One source of transfer functions is from Balance Equations that relate inputs and outputs. Transfer functions are compact representations of dynamic systems and the differential equations become algebraic expressions that can be manipulated or combined with other expressions. The first step in creating a transfer function is to convert each term of a differential equation with a Laplace transform as shown in the table of Laplace transforms. A transfer function, G(s), relates an input, U(s), to an output, Y(s).

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

Do the following, assuming that q and V are constant:

1. Transform each equation in to the Laplace domain.
2. Find the transfer function between the specified variables.
3. Define the time constant and the gain for each transfer function in terms of the parameters given.

The following equations are linearized and variables are in deviation form with `y'(t) = y(t) - y_{0}`.

Exercise 1

Find `{C(s)}/{C_i(s)}`:

$$\frac{dc'(t)}{dt} = \frac{q}{V} c_i'(t) - \frac{q}{V} c'(t) $$

Exercise 2

Find `{C(s)}/{C_i(s)}`:

$$\frac{dc'(t)}{dt} = \frac{q}{V} c_i'(t) - \left(\frac{q}{V} + 2 k_2 \bar c\right) c'(t) $$

Exercise 3

The inlet temperature `T_i(t)` is a process disturbance and the heater `Q(s)` is adjusted to maintain the outlet temperature `T(t)` at a specified set point. Start with the following energy balance and find `{T(s)}/{T_i(s)}` and `{T(s)}/{Q(s)}`.

$$\frac{dT'(t)}{dt} = \frac{q}{V} \left(T_i'(t)-T'(t)\right) + \frac{Q'(t)}{\rho\,V\,C_p} $$

Determine if the two transfer functions are combined in parallel (additive) or series (multiplicative) and input the transfer functions in the appropriate boxes.

Transfer Functions in Parallel

Transfer Functions in Series

Show how the two transfer functions `{T(s)}/{T_i(s)}` and `{T(s)}/{Q(s)}` would fit into an overall control block diagram and label each block and the signal connections between each block. Cross off any blocks not needed for this problem.

What to Turn In

• Exercise 1: Laplace transform of the mass balance equation and derivation of the transfer function `(C(s))/(C_i(s))`, including gain and time constant.
• Exercise 2: Laplace transform and derivation of the modified concentration transfer function `(C(s))/(C_i(s))` with reaction term, including expressions for gain and time constant.
• Exercise 3: Laplace transform of the energy balance equation to find `(T(s))/(T_i(s))` and `(T(s))/(Q(s))`. Determine if they are combined in parallel or series and illustrate in a labeled block diagram.

Generative AI Learning

Use these prompts to test your understanding after completing the exercise. Direct the AI - do the block algebra yourself first.

Tip: Every reduced transfer function has instant checks: set s=0 for steady-state gain, count the denominator order against the number of first-order elements, and confirm limits as controller gain goes to 0 or infinity. Run all three before trusting any algebra - yours or the AI's.

What to Turn In

Submit a short report (PDF, 1-2 pages) that curates your results into a demonstration of what you learned. You may use Generative AI to help write the report, but you must supply the correct algebra, justifications, and checks. Answer these questions:

1. Present the block diagram reduction for the assigned problem(s) step by step to the final transfer function.
2. Show your three sanity checks (DC gain, order, Kc limits) on the final result.
3. Derive both the setpoint-to-output and disturbance-to-output transfer functions for the same loop: what do they share and what differs?
4. From the quiz prompt: one question you missed and the corrected answer.