A stirred tank heater is modeled with two state variables: liquid level (h) and liquid temperature (T). A total mass balance determines how the level changes, while an energy balance determines how the well-mixed liquid temperature changes. Together, these equations predict transient behavior after changes in inlet flow, outlet flow, inlet temperature, or steam heat input, and they define steady-state conditions when both time derivatives are zero.

The liquid level and temperature are the system’s dynamic memory. Conservation of mass governs (h), conservation of energy governs (T), and the resulting differential equations connect pilot-plant inputs and disturbances to measurable process outputs.

Identifying the State Variables

Liquid Level as a State Variable

For a tank with constant cross-sectional area (A), the liquid volume is (V=Ah). The level (h) is therefore sufficient to describe the tank’s stored liquid mass when density is treated as constant.

The mass balance is

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

where (F_i) is the inlet volumetric flow rate and (F) is the outlet volumetric flow rate.

The equation states that liquid level rises when inflow exceeds outflow and falls when outflow exceeds inflow. If (F_i=F), the level is stationary.

Temperature as a State Variable

Because the tank is stirred, the liquid is commonly assumed to be perfectly mixed. The outlet temperature is therefore equal to the tank temperature (T).

Temperature becomes a required state variable whenever the control objective includes heating performance or effluent temperature. It represents the thermal energy stored in the liquid and changes when energy enters or leaves the tank.

Choosing States from the Control Objective

A level-control experiment may require only the mass balance and (h) as the state variable. A temperature-control experiment requires both the mass balance and the energy balance because the tank’s volume affects its thermal capacitance.

This is a practical modeling principle: include the states needed to describe the controlled outputs and their dynamic storage mechanisms, but avoid balance equations that do not affect the objective.

Applying Conservation of Mass

Formulating the Dynamic Balance

The general conservation statement is

[ \text{Accumulation}=\text{Input}-\text{Output} ]

For constant liquid density and a tank with volume (V=Ah),

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

Substituting (V=Ah) gives

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

This is a first-order differential equation for the level state.

Interpreting the Inputs

The inlet flow (F_i) is typically a manipulated variable or a disturbance, depending on the plant configuration. The outlet flow (F) may be controlled directly by a pump or valve, or it may depend on level through gravity-driven discharge.

The mass balance does not directly determine temperature. It determines the amount of liquid available to store thermal energy and therefore influences the temperature dynamics indirectly.

Finding the Level Steady State

At steady state,

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

so the mass balance becomes

[ F_i=F ]

This condition is useful for checking both a mathematical model and a pilot-plant operating point. A persistent mismatch between inlet and outlet flow indicates that the level must continue changing.

Applying Conservation of Energy

Formulating the Thermal Balance

The energy balance accounts for thermal energy entering with the feed, leaving with the outlet, and entering from the steam heater. Under constant density and heat capacity assumptions, the balance is

[ Ah\frac{dT}{dt}

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

where:

(T_i) is the inlet liquid temperature,
(Q) is the heat supplied by steam,
(\rho) is liquid density,
(c_p) is liquid heat capacity.

The left side represents the thermal storage associated with the liquid volume. The term (F_i(T_i-T)) represents the net convective thermal effect of replacing tank liquid with inlet liquid.

Understanding the Heat-Input Term

The heat duty (Q) raises the tank temperature when it is positive. Dividing by (\rho c_p) converts the supplied heat rate into an equivalent volumetric temperature-change contribution.

In a more detailed plant model, the relationship between steam conditions and (Q) may be represented through a heat-transfer coefficient, coil area, steam pressure, or actuator model. The conservation equation remains the process-level description of how that heat changes the liquid temperature.

Recognizing the Coupling with Level

The energy equation contains (h) in the storage term (Ah,dT/dt). A higher liquid level means more liquid thermal mass, so the same heat input generally produces a slower temperature response.

This coupling means that level disturbances can affect temperature dynamics even when the steam input is unchanged. The level and temperature equations should therefore be solved together when both variables vary significantly.

Finding the Temperature Steady State

At a steady state,

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

The energy balance becomes

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

or, equivalently,

[ T=T_i+\frac{Q}{\rho c_p F_i} ]

under the stated assumptions and with consistent units. This expression shows how the steady temperature depends on inlet temperature, flow rate, and heat duty.

Using the Model for Process Dynamics

Defining Inputs, Disturbances, and Outputs

Typical manipulated variables include the steam heat rate (Q), inlet flow (F_i), or outlet flow (F). Common disturbances include changes in inlet temperature (T_i), feed flow, steam pressure, or liquid properties.

Measured outputs commonly include tank level (h), tank temperature (T), outlet temperature, and sometimes flow rates. The state equations provide the internal dynamic model connecting these quantities.

Simulating Transient Responses

Given initial conditions (h(0)) and (T(0)), the two differential equations can be integrated over time after a step or other change in an input.

For example, changing (Q) produces a temperature transient, while changing (F_i) can produce both a level transient and a temperature response. Numerical solvers such as MATLAB’s ode45 or Python’s solve_ivp can calculate the trajectories when the equations are nonlinear or time-varying.

Linearizing Around an Operating Point

For control analysis, variables are often expressed as deviations from steady state:

[ h'=h-h_s,\qquad T'=T-T_s,\qquad T_i'=T_i-T_{i,s},\qquad Q'=Q-Q_s ]

where the subscript (s) denotes the nominal operating point.

With additional assumptions, such as constant level and flow, the temperature model can be reduced to a first-order form:

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

\frac{1}{\tau}T_i'+KQ' ]

The parameters (a), (\tau), and (K) summarize the effects of flow, thermal capacity, and heat-transfer behavior. This form is useful for deriving transfer functions, estimating response time, and designing feedback controllers.

Comparing the Model with Pilot-Plant Data

Students can impose controlled disturbances, record level and temperature measurements, and compare the measured curves with model predictions. Differences may reveal sensor lag, imperfect mixing, heat losses, variable properties, actuator limitations, or an inaccurate estimate of the heat duty.

Model validation should include both transient and steady-state checks. A credible model should reproduce the initial condition, approach the correct steady state, and satisfy the governing balance equations.

Understanding the Trade-offs

Simplifying Assumptions

The basic model usually assumes perfect mixing, constant density, constant heat capacity, negligible heat loss, and a uniform tank temperature. It may also treat the tank area and flow rates as known and constant.

These assumptions make the model suitable for teaching and initial control design, but they may be inadequate for high-accuracy prediction or strongly varying operating conditions.

Coupled and Nonlinear Behavior

The energy equation is coupled to level through (Ah). If level changes substantially, the temperature equation has a time-varying thermal storage term and may not behave like a fixed first-order system.

A model that assumes constant level can be convenient for temperature-control studies, but it should not be used without checking whether level variations are small relative to the operating level.

Heat Losses and Actuator Dynamics

Real tanks lose heat through walls, piping, and the surrounding environment. Steam valves and heating coils also have dynamics that separate the commanded input from the actual heat transferred to the liquid.

Ignoring these effects can make the predicted temperature response faster or more favorable than the measured response. They should be added when model accuracy or controller robustness requires them.

Over- or Under-Modeling

Using only a level balance is insufficient for temperature control because it omits thermal storage. Conversely, adding species, reaction, or detailed enthalpy balances is unnecessary when the pilot plant contains a nonreactive liquid with nearly constant properties.

The appropriate model is determined by the control objective, operating range, and accuracy required from the experiment.

Making the Right Choice for Your Goal

The model should be matched to the experiment’s controlled variables, disturbances, and required prediction accuracy.

If your primary focus is level control: Use the mass balance (A,dh/dt=F_i-F) and treat (h) as the principal dynamic state.
If your primary focus is temperature control: Include both mass and energy balances, because liquid volume determines the thermal storage and (T) is the controlled thermal state.
If your primary focus is controller design: Linearize the energy balance around a steady operating point and use deviation variables to estimate response time and process gain.
If your primary focus is model validation: Compare simulated level and temperature transients with pilot-plant data under known disturbances, then account for heat loss, sensor lag, and actuator behavior where necessary.

Conservation balances turn the stirred tank heater from a collection of equipment into a predictive dynamic system whose level and temperature responses can be analyzed, tested, and controlled.

Summary Table:

Model element	Governing relationship	Process insight
Level state, h	A dh/dt = F_i - F	Inflow and outflow determine liquid-level changes.
Temperature state, T	Ah dT/dt = F_i(T_i - T) + Q/(rho c_p)	Feed conditions and heater duty determine temperature dynamics.
Level steady state	F_i = F	The tank level is constant when inlet and outlet flows match.
Temperature steady state	T = T_i + Q/(rho c_p F_i)	Steady temperature depends on feed temperature, flow, and heat duty.
Typical inputs	F_i, F, Q, T_i	Flow, steam heating, and feed temperature act as inputs or disturbances.
Model applications	Simulation, linearization, control, validation	The model supports transient analysis and comparison with pilot-plant data.
Key assumptions	Perfect mixing, constant properties, negligible heat loss	Additional heat-loss, sensor, and actuator dynamics may be needed for accuracy.

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How are conservation principles and state variables applied to model a stirred tank heater in a process dynamics pilot plant?