**2. Mathematical model**

The mathematical modelling of desiccant wheels is of key importance for equipment developers, so as to provide them with guidelines for improved design. It is also of importance to HVAC engineers, in order to access if the thermal comfort condition can be attained for a typical set of atmospheric conditions. The mathematical model relies on a number of simplifying assumptions, aiming at keeping the model (and its solution) as simple as possible, while retaining the physical meaning. An excellent review of the

Mathematical Modelling of Air Drying by Adiabatic Adsorption 73

1 1

<sup>1</sup> <sup>0</sup> *<sup>m</sup> H H m H w w uT x L t* 

Consider Figure (5.b), which represents a differential control volume which solely encloses the airstream. The mass conservation principle applied to the depicted control volume

> <sup>1</sup> 2 2 *w h <sup>w</sup> H H <sup>H</sup> m h <sup>Y</sup> Y h <sup>T</sup> <sup>T</sup>*

In which the first term on the right hand side stands for the heat transfer between the sorbent and the air, whereas the second term represents the heat released during the

*uT x Y*

\*

1 1 <sup>1</sup>

1 1

2 *h h hdx <sup>x</sup> <sup>H</sup> m T*

*w wr h d xt <sup>t</sup>*

 \* *<sup>w</sup> <sup>Y</sup> Y Y*

*x*

 

\* 2 *h h*

(4)

(3)

1

(5)

*m C* (6)

(7)

1

Fig. 4. Differential control volumes for mass balances

1

adsorption. Defining the following non-dimensional parameters,

After extensive algebra, Equations (1)-(4) can be rewritten as

yields:

mathematical modelling of adsorptive dehumidification can be found in the literature (Ge et al., 2008).


Fig. 3. Schematic of the flow channel with desiccant coating

Assumption (1) relies on symmetry between the cells, which can be represented by adiabatic surfaces. Assumption (6) is adopted in light of the small thickness of the desiccant layer (Shen & Worek, 1992), (Sphaier & Worek, 2006). Consider Figure (4.a), which represents a differential control volume which simultaneously encloses the desiccant layer and the flow channel. The mass conservation principle applied to the depicted control volume yields:

$$
\dot{m} \left[ \frac{1}{u\_1} \frac{\partial Y}{\partial T} + \frac{\partial Y}{\partial \mathbf{x}} \right] + f \frac{m\_w}{L} \frac{\partial \mathcal{W}}{\partial t} = 0 \tag{1}
$$

Consider Figure (4.b), which represents a differential control volume which solely encloses the desiccant layer. The mass conservation principle applied to the depicted control volume yields

$$f\frac{m\_w}{d\_h L}\frac{\partial W}{\partial t} = 2h(Y - Y\_w) \tag{2}$$

Figure (5.a) represents a differential control volume which simultaneously encloses the desiccant layer and the air stream. The energy conservation principle applied to the depicted control volume yields

mathematical modelling of adsorptive dehumidification can be found in the literature (Ge et

5. The heat and mass transfer coefficients are assumed to be uniform along the micro-

6. Temperature and concentration distributions in the direction normal to the flow are

Assumption (1) relies on symmetry between the cells, which can be represented by adiabatic surfaces. Assumption (6) is adopted in light of the small thickness of the desiccant layer (Shen & Worek, 1992), (Sphaier & Worek, 2006). Consider Figure (4.a), which represents a differential control volume which simultaneously encloses the desiccant layer and the flow channel. The mass conservation principle applied to the

> <sup>1</sup> <sup>0</sup> *m f YY W mw uT x L t*

Consider Figure (4.b), which represents a differential control volume which solely encloses the desiccant layer. The mass conservation principle applied to the depicted control volume

2 *<sup>w</sup> <sup>w</sup>*

*<sup>m</sup> <sup>W</sup> <sup>f</sup> hY Y*

Figure (5.a) represents a differential control volume which simultaneously encloses the desiccant layer and the air stream. The energy conservation principle applied to the depicted

(1)

(2)

3. All thermo-physical properties for the fluid and the solid are considered constant.

al., 2008).

channel

1. The micro-channels are perfectly insulated.

2. Heat and humidity transients within the air are negligible.

4. The flow is hydro-dynamically and thermally developed.

Fig. 3. Schematic of the flow channel with desiccant coating

1

*h*

*dL t*

depicted control volume yields:

yields

control volume yields

taken to be uniform (lumped) within the channel and the solid. 7. The adsorption heat is modeled as a heat source within the solid material

Fig. 4. Differential control volumes for mass balances

$$
\dot{m} \left[ \frac{1}{\mu\_1} \frac{\partial H\_1}{\partial T} + \frac{\partial H\_1}{\partial \mathbf{x}} \right] + \frac{m\_w}{L} \frac{\partial H\_w}{\partial t} = \mathbf{0} \tag{3}
$$

Consider Figure (5.b), which represents a differential control volume which solely encloses the airstream. The mass conservation principle applied to the depicted control volume yields:

$$\dot{m}\left[\frac{1}{\mu\_1}\frac{\partial H\_1}{\partial T} + \frac{\partial H\_1}{\partial \mathbf{x}}\right] = 2h(Y\_w - Y)\frac{\partial H\_1}{\partial Y} + 2h\_h(T\_w - T\_1) \tag{4}$$

In which the first term on the right hand side stands for the heat transfer between the sorbent and the air, whereas the second term represents the heat released during the adsorption. Defining the following non-dimensional parameters,

$$\mathbf{x}^\* = \frac{2h\_h d\_h \mathbf{x}}{\dot{m}\frac{\partial H\_1}{\partial T\_1}}\tag{5}$$

$$\mathbf{t}^\* = \frac{2\mathbf{h}\_h d\_h \mathbf{x} t}{m\_w \mathbf{C}\_{wr}} \tag{6}$$

After extensive algebra, Equations (1)-(4) can be rewritten as

$$\frac{\partial \mathcal{Y}}{\partial \mathbf{x}^\*} = \left(Y\_w - Y\right) \tag{7}$$

Mathematical Modelling of Air Drying by Adiabatic Adsorption 75

12400 3500 , 0.05 1400 2950 , 0.05

It shows that the heat release is not constant during the adsorption process, exhibiting a small reduction as the adsorption develops. This could be explained by observing that the first adsorbed molecules are attracted to the most energetically unbalanced sites. As the moisture uptake continues, the remaining spots to be occupied require less bonding energies, approaching ordinary latent heat as the solid becomes saturated. From the mathematical point of view, the problem is still undetermined, since there are five unknowns (T1, Tw, Y, Yw and W) and only four equations, (7) to (10). The missing equation is the adsorption isotherm, which is characteristic of each adsorptive material. For regular

> 3 4 0.0078 0.0579 24.16554

Equations (15) and (16) are auxiliary equations, which relates the partial pressure of the air

3816.44 exp 23.196 46.13 *ws*

0.62188 0.62188 *<sup>w</sup> <sup>w</sup> <sup>w</sup> atm w atm <sup>w</sup> ws*

The periodic nature of the problem implies an iterative procedure. Both initial distributions of temperature and humidity within the solid are guessed, and equations (7) to (10), assume the form of tridiagonal matrices, as a result of the discretization using the finite-volume technique, with a fully implicit scheme to represent the transient terms (Patankar, 1980).. By the end of the cycle, both calculated temperature and moisture fields are compared to the initially guessed. If there is a difference in any nodal point bigger than the convergence

> ( ,0) ( )( ,0) . . ( ,0) *w w temp*

( ,0) ( )( ,0) . . ( ,0) *mass Wx W guess x Crit Conv*

the procedure is repeated, using the calculated fields as new guesses for the initial distributions. Figure (6) shows a simplified fluxogram for the numerical solution. Figures (7) and (8) show mass and temperature distributions along the desiccant felt, at selected angular positions. The curves relative to 0 and 2π are indistinguishable, as the periodic behaviour was attained. The average "hot outlet" enthalpy during a cycle is defined as

> 0 1 *<sup>h</sup> <sup>p</sup> ho ho h H H dt*

*T x T guess x Crit Conv*

*w*

*T x*

*W x*

\*

(17)

(18)

*<sup>P</sup>* (19)

*p p p*

*w*

*p*

*T* 

*<sup>w</sup> W W W W*

kJ/kg (13)

(14)

(15)

(16)

2

*Q WW QW W* 

density silica-gel, the following expression was experimentally obtained,

*P*

criteria established for temperature and moisture content,

layer with the absolute humidity,

124.78 204.2264

*<sup>p</sup> <sup>Y</sup>*

Fig. 5. Differential control volumes for energy balances

$$\frac{\partial \mathcal{W}}{\partial t} = \lambda\_2 \left( Y - Y\_w \right) \tag{8}$$

$$\frac{\partial T\_1}{\partial \mathbf{x}^\*} = \left(T\_w - T\right) \tag{9}$$

$$\frac{\partial T\_w}{\partial t} = \left(T - T\_w\right) + \lambda\_1 \left(Y - Y\_w\right) \tag{10}$$

With

$$
\lambda\_2 = \frac{C\_{wr}}{f \frac{\partial H\_1}{\partial T\_1}} \tag{11}
$$

$$
\lambda\_2 = \frac{\overline{Q}}{\frac{\overline{\mathcal{O}}H\_1}{\overline{\mathcal{O}}T\_1}} \tag{12}
$$

Equation (12) represents the heat of adsoprtion, released as the vapor molecule is adsorbed within the silica-gel. The adsorption heat is comprised of the condensation heat plus the wettability heat, which accounts for reducing the degrees of movement freedom of a gas molecule from three to two, as it is captured by a surface. The current modeling allows different approaches to to the adsoprtion heat, as both analytical and experimentally obtained expressions for Q could be easily fitted to Eq. (12). For regular density siilica-gel, the following expression was experimentally obtained (Peasaran & Mills, 1987),

<sup>2</sup> \* *<sup>w</sup> <sup>W</sup> Y Y*

 <sup>1</sup> \* *w <sup>T</sup> T T*

\* <sup>1</sup> *<sup>w</sup> w w <sup>T</sup> TT YY*

> *Cwr H f T*

*Q H T*

Equation (12) represents the heat of adsoprtion, released as the vapor molecule is adsorbed within the silica-gel. The adsorption heat is comprised of the condensation heat plus the wettability heat, which accounts for reducing the degrees of movement freedom of a gas molecule from three to two, as it is captured by a surface. The current modeling allows different approaches to to the adsoprtion heat, as both analytical and experimentally obtained expressions for Q could be easily fitted to Eq. (12). For regular density siilica-gel,

1 1

1 1

2

2

the following expression was experimentally obtained (Peasaran & Mills, 1987),

(8)

(9)

(11)

(12)

(10)

*t*

*x*

*t*

Fig. 5. Differential control volumes for energy balances

With

$$\begin{bmatrix} Q = -12400 \,\text{W} + 3500 \,\text{J} \,\text{ } \text{ } \text{ } 0.05\\ Q = -1400 \,\text{W} + 2950 \,\text{ } \text{ } \text{ } \text{ } \text{ } 0.05 \end{bmatrix} \text{kJ/kg} \tag{13}$$

It shows that the heat release is not constant during the adsorption process, exhibiting a small reduction as the adsorption develops. This could be explained by observing that the first adsorbed molecules are attracted to the most energetically unbalanced sites. As the moisture uptake continues, the remaining spots to be occupied require less bonding energies, approaching ordinary latent heat as the solid becomes saturated. From the mathematical point of view, the problem is still undetermined, since there are five unknowns (T1, Tw, Y, Yw and W) and only four equations, (7) to (10). The missing equation is the adsorption isotherm, which is characteristic of each adsorptive material. For regular density silica-gel, the following expression was experimentally obtained,

$$\begin{aligned} \phi\_w &= 0.0078 - 0.0579W + 24.16554W^2 \\ -124.78W^3 &+ 204.2264W^4 \end{aligned} \tag{14}$$

Equations (15) and (16) are auxiliary equations, which relates the partial pressure of the air layer with the absolute humidity,

$$P\_{ws} = \exp\left(23.196 - \frac{3816.44}{T\_w - 46.13}\right) \tag{15}$$

$$Y\_w = \frac{0.62188 p\_w}{p\_{atm} - p\_w} = \frac{0.62188 \phi\_w}{\frac{p\_{atm}}{p\_{ws}} - \phi\_w} \tag{16}$$

The periodic nature of the problem implies an iterative procedure. Both initial distributions of temperature and humidity within the solid are guessed, and equations (7) to (10), assume the form of tridiagonal matrices, as a result of the discretization using the finite-volume technique, with a fully implicit scheme to represent the transient terms (Patankar, 1980).. By the end of the cycle, both calculated temperature and moisture fields are compared to the initially guessed. If there is a difference in any nodal point bigger than the convergence criteria established for temperature and moisture content,

$$\text{Crit.}\ \text{Conv}\_{\text{temp}} = \frac{T\_w(\text{x}, 0) - T\_w(\text{guess})(\text{x}, 0)}{T\_w(\text{x}, 0)}\tag{17}$$

$$\text{Crit.}\ \text{Conv}\_{\text{mass}} = \frac{\mathcal{W}(\mathbf{x}, \mathbf{0}) - \mathcal{W}(\text{gauss})(\mathbf{x}, \mathbf{0})}{\mathcal{W}(\mathbf{x}, \mathbf{0})} \tag{18}$$

the procedure is repeated, using the calculated fields as new guesses for the initial distributions. Figure (6) shows a simplified fluxogram for the numerical solution. Figures (7) and (8) show mass and temperature distributions along the desiccant felt, at selected angular positions. The curves relative to 0 and 2π are indistinguishable, as the periodic behaviour was attained. The average "hot outlet" enthalpy during a cycle is defined as

$$\overline{H\_{ho}} = \frac{1}{P\_{\hbar}} \int\_{0}^{p\_{\hbar}} H\_{ho} dt^\* \tag{19}$$

Mathematical Modelling of Air Drying by Adiabatic Adsorption 77

0 0.2 0.4 0.6 0.8 1 non-dimensional position, (x\*)

0 0.2 0.4 0.6 0.8 1 non-dimensional position, x\*

Fig. 8. Temperature distributions at selected angular positions P\*40.0, NTU=16.0, Treg=100°C.

Fig. 7. Mass distributions at selected angular positions, P\*40.0, NTU=16.0, Treg=100°C.

3

0, 2

0, 2

0

100

20

40

60

T

C

W (x\*), 80

0.1

Solid Humidity 0.2

 Content,

 W(x\*)

0.3

0.4

Since the wheel is to store neither energy nor mass after a complete cycle,

$$
\sum \stackrel{\bullet}{m\_i} H\_i = \sum \stackrel{\bullet}{m\_o} \overline{H}\_o \tag{20}
$$

$$
\dot{m}\_h \, H\_{h\bar{l}} + \dot{m}\_c \, H\_{c\bar{l}} = \dot{m}\_h \frac{1}{p\_h} \int\_0^{p\_h} H\_{h\alpha} \, d\mathfrak{f}^\* + \dot{m}\_c \frac{1}{p\_c} \int\_0^{p\_c} H\_{c\alpha} \, d\mathfrak{f}^\* \tag{21}
$$

the normalized difference between the two sides of equation (21) is defined as the Heat Balance Error (HBE), which was found to be of the order of 0.1% for all simulations carried.

$$HBE = \frac{\dot{m}\_h \, H\_{hi} + \dot{m}\_c \, H\_{ci} - \left(\dot{m}\_h \frac{1}{p\_h} \int\_0^{p\_h} H\_{ho} \, dt\right.^{\ast} + \dot{m}\_c \frac{1}{p\_c} \int\_0^{p\_c} H\_{co} \, dt\,^{\ast}\right)}{\dot{m}\_h \, H\_{hi} + \dot{m}\_c \, H\_{ci}} \tag{22}$$

Fig. 6. Fluxogram of the numerical solution

*mH mH ii oo* 

*h hi c ci h ho c co*

*m H m H m H dt m H dt*

the normalized difference between the two sides of equation (21) is defined as the Heat Balance Error (HBE), which was found to be of the order of 0.1% for all simulations carried.

*h hi c ci h ho c co*

*p p HBE*

Fig. 6. Fluxogram of the numerical solution

*m H m H m H dt m H dt*

*mH mH*

*h hi c ci*

0 0 1 1 *h c p p*

(21)

0 0 1 1 ( ) *h c p p*

*h c*

*h c*

*p p*

(20)

\* \*

(22)

\* \*

Since the wheel is to store neither energy nor mass after a complete cycle,

Fig. 7. Mass distributions at selected angular positions, P\*40.0, NTU=16.0, Treg=100°C.

Fig. 8. Temperature distributions at selected angular positions P\*40.0, NTU=16.0, Treg=100°C.

Mathematical Modelling of Air Drying by Adiabatic Adsorption 79

Treg = 120C

0 2 4 6 810 Non-dimensional position, x\*

P\* = 80.0

40 60 80 100 120 Regeneration Temperature, Thi (

C)

P\* = 40.0

P\* = 10.0

Treg = 80C

Treg = 60C

0

Fig. 10. Effectiveness-NTU chart, P\*=80.0

0.2

Fig. 11. Influence of P\*, NTU=10.0, Thi = 100°C

0.3

0.4

0.5

dw

0.6

0.7

0.8

0.2

0.4

dw

0.6

Treg = 100C

0.8
