Engineering task and calculation objective
The TIME module calculates transient heat transfer processes – the time-dependent behavior of systems during heating or cooling. While the steady-state design of a heat exchanger considers only the equilibrium condition, practice is often about time: how long does it take to bring a vessel's contents from the initial temperature to the target temperature? How fast does an apparatus cool down if the heating fails? When does a product fall below its critical temperature, for example its pour point?
The basis is the transient heat balance: the temperature change of the system's heat capacity (mass times specific heat capacity of medium and apparatus) is balanced against the time-dependent heat flow that passes to the heat sink or heat source via the overall heat transfer coefficient and the exchange area. Since the driving temperature difference decreases as the process progresses, the typical exponential temperature-time curve results.
Applications are found throughout plant engineering and process engineering: heating and cooling times of stirred vessels and storage tanks, design of batch processes, assessment of standstill times and freeze protection questions, and the estimate of whether an existing heating or cooling capacity is sufficient for the required process time.
Calculation workflow
- Define the system and its heat capacity: The masses and specific heat capacities of the medium and, where applicable, of the apparatus itself are recorded; from these follows the total heat capacity that must be heated or cooled.
- Describe the heat transfer to the sink/source: The overall heat transfer coefficient and the exchange area to the heat sink or source (heating coil, jacket, surroundings) are determined – either taken over from a separate heat transfer calculation or specified directly.
- Set initial and boundary temperatures: The initial temperature of the system and the temperature of the sink or source (e.g. cooling water, heating medium or ambient temperature) are set; for flowing heating/cooling media, their mass flow also enters the balance.
- Calculate the time history: The transient balance yields the temperature-time curve of the system. The module delivers the time until a target temperature is reached, or the temperature change after a given time.
- Interpret the result: Based on the curve, it is checked whether process times are met and whether critical temperatures (pour point, freezing limit, permissible product temperature) are crossed within the period considered.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Time | t | s |
| Initial temperature fluid | Tflo | °C |
| Temperature fluid | Tfl | °C |
| Initial temperature body | Tko | °C |
| Temperature body | Tk | °C |
| Surface of the body | A | m² |
| Mass of the body | Mk | kg |
| Specific heat capacity of the body | cpk | J/(kg·K) |
| Heat transfer coefficient (body - fluid) | α | W/(m²·K) |
| Inlet temperature heating/cooling medium | Th,Eintritt | °C |
| Fluid mass | Mfl | kg |
| Heat capacity of the system | cpfl | J/(kg·K) |
| Mass flow heating/cooling medium | Mh | kg/s |
| Specific heat capacity heating/cooling medium | cph | J/(kg·K) |
| Heat transfer coefficient | k | W/(m²·K) |
| Temperature after infinite time | Tsys | °C |
| Thickness of insulation | Isolation | m |
| Thermal conductivity of insulation | Isolation | W/(m·K) |
| Start temperature | Behälterinhaltes | °C |
| Vessel mass | Behältermasse | kg |
| Specific eat capacity vessel material | Behältermaterials | J/(kg·K) |
| Thermal conductivity wall section I | I | W/(m·K) |
| Thermal conductivity wall section II | II | W/(m·K) |
| Wall thickness section I | I | m |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Time steps | Zeit | s |
| End temperature | Endtemperatur | °C |
Calculation options
Option
Infinite fluid · Final insulated fluid · Heating / cooling · Transient
Worked example
An ideally stirred vessel contains 5,000 kg of water at 90 °C and cools via a cooling coil whose cooling water temperature is approximately constant at 20 °C. Overall heat transfer coefficient k = 350 W/(m²·K), exchange area A = 8 m². This worked example determines the time until the contents reach 40 °C (heat capacity of the vessel and losses to the surroundings neglected).
Given values
| Mass of water m | 5,000 kg |
| Specific heat capacity cp | 4,186 J/(kg·K) |
| Initial temperature ϑ0 | 90 °C |
| Target temperature ϑ1 | 40 °C |
| Coolant temperature ϑK | 20 °C (constant) |
| Overall heat transfer coefficient k | 350 W/(m²·K) |
| Exchange area A | 8 m² |
Solution
Transient balance
For the ideally mixed system with constant sink temperature:
m · cp · dϑ/dt = −k · A · (ϑ − ϑK)
with the solution t = (m · cp)/(k · A) · ln[(ϑ0 − ϑK)/(ϑ1 − ϑK)]
Time constant
m · cp = 5,000 · 4,186 = 20.93 · 10⁶ J/K
k · A = 350 · 8 = 2,800 W/K
τ = 20.93 · 10⁶ / 2,800 ≈ 7,475 s ≈ 2.08 h
Cooling time
t = τ · ln[(90 − 20)/(40 − 20)] = 7,475 · ln(3.5) = 7,475 · 1.2528 ≈ 9,360 s ≈ 2.6 h
Result
| Time constant τ | ≈ 2.1 h |
| Cooling time 90 → 40 °C | ≈ 9,360 s ≈ 2.6 h |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
What assumption underlies the exponential temperature curve?
The ideally mixed system (stirred tank assumption): the entire contents have a uniform temperature at every instant, and the heat flow is proportional to the current temperature difference. The temperature then follows an exponential function with the time constant τ = m·cp/(k·A). For unstirred, stratified vessels or for thick-walled components with an internal temperature gradient (large Biot number), the real behavior deviates from this.
Does the heat capacity of the apparatus have to be included?
For heavy apparatus with comparatively little contents – for example thick-walled pressure vessels or tanks at low filling level – the steel mass can contribute a substantial share of the total heat capacity and lengthen the heat-up time considerably. As a rule of thumb, it should always be included when the m·cp of the apparatus reaches more than about 10% of the value for the contents.
Why do calculated heat-up times often deviate from practice?
The most frequent causes: the overall heat transfer coefficient was assumed for clean surfaces (fouling lengthens the time), the heating medium temperature is not constant (steam pressure fluctuations, limited heating capacity), heat losses to the surroundings were neglected, or the product is not ideally mixed. For reliable commitments, one should calculate with conservative k values and account for losses explicitly.
What changes when the heating or cooling medium itself changes temperature?
For flowing heating/cooling media with a limited mass flow, it is not the inlet temperature that governs but an effective mean temperature difference; the balance is then carried out via an effectiveness or NTU analysis of the heat exchanger. Condensing steam is the special case with a practically constant heating medium temperature – the simple exponential solution applies best there.