Transfer functions are derived by linearizing the stirred-tank heater’s transient energy balance around a steady operating point. For a perfectly mixed tank with constant liquid mass, inlet temperature (T_i), steam temperature (T_{st}), flow rate (F), heat-transfer area (A), and overall heat-transfer coefficient (U), the resulting model is first order. It produces separate transfer functions from inlet-fluid temperature and steam temperature to tank temperature.

The tank temperature responds to both disturbances through the same thermal storage and heat-removal dynamics. The inlet-temperature path is governed by fluid replacement, while the steam-temperature path is governed by the coil heat-transfer capacity (UA).

Start with the Unsteady Energy Balance

Define the Tank and Flow Assumptions

Assume the tank is perfectly agitated, so the outlet temperature equals the tank temperature (T). Also assume constant liquid volume, density, heat capacity, flow rate, and heat-transfer properties.

The tank contains liquid mass (M), and the process fluid has heat capacity (C_p). Heat enters through the steam coil according to

[ Q = UA(T_{st}-T) ]

where (U) is the overall heat-transfer coefficient and (A) is the effective heat-transfer area.

Write the Accumulation Term

The thermal energy stored in the tank changes at the rate

[ MC_p\frac{dT}{dt} ]

This term represents the dynamic behavior: a larger liquid mass or heat capacity requires more energy to change the tank temperature.

Write the Flow and Heating Terms

The net energy entering through the process stream is

[ FC_p(T_i-T) ]

The first part, (FC_pT_i), represents energy entering with the feed. The second part, (FC_pT), represents energy leaving with the well-mixed outlet.

Adding the steam-coil duty gives the complete balance:

[ MC_p\frac{dT}{dt}

FC_p(T_i-T)+UA(T_{st}-T) ]

Rearrange the Dynamic Model

Expand the Heat-Transfer Terms

Expanding the right-hand side gives

[ MC_p\frac{dT}{dt}

FC_pT_i-FC_pT+UAT_{st}-UAT ]

Collecting the tank-temperature terms produces

[ MC_p\frac{dT}{dt}+(FC_p+UA)T

FC_pT_i+UAT_{st} ]

Normalize the Equation

Dividing by (MC_p) gives

[ \frac{dT}{dt} + \left( \frac{F}{M}+\frac{UA}{MC_p} \right)T

\frac{F}{M}T_i + \frac{UA}{MC_p}T_{st} ]

Define

[ a=\frac{F}{M}+\frac{UA}{MC_p} ]

[ b_i=\frac{F}{M} ]

[ b_{st}=\frac{UA}{MC_p} ]

The model becomes

[ \frac{dT}{dt}+aT=b_iT_i+b_{st}T_{st} ]

The coefficient (a) determines the rate at which the tank approaches a new equilibrium. Its reciprocal,

[ \tau=\frac{1}{a}

\frac{MC_p}{FC_p+UA} ]

is the effective thermal time constant.

Convert to Deviation Variables

Define the Steady Operating Point

Let the initial steady-state values be (T_s), (T_{i,s}), and (T_{st,s}). At steady state,

[ \frac{dT}{dt}=0 ]

so the steady-state balance is

[ aT_s=b_iT_{i,s}+b_{st}T_{st,s} ]

This operating point is the reference used to study disturbances.

Define Temperature Deviations

Define deviation variables as

[ T'=T-T_s ]

[ T_i'=T_i-T_{i,s} ]

[ T_{st}'=T_{st}-T_{st,s} ]

Because the system parameters are assumed constant, subtracting the steady-state equation from the dynamic equation gives

[ \frac{dT'}{dt}+aT'

b_iT_i'+b_{st}T_{st}' ]

This step removes the constant operating-point terms and leaves only the response to changes in the two inputs.

Apply the Laplace Transform

Transform the Differential Equation

Assume zero initial deviation, meaning the tank begins at its steady-state temperature. Taking the Laplace transform gives

[ sT'(s)+aT'(s)

b_iT_i'(s)+b_{st}T_{st}'(s) ]

Factoring the tank-temperature transform yields

[ (s+a)T'(s)

b_iT_i'(s)+b_{st}T_{st}'(s) ]

Therefore,

[ T'(s)

\frac{b_i}{s+a}T_i'(s) + \frac{b_{st}}{s+a}T_{st}'(s) ]

Identify the Two Transfer Functions

The transfer function from inlet-fluid temperature to tank temperature is

[ G_i(s)

\frac{T'(s)}{T_i'(s)}

\frac{b_i}{s+a}

\frac{F/M}{s+F/M+UA/(MC_p)} ]

with the steam-temperature deviation held at zero.

The transfer function from steam temperature to tank temperature is

[ G_{st}(s)

\frac{T'(s)}{T_{st}'(s)}

\frac{b_{st}}{s+a}

\frac{UA/(MC_p)}{s+F/M+UA/(MC_p)} ]

with the inlet-fluid temperature deviation held at zero.

The complete input-output relationship is therefore

[ T'(s)=G_i(s)T_i'(s)+G_{st}(s)T_{st}'(s) ]

Interpret the Model Physically

Inlet-Temperature Response

The inlet-temperature path has numerator (F/M). A higher flow rate causes inlet disturbances to influence the tank more strongly and reduces the liquid residence time.

At steady state, the inlet-temperature gain is

[ K_i=G_i(0)=\frac{b_i}{a} ]

or

[ K_i

\frac{F/M}{F/M+UA/(MC_p)} ]

Steam-Temperature Response

The steam-temperature path has numerator (UA/(MC_p)). Increasing the heat-transfer area or overall heat-transfer coefficient increases the influence of steam-temperature changes on the tank.

Its steady-state gain is

[ K_{st}=G_{st}(0)=\frac{b_{st}}{a} ]

or

[ K_{st}

\frac{UA/(MC_p)}{F/M+UA/(MC_p)} ]

For this simplified model,

[ K_i+K_{st}=1 ]

This reflects the fact that the steady tank temperature is a weighted balance between the inlet-fluid temperature and the steam temperature.

Time Constant and Pole

Both transfer functions have the same denominator and therefore the same pole:

[ s=-a ]

The common time constant is

[ \tau=\frac{1}{a} ]

Using (\tau), the transfer functions can also be written as

[ G_i(s)=\frac{K_i}{\tau s+1} ]

[ G_{st}(s)=\frac{K_{st}}{\tau s+1} ]

The common pole means that both inputs are filtered by the same tank thermal storage. Their steady-state effects differ, but their basic first-order response speed is the same under the stated assumptions.

Predict Step Responses

Step Change in Inlet Temperature

For a step change of magnitude (\Delta T_i) in inlet temperature, with steam temperature unchanged,

[ T'(t)

K_i\Delta T_i \left(1-e^{-t/\tau}\right) ]

The tank temperature approaches its new steady-state value exponentially.

Step Change in Steam Temperature

For a steam-temperature step of magnitude (\Delta T_{st}), with inlet temperature unchanged,

[ T'(t)

K_{st}\Delta T_{st} \left(1-e^{-t/\tau}\right) ]

The form is identical, but the gain is determined by the relative strength of the coil heat transfer.

Simultaneous Changes

If both temperatures change at the same time,

[ T'(t)

K_i\Delta T_i \left(1-e^{-t/\tau}\right) + K_{st}\Delta T_{st} \left(1-e^{-t/\tau}\right) ]

For more general time-varying inputs, the Laplace-domain model is used directly, or the original differential equation can be solved numerically and compared with pilot-plant sensor data.

Connect Parameters to Pilot-Plant Measurements

Estimate the Heat-Transfer Capacity

The quantity (UA) controls how effectively steam temperature changes are transmitted to the tank. It can be estimated from

[ Q=UA\Delta T_m ]

where (\Delta T_m) is the appropriate mean temperature difference, often the log mean temperature difference for exchanger-style calculations.

The measured value of (U) incorporates convection, wall conduction, and fouling resistances.

Evaluate the Effect of Agitation

Agitation improves mixing and can increase the fluid-side heat-transfer coefficient. It also supports the perfectly mixed assumption used in the model.

Poor mixing can cause measured responses to deviate from the single-temperature model because different regions of the tank may respond at different rates.

Use Dimensionless Parameters Carefully

The ratio

[ B=\frac{UA}{FC_p} ]

compares coil heat-transfer capacity with the heat-capacity flow rate of the process stream.

A large (B) indicates that steam heating is strong relative to fluid-throughput effects. A small (B) indicates that the inlet stream has a greater influence on the outlet temperature.

Understanding the Trade-offs

A First-Order Model Is an Approximation

The model assumes constant (U), (A), (F), (M), and (C_p). In a real pilot plant, these parameters can vary with flow rate, temperature, viscosity, fouling, and agitator speed.

The transfer functions are therefore most reliable near the steady operating point used for linearization.

Steam Temperature May Not Equal Heat Duty

The model treats steam temperature as an input and uses

[ Q=UA(T_{st}-T) ]

This is appropriate when the steam condenses at an approximately constant temperature and the coil dynamics are fast relative to the tank dynamics.

If steam pressure, valve position, condensate removal, or coil-wall temperature has significant dynamics, steam temperature alone may not adequately represent the heating input.

Transport and Sensor Dynamics May Matter

The ideal balance has no explicit delay between a change in inlet temperature and its effect on the tank. Real piping, sampling locations, thermowells, and temperature sensors can introduce transport delays and additional time constants.

These effects should be included when fitted transfer functions do not match experimental step-test data.

Linearization Limits the Valid Range

The deviation-variable model assumes small enough changes that the system parameters remain approximately constant. Large temperature or flow changes may require a nonlinear energy balance or numerical integration.

A reliable validation checks the initial condition, the long-time steady state, and whether the predicted transient curve agrees with measured data.

Making the Right Choice for Your Goal

Use the balance-derived model as the starting point for analysis and control design.

• If your primary focus is inlet-temperature disturbances: Use (G_i(s)), emphasizing the flow-to-holdup ratio (F/M) and the inlet step-response gain (K_i).
• If your primary focus is steam-temperature control: Use (G_{st}(s)), emphasizing (UA/(MC_p)), heat-transfer area, agitation, and the overall coefficient (U).
• If your primary focus is pilot-plant validation: Perform step tests, estimate the gains and time constant, and compare the measured response with the first-order predictions.
• If your primary focus is high-fidelity simulation: Retain the nonlinear energy balance and add transport, coil, valve, and sensor dynamics where the data show that the first-order model is insufficient.

A transfer function is the compact dynamic consequence of a physical energy balance, linking measurable process changes to the tank-temperature response.

Summary Table:

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