To simplify the mathematical analysis, the following assumptions were made for the constant thickness film model: the wall is considered isothermal, i. e., the cooling water flow is not considered, the flow is laminar and fully developed for the liquid-desiccant and the air, the desiccant flow is smooth (not wavy), the physical properties are constant, there is thermodynamic equilibrium at the desiccant/air interface, air is an ideal gas, no shear forces are exerted by the air on the desiccant, the body force in the air is negligible, diffusion in the direction of the flow is negligible, species thermo-diffusion and diffusion — thermo effects are negligible, the rate of water vapour absorption is small, i. e., the velocity in the transverse direction is negligible, the solubility of air in the desiccant solution is negligible and the film thickness is constant.

for the liquid-desiccant film,

and for the air,

Fig. 2: schematic diagram of the dehumidifier channel with notation.

d u

Yd’

d

u

dy2 T dx dCA

— + Pdg = 0

■ = ая

d 2T

d

dy2 5 C

"d dx Dd dy2

(1) (2) (3)

The boundary conditions are:

at x=0, Td=Tdi, Ta=T at y=0, T=T C |d=|w, dy at y=Sd, Td=Ta, ud=ua

= 0, ud=0,

a

Ca

dUj_ ‘ dy

dp d2 ua dx = A. y 2 (4) Ua STa = «. SlT; dx a dy2 (5) u C. — d ^ a dx D dy2 (6) Cai, (7) (8) = 0, (9)

Considering the assumptions above and the geometry presented in Fig. 2, the governing momentum, energy and species equations are:

SHAPE * MERGEFORMAT

at y=Sc

(10)

dT dC du a = 0 , ^ = 0 , ^ = 0 . dy dy dy

The velocity profile for the liquid-desiccant is calculated integrating (1) twice across the desiccant film:

ud

2

2

(11)

and considering the continuity across the film and a channel 1 m wide:

md = £’ Pduddy (12)

Solving (12) allows the development of the expressions for film thickness and velocity profile, according to the desiccant mass flow, md:

к

(13)

3mdrd

Pdg j

and

ud

(14)

3md ( 2у _ т! л 2 Pd v^d ^d J

The velocity profile for the air stream can be calculated also through the momentum and continuity equations for the air stream:

(15)

ma = 2‘PaUady

dd

dp dx

3 Pa

ud fo — sc )2

(16)

(17)

m

a

2Pa {Sd ~SC )3 _

2 (y2~5l )+(A — y)

dp 1 dx pa

SHAPE * MERGEFORMAT

The energy and species balances at the interface (y=8d) give the additional matching equations:

_ k k T. П h C

d dy a dy Pa “ ff dy

PdDd

dCj

dy

(19)

PaDa

C dy

SHAPE * MERGEFORMAT

C

* a

(20)

18.0153 pw 28.9645p -10.949pw

The equilibrium water mass fraction in the air at the interface is given by the water vapour pressure of the water in the desiccant solution at the interface, pw(Td, Cd), and Dalton’s law applied for a water vapour/air mixture:

Equations (2),(3),(5),(6),(18) and (19) were approximated using the finite-difference method. Central-differencing was used for the diffusion terms and backward or forward — scheme for the convection terms, depending on whether the air/desiccant flow configuration was co-current or counter-current. The equations were solved simultaneously using the software package Microsoft EXCEL[6]. In the “y” direction, 7 nodes were used for the liquid-desiccant film and 40 for the air (Figure 3). The interface between the last node in the desiccant side and the first node in the air side was represented by an additional node. This interface node has no volume, but provides, through equations (18), (19) and (20), the coupling between the air and liquid streams and between the energy and species equations. The number of nodes in the flow direction varied according to the height of the channel.

representation for one step in flow direction.

INTERFACE -1

WALL

+1

The variable thickness model uses the same equations, assumptions and number of grid nodes of the constant thickness model. The difference is that the thickness of the desiccant film is allowed to vary and is recalculated using equation (13), for every step of the simulation. Once the film thickness is recalculated, it is possible to obtain a new velocity profile for the desiccant using (14), i. e., the profile is still the profile for the fully developed laminar flow, but recalculated to account for the change in

Fig. 4: schematic of grid with thickness change in the desiccant film.

Variable Thickness Model

mass flow and film thickness. This procedure is similar to the one adopted by Jernqvist and Kockum [7] for developing falling film flow. To reduce the computational time, and because the changes in film thickness are small, the change in the air flow velocity profile was neglected.