An input-output model for a continuous stirred-tank heater is obtained by applying an unsteady-state energy balance, defining the inlet temperature and heat duty as inputs, and defining the tank temperature as the output. For a constant-volume tank with constant physical properties, the model is

[ \frac{dT}{dt}

\frac{F}{V}(T_i-T) + \frac{Q}{\rho c_p V}, ]

where (T) is the well-mixed tank temperature, (T_i) is the inlet temperature, (F) is the volumetric flow rate, (V) is liquid volume, and (Q) is the net heat-transfer rate. Linearizing around a steady state produces a first-order input-output model suitable for transfer-function analysis and feedback-controller design.

The essential formulation is a dynamic energy balance expressed in deviation variables. With (T_i) and (Q) as inputs and (T) as the output, the constant-volume CSTH behaves approximately as a first-order process with inlet-temperature and heat-duty contributions.

Define the Control Problem

Select the Controlled Output

The usual controlled variable is the tank liquid temperature, (T(t)), because it determines the thermal condition of the process stream leaving the heater.

The tank is assumed to be perfectly mixed, so the outlet temperature is equal to the tank temperature:

[ T_{\text{out}}=T. ]

Identify the Process Inputs

The principal manipulated input is the heating duty, (Q(t)), supplied through steam, an electrical heater, or a heating jacket.

The inlet temperature (T_i(t)) is normally treated as a disturbance input, although it can also be manipulated in some pilot-plant configurations. Flow rate (F(t)) may also act as a disturbance or manipulated variable because it changes both residence time and energy throughput.

Establish the Modeling Assumptions

A useful first model generally assumes:

• Perfect liquid mixing
• Constant liquid volume
• Constant density (\rho)
• Constant heat capacity (c_p)
• Constant inlet and outlet flow, (F_i=F)
• Negligible heat losses unless they are explicitly modeled
• Negligible kinetic, potential, shaft-work, and expansion-work effects

These assumptions produce a compact model while retaining the dominant thermal dynamics.

Formulate the Dynamic Balances

Write the Total Mass Balance

The general liquid-volume balance is

[ \frac{dV}{dt}=F_i-F. ]

For a constant-volume CSTH,

[ F_i=F, \qquad \frac{dV}{dt}=0. ]

If level dynamics are important, the tank volume must be retained as a state variable rather than treated as constant.

Write the Energy Balance

For a nonreactive stirred-tank heater, the unsteady energy balance is

[ \rho c_p V\frac{dT}{dt}

\rho c_p F(T_i-T)+Q. ]

Dividing by (\rho c_p V) gives the temperature state equation:

[ \frac{dT}{dt}

\frac{F}{V}(T_i-T) + \frac{Q}{\rho c_p V}. ]

Equivalently,

[ \frac{dT}{dt} + \frac{F}{V}T

\frac{F}{V}T_i + \frac{1}{\rho c_p V}Q. ]

The first term on the right represents convective energy entering with the feed, while (Q) represents net heat added to the liquid. A cooling duty can be represented by taking (Q<0).

Include Reaction Effects When Required

If the CSTH also functions as a reacting CSTR, the energy balance may include reaction heat:

[ \frac{dT}{dt}

\frac{F_i}{V}(T_i-T) + \frac{(-\Delta H_r)r_A}{\rho c_p} + \frac{Q}{\rho c_p V}, ]

where (r_A) is the reaction rate.

In that case, temperature is coupled to concentration through the reaction kinetics. For a first-order reaction,

[ r_A=k_0\exp\left(-\frac{E}{RT}\right)c_A, ]

and a complete nonlinear model must also include the component balance for (c_A).

Convert the Model to Deviation Variables

Define the Steady-State Operating Point

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

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

so the operating point satisfies

[ 0= \frac{F}{V}(T_{i,s}-T_s) + \frac{Q_s}{\rho c_p V}. ]

This operating point is the reference around which dynamic changes are analyzed.

Define the Deviations

Define

[ T'=T-T_s, ]

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

[ Q'=Q-Q_s. ]

For constant (F), (V), (\rho), and (c_p), subtracting the steady-state equation from the dynamic equation gives

[ \frac{dT'}{dt} + \frac{F}{V}T'

\frac{F}{V}T_i' + \frac{1}{\rho c_p V}Q'. ]

Using the residence-time definition

[ \tau=\frac{V}{F}, ]

the model becomes

[ \boxed{ \frac{dT'}{dt} + \frac{1}{\tau}T'

\frac{1}{\tau}T_i' + \frac{1}{\rho c_p V}Q' } ]

This is the standard linear first-order input-output model for a constant-volume continuous stirred-tank heater.

Relate the Model to General Parameters

The model can also be written in the generic form

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

\frac{1}{\tau}T_i' + K_QQ', ]

where

[ a=\frac{F}{V}=\frac{1}{\tau}, ]

and

[ K_Q=\frac{1}{\rho c_p V}. ]

If (Q) denotes a steam-flow signal, valve position, or heater command rather than actual heat rate, (K_Q) must include the corresponding actuator and heat-transfer gain.

Derive the Transfer Functions

Apply the Laplace Transform

Taking the Laplace transform and assuming zero initial temperature deviation gives

[ sT'(s)+\frac{1}{\tau}T'(s)

\frac{1}{\tau}T_i'(s) + K_QQ'(s). ]

Therefore,

[ T'(s)

\frac{1/\tau}{s+1/\tau}T_i'(s) + \frac{K_Q}{s+1/\tau}Q'(s). ]

Obtain the Inlet-Temperature Transfer Function

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

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

\frac{1}{\tau s+1} } ]

The inlet temperature therefore affects the tank temperature through a first-order dynamic response with time constant (\tau=V/F).

Obtain the Heat-Duty Transfer Function

The transfer function from heat duty to tank temperature is

[ \boxed{ \frac{T'(s)}{Q'(s)}

\frac{K_Q}{s+1/\tau}

\frac{K_T}{\tau s+1} } ]

where

[ K_T=\frac{\tau}{\rho c_p V} =\frac{1}{\rho c_p F}. ]

Thus, for a direct heat-rate input,

[ \boxed{ \frac{T'(s)}{Q'(s)}

\frac{1}{\rho c_p F}\frac{1}{\tau s+1} } ]

The total temperature response is the sum of the responses to the disturbance and manipulated input:

[ T'(s)

\frac{1}{\tau s+1}T_i'(s) + \frac{K_T}{\tau s+1}Q'(s). ]

Extend the Model for Level Dynamics

Add Liquid Level as a State

If the tank level is not constant, define (h(t)) as the liquid height and (A) as the tank cross-sectional area. The mass balance becomes

[ A\frac{dh}{dt}=F_i-F. ]

The energy balance can be written as

[ Ah\frac{dT}{dt}

F_i(T_i-T) + \frac{Q}{\rho c_p}. ]

The factor (Ah) is the liquid volume. Consequently, changes in level alter the thermal capacitance and therefore the temperature dynamics.

Decide Whether Level Coupling Is Necessary

A temperature-only model is appropriate when level control maintains (V) nearly constant and level variations are small.

A coupled level-temperature model is required when flow changes produce significant volume changes, when level control is part of the control objective, or when the residence time varies substantially during operation.

Understand the Model’s Control Meaning

Interpret the Time Constant

The time constant

[ \tau=\frac{V}{F} ]

is the nominal liquid residence time. A larger tank volume produces slower temperature changes, while a larger flow rate produces faster replacement of the tank contents.

After a step change, a first-order process reaches approximately 63% of its final temperature change after one time constant and approaches its final value asymptotically.

Interpret the Process Gain

For heat-duty changes, the steady-state gain is

[ K_T=\frac{1}{\rho c_p F}. ]

The temperature rise caused by a given heat input decreases as the flow rate, density, or heat capacity increases.

For inlet-temperature changes, the steady-state gain is unity under the ideal constant-property model:

[ \lim_{s\to 0}\frac{T'(s)}{T_i'(s)}=1. ]

At steady state, the tank temperature ultimately follows a sustained inlet-temperature change, subject to the effect of the heat-duty controller.

Use the Model for Feedback Design

The model identifies the key elements required for a temperature-control loop:

• Controlled variable: tank temperature (T)
• Manipulated variable: heating duty (Q)
• Main disturbance: inlet temperature (T_i)
• Process time constant: (\tau=V/F)
• Process gain: (K_T=1/(\rho c_p F))
• Measurement: temperature transmitter
• Final control element: steam valve, heater, or jacket-duty actuator

These parameters support PID tuning, disturbance-rejection analysis, simulation, and comparison between predicted and measured pilot-plant responses.

Understanding the Trade-offs

Do Not Treat a Nonlinear Reactor as First Order Without Checking

The first-order model is valid for a nonreactive or weakly coupled thermal system near a fixed operating point.

For an exothermic reactor, the Arrhenius reaction rate introduces strong temperature dependence. The resulting model may have multiple steady states, nonlinear gains, or thermal instability that a simple first-order transfer function cannot represent.

Do Not Omit Heat Losses When They Matter

If heat is lost through tank walls, insulation, or the vapor space, an additional term may be required:

[ Q_{\text{loss}}=UA(T-T_{\text{amb}}). ]

The energy balance then contains the net heat input

[ Q_{\text{net}}=Q-UA(T-T_{\text{amb}}). ]

Ignoring substantial heat loss causes errors in both the steady-state gain and the controller tuning parameters.

Distinguish Heat Rate from Heater Command

The variable (Q) should represent an actual heat-transfer rate if the gain (1/(\rho c_p V)) is used directly.

If the measured input is valve position, steam flow, electrical power command, or jacket temperature, the actuator and heat-transfer relationship must be modeled separately or incorporated into an experimentally identified process gain.

Account for Measurement and Actuator Dynamics

Real pilot plants often include sensor lag, valve dynamics, transport delays, and steam-condensation dynamics.

If these effects are significant, the process model may take the form

[ G(s)= \frac{K_T e^{-\theta s}} {(\tau s+1)(\tau_m s+1)}, ]

where (\theta) is a delay and (\tau_m) is a measurement or actuator time constant.

Making the Right Choice for Your Goal

Use the model complexity that matches the control question and the operating conditions.

• If your primary focus is basic temperature control: Use the constant-volume first-order model with (T_i) as a disturbance and (Q) as the manipulated input.
• If your primary focus is level and temperature control: Retain both the mass balance and energy balance, with (h) or (V) and (T) as state variables.
• If your primary focus is reactor behavior: Add the component balance and temperature-dependent reaction kinetics, then linearize only around a clearly defined steady state.
• If your primary focus is pilot-plant controller tuning: Include measured actuator, sensor, heat-loss, and delay effects when they are large enough to affect the observed response.
• If your primary focus is simulation or fault detection: Validate the model against step-test data before using it for operator training, optimization, or diagnosis.

A well-formulated CSTH input-output model connects conservation laws to measurable process behavior, giving engineers a defensible basis for control design and safe pilot-plant operation.

Summary Table:

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