# READER FAVOURITE FROM HPAC’S ARCHIVE: Formulas For Success

February 1, 2013 | By Robert Bean

Solve the mean radiant temperature problems to solve building energy problems.

If building codes dropped the reference to controlling air temperatures and switched the requirements to controlling mean radiant temperature (MRT or t_{mr}), building performance specifications would have to change overnight and for the better. Despite its significance in thermal comfort and energy efficiency, many people do not really understand what MRT is.

Why is our understanding of MRT important?

One very good reason is found in the National Building Code of Canada v2010: Section A-5.3.1.2.(1) Use of Thermal Insulation or Mechanical Systems for Environmental Control which states, *“In addition to controlling condensation, interior surface temperatures must be warm enough to avoid occupant discomfort due to excessive heat loss by radiation.”* If the code references interior surface temperatures and comfort we certainly ought to know something about interior surface temperatures and comfort.

To put MRT into practical terms, it means: surfaces cooler than skin temperature will draw energy away from the body via radiation resulting in a sensation of cooling. Likewise it means surfaces warmer than skin temperature will impede the release of energy from the body via radiation resulting in a sensation of heating.

So let’s have a short lesson on MRT by first stating that it is: “the theoretical uniform surface temperature of an enclosure in which an occupant would exchange the same amount of radiant heat as in the actual non-uniform enclosure.” Graphically MRT can be shown as the simplified illustration on the left side image in *Figure 1*, where the globe thermometer would convert all the non-uniform temperatures of the right side image into a homogenous or uniform value. This is an acceptable solution for numerical calculations addressing a general comfort factor such as MRT but the caveat is – it does not address the actual ambiguous or non-uniform conditions influencing localized discomfort due to such things as radiant asymmetry, drafts, air temperature stratification or unacceptable floor surface temperatures.

We are going to set aside localized discomforts for the rest of this discussion and focus exclusively on MRT and how it fits into another thermal indices called, “Operative Temperature (t_{o} or t_{op}).” Operative temperature is the value shown on the x-axis of the frequently referenced psychrometric chart found in ASHRAE Standard 55 – Thermal Environmental Conditions for Human Occupancy (*Figure 2*).

Operative Temperature by definition is, “the uniform temperature of an enclosure in which an occupant would exchange the same amount of heat by radiation plus convection as in the actual non uniform environment.” This is not a trivial element to ignore since the majority in industry still incorrectly contract air temperature as the solitary metric for thermal comfort when research clearly shows that radiant transfer represents anywhere between 50 per cent to 60 per cent of the sensible transfer from occupants in various indoor environments at normal activity levels.^{ii}

Those who specialize in thermal comfort advise that spaces where the occupants are engaged in near sedentary physical activity with metabolic rates between 1.0 and 1.3; not in direct sunlight and not exposed to air velocities greater than 0.20m/s(40fpm); and where the difference between MRT and dry-bulb is less than 4C (7F), the operative temperature (to) can be approximated with sufficient accuracy using this simple formula;

**t _{o}= (t_{mr}+t_{db})/2**

where,

t_{o} = operative temperature °R (K)

t_{mr} = mean radiant temperature (also MRT) °R (K)

t_{db} = drybulb temperature, i.e., the air temperature measured by a dry temperature sensor or thermometer, °R (K)

You should imagine this formula as illustrated in *Figure 3*. Picture these values sitting on a teetertotter just the kind you would find in a school playground. On one side you have the mean radiant and on the other the drybulb. What this tells us is that for a given operative temperature (shown at the fulcrum), the mean radiant temperature in a space needs to go up if the air temperature goes down; likewise the mean radiant temperature needs to go down if the drybulb goes up. This is in fact the principle behind the ability to operate spaces conditioned with radiant based HVAC system at a lower drybulb in heating or higher drybulb temperature in cooling (tdb).

You can see this MRT, to and tdb relationship plotted on a graph as shown in *Figure 4*. Note the resultant operative temperature from any combination of mean radiant and drybulb temperature.

So what you should know at this point is operative temperature is a key component of thermal comfort of which the MRT is intimately coupled; plus you should know MRT is intimately associated with building performance where good to great buildings have cooler interior surfaces in summer and warmer interior surfaces in winter (*Figure 5*).

So how do we calculate MRT? Well that all depends on how accurate you want or need to be…the flow chart in *Figure 6* will get you started in the right direction.

**#1 Easy Method (AUST)**

The easiest but least representative method is the weighted average of all the surface temperatures in the space (AUST) and that formula looks like this^{v};

**t _{mr}=t_{1}A_{1}+t_{2}A_{2}+…+t_{N}A_{N}/(A_{1}+A_{2}+…+A_{N})**

where,

t_{mr} = mean radiant temperature

(MRT≈ equal to AUST), °R (K)

t_{N} = temperature of the inside surface N, °R (K)

A_{N} = Area of surface N, ft^{2} (m^{2})

The area you can obtain from the blueprints leaving you with calculating the inside surface temperatures of each of the six sides (tu) of the enclosure (*Figure 7*). Easy task right? Well not really because of the involved dynamics due to fenestration and transient conditions but ASHRAE does provide a simple solution for very simple problems and this is how it looks;

**t _{u}= t_{a}-U/h(t_{a}– t_{o})**

where,

t_{u} = inside surface temperature, °F (C)

t_{a} = drybulb indoor space design(also tdb), °F (C)

t_{o} = drybulb outdoor space design, °F (C)

U = overall (weighted)heat transfer coefficient,

Btuhft^{2}°F (W/m^{2} K)

h = naturalconvection coefficient of the inside surface, Btuhft^{2}°F (W/m^{2}K)

SEE TABLE 1

Now, if all you were doing is a simple, good to great enclosure with low window to wall ratios and all you wanted was to understand the overall radiant influence on the occupant and space – well then Ms. and Mr. Designer use the method above. It is as simple as it gets while retaining your sanity. Just remember that if you are using a radiant based HVAC system such as radiant floors, substitute the design temperature of the floor instead of calculating the floor surface temperature using formula 3.

**#2 Intermediate Method (Plane Radiant, t _{pr} )**

Unlike method one, method #2 recognizes the occupant in the space and provides for two positions, that of standing or seated; these formulae look like this;

**t _{mr(standing)}= **

**{0.08[t _{pr(up)}+t_{pr(down)}]+0.23[t_{pr(right)}+t_{pr(left)}]**

**+0.35[t _{pr(front)}+t_{pr(back
)}]}/[2(0.08+0.23+0.35)]**

and,

**t _{mr(seated)}= **

**{0.18[t _{pr(up)}+t_{pr(down)}]+0.22[t_{pr(right)}+t_{pr(left)}]**

**+0.30[t _{pr(front)}+t_{pr(back)}]}/[2(0.18+0.22+0.30)]**

where,

t_{pr} = plane radiant temperature, °R (K)^{iii}

and where,

pr(up) = ceiling temperature above the occupant,

pr(down) = floor temperature below occupant,

pr(right) = wall temperature to the right of the occupant

pr(left) = wall temperature to the left of the occupant

pr(front) = wall temperature in front of the occupant

pr(back) = wall temperature behind the occupant

As you can see method #2 adds a little bit more complexity but it is not difficult math. Like method #1 the accuracy will depend more on how one determines the weighted inside surface temperature from each of the six sides of the enclosure.

**#3 Difficult Method (Angle or View Factor, F _{p-N} )**

For greater accuracy in more complex spaces the MRT should be calculated from surfaces above and below the center of gravity of the occupant and from all six sides (*Figure 7*); and the calculation must consider the coordinates of the occupant in relation to each of the surfaces, mathematically it is shown as;

**t _{mr}^{4}= t_{1}^{4}F_{p-1}+t_{2}^{4}F_{p-2}+…+t_{N}^{4}F_{p-N}**

where,

t_{mr} = mean radiant temperature, °R (K)

t_{N} = surface temperature of surface N, °R (K)

F_{p-N} = angle factor between a person and surface N

The angle (view) factor (Fp-N ) (*Figure 8*) describes the geometric relationship an occupant has with each surface. The larger and closer the surface is to a person the more potential it has to thermally influence the occupant. To some degree for example, we intuitively know that as we get closer to some surfaces like cold windows we can sense a corresponding influence on our discomfort. Likewise radiant floors for a seated person in a tall space would have more influence on the occupant than say radiant ceilings of the same temperature simply because of the comparative distance from the occupant to those surfaces.

It is beyond the scope of this article to take the readers through a complete angle factor calculation which requires at a minimum, 24 separate determination of view factors (F_{p-N}) and 24 corresponding surface temperatures (t_{N}) plus the summation of the 24 products(t_{N}^{4}F_{p-N}) to derive a MRT; but I will show you the key component of angle (view) factors in the graphs developed by Fanger.

The late Professor Ole Fanger published a number of formulae and graphs in his seminal doctoral thesis, “Thermal Comfort: Analysis and Applications in Environmental Engineering.” These tools from his 1970 paper are applied today to analyze the influence of surface temperatures on occupants based on the person’s geometric coordinates within a space. Along with the angle factor and hand or computer calculations of surface temperatures, one can define the MRT for any individual within any space. Together with the air temperature one can then also define the operative temperature and thus be able to provide at least one key ingredient in thermal comfort analysis. Although not entirely practical for very simple uniform spaces with low loads where MRT and tdb are essentially equal, it is immensely useful for non-uniform environments of complex geometries (think origami glass buildings).

The challenge with using graphs in general is the error in interpreting intersection points. For simplification and improvement in accuracy, the Federation of European Heating, Ventilation and Air Conditioning Associations (REHVA ) published the following formulas and accompanying Table 2, (abridged for brevity)^{v};

**F _{p-N}= F_{max}(1-e^{-(a/c)/τ}**

**)(1-e**

^{-(b/c)/}**)**

Where,

F_{p-N} = angle factor between a person and surface N

and,

**τ= A+B (a/c)**

**= C+D (b/c) + E(a/c)**

Values for A through E can be found in Table 2:

SEE TABLE 2

The intent of this article was to first introduce readers to mean radiant temperature and how and where it fits into thermal comfort analysis using operative temperature. Secondly I wanted to describe three methods of calculating MRT so design practitioners can choose a method appropriate for the spaces being evaluated. Obviously if this is new to you, you have your work cut out to study the literature and attend classes which cover this content. But once you have a basic grasp of the principles I would encourage you to obtain a copy of the ASHRAE Thermal Comfort Software Tool which has an MRT calculator among other “comfort” features (see *Figure 9* on p.78).

My last thoughts on this topic come from global IEQ expert Dr. B.W. Olesen (Director of the International Centre for Indoor Environment and Energy, Denmark) who stated, “We cannot forget – energy efficiency in buildings is for the benefit of conditioning the occupants.” This is echoed by industry colleague Dr. Fergus Nicol (Professor Emeritus of Architecture, Oxford Brookes University) who said, “An energy declaration without a declaration related to the indoor environment makes no sense.” Both of these statements draw attention to the relationship between MRT (a function of building performance), comfort and energy. It is for this reason that I believe we ought to drop the reference in codes to controlling air temperatures and focus on the real problem in buildings which is surface temperatures. At the end of the day in my 40 grit humble opinion, if we actually focused on solving the mean radiant temperature problems with enclosure performance we would solve the energy and thermal comfort problems in buildings. <>

*Robert Bean, R.E.T., P.L.(Eng.), is a registered practitioner in building construction engineering technology (ASET) and a professional licensee in mechanical engineering (APEGGA). He has over 30 years experience in the construction industry specializing in energy and indoor environmental quality and is an author and lecturer for professional development programs addressing building science, thermal comfort quality, indoor air quality and radiant-based HVAC systems.* www.healthyheating.com

high temperature cooling, REHVA Guidebook 7, 2007