C1hapter 5.3.2.1 - Heat Transfer Model

Although the principle of energy transfer from a radiant tube heater to the surroundings
is the same for all different designs, mathematical modeling of the heater
varies depending on the geometric design and operating characteristics and assumptions.
To provide basic concepts of the modeling scheme, this section illustrates onedimensional
modeling for the straight-through radiant tube heater.

The three basic principles of modeling a radiant heater and most thermal systems
are conservation of (1) mass, (2) momentum, and (3) energy. Provided that there is a
steady-state condition for an axially symmetric flow in the tube, these equations are
(Chapman et al., 1988)

The subscript i stands for inside the tube; Δh_fuel represents the heat of formation of
the fuel; P is the perimeter; and h represents the convective heat transfer coefficient.
The shear stress, τ, due to friction in the momentum equation, could be expressed as
(Chapman et al., 1988)

The parameter f is the friction factor of the tube (refer to Sec. 2, Chap. 6 in this
Handbook for more information on the friction factor). The fuel-air mixture is
assumed to behave as an ideal gas, and the coexistence of air, fuel, and combustion
product is also assumed at any axial location in the tube. To incorporate different
compositions of each constituent at an arbitrary location, the notion of a fuel burnup
ratio is employed.This concept was introduced by Lisenko et al. (1986).The functional
expression for the fuel burn-up ratio is the following:

The parameter D_i is the inner diameter of the tube.This correlation was developed
through an extensive study by Lisenko et al. (1986) on combustion phenomena in
the straight-through-type radiant heater. Using this parameter, the mass flow rates
of each constituent could be expressed as (Chapman et al., 1988)

The parameter Φ is the equivalence ratio.The convective heat transfer coefficient in
Eq. (3.3) and the friction factor in Eq. (3.4) should be evaluated depending on the
flow condition that is either laminar or turbulent as explained in Sec. 2, Chap. 6 of
this Handbook.

With the assumption of absorbing-emitting medium, the radiative heat transfer
between the flowing gases inside the tube and the inner surface of the tube appearing
in Eq. (3.3) can be expressed as (Chapman et al., 1988)

Neglecting axial conduction, the energy balance on the tube wall gives the following
(Chapman et al., 1988):

The convective heat transfer coefficient, h_o, was obtained assuming free convection
at the outer tube surface.

At this point, it is finally possible to calculate the radiant tube surface temperature
for a given flow rate through the tube. One has to simultaneously solve Eqs.
(3.1) through (3.3), along with Eqs. (3.9) and (3.10). The gas emissivity and absorptivity
in Eq. (3.9) can be obtained with the aid of the temperature and the pressurepath
length product that has been studied by Hottel (1985). Calculating the tube
surface temperature almost always requires the use of a computer to solve the complex
equations. Solving these equations would be mandatory in order to design a
new radiant tube heater.The tube manufacturer should be able to provide the external
temperature that can then be used in thermal comfort calculations if one is interested
in sizing a particular type of radiant tube. For the interested reader, Chapman
et al. (1988) obtained the local tube wall temperature by solving Eq. (3.10) with the
underrelaxed Newton-Raphson iteration during each predictor-corrector step.

Although the principle of energy transfer from a radiant tube heater to the surroundings
is the same for all different designs, mathematical modeling of the heater
varies depending on the geometric design and operating characteristics and assumptions.
To provide basic concepts of the modeling scheme, this section illustrates onedimensional
modeling for the straight-through radiant tube heater.

The three basic principles of modeling a radiant heater and most thermal systems
are conservation of (1) mass, (2) momentum, and (3) energy. Provided that there is a
steady-state condition for an axially symmetric flow in the tube, these equations are
(Chapman et al., 1988)

The subscript i stands for inside the tube; Δh_fuel represents the heat of formation of
the fuel; P is the perimeter; and h represents the convective heat transfer coefficient.
The shear stress, τ, due to friction in the momentum equation, could be expressed as
(Chapman et al., 1988)

The parameter f is the friction factor of the tube (refer to Sec. 2, Chap. 6 in this
Handbook for more information...