THERMOCHEMICAL CONVERSION: DIRECT COMBUSTION - Biomass as a Sustainable Energy Source for the Future - page 279

the different simplified char combustion models of Figure 9.2 can be used. The shrink-
ing density model, discussed earlier, assumes that the reactions take place uniformly
inside the char particle, which is true for low values of the Thiele modulus (<1) as can
be seen from Figure 9.4. An alternative simplified model for char combustion is the
shrinking sphere model, or when an ash layer is built up outside of the reacting core,
the char burns like a shrinking core and the shrinking core model can be used. The
assumption for the shrinking sphere and the shrinking core model is that the reactions
take place in a narrow area close to what is defined as the reaction surface. Figure 9.4
shows that this assumption is valid for high values of the Thiele modulus (>10). More
about the use of the dimensionless form in combustion and in fluid dynamics is given
by Thunman and Leckner (2007).

9.3.2 Shrinking Sphere Model

The theory of surface combustion of isolated fuel droplets goes back to 1953 with a
publication by Spalding (1953). This theory is used as a basis for the present analysis,
supplemented by information as found on the website of Professor Dryer from Prince-
ton University (tinyurl.com/o7cacsn). In this section, the general theory of surface
conversion is explained by first looking at simple evaporation of a droplet. At the
end of the chapter, in Problem 9.5, the extension to combustion of fuel droplets is
treated.
Because of the simplicity of analyzing evaporation processes, we derive the model
for evaporation in order to understand the approach. In this derivation, the so-called
Spalding number pops up, which can be arranged in such a way that other processes
like surface combustion and combustion with a standoff distance can be treated as
well. A good reference is the topic of Turns (2000). Models of solid-sphere combus-
tion under the three simplified conditions of shrinking sphere, shrinking core, and
shrinking density behavior are widely available in the literature.
We consider an evaporating, spherical droplet that is heated up to its boiling tem-
perature and assume it to be present in a quiescent surrounding with an even higher
temperature sufficiently far away from it. In this case, the droplet acts as a heat sink,
associated simultaneously with the release of vapor to the surroundings. By this mech-
anism, the diameter of the droplet decreases at a certain rate. Hereby, the lifetime
(or burnout time) of the droplet, t d , is defined as the time at which its diameter reaches
zero, starting from an initial diameter of d d (t = 0) = d d,0 . It is found that the lifetime of
such a droplet is proportional to a rate constant, 1/K (K is called the evaporation con-
stant), times the initial diameter squared, t d =d d , 0 = K. The constant is determined by

the ratio of available heat transported to the droplet and the heat of vaporization of the
substance in the droplet, which is referred to as the Spalding number or transfer num-
ber. Therefore, the governing law is called the d 2 -law. This law can be used for a set of
shrinking processes like evaporation, surface gasification, surface combustion, and
sheath combustion. It is important to stress that the principal change in the description
of these processes is the definition of the Spalding number and therefore the rate
constant K.