Engineering task and calculation objective
This module calculates the thermal output of heating appliances (radiators) operating with hot water according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019). Radiators release their output to the room by natural convection and radiation; their heat output is therefore described not by individual correlations but by the standardized radiator equation: the output follows a power law of the excess temperature between the heating medium and the room air, with a radiator exponent that depends on the type of construction.
Engineers need to calculate the thermal output of radiators whenever heating surfaces are to be assessed for conditions other than the standard test point – a recurring topic in the conversion of existing systems to lower system temperatures, for example with heat pumps or condensing boilers: from the rated output stated at the standard point (test conditions per EN 442), the actual output at the real flow and return temperatures and the real room temperature is determined.
The type selection defines the radiator design (e.g., sectional radiator, panel radiator, convector, tubular register), which shapes the part-load behavior via its exponent: convectors react more sensitively to falling excess temperatures than radiation-dominated designs.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the type and rated data: First, the type of the radiator and its rated output at the standard point are specified – usually the test conditions 75/65/20 °C (flow/return/room). The type includes the radiator exponent, which describes the combined effect of convection and radiation of the respective design.
- Define the operating conditions: For the operating point to be assessed, the flow and return temperatures of the hot water and the room temperature are entered; alternatively, the mass flow and one of the temperatures can be specified and the missing quantity determined from the balance.
- Form the governing excess temperature: From the temperatures, the logarithmic excess temperature between the heating water and the room air is formed. It is the correct averaging of the temperature drop across the radiator; the arithmetic excess temperature is a permissible approximation only when the temperature spread is small relative to the excess temperature.
- Convert the output via the power law: The operating output results from the rated output multiplied by the ratio of the excess temperatures raised to the radiator exponent. This allows both the output at given temperatures to be calculated and, conversely, the excess temperature required for a demanded output.
- Check the water-side balance: Finally, the consistency with the water side is checked: output, mass flow, and temperature spread must satisfy the energy balance. At strongly reduced mass flow, the spread grows, the logarithmic excess temperature drops, and the output falls disproportionately – important for hydraulic balancing and part-load operation.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Standard excess temperature | Δϑn | °C |
| Excess temperature | Δϑ | °C |
| Inlet temperature | ϑV | °C |
| Return temperature | ϑR | °C |
| Air temperature | ϑL | °C |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Existing thermal performance | Wärmeleistung | W |
| Construction height | BH | mm |
| Construction height | BH | mm |
| Construction height | BH | mm |
| Construction height | BH | mm |
| Construction height | BH | mm |
| Construction height | BH | mm |
Calculation options
Type
Thermal performance of sectional radiators · Thermal performance of plate radiators · Thermal performance of tube radiators
Worked example
A panel radiator has a rated output of 1,000 W at the standard point 75/65/20 °C; the radiator exponent is n = 1.3. What output does the radiator deliver at lowered system temperatures of 55/45 °C and an unchanged room temperature of 20 °C?
Given values
| Rated output Q̇N (75/65/20 °C) | 1,000 W |
| Radiator exponent n | 1.3 |
| Operating point flow/return/room | 55 °C / 45 °C / 20 °C |
Solution
Logarithmic excess temperature at the standard point
Δϑln,N = (ϑV − ϑR) / ln[(ϑV − ϑL)/(ϑR − ϑL)]
Δϑln,N = (75 − 65) / ln(55/45) = 10 / 0.2007 = 49.83 K
Logarithmic excess temperature at the operating point
Δϑln,B = (55 − 45) / ln[(55 − 20)/(45 − 20)] = 10 / ln(35/25) = 10 / 0.3365 = 29.72 K
Output according to the radiator equation
Q̇ = Q̇N · (Δϑln,B / Δϑln,N)n = 1,000 W · (29.72 / 49.83)1.3
Q̇ = 1,000 W · 0.59641.3 ≈ 511 W
When lowering to 55/45 °C, only about half of the rated output therefore remains – a typical result when checking existing radiators for heat pump operation.
Result
| Excess temperature at standard point Δϑ_ln,N | 49.83 K |
| Excess temperature at operating point Δϑ_ln,B | 29.72 K |
| Heat output at operating point Q̇ | ≈ 511 W |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
What does the radiator exponent mean physically?
It describes how strongly the heat output grows with the excess temperature. Pure radiation and natural convection are both nonlinear; in combination, an exponent of about 1.3 results for usual sectional and panel radiators, up to about 1.4 for convectors, and considerably lower (about 1.1) for surface heating systems. The larger the exponent, the more the output collapses when the system temperatures are lowered.
Why is the logarithmic rather than the arithmetic excess temperature used?
The heating water cools across the radiator from the flow to the return temperature, so the driving difference to the room air varies locally. The logarithmic excess temperature is the exact averaging of this profile. For large spreads relative to the excess temperature – typical of low-temperature operation – the arithmetic mean noticeably overestimates the output.
How much output does a radiator lose when switching from 75/65 to 55/45 °C?
The logarithmic excess temperature then drops (at a room temperature of 20 °C) from about 49.8 K to about 29.7 K. With an exponent of 1.3, about 51 % of the rated output remains. For heat pump conversions, this means: either the existing heating surface is still sufficient because the heat load has decreased, or larger radiators or designs with more surface area must be installed.
Does the calculation also apply to steam or electric radiators?
The module is designed for operation with hot water. With steam heating, the heating medium temperature is constant over the surface (saturation temperature), so the excess temperature is formed differently; electric heating surfaces release an imposed output. For such cases, the approaches must be adapted accordingly, and the manufacturer's data are decisive.