R/OperativeTemperatureFunctions.R
In trenchproject/TrenchR: Tools for Microclimate and Biophysical Ecology
#' @title Operative Environmental Temperature of an Ectotherm based on Campbell and Norman (1988)
#'
#' @description The function estimates body temperatures (C, operative environmental temperature) of an ectotherm using an approximation based on \insertCite{Campbell1998;textual}{TrenchR} and \insertCite{Mitchell1976;textual}{TrenchR}.
#'
#' @param T_a \code{numeric} air temperature (C).
#'
#' @param T_g \code{numeric} ground temperature (C).
#'
#' @param S \code{numeric} flux density of solar radiation (\ifelse{html}{\out{W m<sup>-2</sup>}}{\eqn{W m^-2}{ASCII}}), combining direct, diffuse, and reflected radiation accounting for view factors.
#'
#' @param a_s \code{numeric} organismal solar absorptivity (proportion).
#'
#' @param a_l \code{numeric} organismal thermal absorptivity (proportion); 0.965 for lizards \insertCite{Bartlett1967}{TrenchR}.
#'
#' @param epsilon \code{numeric} longwave infrared emissivity of skin (proportion), 0.95 to 1 for most animals \insertCite{Gates1980}{TrenchR}.
#'
#' @param c_p \code{numeric} specific heat of air (\ifelse{html}{\out{J mol<sup>-1</sup> C<sup>-1</sup>}}{\eqn{J mol^-1 C^-1}{ASCII}}).
#'
#' @param D \code{numeric} characteristic dimension of the animal (m).
#'
#' @param u \code{numeric} wind speed (\ifelse{html}{\out{m s<sup>-1</sup>}}{\eqn{m s^-1}{ASCII}}).
#'
#' @details Boundary conductance uses a factor of 1.4 to account for increased convection \insertCite{Mitchell1976}{TrenchR}. The function assumes forced conduction.
#'
#' @return \code{numeric} operative environmental temperature, \code{T_e} (C).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Tb_CampbellNorman (T_a = 30,
#' T_g = 30,
#' S = 823,
#' a_s = 0.7,
#' a_l = 0.96,
#' epsilon = 0.96,
#' c_p = 29.3,
#' D = 0.17,
#' u = 1)
#'
Tb_CampbellNorman <- function (T_a,
T_g,
S,
a_s = 0.7,
a_l = 0.96,
epsilon = 0.96,
c_p = 29.3,
D,
u) {
stopifnot(T_a > -50,
T_a < 100,
epsilon >= 0.5,
epsilon <= 1,
a_s >= 0,
a_l >= 0,
c_p >= 0,
D > 0,
u >= 0)
sigma <- stefan_boltzmann_constant()
#convert temperatures to Kelvin
T_a = T_a +273.15
T_g = T_g +273.15
# solar and thermal radiation absorbed
# (10.7) long wave flux densities from atmosphere, ground
L_a <- sigma * T_a^4
L_g <- sigma * T_g^4
# proportion of organism exposure to air, ground
F_a <- 0.5
F_g <- 0.5
# (11.14) Absorbed radiation
R_abs <- a_s * S + a_l * (F_a * L_a + F_g * L_g)
# thermal radiation emitted
Qemit <- epsilon * sigma * T_a^4
# conductance: boundary, ground
# boundary conductance, factor of 1.4 to account for increased convection (Mitchell 1976), assumes forced conduction
g_Ha <- 1.4 * 0.135 * sqrt(u / D)
g_r <- 4 * epsilon * sigma * T_a^3 / c_p
# operative environmental temperature in C
T_a + (R_abs - Qemit) / (c_p * (g_r + g_Ha)) -273.15
}
#' @title Net Energy Exchange Between an Animal and the Environment
#'
#' @description The function estimates the net energy exchange (W) between an animal and the environment. The function follows \insertCite{Gates1980;textual}{TrenchR} and others.
#'
#' @param Qabs \code{numeric} solar radiation absorbed (W).
#'
#' @param Qemit \code{numeric} thermal radiation emitted (W).
#'
#' @param Qconv \code{numeric} energy exchange due to convection; Energy exchange from an animal to its surrounding environment (air or water) (W).
#'
#' @param Qcond \code{numeric} energy exchange due to conduction; Energy exchange from animal to a surface if they are in contact (W).
#'
#' @param Qmet \code{numeric} energy emitted due to metabolism (W).
#'
#' @param Qevap \code{numeric} energy emitted due to evaporative water loss (W).
#'
#' @return \code{numeric} net energy exchange (W).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Qnet_Gates(Qabs = 500,
#' Qemit = 10,
#' Qconv = 100,
#' Qcond = 100,
#' Qmet = 10,
#' Qevap = 5)
#'
Qnet_Gates <- function (Qabs,
Qemit,
Qconv,
Qcond,
Qmet,
Qevap) {
stopifnot(Qabs >= 0,
Qmet >= 0)
Qabs - Qemit - Qconv - Qcond - Qmet - Qevap
}
#' @title Operative Environmental Temperature of an Ectotherm Based on Gates (1980)
#'
#' @description The function predicts body temperatures (C, operative environmental temperature) of an ectotherm using the approximation in \insertCite{Gates1980;textual}{TrenchR}. The functions omits evaporative and metabolic heat loss \insertCite{Mitchell1976,Kingsolver1983}{TrenchR}.
#'
#' @param A \code{numeric} surface area (\ifelse{html}{\out{m<sup>2</sup>}}{\eqn{m^2}{ASCII}}).
#'
#' @param D \code{numeric} characteristic dimension for conduction (meters).
#'
#' @param psa_dir \code{numeric} view factor for the proportion surface area exposed to direct radiation from the sky (or enclosure) (0-1).
#'
#' @param psa_ref \code{numeric} view factor for proportion surface area exposed to reflected radiation from the ground (0-1).
#'
#' @param psa_air \code{numeric} proportion surface area exposed to air (0-1).
#'
#' @param psa_g \code{numeric} proportion surface area in contact with substrate (0-1).
#'
#' @param T_g \code{numeric} ground surface temperature (C).
#'
#' @param T_a \code{numeric} ambient air temperature (C).
#'
#' @param Qabs \code{numeric} Solar radiation absorbed (W).
#'
#' @param epsilon \code{numeric} longwave infrared emissivity of skin (proportion), 0.95 to 1 for most animals \insertCite{Gates1980}{TrenchR}.
#'
#' @param H_L \code{numeric} Convective heat transfer coefficient (\ifelse{html}{\out{W m<sup>-2</sup> K<sup>-1</sup>}}{\eqn{W m^-2 K^-1}{ASCII}}).
#'
#' @param ef \code{numeric} enhancement factor used to adjust H_L to field conditions using h_L approximation from \insertCite{Mitchell1976;textual}{TrenchR}. Approximated as 1.23 by default, but see \insertCite{Mitchell1976;textual}{TrenchR} for relationship.
#'
#' @param K \code{numeric} Thermal conductivity (\ifelse{html}{\out{W K<sup>-1</sup> m<sup>-1</sup>}}{\eqn{W K^-1 m^-1}{ASCII}}), K = 0.5 \ifelse{html}{\out{W K<sup>-1</sup> m<sup>-1</sup>}}{\eqn{W K^-1 m^-1}{ASCII}} for naked skin, K = 0.15 \ifelse{html}{\out{W K<sup>-1</sup> m<sup>-1</sup>}}{\eqn{W K^-1 m^-1}{ASCII}}for insect cuticle \insertCite{Galushko2005;textual}{TrenchR}; conductivity of the ground is generally greater than that of animal tissues, so animal thermal conductivity is generally the rate limiting step.
#'
#' @return \code{numeric} operative environmental temperature, \code{T_e} (C).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Tb_Gates (A = 0.1,
#' D = 0.025,
#' psa_dir = 0.6,
#' psa_ref = 0.4,
#' psa_air = 0.5,
#' psa_g = 0.1,
#' T_g = 30,
#' T_a = 37,
#' Qabs = 2,
#' epsilon = 0.95,
#' H_L = 10,
#' ef = 1.23,
#' K = 0.5)
#'
Tb_Gates <- function (A,
D,
psa_dir,
psa_ref,
psa_air,
psa_g,
T_g,
T_a,
Qabs,
epsilon,
H_L,
ef = 1.3,
K) {
stopifnot(A > 0,
D > 0,
psa_dir >= 0,
psa_dir <= 1,
psa_ref >= 0,
psa_ref <= 1,
psa_air >= 0,
psa_air <= 1,
psa_g >= 0,
psa_g <= 1,
T_g > -50,
T_g < 100,
T_a > -50,
T_a < 100,
Qabs >= 0,
epsilon > 0.5,
epsilon <= 1,
H_L > 0,
K > 0)
sigma <- stefan_boltzmann_constant()
# Areas
A_s <- A * psa_dir
A_r <- A * psa_ref
# skin area exposed to air
A_air <- A * psa_air
# area of contact
A_contact <- A * psa_g
# effective radiant temperature of sky
# K, Gates 1980 Biophysical ecology based on Swinback 1960, Kingsolver (1983) estimates using Brunt equation
T_sky <- celsius_to_kelvin(1.22 * T_a - 20.4)
#convert to Kelvin
T_a= celsius_to_kelvin(T_a)
T_g= celsius_to_kelvin(T_g)
# energy balance function for steady state conditions
# 0 = Qabs - Qemit - Qconv - Qcond
Qfn <- function(T_b, Qabs, epsilon, sigma, A_s, T_sky, A_r, T_g, H_L, A_air, T_a, A_contact, K, D) {
Qemit <- epsilon * sigma * (A_s * (T_b^4 - T_sky^4) + A_r * (T_b^4 - T_g^4))
Qconv <- ef * H_L * A_air * (T_b - T_a)
Qcond <- A_contact * K * (T_b - T_g) / D
Qabs - Qemit - Qconv - Qcond
}
T_e <- tryCatch(expr = uniroot(f = Qfn,
interval = c(273, 350),
Qabs = Qabs,
epsilon = epsilon,
sigma = sigma,
A_s = A_s,
T_sky = T_sky,
A_r = A_r,
T_g = T_g,
H_L = H_L,
A_air = A_air,
T_a = T_a,
A_contact = A_contact,
K = K,
D = D,
tol = 0.0001)$root,
error = function(e) {
message("Unable to balance energy budget. One issue to check is whether absorbed solar radiation exceeds energy potentially lost to thermal radiation, convection, and conduction.")
NA})
kelvin_to_celsius(T_e)
}
#' @title Operative Environmental Temperature of an Ectotherm Based on a Variant of Gates (1980)
#'
#' @description The function predicts body temperatures (C, operative environmental temperature) of an ectotherm using the approximation in \insertCite{Gates1980;textual}{TrenchR}. The function omits evaporative and metabolic heat loss.
#'
#' @param A \code{numeric} surface area (\ifelse{html}{\out{m<sup>2</sup>}}{\eqn{m^2}{ASCII}}).
#'
#' @param D \code{numeric} characteristic dimension for conduction (meters).
#'
#' @param T_g \code{numeric} ground surface temperature (C).
#'
#' @param T_a \code{numeric} ambient air temperature (C).
#'
#' @param Qabs \code{numeric} Solar radiation absorbed (W).
#'
#' @param u \code{numeric} Wind speed (\ifelse{html}{\out{m s<sup>-1</sup>}}{\eqn{m s^-1}{ASCII}}).
#'
#' @param epsilon \code{numeric} longwave infrared emissivity of skin (proportion), 0.95 to 1 for most animals \insertCite{Gates1980}{TrenchR}.
#'
#' @return \code{numeric} operative environmental temperature (C).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Tb_Gates2(A = 1,
#' D = 0.001,
#' T_g = 27,
#' T_a = 37,
#' Qabs = 2,
#' u = 0.1,
#' epsilon = 1)
#'
Tb_Gates2 <- function (A,
D,
T_g,
T_a,
Qabs,
u,
epsilon) {
sigma <- stefan_boltzmann_constant()
stopifnot(A > 0,
D > 0,
T_g > -50,
T_g < 100,
T_a > -50,
T_a < 100,
Qabs >= 0,
epsilon > 0.5,
epsilon <= 1,
u > 0)
# skin area exposed to air
A_air <- A
# Convection coefficient from Gates (1980)
H_L <- 3.49 * (u^0.5 / D^0.5)
# effective radiant temperature of sky
# K, Gates 1980 Biophysical ecology based on Swinback 1960, Kingsolver (1983) estimates using Brunt equation
T_sky <- celsius_to_kelvin(1.22 * (T_a - 20.4))
#convert to Kelvin
T_a= celsius_to_kelvin(T_a)
T_g= celsius_to_kelvin(T_g)
# Thermal radiation absorbed
QIR <- (A_air / 2) * (sigma * epsilon * T_g^4 + sigma * epsilon * T_sky^4)
# energy balance function for steady state conditions
# 0 = Qabs - Qemit - Qconv - Qcond
Qfn <- function(T_b, Qabs, QIR, epsilon, sigma, T_sky, H_L, A_air, T_a) {
Qemit <- A_air * sigma * epsilon * T_b^4
Qconv <- H_L * A_air * (T_b - T_a)
Qabs + QIR - Qemit - Qconv
}
T_e <- tryCatch(expr = uniroot(f = Qfn,
interval = c(273, 353),
Qabs = Qabs,
QIR = QIR,
epsilon = epsilon,
sigma = sigma,
T_sky = T_sky,
H_L = H_L,
A_air = A_air,
T_a = T_a,
tol = 0.0001)$root,
error = function(e) {
message("Unable to balance energy budget. One issue to check is whether absorbed solar radiation exceeds energy potentially lost to thermal radiation, convection, and conduction.")
NA})
kelvin_to_celsius(T_e)
}
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#' @title Operative Environmental Temperature of an Ectotherm based on Campbell and Norman (1988)
#'
#' @description The function estimates body temperatures (C, operative environmental temperature) of an ectotherm using an approximation based on \insertCite{Campbell1998;textual}{TrenchR} and \insertCite{Mitchell1976;textual}{TrenchR}.
#'
#' @param T_a \code{numeric} air temperature (C).
#'
#' @param T_g \code{numeric} ground temperature (C).
#'
#' @param S \code{numeric} flux density of solar radiation (\ifelse{html}{\out{W m<sup>-2</sup>}}{\eqn{W m^-2}{ASCII}}), combining direct, diffuse, and reflected radiation accounting for view factors.
#'
#' @param a_s \code{numeric} organismal solar absorptivity (proportion).
#'
#' @param a_l \code{numeric} organismal thermal absorptivity (proportion); 0.965 for lizards \insertCite{Bartlett1967}{TrenchR}.
#'
#' @param epsilon \code{numeric} longwave infrared emissivity of skin (proportion), 0.95 to 1 for most animals \insertCite{Gates1980}{TrenchR}.
#'
#' @param c_p \code{numeric} specific heat of air (\ifelse{html}{\out{J mol<sup>-1</sup> C<sup>-1</sup>}}{\eqn{J mol^-1 C^-1}{ASCII}}).
#'
#' @param D \code{numeric} characteristic dimension of the animal (m).
#'
#' @param u \code{numeric} wind speed (\ifelse{html}{\out{m s<sup>-1</sup>}}{\eqn{m s^-1}{ASCII}}).
#'
#' @details Boundary conductance uses a factor of 1.4 to account for increased convection \insertCite{Mitchell1976}{TrenchR}. The function assumes forced conduction.
#'
#' @return \code{numeric} operative environmental temperature, \code{T_e} (C).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Tb_CampbellNorman (T_a = 30,
#' T_g = 30,
#' S = 823,
#' a_s = 0.7,
#' a_l = 0.96,
#' epsilon = 0.96,
#' c_p = 29.3,
#' D = 0.17,
#' u = 1)
#'
Tb_CampbellNorman <- function (T_a,
T_g,
S,
a_s = 0.7,
a_l = 0.96,
epsilon = 0.96,
c_p = 29.3,
D,
u) {
stopifnot(T_a > -50,
T_a < 100,
epsilon >= 0.5,
epsilon <= 1,
a_s >= 0,
a_l >= 0,
c_p >= 0,
D > 0,
u >= 0)
sigma <- stefan_boltzmann_constant()
#convert temperatures to Kelvin
T_a = T_a +273.15
T_g = T_g +273.15
# solar and thermal radiation absorbed
# (10.7) long wave flux densities from atmosphere, ground
L_a <- sigma * T_a^4
L_g <- sigma * T_g^4
# proportion of organism exposure to air, ground
F_a <- 0.5
F_g <- 0.5
# (11.14) Absorbed radiation
R_abs <- a_s * S + a_l * (F_a * L_a + F_g * L_g)
# thermal radiation emitted
Qemit <- epsilon * sigma * T_a^4
# conductance: boundary, ground
# boundary conductance, factor of 1.4 to account for increased convection (Mitchell 1976), assumes forced conduction
g_Ha <- 1.4 * 0.135 * sqrt(u / D)
g_r <- 4 * epsilon * sigma * T_a^3 / c_p
# operative environmental temperature in C
T_a + (R_abs - Qemit) / (c_p * (g_r + g_Ha)) -273.15
}
#' @title Net Energy Exchange Between an Animal and the Environment
#'
#' @description The function estimates the net energy exchange (W) between an animal and the environment. The function follows \insertCite{Gates1980;textual}{TrenchR} and others.
#'
#' @param Qabs \code{numeric} solar radiation absorbed (W).
#'
#' @param Qemit \code{numeric} thermal radiation emitted (W).
#'
#' @param Qconv \code{numeric} energy exchange due to convection; Energy exchange from an animal to its surrounding environment (air or water) (W).
#'
#' @param Qcond \code{numeric} energy exchange due to conduction; Energy exchange from animal to a surface if they are in contact (W).
#'
#' @param Qmet \code{numeric} energy emitted due to metabolism (W).
#'
#' @param Qevap \code{numeric} energy emitted due to evaporative water loss (W).
#'
#' @return \code{numeric} net energy exchange (W).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Qnet_Gates(Qabs = 500,
#' Qemit = 10,
#' Qconv = 100,
#' Qcond = 100,
#' Qmet = 10,
#' Qevap = 5)
#'
Qnet_Gates <- function (Qabs,
Qemit,
Qconv,
Qcond,
Qmet,
Qevap) {
stopifnot(Qabs >= 0,
Qmet >= 0)
Qabs - Qemit - Qconv - Qcond - Qmet - Qevap
}
#' @title Operative Environmental Temperature of an Ectotherm Based on Gates (1980)
#'
#' @description The function predicts body temperatures (C, operative environmental temperature) of an ectotherm using the approximation in \insertCite{Gates1980;textual}{TrenchR}. The functions omits evaporative and metabolic heat loss \insertCite{Mitchell1976,Kingsolver1983}{TrenchR}.
#'
#' @param A \code{numeric} surface area (\ifelse{html}{\out{m<sup>2</sup>}}{\eqn{m^2}{ASCII}}).
#'
#' @param D \code{numeric} characteristic dimension for conduction (meters).
#'
#' @param psa_dir \code{numeric} view factor for the proportion surface area exposed to direct radiation from the sky (or enclosure) (0-1).
#'
#' @param psa_ref \code{numeric} view factor for proportion surface area exposed to reflected radiation from the ground (0-1).
#'
#' @param psa_air \code{numeric} proportion surface area exposed to air (0-1).
#'
#' @param psa_g \code{numeric} proportion surface area in contact with substrate (0-1).
#'
#' @param T_g \code{numeric} ground surface temperature (C).
#'
#' @param T_a \code{numeric} ambient air temperature (C).
#'
#' @param Qabs \code{numeric} Solar radiation absorbed (W).
#'
#' @param epsilon \code{numeric} longwave infrared emissivity of skin (proportion), 0.95 to 1 for most animals \insertCite{Gates1980}{TrenchR}.
#'
#' @param H_L \code{numeric} Convective heat transfer coefficient (\ifelse{html}{\out{W m<sup>-2</sup> K<sup>-1</sup>}}{\eqn{W m^-2 K^-1}{ASCII}}).
#'
#' @param ef \code{numeric} enhancement factor used to adjust H_L to field conditions using h_L approximation from \insertCite{Mitchell1976;textual}{TrenchR}. Approximated as 1.23 by default, but see \insertCite{Mitchell1976;textual}{TrenchR} for relationship.
#'
#' @param K \code{numeric} Thermal conductivity (\ifelse{html}{\out{W K<sup>-1</sup> m<sup>-1</sup>}}{\eqn{W K^-1 m^-1}{ASCII}}), K = 0.5 \ifelse{html}{\out{W K<sup>-1</sup> m<sup>-1</sup>}}{\eqn{W K^-1 m^-1}{ASCII}} for naked skin, K = 0.15 \ifelse{html}{\out{W K<sup>-1</sup> m<sup>-1</sup>}}{\eqn{W K^-1 m^-1}{ASCII}}for insect cuticle \insertCite{Galushko2005;textual}{TrenchR}; conductivity of the ground is generally greater than that of animal tissues, so animal thermal conductivity is generally the rate limiting step.
#'
#' @return \code{numeric} operative environmental temperature, \code{T_e} (C).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Tb_Gates (A = 0.1,
#' D = 0.025,
#' psa_dir = 0.6,
#' psa_ref = 0.4,
#' psa_air = 0.5,
#' psa_g = 0.1,
#' T_g = 30,
#' T_a = 37,
#' Qabs = 2,
#' epsilon = 0.95,
#' H_L = 10,
#' ef = 1.23,
#' K = 0.5)
#'
Tb_Gates <- function (A,
D,
psa_dir,
psa_ref,
psa_air,
psa_g,
T_g,
T_a,
Qabs,
epsilon,
H_L,
ef = 1.3,
K) {
stopifnot(A > 0,
D > 0,
psa_dir >= 0,
psa_dir <= 1,
psa_ref >= 0,
psa_ref <= 1,
psa_air >= 0,
psa_air <= 1,
psa_g >= 0,
psa_g <= 1,
T_g > -50,
T_g < 100,
T_a > -50,
T_a < 100,
Qabs >= 0,
epsilon > 0.5,
epsilon <= 1,
H_L > 0,
K > 0)
sigma <- stefan_boltzmann_constant()
# Areas
A_s <- A * psa_dir
A_r <- A * psa_ref
# skin area exposed to air
A_air <- A * psa_air
# area of contact
A_contact <- A * psa_g
# effective radiant temperature of sky
# K, Gates 1980 Biophysical ecology based on Swinback 1960, Kingsolver (1983) estimates using Brunt equation
T_sky <- celsius_to_kelvin(1.22 * T_a - 20.4)
#convert to Kelvin
T_a= celsius_to_kelvin(T_a)
T_g= celsius_to_kelvin(T_g)
# energy balance function for steady state conditions
# 0 = Qabs - Qemit - Qconv - Qcond
Qfn <- function(T_b, Qabs, epsilon, sigma, A_s, T_sky, A_r, T_g, H_L, A_air, T_a, A_contact, K, D) {
Qemit <- epsilon * sigma * (A_s * (T_b^4 - T_sky^4) + A_r * (T_b^4 - T_g^4))
Qconv <- ef * H_L * A_air * (T_b - T_a)
Qcond <- A_contact * K * (T_b - T_g) / D
Qabs - Qemit - Qconv - Qcond
}
T_e <- tryCatch(expr = uniroot(f = Qfn,
interval = c(273, 350),
Qabs = Qabs,
epsilon = epsilon,
sigma = sigma,
A_s = A_s,
T_sky = T_sky,
A_r = A_r,
T_g = T_g,
H_L = H_L,
A_air = A_air,
T_a = T_a,
A_contact = A_contact,
K = K,
D = D,
tol = 0.0001)$root,
error = function(e) {
message("Unable to balance energy budget. One issue to check is whether absorbed solar radiation exceeds energy potentially lost to thermal radiation, convection, and conduction.")
NA})
kelvin_to_celsius(T_e)
}
#' @title Operative Environmental Temperature of an Ectotherm Based on a Variant of Gates (1980)
#'
#' @description The function predicts body temperatures (C, operative environmental temperature) of an ectotherm using the approximation in \insertCite{Gates1980;textual}{TrenchR}. The function omits evaporative and metabolic heat loss.
#'
#' @param A \code{numeric} surface area (\ifelse{html}{\out{m<sup>2</sup>}}{\eqn{m^2}{ASCII}}).
#'
#' @param D \code{numeric} characteristic dimension for conduction (meters).
#'
#' @param T_g \code{numeric} ground surface temperature (C).
#'
#' @param T_a \code{numeric} ambient air temperature (C).
#'
#' @param Qabs \code{numeric} Solar radiation absorbed (W).
#'
#' @param u \code{numeric} Wind speed (\ifelse{html}{\out{m s<sup>-1</sup>}}{\eqn{m s^-1}{ASCII}}).
#'
#' @param epsilon \code{numeric} longwave infrared emissivity of skin (proportion), 0.95 to 1 for most animals \insertCite{Gates1980}{TrenchR}.
#'
#' @return \code{numeric} operative environmental temperature (C).
#'
#' @family biophysical models
#'
#' @export
#'
#' @references
#' \insertAllCited{}
#'
#' @examples
#' Tb_Gates2(A = 1,
#' D = 0.001,
#' T_g = 27,
#' T_a = 37,
#' Qabs = 2,
#' u = 0.1,
#' epsilon = 1)
#'
Tb_Gates2 <- function (A,
D,
T_g,
T_a,
Qabs,
u,
epsilon) {
sigma <- stefan_boltzmann_constant()
stopifnot(A > 0,
D > 0,
T_g > -50,
T_g < 100,
T_a > -50,
T_a < 100,
Qabs >= 0,
epsilon > 0.5,
epsilon <= 1,
u > 0)
# skin area exposed to air
A_air <- A
# Convection coefficient from Gates (1980)
H_L <- 3.49 * (u^0.5 / D^0.5)
# effective radiant temperature of sky
# K, Gates 1980 Biophysical ecology based on Swinback 1960, Kingsolver (1983) estimates using Brunt equation
T_sky <- celsius_to_kelvin(1.22 * (T_a - 20.4))
#convert to Kelvin
T_a= celsius_to_kelvin(T_a)
T_g= celsius_to_kelvin(T_g)
# Thermal radiation absorbed
QIR <- (A_air / 2) * (sigma * epsilon * T_g^4 + sigma * epsilon * T_sky^4)
# energy balance function for steady state conditions
# 0 = Qabs - Qemit - Qconv - Qcond
Qfn <- function(T_b, Qabs, QIR, epsilon, sigma, T_sky, H_L, A_air, T_a) {
Qemit <- A_air * sigma * epsilon * T_b^4
Qconv <- H_L * A_air * (T_b - T_a)
Qabs + QIR - Qemit - Qconv
}
T_e <- tryCatch(expr = uniroot(f = Qfn,
interval = c(273, 353),
Qabs = Qabs,
QIR = QIR,
epsilon = epsilon,
sigma = sigma,
T_sky = T_sky,
H_L = H_L,
A_air = A_air,
T_a = T_a,
tol = 0.0001)$root,
error = function(e) {
message("Unable to balance energy budget. One issue to check is whether absorbed solar radiation exceeds energy potentially lost to thermal radiation, convection, and conduction.")
NA})
kelvin_to_celsius(T_e)
}
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