Non-Isothermal Reactors

What defines a non-isothermal reactor mathematically?

$$T = T(z,t) \neq \text{constant}$$

Why does temperature strongly affect reaction rate?

$$k = A e^{-E_a/(RT)}$$

Why do small temperature changes cause large rate changes?

$$\frac{dk}{dT} \propto \frac{E_a}{RT^2}$$

What additional balance is required in non-isothermal design?

$$\text{Mole balance} + \text{Energy balance}$$

Why are mole and energy balances coupled?

$-rA = f(T), \quad T = f(-rA)$

What balances are required for isothermal reactor design?

$$\text{Mole balance only}$$

What balances are required for non-isothermal reactor design?

$$\text{Mole balance} + \text{Energy balance}$$

What does a general energy balance account for?

Energy in, energy out, heat of reaction, and heat transfer

Why does reaction enthalpy not appear as a separate term in the energy balance?

$$\Delta H_r \text{ is embedded in composition changes}$$

What defines an adiabatic reactor?

$$Q = 0$$

How does temperature vary in an adiabatic exothermic reactor?

\Delta H < 0 \Rightarrow T \uparrow \text{ as } X_A \uparrow

How does temperature vary in an adiabatic endothermic reactor?

\Delta H > 0 \Rightarrow T \downarrow \text{ as } X_A \uparrow

What is the key coupling in adiabatic reactors?

$$T = T(X_A)$$

Why can temperature and conversion not be chosen independently in adiabatic reactors?

$$Q = 0 \Rightarrow \text{single } T\text{–}X_A \text{ trajectory}$$

How does equilibrium behave in an adiabatic exothermic reactor?

$$T \uparrow \Rightarrow K \downarrow \Rightarrow X_{eq} \downarrow$$

How does equilibrium behave in an adiabatic endothermic reactor?

$T \downarrow \Rightarrow -rA \downarrow \Rightarrow XA \text{ stalls}$

Why can long reactors fail to increase conversion in non-isothermal systems?

$XA \to X{eq} \Rightarrow -r_A \to 0$

What is a common design strategy to overcome equilibrium limitations?

Multiple reactors with interstage cooling or heating

What heat-transfer term appears in a non-isothermal CSTR energy balance?

$$Q = UA(T_a - T)$$

$U$ is overall heat-transfer coefficient
$A$ is heat-transfer area
$T_a$ is jacket temperature
$T$ is reactor temperature

Why is heat exchange used in non-isothermal reactors?

$$T \text{ control} \Rightarrow X_A \uparrow \text{ and safety}$$

How does temperature vary in a PFR with heat exchange?

$$T = T(V)$$

How many equations must be solved for a non-isothermal PFR?

$$\frac{dX_A}{dV}, \quad \frac{dT}{dV}$$

Why must non-isothermal PFR equations be solved simultaneously?

$-rA(XA,T) \;\text{and}\; T(X_A)$

What is the key advantage of counter-current heat exchange?

$$\Delta T_{\text{driving}} \approx \text{uniform}$$

Which heat-exchange configuration is usually more efficient?

Counter-current

Why do multiple steady states occur in non-isothermal CSTRs?

$Q{\text{gen}}(T) \text{ intersects } Q{\text{rem}}(T)$

What causes multiple intersections in heat generation and removal curves?

$Q{\text{gen}} \propto e^{-Ea/(RT)}$

How many steady states can a non-isothermal CSTR exhibit?

$$1,\;2,\;\text{or }3$$

Which steady state is thermally unstable?

The intermediate temperature steady state

What defines thermal runaway?

Q{\text{gen}} > Q{\text{rem}} \Rightarrow T \uparrow\uparrow

Why are high-temperature steady states dangerous?

$T \uparrow \Rightarrow -rA \uparrow \Rightarrow Q{\text{gen}} \uparrow$

What disturbances can trigger runaway?

Changes in feed temperature, flow rate, or cooling rate

What must always be included in non-isothermal reactor design?

$$\text{Mole balance} + \text{Energy balance} + Q \text{ terms}$$

Why are numerical methods often required?

$$\text{Coupled nonlinear equations}$$

What three aspects are most affected by temperature in reactors?

$$\text{Rate},\; \text{Equilibrium},\; \text{Safety}$$

Why do ideal isothermal models over-predict performance?

$$T = \text{constant assumed} \Rightarrow -r_A \text{ overestimated}$$

What is the central danger of non-isothermal operation?

$$\text{Multiple steady states} \Rightarrow \text{runaway risk}$$

One-sentence exam answer for non-isothermal reactors

In non-isothermal reactors, temperature variations strongly affect reaction rate, equilibrium conversion, and safety, requiring coupled mole and energy balances for accurate reactor design