Batch Vessel Heating / Cooling Time
Finds how long a jacket or coil takes to bring a well-stirred batch of liquid from one temperature to another, when the heating or cooling medium stays at one temperature. Process engineers use it to plan batch cycle times, to size a jacket or coil, and to check a steam pressure.
Equation
θ = (M·cp / (U·A)) · ln((Ts − T1) / (Ts − T2))
This is Kern’s result for a batch with an isothermal heating medium [1]. θ is the time in seconds. M is the batch mass in kg and cp its specific heat, entered in kJ/(kg·K) and used in J/(kg·K). U is the overall heat transfer coefficient in W/(m²·K) and A the wetted heat transfer area in m². Ts is the medium temperature, T1 the starting batch temperature and T2 the target. The same equation covers cooling, with Ts below both. The page also shows the initial rate, U·A·|Ts − T1|, the peak load the utility must supply.
Inputs
- Choosing a Batch Liquid fills cp from the shared fluid table [2]–[4]. Pick the entry nearest the batch’s mean temperature.
- With Heating/Cooling Medium set to saturated steam, enter the jacket pressure. The page finds its saturation temperature from the IAPWS-IF97 steam table [2] and uses that as Ts. Say whether the pressure is gauge or absolute.
Limits and common mistakes
- Ts must be constant. Condensing steam is, but cooling water or hot oil warms or cools as it passes through the jacket, and then the page overstates the rate. Use the medium’s mean temperature only as a rough estimate.
- The batch must be well mixed. An unstirred vessel heats far more slowly.
- U and A are for the wetted surface only. A part-full vessel has less area.
- The time ignores the heat taken by the vessel wall and agitator, and any heat lost to the room.
Worked example
A jacketed tank holds 3000 kg of water at 15 °C. It must reach 80 °C using 3 bar gauge steam in the jacket. The jacket has 8 m² of wetted area, and U is about 600 W/(m²·K). How long will it take?
- Enter Batch Mass 3000 kg.
- Choose Batch Liquid Water, 50°C. cp fills in as 4.18 kJ/(kg·K).
- Enter Overall Coefficient 600 and Heat Transfer Area 8.
- Choose Heating/Cooling Medium Saturated steam at a pressure. Enter Steam Pressure 3 bar and leave Steam Pressure Is at Gauge.
- Enter Initial Batch Temperature 15 °C and Target Batch Temperature 80 °C, and press Solve. The Solution shows:
Steam saturation temperature Ts: 143.732 °C
UA: 4,800 W/K
Mode: Heating
Time required: 1,836.7 s
Time required: 30.6117 min
Time required: 0.510195 hr
Initial heat-transfer rate: 617.915 kW
Check by hand: M·cp/(U·A) = 3000 × 4180/4800 = 2612.5 s, and ln(128.73/63.73) = 0.7030, so θ = 1837 s, or about half an hour. 3 bar gauge is 4.01 bar absolute, and steam tables give 143.6 °C at 4 bar absolute, in line with the saturation temperature above.
