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