**CHAPTER 2: LITERATURE REVIEW**

**2.3 Development and analysis of a mathematical model**

**2.3.3 Energy balance**

considered both macropore and micropore diffusion by LDF approximations taken in series (model 28). A comparison of the model predictions with the experimental data revealed that the approximated model proposed in this study yields a good representation of the intraparticle mass transfer.

and the energy transferred to the wall of the column, may be written as the following (Rezaei & Grahn, 2012; Ruthven, 1984):

2 2

int

( ) 1 4

( ) ( ) ( ) 0 (2 35)

*g* *g* *g* *b* *w*

*L* *g* *g* *g* *g* *f* *s* *g* *s* *g* *w*

*b* *b*

*T* *uT* *T* *h*

*C* *C* *h a T* *T* *T* *T*

*z* *z* *t* *d*

The above equation is used to find the distribution of gas temperature along the bed,
where T*g** represents the bulk gas temperature; C**g* is the heat capacity of the gas; ρ*g* is the
bulk density of the gas; h*f* denotes the film heat transfer coefficient between the gas and
the adsorbent; a*s *expresses the ratio of the particle external surface area to volume; *T**s*

denotes the solid temperature; *h**w* is the internal convective heat transfer coefficient
between the gas and the column wall; d*int* is the column internal diameter; T*w* denotes the
wall temperature; and λ*L* is the effective axial heat dispersion, which can be estimated
using the following correlation (Grande & Rodrigues, 2005; Wakao & Funazkri, 1978;

Yang, 1987):

= 7 + 0.5 (2 36)

*L*
*g*

*k* *PrRe*

_{}

where k*g** is the thermal conductivity of the gas mixture and Pr is the Prandtl number. *

The Chilton–Colburn analogy can be applied (in analogy with Eq. (2-11)) to estimate the
convective film heat transfer coefficient between the gas and the adsorbent, h*f* (Chilton &

Colburn, 1934). It can be estimated through the following correlation, which is particularly applicable at higher Reynolds numbers (Wakao, Kaguei, & Funazkri, 1979):

1/3 0.6

2 ^{f}* ^{p}* 2 1.1 (2 37)

*g*

*Nu* *h R* *Pr Re*

*k*

The following correlation can be applied for the estimation of the internal convective heat
transfer coefficient, h*w* (Dantas et al., 2009; Dantas et al., 2011; Dantas et al., 2011):

int 12.5 0.048 (2 38)

*w*
*w*

*g*

*Nu* *h d* *Re*

*k*

The following boundary conditions are assumed:

0 0 0

0

( ) (2 39)

*g*

*L* *g* *g* *z* *g* *z* *g* *z*

*z*

*T* *C u* *T* *T*

*z* _{}

0 (2 40)

*g*
*z L*

*T*
*z* _{}

where

*g z* 0

*T* _{}

is feed temperature.

**2.3.3.2 Solid phase energy balance **

For the solid phase, a separate energy balance equation can be assumed that considers the accumulation term, the film heat transfer term, and the heat generated by the adsorption of the adsorbate, as shown by the following (Do, 1998b):

1

( ) ( ) (2 41)

*n*

*s* *i*

*p* *s* *f* *s* *g* *s* *i*

*i*

*T* *q*

*C* *h a T* *T* *H*

*t* *t*

###

where ρ*p** is the particle density, C**s* represents the heat capacity of the adsorbent and

*(-∆H**i*) is the isosteric heat of adsorption for the i component at zero coverage, which can
be calculated by using the Clausius-Clapeyron equation (Yang, Lee, & Chang, 1997).

When the external heat transport limitations are negligible, it is a reasonable approximation to neglect the occurrence of temperature gradients in the particles and consider the gas phase and the surface of the adsorbing particles to be isothermal (Ribeiro, Grande, Lopes, Loureiro, & Rodrigues, 2008). In this case, a single temperature equation, which is obtained from the overall local balance in the bed (combining Eqs. (2-35) and (2-41) into one), is sufficient to describe the energy transport in the bed (Hu & Do, 1995;

Atanas Serbezov & Sotirchos, 1998). However, some experimental studies have demonstrated the occurrence of temperature differences between the gas phase and the

surface of the adsorbent particles (Haul & Stremming, 1984; Lee & Ruthven, 1979;

Ruthven, Lee, & Yucel, 1980). This situation is particularly relevant when an adsorption process occurs at a relatively high rate such that the time needed for the released heat to be transported to the bulk phase may not be sufficient and, consequently, significant temperature gradients are encountered in the interior of the particle (Atanas Serbezov &

Sotirchos, 1998).

**2.3.3.3 Wall energy balance **

Finally, the energy balance for the column wall, which includes the wall heat-transfer to the external environment and to the gas phase inside the column, can be expressed as follows (Da Silva et al., 1999):

( ) ( ) (2 42)

*w*

*w* *w* *w* *w* *g* *w* *a* *w*

*C* *T* *h a T* *T* *Ua T* *T*

^{} *t* _{}

where C*w* and *ρ**w* represent the heat capacity and the density of the column wall,
respectively; *a**w *represents the ratio of the internal surface area to the volume of the
column wall (Da Silva et al., 1999; Dantas et al., 2009; Huang & Fair, 1988); a*a *denotes
the ratio of the external surface area to the volume of the column wall; U is the external
overall heat transfer coefficient from the wall to ambient air; and *T**∞* is the ambient
temperature.

The external overall heat transfer coefficient, *U, can be estimated through following *
correlation (Incropera & Witt, 1996):

int int

int

1 1

ln( * ^{ext}*) (2 43)

*w* *w* *ext ext*

*d* *d* *d*

*U* *h* *k* *d* *d h*

where k*w* is the column wall conductivity; d*ext* is the column external diameter; and h*ext* is
the external convective heat transfer coefficient that can be estimated using the following
correlation (Incropera & Witt, 1996):

1/ 4 9/12 4/9

0.68 0.67 (2 44)

[1 (0.492 / ) ]

*ext*
*ext*

*h L* *Ra*

*k* *Pr*

In the above equation, *k**ext* is the column external air conductivity and
(*T*_{w}*T* ) 3

*Ra* *g* *L*

*v*

^{}

is the Rayleigh number, where g is the gravity acceleration; β is the

thermal expansion coefficient; and ν and α are the air kinematic viscosity and thermal
diffusivity at the film temperature ((T*w**+T**∞*)/2), respectively.

If the area of heat transfer from the fluid to the wall is an order of magnitude larger than the area in the axial direction, the contribution of the axial heat conduction along the column wall can be neglected (Ahn, Lee, Seo, Yang, & Baek, 1999; Ahn et al., 2001;

Mohamadinejad et al., 2000). In the above wall energy balance equation (Eq. 2-42), the
resistance of the metal wall to radial heat transfer has been considered in the overall heat
transfer coefficient, *U, which makes this value lower than that of the individual *
convection heat transfer coefficient between the wall and the surroundings (Grande &

Rodrigues, 2005).

In an industrial-scale process in which the column length-to-diameter ratio is not large,
the heat loss through a wall and heat accumulation in the wall are considered negligible
in comparison to the amount of heat caused by the heat of adsorption, resulting in
operation close to adiabatic behavior (Lee et al., 1999). In such a situation, the overall
heat transfer coefficient, *U, and therefore the last term in Eq. (2-42) can be dropped *
(Bastos-Neto, Moeller, Staudt, Bohm, & Glaser, 2011). In the case of an isothermal
system, an instantaneous thermal equilibrium is assumed to exist between the gas and
solid phases (T*g**=T**s*) or between the gas, solid, and column wall (T*g**=T**s**=T**w*), depending
on the system’s conditions. Under such assumptions, the original three energy balances
of the complete model discussed above (Eqs. 2-35, 2-41, and 2-42) are reduced to two
equations and one equation, respectively.