Study notes

Qwestrum Engineering360 · Chemical Engineering · Process Dynamics & Control

Dynamic Modelling of Processes

Dynamic modelling turns unsteady mole and energy balances into differential equations, then linearises them about the steady state and Laplace-transforms into transfer functions G(s) that relate output to input deviations.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Linearise nonlinear balances about the steady state for small deviations
Dead time θ appears as an e^(−θs) factor
Holdup provides capacitance — it integrates any flow imbalance

Topic details

Introduction

This Coughanowr topic is the foundation of process control. You write the transient balance for a tank, reactor or exchanger, express variables as deviations from steady state, linearise any nonlinear terms, and convert to a transfer function so the process can be analysed and a controller designed in the frequency domain.

Key relations & formulas

C \frac{dT}{dt} = Q_{in} - Q_{out} + Q_{gen}

(lumped energy balance)

\tau \frac{dx}{dt} + x = K u

(linear first-order deviation form)

G(s) = Y\frac{(s)}{U}(s)

(transfer function, Laplace domain)

Notation and sign conventions

Relation 1 —

C \frac{dT}{dt} = Q_{in} - Q_{out} + Q_{gen}

C \frac{dT}{dt} = Q_{in} - Q_{out} + Q_{gen}

(lumped energy balance)

Write this relation with symbols exactly as in Process Systems Analysis & Control — Coughanowr & LeBlanc before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

\tau \frac{dx}{dt} + x = K u

\tau \frac{dx}{dt} + x = K u

(linear first-order deviation form)

Relation 3 —

G

G(s) = Y\frac{(s)}{U}(s)

(transfer function, Laplace domain)

Concept in depth

Control design needs a model of how the output responds to input changes over time. Starting from a dynamic balance, the accumulation term (capacitance times rate of change) captures the process’s natural sluggishness — a large holdup filters disturbances slowly. Because most balances are nonlinear, they are linearised with a Taylor expansion about the operating point, valid for the small excursions a controller must handle. The Laplace transform then replaces calculus with algebra, giving a transfer function whose gain sets the steady-state sensitivity, whose time constant sets the speed, and whose dead-time term captures transport delay that is especially destabilising.

Assumptions and validity limits

State assumptions explicitly before using any relation for dynamic modelling of processes — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Process Dynamics & Control viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Process Dynamics & Control papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to dynamic modelling of processes.
4. Use equation 1:

C \frac{dT}{dt} = Q_{in} - Q_{out} + Q_{gen}

.
5. Use equation 2:

\tau \frac{dx}{dt} + x = K u

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Dynamic Modelling of Processes appears in DCS and plant automation. In Indian chemical curricula this topic is tested because it connects theory to dynamic models and loop tuning.

GATE and semester exams often combine dynamic modelling of processes with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use dynamic modelling of processes?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Students skip the deviation-variable step and carry absolute values into the transfer function, linearise incorrectly (wrong partial derivatives), and omit dead time even when transport delay is obvious. Confusing the time constant with the dead time is a common conceptual slip.

Quick revision checklist

Before attempting dynamic modelling of processes problems, confirm you can:

1. Linearise nonlinear balances about the steady state for small deviations
2. Dead time θ appears as an e^(−θs) factor
3. Holdup provides capacitance — it integrates any flow imbalance

Revise the solved examples in Process Systems Analysis & Control — Coughanowr & LeBlanc and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Tank time constant

Problem

A stirred tank of volume 2 m³ has an outlet flow of 0.5 m³/min proportional to level. Estimate its first-order time constant.

Solution

τ = holdup/throughput = V/Q = 2/0.5 = 4 min. The output reaches 63.2% of a step change in 4 minutes.

Conceptual check — Dynamic Modelling of Processes

Problem

In a Process Dynamics & Control semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of dynamic modelling of processes." What should a complete answer include?

Exams & GATE

Coughanowr Ch. 3 — derive the model from a mole/energy balance, then linearise.

📖 Standard books (India)

Process Systems Analysis & Control — Coughanowr & LeBlanc
Read: Syllabus unit
Dynamic modelling and control loops

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📚Key formulas & points