** ![Water draining from a bucket](../images/cstr1.png) $$ -\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + g_{gen} - g_{cons} +\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + r_{gen} - r_{cons} $$ For **mass balance**: $$ -\frac{dm}{dt} = \dot{m}_{in} - 10 +\frac{dm}{dt} = \dot{m}_{in} - 16 $$ ______________________________________________________________________ -**Example: Continuous Stirred Tank Reactor (CSTR)** +**Example 2: Continuous Stirred Tank Reactor (CSTR)** ![CSTR Diagram](../images/CSTR2.png) From 0ce844c00df207ed455207500cc1d1baa03cc11f Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Wed, 26 Mar 2025 14:01:31 -0500 Subject: [PATCH 28/33] corrections --- _toc.yml | 1 + {calculus => conservation}/index.md | 0 conservation/unsteady_mass_balance.md | 21 ++++++++++++++ conservation/unsteady_mole_balance.md | 41 +++++++++++++++++++++++++++ 4 files changed, 63 insertions(+) rename {calculus => conservation}/index.md (100%) create mode 100644 conservation/unsteady_mass_balance.md create mode 100644 conservation/unsteady_mole_balance.md diff --git a/_toc.yml b/_toc.yml index b77ecb462..fddbe2dbb 100644 --- a/_toc.yml +++ b/_toc.yml @@ -3,6 +3,7 @@ root: index options: numbered: true chapters: + - file: conservation/index - file: calculus/index - file: multivariable-calculus/index - file: linear-algebra/index diff --git a/calculus/index.md b/conservation/index.md similarity index 100% rename from calculus/index.md rename to conservation/index.md diff --git a/conservation/unsteady_mass_balance.md b/conservation/unsteady_mass_balance.md new file mode 100644 index 000000000..e87a1a8e9 --- /dev/null +++ b/conservation/unsteady_mass_balance.md @@ -0,0 +1,21 @@ +### **Mass Balance** + +$$ +\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 +$$ + +______________________________________________________________________ +```{tableofcontents} +``` \ No newline at end of file diff --git a/conservation/unsteady_mole_balance.md b/conservation/unsteady_mole_balance.md new file mode 100644 index 000000000..a0336a8b5 --- /dev/null +++ b/conservation/unsteady_mole_balance.md @@ -0,0 +1,41 @@ +### **Mole Balance** + +$$ +\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} +$$ From 35acdce9c182c75a4ea17d283120efaa394b6fe2 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 26 Mar 2025 19:02:20 +0000 Subject: [PATCH 29/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- conservation/unsteady_mass_balance.md | 43 +++++++------- conservation/unsteady_mole_balance.md | 82 +++++++++++++-------------- 2 files changed, 63 insertions(+), 62 deletions(-) diff --git a/conservation/unsteady_mass_balance.md b/conservation/unsteady_mass_balance.md index e87a1a8e9..83d0859e2 100644 --- a/conservation/unsteady_mass_balance.md +++ b/conservation/unsteady_mass_balance.md @@ -1,21 +1,22 @@ -### **Mass Balance** - -$$ -\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 -$$ - -______________________________________________________________________ -```{tableofcontents} -``` \ No newline at end of file +### **Mass Balance** + +$$ +\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 +$$ + +______________________________________________________________________ + +```{tableofcontents} +``` diff --git a/conservation/unsteady_mole_balance.md b/conservation/unsteady_mole_balance.md index a0336a8b5..643cd1795 100644 --- a/conservation/unsteady_mole_balance.md +++ b/conservation/unsteady_mole_balance.md @@ -1,41 +1,41 @@ -### **Mole Balance** - -$$ -\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} -$$ +### **Mole Balance** + +$$ +\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} +$$ From e3ccef4782e6fa2527777ab6a4c24061e721a3b0 Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Wed, 26 Mar 2025 14:12:23 -0500 Subject: [PATCH 30/33] correct --- conservation/index.md | 81 -------------------------- conservation/unsteady_mass_balance.md | 44 +++++++------- conservation/unsteady_mole_balance.md | 82 +++++++++++++-------------- 3 files changed, 63 insertions(+), 144 deletions(-) diff --git a/conservation/index.md b/conservation/index.md index 889169912..9698dfe6e 100644 --- a/conservation/index.md +++ b/conservation/index.md @@ -16,84 +16,3 @@ $$ \frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + r_{gen} - r_{cons} $$ -For **mass balance**: - -$$ -\frac{dm}{dt} = \dot{m}_{in} - 16 -$$ - -______________________________________________________________________ - -**Example 2: Continuous Stirred Tank Reactor (CSTR)** - -![CSTR Diagram](../images/CSTR2.png) - -**Reaction:** \$$ -r_A = -k C_A V -$\$ - -**Assume mass density is constant.** - -______________________________________________________________________ - -### **Mass Balance** - -$$ -\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 -$$ - -______________________________________________________________________ - -### **Mole Balance** - -$$ -\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} -$$ diff --git a/conservation/unsteady_mass_balance.md b/conservation/unsteady_mass_balance.md index 83d0859e2..28794d3e1 100644 --- a/conservation/unsteady_mass_balance.md +++ b/conservation/unsteady_mass_balance.md @@ -1,22 +1,22 @@ -### **Mass Balance** - -$$ -\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 -$$ - -______________________________________________________________________ - -```{tableofcontents} -``` +# **Mass Balance** + +$$ +\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 +$$ + +______________________________________________________________________ + +```{tableofcontents} +``` diff --git a/conservation/unsteady_mole_balance.md b/conservation/unsteady_mole_balance.md index 643cd1795..1e99d63b5 100644 --- a/conservation/unsteady_mole_balance.md +++ b/conservation/unsteady_mole_balance.md @@ -1,41 +1,41 @@ -### **Mole Balance** - -$$ -\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} -$$ +# **Mole Balance** + +$$ +\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} +$$ From 025b18771e3d2afe975ed88adab21c8fba31c2ad Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 26 Mar 2025 19:12:31 +0000 Subject: [PATCH 31/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- conservation/index.md | 1 - conservation/unsteady_mass_balance.md | 44 +++++++------- conservation/unsteady_mole_balance.md | 82 +++++++++++++-------------- 3 files changed, 63 insertions(+), 64 deletions(-) diff --git a/conservation/index.md b/conservation/index.md index 9698dfe6e..adcc79091 100644 --- a/conservation/index.md +++ b/conservation/index.md @@ -15,4 +15,3 @@ ______________________________________________________________________ $$ \frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + r_{gen} - r_{cons} $$ - diff --git a/conservation/unsteady_mass_balance.md b/conservation/unsteady_mass_balance.md index 28794d3e1..c0a4f60d8 100644 --- a/conservation/unsteady_mass_balance.md +++ b/conservation/unsteady_mass_balance.md @@ -1,22 +1,22 @@ -# **Mass Balance** - -$$ -\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 -$$ - -______________________________________________________________________ - -```{tableofcontents} -``` +# **Mass Balance** + +$$ +\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 +$$ + +______________________________________________________________________ + +```{tableofcontents} +``` diff --git a/conservation/unsteady_mole_balance.md b/conservation/unsteady_mole_balance.md index 1e99d63b5..196f01b51 100644 --- a/conservation/unsteady_mole_balance.md +++ b/conservation/unsteady_mole_balance.md @@ -1,41 +1,41 @@ -# **Mole Balance** - -$$ -\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} -$$ +# **Mole Balance** + +$$ +\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} +$$ From 2833207bf570ddb4d567e87158b0ed5730453f6a Mon Sep 17 00:00:00 2001 From: josholaniyan <72916041+josholaniyan@users.noreply.github.com> Date: Wed, 26 Mar 2025 14:25:45 -0500 Subject: [PATCH 32/33] c --- conservation/index.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/conservation/index.md b/conservation/index.md index adcc79091..5066722eb 100644 --- a/conservation/index.md +++ b/conservation/index.md @@ -15,3 +15,6 @@ ______________________________________________________________________ $$ \frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + r_{gen} - r_{cons} $$ + +```{tableofcontents} +``` \ No newline at end of file From e1ff1b57fda1787cf3202544dd11a7a0791c6daa Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 26 Mar 2025 19:26:38 +0000 Subject: [PATCH 33/33] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- conservation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/conservation/index.md b/conservation/index.md index 5066722eb..8912d7a6a 100644 --- a/conservation/index.md +++ b/conservation/index.md @@ -17,4 +17,4 @@ $$ $$ ```{tableofcontents} -``` \ No newline at end of file +```