Heat transfer by radiation (ht.radiation)

ht.radiation.blackbody_spectral_radiance(T, wavelength)[source]

Returns the spectral radiance, in units of W/m^2/sr/µm.

\[I_{\lambda,blackbody,e}(\lambda,T)=\frac{2hc_o^2} {\lambda^5[\exp(hc_o/\lambda k T)-1]}\]
T : float

Temperature of the surface, [K]

wavelength : float

Length of the wave to be considered, [m]

I : float

Spectral radiance [W/(m^2*sr*m)]


Can be used to derive the Stefan-Boltzman law, or determine the maximum radiant frequency for a given temperature.


[1]Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.
[2](1, 2) Spectral-calc.com. Blackbody Calculator, 2015. http://www.spectralcalc.com/blackbody_calculator/blackbody.php


Checked with Spectral-calc.com, at [2].

>>> blackbody_spectral_radiance(800., 4E-6)

Calculation of power from the sun (earth occupies 6.8E-5 steradian of the sun):

>>> from scipy.integrate import quad
>>> rad = lambda l: blackbody_spectral_radiance(5778., l)*6.8E-5
>>> quad(rad, 1E-10, 1E-4)[0]
ht.radiation.q_rad(emissivity, T, T2=0)[source]

Returns the radiant heat flux of a surface, optionally including assuming radiant heat transfer back to the surface.

\[q = \epsilon \sigma (T_1^4 - T_2^4)\]
emissivity : float

Fraction of black-body radiation which is emitted, [-]

T : float

Temperature of the surface, [K]

T2 : float, optional

Temperature of the surrounding material of the surface [K]

q : float

Heat exchange [W/m^2]


Emissivity must be less than 1. T2 may be larger than T.


[1]Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.


>>> q_rad(emissivity=1, T=400)
>>> q_rad(.85, T=400, T2=305.)
ht.radiation.grey_transmittance(extinction_coefficient, molar_density, length, base=2.718281828459045)[source]

Calculates the transmittance of a grey body, given the extinction coefficient of the material, its molar density, and the path length of the radiation.

\[\tau = base^{(-\epsilon \cdot l\cdot \rho_m )}\]
extinction_coefficient : float

The extinction coefficient of the material the radiation is passing at the modeled frequency, [m^2/mol]

molar_density : float

The molar density of the material the radiation is passing through, [mol/m^3]

length : float

The length of the body the radiation is transmitted through, [m]

base : float, optional

The exponent used in calculations; e is more theoretically sound but 10 is often used as a base by chemists, [-]

transmittance : float

The fraction of spectral radiance which is transmitted through a grey body (can be liquid, gas, or even solid ex. in the case of glasses) [-]


For extinction coefficients, see the HITRAN database. They are temperature and pressure dependent for each chemical and phase.


[1]Modest, Michael F. Radiative Heat Transfer, Third Edition. 3rd edition. New York: Academic Press, 2013.
[2](1, 2) Eldridge, Ralph G. “Water Vapor Absorption of Visible and Near Infrared Radiation.” Applied Optics 6, no. 4 (April 1, 1967): 709-13. https://doi.org/10.1364/AO.6.000709.


Overall transmission loss through 1 cm of precipitable water equivalent atmospheric water vapor at a frequency of 1.3 um [2]:

>>> grey_transmittance(3.8e-4, molar_density=55300, length=1e-2)

Returns the solar spectrum of the sun according to the specified model. Only the ‘SOLAR-ISS’ model is supported.

model : str, optional

The model to use; ‘SOLAR-ISS’ is the only model available, [-]

wavelengths : ndarray

The wavelengths of the solar spectra, [m]

SSI : ndarray

The solar spectral irradiance of the sun, [W/(m^2*m)]

uncertainties : ndarray

The estimated absolute uncertainty of the measured spectral irradiance of the sun, [W/(m^2*m)]


The power of the sun changes as the earth gets closer or further away.

In [1], the UV and VIS data come from observations in 2008; the IR comes from measurements made from 2010-2016. There is a further 28 W/m^2 for the 3 micrometer to 160 micrometer range, not included in this model. All data was corrected to a standard distance of one astronomical unit from the Sun, as is the resultant spectrum.

The variation of the spectrum as a function of distance from the sun should alter only the absolute magnitudes.

[2] contains another dataset.


[1](1, 2) Meftah, M., L. Damé, D. Bolsée, A. Hauchecorne, N. Pereira, D. Sluse, G. Cessateur, et al. “SOLAR-ISS: A New Reference Spectrum Based on SOLAR/SOLSPEC Observations.” Astronomy & Astrophysics 611 (March 1, 2018): A1. https://doi.org/10.1051/0004-6361/201731316.
[2](1, 2) Woods Thomas N., Chamberlin Phillip C., Harder Jerald W., Hock Rachel A., Snow Martin, Eparvier Francis G., Fontenla Juan, McClintock William E., and Richard Erik C. “Solar Irradiance Reference Spectra (SIRS) for the 2008 Whole Heliosphere Interval (WHI).” Geophysical Research Letters 36, no. 1 (January 1, 2009). https://doi.org/10.1029/2008GL036373.


>>> wavelengths, SSI, uncertainties = solar_spectrum()

Calculate the minimum and maximum values of the wavelengths (0.5 nm/3000nm) and SSI:

>>> min(wavelengths), max(wavelengths), min(SSI), max(SSI)
(5e-10, 2.9999000000000003e-06, 1330.0, 2256817820.0)

Integration - calculate the solar constant, in untis of W/m^2 hitting earth’s atmosphere.

>>> np.trapz(SSI, wavelengths)