Dynamic energy balances for a stirred-tank heater are written by subtracting the steady-state balance from the transient balance. For a constant-volume, well-mixed tank with constant flow and properties, the temperature model becomes

[ \frac{dT'}{dt}+aT'=\frac{1}{\tau}T_i'+KT_{st}' ]

where (T'), (T_i'), and (T_{st}') are deviations of tank temperature, inlet temperature, and heating input from their nominal steady-state values. This form isolates only the dynamic changes that matter for response analysis and control design.

Core takeaway: Define every variable as its deviation from a steady-state operating point, substitute these definitions into the dynamic energy balance, and use the steady-state equation to cancel the constant terms. The resulting deviation model describes transient temperature behavior without carrying the nominal operating-point offsets.

Start with the Transient Energy Balance

Define the control volume and assumptions

For a well-mixed stirred tank heater, the liquid temperature is assumed uniform throughout the tank. A common pilot-plant model assumes constant liquid volume (V), constant volumetric flow rate (F), constant density (\rho), and constant heat capacity (c_p).

Under these assumptions, kinetic energy, potential energy, shaft work, and volume-change effects are neglected. The dominant terms are energy accumulation, inlet and outlet enthalpy flow, and externally supplied heat.

Write the dimensional energy balance

For a nonreactive liquid tank with heat input (Q),

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

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

where:

• (T) is the tank liquid temperature,
• (T_i) is the inlet temperature,
• (F) is the volumetric flow rate,
• (V) is the liquid volume,
• (Q) is the heater duty.

The outlet temperature is (T) because the tank is perfectly mixed.

Put the balance in standard dynamic form

Dividing by (\rho c_p V) gives

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

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

The coefficient (F/V) is the inverse residence time. Defining

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

gives

[ \frac{dT}{dt} + \frac{1}{\tau}T

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

If the heater input is represented by a variable (T_{st}), the heat-input term may instead be written as (KT_{st}). The resulting model is commonly expressed as

[ \frac{dT}{dt}+aT=\frac{1}{\tau}T_i+KT_{st} ]

where (a), (1/\tau), and (K) collect the relevant flow, volume, heat-capacity, and heat-transfer parameters.

Convert the Model to Deviation Variables

Define the nominal operating point

Let the steady-state operating conditions be

[ T_s,\qquad T_{i,s},\qquad T_{st,s} ]

At steady state, the tank temperature does not change, so

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

The steady-state equation is therefore

[ aT_s=\frac{1}{\tau}T_{i,s}+KT_{st,s} ]

This equation describes the operating point around which the dynamic model is developed.

Define deviations from steady state

The deviation variables are

[ T'=T-T_s ]

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

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

Equivalently,

[ T=T_s+T' ]

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

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

Because (T_s) is constant,

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

Substitute into the dynamic balance

Starting with

[ \frac{dT}{dt}+aT=\frac{1}{\tau}T_i+KT_{st} ]

substitution gives

[ \frac{dT'}{dt} +a(T_s+T')

\frac{1}{\tau}(T_{i,s}+T_i') + K(T_{st,s}+T_{st}') ]

Rearranging,

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

\frac{1}{\tau}T_{i,s} + \frac{1}{\tau}T_i' + KT_{st,s} + KT_{st}' ]

The constant terms satisfy the steady-state relationship

[ aT_s=\frac{1}{\tau}T_{i,s}+KT_{st,s} ]

and therefore cancel. The deviation-variable equation is

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

\frac{1}{\tau}T_i' + KT_{st}' } ]

This is the desired dynamic energy balance in deviation form.

Interpret the Deviation Model

Separate manipulated and disturbance inputs

The heating input (T_{st}'), or equivalently (Q'), is typically the manipulated-variable deviation. The inlet-temperature deviation (T_i') is commonly treated as a disturbance.

The tank-temperature deviation (T') is the process output. The equation therefore describes how both heater changes and inlet disturbances affect the outlet or tank temperature.

Use the physical heat-duty form when appropriate

If the manipulated input is heater duty rather than a heating-medium temperature, define

[ Q'=Q-Q_s ]

The constant-property deviation model becomes

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

\rho c_p F(T_i'-T')+Q' ]

or

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

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

This form is often more transparent for pilot-plant calculations because (Q') has the direct physical meaning of a change in heater duty.

Recognize the state variable

For a constant-volume heater, (T') is the principal thermal state variable. If liquid level or volume changes significantly, the mass balance must also be included:

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

and the energy balance may be written as

[ Ah\frac{dT}{dt}

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

In that case, the temperature dynamics are coupled to the level dynamics, and a single first-order temperature equation may no longer be adequate.

Obtain Transfer Functions for Dynamic Analysis

Apply the Laplace transform

Taking the Laplace transform of

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

\frac{1}{\tau}T_i'+KT_{st}' ]

gives

[ s\bar T'(s)-T'(0)+a\bar T'(s)

\frac{1}{\tau}\bar T_i'(s) + K\bar T_{st}'(s) ]

With zero initial temperature deviation,

[ T'(0)=0 ]

so

[ (s+a)\bar T'(s)

\frac{1}{\tau}\bar T_i'(s) + K\bar T_{st}'(s) ]

Derive the inlet-temperature transfer function

Setting the heater deviation to zero gives

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

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

This transfer function describes the response of tank temperature to inlet-temperature disturbances.

Derive the heater-input transfer function

Setting the inlet-temperature deviation to zero gives

[ \boxed{ \frac{\bar T'(s)}{\bar T_{st}'(s)}

\frac{K}{s+a} } ]

If heater duty (Q') is used directly, the corresponding transfer function is

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

\frac{1}{\rho c_p V} \frac{1}{s+F/V} } ]

These relationships support step, pulse, and ramp-response analysis and provide a basis for feedback-controller design.

Extend the Balance When the Heater Is a Jacket or Coil

Include heat-transfer driving force

If heat is supplied through a jacket or coil, the duty can be modeled as

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

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

The transient energy balance becomes

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

\rho c_p F(T_i-T) + UA(T_{st}-T) ]

Rearranging,

[ \frac{dT}{dt} + \left( \frac{F}{V}+\frac{UA}{\rho c_p V} \right)T

\frac{F}{V}T_i + \frac{UA}{\rho c_p V}T_{st} ]

Thus, in this representation,

[ a=\frac{F}{V}+\frac{UA}{\rho c_p V} ]

and

[ K=\frac{UA}{\rho c_p V} ]

The deviation equation retains the same structure:

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

\frac{F}{V}T_i' + \frac{UA}{\rho c_p V}T_{st}' ]

Account for reaction effects when necessary

For a reacting stirred tank, the energy balance must also include heat generated or consumed by reactions. A general enthalpy balance can be represented as

[ V\frac{d}{dt} \left(\sum_i C_i\bar H_i\right)

F\left[ \sum_i(C_i\bar H_i)_{\text{in}}

\sum_i C_i\bar H_i \right] +\dot Q ]

Reaction heat may make the balance nonlinear because reaction rates and heats can depend on temperature and composition.

A linear deviation model is then obtained by linearizing the nonlinear balance around the chosen steady state. The coefficients in the deviation equation represent local sensitivities at that operating point.

Understanding the Trade-offs

Deviation models are local models

The deviation equation is usually accurate near the selected steady state. Large temperature changes, changing physical properties, variable flow, phase changes, or strongly temperature-dependent reaction rates can invalidate the constant-coefficient approximation.

A new operating point may require a new linearization and new model parameters.

The steady state must be consistent

The constant-term cancellation works only when (T_s), (T_{i,s}), and (T_{st,s}) satisfy the actual steady-state energy balance. If the baseline values are inconsistent, residual constant terms remain in the deviation equation.

This is a common source of apparent modeling errors.

A constant-volume assumption may hide level effects

If inlet and outlet flow rates differ, the tank level changes. Since the liquid volume determines thermal capacitance, level dynamics alter the temperature response.

For level-controlled pilot plants, the constant-volume model is often appropriate. During level transients, however, both mass and energy balances should be retained.

Heater temperature and heater duty are not identical inputs

Using (T_{st}') assumes that the heating-medium temperature is the selected input and that the heat-transfer relationship has been incorporated into (K). Using (Q') treats heat duty as the direct manipulated variable.

The correct choice depends on the physical actuator and the measurements available on the pilot plant.

How to Apply This to Your Project

The appropriate formulation depends on whether the pilot plant is being modeled for basic response analysis, control design, or detailed equipment simulation.

• If your primary focus is temperature response to inlet disturbances: Use (\frac{dT'}{dt}+aT'=\frac{1}{\tau}T_i'+KT_{st}'), treating (T_i') as the disturbance input.
• If your primary focus is heater control: Use heater-duty deviation (Q') and write (\frac{dT'}{dt}+\frac{F}{V}T'=\frac{F}{V}T_i'+\frac{Q'}{\rho c_pV}).
• If your primary focus is jacket or coil design: Retain (UA) explicitly so the model shows how heat-transfer area and coefficient affect both gain and response time.
• If your primary focus is level and temperature control together: Formulate coupled deviation mass and energy balances with (h') and (T') as state variables.
• If your primary focus is a reacting pilot plant: Start with the full enthalpy balance and linearize reaction, heat-transfer, and property terms around the operating point.

A correctly formulated deviation model turns the stirred-tank heater’s operating-point problem into a clear, tractable description of how disturbances and control actions change temperature.

Summary Table:

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