Batch Vessel Heating Time: Comparing Constant-Power and Constant-Temperature Models

Estimating how long a batch of liquid needs to reach its target temperature is a routine calculation for jacketed reactors, stirred tanks, and other heated vessels. Two idealized models dominate the analysis, and they produce visibly different temperature curves and different time formulas. The constant-temperature model assumes the heating medium — typically condensing steam or a high-flow jacket fluid — stays at one temperature, so the driving force shrinks as the batch warms and the temperature rises along an exponential curve. The constant-power model assumes heat arrives at a fixed rate, as from an electric heater, so the batch temperature climbs in a straight line. This article compares the two models, sets out their governing equations and heating-time expressions, and explains when each describes real equipment.

The energy balance behind every batch heat-up

All batch heating models start from one statement: how fast the batch temperature changes depends on how quickly heat enters relative to the batch heat capacity. A widely used form of the batch energy balance is

m c_p dT/dt = Q_jacket + P_agitator − UA_loss(T − T_amb)

where m is the batch mass, c_p its specific heat, T its temperature, Q_jacket the heat delivered through the jacket or coil, P_agitator the mechanical power dissipated into the batch by the mixer, and the final term the heat lost to the ambient surroundings. This formulation underlies a published batch heating and cooling time simulator that steps through time and checks its numerical results against closed-form solutions for the simple cases. The two models compared here are simply two assumptions about Q_jacket: proportional to the remaining temperature difference, or a fixed value. Agitator power and ambient losses can be layered onto either.

Constant-temperature model: the isothermal medium

When heat enters through a jacket or coil from a medium at a fixed temperature T_u, the instantaneous heat rate is Q = UA (T_u − T), where UA is the product of the overall heat transfer coefficient and the effective heat transfer area. A fixed medium temperature is a good assumption for condensing steam, whose temperature is fixed by its pressure, and for circulating utilities at high flow where the fluid temperature barely changes across the jacket; the classic treatment for steam-jacketed agitated vessels is set out in a PDHonline course on isothermal-medium batch heating.

Substituting Q = UA (T_u − T) into the energy balance, ignoring agitator power and losses for clarity, gives m c_p dT/dt = UA (T_u − T). The solution shows the batch temperature approaching T_u exponentially:

T(t) = T_u − (T_u − T₀) e^−t/τ, with time constant τ = m c_p/UA,

and the time needed to reach a target temperature is the logarithmic expression used as a check in the PharmaChemE simulator:

t = (m c_p/UA) ln[(T_u − T₀)/(T_u − T_target)]

Three practical consequences follow from the exponential shape. First, the batch never quite reaches T_u: after one time constant the remaining gap is about 37% of the starting gap (roughly 63% of the journey complete), after three time constants about 5%, and only after about 4.6 time constants does it fall to 1%. Second, the final degrees are the most expensive; halving an already small remaining gap costs a fixed τ ln 2 of extra time no matter how close the batch already is. Third, because the formula depends on m c_p/UA, a heavier batch, a higher specific heat, or a weaker jacket slows the heat-up in direct proportion.

Constant-power model: fixed heat input

The constant-power model assumes Q is fixed and independent of the batch temperature. The energy balance reduces to m c_p dT/dt = Q, so the temperature rises at a constant rate:

T(t) = T₀ + (Q/m c_p) t, and t = m c_p(T_target − T₀)/Q.

The curve is a straight line, and doubling the available duty halves the heat-up time. The assumption fits electrically heated vessels with a controlled power supply. It also appears in jacketed service when the duty is deliberately held constant: for some glass-lined vessels the jacket fluid temperature must stay within a set difference (ΔT) of the batch temperature, so the jacket is run along just above the batch at that allowed difference, and the duty remains constant until the jacket reaches the utility temperature — a regime represented explicitly in the PharmaChemE simulator.

The model carries a caution. Constant power delivered through a wall implies the temperature difference across that wall grows as the batch warms, so the hot-side temperature climbs too; wall temperature limits and, for glass-lined steel, thermal stress cap how far this can go. Constant power is a statement about the heat supply, not a license for an unbounded jacket temperature.

Side-by-side comparison

The curve shapes drive the practical differences. In the isothermal case, heating time grows logarithmically as the target closes on T_u, so the chosen target matters as much as the equipment; in the constant-power case only the total temperature span and the available duty matter. The same pair of models applies, with signs reversed, to batch cooling against a constant-temperature coolant or a fixed refrigeration duty.

Where real equipment falls between the models

• Limited utility flow. When the jacket fluid passes through at modest flow, it warms as it goes, and the effective driving force is lower than the inlet temperature suggests. The heat rate becomes Q = ṁ c_p,u(T_u − T)(1 − e^{−UA/(ṁ c_p,u)}), where ṁ c_p,u is the utility flow rate times its heat capacity; low flow, not area, can be the real bottleneck. With a constant inlet temperature this still has the isothermal form Q = UA_eff(T_u − T), so the logarithmic time formula remains valid using the reduced conductance UA_eff = ṁ c_p,u(1 − e^{−UA/(ṁ c_p,u)}).
• Agitator power. The P_agitator term adds a small fixed heat input — a linear-ramp component on top of the exponential approach. It is negligible for many steam-heated batches but can matter for viscous, high-power mixing duty; the simulator carries the term explicitly.
• Ambient losses. The loss term −UA_loss(T − T_amb) grows as the batch warms, so the net heat input falls faster than the jacket input alone. Heat-up then ends not at T_u but at the lower temperature where jacket input balances loss to the surroundings — one reason real vessels settle below their steam temperature.
• Regime changes mid-batch. A ΔT-limited glass-lined jacket can start the heat-up in constant-duty (constant-power-like) operation and then transition to the exponential approach once the jacket has climbed to the utility temperature. One vessel can show both curves in a single batch.

Common mistakes and limitations

• Planning to reach the steam temperature. Under the isothermal model the batch approaches T_u only asymptotically, and with ambient losses the achievable temperature is lower still. Schedule with the logarithmic formula and target a margin below T_u.
• Extrapolating early data as a straight line. An isothermal heat-up bends away from a linear trend almost immediately; fitting a line to the first minutes overpredicts later progress.
• Treating UA as a constant. Only the wetted portion of the jacket transfers heat, so the effective area changes with batch level; fouling and agitation further shift the coefficient. Use effective values from data where available.
• Ignoring scale-up. Time scales with m c_p/UA, and volume grows with the cube of vessel size while jacket area grows much more slowly, so plant batches heat and cool slower than lab batches even with similar utilities.
• Forgetting the well-mixed assumption. The closed forms describe agitated liquid batches with a uniform temperature; a poorly mixed vessel develops internal gradients and will not follow the predicted curve.

Practical takeaways

Choose the model by looking at how heat actually arrives at the batch:

• Condensing steam, or a circulating utility at high flow with a stable inlet temperature: use the isothermal model and its logarithmic time formula.
• Electric heating or any genuinely fixed duty: use the constant-power model and its linear formula.
• A circulating utility at modest flow: apply the flow-limited duty expression, or equivalently reduce UA to its effective value.
• Check whether agitator power and ambient losses are large enough to include.

For combined effects, a time-stepped model is the standard tool. Spreadsheet walkthroughs of exactly this kind — tracking temperature, duty, and resource use batch by batch — have been published for working engineers, such as Stephen Hall’s 2023 Excel series in The Chemical Engineer, and web simulators step through the mixed cases numerically while checking them against the closed forms. The underlying mathematics is old and stable: a 1959 Newark College of Engineering thesis assembled batch heating and cooling nomographs — alignment charts that let engineers read approximate answers without solving the equations — for precisely these calculations. What the formulas cannot supply is the judgment call of deciding which heat-supply behavior the vessel in front of you actually follows, and that judgment is what separates a schedule that holds from one that slips.