Article Request Page ASABE Journal Article ## Dairy Cow Thermal Balance Model During Heat Stress: Part 1. Model Development

Chad R. Nelson^{1}, Kevin A. Janni^{1,*}## Published in

Journal of the ASABE66(2): 441-460 (doi: 10.13031/ja.15190). Copyright 2023 American Society of Agricultural and Biological Engineers.

^{1}Bioproducts and Biosystems Engineering, University of Minnesota, St. Paul, Minnesota, USA.

^{*}Correspondence: kjanni@umn.eduThe authors have paid for open access for this article. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License https://creative commons.org/licenses/by-nc-nd/4.0/

Submitted for review on 16 May 2022 as manuscript number PAFS 15190; approved for publication as a Research Article by Associate Editor Dr. Lingjuan Wang-Li and Community Editor Dr. Jun Zhu of the Plant, Animal, & Facility Systems Community of ASABE on 7 December 2022.

Highlights

Equations for a modified steady-state thermal balance model solved with a spreadsheet are described.The modified model describes heat exchange between lactating cows and the surrounding environment.New relations were used for tissue insulation, sweat rate, longwave radiation, and convective heat exchange.A companion paper compares model results to published body temperatures, respiration rates, and skin temperatures.

Abstract.Dairy cow heat stress impacts cow well-being, reduces milk yield, and leads to economic losses. Understanding heat stress mechanics supports ongoing and future efforts to mitigate heat stress. The purpose of this project was to modify a steady-state heat transfer model developed by McGovern and Bruce (2000) by incorporating work by Berman (2005), McArthur (1987), Turnpenny et al. (2000a,b), Thompson et al. (2014), Gwadera et al. (2017), two new empirical relations for tissue insulation and sweat rate, and a new solution method that allowed for overlapping changes in heat exchange. The modified model describes heat exchange between a lactating cow and the environment through respiration, convection, sweating, and shortwave and longwave radiation. This article describes the process-based model equations, compares results from the two new empirical relations used to published work, and presents the inputs and results for a cow on pasture in sunlight. The modified model, which can be solved with a spreadsheet, provides insight into factors and processes that affect lactating cow heat exchange. A companion paper compares the modified model results with published average measured body temperatures, respiration rates, and skin temperatures and unpublished body temperature data for cows on pasture in the sunshine.

Body temperature, Dairy, Heat stress, Lactating cow, Respiration rate, Thermal balance model. Keywords.Dairy cows are homeothermic, which means that they make behavioral and physiological adjustments to maintain their core body temperature within a narrow range. The behavioral and physiological adjustments that cows make can modify metabolic heat production, reduce heat gains from the environment, or increase heat losses to the environment. Animal caregivers can better help cows deal with heat stress when they understand and can quantitatively describe heat exchange between lactating cows and the surrounding environment.

Dairy cow heat stress is an active research topic. Several studies have reported on heat stress causes and impacts on dairy cows (West, 2003; Berman, 2005; Gebremedhin et al., 2010). Shoshani and Hetzroni (2013) assessed the impact of naturally ventilated barn design factors on high-producing cows in hot weather using a threshold temperature that was a function of ambient temperature, relative humidity, and wind velocity. Chen et al. (2015) reported physiological responses (i.e., body temperatures, respiration rates, and skin temperatures) to water sprayed on lactating Holstein cows in hot weather. Gorczya and Gebremedhin (2020) used individual cow data collected by Chen et al. (2015) to develop machine learning algorithms to rank heat stressor factors. Wang et al. (2018a, 2019) developed an empirical equivalent temperature index for cattle that calculates a temperature that considers dry-bulb temperature, relative humidity, air velocity, and solar radiation. Li et al. (2020) presented relations between Holstein dairy cows' mean rectal temperature and mean respiration rate and developed an empirical equation for finding the mean respiration rate based on air temperature, relative humidity, wind speed, and milk yield.

Work has also been done to assess the expected impact of heat stress on dairy cows. St-Pierre et al. (2003) and Abuajamieh et al. (2013) reported on economic losses due to heat stress on dairy cows. Additional studies by Key et al. (2014) and Mauger et al. (2015) described expected climate change impacts on U.S. milk production.

Numerous researchers have developed models that describe how dairy and beef cows are expected to balance metabolic heat production, shortwave radiation, respiration, convection, longwave radiation, and evaporation to maintain body temperatures in acceptable ranges (McArthur, 1987; McGovern and Bruce, 2000; Turnpenny et al., 2000a; Thompson et al., 2014). These researchers neglected conductive heat loss to the surrounding environment, which corresponds to standing cows (McArthur, 1987; McGovern and Bruce, 2000; Turnpenny et al., 2000a; Thompson et al., 2014).

Mondaca et al. (2013) developed a transient conjugate heat and mass transfer model of conductive, longwave radiation, convective, and evaporative heat exchange for a cow lying down on a cooling surface. They assumed that the cow had a constant body core temperature, did not include respiration heat exchange, and used a control volume-based approach to model metabolic heat generation and transfer within the cow’s body to maintain a constant core body temperature (Mondaca et al., 2013). The modeled cow’s metabolic heat production was not related to either cow mass or daily milk yield (Mondaca et al., 2013). The model was validated using data from a fiberglass calf with a heater placed in the body to mimic actual metabolic heat production (Mondaca et al., 2013).

Wang et al. (2018a, 2019) developed an empirical temperature index based on air temperature, relative humidity, air velocity, and solar radiation for high-producing Holstein cows using a data set with over 845 observations. The resulting Equivalent Temperature Index for Cattle (ETIC) did not consider cow mass or daily milk yield. Wang et al. (2018a) reported that the ETIC was able to account for approximately half of the variance in observed respiration rates.

Wang et al. (2018b) used computational fluid dynamics to assess convective heat transfer from cows at five air velocities either standing or reclining and exposed to either cross, front, or downward airflows. Body heat exchange through conduction, radiation, and evaporation was not modeled (Wang et al., 2018b). Zhou et al. (2019) used computational fluid dynamics and the empirical ETIC developed by Wang et al. (2018a, 2019) to assess baffle locations in cross-ventilated dairy barns. Zhou et al. (2019) modeled heat exchange through a cow’s skin and hair coat via convection, radiation, and evaporation of sweat assuming a constant body temperature but the study neglected to include respiration heat exchange in the model. Both Wang et al. (2018b) and Zhou et al. (2019) conducted steady-state analyses.

Existing models describing heat exchange between lactating cows and the surrounding environment include numerous assumptions and simplifications. Model assumptions and simplifications are acceptable if noted, and the model results provide useful information.

Engineers use models to help them design systems, processes, and facilities that are effective and economical. A steady-state process-based model describing heat exchange between a lactating cow and the environment through respiration, convection, sweating, and shortwave and longwave radiation heat exchange that produced useful results and could be solved using a spreadsheet will help engineers assess cooling system design factors (e.g., air velocity and evaporative cooling) for different environmental conditions (e.g., hot and dry, hot and humid, sunny, and shaded).

This project aimed to modify a steady-state heat transfer model that McGovern and Bruce (2000) developed to describe a cow’s heat exchange during heat stress and that can be solved using a spreadsheet. The McGovern and Bruce (2010) model was modified by incorporating work by Berman (2005), McArthur (1987), Turnpenny et al. (2000a, b), Thompson et al. (2014), and Gwadera et al. (2017). The modified model incorporated new empirical relations for finding tissue insulation and sweat rate and used a solution procedure that allowed all heat exchange methods to adjust concurrently rather than sequentially, as done by McGovern and Bruce (2010).

This is the first of two articles. This article describes the modified model’s development and the equations used. Select intermediate values are compared with published studies. An example set of inputs and outputs is presented. A second article compares the modified model respiration rate, sweating rate, and body temperatures to published data for heat-stressed lactating cows from Gebremedhin et al. (2010) and Chen et al. (2015). The second article also compares unpublished body temperatures of grazing cows in the sun at the University of Minnesota West Central Research and Outreach Center (WCROC) that were part of a shade study (Sharpe et al., 2021). The results of the modified model were compared to averaged data corresponding to steady-state conditions.

Model DevelopmentMcGovern and Bruce (2000) developed a steady-state energy balance model that described the heat exchange between a standing cow and the surrounding environment, which described a cow’s metabolic heat generation (M) and heat exchange to the surrounding environment. The heat exchanges were respiratory latent and sensible heat loss (E

_{r}), evaporative heat loss from the skin from sweating (E_{s}), shortwave solar radiation heat gain (R_{n}), and convective (C) and longwave radiative (LW_{n}) heat exchanges (gain or loss), and body heat storage (G_{b}) (fig. 1). Conductive heat exchange was neglected.The heat balance model by McGovern and Bruce (2000) achieved a steady-state heat balance by adjusting tissue insulation, sweat rate, and respiration rate in sequential order. The sequence to increase heat exchange from the cow to the environment was: (1) reduce tissue insulation (I

_{b}) to a minimum to account for vasodilation and increased conduction through the cow’s skin, (2) increase sweat rate to a maximum physiological rate, and (3) increase respiration rate to a maximum limit. If the metabolic heat could not be dissipated to the cow's surroundings as a result of those changes, the excess heat was stored in the body (an unsteady-state condition) and raised the cow's body temperature.The sequential initiation of heat loss regulating mechanisms used by McGovern and Bruce (2000) (i.e., reduced tissue insulation, increased sweat rate, and increased respiration) was not used. The modified model allowed overlapping changes in heat exchange in response to heat stress, as described by McArthur (1987) and Thompson et al. (2011). Tissue insulation, sweat rate, and body temperature were calculated using functions (i.e., eqs. 26, 38, and 61) that allowed all methods of heat exchange to be adjusted at the same time in the modified model described here.

The modified model also incorporated modifications and empirical equations for heat production, coat insulation, tissue insulation, body surface area, and respiratory cooling using exhaled air temperature and tidal volume developed by Berman (2003, 2004, 2005).

Figure 1. Schematic diagram of modeled heat flows following McGovern and Bruce (2000). Variables, inputs, and parameters with their corresponding values used in the modified model are listed and defined in table 1. The value column indicates whether the number was calculated using the equation listed in the equation column, was a model input, or was the constant given.

Model SchematicFigure 2 is a schematic outlining the thermal processes and equations used in the modified spreadsheet model. The equations describe the heat exchanges shown in figure 1. The thermal balance and heat exchange equations are described in the following sections. A spreadsheet add-in program, Solver, was used to solve the system of equations using the given constants in table 1 and the inputs used. Arrows in figure 2 indicate the direction of information flow to solve the model. Bidirectional arrows indicate where information flows in both directions using Solver. The modified spreadsheet model adjusted the assumed inspired volumetric respiration rate, V

_{resp}, and other variables to find intermediate calculated values and the cow body temperature, T_{b}, skin temperature, T_{s}, coat temperature, T_{c}, and respiration rate, rr, that balanced the heat balance given by equation 4.

Overall Heat Balance and ExchangesThe modified model describes heat exchange through the body core, skin surface, and coat surface using heat balance equations from McGovern and Bruce (2000) (fig. 1). The flux units used were watts per square meter of cow surface area (W m

^{-2}). The body core equation (eq. 1) (McGovern and Bruce, 2000) balances metabolic heat production, M, conductive heat exchange from the body to the skin surface, Q_{b}, sensible and latent heat exchange due to respiration, Q_{r,n}, and body heat storage, G_{b}:(1)

where

Q

_{b}=Heat flux from body core to skin surface (W m^{-2})M

=Metabolic heat production flux (W m^{-2})Q

_{r,n}=Net latent and sensible heat exchange flux by respiration (W m^{-2})G

_{b}=Body heat storage flux (W m^{-2}).The skin surface equation (eq. 2) (McGovern and Bruce, 2000) balances conductive heat transfer from the body to the skin surface, Q

_{b}, and the evaporative heat exchange from the skin through the coat, E_{s}, and sensible heat exchange from the skin surface to the coat surface, Q_{s,c}:(2)

where

Q

_{s,c}=Heat flux from skin surface to coat surface (W m^{-2})Q

_{b}=Heat flux from body core to skin surface (W m^{-2})E

_{s }=Latent heat loss from skin (W m^{-2})The coat surface equation (eq. 3) (McGovern and Bruce, 2000) balances the sensible heat exchange from the coat surface to the surrounding environment, Q

_{c,e}, the convective heat exchange between the coat surface and the surrounding air, C, longwave radiation between the coat surface and surrounding surfaces, LW_{n}, and shortwave radiation from the sun if in sunshine or under solar lamps, R_{n}:(3)

where

Q

_{c,e}=Outbound heat flux from coat surface to the surrounding environment (W m^{-2})C = Convective heat flux from coat surface to the surrounding air (W m

^{-2})

Table 1. Variable symbol, description, value or source, units, and equation where calculated. Symbol Description Value ^{[a]}Units Equation ^{[b],[c]}a _{1}Parameter used in equation 26 Calculated bpm m ^{2}K W^{-1}27 a _{2}Parameter in sweat rate function equation 38 Calculated °C 39 A Animal surface area Calculated m ^{2}5 A _{e}Percent animal surface exposed to air Input percent A _{h}/ AHorizontal area absorbing shortwave radiation per unit area of cow Calculated m ^{2}m^{-2}57 or 58 b Parameter used in equation 26 Calculated m ^{2}K W^{-1}29 C Convective heat flux from coat surface to air Calculated W m ^{-2}43 c _{b}Heat capacity of the cow’s body 3,400 J kg ^{-1}K^{-1}CI Coat insulation for Holsteins 0.30 °C m ^{2}d M_{cal}^{-1}mm^{-1}CC _{f}Cloud cover factor Calculated Decimal 53 or 54 Cp _{a}Specific heat of air Calculated J kg ^{-1}K^{-1}P CS _{f}Clear sky fraction Input Decimal Cv _{a}Volumetric specific heat of air Calculated J m ^{-3}K^{-1}P d _{t}Effective cow trunk diameter Calculated m 6 D Diffusivity of water in air adjusted for temperature and pressure Calculated m ^{2}s^{-1}34 E _{s}Latent heat loss from skin Calculated W m ^{-2}40 E _{vap}Evaporation flux for finding ground temperature 52 W m ^{-2}f Longwave exposure factor 1 Dimensionless Fat% Butterfat content Input % G _{b}Body heat storage flux Calculated W m ^{-2}23 Gr _{coat}Grashof number at coat and air interface Calculated Dimensionless 48 h Convective coefficient for finding ground temperature 33 W m ^{-2}K^{-1}h _{c}Coat conductance Calculated W m ^{-2}°C42 h _{m}Mid cow height Input m HP _{lact}Lactation heat production Calculated M _{cal}d^{-1}9 HP _{maint}Maintenance heat production Calculated M _{cal}d^{-1}8 I _{b}Tissue resistance Calculated m ^{2}K W^{-1}26 I _{b,max}Maximum tissue resistance Calculated m ^{2}K W^{-1}25 I _{b,min}Minimum tissue resistance 0.016 m ^{2}K W^{-1}k Thermal conductivity of air using average of air and coat surface temperatures Calculated W m ^{-1}K^{-1}43 l _{t}Effective cow trunk length Calculated m 7 l _{c}Coat thickness 3.0 mm l _{d}Coat depth = l _{c}/ (1000 mm/m)Calculated m 35 ?l _{w}Coat depth decrease due to wind Calculated m 35 LW _{n}Longwave radiant heat flux exchange between cow’s coat and surrounding surfaces Calculated W m ^{-2}50 LW _{g}Ground longwave radiation flux for finding T _{ground}63 W m ^{-2}M Metabolic heat production flux Calculated W m ^{-2}4, 13 M _{maint+lact}Maintenance and lactation heat production flux Calculated W m ^{-2}10 ?M _{b}Metabolic heat production adjustment for body temperature change Calculated W m ^{-2}11 ?M _{rr}Metabolic heat production adjustment for respiration rate change Calculated W m ^{-2}12 Nu Nusselt number Calculated Dimensionless 44, 45 P Atmospheric pressure Input Pa P _{w}Partial vapor pressure Calculated Pa P Pw _{coat}Average of P _{ws,Ts}and P_{w}Calculated Pa 49 P _{ws}Saturation vapor pressure Calculated Pa P P _{ws,Ts}Saturation vapor press at skin surface temperature Calculated Pa P Q _{lw rad}Longwave radiation flux between coat and surroundings when outside Calculated W m ^{-2}51 Q _{b}Heat flux from body core to skin surface Calculated W m ^{-2}1 Q _{c,e}Heat flux from coat surface to surrounding environment Calculated W m ^{-2}3 Q _{s,c}Heat flux from skin surface to coat surface Calculated W m ^{-2}2, 41 Q _{r,min}Net latent and sensible heat exchange by respiration at minimum respiration for finding sweat rate Calculated W m ^{-2}14 Q _{r,n}Net latent and sensible heat exchange by respiration Calculated W m ^{-2}14 r _{coat}Coat reflectivity 0.3 Dimensionless Re _{coat}Reynolds number at the coat and air interface Calculated Dimensionless 46 r _{v}Resistance to vapor transfer in the coat Calculated s m ^{-1}33 rr Respiration rate Interpolated bpm 17 - 22 rr _{50%}_{resp}Respiration rate at 50% of first phase respiratory cooling Calculated bpm 28 rr _{50%totresp}Respiration rate at 50% total respiratory cooling capacity Calculated bpm 37 rr _{max}Respiration rate at maximum respiration rate limit Calculated bpm 18 rr _{severe}Respiration rate at severe heat limit Calculated bpm 19 rr _{warm}Respiration rate at warm conditions limit Calculated bpm 17 R _{n}Radiant heat flux incoming from sun or solar lamps Calculated W m ^{-2}60 RH _{a}Ambient air relative humidity Calculated % P

^{[a]}Value indicates whether the number was calculated using the equation listed, was a model input or the constant given.

^{[b]}S = Solar equations from ASHRAE (2021).

^{[c]}P = Psychrometric equations from ASHRAE (2021).

Table 1 (continued). Variable symbol, description, value or source, units and equation where calculated. Symbol Description Value ^{[a]}Units Equation ^{[b],[c]}S _{nh}Solar irradiance on a horizontal surface. Calculated using equation 59 or inputted. Input or

calculatedW m ^{-2}59 S _{0}Extraterrestrial radiant flux at 21 ^{st}of the monthInput W m ^{-2}S s _{s}Sun shading factor Input Decimal t _{1}Heat storage time 3 h t _{2}Apparent solar time (0 to 24) Input h T _{a}Ambient air dry-bulb temperature Input C T _{a,K}Ambient air dry-bulb temperature Input K T _{b}Cow’s body temperature Calculated °C 61 T _{c}Coat surface temperature Calculated °C 41 T _{coat}_{ave}Average of coat surface and air temperature Averaged °C T _{dp}Ambient air dew-point temperature Input °C T _{ground}Ground surface temperature Calculated K 56 T _{ex}Exhaled air temperature Calculated °C 15 T _{k}Absolute temperature (i.e., ambient air, exhaust air or skin) Input or

calculatedK T _{MRT}Surrounding surfaces mean radiant temperature Input °C T _{N}Normal cow body temperature 38.0 °C T _{s}Skin temperature Calculated °C 24 T _{sky,K}Surrounding sky temperature Calculated K 55 T _{surrounding}Temperature of the surroundings which was an average of

the ground temperature and the sky temperatureCalculated K 53,54 T _{s,LCT}Lower critical skin temperature for finding sweat rate function Calculated °C 24 T _{s, 50%Qr,n}Skin temperature at 50% respiratory cooling Calculated °C 24 Ambient air virtual temperature Calculated K 16 Exhaled air virtual temperature at saturated vapor pressure Calculated K 16 Air virtual temperature used in equation 33 Calculated K 16 Skin virtual temperature used in equation 33 Calculated K 16 u _{cow}Air speed at cow level Input or

calculatedm s ^{-1}u _{W}Wind speed Input m s ^{-1}V _{resp}Volumetric respiration rate Variable L min ^{-1}V _{t,max}Tidal volume at maximum respiration Calculated L breath ^{-1}21 V _{t,severe}Maximal tidal volume based on body mass Calculated L breath ^{-1}20 V _{t,warm}Tidal volume at warm conditions Calculated L breath ^{-1}22 V _{t,50%resp}Tidal volume at 50% total respiratory cooling capacity Calculated L breath ^{-1}36 W Cow mass Input kg W _{a}Ambient air saturation humidity ratio Calculated decimal P W _{s}Saturation humidity ratio at exhaled air temperature Calculated decimal P x Animal size ratio = (2· l) /_{t}d_{t}Calculated Dimensionless Y _{milk}Milk yield Input kg day ^{-1}a _{a}Atmospheric dispersion factor 0.1 Dimensionless ß Solar altitude Calculated deg S d Solar declination angle Calculated deg S e _{coat}Emissivity of cow’s coat 0.98 Dimensionless e _{ground}Ground emissivity 0.9 Dimensionless e _{sky}Cloudy sky emissivity Calculated Dimensionless 52 ? _{cow}Azimuth of cow based on orientation Input deg ? _{s}Solar azimuth Calculated deg S ? _{T}Angular difference between the solar azimuth and azimuth of the cow Calculated deg S ? Psychrometric constant Calculated Pa K ^{-1}32 ? _{ave}Average psychrometric constant value using ambient and skin temperatures Averaged Pa K ^{-1}Sweat rate as function of skin temperature, T _{s}Calculated g m ^{-2}h^{-1}38 Minimum sweat rate 14.4 g m ^{-2}h^{-1}Maximum potential environmental sweat rate Calculated g m ^{-2}h^{-1}31 Maximum physiological sweat rate 660 g m ^{-2}h^{-1}Sweat rate at 50% total respiratory cooling capacity Calculated g m ^{-2}h^{-1}30 ? Latent heat of vaporization Calculated J kg ^{-1}P ? _{ave}Average latent heat of vaporization using ambient and skin temperatures Averaged J kg ^{-1}P ^{[a]}Value indicates whether the number was calculated using the equation listed, was a model input or the constant given.^{[b]}S = Solar equations from ASHRAE (2021).^{[c]}P = Psychrometric equations from ASHRAE (2021).

Figure 2. Schematic of model thermal processes, equations, inputs, and constants used in the spreadsheet model. LW

_{n}=Longwave radiation heat flux exchange with surrounding surfaces (W m^{-2})R

_{n}=Shortwave radiation heat flux from the sun or solar lamps (W m^{-2})When fluxes Q

_{s,c}and Q_{c,e}are equal, equations 1, 2, and 3 can be combined and reorganized to describe the overall heat balance of a cow (eq. 4) with the surrounding environment. Equation 4 neglects conductive heat exchange:(4)

Calculated Cow DimensionsThe cow dimensions needed in the modified model were based on an input body mass. A total surface area equation (eq. 5) used by Thompson et al. (2014) from Brody (1945) was used:

(5)

Table 1 (continued). Variable symbol, description, value or source, units and equation where calculated. Symbol Description Value ^{[a]}Units Equation ^{[b],[c]}? _{s}Latent heat of vaporization at skin temperature Calculated J kg ^{-1}P O Cloud cover factor (okta 0 to 8) Input f Albedo 0.25 Dimensionless ? _{a}Air density Calculated kg m ^{-3}P v_{coat air}Kinematic viscosity at the coat and air interface Calculated m ^{2}s^{-1}47 s Stefan-Boltzmann constant 5.67 x 10 ^{-8}W m ^{-2}K^{-4}

^{[a]}Value indicates whether the number was calculated using the equation listed, was a model input or a constant.

^{[b]}S = Solar equations from ASHRAE (2021).

^{[c]}P = Psychrometric equations from ASHRAE (2021).where

A = Calculated cow surface area (m

^{2})W = Cow mass (kg).

Expressions from McGovern and Bruce (2000) were used to find an effective cow trunk diameter, d

_{t}(eq. 6), and length l_{t}(eq. 7), assuming that the total cow surface area could be represented by a cylinder:(6)

(7)

where

d

_{t}= Effective cow trunk diameter (m)l

_{t}= Effective cow length (m).

Exposed Surface AreaThe heat exchange flux terms used in the modified model were given as W m

^{-2}of cow surface area. The percent of cow surface area assumed to be available for convective, longwave, and skin evaporative heat exchange was adjustable, A_{e}. An A_{e}value of 100% represented a standing cow with 100% of her surface area available for heat exchange.

Metabolic Heat ProductionHeat production was calculated using procedures developed by Berman (2005). Maintenance heat production was found using equation 8:

(8)

where

HP

_{maint}= Maintenance heat production (M_{cal}d^{-1}).Lactation heat production was found using milk yield and percent fat in the milk:

(9)

where

HP

_{lact}Lactation heat production (Mcal d^{-1})Y

_{milk}= Milk yield (kg d^{-1})Fat% = Milk fat (%).

The maintenance and lactation heat production flux per m

^{2}skin surface was found using equation 10:(10)

where

M

_{maint+lact}= Maintenance and lactation heat production flux (W m^{-2})Unit conversion

The modified model used equations from McGovern and Bruce (2000) to increase metabolic heat production for both increases in body temperature and respiration rate. The body temperature adjustment equation was originally from Blaxter (1967). The increase in metabolic heat production per unit m

^{2}skin area from an increase in body temperature, T_{b}- T_{N}, was found using equation 11:(11)

where

?M

_{b}= Adjustment in metabolic heat production due to change in body temperature (W m^{-2})T

_{b}= Cow’s body temperature (°C)T

_{N}= Normal cow’s body temperature (°C).The increase in metabolic heat production due to the respiration rate relation from McGovern and Bruce (2000) was calculated using equation 12:

(12)

where

?M

_{rr}= Adjustment in metabolic heat production due to changing respiration rate (W m^{-2})rr = Respiration rate in breaths per minute (bpm).

Total metabolic heat production flux was the sum of the maintenance and milk production heat production flux and the metabolic heat production flux adjustments for increased body temperature, ?M

_{b}, and respiration rate, ?M_{rr}:(13)

Cows commonly reduce dry matter intake and daily milk production during heat stress (St-Pierre et al., 2003; Abuajamieh et al., 2013). This process-based model does not adjust metabolic heat production and assumes no change in dry matter intake or milk production by the cow in response to heat stress.

Psychrometric and Solar ValuesThe modified model uses numerous psychrometric values (e.g., air density, specific volume, enthalpy, and partial pressure) for given or calculated dry-bulb and dew-point temperatures at different locations given in table 1. These psychrometric values were calculated using equations from ASHRAE (2021) for given or calculated dry-bulb and dew-point temperatures and a given atmospheric pressure.

Similarly, several solar values that depend on latitude, day of the year, and apparent solar hour (i.e., solar altitude and solar azimuth) were calculated using equations from ASHRAE (2021). These solar values are used to describe the shortwave radiation flux from the sun.

Respiration Heat ExchangeThe respiratory heat flux was calculated using virtual temperatures and an equation from McGovern and Bruce (2000). The air density, ?

_{a}, specific heat, Cp_{a}, and latent heat of vaporization, ?, values used in equation 14 were average values at the ambient, T_{a}, and exhaled, T_{ex}, air temperatures:(14)

where

V

_{rep}= Volumetric respiration rate, (L min^{-1})?

_{a}= Average air density of ambient air and exhaled air conditions, (kg m^{-3})Cp

_{a}= Average specific heat of ambient air and exhaled air, (J kg^{-1}K^{-1})= Exhaled air virtual temperature at saturated vapor pressure, (K)

= Ambient air virtual temperature (K)

? = Average latent heat of vaporization using ambient air and exhaled air temperatures (J kg

^{-1})W

_{s}= Saturation humidity ratio at exhaled air temperature (decimal)W

_{a}= Ambient air humidity ratio (decimal)The exhaled respiration air temperature was calculated using equation 15 from Berman (2005):

(15)

where

T

_{a}= Ambient air temperature (°C)RH

_{a}= Ambient air relative humidity (percent)Virtual temperatures were adjusted using either the air saturation vapor pressure, P

_{ws}, or partial vapor pressure, P_{w}, for either ambient air or exhaled air conditions. Virtual temperatures were calculated for corresponding temperatures using equation 16 from McGovern and Bruce (2000):(16)

where

T* = Virtual temperature (K)

T

_{k}= Corresponding temperature. Ambient air or exhaled air temperature for heat exchange by respiration (K)P

_{w}= Partial vapor pressure for ambient air or saturation vapor pressure (P_{ws}) for exhaled air (Pa)P = Atmospheric pressure (Pa).

Respiration Rate, Tidal Volume, and Inspired Volumetric Rate RelationsRespiration rate in breaths per minute (bpm) times the tidal volume (L per breath) gives the inspired volumetric rate by respiration in L min

^{-1}. Respiration rate relations by McGovern and Bruce (2000) and the maximal tidal volume relationship by Berman (2005) based on cow mass were used to describe the change in inspired volumetric rate as the cow went from warm conditions to maximum respiration to severe heat.

Respiration Rate RelationsThree respiration rate relations (eqs. 17, 18, and 19) from McGovern and Bruce (2000), depend on cow mass, W:

Warm conditions

(17)

Maximum respiration rate

(18)

Severe heat

(19)

where

rr

_{warm}= Respiration rate at warm conditions limit (bpm)rr

_{max}= Respiration rate at maximum respiration rate limit (bpm)rr

_{severe}= Respiration rate at severe heat limit (bpm).

Tidal Volume RelationsThree maximal tidal volumes, like the respiration rate relations (eqs. 17, 18 and 19), depend on cow mass. The severe heat tidal volume, V

_{t,severe}, relation for Holstein cows is from Berman (2005) (eq. 20). The V_{t,severe}result is used to find the tidal volumes at maximum respiration (eq. 21) (McGovern and Bruce, 2000). In turn, the V_{t,max}is used to calculate the tidal volume at warm conditions (eq. 22) given by McGovern and Bruce (2000):Severe heat tidal volume:

(20)

Tidal volume at maximum respiration:

(21)

Tidal volume at warm conditions:

(22)

where

V

_{t,severe}= Maximal tidal volume based on body mass from Berman (2005) (L breath^{-1})V

_{t,max}= Tidal volume at maximum respiration (L breath^{-1})V

_{t,warm}= Tidal volume at warm conditions (L breath^{-1}).Intermediate respiration rates and tidal volumes were found assuming linear relations between either the warm condition limit and maximum respiration limit or between the maximum respiration limit and severe heat limit. Respiration rates were described as rapid shallow panting when they were between the warm condition limit and maximum respiration limit and as deep slower panting when between the maximum respiration limit and severe heat limit (McGovern and Bruce, 2000).

The maximum respiration rate found using equation 18 creates an upper respiration rate limit for the modified model based on cow mass. For example, maximum respiration rates decrease from 138 bpm for a 400 kg cow to 120 bpm for a 650 kg cow.

Body Heat StorageMcGovern and Bruce (2000) included body heat storage to account for the energy used to raise a cow’s body temperature above a specified reference value, T

_{N}. The heat storage flux, G_{b}, depended on both the magnitude of the body temperature rise and the time over which the temperature rise occurred. The body temperature and heat storage equation in McGovern and Bruce (2000) was rearranged to find body heat storage given the body temperature rise above the reference value (eq. 23):(23)

where

G

_{b}= heat storage flux (W m^{-2}) over time, t (h)c

_{b}=Heat capacity of the cow’s body (3,400 J kg^{-1}°C^{-1})t

_{1}= Heat storage time (h).The heat storage time, t

_{1}, impacts heat storage flux and other model results. Increasing heat storage time reduces the heat storage flux value and causes the respiration volume and rate to increase, which causes the body temperature to increase too. At sufficiently long heat storage times, the respiration rate exceeds the maximum allowed (rr_{max}=120 bpm for a 650 kg cow or 138 bpm for a 400 kg cow), and the modified model cannot balance the system of equations. Decreasing the heat storage time increases the heat storage flux and causes the respiration volume and rate to decrease, which causes a decrease in body temperature. Gebremedhin et al. (2010) reported large body temperature rises within 3 h when solar loading was 850 W m^{-2}. The heat storage time used was 3 h.

Sensible Heat Flux from Body Core to Skin SurfaceThe sensible heat flux, Q

_{b}, found using equation 1, can also describe heat flow from the cow’s core through her skin using a conductive heat transfer relation using the temperature difference between the skin surface temperature, T_{s}, and the body temperature, T_{b}, and an average tissue insulation value, I_{b}, for the entire modeled cow (McGovern and Bruce, 2000). The conductive heat relation can be used to calculate the skin temperature using equation 24:(24)

Figure 3. Tissue insulation as function of respiration rate for 600 kg cow. where

T

_{s}= Skin temperature (°C)I

_{b}= Tissue insulation (m^{2}K W^{-1}).The tissue insulation, I

_{b}, is a proxy for the cow’s peripheral blood flow, which is under the cow’s physiological control. Tissue insulation is reduced when peripheral blood flow is increased through vasodilation to increase sensible heat transfer between the body core and the skin surface. As air dry-bulb temperature rises, skin temperature rises resulting in a reduced temperature gradient between body core and the skin surface (i.e., T_{b}-T_{s}). When the temperature gradient is reduced, the effective tissue insulation must also be reduced, for metabolic heat to be transferred to the skin surface and dissipated to the surrounding environment.McArthur (1987) related decreased tissue insulation to increased skin temperature. McGovern and Bruce (2000) found skin temperature using tissue insulation as an independent variable. Iterative techniques would have been needed to solve these equations, while possible, it was deemed to be too complex an approach.

Berman (2005) related linear tissue insulation reductions to air temperature increases when above a lower critical temperature. This linear tissue insulation relation was not selected for use.

For this model, tissue insulation was found using a non-linear function dependent on respiration rate (fig. 3). The non-linear inverse relation was used between the specified maximum tissue resistance value at the warm conditions respiration rate, rr

_{warm}, (i.e., approximately 12 bpm for a 600 kg cow [McGovern and Bruce, 2000]) and the minimum tissue resistance at 50% of the rapid shallow panting phase respiration rate (i.e., approximately 68 bpm for a 600 kg cow [McGovern and Bruce, 2000]). The 50% respiration rate in the rapid shallow panting phase was the average of the respiration rate at warm conditions, rr_{warm}, and the maximum respiration rate, rr_{max}.The maximum tissue insulation was calculated following Berman (2004) using equation 25:

(25)

where

I

_{b,max}= Maximum tissue insulation, (m^{2}K W^{-1})The minimum tissue insulation, I

_{b,min}, was set at 0.016 m^{2}K W^{-1}(McGovern and Bruce, 2000).The nonlinear tissue insulation function used in this model, based on respiration rate and the maximum and minimum tissue insulation values, is shown in figure 3 and found using equation 26:

(26)

where

I

_{b}= tissue insulation at respiration rate rr (m^{2}K W^{-1})a

_{1}and b = constants found using equations 27 and 28, respectively:(27)

where

a

_{1}= Parameter in equation 26 to find tissue insulation as a function of respiration rate (m^{2}K W^{-1}bpm^{-1})I

_{b,min}= 0.016 m^{2}K W^{-1}rr

_{50%}_{resp}= Respiration rate at 50% of first phase respiratory cooling (bpm).(28)

(29)

where

b = Parameter in equation 26 to find tissue insulation as a function of respiration rate (m

^{2}K W^{-1})The minimum tissue insulation value was commonly used during heat stress conditions.

Latent Heat Loss from SkinThe latent heat loss flux due to sweat evaporation from the skin depends on sweat rate, skin temperature, and air temperature. A modified sweat rate relation was developed for this model.

Sweat rate is important because skin evaporative cooling can account for up to 85% of the total heat loss when air temperatures exceed 30°C (Maia et al., 2005). Thompson et al. (2011) compared four empirical relationships relating sweat rate to skin temperature for

Bos taurus(Gatenby, 1986; McArthur, 1987; Maia et al., 2005). Data scatter around the best-fit empirical relations was significant and showed large variation within breeds and between animals (Gatenby, 1986; McArthur, 1987; Maia et al., 2005). McArthur (1987) used linear relations, while Gatenby (1986), Maia et al. (2005), and Thompson et al. (2011) used exponential relations. For this model, the cow’s sweat rate was described with a new exponential function based on skin temperature, like Gatenby (1986), McArthur (1987), Maia et al. (2005), and Thompson et al. (2011).

Minimum Sweat Rate and Corresponding Lower Critical Skin TemperatureThe first point needed to find the exponential relation between sweat rate and skin temperature was the minimum sweat rate due to diffusion, 14.4 g m

^{-2}h^{-1}(McArthur, 1987), and a corresponding skin temperature. The skin temperature was called the lower critical skin temperature, T_{s,LCT}, and was found using equation 24. The T_{s,LCT}is the skin temperature when the metabolic heat production, M, was dissipated to the surroundings with maximum tissue insulation, I_{b,max}, minimum respiratory cooling, Q_{r,min}, body storage set to zero (G_{b}= 0) and the body temperature was normal (T_{b}= T_{N}). The metabolic heat production value used for the T_{s,LCT}calculation was the maintenance and milk production, M_{maint+lact}, heat production found using equation 10. Maximum tissue insulation was calculated using equation 25. The minimum respiratory cooling value was found using equation 14 for a lower critical dry-bulb temperature of 0°C, a lower critical dew-point temperature of -8°C, an exhaled air temperature found using equation 15, and the warm respiration rate, rr_{warm}. The warm respiration rate was found using equation 17, and the tidal volume at warm conditions was found using equation 22.

Sweat Rate at 50% Total Respiratory Cooling Capacity and Corresponding Skin TemperatureThe second sweat rate and corresponding temperature needed to find coefficients for the exponential sweat rate relation were at 50% total respiratory cooling capacity and were a function of a maximum physiological sweat rate, , and a maximum potential environmental sweat rate, (eq. 31). The maximum physiological sweat rate was assumed to be = 660 g m

^{-2}h^{-1}based on Gebremedhin et al. (2010). The sweat rate at 50% of total respiratory cooling capacity was found using equation 30:(30)

where

= Sweat rate at 50% total respiratory cooling capacity (g m

^{-2}h^{-1})= Maximum physiological sweat rate, (660 g m

^{-2}h^{-1})= Maximum potential environmental sweat rate, (g m

^{-2}h^{-1}).The maximum potential environmental sweat rate relation, equation 31, was from Turnpenny et al. (2000b):

(31)

where

Cv

_{a}= Average air volumetric specific heat using ambient air and skin temperature, (J m^{-3}K^{-1})P

_{ws,Ts}= Saturation vapor press at skin surface temperature (Pa)P

_{w}= Partial vapor pressure at ambient temperature (Pa)?

_{ave}= Average psychrometric constant value using ambient and skin temperatures (Pa K^{-1})?

_{ave}= Average latent heat of vaporization using ambient air and skin temperatures (J kg^{-1})r

_{v}= Resistance to vapor transfer in the coat (s m^{-1})3.6 = unit conversion (3.6 s kg h

^{-1}g^{-1}) (Turnpenny et al., 2000b).The psychrometric constant, ?, was calculated using equation 32:

(32)

where

P = Atmospheric pressure, (Pa)

? = Latent heat of vaporization (J kg

^{-1})0.622 = Ratio of the molecular weight of water vapor to dry air.

The resistance to vapor transfer, r

_{v}, relation needed in equation 31, was taken from Turnpenny et al. (2000b), who had obtained it from Cena and Monteith (1975) (eq. 33):(33)

where

r

_{v}= Resistance to vapor transfer in the coat (s m^{-1})l

_{c}= Coat thickness (mm)?l

_{w}= Wind penetration depth (m)D = Averaged diffusivity of water vapor in air at air and skin conditions (m

^{2}s^{-1})= Skin virtual temperature (K)

= Ambient air virtual temperature (K).

Virtual temperatures, . and for equation 33, were calculated using equation 16 for the corresponding skin temperature and saturated vapor pressure and the air temperature and partial vapor pressure, respectively. Water vapor mass diffusivity values were adjusted for air and skin temperatures and pressure using equation 34 based on a water diffusivity value of 0.26 x 10

^{-4}m^{2}s^{-1}at 101,325 Pa pressure and 298 K temperature (Welty et al., 2008):(34)

where T

_{K}was the Absolute air or skin temperature (K).Coat depth reduction due to wind was found using equation 35 from McGovern and Bruce (2000) as specified by McArthur (1987):

(35)

where

?l

_{w}= Coat depth decrease due to wind (m)u = Wind speed at cow height (m s

^{-1})l

_{d}= l_{c}/ (1000 mm/m) = Coat depth (m)19.0 · 10

^{-6}m = Given constant11,000 s m

^{-2}= Given constant.The corresponding skin temperature at 50% total respiratory cooling capacity, T

_{s,50%Qr,n}, was calculated using equation 24, which requires a body temperature. For this model, body temperature was a function of respiration rate and is discussed later. The respiration rate at 50% total respiratory cooling capacity was found using equation 36, the tidal volume at warm conditions, V_{t,warm}, (eq. 22), and the maximal tidal volume, V_{t,severe}, (eq. 20):(36)

where V

_{t,50%resp}= Tidal volume at 50% total respiratory cooling capacity (L breath^{-1}).The corresponding respiration rate at 50% total respiratory cooling capacity, rr

_{50%totresp}, was found using equation 37 which uses the corresponding respiration at V_{t,50%resp}, and the calculated rr_{warm}and rr_{max}:(37)

where

rr

_{50%totresp}= Respiration rate at 50% total respiratory cooling capacity (bpm)With rr

_{50%totresp}the corresponding body temperature at 50% total respiratory cooling capacity needed in equation 24 was calculated using equation 58, which is discussed later. The minimum tissue insulation, I_{b,min}, value used was 0.016 m^{2}K W^{-1}(McGovern and Bruce, 2000) and body heat storage was set to zero.

Sweat Rate Relation as a Function of Skin TemperatureThe exponential sweat rate relation as a function of skin temperature used was similar to those used by Maia et al. (2005) and Gatenby (1986). The relation used uses the minimum sweat rate and its corresponding skin temperature and the sweat rate at 50% total respiratory cooling capacity and its corresponding skin temperature, T

_{s,LCT}, found in the previous sections. Coefficient a_{2}in equation 38 satisfies the boundary conditions given by the two points. The sweat rate relation is:(38)

where

= Sweat rate as a function of skin temperature, T

_{s}. (g m^{-2}h^{-1})= Minimum sweat rate (14.4 g m

^{-2}h^{-1}) (McArthur, 1987)T

_{s,LCT}= Lower critical skin temperature (°C)The parameter a

_{2}was computed using equation 39:(39)

where

= Skin temperature at 50% total respiratory cooling (°C)

= Sweat rate at 50% total respiratory cooling (g m

^{-2}h^{-1})= Minimum sweat rate = 14.4 g m

^{-2}h^{-1}.The sweat rate relation given by equation 38 includes the impact of the surrounding environment on sweat rate, including higher air humidity, and reduced wind speed that leads to reduced sweat rates, as observed by Gebremedhin et al. (2010).

Equation 40 was used in this model to calculate the latent heat loss flux from the skin given the sweat rate, , and exposed cow surface area, A

_{e}:(40)

where

E

_{s}=latent heat loss flux from skin (W m^{-2})?

_{s}= Latent heat of vaporization at the skin temperature (J kg^{-1})A

_{e}= Percent surface area exposed (%).

Convective Heat Flux from Coat to Surface to Surrounding Air

Sensible Heat Flux from Skin Surface to Coat SurfaceFor this model the sensible heat flow through the coat, Q

_{s,c}, was written in terms of the skin to coat temperature differential (T_{s}- T_{c}):(41)

where

h

_{c}= Coat conductance (W m^{-2}°C)T

_{c}= Coat surface temperature (°C).Coat conductance was found using the recommended hair coat insulation, CI, value per mm of coat depth for Holsteins of 0.30 °C m

^{2}d M_{cal}^{-1}mm^{-1}(Berman, 2004) and adjusting for coat depth reduction due to wind using equation 35.Coat conductance was found using equation 42:

(42)

where

CI = Coat insulation value for Holsteins (0.30 °C m

^{2}d M_{cal}^{-1}mm^{-1}) (Berman, 2004)Unit conversion

Convective Heat Flux from Coat Surface to Surrounding AirThe boundary layer convective heat flow from the coat surface to the surrounding air was written in terms of the coat surface temperature to air temperature differential (T

_{c}- T_{a}). Equation 43 was consistent with Turnpenny et al. (2000a) and Thompson et al. (2014):(43)

where

C = Convective sensible heat flux from coat surface to surrounding air (W m

^{-2})k = Thermal conductivity of air using average of air and coat surface temperatures (W m

^{-1}K^{-1})Nu = Nusselt number (dimensionless).

The Nusselt number relation used depended on whether the air velocity past the cow led to forced convection, natural convection, or the transition between them based on the Grashof number, Gr, and the Reynolds number, Re. For natural convection, when Gr > 16 · Re

^{2}, the Nusselt number relation used was:(44)

For forced convection conditions (when Gr < 0.1 · Re

^{2}) the Nusselt number relation used was:(45)

For intermediate conditions (0.1 · Re

^{2}< Gr < 16 · Re^{2}) the larger Nu value was used.The Reynolds number at the coat and air interface was calculated using:

(46)

where

Re

_{coat}= Reynolds number at the coat and air interface (dimensionless)

v_{coat air}= Kinematic viscosity at the coat and air interface (m^{2}s^{-1}).The air’s kinematic viscosity at the coat and air interface is the dynamic viscosity divided by the air density. The dynamic viscosity was adjusted for temperature using Sutherland’s formula (Sutherland, 1893) and divided by air density using equation 47. The reference temperature used was 30°C and the corresponding reference dynamic viscosity was 18.6 x 10

^{-6}N s m^{-2}:(47)

where

T

_{coat}_{ave}= Average of coat surface and air temperature (°C)?

_{a}= Average air density of coat surface and air temperature (kg m^{-3})The Grashof number needed to find the Nu (using either equation 44 or 45) was found using equation 48 (Thompson et al., 2014) and an average coat partial pressure between the skin and the coat (eq. 49):

(48)

where

Gr

_{coat}= Grashof number at coat air interface.(49)

where

Pw

_{coat}= Average coat partial pressure using partial air pressure and skin saturation pressure (Pa).

Longwave Radiation Heat Flux Exchange with Surrounding SurfacesThe longwave radiant heat exchange for cows located outdoors without shade is between the cow’s coat and the sky and ground. For cows in barns or chambers, the longwave radiation is between the cow’s coat and surfaces such as the underside of the barn roof, walls, or adjacent cows.

Longwave Radiation in Barn or ChamberThe two temperatures used to describe the longwave radiation for cows in a barn or chamber were the coat temperature, T

_{c}, and a mean radiant temperature, T_{mrt}, which represented the temperature of surfaces surrounding the cow (i.e., floor, walls, ceiling, or the underside of the roof, or adjacent animals). The cow’s coat temperature was calculated by the modified model. The mean radiant temperature was an assumed input and constant. The longwave radiation flux was adjusted based on the amount of exposed cow surface, A_{e}, in equation 50:(50)

where

LW

_{n}=longwave radiation heat flux between cow’s coat and surrounding surfaces when inside a barn or chamber (W m^{-2})f = Longwave exposure factor, f =1 (Dimensionless)

e

_{coat}= Emissivity of cow’s coat, 0.98 (Dimensionless)s = Stefan-Boltzmann constant 5.67 x 10

^{-8}W m^{-2}K^{-4}T

_{mrt}= Surrounding surfaces mean radiant temperature (°C).

Longwave Radiation for Cows OutsideThe longwave radiation exchange between the cow’s coat and the surrounding ground and sky when the cow is outside was described using equation 51 (Thompson et al., 2014):

(51)

where

q

_{lw}_{rad}= Longwave radiation flux between cow’s coat and surroundings when outside (W m^{-2})e

_{ground}= Ground emissivity (0.9)e

_{sky}= Sky emissivity (Dimensionless)T

_{c}= Coat surface temperature (K)T

_{surrounding}= Temperature of the surroundings, which was an average of the ground temperature and the sky temperature (K).Thompson et al. (2014) used equations from Brutsaert (1975) and Crawford and Duchon (1999) to calculate the sky emissivity. The equations were combined in equation 52:

(52)

where

e

_{sky}= Cloudy sky emissivity, (Dimensionless)CC

_{f}= Cloud cover factor (Dimensionless)P

_{ws}= Saturation vapor pressure at air temperature (Pa)T

_{a,K}= Air dry-bulb temperature (K).The CC

_{f}value used depended on whether the horizontal solar load, S_{nh}, was an input or calculated value. When S_{nh}was calculated using equation 59 based on location latitude, day of the year, and hour of the day, the CC_{f}value was calculated using equation 53 as described by McGovern and Bruce (2000) using an input okta value, CS_{f}(e.g., CS_{f}= 0 for a clear sky to 8 for a completely cloudy sky):(53)

When S

_{nh }was used as an input, the CC_{f}was calculated using equation 54 from Crawford and Duchon (1999), with the theoretical solar load calculated using equation 59 and the input solar load. If the input solar load was greater than the theoretical solar load, CC_{f}was set to 0, which corresponds to clear skies. The theoretical solar load was calculated using equation 59 and is discussed later:(54)

The surrounding temperature for a cow out of doors, T

_{surround}, was the average of the sky temperature and the ground temperature (Thompson et al., 2014). The sky temperature was calculated using equation 55 from Gwarda et al. (2017), which uses a total clear sky emissivity relation from Berdahl and Martin (1984) based on the air dew-point temperature and the solar time:(55)

where

T

_{sky,K}= Surrounding sky temperature (K)T

_{a,K}= Air dry-bulb temperature (K)T

_{dp}= Air dew-point temperature (°C)t

_{2}= Solar hour (h).The ground temperature was calculated using equation 56 from Gwadera et al. (2017) which included terms for solar irradiation and water evaporation from the ground or plants:

(56)

where

T

_{ground}= Ground surface temperature (K)S

_{nh}= Solar irradiance (shortwave radiation) on a horizontal surface (W m^{-2})h

_{g}= Ground convective heat transfer coefficient (33 W m^{-2}K^{-1})E

_{vap}= Evaporation flux (W m^{-2})e

_{ground}= Ground emissivity (0.9 dimensionless)LW

_{g}= Ground longwave radiation flux (63 W m^{-2})The constants for h

_{g}, e, and LW_{g}were taken from Gwarda et al. (2017) for the simplest model of ground temperature. The evaporation flux was an input. The coat surface temperature was an unknown found by the modified model. The shortwave irradiation, S_{nh}, is discussed in the next section.

Shortwave Radiation Heat Flux from the SunShortwave radiation in this modified model describes radiant energy from the sun or solar lamps used in chamber studies. The modified model shortwave solar load, R

_{n}(eq. 60), is the product of the solar load applied times the amount of the cow’s surface area absorbing shortwave radiation per cow total exposed surface area, A_{h}/A, one minus the coat reflectivity, e_{coat}, and one minus the fraction of cow shaded, s_{s}. For cows outside, the A_{h}/A value was calculated using the area of the shadow cast by the cow (eq. 57) as a function of the solar altitude, hour angle, solar declination angle, solar azimuth, and the azimuth angle of the animal (McGovern and Bruce, 2000):(57)

where

A

_{h}/A = Horizontal area absorbing shortwave radiation per unit area of cow (m^{2}m^{-2})ß = Solar altitude which is a function of date, declination angle, latitude, and solar hour (degrees) calculated using equations in the ASHRAE (2021)

?

_{T}= Angular difference between the solar azimuth, ?_{s}, and the azimuth of the cow, ?_{cow}(degrees)x =(2·l

_{t})/d_{t}where l_{t}is the cow trunk length and d_{t}is the trunk diameter (dimensionless).When the shortwave radiant source was directly overhead, as in chamber studies, A

_{h}/A simplified to:(58)

The horizontal shortwave load, S

_{nh}, needed to calculate the shortwave radiation heat flux, R_{n}, can be either measured or estimated by calculating the total irradiance as a function of date, latitude, solar hour, cloud cover, albedo, and an atmospheric dispersion factor using equation 59 from McGovern and Bruce (2000) and extraterrestrial normal irradiance values for the 21^{st}of each month and solar angles needed from ASHRAE (2021):(59)

where

S

_{nh}= Direct and diffuse horizontal shortwave radiation heat flux (W m^{-2})S

_{0}= Extraterrestrial normal irradiance (W m^{-2})f = Albedo (Dimensionless)

CC

_{f}= Cloud cover factor (Dimensionless)ß = Solar altitude (degrees)

a

_{a}= Atmospheric dispersion factor (Dimensionless).This model was also set up to use input horizontal solar radiation or sun lamp values (W m

^{-2}). Whether using a calculated or given measured S_{nh}, the R_{n}was found using equation 60:(60)

where

R

_{n}= Shortwave radiation heat flux (W m^{-2})r

_{coat}= Animal coat reflectivity (0.3 dimensionless)s

_{s}= Sun shading factor (Dimensionless decimal value).

ShadingThe amount of shading was adjustable using a decimal value, s

_{s}, to describe the fraction of solar shortwave radiation that reached the cow as one minus the decimal fraction blocked. In full sun, the fraction blocked was zero in equation 60. When the cow was assumed to be completely in a barn, the fraction of shading was set to 1, and no shortwave radiation reached the cow. When shading was set to 1, longwave radiant energy exchange was calculated using equation 50 and the surrounding surfaces’ mean radiant temperature, T_{mrt}.

Body Temperature and Respiration RateThe body temperature relation used in the modified model was a linear function of respiration rate. After nearly complete vasodilation, the body temperature begins to rise (McArthur, 1987). Thompson et al. (2011) reported that elevated respiration rates and body temperatures were well correlated for

Bos taurus, crossbreds, andBos indicusin three studies (Allen, 1962; Thomas and Pearson, 1986; and Brown-Brandl et al., 2003). Some studies reported correlations between body temperatures and respiration rates of r = 0.046 (Kabuga, 1992) and r = 0.55 (Martello et al., 2010).Recently, Li et al. (2020) collected data from 45 high-producing Holstein cows on 26 days between 1000-1100 h to develop a linear relation between mean rectal temperature, MRT, and mean respiration rate, MRR. The relation Li et al. (2020) reported was MRT = 0.021 · MRR + 37.6. They also determined that heat stress was triggered when MRT = 38.6°C, which corresponded to a respiration rate of 48 bpm (Li et al., 2020).

For this model, the regression relation relating respiration rate to body temperature for

Bos taurusfrom Thompson et al. (2011) was rearranged to find body temperature given the respiration rate, rr, in bpm (eq. 61):(61)

The reference body temperature was set at 38.0°C, which corresponded to a respiration rate of 37 bpm. When respiration rates were calculated to be below 37 bpm, the body temperature was set at the reference value of 38.0°C because the modified model was designed for warm and hot weather conditions, not cool conditions, and 38.0°C was at the lower range of rectal temperatures for dairy cows (Andersson and Jonasson, 1993; Collier and Gebremedhin, 2015). At higher respiration rates above 37 bpm, the body temperature was found using equation 61.

Model ImplementationA spreadsheet was developed to solve the system of steady-state equations describing heat exchange between a cow and the surrounding environment by convection, shortwave radiation, longwave radiation, evaporation from the skin, and sensible and latent respiratory heat loss. The model inputs and coefficients listed in table 1 were adjustable. For this article, the steady state analysis was done over either 1 or 3 h time steps in which the weather, solar, cow, and other characteristics were assumed to be constant.

In the spreadsheet, the inputs and equation coefficients were used to calculate intermediate constants not dependent on respiration rate or body, skin, or coat temperatures (fig 2). The calculated constants include cow surface area (eq. 5), trunk diameter and length (eqs. 6 and 7), heat production for maintenance and milk production (eqs. 8, 9, and 10), respiration rate and tidal volume indices (eqs. 17-22), maximum tissue insulation (eq. 25), and exhaled air temperature (eq. 15). A spreadsheet add-in program, Solver, was used to adjust the inspired volumetric respiration rate, V

_{resp}, to find the modified model cow body, skin, and coat temperatures, respiration rate, and numerous intermediate values that balanced the overall heat balance given by equation 4.

Model Relations AssessedThe sweat rate and tissue insulation relations used were compared to published relations. The sweat rate relation developed for this model was compared to empirical relations relating sweat rate to skin temperature (Gatenby, 1986; McArthur, 1987; Maia et al., 2005; Thompson et al., 2011). Then the model tissue resistance, body temperature, and skin temperature results were compared to equations by McArthur (1987) relating tissue resistance to skin temperature and body temperature to skin temperature.

For each analysis, the modified model was run for a 560 kg cow on pasture at the WCROC with a 32.9 kg d

^{-1}milk yield and 3.7% milk fat. Weather data was recorded automatically every 3 to 5 min at a University of Minnesota Morris weather station (Vantage Pro, Davis Instruments Corp., Hayward, CA) located at 45°35'20? N and 95° 54'7? W, less than 1 km from the WCROC pasture. The weather data was averaged over 3-h time blocks (i.e., 04:45-07:30, 07:45-10:30, 10:45-13:30, 13:45-16:30, 16:45-19:30) over six days (i.e., 10 to 15 July 2015). The recorded weather data used included the air dry-bulb temperature, dew-point temperature, air velocity at 10 m, horizontal solar insolation, and atmospheric pressure. Changes in milk production were not recorded or modeled. Finally, an example case is presented with inputs and results to illustrate the modified model’s use.A subsequent article assesses thermal balance model results against published body temperatures, respiration rates, skin temperatures, and unpublished body temperatures of grazing cows for heat-stressed lactating cows.

Results and DiscussionTable 2 summarizes the weather conditions based on 360 readings recorded every 15 min from 04:45 to 19:30 each day from 10 to 15 July 2015. Averaged values from 3-h periods were used as model inputs for the sweat rate and tissue insulation analyses.

Table 2. Summary of weather conditions at the WCROC between 04:45 and 19:30 from July 10-15, 2015. Weather variable Average Standard Deviation Range Air dry-bulb temperature (°C) 26.4 4.2 14.6-33.4 Air dew-point temperature (°C) 20.7 2.6 12.2-27.8 Temperature Humidity Index 75 4.9 60 - 85 Wind speed at 10 m height (m s ^{-1})3.5 1.6 0.0-7.1 Wind speed at mid cow height h _{m}= 1 m (m s^{-1})^{[a]}2.8 1.2 0.0-5.8 Solar insolation (W m ^{-2})436 312 0-1,074 Atmospheric pressure (Pa) 100,766 385 99,976-101,331

^{[a]}Adjusted for mid cow height using Power law, u_{cow}= u_{w}· (h_{m}/10)^{p}, for wind speed, u_{w}, measured at 10 m height, input mid cow height, h_{m}= 1 m, and exponent, p = 0.1, for stability class C and smooth surface (Cooper and Alley, 2011).Based on calculated Temperature Humidity Index (THI) values, the weather conditions observed were not considered too heat stressful, but solar loads added to the heat stress experienced by cows on pasture in the sun.

Sweat Rate AnalysisFigure 4 is a plot of sweat rate versus skin temperature for four models (Gatenby, 1986; McArthur, 1987; Maia et al., 2005; Thompson et al., 2011) and results from the modified model for average weather conditions for the 30 3-h periods in July 2015. McArthur (1987) used a linear relation, while Gatenby (1986), Maia et al. (2005), and Thompson et al. (2011) used exponential relations. Thompson et al. (2011) compared the four empirical relationships against three data sets (Allen, 1962; Thomas and Pearson, 1986; Brown-Brandl et al., 2003) and concluded that their exponential model was an improvement over the other sweat rate to skin temperature relations for

Bos taurus.The sweating rates obtained using the modified model were consistently above those predicted by the relation developed by Thompson et al. (2011) but followed a very similar trend. The modified model sweat rates were well above those predicted by McArthur (1987) and Gatenby (1986) at skin temperatures below 34°C and below those at skin temperatures above 36.5°C. The modified model sweat rate calculations include the impact of several other factors in addition to skin temperature and show some scatter. Additional data and analyses will be needed to assess the impact of air humidity and air speed past the cow on sweat rates.

Figure 4. Empirical models relating sweat rate to skin temperature (Gatenby, 1986; McArthur, 1987; Maia et al., 2005; Thompson et al., 2011).

Tissue Resistance AnalysisModel tissue resistance and skin temperature results (open circles) for 30 3-hour time blocks over six days (i.e., 10 to 15 July) in 2015 were compared to McArthur’s equation (solid line) in figure 5. The modified model tissue resistance values found using the non-linear relation describing tissue insulation as a function of respiration rate were consistently lower than tissue resistance values found using the empirical equation McArthur (1987) developed. The skin temperature where tissue resistance reaches a minimum was found to be around 36.5°C, which was roughly 0.5°C higher than the relation found by McArthur (1987).

McArthur (1987) also computed body temperature based on skin temperature above a minimum value using data for Holsteins. In this spreadsheet model, body temperature was related to respiration rate, following Thompson et al. (2011). Figure 6 shows the modified model body temperature versus skin temperature results (open circles) and the relation from McArthur (1987). The body temperatures from the modified model increased smoothly as the skin temperature increased. The modified model body temperatures were consistently above the body temperatures predicted by McArthur (1987) for skin temperatures below 36.5°C but were very close above 36.5°C skin temperatures. This threshold skin temperature was below the skin temperature at which tissue insulation stabilizes at a minimum at approximately 36.5°C (fig. 5). McArthur (1987) used a minimum body temperature of 38.2°C, while the modified model used 38°C. The overlap in physiological responses (i.e., tissue resistance decline [fig. 5] and body temperature rise [fig. 6]) with rising heat load agrees with the discussion by McArthur (1987).

Example Model Inputs and ResultsSpreadsheet model inputs in tables 1 and 3 were used to illustrate model use for a hypothetical cow standing in a pasture in the sunshine during the 3-h period between 13:45 and 16:30 on 12 July 2015. This 3-h period had the highest dry-bulb temperature and THI between 10-15 July 2015 at the WCROC. Stability class C was selected because the incoming solar radiation was strong and the wind speed at 10 m was greater than 6 m s

^{-1}(Cooper and Alley, 2011).Table 4 lists spreadsheet model outputs and select calculated intermediate values. Based on the THI value the cow was experiencing moderate-severe heat stress and would be expected to have a respiration rate greater than 85 bpm and rectal temperature greater than 40°C (Renaudeau et al., 2012). The modified model results were a body temperature of 39.8°C and a respiration rate of 86 bpm.

The heat production values in table 4 demonstrate the important role that milk yield plays in total metabolic heat production flux. For this case, milk yield accounted for 141 W m

^{-2}(58%) of the total metabolic heat production. A cow with a lower milk yield would have a lower amount of metabolic heat to dissipate in warm and hot weather to maintain homeothermic conditions. Milk yield is not considered when assessing heat stress using the temperature-humidity-index (THI) or the equivalent temperature index for dairy cattle (ETIC) (Wang et al., 2018a, 2019).The heat flux results indicate the importance of the different methods of heat exchange. For this case, the total flux the cow must dissipate was 386 W/m

^{2}, the sum of the shortwave radiant flux absorbed (144 W/m^{2}) and the metabolic heat flux generated (242 W/m^{2}). The shortwave radiation added 37% to the total flux to be dissipated. The evaporative heat loss flux dissipated over 43% of the total, while the convective heat loss flux dissipated over 21% of the total heat dissipated. The remaining heat flux was either dissipated by convection, respiration, and longwave radiation or stored in the body.

Figure 5. Modified model tissue resistance (dots) versus skin temperature and equation from McArthur (1987) (line). Figure 6. Model body temperatures (dots) versus skin temperature and body temperatures (line) computed using empirical equation from McArthur (1987).

ConclusionA modified steady-state process-based heat transfer model was developed, building on work by McArthur (1987), McGovern and Bruce (2000), Turnpenny et al. (2000a,b), Berman (2005), Thompson et al. (2014), and Gwadera et al. (2017) to describe the heat exchange between a lactating cow and the surrounding environment needed to maintain homeothermic conditions under warm or hot conditions. New and modified relations were used to describe tissue insulation as a function of respiration rate, sweat rate as a function of skin temperature, body temperature as a function of respiration rate, heat storage as a function of body temperature, longwave radiation exchange, and coat and convective boundary heat fluxes as functions of skin temperature, coat temperature, and air temperature. The modified model sweat rate, tissue resistance, and body temperature results versus skin temperatures were similar to those published by McArthur (1987) and Thompson et al. (2011).

Table 3. Model inputs for the 15:00 solar hour on July 12, 2015, not listed in table 1. Variable description Value Variable description Value Cow mass 560 kg Atmospheric pressure 100,230 Pa Milk production 32.9 kg d ^{-1}Air dry-bulb temperature 32.1°C Milk fat 3.7% Air dew-point temperature 26.9°C Mid cow height 1.0 m Solar load (measured) 535 W m ^{-2}Percent cow surface area exposed 100% Date and Solar time July 12 @ 15:00 Cow azimuth relative to south 0 deg Solar shading factor 0 Wind speed at 10 m height 6.5 m s ^{-1}Latitude for the WCROC 45.35°N Wind speed at mid cow height of 1.0 m ^{[a]}5.1 m s ^{-1}Evaporation flux for finding ground temperature 52 W m ^{-2}^{[a]}Adjusted for mid cow height using Power law, u_{cow}= u_{w}· (h_{m}/10)^{p}, for wind speed, u_{w}, measured at 10 m height, input mid cow height, h_{m}= 1 m, and exponent, p = 0.1, for an assumed stability class C and smooth rural surface (Cooper and Alley, 2011).

Table 4. Model outputs. Variable description Value Variable description Value Cow characteristics Ambient air conditions Cow surface area 5.2 m ^{2}Relative humidity 74% Cow trunk diameter 0.71 m Temperature Humidity Index 83 Cow trunk length 1.98 m Air density 1.13 kg m ^{-3}Coat depth decrease 1.56 mm Humidity ratio Maximum tissue insulation 0.112 m ^{2}K W^{-1}Specific heat of air 1006 J kg ^{-1}K^{-1}Tissue insulation 20.5 s m ^{-1}Volumetric specific heat of air 1136 J m ^{-3}K^{-1}Coat conductance 112 W m ^{-2}C^{-1}Sweat rate 252 g m ^{-2}h^{-1}Temperatures Body temperature 39.8°C Respiration values Skin temperature 37.6°C Warm respiration rate limit 12 bpm Coat temperature 37.9°C Transition respiration rate limit 126 bpm Exhaled air temperature 38.1°C Severe respiration rate limit 88 bpm Lower critical skin temperature 14°C Tidal volume at warm conditions 3.79 L breath ^{-1}Skin temperature at 50% respiration rate 36°C Tidal volume at maximum respiration 2.12 L breath ^{-1}Ground temperature 315 K Maximal tidal volume limit based on cow mass 3.98 L breath ^{-1}Sky temperature 298 K Inspiration volumetric rate 193 L min ^{-1}Ambient air virtual temperature 309 K Respiration rate 88 bpm Exhaled air virtual temperature 319 K Skin virtual temperature 318 K Heat production Maintenance 9.2 Mcal d ^{-1}Heat flux values Lactation 15.1 Mcal d ^{-1}Heat flux from body core to skin surface 137 W m ^{-2}Maintenance plus lactation production flux 227 W m ^{-2}Respiratory heat flux 43 W m ^{-2}Flux increase due to body temperature rise 12 W m ^{-2}Convective heat flux from coat to air 84 W m ^{-2}Flux increase due to increased respiration 3 W m ^{-2}Evaporative (sweat) heat flux 169 W m ^{-2}Total metabolic heat production flux 242 W m ^{-2}Body storage 61 W m ^{-2}Shortwave radiation absorbed 144 W m ^{-2}Solar characteristics Longwave radiation outbound 29 W m ^{-2}Hour angle 45 deg Outbound skin to coat sensible heat flux -31 W m ^{-2}Solar declination angle 22 deg Outbound coat surface to surroundings -31 W m ^{-2}Solar altitude 46.6 deg Solar azimuth 72.8 deg Sweat rate values Extraterrestrial irradiance for July 1324 W m ^{-2}Parameter a _{2}in sweat rate function (eq. 38)8.37 Cow’s azimuth angle 0 deg Sweat rate at 50% respiratory cooling 218 g m ^{-2}h^{-1}Azimuth angle of cow to sun 72.8 deg Maximum environmental sweat rate 1280 g m ^{-2}h^{-1}Horizontal area absorbing shortwave radiation 0.385 m ^{2}m^{-2}Sweat rate at skin temperature 252 g m ^{-2}h^{-1}Solar irradiance on a horizontal surface 206 W m ^{-2}Tidal volume at 50% total respiration cooling 198 L min ^{-1}Net shortwave radiation absorbed 144 W m ^{-2}Resistance to vapor transfer in coat 51 s m ^{-1}Sun shading factor 0.85 Cloud cover factor 0.15 Additional psychrometric values Parameter a _{1}used in equation 261.405 Exhaled air saturation humidity ratio Parameter b used in equation 26 -0.00441 Density exhaled air 1.11 kg m ^{-3}Cloudy sky emissivity 0.96 Latent heat of vaporization – ambient air 2.42·10 ^{6}J kg^{-1}Latent heat of vaporization – exhaled air 2.41·10 ^{6}J kg^{-1}Convection related values Latent heat of vaporization – skin temperature 2.41·10 ^{6}J kg^{-1}Kinematic viscosity at coat and air interface 1.68·10 ^{-5}m^{2}s^{-1}Ambient air partial vapor pressure 3,546 Pa Nusselt Number 384 Saturation vapor pressure at skin temperature 6,490 Pa Reynolds Number 2.19·10 ^{5}Psychrometric constant at air temperature 66.9 Grashof Number 2.64·10 ^{8}Water vapor diffusivity in air at air temperature 2.72·10 ^{-5}m^{2}s^{-1}Water vapor diffusivity in air at skin temperature 2.80·10 ^{-5}m^{2}s^{-1}Air thermal conductivity at air temperature 0.0265 W m ^{-2}K^{-1}Air thermal conductivity at coat surface temperature 0.0269 W m ^{-2}K^{-1}The modified model, which can be solved with a spreadsheet, provides insight into factors that affect lactating cow heat exchange in addition to dry-bulb and dew-point temperature. A companion paper compares model results with published measured body temperatures, respiration rates, and skin temperatures. The companion paper also compares model results with unpublished reticular temperatures of cows on pasture. It is expected that the modified model can help assess the relative importance of factors affecting heat exchange (i.e., body mass, daily milk yield, coat thickness, coat reflectivity, tissue insulation, solar load, exposed surface area, and air velocity) and their impact on heat exchange, cow respiration rate, and body temperature.

AcknowledgmentsThe authors thank Kirsten Sharpe and Brad Heins for sharing weather and cow data from the University of Minnesota West Central Research and Outreach Center.

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