Mean Radiant Temperature (MRT) Calculator

An expert tool for the calculation of the mean radiant temperature directly using radiant intensities (surface temperatures and angle factors).

MRT Calculator

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What is the Calculation of the Mean Radiant Temperature?

The Mean Radiant Temperature (MRT) is one of the most important factors in determining human thermal comfort. It represents the uniform temperature of an imaginary enclosure where the radiant heat transfer from the human body is equal to the radiant heat transfer in the actual, non-uniform environment. In simpler terms, it’s the average temperature of all the surfaces surrounding you, weighted by how much of your view they take up. This calculation is crucial because a significant portion of your body’s heat exchange with its environment (around 60%) occurs through thermal radiation. If you’re standing in a room with a 70°F air temperature but next to a large, cold single-pane window, you will feel chilly because your body is radiating more heat towards the window than it receives. The calculation of the mean radiant temperature directly using radiant intensities quantifies this effect.

This calculator is used by building scientists, HVAC engineers, architects, and thermal comfort researchers to design and evaluate indoor and outdoor spaces. Unlike simple air temperature, MRT provides a more holistic measure of how a space “feels” thermally. Understanding it is key for designing energy-efficient buildings with high levels of occupant comfort, such as those employing radiant heating design.

MRT Formula and Explanation

The calculation of the mean radiant temperature is based on the Stefan-Boltzmann law, which describes the power radiated from a black body in terms of its temperature. When considering a person in an enclosure, the MRT can be calculated by taking the fourth power of the surrounding surface temperatures, weighting them by their respective angle factors, summing them up, and then taking the fourth root of the result.

The formula is as follows:

MRT⁴ = T₁⁴Fₚ₋₁ + T₂⁴Fₚ₋₂ + … + Tₙ⁴Fₚ₋ₙ

Or, more concisely:

MRT = [ Σ (Tₙ⁴ × Fₚ₋ₙ) ]¹/⁴

Description of Variables for the MRT Formula

Variable	Meaning	Unit (for calculation)	Typical Range
MRT	Mean Radiant Temperature	Kelvin (K)	283 K to 303 K (10°C to 30°C) for indoor comfort
Tₙ	Absolute temperature of surface ‘n’	Kelvin (K)	Varies widely based on surface type (e.g., window, heated floor)
Fₚ₋ₙ	Angle factor (or view factor) between the person and surface ‘n’	Unitless	0 to 1 (The sum of all Fₚ₋ₙ for a complete enclosure must equal 1)

This formula accurately captures the non-linear nature of radiant heat exchange. For precise thermal comfort analysis, like that specified by ASHRAE 55 standards, using this fourth-power equation is essential.

Practical Examples

Example 1: Office with a Cold Window

Imagine a person sitting in an office in winter. The air temperature is 22°C, but they are next to a large window with a surface temperature of 15°C.

• Input – Surface 1 (Window): Temp = 15°C, Angle Factor = 0.3
• Input – Surface 2 (Opposite Wall): Temp = 22°C, Angle Factor = 0.3
• Input – Surface 3 (Ceiling): Temp = 21°C, Angle Factor = 0.2
• Input – Surface 4 (Floor): Temp = 20°C, Angle Factor = 0.2
• Result: The calculated MRT would be approximately 19.6°C. Even though the air is 22°C, the cold window significantly lowers the mean radiant temperature, making the person feel cooler than the air temperature suggests.

Example 2: Room with Radiant Floor Heating

Now consider a room with a heated floor. The air temperature is slightly lower at 20°C, but the floor is warmed.

• Input – Surface 1 (Heated Floor): Temp = 28°C, Angle Factor = 0.4
• Input – Surface 2 (Ceiling): Temp = 20°C, Angle Factor = 0.4
• Input – Surface 3 (Walls): Temp = 19°C, Angle Factor = 0.2
• Result: The calculated MRT would be approximately 23.5°C. In this case, the radiant warmth from the floor makes the occupant feel much warmer than the air temperature alone would indicate. This highlights the power of controlling surface temperatures for comfort, a key aspect of passive solar design.

How to Use This MRT Calculator

1. Select Temperature Unit: Choose between Celsius (°C) and Fahrenheit (°F) for your inputs and results.
2. Add Surfaces: The calculator starts with four surfaces. Use the “Add Surface” button to add more surfaces to model your environment accurately. For a complete enclosure, you would typically model at least six surfaces (four walls, floor, ceiling).
3. Enter Surface Temperatures: For each surface, enter its measured or estimated surface temperature.
4. Enter Angle Factors: Enter the angle factor (Fₚ₋ₙ) for each corresponding surface. The angle factor represents the proportion of the person’s total radiant field of view that the surface occupies. The sum of all angle factors should be 1.0 for an enclosed space. The calculator will warn you if the sum deviates from 1.0.
5. Calculate and Interpret: Click “Calculate MRT”. The primary result is the Mean Radiant Temperature in your selected unit. You can also review the intermediate results and the chart, which visualizes the contribution of each surface temperature relative to the final MRT. A proper operative temperature calculation would use this MRT value as a key input.

Key Factors That Affect Mean Radiant Temperature

• Surface Temperatures: This is the most direct factor. Hot surfaces (like a sunny window, radiator, or heated floor) will increase the MRT, while cold surfaces (like a poorly insulated wall or single-pane window in winter) will decrease it.
• Angle (View) Factor: A surface’s influence on MRT depends on its size and proximity to the person. A large, close surface has a much higher angle factor than a small, distant one.
• Solar Radiation: Direct sunlight entering a space can dramatically increase the surface temperature of floors and walls, thereby raising the MRT. This is a primary consideration in building energy modeling.
• Surface Emissivity: This property describes how effectively a surface radiates energy. Most common building materials (wood, plaster, brick) have high emissivity (around 0.9), meaning they are very effective at radiating heat. The formula in this calculator assumes high emissivity.
• Building Insulation: Well-insulated walls, roofs, and high-performance windows maintain interior surface temperatures closer to the indoor air temperature, leading to a more stable and comfortable MRT.
• Outdoor Conditions: In outdoor spaces, factors like tree cover, pavement materials, and the “view” of the cold open sky significantly impact MRT.

Frequently Asked Questions (FAQ)

1. What is the difference between air temperature and mean radiant temperature?

Air temperature measures the heat in the air (convective heat), while mean radiant temperature measures the average heat radiating from surrounding surfaces (radiant heat). Human comfort depends on both, and ignoring MRT can lead to uncomfortable spaces even when the thermostat is set to a “comfortable” temperature.

2. Why does the calculation use the fourth power of temperature?

This comes from the Stefan-Boltzmann law of thermal radiation, which states that the energy radiated by an object is proportional to the fourth power of its absolute temperature. This non-linear relationship means that hotter surfaces have a disproportionately larger impact on radiant heat exchange.

3. How do I determine the angle factors (Fₚ₋ₙ)?

Angle factors are complex to calculate precisely as they depend on the geometry between the person and the surfaces. However, for a simple rectangular room, you can approximate them based on the proportion of the visual field each surface takes up. For example, for a person standing in the middle of a cube-shaped room, each of the six surfaces would have an angle factor of roughly 1/6 or ~0.167. A person sitting down will have a larger angle factor with the floor than with the ceiling.

4. What happens if my angle factors don’t add up to 1.0?

If the sum is less than 1, it implies an opening in the enclosure, and the calculation will be less accurate as it’s missing a radiant source/sink. If it’s greater than 1, the geometric assumptions are incorrect. This calculator will still compute a result but will show a warning, as the calculation’s physical basis assumes a complete enclosure where the sum is exactly 1.

5. Can I use this for outdoor spaces?

Yes, but it’s more complex. For an outdoor calculation, you would need to include surfaces like the ground, nearby buildings, trees, and critically, the sky. The “sky temperature” can be significantly lower than the air temperature, acting as a massive radiant heat sink.

6. What is a “good” MRT value?

A “good” MRT is one that, combined with the air temperature, humidity, and air speed, leads to thermal comfort. Generally, you want the MRT to be close to the desired indoor air temperature. A large difference between air temperature and MRT often causes discomfort.

7. How does clothing affect this?

Clothing acts as an insulator, slowing the rate of radiant heat exchange between your body and the surrounding environment. While clothing doesn’t change the MRT of the environment itself, it changes your body’s response to it, which is a key part of overall thermal comfort analysis.

8. Why do units need to be converted to Kelvin for the calculation?

The Stefan-Boltzmann law is based on absolute temperature, which is measured in Kelvin. The formula T⁴ is only physically meaningful when T is an absolute temperature. Using Celsius or Fahrenheit directly in the T⁴ calculation would produce incorrect results.

Related Tools and Internal Resources

Explore these resources for a deeper understanding of thermal comfort and building science: