CHEN90032 · Process Simulation and Control
Dynamic Process Models
Dynamic Process Models is the Week-2 core of University of Melbourne CHEN90032 Process Simulation and Control: it turns an unsteady-state mass or energy balance on a process (a tank, reactor or heat exchanger) into the differential equation that describes how it moves in time. You recast that ODE in deviation variables so it has zero initial conditions and Laplace-transforms into a transfer function, and you learn to tell a linear first-order model (constant time constant τ and gain K) from a nonlinear one such as the triangular-trough level, where a state-dependent coefficient means no single transfer function exists. Everything later in the subject — transfer functions, stability, tuning — is built on the models you write here.
What this chapter covers
- 01The conservation law accumulation = in − out (+ generation) for mass, a component and energy
- 02Writing the stirred-tank energy balance ρV Cp dT/dt = w Cp(Ti − T) + Q with correct units
- 03The five-step recipe: balance → relations → steady state → deviations → Laplace
- 04Deviation variables y′ = y − y̅ and why they give zero initial conditions
- 05Collapsing a balance to first-order form G(s) = K/(τs + 1); identifying τ = ρV/w and the gain K
- 06The gravity-drain tank as a second linear first-order example (K = R, τ = A R)
- 07Recognising nonlinearity: products, powers and geometry that break the constant-coefficient rule
- 08Deriving the triangular-trough level model dh/dt = (qi − qo)/(l h) via the chain rule
- 09Why the trough is not a transfer function, and linearising about an operating level h̅
- 10Discussing the qualitative response and the sensitivity of level to flow at low vs high level
Derive the triangular-trough level model and evaluate its rate
- +2Write the unsteady-state total-volume balance (constant density): accumulation = in − out, so dV/dt = qi − qo.
- +2Close it with the geometry using the chain rule: V = ½ l h² gives dV/dt = l h (dh/dt). Equate to the balance to get dh/dt = (qi − qo)/(l h).
- +2Net inflow: qi − qo = 0.030 − 0.024 = +0.006 m³/s (positive, so the level rises). At h = 0.5 m: dh/dt = 0.006 / (6.0 × 0.5) = 0.006 / 3.0 = 2.0 × 10⁻³ m/s = 2.0 mm/s.
- +1At h = 2.0 m: dh/dt = 0.006 / (6.0 × 2.0) = 0.006 / 12.0 = 5.0 × 10⁻⁴ m/s = 0.5 mm/s. Same net inflow, but the level climbs 4× faster when nearly empty — the rate depends on the state h.
Key terms
- Unsteady-state (dynamic) balance
- A conservation balance in which the accumulation term is not zero: rate of accumulation = rate in − rate out (+ rate generated). It produces an ODE in time rather than an algebraic steady-state equation.
- Deviation variable
- The departure of a quantity from its steady-state value, y′(t) = y(t) − y̅. Subtracting the steady-state equation cancels the constant terms and gives zero initial conditions, which is what a transfer function requires.
- Time constant τ
- The coefficient in first-order form τ(dy′/dt) + y′ = K u′, with units of time; it sets how fast the process responds. For a well-mixed tank τ = ρV/w, the residence time. A step reaches 63% of its final change at t = τ and 95% at 3τ.
- Steady-state gain K
- The ratio of the eventual output change to a sustained input change, K = Δy_ss/Δu_ss, carrying the units [output]/[input]. Its sign tells the direction of the response; for the heater input K_Q = 1/(w Cp).
- Transfer function G(s)
- The Laplace-domain ratio of output deviation to input deviation, G(s) = Y′(s)/U′(s). A linear first-order process has G(s) = K/(τs + 1); it acts on deviations only and carries no absolute-value information.
- Linear vs nonlinear model
- A model is linear when constant coefficients multiply the states and their derivatives, so it Laplace-transforms into a fixed transfer function. Products, powers, or a geometry like V = ½ l h² make a coefficient depend on the state — the model is then nonlinear.
- Triangular-trough level model
- The level equation for a V-shaped vessel, dh/dt = (qi − qo)/(l h). The h in the denominator makes it nonlinear: the level is very sensitive to a flow imbalance when nearly empty and insensitive when nearly full.
- Linearisation about an operating point
- Approximating a nonlinear model near a chosen steady state (e.g. replacing l h by the constant l h̅) to obtain a local first-order model valid only for small excursions around that point.
Dynamic Process Models FAQ
Why do we shift to deviation variables before finding a transfer function?
Because a transfer function needs zero initial conditions. Subtracting the steady-state equation from the dynamic ODE cancels every constant term and leaves an equation in y′ = y − y̅ that starts from zero, so the Laplace derivative rule ℒ{dy′/dt} = s Y′(s) applies cleanly. Remember to add the steady state back, T(t) = T̅ + T′(t), when you report a real value.
Why is the triangular-trough level model nonlinear when the stirred-tank one is linear?
In the stirred tank every term is proportional to a state to the first power (w·T, Q), so τ and K are constants and the model is a transfer function. In the trough the volume is V = ½ l h², so the balance becomes dh/dt = (qi − qo)/(l h): the coefficient 1/(l h) depends on the level h. A state-dependent coefficient is the signature of a nonlinear model, so no single time constant or gain describes it.
Can AI help me with dynamic process models in CHEN90032?
Yes — used as a study aid. Sia can walk you through the modelling recipe step by step: how to write an unsteady-state balance, apply the chain rule to the trough geometry, move to deviation variables, and check units on τ and K. Treat it as a tutor that explains method and checks your reasoning on practice problems; it does not sit your assessment or hand you final answers, and building the derivations yourself is what earns the marks and passes the exam hurdle.
Studying with AI? Sia — free AI chemical engineering tutor works through CHEN90032 step by step.
Exam move
Drill the five-step recipe until it is automatic: balance, closing relations, steady state, deviation variables, Laplace. Practise both archetypes side by side — the linear stirred tank (get τ = ρV/w and G = K/(τs + 1) with units) and the nonlinear triangular trough (chain rule to dh/dt = (qi − qo)/(l h), then explain the state-dependent sensitivity). At about 1.8 minutes per mark on the final, a 15-mark model-derivation question is roughly 27 minutes, so spend the first few writing the balance and stating assumptions rather than rushing to a transfer function. Every practice attempt: finish with a unit check on τ and K, and a one-line comment on whether the model is linear or nonlinear.
