Isothermal Reactor Design

What does isothermal operation imply mathematically?

$$T = \text{constant} \;\;\Rightarrow\;\; k = \text{constant}$$

$T$ is temperature
$k$ is the rate constant

Why does reaction rate still change under isothermal conditions?

$-rA = f(CA), \quad CA = CA(X_A)$

What is the general logic of isothermal reactor design?

$\text{Mole balance} + \text{rate law} + CA(XA) \Rightarrow \text{design equation}$

What variables are solved for in isothermal reactor design?

Batch: time
CSTR / PFR: volume

What assumptions define an isothermal batch reactor?

$F{i,\text{in}} = F{i,\text{out}} = 0$

What is the batch reactor design equation in concentration form?

$t = \int{C{A,0}}^{CA} \frac{dCA}{-r_A}$

$t$ is reaction time
$C_{A,0}$ is initial concentration

What is the batch reactor design equation in conversion form?

$t = C{A,0}\int0^{XA} \frac{1}{-rA}\, dX_A$

What variables determine batch reaction time?

$t = f(-rA, XA)$

What key assumption defines an isothermal CSTR?

$C{A,\text{reactor}} = C{A,\text{outlet}}$

What is the isothermal CSTR design equation?

$V = \frac{F{A,0} XA}{-rA|{\text{outlet}}}$

$V$ is reactor volume
$F_{A,0}$ is inlet molar flow of A

Why is the CSTR design equation algebraic?

$$-r_A = \text{constant throughout reactor}$$

Where is the reaction rate evaluated in a CSTR?

$-rA = -rA|_{\text{outlet}}$

What key assumption defines an isothermal PFR?

$CA = CA(z), \quad \text{no axial mixing}$

What is the isothermal PFR design equation?

$V = \int0^{XA} \frac{F{A,0}}{-rA}\, dX_A$

Why is PFR design an integral equation?

$-rA = -rA(X_A)$

How does reaction rate vary inside a PFR?

-rA|{XA \approx 0} > -rA|{XA \to X_{A,\text{out}}}

Why does a PFR require less volume than a CSTR for the same conversion?

\langle -rA \rangle{\text{PFR}} > -rA|{\text{CSTR}}

What quantity is plotted in a Levenspiel plot?

$\frac{1}{-rA} \;\text{vs}\; XA$

What does the area under a Levenspiel plot represent for a batch reactor?

$t = C{A,0}\int0^{XA} \frac{1}{-rA}\, dX_A$

What does the area under a Levenspiel plot represent for a PFR?

$V = F{A,0}\int0^{XA} \frac{1}{-rA}\, dX_A$

How is a CSTR represented on a Levenspiel plot?

$V = F{A,0} XA \left(\frac{1}{-rA}\right){\text{exit}}$

Why do Levenspiel plots show PFR superiority visually?

\text{Area}{\text{PFR}} < \text{Area}{\text{CSTR}}

What is the definition of the Damköhler number?

$$\text{Da} = \frac{\text{reaction rate}}{\text{flow rate}}$$

What is Da for a first-order reaction?

$\text{Da} = k\tau$

$\tau = \frac{V}{\dot V}$ is space time

What is the CSTR conversion for a first-order reaction in terms of Da?

$$X_A = \frac{\text{Da}}{1 + \text{Da}}$$

What does a small Damköhler number imply?

$$\text{Da} \ll 1 \Rightarrow X_A \approx 0$$

What does a large Damköhler number imply?

$$\text{Da} \gg 1 \Rightarrow X_A \to 1$$

Why is the Damköhler number useful?

Quick estimation of conversion and reaction vs flow limitation

What effects are excluded from isothermal reactor design?

Temperature change, heat transfer, equilibrium shifts due to temperature

Why is isothermal design considered foundational?

$$\text{Non-isothermal design} \supset \text{isothermal framework}$$

One-sentence exam definition of isothermal reactor design

Isothermal reactor design combines mole balances and kinetics at constant temperature to determine the time or reactor volume required to achieve a specified conversion