Boltzmann number, Bo, is a dimensionless parameter used in the problem of heat transfer by radiation and convection formally showing the radiation contribution to the overall heat transfer. It appears when dimensionless values are introduced into the equation of energy transfer in the radiating and absorbing gas. For instance, for a stationary subsonic flow in one-dimensional channel, the energy equation written for simplicity in the “narrow channel” approximation is of the form:
If we introduce the dimensionless values x' = x/D; y' = y/D; u' = u/u0, c' = cp0, λ' = λ/λ0; ρ = ρ/ρ0; μ' = μ/μ0; T' = T/T0; where D is the characteristic dimension (the height of the channel) and the zero subscript denotes characteristic conductions, e.g., at the inlet to the channel, then the equation takes the form (the primes are omitted):
where Pe = RePr is the Peclet number, Bo = ρ0cp0y0/σT30 is the Boltzmann number.
A high value of Boltzmann number shows a weak effect of the radiation on gas temperature.
To evaluate the role heat transfer by radiation plays in the flow of optically “grey” medium (this is a rarely applicable approximation in which the optical properties are taken to be independent of frequency), it is necessary, apart from Boltzmann number, to consider an additional parameter — optical thickness τ = kD (where k is the volume coefficient of absorption of the medium in the channel), which is the ratio of the characteristic dimension D to the mean path length of radiation k−1. In the case of optically-thin gas layer (τ << 1), the density of radiation heat flux from the medium to the channel walls can be estimated to be 2τ0σT40, and in the case of optically-thick layer (τ >> 1) to be 16σT30/3τ0. The density of convective heat flux is proportional to ρ0, u0, cp0, T0. Then the effect of heat transfer by radiation is characterized by the parameters:
Therefore, even for low Bo the impact of heat transfer by radiation may not be strong if for τ << 1, τ0/Bo << 1 and for τ0 >> 1, (τ0Bo)−1 << 1.
In the general case of selective (nongrey) gas when k = k(λ), where λ is the radiation wave length, the degree of radiation effect on heat transfer generally cannot be described by only two parameters, Bo and τ0. It also depends on optical thickness τ0λ of the medium in energy-carrying wavelength bands, the values of the latter and their positioning with respect to the maximum point of the intensity function of the black body radiation.
Sometimes, in problems of heat transfer by radiation and convection or by radiation and conduction, a dimensionless parameter
is used, known as a conduction-radiation parameter or the Stark number.
Özisik, M. N. (1973) Radiative Transfer and Interactions with Conduction and Convection, J. Wiley & Sons, New York, London, Sydney, Toronto.
Siegel, R. and Howell, J. R. (1972) Thermal Radiation Heat Transfer, McGraw Hill Book Comp., New York et al.