# coding=utf-8
"""Utility functions for calculating Predicted MEan Vote (PMV).
PMV is a thermal comfort model for use on the interior of buildings where a
heating or cooling system is operational. For naturally ventilated buildings,
the Adaptive thermal comfort model is recommended and, for outdoor conditions,
models such as Universal Thermal Climate Index (UTCI) or Physiological
Equivalent Temperature (PET) are recommended.
"""
from __future__ import division
from ladybug.rootfinding import secant
from ladybug.rootfinding import bisect
import math
FAILURE_MESSAGE = 'The following conditions caused a failure of PMV model ' \
'convergence: Ta = {}; Tr = {}; Vel = {}; RH = {}; Met = {}; Clo= {}'
[docs]def predicted_mean_vote(ta, tr, vel, rh, met, clo, wme=0, still_air_threshold=0.1):
"""Calculate PMV using Fanger's original model and Pierce SET.
This method uses the officially correct way to calculate PMV comfort according to
the 2017 ASHRAE-55 Thermal Comfort Standard. This function will return
valid values even if the air speed is above the sill air threshold of
Fanger's original model (> 0.1 m/s).
However, because this function always returns the Standard Effective Temperature
(SET), it will run much more slowly for cases that are below the still air threshold
(roughly 10 times as long). For cases where a SET output is not required, it
is recommended that the predicted_mean_vote_no_set method be used.
Note:
[1] ASHRAE Standard 55 (2017). "Thermal Environmental Conditions
for Human Occupancy". Atlanta Georgia: American Society of Heating,
Refrigerating and Air Conditioning Engineers.
[2] Hoyt Tyler, Schiavon Stefano, Piccioli Alberto, Cheung Toby, Moon Dustin,
and Steinfeld Kyle, 2017, CBE Thermal Comfort Tool. Center for the Built
Environment, University of California Berkeley,
http://comfort.cbe.berkeley.edu/
[3] Doherty, T.J., and E.A. Arens. (1988). Evaluation of the physiological
bases of thermal comfort models. ASHRAE Transactions, Vol. 94, Part 1, 15 pp.
https://escholarship.org/uc/item/6pq3r5pr
Args:
ta: Air temperature [C]
tr: Mean radiant temperature [C]
vel: Relative air velocity [m/s]
rh: Relative humidity [%]
met: Metabolic rate [met]
clo: Clothing [clo]
wme: External work [met], normally around 0 when seated
still_air_threshold: The air velocity in m/s at which the Pierce
Standard Effective Temperature (SET) model will be used
to correct values in the original Fanger PMV model.
Default is 0.1 m/s per the 2015 release of ASHRAE Standard-55.
Returns:
A dictionary containing results of the PMV model with the following keys
- pmv -- Predicted mean vote (PMV)
- ppd -- Percent predicted dissatisfied (PPD) [%]
- set -- Standard effective temperature (SET) [C]
- ta_adj -- Air temperature adjusted for air speed [C]
- ce -- Cooling effect. The difference between the air temperature
and the adjusted air temperature [C]
- heat_loss -- A dictionary with the 6 heat loss terms of the PMV model.
The dictionary keys are as follows:
- cond -- heat loss by conduction [W]
- sweat -- heat loss by sweating [W]
- res_l -- heat loss by latent respiration [W]
- res_s -- heat loss by dry respiration [W]
- rad -- heat loss by radiation [W]
- conv -- heat loss by convection [W]
"""
se_temp = pierce_set(ta, tr, vel, rh, met, clo, wme)
if vel <= still_air_threshold: # use the original Fanger model
pmv, ppd, heat_loss = fanger_pmv(ta, tr, vel, rh, met, clo, wme)
ta_adj, ce = ta, 0.
else: # use the SET model to correct the cooling effect in Fanger model
ce_l = 0.
ce_r = 40.
eps = 0.001 # precision of ce
def fn(ce):
return se_temp - pierce_set(ta - ce, tr - ce, still_air_threshold,
rh, met, clo, wme)
try:
ce = secant(ce_l, ce_r, fn, eps)
except OverflowError:
ce = None
if ce is None: # ce can be None because OverflowError or max secant iterations
ce = bisect(ce_l, ce_r, fn, eps, 0)
pmv, ppd, heat_loss = fanger_pmv(
ta - ce, tr - ce, still_air_threshold, rh, met, clo, wme)
ta_adj = ta - ce
result = {}
result['pmv'] = pmv
result['ppd'] = ppd
result['set'] = se_temp
result['ta_adj'] = ta_adj
result['ce'] = ce
result['heat_loss'] = heat_loss
return result
[docs]def predicted_mean_vote_no_set(
ta, tr, vel, rh, met, clo, wme=0, still_air_threshold=0.1):
"""Calculate PMV using Fanger's model and Pierce SET model ONLY WHEN NECESSARY.
This method uses the officially correct way to calculate PMV comfort according to
the 2015 ASHRAE-55 Thermal Comfort Standard. This function will return
correct values even if the air speed is above the sill air threshold of
Fanger's original equation (> 0.1 m/s).
However, because this function does not return the Standard Effective Temperature
(SET), it will run much faster for cases that are below the still air threshold
(roughly 1/10th the time).
Args:
ta: Air temperature [C]
tr: Mean radiant temperature [C]
vel: Relative air velocity [m/s]
rh: Relative humidity [%]
met: Metabolic rate [met]
clo: Clothing [clo]
wme: External work [met], normally around 0 when seated.
still_air_threshold: The air velocity in m/s at which the Pierce
Standard Effective Temperature (SET) model will be used
to correct values in the original Fanger PMV model.
Default is 0.1 m/s per the 2015 release of ASHRAE Standard-55.
Returns:
A dictionary containing results of the PMV model with the following keys
- pmv -- Predicted mean vote (PMV)
- ppd -- Percent predicted dissatisfied (PPD) [%]
- ta_adj -- Air temperature adjusted for air speed [C]
- ce -- Cooling effect. The difference between the air temperature
and the adjusted air temperature [C]
- heat_loss -- A dictionary with the 6 heat loss terms of the PMV model.
The dictionary keys are as follows:
- cond -- heat loss by conduction [W]
- sweat -- heat loss by sweating [W]
- res_l -- heat loss by latent respiration [W]
- res_s -- heat loss by dry respiration [W]
- rad -- heat loss by radiation [W]
- conv -- heat loss by convection [W]
"""
if vel <= still_air_threshold: # use the original Fanger model
pmv, ppd, heat_loss = fanger_pmv(ta, tr, vel, rh, met, clo, wme)
ta_adj, ce = ta, 0.
else: # use the SET model to correct the cooling effect in Fanger model
ce_l = 0.
ce_r = 40.
eps = 0.001 # precision of ce
se_temp = pierce_set(ta, tr, vel, rh, met, clo, wme)
def fn(ce):
return se_temp - pierce_set(ta - ce, tr - ce, still_air_threshold,
rh, met, clo, wme)
try:
ce = secant(ce_l, ce_r, fn, eps)
except OverflowError:
ce = None
if ce is None: # ce can be None because OverflowError or max secant iterations
ce = bisect(ce_l, ce_r, fn, eps, 0)
pmv, ppd, heat_loss = fanger_pmv(
ta - ce, tr - ce, still_air_threshold, rh, met, clo, wme)
ta_adj = ta - ce
result = {}
result['pmv'] = pmv
result['ppd'] = ppd
result['ta_adj'] = ta_adj
result['ce'] = ce
result['heat_loss'] = heat_loss
return result
[docs]def fanger_pmv(ta, tr, vel, rh, met, clo, wme=0):
"""Calculate PMV using only Fanger's original equation.
Note that Fanger's original experiments were conducted at
low air speeds (<0.1 m/s) and the 2015 ASHRAE-55 thermal comfort
standard states that one should use standard effective temperature
(SET) to correct for the cooling effect of air speed in cases
where such speeds exceed 0.1 m/s. The pmv() function in this module
will apply this SET correction in cases where it is appropriate.
Note:
[1] Fanger, P.O. (1970). Thermal Comfort: Analysis and applications
in environmental engineering. Copenhagen: Danish Technical Press.
Args:
ta: Air temperature [C]
tr: Mean radiant temperature [C]
vel: Relative air velocity [m/s]
rh: Relative humidity [%]
met: Metabolic rate [met]
clo: Clothing [clo]
wme: External work [met], normally around 0 when seated
Returns:
A tuple with three elements
- pmv: Predicted mean vote (PMV)
- ppd: Percentage of people dissatisfied (PPD) [%]
- heat_loss: A dictionary with the 6 heat loss terms of the PMV model.
The dictionary items are as follows:
- 'cond': heat loss by conduction [W]
- 'sweat': heat loss by sweating [W]
- 'res_l': heat loss by latent respiration [W]
- 'res_s' heat loss by dry respiration [W]
- 'rad': heat loss by radiation [W]
- 'conv' heat loss by convection [W]
"""
pa = rh * 10. * math.exp(16.6536 - 4030.183 / (ta + 235.))
icl = 0.155 * clo # thermal insulation of the clothing in M2K/W
m = met * 58.15 # metabolic rate in W/m2
w = wme * 58.15 # external work in W/m2
mw = m - w # internal heat production in the human body
if icl <= 0.078:
fcl = 1 + (1.29 * icl)
else:
fcl = 1.05 + (0.645 * icl)
# heat transfer coefficient by forced convection
hcf = 12.1 * math.sqrt(vel)
taa = ta + 273.
tra = tr + 273.
tcla = taa + (35.5 - ta) / (3.5 * icl + 0.1)
p1 = icl * fcl
p2 = p1 * 3.96
p3 = p1 * 100.
p4 = p1 * taa
p5 = 308.7 - 0.028 * mw + (p2 * ((tra / 100.) ** 4))
xn = tcla / 100.
xf = tcla / 50.
eps = 0.00015
n = 0
while abs(xn - xf) > eps:
xf = (xf + xn) / 2.
hcn = 2.38 * (abs(100.0 * xf - taa) ** 0.25)
if hcf > hcn:
hc = hcf
else:
hc = hcn
xn = (p5 + p4 * hc - p2 * (xf ** 4)) / (100. + p3 * hc)
n += 1
if n > 150:
print(FAILURE_MESSAGE.format(ta, tr, vel, rh, met, clo))
return 0.0, 5.0, \
{'cond': 0, 'sweat': 0, 'res_l': 0, 'res_s': 0, 'rad': 0, 'conv': 0}
tcl = 100. * xn - 273.
# heat loss conduction through skin
hl1 = 3.05 * 0.001 * (5733. - (6.99 * mw) - pa)
# heat loss by sweating
if mw > 58.15:
hl2 = 0.42 * (mw - 58.15)
else:
hl2 = 0
# latent respiration heat loss
hl3 = 1.7 * 0.00001 * m * (5867. - pa)
# dry respiration heat loss
hl4 = 0.0014 * m * (34. - ta)
# heat loss by radiation
hl5 = 3.96 * fcl * (math.pow(xn, 4) - math.pow(tra / 100., 4))
# heat loss by convection
hl6 = fcl * hc * (tcl - ta)
ts = 0.303 * math.exp(-0.036 * m) + 0.028
pmv = ts * (mw - hl1 - hl2 - hl3 - hl4 - hl5 - hl6)
ppd = ppd_from_pmv(pmv)
# collect heat loss terms.
heat_loss = {
'cond': hl1,
'sweat': hl2,
'res_l': hl3,
'res_s': hl4,
'rad': hl5,
'conv': hl6
}
return pmv, ppd, heat_loss
[docs]def pierce_set(ta, tr, vel, rh, met, clo, wme=0.):
"""Calculate Standard Effective Temperature (SET).
This function uses the J.B. Pierce two-node model of human thermoregulation.
Note:
[1] Nishi, Y; Gagge, A.P. (1977). "Effective temperature scale useful for
hypo-and hyperbaric environments". Aviation, Space, and Environmental Medicine.
48 (2): 97–107.
Args:
ta: Air temperature [C]
tr: Mean radiant temperature [C]
vel: Relative air velocity [m/s]
rh: Relative humidity [%]
met: Metabolic rate [met]
clo: Clothing [clo]
wme: External work [met], normally around 0 when seated
Returns:
se_temp -- Standard effective temperature [C]
"""
# key initial variables
vapor_pressure = (rh * saturated_vapor_pressure_torr(ta)) / 100.
air_velocity = max(vel, 0.1)
kclo = 0.25
bodyweight = 69.9
bodysurfacearea = 1.8258
metfactor = 58.2
sbc = 0.000000056697 # Stefan-Boltzmann constant (W/m2-K4)
csw = 170.
cdil = 120.
cstr = 0.5
temp_skin_neutral = 33.7 # setpoint (neutral) value for Tsk
temp_core_neutral = 36.8 # setpoint value for Tcr
temp_body_neutral = 36.49 # setpoint for Tb
skin_blood_flow_neutral = 6.3 # neutral value for skin_blood_flow
# INITIAL VALUES - start of 1st experiment
temp_skin = temp_skin_neutral
temp_core = temp_core_neutral
skin_blood_flow = skin_blood_flow_neutral
mshiv = 0.0
alfa = 0.1
esk = 0.1 * met
# Start new experiment here (for graded experiments)
# UNIT CONVERSIONS (from input variables)
p = 101.325 # pressure of the atmosphere in kPa
pressure_in_atmospheres = p * 0.009869
ltime = 60
rcl = 0.155 * clo
facl = 1.0 + 0.15 * clo # % INCreaSE IN BODY SURFACE Area DUE TO CLOTHING
LR = 2.2 / pressure_in_atmospheres # Lewis Relation is 2.2 at sea level
RM = met * metfactor
M = met * metfactor
if clo <= 0:
wcrit = 0.38 * pow(air_velocity, -0.29)
icl = 1.0
else:
wcrit = 0.59 * pow(air_velocity, -0.08)
icl = 0.45
chc = 3.0 * pow(pressure_in_atmospheres, 0.53)
chcV = 8.600001 * pow((air_velocity * pressure_in_atmospheres), 0.53)
chc = max(chc, chcV)
# initial estimate of Tcl
chr = 4.7
ctc = chr + chc
ra = 1.0 / (facl * ctc) # resistance of air layer to dry heat transfer
top = (chr * tr + chc * ta) / ctc
tcl = top + (temp_skin - top) / (ctc * (ra + rcl))
# ======================== BEGIN ITERATION
#
# Tcl and chr are solved iteratively using: H(Tsk - To) = ctc(Tcl - To),
# where H = 1 / (ra + Rcl) and ra = 1 / Facl * ctc
tcl_old = False
flag = True
for i in range(ltime - 1):
while abs(tcl - tcl_old) > 0.01:
if flag:
tcl_old = tcl
chr = 4.0 * sbc * pow(((tcl + tr) / 2.0 + 273.15), 3.0) * 0.72
ctc = chr + chc
ra = 1.0 / (facl * ctc) # resistance of air layer to dry heat transfer
top = (chr * tr + chc * ta) / ctc
tcl = (ra * temp_skin + rcl * top) / (ra + rcl)
flag = True
flag = False
dry = (temp_skin - top) / (ra + rcl)
hfcs = (temp_core - temp_skin) * (5.28 + 1.163 * skin_blood_flow)
eres = 0.0023 * M * (44.0 - vapor_pressure)
cres = 0.0014 * M * (34.0 - ta)
scr = M - hfcs - eres - cres - wme
ssk = hfcs - dry - esk
tcsk = 0.97 * alfa * bodyweight
tccr = 0.97 * (1 - alfa) * bodyweight
dtsk = (ssk * bodysurfacearea) / (tcsk * 60.0) # deg C per minute
dtcr = scr * bodysurfacearea / (tccr * 60.0) # deg C per minute
temp_skin = temp_skin + dtsk
temp_core = temp_core + dtcr
TB = alfa * temp_skin + (1 - alfa) * temp_core
sksig = temp_skin - temp_skin_neutral
warms = (sksig > 0) * sksig
colds = ((-1.0 * sksig) > 0) * (-1.0 * sksig)
crsig = (temp_core - temp_core_neutral)
warmc = (crsig > 0) * crsig
coldc = ((-1.0 * crsig) > 0) * (-1.0 * crsig)
bdsig = TB - temp_body_neutral
warmb = (bdsig > 0) * bdsig
skin_blood_flow = (skin_blood_flow_neutral + cdil * warmc) / (1 + cstr * colds)
if skin_blood_flow > 90.0:
skin_blood_flow = 90.0
if skin_blood_flow < 0.5:
skin_blood_flow = 0.5
regsw = csw * warmb * math.exp(warms / 10.7)
if regsw > 500.0:
regsw = 500.0
ersw = 0.68 * regsw
rea = 1.0 / (LR * facl * chc) # evaporative resistance of air layer
recl = rcl / (LR * icl) # evaporative resistance of clothing (icl=.45)
emax = (saturated_vapor_pressure_torr(
temp_skin) - vapor_pressure) / (rea + recl)
prsw = ersw / emax
pwet = 0.06 + 0.94 * prsw
edif = pwet * emax - ersw
esk = ersw + edif
if pwet > wcrit:
pwet = wcrit
prsw = wcrit / 0.94
ersw = prsw * emax
edif = 0.06 * (1.0 - prsw) * emax
esk = ersw + edif
if emax < 0:
edif = 0
ersw = 0
pwet = wcrit
prsw = wcrit
esk = emax
esk = ersw + edif
mshiv = 19.4 * colds * coldc
M = RM + mshiv
alfa = 0.0417737 + 0.7451833 / (skin_blood_flow + .585417)
# Define new heat flow terms, coefficients, and abbreviations
hsk = dry + esk # total heat loss from skin
RN = M - wme # net metabolic heat production
ecomf = 0.42 * (RN - (1 * metfactor))
if ecomf < 0.0:
ecomf = 0.0 # from Fanger
emax = emax * wcrit
W = pwet
pssk = saturated_vapor_pressure_torr(temp_skin)
# Definition of ASHRAE standard environment... denoted "S"
chrS = chr
if met < 0.85:
chcS = 3.0
else:
chcS = 5.66 * pow((met - 0.85), 0.39)
if chcS < 3.0:
chcS = 3.0
ctcs = chcS + chrS
rclos = 1.52 / ((met - wme / metfactor) + 0.6944) - 0.1835
rcls = 0.155 * rclos
facls = 1.0 + kclo * rclos
fcls = 1.0 / (1.0 + 0.155 * facls * ctcs * rclos)
ims = 0.45
icls = ims * chcS / ctcs * (1 - fcls) / (chcS / ctcs - fcls * ims)
ras = 1.0 / (facls * ctcs)
reaS = 1.0 / (LR * facls * chcS)
reclS = rcls / (LR * icls)
hd_s = 1.0 / (ras + rcls)
he_s = 1.0 / (reaS + reclS)
# SET* (standardized humidity, clo, Pb, and chc)
# determined using Newton's iterative solution
# FNERRS is defined in the GENERAL SETUP section above
delta = .0001
dx = 100.0
x_old = temp_skin - hsk / hd_s # lower bound for SET
while abs(dx) > .01:
be1 = pssk - 0.5 * saturated_vapor_pressure_torr(x_old)
err1 = hsk - hd_s * (temp_skin - x_old) - W * he_s * be1
be2 = pssk - 0.5 * saturated_vapor_pressure_torr((x_old + delta))
err2 = hsk - hd_s * (temp_skin - (x_old + delta)) - W * he_s * be2
x = x_old - delta * err1 / (err2 - err1)
dx = x - x_old
x_old = x
se_temp = x
return se_temp
[docs]def saturated_vapor_pressure_torr(db_temp):
"""Calculate saturated vapor pressure (Torr) at temperature (C)
This is used to synchronize the results of the Standard Effective temperature (SET)
model with the results of the original Fanger model.
"""
return math.exp(18.6686 - 4030.183 / (db_temp + 235.0))
[docs]def ppd_from_pmv(pmv):
"""Calculate the Percentage of People Dissatisfied (PPD) from PMV.
Args:
pmv: The predicted mean vote (PMV) for which you want to know the PPD.
Returns:
ppd -- The percentage of people dissatisfied (PPD) for the input PMV.
"""
return 100.0 - 95.0 * math.exp(-0.03353 * pow(pmv, 4.0) - 0.2179 * pow(pmv, 2.0))
[docs]def pmv_from_ppd(ppd, pmv_up_bound=3, ppd_tolerance=0.001):
"""Calculate the two possible Predicted Mean Vote (PMV) values for a PPD value.
Args:
ppd: The percentage of people dissatisfied (PPD) for which you want to know
the possible PMV.
pmv_up_bound: The upper limit of PMV expected for the input PPD. Putting
in a good estimate here will help the model converge on a
solution faster. Defaut = 3
ppd_tolerance: The acceptable error in meeting the target PPD. Default = 0.001.
Returns:
A tuple with two elements
- pmv_lower: The lower (cold) PMV value that will produce the input ppd.
- pmv_upper: The upper (hot) PMV value that will produce the input ppd.
"""
assert ppd > 5 and ppd < 100, \
'PPD value {}% is outside acceptable limits of the PMV model.'.format(ppd)
def fn(pmv):
return (100.0 - 95.0 * math.exp(
-0.03353 * pow(pmv, 4.) - 0.2179 * pow(pmv, 2.0))) - ppd
# Solve for the missing higher PMV value.
pmv_upper = secant(0, pmv_up_bound, fn, ppd_tolerance)
if pmv_upper is None:
pmv_upper = bisect(0, pmv_up_bound, fn, ppd_tolerance, 0)
pmv_lower = pmv_upper * -1
return pmv_lower, pmv_upper
[docs]def ppd_threshold_from_comfort_class(comf_class):
"""Get acceptable PPD comfort threshold given the EN-15251 comfort class.
Args:
comf_class: An integer representing the EN-15251 comfort class.
Choose from: 1, 2, 3 (the higher the class, the greater the PPD threshold)
Returns:
ppd_thresh -- Acceptable PPD threshold.
"""
if comf_class == 1:
ppd_thresh = 6
elif comf_class == 2:
ppd_thresh = 10
elif comf_class == 3:
ppd_thresh = 15
else:
raise ValueError('Comfort class {} is not an acceptable value. '
'Choose from: 1, 2, 3'.format(comf_class))
return ppd_thresh