Basic Laws of Radiative Heat Transfer

Please find below some of the basic laws of radiative heat transfer. We will be adding to and developing this information over time so please keep an eye out. This information is an extract from “Infrared Heating for Food and Agriculture Processing” as edited by Zhongli Pan and Griffiths Gregory Atungulu which can be purchased from http://www.crcpress.com/product/isbn/9781420090970.

IR radiation is the part of the electromagnetic spectrum that is predominately responsible for the heating effect of the sun, as show in figure 1.1 (modest , 1993). IR radiation can be divided into three different categories: near-IR (NIR/Short wave), mid-IR radiation (MIR/Medium wave), and far-IR radiation (FIR / Long wave) (Table 1.1; Sakai and Hanzawa, 1994). Since IR radiation is an electromagnetic wave, it has both a spectral and directional dependence. Spectral dependence of IR heating needs to be considered because energy coming out of an emitter is composed of different wavelengths, and the fraction of the radiation in each band is dependent on a number of factors such as the temperature of the emitter, emissivity of the emitter, etc. Radiation phenomena become more complicated because the amount of radiation that is incident on any surface does not have only a spectral dependence but also a directional dependence.

The wavelength at which the maximum radiation occurs is determined by the temperature of the heater. This relationship is described by the basic laws for blackbody radiation, such as Planck’s law, Wien’s displacement law, and Stefan-Boltzmann’s law (Sakai and Hanzawa, 1994; Dangerskog and Osterstrom, 1979)

Figure 1.1, Electromagnetic wave spectrum

Planck’s Law

Planck’s law presents the spectral distribution of radiation from a blackbody source that emits 100% IR radiation at a given single temperature (Modest, 1993).
IR sources are made up of thousands of point sources at different temperatures. By combining the point sources, an entire spectral distribution for specific regions can be obtained. The theory outlined here uses an approximation of the spectral distribution using an average surface temperature and emissivity value to characterize the IR radiation. In practice however, no infrared radiation can be averaged as the radiation output changes with the source temperature as does the absorption.
Max Planck (1901) reported the spectral blackbody emissive power distribution now commonly known as Planck’s law, for a black surface bounded by a transparent medium with refractive index n as

E^bλ(T,λ)= (2πhc 2/0)/(n²λ⁵[e^{(hc₀⁄nλkT)}-1)                           (1.1)

Where k is known as Boltzmann’s constant (1.3806 X 10^-23J/K), and n is the refractive index of the medium. By definition, the refractive index of a vacuum is n=1. For most gases, the refractive index is very close to unity. λ is the wavelength (μm), T is the source temperature (K), c_0 is the speed of light (Km/s), and h is Planck’s constant (6.626 X 10^-34J-s).

Figure 1.2(a) shows Planck’s curve based on equation 1.1 for a number of black body temperatures. Overall, the level of emissive power rises with an increase of temperature, while the wavelength of the corresponding maximum emissive power shifts toward shorter wavelengths. The total amount of IR emissive power within a specific region considered can be estimated by integration of Planck’s law at a given temperature with respect to the wavelength.
Planck’s law can be applied to estimate the total amount of radiative heat flux when a specific surface temperature of the heating element is known. An energy balance, to assess the amount of energy emitted from the IR source, proportionally directed through a conveying chamber known as view factor. Hence, the actual amount of heat flux absorbed by a target material can be estimated by calculating the total emissive power and view factors from the source to the target.

Wien’s Displacement Law

Wien’s displacement law gives the wavelength (denoted as peak wavelength) where the spectral distribution of radiation emitted by a blackbody reaches a maximum emissive power. The maximum of the curves (Figure 1.2) can be determined differentiating Equation 1.1:

[d/d(nλT)](E_bλ/n³λ⁵)=0 (1.2)

Source temperatures of IR lamps needed for a desired spectral distribution can be estimated by (Modest, 1993)

λ_max=2898/T (1.3)

Where T is the source temperature and λ_max  is the peak wavelength. If the source temperature is known, the peak wavelength can be derived from Equation 1.3. The dotted line in Figure 1.2(a) demonstrates the relationship between the source temperature and the peak wavelength. As an example, the emissive power spectrum of the original IR source with unknown surface temperature can be measured and recorded using the Fourier transform IR (FTIR) spectrometer (Figure 1.2(b)). Based on the plot and Equation 1.3, a peak wavelength of 2.92μm and correspondent IR source temperature of 720⁰C are obtained.

Figure 1.2 (a), Blackbody emissive power spectrum.

Figure 1.2 (b), Measured emissive power spectrum of IR heating elements.

Stefan-Boltzmann’s Law

Stefan – Boltzmann’s law gives the total power radiated at a specific temperature from an IR source. The entire amount of heat flux estimated using this law should be consistent with integration of the spectral amount of heat flux estimated using Planck’s law given in Sakai and Hanzawa (1994):

Where C1 = 2πhc²₀= 3.7419 X10^-16Wm², C2 = hc_olk=14,388μmK and σ is known as the Stefan-Boltzmann constant (5.670 X 10-8 W/m^2 K^4). Stefan-Boltzmann’s law is available for prompt estimation of the total amount of heat flux at a given source temperature.

Extinction of Radiation, Transmission, Absorption and Reflection

The mechanisms to explain the attenuation of electromagnetic radiation as it propagates through a medium are absorption and scattering. Converting the radiation to some other forms of energy (or some spectral distribution) is called absorption phenomena, whereas scattering mechanisms redirect the radiant energy from its original direction of propagation due to the combined effect of reflection, refraction and diffraction. The sum of the mechanisms of attenuation of electromagnetic radiation as it passes through a medium (absorption plus scattering) is generally called extinction of radiation (Sandu, 1986; Modest, 1993).
When the extinguishing material is agglomerated into particles, separated by regions of different transmissivities (such as emulsions and dispersions), or when variations occur in the density of the samples (as in capillary-porous bodies or in bodies subject to a temperature or moisture gradient or in solid bodies that contain a liquid free phase inside), Beer’s law should be formally adjusted for nonhomogeneous systems using

H_λ=H_λ0 exp(-σ_λ*u) (1.5)

Where H_λ is the transmitted spectral irradiance (W⁄(m²μm)),H_λ0 is the incident spectral irradiance (W⁄(m²μm)),u, is the mass of absorbing medium per unit area (kg⁄m² ) and σ_λ* is the spectral extinction coefficient (m²/kg).

Beer’s law states that the amount of light absorbed by a solution varies exponentially with the concentration of the solution and the length of the light path in the solution. The spectral extinction coefficient, σ_λ* (m²/kg) for a nonhomegeneous system is a complex function of the chemical composition of the radiated medium, the physiochemical state of the radiated medium, and the physiochemical parameters defining the radiated medium (density, porosity, diameter of particles, water content, etc.)

In radiative heating, an energy balance can be defined in relation to the extinction of radiation by a physical body. Assuming that this body is an infinite slab of given physicochemical composition and absorbed energy is the total radiation converted into heat inside the body, the entire process of extinction can be defined in terms of reflection, absorption, and transmission of radiation. The three fundamental radiative properties are reflectivity (ρ), absorptivity (α) as the ration of absorbed part of incoming radiation to the total incoming radiation and transmissivity (τ) as the ratio of transmitted part of incoming radiation to the total incoming radiation.

Under these terms, the energy balance leads to the well-known relation given by

ρ+α+τ= 1 (1.6)

Understanding the extinction of radiation is crucial because most IR heat transfer models count on the amount of local heat flux imparted to the food material in relation to the penetration depth.

Figure 1.3, Extinction of radiation (absorption, transmission, and reflection).

Copyright (2011) From (Infrared Heating For Food and Agriculture Processing) by (Zhongli Pan and Griffiths Gregory Atungulu). Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.