Thermal comfort models

Theoretical models

PMV-PPD

PMV represents the 'predicted mean vote' (on the thermal sensation scale) of a large population of people exposed to a certain environment. PMV is derived from the physics of heat transfer combined with an empirical fit to sensation. PMV establishes a thermal strain based on steady-state heat transfer between the body and the environment and assigns a comfort vote to that amount of strain. PPD is the predicted percent of dissatisfied people at each PMV. As PMV changes away from zero in either the positive or negative direction, PPD increases.

The PMV equation for thermal comfort is a steady-state model. It is an empirical equation for predicting the mean vote on a ordinal category rating scale of thermal comfort of a population of people. The equation uses a steady-state heat balance for the human body and postulates a link between the deviation from the minimum load on heat balance effector mechanisms, eg sweating, vaso-constriction, vaso-dilation, and thermal comfort vote. The greater the load, the more the comfort vote deviates from zero. The partial derivative of the load function is estimated by exposing enough people to enough different conditions to fit a curve. PMV (Predicted Mean Vote), as the integrated partial derivative is now known, is arguably the most widely used thermal comfort index today. The ISO (International Standards O rganization) Standard 7730 (ISO 1984), "Moderate Thermal Environments -- Determination of the PMV and PPD Indices and Specification of the Conditions for Thermal Comfort," uses limits on PMV as an explicit definition of the comfort zone.

The PMV equation only applies to humans exposed for a long period to constant conditions at a constant metabolic rate. Conservation of energy leads to the heat balance equation:
H-Ed-Esw-Ere-L=R+C

Where:
H = internal heat production
Ed = heat loss due to water vapour diffusion through the skin
Esw = heat loss due to sweating
Ere = latent heat loss due to respiration
L = dry respiration heat loss
R = heat loss by radiation from the surface of the clothed body
C = heat loss by convection from the surface of the clothed body

The equation is expanded by substituting each component with a function derivable from basic physics. All of the functions have measurable values with exception of clothing surface temperature and the convective heat transfer coefficient which are functions of each other. To solve the equation, an initial value of clothing temperature is estimated, the convective heat transfer coefficient computed, a new clothing temperature calculated etc., by iteration until both are known to a satisfactory degree.

Now let us assume the body is not in balance and write the heat equation as:
L = H-Ed-Esw-Ere-L-R-C,
where L is the thermal load on the body.

Define thermal strain or sensation, Y, as some unknown function of L and metabolic rate. Holding all variables constant except air temperature and metabolic rate, we use mean votes from climate chamber experiments to write Y as function of air temperature for several activity levels. Then substituting L for air temperature, determined from the heat balance equation above, evaluate the partial derivative of Y with respect to L at Y=0 and plot the points versus metabolic rate. An exponential curve is fit to the points and integrated with respect to L. L is simply renamed "PMV" and we have (in simplified form):

PMV = exp[met]*L.
Where:
L=F(Pa,Ta,Tmrt,Tcl)

PMV is "scaled" to predict thermal sensation votes on a seven point scale (hot, warm, slightly warm, neutral, slightly cool, cool, cold) by virtue of the fact that for each physical condition, Y is the mean vote of all subjects exposed to that condition. The major limitation of the PMV model is the explicit constraint of skin temperature and evaporative heat loss to va lues for comfort and "neutral" sensation at a given activity level.

ET*-DISC

ET*- DISC also uses a heat balance model to predict thermal comfort, but the model evolves with time rather than being steady-state like PMV. ET* stands for New Effective Temperature where "effective temperature" is an temperature index that accounts for radiative and latent heat transfers. ET* can be calculated using the '2-Node' model. The 2-node model determines the heat flow between the environment, skin and core body areas on a minute by minute basis. Starting from an initial condition at time=0, the model iterates until equilibrium has been reached (60 minutes is a typical time). The final mean skin temperature and skin wettedness are then associated with an effective temperature. DISC predicts thermal discomfort using skin temperature and skin wettedness.

The 2-node model was introduced in 1970 specifically to formulate a new effective temperature scale. The purpose was to determine particular combinations of physical conditions producing equal physiological strain. Backed by extensive data from climate chamber experiments, it was determined that while skin temperature is a good indicator of thermal comfort sensation in cold environments, skin wettedness is a better indicator in warm environments where sweating occurs because skin temperature changes are small by comparison. The model represents the human body as two concentric cylinders, a core cylinder and a thin skin cylinder surrounding it. Clothing and sweat are assumed to be evenly distributed over the skin surface. At time "zero", the cylinder is exposed to a uniform environment, and the model produces a minute-by-minute simulation of the human thermoregulatory system. After the user-specified time period is reached, the final surface temperature and surface skin wettedness of the cylinder are used to calculate ET*, SET*, and other indices. ET* is the temperature of an environment at 50% relative humidity in which a person experiences the same amount heat loss as in the actual environment.

SET*

SET* numerically represents the thermal strain experienced by the cylinder relative to a "standard" person in a "standard" environment. SET* has the advantage of allowing thermal comparisons between environments at any combination of the physical input variables, but the disadvantage of also requiring "standard" people.

Based on a laboratory study with a large number of subjects, empirical functions between two comfort indices, and skin temperature and skin wettedness, were developed. These functions (both linear) are used in the 2-Node model to produce predicted values of the votes of populations exposed to the same conditions as the cylinder.

TSENS, DISC

TSENS, the first index, represents the model's prediction of a vote on the seven point thermal sensation scale. DISC, the second index, predicts a vote on a scale of thermal discomfort:

DISC:
Intolerable
Very uncomfortable
Uncomfortable
Slightly uncomfortable
Comfortable

The 2-Node model has undergone many iterations and refinements. In the most recent iteration, a new temperature index, PMV*, that incorporates skin wettedness into the PMV equation using SET* or ET* to characterize the environment.

Empirical Models

Apart from the thermal comfort models described above, there are many more theoretical models, both deterministic and empirical. Some empirical models with application to building design and/or environmental engineering are outlined below.

PD

PD or "predicted percent dissatisfied due to draft", is a fit to data of persons expressing thermal discomfort due to drafts. The inputs to PD are air temperature, air velocity , and turbulence intensity. PS is a fit to data of comfortable persons choosing air velocity levels. The inputs to PS are operative temperature and air velocity. TS is a fit to data of thermal sensation as a linear function of air temperature and partial vapour pressure.

A 'draft' is unwanted local cooling. The draft risk (or PD) equation is:
PD=3.413(34-Ta)(v-0.05)0.622+0.369vTu(34-Ta)(v-0.05)0.622

Tu is the turbulence intensity expressed as a percent. 0 represents laminar flow and 100% means that the standard deviation of the air velocity over a certain period is of the same order of magnitude as the mean air velocity. v is the air velocity (in meters per second) and Ta is the air temperature in degrees Celsius. The PD equation arises from two studies in which 100 people were exposed to various combinations of air temperature, air velocity, and turbulence intensity. For each combination of conditions, the people were asked if they felt a draft. PD represents the percent of subjects who voted that they felt a draft for the selected conditions.

PS

The PS equation predicts the air velocity that will be chosen by a person exposed to a certain air temperature when the person has control of the air velocity source. The PS equation is
PS=1.13SQRT(Top)-0.24Top+2.7SQRT(v)-0.99v

Top is operative temperature (in degrees Celsius) and v is the air velocity in meters per second. The PS equation arises from a study in which 50 people were asked to adjust an air velocity source as they pleased when exposed to a specific air temperature. PS represents the cumulative percent of people choosing a particular air velocity at the specific temperatures tested in this experiment

TS

TS is an equation that predicts thermal sensation vote using a linear function of air temperature and partial vapour pressure. The TS equation is:
TS=0.245Ta+0.248p-6.475

Ta is the air temperature in degrees Celsius and p is the partial vapour pressure in kilo-pascals. The TS equation arises from a study similar to the PMV-PPD study described above.

Adaptive models include in some way the variations in outdoor climate for determining thermal preferences indoors.

Auliciems

An adaptive model developed by Auliciems fits sensation data based on field investigations of thermal comfort in Australia spanning several climates. Auliciems equation is:
Tn=9.22+0.48Ta+0.14Tmmo

Humphreys

Humphreys equation is a fit to considerable data for climate-controlled and non-climate controlled buildings:

For both the Auliciems and Humphreys models, Tn is the neutral temperature, Ta is the air temperature, and Tmmo is the mean monthly outdoor temperature.