2.1. Heat Balance of the Body
The complex system of heat transfer within the human body and with the environment is often simplified for evaluation using a two-node model, including in ASHRAE [
17] and ISO algorithms. The body core and the skin comprise the 2 nodes, with a single layer of clothing, each with its own average temperature to represent the whole body. The heat generation of the body is expressed in terms of metabolic rate units (met), or as an equivalent heat flux at the skin surface. The metabolic rate while seated and quiet is defined as 1 met corresponding to 58W/m
2 of skin surface area. Workers have increased metabolic rates, expressed as larger met values; such as 1.7 met for “walking about” in office work, 2.0-2.4 for light machine work, or 4.0 met for handling 50kg bags [
18]. These metabolic rates are used along with the environment WBGT (wet bulb globe temperature) to determine any restrictions on activity to help reduce heat stress, such as in the JIS Standard Z8504 [
19] (based on ISO 7243), where an increase in metabolic rate of about 30W/m
2 reduces the acceptable WBGT by 1ºC.
The heat flux from the skin surface
Qsk for a person wearing clothing with an insulation value of
Rcl in an environment with operative temperature
To can be found by [
20],
where
Tsk is the skin surface temperature and
Ra,cl is the thermal resistance of the outermost boundary (skin or clothing), which can be taken as the inverse of the heat transfer coefficient with the environment, thus
Ra,cl = 1/
h. In the characterization experiments in still for the unclothed mannequin used here,
Ra,cl had a whole-body average of 0.115m
2K/W, which corresponds to a net heat transfer coefficient of 8.7W/m
2K.
The above equation incorporates both convection and radiant heat transfer through use of operative temperature. Separate evaluation of these two components at the bare skin surface is the sum of the convective heat flux
Qc and the radiant heat flux
Qr
where
hc and
hr are the convective and radiant heat transfer coefficients,
Tair is the environment air temperature,
Ts is the skin surface temperature and
Tr is the mean radiant temperature. The radiant heat transfer coefficient for a standing human is taken as 4.7 W/m
2K as per ASHRAE. If net surface heat flux, skin surface temperature, and environment temperatures are measured, they can be substituted into Eq (2) to find the convective heat transfer coefficient during each experiment trial or the total thermal insulation value of clothing plus the surface air layer
Itot [
21],
Evaporative cooling at the skin surface
Esk for a person wearing clothes with an evaporative heat transfer resistance of
Re,cl in an environment with water vapor pressure of
Pair can be found by
where
Psk is the water vapor pressure at the skin (normally assumed as that of the saturated state), w is the skin wettedness,
fcl is the clothing area factor (the ratio of clothing area to skin area), and
he is the evaporative heat transfer coefficient. However, this formulation assumes one value of wettedness, while misting onto PPE would yield a second wet layer, possibly at different locations than natural sweat. Evaporation due to mist wetting may in part be evaluated by a method explained below.
2.2. Evaporation Cooling Effect on Air
The cooling effect of water evaporation
Qev is from the latent heat of evaporation
L, plus any sensible heat exchange as the water temperature
Tw changes before evaporation by
where
mev is the evaporated mass,
mw is the mass of water interacting with air, and C
P,w is the specific heat of liquid water taken as 4.184kJ/kg-K. The latent heat of water evaporating in air at 25ºC is about 2450kJ/kg. When a packet of air of mass
mair at initial temperature
Ta,i into which water evaporates yielding a cooling of
Qev, the average cooled air temperature
Tc can be found substituting into
where C
P,air is the specific heat of air, taken as 1.005kJ/kg-K. Generally, the lowest possible evaporation-cooled air temperature in this adiabatic process is the wet bulb temperature. The ratio of change in air temperature to this maximum possible air temperature change (called the “wet bulb depression”) is the evaporation efficiency
ηev.
As water evaporates, it also increases the amount of water vapor in air. The change from the initial vapor pressure
Pa,i to that in the cooled air
pc is proportional to the evaporation efficiency, where the maximum potential water vapor pressure is to the saturated state
psat at the same specific enthalpy in the initial state in the adiabatic process.
In a case with well-defined boundaries, such as an air duct, with complete evaporation of all water, this calculation can be relatively simple. In the case of mist evaporation cooling in open spaces, it is difficult to evaluate the amount of air being cooled, and its ratio to the amount of water evaporated at any given point.
2.4. Maximum Convection and Radiation Heat Transfer without Wetting
The maximum possible convection heat transfer due to mist cooling without surface/skin evaporation wetting effects at the outer surface can be found using Eq.(10), taking
Ts as the initial surface/skin temperature with the air temperature reduced to the wet bulb temperature. This would represent a case with air reaching the minimum possible temperature and the surface not having yet cooled.
The radiant heat transfer could reach a maximum with the mist cloud completely blocking any environmental thermal radiation and solely producing a radiant temperature of the wet bulb temperature cloud,
The convection heat transfer coefficient
hc needed in Eq.(10) can be taken from the literature, or found for each body location in each experiment trial from the net heat flux
Qc +
Qr measured during forced convection cooling with only the fan while taking
hr as 4.7 and substituting into Eq.(2), such that
Any heat transfer above the total of these amounts (Qc,max + Qr,max) may be due to wetting yielding evaporation cooling at the outer surface.
The heat flux from the skin surface can rapidly change when mist cooling occurs. In our previous research with misting fans, it was found that human skin, or a heated silicone rubber surface similar to human skin, showed temperature drops more than 2K in under 5 seconds even when exposed to short (about 3s) passes of an oscillating mist fan with a 45s cycle [
23]. When mist-cooled air reaches the outer surface, and the skin surface rapidly cools, the temperature differences in the above equations change rapidly, resulting in change in total heat flux over time. In general, the initial arrival of mist cooled air should have the highest heat flux (cooling) as the difference between cooled air temperature and warm skin is likely large. The heat flux should then drop as the skin temperature decreases while the mist-cooled air temperature is roughly constant. This trend should appear in the experiment data.
Two theoretical examples of the change in skin temperature and heat flux are created in
Figure 1 and
Figure 2. In the first case, the air temperature
Tair and radiant temperature
Tr are initially 33ºC, lower than the skin temperature
Tskin is 35ºC. Mist fan cooling causes a 5K drop in air temperature for 5 minutes, then stops. Heat transfer coefficients are taken from the above system of equations. For this simple simulation case, the skin compartment reaction is set as a lumped sum thermal capacitance model with internal heating. The cooling effect,
Qcool, starts at a peak and slowly decreases as the skin temperature decreases, thus the temperature difference driving the cooling effect becomes smaller. When misting stops, the skin temperature is lower than the environment temperature, so the heat flux at skin becomes negative (heating) and the skin returns toward the initial temperature.
In the second case, the air and radiant temperatures are set at 36ºC, higher than the skin temperature at 35ºC. There is a similar initial peak in mist cooling, but when mist is not active, the skin is being heated by the environment.
As an example case of heat flux values, a hot summer day in Osaka of 35°C and 45%RH has a
Twb of about 25°C. According to ASHRAE, the convection heat transfer coefficient
hc for a standing person in air moving at speed
v is 14.8v
0.69 W/m
2K, while in still air it is 4.0 W/m
2K [
24].
If skin temperature is initially 34°C, the globe temperature in shade is the same as air temperature, hc is the value for a human standing in 1m/s moving air 14.8 W/m2K, and hr is 4.7, the maximum non-wet cooling effect from a misting fan (Qc,max + Qr,max) would be about 175W/m2, while in still air it would be about 78W/m2. At the highest air velocity examined in these experiments, where v = 2.6m/s, the maximum non-wet cooling effect of the misting fan would be about 300 W/m2.
High cooling heat flux values above this theoretical maximum, which is determined from the measured heat transfer with fan convection alone in each trial, would likely indicate that wetting of the outer surface has occurred. Wetting the skin surface or clothing would add evaporative cooling to the case where the natural evaporative cooling from sweat is restricted.
The cooling due to evaporation from a wet surface depends on the difference in vapor pressure between the wet surface, taken as the saturated vapor pressure
Psat, and the environment vapor pressure
Pair and the fraction of the surface that is wet,
w.
The evaporation heat transfer coefficient
he is related to the convection heat transfer coefficient by the Lewis relation.
If a surface becomes wetted by the misting fan, it will tend to drop toward the wet bulb temperature, both due to the mist droplets already being at the wet bulb temperature
Twb, and added forced convection of the fan bringing conditions similar to an aspirated wet bulb thermometer. The net cooling effect can no longer be explained by Eqs (2), (10) and (11). Evaporation cooling
E is added, and surface temperature
Ts tends to fall toward
Twb, while
Tair is at some value between the initial environment air temperature and the wet bulb temperature, depending on the density of the mist.
Accurate evaluation of this process requires sensors that can reliably measure air and surface temperature while wet as well as the wettedness ratio of skin and clothing. This is beyond the scope of this experiment.