Control of distillation columns, heat exchangers and reactors

Why it matters

Distillation columns, heat exchangers and reactors are where control theory meets real equipment. Choosing which variable to control with which valve (the pairing), when to add cascade or feedforward, and how to keep an exothermic reactor from running away are everyday decisions for a process engineer and common interview and GATE topics.

Key ideas

Degrees of freedom and pairing. Each independent manipulated variable (usually a valve) can control one variable. Inventory variables (levels, pressures) are controlled first because they must be held for the unit to operate at all; quality variables (compositions, temperatures) use what remains. With several loops interacting, the steady-state relative gain array (RGA) guides pairing: for a 2 × 2 system with gains K_ij (output i, input j), λ₁₁ = 1/(1 − K₁₂K₂₁/(K₁₁K₂₂)). Pair on elements of the RGA close to 1; avoid negative values; λ₁₁ much greater than 1 means strong interaction.

Distillation column. A simple column has five manipulated flows: distillate D, bottoms B, reflux L, boil-up V (reboiler steam) and condenser duty (coolant).

• Pressure: usually condenser coolant or vapour bypass, or a vent for non-condensables. Pressure must be steady because tray temperatures infer composition only at constant pressure.
• Reflux-drum level: by D (with L used for composition: the LV configuration) or by L.
• Base level: by B (or by V).
• Compositions: often inferred from a sensitive tray temperature; top composition by L (or D), bottom by V (steam).
• Feed-rate changes are handled by feedforward or ratio schemes (L/F, V/F), and reflux ratio L/D affects purity and energy.
• Controlling both compositions (dual composition control) gives strong interaction, measured by the RGA; often one composition is controlled and the other left floating or controlled by a ratio.

Heat exchangers.

• Outlet temperature is controlled by the utility (steam or coolant) flow, often in cascade with a utility-flow or steam-pressure controller to reject supply disturbances.
• For a steam heater the valve on the steam line changes the shell pressure and so the condensing temperature; a valve on the condensate line instead changes the flooded area (slower).
• A bypass of the process stream gives very fast temperature control and lets the exchanger run at full utility flow.
• Feedforward from process-flow changes is common, because process flow is the main load disturbance.
• Gains vary with flow (the process is non-linear), which favours equal-percentage valves.

Reactors.

• Temperature is the key variable. For a jacketed CSTR, reactor temperature is cascaded to jacket temperature or coolant flow.
• Exothermic CSTR stability: heat generated rises exponentially with temperature (Arrhenius) while heat removed rises linearly. An operating point is open-loop stable only if the slope of the heat-removal line exceeds the slope of the heat-generation curve; the middle of three steady states is open-loop unstable but can be held by feedback control.
• Coolant valves fail open; split-range control handles heating at start-up and cooling during reaction.
• Feed rates are ratioed; pressure and level loops hold inventory; composition is often inferred or measured with an analyser (adding dead time).
• Batch reactors need set-point programming and anti-windup because of large set-point changes.

Formulas

Reflux ratio R = L/D

• L reflux flow and D distillate flow (mol/s or kg/h).

λ₁₁ = 1/(1 − K₁₂·K₂₁/(K₁₁·K₂₂)), λ₁₂ = λ₂₁ = 1 − λ₁₁, λ₂₂ = λ₁₁

• RGA of a 2 × 2 system; K_ij steady-state gain of output i to input j.

m_s·λ = m·C_p·(T_out − T_in)

• Steady-state energy balance of a steam heater; m_s steam flow (kg/s), λ latent heat (J/kg), m process flow (kg/s), C_p (J/(kg·K)), T in K or °C.

K_s = ∂T_out/∂m_s = λ/(m·C_p), K_m = ∂T_out/∂m = −(T_out − T_in)/m

• Process gains to steam flow and to process flow.

dQ_rem/dT > dQ_gen/dT

• Open-loop stability condition for an exothermic CSTR steady state.

Worked examples

Example 1 (standard): gains of a steam heater. Water (C_p = 4.18 kJ/(kg·K)) at 5 kg/s is heated from 30 °C to 80 °C by condensing steam with latent heat λ = 2100 kJ/kg (take from steam tables in practice; given here). Find the steam flow, the gain of outlet temperature to steam flow, the outlet-temperature change for a +0.05 kg/s steam step, and the gain to process flow.

1. Duty: m·C_p·ΔT = 5·4.18·50 = 1045 kW, so m_s = 1045/2100 = 0.498 kg/s.
2. K_s = λ/(m·C_p) = 2100/(5·4.18) = 100.5 K per kg/s.
3. ΔT_out = 100.5·0.05 = 5.02 K.
4. K_m = −(80 − 30)/5 = −10 K per kg/s.

m_s ≈ 0.498 kg/s; K_s ≈ 100.5 K/(kg/s); ΔT_out ≈ +5.0 K; K_m = −10 K/(kg/s). The process-flow gain is what a feedforward controller must cancel: G_f = −K_m/K_s = 10/100.5 ≈ 0.0995 kg/s of steam per kg/s of water.

Example 2 (GATE level): pairing in dual composition control. A column's steady-state gains (LV configuration) are: x_D responds to L with 0.8 and to V with −0.6; x_B responds to L with 0.9 and to V with −1.2 (consistent mole-fraction units per unit flow). Find the RGA and recommend the pairing.

1. K₁₁ = 0.8, K₁₂ = −0.6, K₂₁ = 0.9, K₂₂ = −1.2.
2. K₁₂K₂₁/(K₁₁K₂₂) = (−0.6·0.9)/(0.8·(−1.2)) = (−0.54)/(−0.96) = 0.5625.
3. λ₁₁ = 1/(1 − 0.5625) = 2.286; λ₁₂ = 1 − 2.286 = −1.286.
4. RGA = [[2.286, −1.286], [−1.286, 2.286]].

Pair x_D–L and x_B–V (λ = 2.29); the off-diagonal pairing would be negative and must be avoided. A λ well above 1 warns of strong interaction, so the loops need detuning, decoupling or ratio schemes.

Common mistakes

• Defining reflux ratio as L/F or L/V instead of L/D.
• Controlling tray temperature as a composition proxy while letting column pressure float.
• Pairing on the largest gain instead of on the RGA.
• Putting a fail-closed valve on reactor cooling water.
• Assuming an exothermic CSTR that is stable at one temperature is stable at all operating points.
• Ignoring the change of exchanger gain with throughput when tuning.

For GATE CH

Questions here are mainly conceptual (which variable to manipulate, valve fail positions, cascade and feedforward applied to columns and exchangers, reactor runaway) with short numericals: RGA for a 2 × 2 system, reflux ratio, steady-state gains from energy balances and feedforward gains. Practise building the RGA and reading pairing rules from it.

Quick check

1. Reflux is 450 kmol/h and distillate 150 kmol/h. What is the reflux ratio?
2. For K = [[1, 1.5], [2, 4]], what is λ₁₁?
3. What usually manipulates column pressure in a total-condenser column?
4. Why is reactor temperature often cascaded to jacket temperature?

Answers: 1. 3. 2. 1/(1 − 3/4) = 4. 3. Condenser coolant flow (condenser duty). 4. The jacket loop rejects coolant-supply disturbances quickly and makes the outer loop faster and more robust.