# PMV/PPD

The **PMV Model** (=Predicted Mean Vote Model) is probably the best know human thermal comfort model especially for indoor applications. It is based on Fangers (1972) comfort model and relates the energy balance of the human body with the humans thermal impression using a straight empirical function (see below). PMV was originally developed for steady-state indoor situations, but by extending the energy flux related parts of the model with solar and longwave radiation and allowing wind speeds above an indoor room situation, PMV can also be applied -with limits- to outdoor situations (see e.g. German VDI 3787 Part 2, 2008). As the original PMV/PPD uses an empirical function to relate the bodys' energy balance to a thermal sensation, it can be discussed, whether it is valid to extend this model above its original context or not. In addition, the clothing temperature is the only parameter of the PMV model that reacts on the environmental conditions, the skin temperature only depends on the activity of the person (see below).

PMV is, like most other thermal comfort indices, a stationary value. This means that the assessed person is assumed to be exposed long enough to a constant climate situation until all energy exchange processes at the human body have become stationary (if possible). While this is the normal case for a person in an indoor environment, it is the exception for most people in an outdoor environment. The validity of stationary indices in an outdoor setting is therefore limited to either non-moving persons or to large isothermal environments where the microclimate stays constant even if moving.

In addition to the PMV value, ENVI-met BioMet provides the associated **PPD value** (*=Predicted Percentage of Dissatisfied*) which tells the percentage of people who would be dissatisfied with the climate conditions found. PMV and PPD have a linear relationship (=they can be directly transformed into one another). Therefore the PPD maps have the same spatial structure as the associated PMV maps, but different units.

### Definition range

Normally, the PMV scale is defined between -4 (very cold) and +4 (very hot) where 0 is the thermal neutral (comfort) value:

But as the PMV value is a mathematical function of the local climate, in most applications it can reach also values above or below the [-4] - [+4] values, althouth these are off scale of the original Fanger experimental data.

### Calculation method

The basic PMV equation for all cases, indoor and outdoor, is given by

$$ PMV=\left[ 0.028+0.303\cdot \exp \left( -0.036\cdot M/A_{Du}\right) \right] \cdot \left( H/A_{Du}-E_{d}-E_{sw}-E_{re}-L-R-C\right) $$

#### Required input

**Meteorological variables**, all assumed to be defined at the biometeorolgical reference height of 1.6 m or the next closest level in case of model data:

- Air temperature $T_a$
- Mean radiant temperature $T_{mrt}$
- Vapour pressure $e$
- Local wind speed $\mathbf{u}$

**Personal settings** human body

- Clothing insulation $I_{clo}$
- $M$: Mechanical energy production of the body
- $\eta$: Mechanical work factor (0 most of the time)

The PMV/PPD reference person is always 35 year old, male, with a height of 1.75 m and a weight of 75 kg. These assumptions cannot be modified in the PMV/PPD calculations.

#### Components of the PMV equation

Let's look at the different terms in details.

##### Scaling terms

- $\mathbf{0.028+0.303\cdot (...)}$:

Empirical based fitting coefficients to transfer the energy balance of the body to the PMV scale range.

##### Body energy production

- $\mathbf{M/A}_{Du}$: Mechanical energy production of the body

Gross energy production of the body related related to 1 m² of skin. Mechanical energy production depends on the persons activity.

- $\mathbf{H/A}_{Du}:$ Internal remaining energy

Produced energy not used for mechanical work $H/A_{Du}=M/A_{Du}\cdot \left( 1-\eta \right)$ with $\eta $ being the mechanical work factor.

($A_{du}$= Dubois-Area, Skin surface area).

##### Skin water/ vapour exchange

- $\mathbf{E}_{d}\mathbf{:}$ Vapour diffusion through the skin

Amount of vapour diffusing directly through the skin $E_{d}=0.305\cdot (57.3-0,07\cdot H/A_{Du}-e)$ with $e$: Vapor pressure of the air in [hPa] - $\mathbf{E}_{sw}$: Evaporation of sweat on the skin

Cooling effect of liquid sweat evaporating from skin $E_{sw}=0.42\cdot \left( H/A_{Du}-58\right) $

##### Energy exchange through breathing

- $\mathbf{E}_{re}$: Latent heat lost through breathing

Energy lost by humidifying the air in the respiratory system $E_{re}=0.0017\cdot M/A_{Du}\cdot \left( 58.7-e\right) $ - $\mathbf{L}$: Sensible heat exchange through breathing

Energy lost/gained directly through heat exchange with the breathed air withing the body $L=0.0014\cdot M/A_{Du}\cdot \left( 34-t_{a}\right) $ with $t_{a}$: Air temperature deg C

##### Energy exchange at body (cloths) surface

- $\mathbf{R}$ : Radiative energy balance of the body (cloths)

In the PMV model, a more or less fully clothed perosn is assumed. As a consequence, the energy interactions between the body surface and the environment are assumed to take only place at the cloths surface.

Net longwave heat exchange with envrionment, $R=3.95\cdot 10^{-8}\cdot f_{cl}\cdot \left( T_{cl}^{4}-T_{mrt}^{4}\right) $

- $f_{cl}$: Enlargement factor of body area due to clothing layer with $f_{cl}=1.0+I_{cl}\cdot 0.15$ with $I_{cl}$: clothing heat insulation in [clo].

- $T_{cl}$ is the surface temperature of the clothing layer given in [K]
- $T_{mrt}$ is the Mean Radiative Temperature of the surrounding environment in [K].

- $\mathbf{C}$: Energy exchange through convection

Heat directly exchanged with the ambient air through turbulent convection: $C=f_{cl}\cdot h_{c}\cdot (t_{cl}-t_{l})$

- $h_{c}$ :turbulent heat transfer coefficient for heat between clothing and air $h_{c}=\max \left( 2.05\cdot(t_{cl}-t_a),12.1\cdot \sqrt{W}\right)$.

Here, $W$ is the relative wind speed at the body surface. In BioMet, we assume that for outdoor conditions $W=\max(\mathbf{u}(x,y,z),v_{p})$ where $\mathbf{u}(x,y,z)$ is the local wind speed and $v_{p}$ is the walking velocity of the person.

##### Main variables skin and clothing temperature

As mentioned before, in the PMV Model, the clothing temperature is the only parameter of the energy balance equation that interacts with the environment. Hence, the average skin temperature only depends on the activity level of the person.

- $\mathbf{t}_{sk}$: Average skin temperature in [deg C]

The average skin temperature is calculated using the simple equation $t_{sk}=35.7-0.0275\cdot H/A_{Du}.$

- $\mathbf{t}_{cl}$: Average temperature clothing in [deg C]

The clothing temperature is calculated based on the skin temperature and the convective and radiative energy fluxes at the clothing surface : $t_{cl}=t_{s}-0.155\cdot I_{cl}\cdot (R+C)$

#### Numerical solution

Solving the PMV equation is simple once the meteorological parameters and the personal settings are known and defined. The only non-linear term in the equation is the estimation of the clothing temperature $T_{cl}$ as $R$ and $C$ are depending on $T_{cl}$ while $T_{cl}$ itself is defined using both $R$ and $C$. This recursive dependency must be solved iteratively, in which we iterate for a $T_{cl}$ that satisfies $0=R(T_{cl})+C(T_{cl})$. Once this clothing temperature is found, all other terms can be calculated and PMV can be estimated.

### Extension to outdoor conditions

As mentioned above, the concept of PMV/PPD as established by Fanger (1972) was designed for indoor applications. This does affect two fundamental aspects of the PMV mode: the design of the equations and the transition from energy balance units to comfort votes.

First, using the clothing surface temperature as the only environment sensitive variable may be acceptable under office conditions in the absence of direct sun light and almost no ventilation. In an outdoor setting in warm to hot climates, relevant fractions of the human body are not covered by clothing but are exposed to the outdoor environment with varying radiative loads and higher wind speeds. For those parts of the body, the skin temperature will be substantially different to the values estimated by the equations above.

Secondly, the PMV equation relates physical values (energy balance) to a personal comfort assessment. This has been achieved through tests using volunteering persons in a climate camber. As a matter of fact and humanity, the PMV range was only checked within the range of -4 to +4, so the volunteers were neither deep frozen nor grilled. From a scientific point of view, the transition from an energy balance to a comfort vote based on an an empirical study is only valid within the range of the original study. Applying the PMV equation to outdoor conditions in summer heat stress situations can easily produce PMV values high above +4 (+8 and more). While this result is numerically correct, it violates the range of the original PMV system.

Despite these problems, PMV in its outdoor version is able summarize the effects of air temperature, radiation, humidity and wind on the persons energy balance in one value each of it weighted with level of influence.

For more waterproof studies, we suggest to use PET as a thermal comfort scale.

### References

- Fanger, P. O. (1982). Thermal Comfort. Analysis and Application in Environment Engineering. McGraw Hill Book Company, New York.
- VDI (2008): VDI 3787. Environmental meteorology. Methods for the human biometeorological evaluation of climate and air quality for urban and regional planning at regional level. Part I: Climate, Blatt 2/ Part 2