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International Journal of Biometeorology
https://doi.org/10.1007/s00484-022-02321-2
ORIGINAL PAPER
Thermodynamic assessment ofheat stress indairy cattle: lessons
fromhuman biometeorology
SepehrForoushani1 · ThomasAmon1,2
Received: 7 January 2022 / Revised: 12 May 2022 / Accepted: 22 June 2022
© The Author(s) 2022
Abstract
A versatile meteorological index for predicting heat stress in dairy cattle remains elusive. Despite numerous attempts at
developing such indices and widespread use of some, there is growing skepticism about the accuracy and adequacy of the
existing indices as well as the general statistical approach used to develop them. At the same time, precision farming of
high-yielding animals in a drastically changing climate calls for more effective prediction and alleviation of heat stress. The
present paper revisits classical work on human biometeorology, particularly the apparent temperature scale, to draw inspira-
tion for advancing research on heat stress in dairy cattle. The importance of a detailed, mechanistic understanding of heat
transfer and thermoregulation is demonstrated and reiterated. A model from the literature is used to construct a framework
for identifying and characterizing conditions of potential heat stress. New parameters are proposed to translate the heat flux
calculations based on heat-balance models into more tangible and more useful meteorological indices, including an apparent
temperature for cattle and a thermoregulatory exhaustion index. A validation gap in the literature is identified as the main
hindrance to the further development and deployment of heat-balance models. Recommendations are presented for system-
atically addressing this gap in particular and continuing research within the proposed framework in general.
Keywords Heat stress· Mechanistic model· Thermoregulation· Apparent temperature· Thermoregulatory exhaustion
index
Introduction
Overview
Heat stress in dairy cattle has been the subject of contin-
ued interest for several decades. A significant portion of
the literature can be categorized as attempts to develop heat
stress indices through regression analysis of meteorologi-
cal parameters such as ambient temperature, humidity, wind
speed, and solar radiation and animal responses such as body
temperature, respiration rate, and milk yield, or select the
“appropriate” index from the literature. In the latter case,
meteorological data are used to evaluate various available
indices, in search of indices that show significant statistical
correlation with animal response data. Despite numerous
attempts, a versatile index remains elusive as evidenced by
continual revisions to existing indices and development of
new indices. A recent review (Ji etal. 2020a) lists as many
as 20 heat stress indices for dairy cattle.
An alternative, possibly complementary, approach is to
use mechanistic models of heat generation and dissipation
to identify conditions of potential stress based on the heat
balance of the animal. Examples include the work of McAr-
thur (1987), Ehrlemark and Sällvik (1996), Turnpenny etal.
(2000a, b), McGovern and Bruce (2000), and Thompson
etal. (2014). Despite its fundamental robustness, the heat-
balance approach has attracted much less attention than
the statistical approach and its application remains limited.
Very few studies deal with the application and assessment
of heat-balance models, including for instance the papers by
Bloomberg and Bywater (2007) and van der Linden etal.
(2018). Even fewer attempts have been made at systematic
application of a heat-balance model to identify conditions
* Sepehr Foroushani
sforoushani@atb-potsdam.de; sforoushani@outlook.com
1 Engineering forLivestock Management, Leibniz Institute
forAgricultural Engineering andBioeconomy, Potsdam,
Germany
2 Institute ofAnimal Hygiene andEnvironmental Health,
College ofVeterinary Medicine, Free University Berlin,
Berlin, Germany
International Journal of Biometeorology
1 3
of potential heat stress and to develop relevant indices
accordingly. The work of Berman (2004, 2005, 2006) is a
notable exception, which nevertheless also resorts to linear
regression of modeling results to develop simplified indices,
and in some cases, linear fits stretched beyond the range
of the underlying empirical data. The relative unpopular-
ity of mechanistic models can be attributed to their formal
complexity, large number of input parameters and the need
for iterative solutions as well as lack of experimental data
that can be used for reliable estimation of the parameters or
validation of the models, especially for the modern high-
production dairy cow.
The present paper is an attempt to outline a path forward
based on mechanistic models for predicting conditions of
potential heat stress in dairy cattle. The many statistical
indices are not reviewed here. The reader is instead referred
to the recent publications by Ji etal. (2020a) and dos San-
tos etal. (2021). In the absence of similar reviews of heat-
balance models, the few existing models are briefly dis-
cussed here. Adopting classical theoretical work on human
thermal comfort and perception, particularly the apparent
temperature scale (Steadman 1984), two basic questions
are examined: first, is the hitherto shortcoming of the con-
ventional (statistical) approach methodological? Second, is
there inspiration to be drawn from human biometeorology?
More specifically, how can thermodynamic modeling and
prediction of heat stress in cattle offer new perspectives and
possibilities?
Thermoregulation
As homeotherms, cattle maintain a relatively constant core
body temperature in a range known as the “thermoneutral”
zone, where minimal energy is spent on thermoregulation
and maximal energy is devoted to metabolism and produc-
tion (Mount 1974; Godyń etal. 2019). Beyond this zone,
thermoregulatory mechanisms are activated to return to the
thermoneutral zone (dos Santos etal. 2021). There is a limit
to the effectiveness of the thermoregulatory mechanisms
beyond which thermal stress, hyperthermia or hypothermia,
occurs (Kamal etal. 2016; Sejian etal. 2018). In this zone,
behavioral changes such as reduced feed intake, increased
water intake, and reduced lying time and seeking shade
are triggered as secondary coping mechanisms, effectively
assisting the physiological thermoregulatory responses (Rat-
nakaran etal. 2017; Madhusoodan etal. 2019). In thermo-
dynamic terms, the response to heat stress can be divided
into (1) increasing heat dissipation (total heat transfer coef-
ficient), e.g. through enhanced perspiration and increased
surface (skin) temperature, and (2) reducing the endogenous
heat generation, e.g. by reducing feed intake and activity.
Hyperthermia (heat stress) occurs when the heat dissipation
cannot be adequately modulated to meet the thermoneutral
heat generation and maintain the basal (normal) body tem-
perature (Spiers 2012). For a detailed discussion of ther-
moregulation and thermal stress in cattle, see the review by
dos Santos etal. (2021) and the sources therein cited.
Heat stress: indicators andpredictors
As pointed out by West (2003), the term heat stress is used
rather loosely to signify the climate, climatic effects, or the
animal’s response. Alternative terminologies are also used in
the literature, e.g. “heat load” by Heinicke etal. (2018), with
the same meanings. Here, the definition put forth by Lee
(1965) is adopted where heat stress means “the conditions
that displace the animal’s thermoregulation system out of
the thermoneutral zone,” and heat strain is accordingly “the
displacement or deviation of the physiological, behavioral,
or productive parameters from the corresponding base val-
ues in the thermoneutral zone.”
Heat strains are “indicators” of heat stress, i.e. signs that
heat stress is occurring (e.g. increased respiration rate) or
has already occurred (e.g. reduced milk yield). Direct reli-
ance on such indicators for environmental control in live-
stock management would be challenging as it requires close,
real-time monitoring of various physiological parameters
and operation of the environmental control systems based
on such observations. Adding to this challenge is the fact
that some thermoregulatory responses (indicators) such as
sweating or breathing tend to be highly variable from animal
to animal, even within the same genotype or herd. See for
instance the work of Maia etal. (2005a,b) and Gebreme-
dhin etal. (2010) on dairy cattle and similarly the work
of Gaughan etal. (2010) on beef cattle. Meteorological
parameters, on the other hand, are readily available either
from on-farm measurements or nearby weather stations.
Furthermore, some indicators can only be observed when
stress has already occurred [milk yield reduction (dos Santos
etal.2021)] or is well underway (core body temperature
rise). Therefore, indices based on meteorological param-
eters, primarily ambient temperature, humidity, and wind
speed, are sought to identify and categorize the conditions
that are likely to cause heat stress. In this sense, a heat stress
index serves as a “predictor.”
Despite more than 60 years of research, the observation
by Berman (2005) that no clear criteria exist for conditions
in which heat stress relief is needed remains the case. Most
existing heat stress indices are applicable to temporally aver-
aged, herd-level indicators. Even at this resolution, correla-
tions are highly variable. See, for instance, the correlations
of daily milk yield and milk temperature with ten differ-
ent environmental indices presented in the paper by Ji etal.
(2020b). Such levels of resolution and accuracy can be inad-
equate to address the needs of precision livestock farming in
a rapidly and drastically changing climate.
International Journal of Biometeorology
1 3
As shown in a recent review (Ji etal. 2020a), the 40 years
following the introduction of the temperature-humidity
index [THI (Thom and Bosen 1959; Bianca 1962)] can be
roughly summarized as readjustments and reformulations
of essentially the same index (eight variants of THI and
several similar indices). The alternative indices proposed
in the last 20 years, e.g. the Comprehensive Climate Index
[CCI (Mader etal. 2010)], are more complex in form, but
were derived following the same general methodology, i.e.
starting with some variant of THI and introducing incre-
mental regression-based adjustments for air speed or solar
radiation. See, for example, the development of the Heat
Load Index [HLI (Gaughan etal. 2008)] and its expansion
to CCI (Mader etal. 2010). Moreover, some of the adjust-
ments demonstrate non-physical features. For example, the
wind speed adjustments in the high-radiation variant of HLI
(Gaughan etal. 2008) and CCI (Mader etal. 2010) have
non-zero intercepts (respectively -11°C and +3°C at u=0)
and non-zero slopes even at wind speeds as high as 25 m/s,
i.e. non-asymptotic behavior.1 Finally, the adjustments, par-
ticularly the wind-speed term in CCI (Mader etal. 2010), are
rather complex in form.
More recently, indices such as the Dairy Heat Load Index
[DHLI (Lees etal. 2018)] and the Equivalent Temperature
Index for Cattle [ETIC (Wang etal. 2018)] have presented a
breakaway from the practice of incremental improvements to
THI, although they too rely almost exclusively on regression
analyses of meteorological parameters and animal responses.
A recent study (Ji etal. 2020b) has found the new indices
(DHLI, ETIC) to predict heat stress no better than the older
indices (THI, HLI, CCI). A more recent study (Lees etal.
2022) concluded the relative success of DHLI and THI in
predicting heat stress depends on the physiological/behav-
ioral indicator of interest (panting, drinking, standing) as
well as the animals’ access to shade. The statistical indices
discussed above consider the intensity of heat stress only. In
other words, the duration of heat exposure (or relief) is not
considered. Gaughan etal. (2008) have called this a “one-
dimensional” approach. A few studies have attempted to
incorporate the transient nature of the thermal interaction
between animals and their surrounding, and specifically the
effects of prolonged heat exposure and intermittent relief
(e.g. at night). Relying on the classical THI index, Hahn and
Mader (1997) proposed the hours above established THI
thresholds (“THI-hours”) to be considered in the forecast
of heat waves. Heinicke etal. (2018) have used a similar
approach, based on THI and lying/standing behavior, to
examine the effects of the duration of heat exposure in terms
of a heat load duration (HLD) index. Similarly, Gaughan
etal. (2008) used the length of the periods when the heat
load index (HLI; see discussion in preceding paragraphs) is
above or below a critical threshold to develop the Accumu-
lated Heat Load (AHL) index. Methodologically, the AHL
model offers notable advancement as it includes the effects
of wind speed and solar radiation, which are absent from
THI, as well as the effects of heat relief (HLI<threshold).
Although the premise of the “two-dimensional” heat
load duration indices mentioned above is the considera-
tion of heat balance (and heat accumulation in the case of
bodily heat surplus), they are no different than the “one-
dimensional" indices in their reliance on purely statistical
correlations as a proxy for the thermal interaction between
the animal and the environment. As pointed out by Ehrle-
mark and Sällvik (1996), it is a major shortcoming of the
statistical approach that it ignores the thermodynamics of
thermoregulation and heat dissipation. Ehrlemark and Säll-
vik (1996) found it therefore “not surprising” that practical
experience from livestock management shows significant
deviations from the predictions of the statistical models.
Heat-balance models based on thermodynamic principles
have the potential to address that shortcoming and comple-
ment statistical correlations between meteorological and
physiological observations.
Heat‑balance models forcattle
Research on heat-balance models for livestock has a history
of more than three decades. Finding earlier thermal models,
e.g. Porter and Gates (1969), limited in their incorporation of
the thermoregulatory responses, McArthur (1987) developed
a detailed steady-state heat-balance model for homeother-
mic vertebrates, which entailed the physiological responses.
Despite the formal simplicity of the basic equation, the sub-
models used by McArthur (1987) to describe the underlying
physical and physiological phenomena are rather compli-
cated, entailing a high degree of non-linearity and coupling.
Following the same general approach, Ehrlemark and Sällvik
(1996) developed a steady-state heat-balance model. The
ANIBAL (ANImal heat BALance) model (Ehrlemark and
Sällvik 1996) is much simpler than the model by McArthur
(1987) with potentially greater utility. Nevertheless, ANI-
BAL was not used to identify conditions of potential heat
stress, but rather to predict heat generation at low (ambient)
temperatures and evaporative heat loss at high (ambient)
temperatures (Ehrlemark and Sällvik 1996). Moreover, while
criticizing the traditional statistical approach for neglecting
the effects of air speed, Ehrlemark and Sällvik (1996) used a
similar index [the thermal load index (TLI); Ehrlemark and
Sällvik (1996)] for normalizing the environmental condi-
tions and comparison with experimental data and thus failed
to address the shortcoming they had identified.
1 The empirical wind speed data used to derive the correction were in
the range 0.6–15.5 m/s (Mader etal. 2010).
International Journal of Biometeorology
1 3
Acknowledging the practical difficulties arising from the
complexity of McArthur’s model, Turnpenny etal. (2000a)
presented a simplified version of the model, dubbed “par-
simonious,” which was applied to various livestock [cattle,
sheep, pigs, chickens (Turnpenny etal.2000b)]. Following
the same general approach, McGovern and Bruce (2000)
developed a transient model which also included thermoreg-
ulatory mechanisms, namely reducing the thermal resistance
of body tissue (vasodilation), sweating to increase latent
heat loss and panting to increase respiratory heat loss. An
accompanying algorithm for time-step simulations was also
presented (McGovern and Bruce 2000). Similarly, Thomp-
son etal. (2014) developed a comparable transient model,
including relatively detailed climate submodels for calculat-
ing the ambient temperature, wind speed, and solar radia-
tion, to be used when hourly weather data is not available.
Li etal. (2021) added a submodel for conduction between a
lying animal and the ground to the model by Thompson etal.
(2014), further increasing its complexity. The model has so
far only been used to perform a sensitivity analysis (Li etal.
2021). As mentioned above, none of these models has been
systematically applied to identify conditions of potential
heat stress, especially in terms of common meteorological
parameters.
A general shortcoming of the mentioned heat-balance
models is lack of validation against experimental data.
Although most submodels used to describe individual ther-
mal or physiological phenomena are well established, none
of the whole models is thoroughly validated by comparison
with reliable measurements. For instance, Ehrlemark and
Sällvik (1996) compared the predictions of their model with
experimental data pooled from several sources. Notably, the
model predictions were in poor agreement with the experi-
mental data at higher ambient temperatures, namely condi-
tions of potential heat stress. Moreover, as mentioned above,
the results were only presented in terms of a thermal load
index (TLI) that obscures important boundary conditions
such as the air speed. Recently, more attention has been paid
to addressing the validation gap. Li etal. (2021), for exam-
ple, compared their modeling results with experimental data
from the literature, although with unsatisfactory results. The
model overpredicts the core body temperature by up to 3°C
which, given the physiological thresholds for heat stress,
seems too large. On the other hand, the respiration rate, esti-
mated based on linear regression with the body temperature,
was underpredicted almost systematically, by up to 40% (60
br/min discrepancy at 140 br/min).
Turnpenny etal. (2000b) observed that the limited data
available on the partition of heat loss, heat generation, and
the thermophysical characteristics of the livestock hinders
further development and refinement of heat-balance mod-
els. Two decades later, that limitation remains the case.
Despite calorimetric methods, specifically measurements
in respiration chambers, being well established and widely
used for several decades, detailed measurements of heat
transfer and thermoregulation in dairy cattle is scarce. In
many widely used sources, the thermoneutral metabolic heat
generation rate, a key boundary condition in heat-balance
models, is calculated and reported as the residual value
of energy-portioning calculations focusing on nutrition
and productivity, e.g. Coppock (1985); van Knegsel etal.
(2007); Talmón etal. (2020). On the other hand, thermody-
namic measurements of thermoregulation which deal with
the details of heat partition were mostly conducted decades
ago, on animals with much lower yield than the present-day
high-yielding cow, with presumably lower metabolic heat
generation and possibly different thermophysiological char-
acteristics. The seminal study by Worstell and Brody (1953),
for example, was conducted on Holstein cows whose mean
milk yield was less than 20 kg/day. The similarly widely
referenced study by Purwanto etal. (1990) was conducted
on cows among which the “high-yielding” group had a
mean milk yield of about 30 kg/day. More recent measure-
ments of respiratory and cutaneous heat losses by Maia etal.
(2005a,b) were performed on pasture-fed Holsten cows with
even lower yield, 15 kg/day on average. The existing models,
including those referenced above, all rely on such historical
data, while the modern Holstein-Friesian dairy cow can have
yields averaging around 40 kg/day (Pinto etal. 2020) and
exceeding 50 kg/day during peak lactation.
Perhaps driven by an increasing awareness of the endemic
validation gap, there has recently been a slow resurgence of
attention to the measurement of heat transfer and thermoreg-
ulatory responses in cattle. The recent thermographic study
of the skin temperature by Yan etal. (2021) is a promising
step forward, although the presentation of the results follows
the THI orthodoxy. Another important development is the
work of Zhou etal. (2021) where the physiological and pro-
ductive responses of dairy cattle to various combinations of
ambient temperature, humidity, and air speed were measured
in a respiration chamber. Nevertheless, Zhou etal. (2021) do
no present the partitioning of heat dissipation into different
modes of heat transfer.
Another general disadvantage of the mechanistic heat-
balance models is their formal complexity. Mechanistic
models such as the heat-balance models discussed here
have been criticized as unsuitable for use in precision farm-
ing due to complexity and presence of many parameters
that often need to be re-evaluated or adjusted for each
application (Wathes etal. 2008). Nevertheless, formal
complexity and the resulting computational intensity are
not necessarily as big a barrier to wide application as they
were 20 or even 10 years ago. As pointed out by Stiehl and
Marciniak-Czochra (2021), the present day’s computational
power allows the investigation of rather complex issues
based on mechanistic models, affording a deep quantitative
International Journal of Biometeorology
1 3
understanding of a wide range of topics. Aside from the
simplified general model by Turnpenny etal. (2000a,b),
no effort has been made to address the implementation gap
for heat-balance models of cattle, particularly to facilitate
implementation in predictive-model control of the barn
climate. Furthermore, as mentioned above, no attempt
has been made at systematic application of such models
to identify conditions of potential heat stress and develop
meteorological indices.
Parallels withhuman biometeorology
Historical development
The ongoing research on heat stress in cattle exhibits sev-
eral parallels with human biometeorology, specifically the
development of human thermal comfort indices. Given the
longer history and relatively greater success of the latter,
there are lessons to be learned for a more effective pursuit
of the former. After all, the most widely used index, THI,
was imported from human meteorology, virtually shaping
the trajectory of research on heat stress in cattle for more
than half a century. The striking parallels between research
on thermal comfort and heat stress in humans and cattle are
therefore no coincidence. In both cases, researchers started
with a focus on the ambient temperature, then developing
statistical constructs that incorporated humidity, air speed,
and finally solar radiation as wider ranges of climates and
production facilities were considered. The parallels are dem-
onstrated, for instance, by a recent study (Kovács etal. 2018)
where the apparent temperature (for humans) was found to
be a better predictor of heat stress in dairy calves than sev-
eral variants of THI.
In 1962, about the time THI was imported into animal
science, Macpherson (1962) published a comprehensive
review of the metrics devised to assess thermal comfort
in humans. Interestingly, Macpherson’s 1962 review enu-
merates 19 indices for human thermal comfort; Ji etal.’s
2020a review lists 20 heat stress indices for dairy cattle.
As documented in by Macpherson (1962), the pioneering
work on human thermal comfort was by large inspired and
motivated by the proliferating industrial plant; industrial
and mechanized animal husbandry has likewise enhanced
the need for effective prediction and remediation of heat
stress.
Judging from the large number of heat stress indices
developed during the half century prior, Macpherson (1962)
concluded that there was simultaneously a great need for
quantifying and categorizing conditions of thermal comfort,
and little success through the means thitherto devised. Most
notably, Macpherson (1962) concluded:
“[A]ssessment of the thermal environment is not pri-
marily a matter of the selection of some thermal index
in which to express the results. Expressing the results
in the form of an index may be a convenience, but
the assessment of the environment is essentially the
measurement of all the factors concerned… Indices of
thermal stress do not provide a substitute for a sound
knowledge of the mechanisms of heat exchange and of
the physiological adjustments to the thermal environ-
ment.”
The current state of the art in dairy science suggests a
similar conclusion.
Apparent temperature revisited
Inspired by the conclusion that a suitable heat index would
be based on the heat balance of the human body (Macpher-
son 1962), Steadman (1979a,b, 1984) presented a metic-
ulous analysis of heat transfer from the human body to
derive an equivalent temperature. Dubbed the “apparent”
temperature, this equivalent temperature is defined as the
dry-bulb temperature at standardized humidity, wind speed,
and radiation, which would require the same thermal resist-
ance for a walking adult to feel thermal comfort under a
given set of meteorological conditions. The procedure used
to derive the apparent temperature, Tap, is as follows. The
heat-balance equation is first solved iteratively for various
sets of environmental conditions to obtain the thermal resist-
ance of clothing, Rf, required for thermal balance. This first
solution is based on the idea that, in equilibrium, postulated
by Steadman (1979a) as a condition of thermal comfort, the
total heat dissipation rate should equal the heat generation
rate, less the net effect of the incoming solar, albedo and
terrestrial radiation and the outgoing sky radiation (Stead-
man 1979a,b). Moreover, the core body temperature, Tb, is
assumed constant at 37°C as a condition of comfort. After
obtaining Rf, Tap is determined by solving the heat-balance
equation for T, this time with Rf known from the first solu-
tion, and with the other meteorological parameters (solar
radiation, air speed, and humidity) set as “standard,” i.e.
reference, values.
Perhaps most relevant to the topic of heat stress in cat-
tle and the slew of statistical indices is the demonstration
(Steadman 1979b) that for any heat index, Tx, constructed
as a linear combination of the dry-bulb temperature (Tdb),
wet-bulb temperature (Twb), and globe temperature (Tgt), i.e.2
2 Note that indices defined in terms of T and RH, e.g. THI, can be
recast in terms of Tdb and Twb (c3=0) using psychrometric relations or
curve-fits.
International Journal of Biometeorology
1 3
the coefficients c1, c2, and c3 are not constant, but highly
dependent on the ambient temperature and considerably
dependent on activity level (heat generation), humidity,
and wind speed (Steadman 1979b). This key observation
explains the failure of indices constructed using regression
analysis of environmental and physiological, behavioral, or
productive parameters. The constant coefficients typically
obtained from such analyses confine the proposed index
to the environmental conditions and physiological char-
acteristics covered in the original study. See for instance
the recent work of Lees etal. (2022) which concluded that
DHLI, which was developed to include the effects of solar
radiation, is a better predictor of heat stress than THI (no
radiation term), but only for unshaded cows. Ehrlemark and
Sällvik (1996) duly observed that the validity of the statis-
tical models is limited by the range of conditions covered
by the underlying experimental data. Compare the general
analytical expressions for c1, c2, and c3 derived by Steadman
(1979b) and the definitions of common heat stress indices
for cattle, summarized by Ji etal. (2020a).
Recognizing the need for simplified calculation of Tap,
Steadman (1984) also presented linear equations based on
multiple-regression analysis of computed values of Tap.
There have subsequently been other efforts to develop sim-
pler approximations that entail fewer independent variables,
e.g. only Tdb and RH (Rothfusz 1990). It is important to note,
however, that such regression-based equations are ultimately
based on thermodynamic models, i.e. the heat balance of the
animal, rather than on purely statistical correlations between
observations of arbitrary predictors and indicators. Moreo-
ver, in constructing Tap, several conditions were imposed
such that the result would have physical meaning and sig-
nificance. For instance, it was observed that an equivalent
temperature used to express comfort “must have the familiar
properties of temperature” (Steadman 1984). As mentioned
above, some of the cattle heat stress indices do not satisfy
such criteria.
In general, Steadman’s model for humans is simpler than
heat-balance models for cattle due to several underlying
simplifying assumptions that are not applicable to cattle. In
Steadman’s model, the basic link between the actual envi-
ronmental conditions and the standard conditions at which
Tap is evaluated is the thermal resistance of clothing (Rf).
The fundamental assumption of the model is that, outdoors,
humans seek (achieve) thermal comfort by adjusting cloth-
ing, more precisely the heat and moisture transfer resistance
of clothing. This fundamental feature cannot be extended
to animals, e.g. dairy cattle, since the haircoat is relatively
constant, despite seasonal variations that decrease the heat
and moisture transfer resistance of the coat in summer and
vice versa in winter (Façanha etal. 2010). Furthermore, the
Tx=c1Tdb +c2Twb +c3Tgt +c4
assumptions and submodels used by Steadman (1979a,b)
for evaluating the various heat and vapor transfer resistances
do not apply to cattle. Lastly, it is important to note that the
apparent temperature scale was developed based on a con-
stant activity level, representing an adult Caucasian walk-
ing at 1.4 m/s (Steadman 1984). It must also be noted that
research on human thermal comfort and perception goes on,
with heat-balance models continuing to provide the frame-
work for many influential developments. See, for instance,
the review by Rupp etal. (2015).
Thermodynamic assessment ofheat stress
indairy cattle
In this section, a model from the literature is applied follow-
ing the general approach used to develop Tap to demonstrate
the utility of heat-balance models in predicting heat stress,
to derive meteorological indices, and to develop a general
framework for systematic application of such models.
Assumptions andprocedure
The general livestock heat-balance model developed by
Turnpenny etal. (2000a,b) was used where the total heat
dissipation from the animal, Ge, is estimated and compared
with the thermoneutral metabolic heat generation rate, M,
both expressed in terms of heat flux, per unit skin surface
area. In general, Ge is comprised of sensible and latent heat
loss from the skin and through respiration. With no solar
radiation, thermal balance (equilibrium) is maintained when
Ge=M.
The thermoregulatory responses are iteratively adjusted
to find conditions where Ge=M, following the “principle of
least metabolic cost” (Mount 1974; Turnpenny etal. 2000a),
namely that an animal will use vasomotor control before
increasing evaporative heat loss which involves water loss
and/or an increase in metabolic rate. This means that, for
given boundary conditions, thermoregulation is simulated
by:
1) Decreasing the tissue resistance to heat transfer (rs), sim-
ulating vasodilation, until thermal balance is achieved
(Ge=M). There is a physiological lower limit to this
resistance.
2) If the minimum tissue resistance is not sufficient for
thermal balance, i.e. Ge<M, the cutaneous latent heat
loss (sweating) is increased until thermal balance is
achieved. There is an upper limit to this heat loss mecha-
nism, dictated by either physiology or the environment.
3) In the present model, the respiration rate was indepen-
dently calculated, based on the ambient temperature and
humidity and using empirical correlations.
International Journal of Biometeorology
1 3
4) If the maximum sweating rate is not sufficient to main-
tain balance, i.e. Ge<M, heat will accumulate in the
body and the core body temperature will increase.
Note that, as pointed out by McArthur (1987), there is
evidence that sweating starts before tissue resistance has
been minimized, i.e. thermoregulation through sweating
and vasodilation may occur simultaneously and not neces-
sarily in succession. Nevertheless, the step-by-step model
of Turnpenny etal. (2000a) provides a reasonable first
approximation.
Further note that:
– Metabolic heat generation was assumed constant at the
thermoneutral rate. In reality, the metabolic rate declines
with prolonged exposure to heat, thereby increasing the
animal’s tolerance to heat. This adjustment is achieved
by reduced food intake and thyroid gland activity. See
the paper by McArthur (1987) and the references therein
cited for details.
– Similarly, the core body temperature, Tb, was assumed
constant at the thermoneutral level (39°C).
– As will be shown, the skin temperature is a dependent
variable, i.e. not directly adjusted as a thermoregulatory
response.
– Only the onset of bodily heat accumulation is sought.
Therefore, the increase in Tb and reduction of M as sec-
ondary coping mechanisms were not modeled.
Some of the sub-models used here are slightly different
from those used by Turnpenny etal. (2000a, b). Details of
the model implementation including boundary conditions,
submodels, and validation against experimental data are pre-
sented in the Appendix.
Sample modeling results
Figure1 shows sample results obtained from running the
heat-balance model at sample constant air speed and relative
humidity, and for various values of the ambient tempera-
ture, Ta. The main output to observe is Ge, specifically its
magnitude relative to M. As mentioned above, bodily heat
accumulation starts when Ge<M. Latent heat flux from the
skin is denoted by Ec while the respiratory heat flux (domi-
nantly latent) is denoted by Er. In the present model, Er is a
function of the Ta and RH only, i.e. adjusted independently
of vasodilation and sweating. See the Appendix for details.
The third component of Ge, not shown in Fig.1, is the sensi-
ble heat flux from skin (convection and long-wave radiation).
Figure1a suggests that, for u=2 m/s and RH=40%, the
thermoregulatory responses are sufficient to maintain the
heat balance up to just above 20°C. In other words, increas-
ing Ta to ~20°C, vasodilation and sweating are able to reduce
the overall heat transfer resistance between the body core
and the ambient in order to compensate for the reduction in
Tb-Ta. The thermoregulatory responses in this region (Ta ≲
20.5°C) can be divided to two phases:
1) For Ta ≲ 8°C, vasodilation is sufficient for maintaining
Ge=M, as seen from the increase in the skin temperature,
Ts, plotted against the vertical axis on the right. (Because
M and Tb are constant, reducing rs increases Ts; see Eq.
(A4) in the Appendix.)
2) After rs reaches its physiological minimum, i.e. vaso-
dilation exhausted, sweating is enhanced to dissipate
more latent heat from the skin; this is reflected by the
monotonic increase of Ec for 8°C ≲ Ta ≲ 20.5°C. The
linear increase in Ec counteracts the linear decrease of
the sensible heat flux, caused by the decrease in the heat
transfer potential (Tb-Ta). The slope of both lines is the
convection heat transfer coefficient, proportional to a
power of the air speed. See Eq. (A6) in the Appendix.
For u=2 m/s and RH=40%, the onset of heat accumula-
tion at Ta≈20.5°C is dictated by Ec reaching its limit, in this
case Ec,max=120 W/m2. As discussed in the Appendix, this
limit can be physiological or environmental.
In reality, once rs=rs,min and Ec=Ec,max, Tb increases, fol-
lowed by a decrease in M due to reduced feed intake and
metabolic activity3. The results shown in Fig.1a are there-
fore only qualitatively valid for Ta ≳ 20.5°C, nonetheless
instructive. The shaded area denotes Ge<M.
The effect of air speed can be seen from comparison of
Fig.1a, b, and d. Higher air speed, corresponding to a higher
convective heat loss from the skin, shifts the onset of heat
accumulation to a higher Ta. In other words, vasodilation
(reflected by the rise in Ts) and sweating (reflected by the
rise in Ec) are triggered and therefore exhausted at higher Ta,
meaning Ge=M can be maintained for higher values of Ta.
Most notably, Fig.1 suggests that humidity has lit-
tle effect on the onset of heat accumulation. Compare
Fig. 1b and c, representing moderate (RH=40%) and
extremely high (RH=90%) humidity, respectively. Note how
the various heat fluxes are virtually identical up to Ta≈27°C,
well beyond the onset of heat accumulation (Ta≈15°C in
both cases). Even at u=0.5 m/s, corresponding to “clam”
air (WMO 1970), Ec reaching its physiological limit is the
determining factor. The adverse effect of excessive humidity
(RH=90%; Fig.1c) on heat dissipation becomes apparent
only at Ta ≈27°C, some 12°C above the onset of heat accu-
mulation, where Ec is suppressed below the physiological
limit, leading to a drastic drop in Ge. This is in agreement
3 M may initially increase due to panting but will eventually drop in
response to prolonged heat exposure (McArthur 1987).
International Journal of Biometeorology
1 3
with the conclusion by Turnpenny etal. (2000b) that evapo-
rative heat loss from cattle is restricted by ambient vapor
pressure only at high ambient temperatures (Ta>30°C),
while at lower ambient temperatures, sweating is limited by
water supply rather than environmental conditions. Simi-
larly, the effect of humidity on Er is only significant for Ta ≳
30°C, as seen from Fig.1aand c.
From heat uxes totemperatures
As shown above, the main outcome of heat-balance models
is heat dissipation fluxes which can then be compared to
fluxes from the heat sources (metabolic heat generation and
solar irradiation) to assess thermal balance. Nevertheless,
heat fluxes are not as tangible as meteorological parameters
(ambient temperature, humidity, air speed). Moreover, as
discussed in Heat stress: Indicators and predictors, effec-
tive implementation for heat stress prevention/alleviation
depends on indices and thresholds in terms of the readily
available meteorological parameters. This section presents
three proposals for translating the heat-flux results into sim-
plified indices for the state of thermoregulation and the onset
of heat accumulation.
Onset ofheat accumulation: thecritical temperature
In order to express the heat-balance results in terms of
the more familiar meteorological parameters, a critical
Fig. 1 Equilibrium heat fluxes and skin temperature of a Holstein
dairy cow as a function of the ambient temperature and for various air
speed and relative humidities and no solar radiation, estimated based
on the heat-balance model of Turnpenny et al. (2000a). M, meta-
bolic heat generation; Ge, total heat dissipiation; Ec, cutaneous latent
heat loss; Er, repsiratory heat loss; Ts, skin temperature. Shaded area
denotes Ge<M where heat accumulates in the body and Tb=const and
M=const assumptions may no longer be valid
International Journal of Biometeorology
1 3
temperature, Tcr, may be defined as the ambient tempera-
ture corresponding to the onset of heat accumulation4 for
given u and RH. Here, Tcr was defined as Ta for which Ge
falls to 99% of M, an arbitrary threshold.
In Fig.2, Tcr is plotted as a function of u for RH=40%
and three representative values of the physiological limit on
Ec, denoted by
Ec,max
:
1)
Ec,max
=120 W∕m
2
corresponding to historical data for
Holstein cows, used by Turnpenny etal. (2000b), also
the default value in the present model.
2)
Ec,max
=138 W∕m
2
corresponding to the average of the
measurements by Gebremedhin etal. (2010) for Holstein
cows.
3)
Ec,max
=200 W∕m
2
corresponding to the upper end of
the measurements by Gebremedhin etal. (2010). This
is an extremely high heat flux, unlikely to be sustained
over the entire skin area, especially at M = 240 W/m2.
This value was nonetheless included for demonstration
and comparison purposes.
Recall that, as discussed in the “Sample modeling
results” section, humidity hardly has any effect on the
onset of heat accumulation. The significant effect of air
speed (u), on the other hand, can be seen in Fig.2, espe-
cially for
Ec,max
=120 W∕m
2
and
Ec,max
=138 W∕m
2
:
increasing the air speed from 0.5 to 6 m/s, shifts Tcr by
as much as 10°C. For
Ec,max
=200 W∕m
2
, the excessive
sweating capacity can compensate reduced convective
heat loss at lower air speeds to a large extent, diminish-
ing the difference between Tcr at u=0.5 m/s and u=6 m/s.
Noteworthy is that, according to Fig.2, heat accumulation
may start well below the established values for the upper
critical temperature (UCT), e.g. 25°C according to Ber-
man etal. (1985), especially for u < 4 m/s. The Tcr results
are in agreement with UCT=25°C only for animals with
extremely high sweating capacity (
Ec,max
=200 W∕m
2
) or
at high air speeds (u > 4 m/s).
Apparent temperature forcattle
The next step in consolidating the results would be to
integrate T and u into a single parameter. Following the
general procedure used by Steadman (1979a, b, 1984)
to develop Tap, an “apparent temperature” for cattle,
Tap
,
may be defined for any combination of Ta and u as the
ambient temperature which would require the same Ec for
Fig. 2 Critical tempearture as
function of air speed for various
representative phsyiological
limits on sweating (M = 240 W/
m2, no solar radiation)
4 Blaxter and Wainman (1962) have similarly used the term “criti-
cal temperature” in assessing cold stress in steers to denote the tem-
perature below which metabolism is increased to maintain the normal
body temperature.
~
International Journal of Biometeorology
1 3
thermal equilibrium at a reference air speed.5 The main
difference with Tap (Steadman 1984) is that, instead of
the required clothing insulation, Ec is the link between
the actual and apparent temperatures. Since air is rarely
still in naturally ventilated dairy barns, u0=1.4 m/s was
chosen as the reference air speed, corresponding to the
upper end of the “light air” range in the Beaufort scale
(WMO 1970) and in accordance with the original Tap
(Steadman 1984; see the “Apparent temperature revis-
ited” section).
The procedure for calculating
∼
Tap
is demonstrated in
Fig.3 where the equilibrium Ec is plotted as a function of Ta
and for various values of u. For any given combination of Ta
and u, there is a unique equilibrium Ec so long as Ta<Tcr. To
find
∼
Tap
, the corresponding line of constant Ec is intersected
with the u0 curve (dashed curve in Fig.3);
∼
Tap
is the abscissa
of the intersection.
Even in the absence of strong winds, there is air move-
ment around caused by free convection and the movements
of the animals. The u=0 curve was therefore included in
Fig.3 only as a reference for comparison. On the other hand,
u=16 m/s represents “near gale” high winds (WMO 1970),
also included for comparison. In most applications, the air
speed is between 0.5 and 6 m/s.
Thermoregulatory exhaustion index
From Fig. 2, it is known that Tcr =20°C for u=1.4
m/sandEc,max = 120 W/m2. Transformed into
∼
Tap
, any
combination of Ta and u can simply be compared against
this threshold,
∼
Tcr
=T
cr (
u
0)
=20
◦
C
, to determine whether
thermonetural heat balance can be maintained as well as how
far the conditions from the onset of heat accumulation are. A
second integrated parameter can be defined to characterize
the extent to which the thermoregulatory mechanisms (par-
ticularly sweating) have been “exhausted” and as a measure
of how far the conditions from the upper limit of the ther-
moneutral zone are. Here, the thermoregulatory exhaustion
index (TEI) is defined as:
where Ec,min and Ec,max denote the minimum and maximum
of Ec, respectively.6
One advantage of TEI over Tcr is that it entails two uni-
versal limits, TEI=0 and TEI=1.0 correspoding to Ec= Ec,min
and Ec= Ec,max respectively. In other words, TEI consolil-
dates the effect of the most important enviornmental (Ta,
u) and phsyiological parameters (Ec,max). TEI=1.0 signifies
the exhaustion of the “primiary” thermoregulatory respnse
mechanisms and the onset of heat accumulation.
TEI =
E
c
−E
c,min
E
c,max
−E
c,min
Fig. 3 Equilibrium cutaneous
latent heat flux (Ec) and ther-
moregulatory exhaustion index
(TE) as functions of ambient
temperature and at various air
speeds; graphical evaluation of
apparent temperature for cattle
(
∼
Tap
) [M = 240 W/m2,
Ec,max
=
120 W/m2, no solar radiation]
6 See the Appendix for more details about each limit.
5 Mader et al. (2010) have similarly used “apparent temperature”
as an alternative term for the Comprehensive Climate Index (CCI),
defined as the dry-bulb temperature with three linear corrections for
the effects of humidity, wind speed and radiation.
International Journal of Biometeorology
1 3
In Fig.3, TEI is plotted against the vertical axis on the
right. Two examples are also shown:
In Example I,
∼
Tap,I
<
∼
T
cr
, meaning Ge=M can be sus-
tained and therefore thermoneutrality maintained, reflected
by TEII<1.0. On the other hand, in Example II, Ec = Ec,max,
meaning the thermoregulatory reponses are exausted (TEIII
= 1.0) beyond thermoneutrality and heat accumulation is
underway.
Note that the results shown in Fig.3 apply to the repre-
sentative values of the physiological parameters, particularly
M and
Ec,max
, used in the present model. Similar graphs can
be generated for animals with different characteristics or
other weather conditions of interest, including with solar
radiation.
Conclusion
Indices constructed through regression of meteorological
parameters and animal responses have dominated research
on heat stress in cattle for more than six decades. Never-
theless, there is increasing skepticism about the effective-
ness and adequacy of such indices. As suggested by the
parallels with human biometeorology drawn in this paper,
forging new paths forward to meet the needs of modern
livestock management in the times of climate change
requires a renewed attention to physics-based heat-bal-
ance models. Although several heat-balance models have
been developed, no attempt has been made at systematic
application of the models to predict conditions of potential
heat stress.
In that context, the present work revisited classical work
in human biometeorology to develop a framework for iden-
tifying heat stress based on thermodynamic models of
thermoregulation and heat dissipation. A model from the
literature was used to assess the heat balance of a typical
Holstein dairy cow under various combinations of ambient
temperature, humidity, and air speed. It was shown that the
onset of heat accumulation strongly depends on temperature
and air speed, but hardly on humidity. While many studies
have paid little attention to the effect of air speed on heat
stress, especially in the vicinity of the animal, the results
of the present study underline the importance of systematic
collection and reporting of air speed data.
As evidenced by the dominance and resurgence of THI
and its many variants, an easy-to-use index presented in
Ta
=25
◦
C, u=6m∕s⇒E
c
≈106 W∕m
2
⇒T
ap,I
≈18
◦
C<T
cr
, TEI
I
=
0.88
Ta
=12
◦
C, u=0.1 m∕s⇒E
c
=120 W∕m2⇒T
ap,II
>T
cr
, TEI
II
=
1.0
tables or simple graphs is extremely useful. Simplicity
and ease-of-use compel practitioners and researchers to be
surprisingly forgiving of fundamental and methodologi-
cal shortcomings. Therefore, the present work introduced
new parameters to translate the modeling results (heat
fluxes) into meteorological indices. The critical tempera-
ture denotes the onset of heat accumulation at given air
speed. The apparent temperature for cattle maps the ambi-
ent temperature at any given air speed onto a thermophysi-
ologically equivalent temperature at a reference air speed.
Finally, the thermoregulatory exhaustion index (TEI) is
a measure of the extent to which the thermoregulatory
responses have been mobilized and how far the conditions
from the upper limit of the thermoneutral zone are.
The general framework developed in this paper can
serve as a roadmap for future work. To further establish
the thermodynamic approach, the endemic validation gap
must be first addressed through detailed measurements
of the thermophysiological characteristics of the modern
high-yielding cow, most importantly the thermoneutral
metabolic heat generation rate and core boy tempera-
ture, and the maximum sweating rate. Furthermore, con-
crete definitions for what constitutes heat stress must be
developed in physiological or productive terms. More
precisely, thresholds for critical heat strain, e.g. in terms
of increase in the core body temperature or decrease in
milk yield, must be established. Finally, metabolism and
productivity must be integrated in the thermodynamic
heat dissipation models in order to increase the utility
and accuracy of the models for analysis at animal-indi-
vidual level.
Appendix: Details oftheheat balance model
General approach andformulation
The heat-balance model by Turnpenny etal. (2000a,b) was
implemented where the total heat dissipation from the ani-
mal, Ge, is estimated, and compared with the thermoneu-
tral metabolic heat generation, M. When the skin surface is
in thermal equilibrium, the total heat flux dissipated to the
environment equals the heat flux through the body tissue, Gs,
as well as the heat flux through the haircoat, Gc;
For given environmental conditions (ambient tempera-
ture, humidity, air speed, solar radiation), heat-balance
equations are solved iteratively to find the equilibrium skin
and coat temperatures. The thermoregulatory responses are
incrementally increased during iterations until converged.
Ge=Gs=Gc
~~
~~
International Journal of Biometeorology
1 3
See the papers by Turnpenny etal. (2000a, b) for more
details and general solution algorithm.
Cattle dissipate bodily heat mainly through respiration
and from skin. The total heat dissipation rate on a flux (per
unit area) basis can be written as shown in Eq. (A1) (Turn-
penny etal. 2000a).
where Ge is the heat flux per unit skin surface area; the first
term on the right-hand side represents convective heat trans-
fer from the haircoat to the ambient air; the second term
represents the net long-wave radiant exchange at the external
surface of the haircoat; Sabs is the absorbed solar radiation;
Er is the dominantly latent heat flux through respiration; and
Ec is the latent heat flux from the skin, both normalized by
the skin surface are, As.
Following McArthur (1987), the formulation of Monteith
and Unsworth (1990) was used for the sensible components
of heat loss:
where ΔT is the driving temperature difference, r [s/m] is the
resistance to heat transfer, and ρcp is constant, here taken as
1220 J/(m3K), corresponding to 20°C and 1 atm.
Animal data
The animal size characteristics were chosen to represent a
typical mature Holstein cow: m=670 kg, dt=0.5 m. A hair-
coat length of l =9 mm was chosen to represent the summer
conditions (Façanha etal. 2010).
The skin surface area, As [m2], was calculated using the
regression proposed by Webster (1974):
The result is in agreement with the measurements of
Le Cozler etal. (2019) though not with the conclusion by
Berman (2003).
The model by Turnpenny etal. (2000b) was originally
developed as a generic model for livestock, including
sheep, pigs, cattle, and poultry. Consequently, the model
entails parameters to account for the difference between
the skin surface area and the haircoat surface area as well
as the haircoat’s effect on the curvature of the exposed
surface. Given that l is small for cattle, those two effects
were ignored in the present work. Furthermore, the entire
skin area was assumed to be exposed, representing a
cow in standing position and thermally isolated from
other animals. The latter assumption is supported by the
(A1)
G
e=
𝜌c
p
r
H
(
Tc−Ta
)
+
𝜌c
p
r
R
(
Tc−Tr
)
−Sabs +Er+E
c
(A2)
G
=
𝜌c
p
ΔT
r
(A3)
As=0.09 m0.67
conclusion by Wiersma and Nelson (1967) that convec-
tive heat loss is not affected by the presence of animals
farther than two inches (approximately five centimeters).
The metabolic heat flux, M [W/m2], was calculated based
on the regression proposed by van Knegsel etal. (2007) for
daily heat production in lactating cows, 1110m0.75 [kJ],
resulting in an average metabolic heat flux of M=240 W/m2.
More recent measurements by Talmón etal. (2020) yield
comparable results for pasture-fed Holstein cows.
Following Turnpenny etal. (2000b), the core body tem-
perature was assumed constant at Tb=39°C.
Ambient conditions
Ambient conditions were defined in terms of pressure,
p [kPa], temperature, Ta [°C], mean radiant temperature,
Tr [°C], relative humidity, RH [%], wind speed, u [m/s],
and assuming no solar irradiation. For simplicity it was
assumed Tr=Ta, though it is likely that Tr >Ta during hot
sunny days.
Heat transfer throughbody tissue
The rate of heat transfer through body tissue is given by:
where rs [s/m] is the resistance of the body tissue to heat
transfer, Tb is the core body temperature, and Ts is the mean
temperature of the skin surface.
The vasomotor response is simulated by adjusting rs,
starting at 100 s/m at the start of each iteration and decreas-
ing to a minimum of 30 s/m, before the sweating rate is
adjusted. The maximum and minimum values of rs were
chosen based on the work of Webster (1974) and Turnpenny
etal. (2000b).
Heat loss fromskin
Convective heat loss
Convective heat transfer between the outer surface of the
haircoat and the ambient air can be written as:
Here, the convective resistance was estimated using the
correlation by Wiersma and Nelson (1967), reproduced in
Eq. (A6), with both the Reynolds number (Re) and Nusselt
(A4)
G
s=
𝜌c
p
rs
(
Tb−Ts
)
(A5)
C
=
𝜌c
p
r
H
(
Tc−Ta
)
International Journal of Biometeorology
1 3
number (Nu) based on the trunk diameter, set in the present
model to dt=0.5 m.
Radiant heat exchange
Since heat stress in cattle sheltered in naturally ventilated
barns is of main interest to the present study, the radiant
exchange calculations were simplified by assuming no
solar radiation (Sabs=0) and considering only the long-
wave radiant exchange with the surrounding surfaces.
The net long-wave radiant exchange between the outer
surface of the haircoat and the surroundings, L, can be
written as shown in Eq. (A7) with the radiation resistance
rR obtained from the linearized Stefan’s Law.
Latent heat loss fromskin
Heat loss through the evaporation of sweat on the skin
(Ec [W/m2]) is a crucial heat dissipation mechanism, and
its maximum a major determinant of the onset of heat
stress. There is a physiological limit on the sweating rate,
dictated by water availability and the activity of sweat
glands (McArthur 1987). As pointed out by McArthur
(1987) and Turnpenny etal. (2000b), Ec,max is normally
determined by this physiological limit. Turnpenny etal.
(2000b) used
Ec,max
=120 W∕m
2
as the physiological
limit, derived from studies conducted in the 1970s. It is
likely that the modern high-yield cow has a higher sweat-
ing capacity. A 2010 study (Gebremedhin etal. 2010),
for instance, reports Ec,max as high as 280 W/m2 from cat-
tle subject to hot and dry conditions in the shade, though
the reported mean is less than half the maximum (138 W/
m2). Maia etal. (2005a) have similarly reported meas-
urements of Ec well exceeding 300 W/m2 for pasture-fed
Holstein cows in a tropical climate, which is difficult to
reconcile with typical metabolic heat generation rates
(M<300 W/m2), especially for the relatively low-yielding
cows (~15 kg/day) studied in that work. A 2011 study
of Holstein cows (da Silva and Maia 2011), on the other
hand, reports Ec < 100 W/m2. In the present work, unless
otherwise specified,
Ec,max
=120 W∕m
2
was assumed, in
accordance with Turnpenny etal. (2000b).
In high humidity, Ec may be supressed below the physi-
ological limit due to low evaporation potential from the skin.
As suggested in by Turnpenny etal. (2000b), this environ-
mental limit can be estimated as:
(A6)
Nu = 0.65 Re0.53
(A7)
L
=
𝜌c
p
r
R
(
Tc−Tr
)
where e [Pa] is the vapor pressure, γ is the psychrometric
constant (0.066 kPa/K), and rv [s/m] is the lower limit of
the resistance to mass transfer, namely the vapor transfer
resistance due to diffusion and free convection through the
haircoat, given by Cena and Monteith (1975) as:
where D [m2/s] is the diffusivity of water vapor in air
(2.5×10−5 m2/s at 20°C), and Ts
* and Ta
* [K] are the virtual
temperature of skin and air, respectively. The virtual tem-
perature is defined in Eq. (A10) (McArthur 1987).
To account for the effect of wind on the evaporation of
sweat, l in Eq. (A9) is replaced with l–lw where lw is the
“wind penetration depth” (Turnpenny etal. 2000a).
In general, Ec,max must be re-evaluated in each iteration
as the smaller of the physiological and environmental limits;
Under conditions of primary interest to the present study
(Ta ≤ 40°C, RH<60%),
∼
Ec,max
>
120
W/m2 and the physi-
ological limit of 120 W/m2 prevails.
As suggested by McArthur (1987), the evaporation of
moisture from an animal’s body can take place below the
skin surface. In cold, for instance, when the sweat glands are
inactive and the skin surface is dry, there is water vapor loss
by diffusion through the skin. Therefore, Ec,min>0. Accord-
ing to McArthur (1987), Ec,min ≈ 0.04Ec,max in homeotherms.
This baseline was implemented in the present model by ini-
tializing Ec at 0.04Ec,max.
Respiratory heat loss
Respiratory heat loss was calculated based on an energy
balance between the inspired and expired air streams. The
inspired air was assumed to be at the ambient conditions,
i.e. re-inhalation of the expired air was ignored. The expired
air was assumed to be saturated at a temperature calculated
based on the empirical correlation proposed by Stevens
(1981):
As noted by Stevens (1981), the correlation above has
two important implications:
(A8)
∼
E
c,max =
𝜌cp
𝛾
esat
Ts
−ea
rv
(A9)
r
v=
l
D
[
1+1.54 l
d
t(
T∗
s−T∗
a
)
0.7
]
(A10)
T∗=(T+273.15)(1+0.38e∕p)
(A11)
Ec,max
=min
{
E
c,max
,E
c,max}
(A12)
Tex
=17 +0.3T
a
+exp
(
0.01611RH + 0.0387T
a)
~
International Journal of Biometeorology
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1) The exhaled air is not at the body temperature, as
assumed in many studies, e.g. McArthur (1987), Turn-
penny etal. (2000a, b) and McGovern and Bruce (2000).
2) Tex is a function of the ambient conditions only.
The respiration rate, R [br/min], and tidal volume, Vt [m3/
br], were also calculated using correlations proposed by Ste-
vens (1981):
Note the reverse trends of Vt and R as Ta increases, i.e.
rapider but shallower respiration at higher Ta. As noted by
Stevens (1981), the inspired air cools the upper respiratory
tract as it passes through. When air is expired, on the other
hand, it is cooled by the respiratory passages. At a given Ta,
(A13)
R
=exp
(
2.966 +0.0218T
a
+0.00069T
2
a)
(A14)
Vt
=0.0189R
−0.463
Table A1 Summary of experimental data from Blaxter and Wainman (1964) for steers, used in the validation study (μ: mean, σ: standard devia-
tion, E: evpoartive heat loss)
Animal Ta [°C] u [m/s] l [mm] m [kg] Emin [W/m2]rs,max [s/m] Tb [°C] M [W/m2]Ts [°C] E[W/m2]
J 18.9 0.72 22 359 16 186 38.3 89 30.8 40
I 18.6 0.72 23 343 15 224 38.6 87 32.8 40
G 19.3 0.72 19 357 20 194 38.2 87 30.2 36
μ 18.9 0.72 21 353 17 202 38.4 88 31.3 39
σ 0.4 0.00 2.1 8.7 3 20 0.2 1 1.4 2.5
I 19.0 0.72 6 324 15 224 38.1 91 28.3 26
J 19.4 0.72 12 348 16 186 37.9 95 29.1 24
G 19.5 0.72 11 341 20 194 38.3 97 29.9 28
μ 19.3 0.72 10 338 17 202 38.1 94 29.1 26
σ 0.3 0.00 3.2 12.3 3 20 0.2 3 0.8 1.8
I 17.9 0.18 23 343 15 224 38.7 87 32.6 41
G 18.3 0.18 18 345 20 194 38.5 93 33.4 39
J 18.8 0.18 22 362 16 186 38.2 88 32.3 40
μ 18.3 0.18 21 350 17 202 38.5 89 32.8 40
σ 0.5 0.00 2.6 10.4 3 20 0.3 3 0.6 0.8
G 0.3 0.18 24 358 20 194 38.4 97 24.4 21
J −0.5 0.18 31 361 16 186 38.4 94 24.9 15
I −1.1 0.18 29 337 15 224 38.6 88 23.3 15
μ −0.4 0.18 28 352 17 202 38.5 93 24.2 17
σ 0.7 0.00 3.6 13.1 3 20 0.1 5 0.8 3.2
J 19.0 0.18 14 348 16 186 38.0 91 31.4 25
I 18.8 0.18 5 325 15 224 38.7 93 31.2 23
G 19.0 0.18 9 342 20 194 38.3 96 30.5 28
μ 18.9 0.18 9 338 17 202 38.3 93 31.0 25
σ 0.1 0.00 4.5 11.9 3 20 0.4 2 0.5 2.6
I 0.9 0.72 4 323 15 224 38.1 145 22.0 13
J 0.6 0.72 28 360 16 186 38.4 99 22.7 16
G 0.9 0.72 24 356 20 194 38.6 101 22.2 20
μ 0.8 0.72 19 346 17 202 38.4 115 22.3 17
σ 0.2 0.00 12.9 20.3 3 20 0.3 26 0.4 3.4
I 1.0 0.18 4 323 15 224 38.2 139 23.7 16
J 1.0 0.18 5 350 16 186 37.4 144 17.7 16
G 2.9 0.18 5 345 20 194 38.6 146 21.0 20
μ 1.6 0.18 5 339 17 202 38.1 143 20.8 17
σ 1.1 0.00 0.6 14.4 3 20 0.6 4 3.0 2.6
International Journal of Biometeorology
1 3
increasing RH reduces the cooling potential of the inhaled
air. Therefore, the upper respiratory tract is cooled by the
inhaled air to a lesser extent, leading to higher Tex.
The correlations above apply to first-phase panting only
(Stevens 1981; Berman 2004). Furthermore, the correlations
were obtained based on environmental chamber measure-
ments with no induced air flow. Equation (A13) is therefore
expected to overestimate R in open barns where there is
virtually always air flow. An overestimated R will however
only lead to a more conservative estimation of heat stress.
Moreover, Er is by far dominated by Ec, making the overes-
timation of Er even less problematic.
Alternatively, the comparable correlations proposed by
Maia etal. (2005b) can be used to evaluate the respiratory
heat loss as a function the ambient temperature only.
Validation
In the absence of detailed heat-transfer measurements on the
modern, high-yielding dairy cattle, data from the seminal
work of Blaxter and Wainman (1964) on steers was used
for validating the present implementation of the model by
Turnpenny etal. (2000a,b). The measurements by Blaxter
and Wainman (1964) were divided into seven groups of three
trials based on the air temperature and speed. To simulate
each group, the mean values were used to define the bound-
ary conditions (Ta, u) and animal characteristics (m, l, M, Tb,
rs,min) in the model. See Table1. The ambient humidity and
surface temperatures were not reported by Blaxter and Wain-
man (1964). In the model, a moderate humidity of RH=40%
was assumed. Based on data from an earlier study using the
same respiration chamber (Blaxter and Wainman 1962), it
was assumed that Tr=Ta.
In Fig.A1, the model predictions for the skin temperature
(Ts) and total evaporative (latent) heat loss (E) are compared
to the measurements (Blaxter and Wainman 1964). The hori-
zontal error bars represent the two-standard-deviation band
of the three trials in each group. The model predictions are
in overall agreement with the measurements, particularly in
the case of Ts. In the absence of data that can be used for a
more thorough validation, the comparison shown in Fig.A1
was taken as sufficient validation of the model.
Nomenclature Ec:Cutaneous latent heat flux [W/m2];
Ec,max
:Physio-
logical limit on cutaneous latent heat flux [W/m2]; Er:Respiratory heat
flux [W/m2]; Ge:Total heat dissipation flux [W/m2]; M:Metabolic heat
generation flux [W/m2]; RH:Relative humidity [%]; Ta:Ambient tem-
perature [°C]; Tb:Body temperature [°C]; Tap:Apparent temperature
[°C];
Tap
:Apparent temperature for cattle [°C]; Tcr:Critical tempera-
ture [°C]; Ts:Skin temperature [°C]; TEI:Thermoregulatory exhaus-
tion index [-]; THI:Temperature humidity index [-]; u:Air speed [m/s]
Funding This work was supported by funding received from the Euro-
pean Union’s Horizon 2020 research and innovation programme under
grant agreement No. 101000785. Open Access funding enabled and
organized by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
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the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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