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Influence of the atmospheric pressure on the mass transfer rate of desiccant wheels
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
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Influence of the atmospheric pressure on the mass transfer rate of desiccant wheels C.R. Ruivo a,b,*, J.J. Costa b, A.R. Figueiredo b a b
Dep. Eng. Mec., Instituto Superior de Engenharia, University of Algarve, Campus da Penha, 8005-139 Faro, Portugal ADAI, Department of Mechanical Engineering, University of Coimbra, Rua Luı´s Reis Santos, 3030-788 Coimbra, Portugal
article info
abstract
Article history:
The performance of a desiccant wheel is evaluated by modelling a representative channel.
Received 17 December 2009
The hypothesis of negligible resistances to heat and mass transfer in the cross-direction is
Received in revised form
assumed in the thin porous desiccant wall of the channels and the airflow is treated as
16 October 2010
a bulk flow. Parametric studies were conducted to investigate the influence of the atmo-
Accepted 26 November 2010
spheric pressure decrease from 101,325 Pa to 60,000 Pa (0e4217 m of altitude) on the mass
Available online 3 December 2010
transfer rate of desiccant wheels considering distinct channel lengths and different inlet airflow rates, a large range of values of the rotation speed, as well as three alternative ways
Keywords:
to specify the inlet conditions of the regeneration and of the process airflows. A procedure
Adsorption
to derive correlations based on the numerical results is presented for the correction factor
Desiccant wheel
of the mass transfer rate when a desiccant wheel is operating at non-standard atmospheric
Modelling
conditions. Four parametric studies were performed, the derived correlations were tested
Steady state
and a good agreement was found between the estimated correction factor and the
Pressure
correction factor calculated after the numerical results. ª 2010 Elsevier Ltd and IIR. All rights reserved.
Influence de la pression atmosphe´rique sur la vitesse de transfert de masse des roues de´shydratantes Mots cle´s : Adsorption ; Roue deshydratante ; Modelisation ; Regime permanent ; Pression
1.
Introduction
Usually the available performance data of dehumidification and cooling systems integrating desiccant wheels are based on atmospheric conditions at sea level. The use of specific manufacturer software permits to select or to rate the equipment, but an arduous task still remains: to perform the
energy dynamic simulation of the complete system and building or of the process during a year or a season. The existing standards and guidelines do not address the methods that should be used to predict the performance of a desiccant wheel at high altitudes. Manufacturers of desiccant wheels have devised methods for restating performance, but there is no consensus on the better
* Corresponding author. Dep. Eng. Mec., Instituto Superior de Engenharia, University of Algarve, Campus da Penha, 8005-139 Faro, Portugal. Tel.: þ351 289 800100; fax: þ351 289 888405. E-mail address: [email protected] (C.R. Ruivo). 0140-7007/$ e see front matter ª 2010 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2010.11.013
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
Nomenclature a as b cp;f
cp;f cpwpm Df DK,eff Ds,eff dhyd Fm Fm,in Hc Hp hads hfg0 hh h[ hlig hm hv J_m 0 J_m
J_v;gs jh,gs jh,x j[;x jv,gs jv,sx kp kp kp,cyc kp,cycref Lc Le Nu p pvs
coefficient of the linear regression specific transfer area of the hygroscopic matrix (m2 m3) term of the linear regression specific heat of the moist air referred to the unit of mass of mixture (J kg1 K1) mean specific heat of the moist air referred to the unit of mass of mixture (J kg1 K1) mean specific heat of the wet porous medium (J kg1 K1) diffusion coefficient of water vapour in the airflow (m2 s1) effective coefficient of Knudsen diffusion (m2 s1) effective coefficient of surface diffusion of adsorbed water (m2 s1) hydraulic diameter (m) airflow mass velocity (kg s1 m2) airflow mass velocity at the inlet of the channel (kg s1 m2) channel half-height (cf. Fig. 1) (m) half-thickness of the desiccant wall (cf. Fig. 1) (m) heat of adsorption (J kg1) latent heat of vaporization (J kg1) convection heat transfer coefficient (W m2 K1) enthalpy of adsorbed water (J kg1) heat of wetting (J kg1) convection mass transfer coefficient (m s1) enthalpy of vapour in the desiccant wall (J kg1) global mass transfer rate per unit of transfer area of matrix (kg s1 m2) global mass transfer rate per unit of transfer area of matrix at sea-level atmospheric pressure (kg s1 m2) gas side mass transfer rate at the interface per unit of transfer area (kg s1 m2) gas side convective heat flux at the interface (W m2) axial heat conduction flux in the desiccant (W m2) axial surface diffusion flux in the desiccant (kg s1 m2) gas side convective mass flux at the interface (kg s1 m2) axial Knudsen diffusion flux in the desiccant (kg s1 m2) correction factor for the effect of atmospheric 0 pressure on the mass transfer rate (J_m =J_m ) correction factor kp predicted by a correlation expressed by Eq. (15) correction factor kp predicted by the Eq. (19) reference correction factor kp used in Eq. (19) channel length (cf. Fig. 1) (m) Lewis number Nusselt number pressure of the airevapour mixture (Pa) saturation pressure (Pa)
Rg Rg$v Rv R2 RMSD Sh Se,d Sv[;d Sf Td Tf Tin t uf uin uf wv wv,in
X[ x y
gas constant of the air (J kg1 K1) gas constant of the airevapour mixture (J kg1 K1) gas constant of the water vapour (J kg1 K1) coefficient of the correlation root mean square deviation Sherwood number source-term in the energy conservation equation source-term in the water mass conservation equation generic source-term in Eq. (1) temperature of the desiccant ( C) temperature of the airflow ( C) bulk airflow temperature at the inlet of the channel ( C) time (s) bulk airflow velocity (m s1) bulk airflow velocity at the channel inlet (m s1) velocity (m s1) water vapour content (kg kg1, i.e., kg of water vapour per kg of dry air) bulk airflow water vapour content at the channel inlet (kg kg1, i.e., kg of water vapour per kg of dry air) adsorbed water content in the desiccant (kg kg1, i.e., kg of adsorbed water per kg of dry desiccant) spatial coordinate (m) spatial coordinate (m)
Greek symbols volume fraction of the gaseous mixture within the 3g$v porous medium porosity of the hygroscopic matrix 3m f generic variable generic diffusion coefficient in Eq. (1) Gf vapour mass fraction in the desiccant (kg kg1, i.e., 4v;d kg of water vapour per kg of moist air) vapour mass fraction in the airflow (kg kg1, i.e., 4v;f kg of water vapour per kg of moist air) bulk airflow vapour mass fraction at the inlet of 4v;in the channel (kg kg1, i.e., kg of water vapour per kg of moist air) thermal conductivity of the airflow (W m1 K1) lf thermal conductivity of wet porous medium lwpm (W m1 K1) density of the airflow (kg m3) rf density of the gas mixture (kg m3) r[ apparent density of the gas mixture inside the rg$v porous medium (kg m3) apparent density of the dry desiccant (kg m3) rsd generic term in Eq. (1) rf s time coordinate in an adsorption/desorption cycle (s) duration time of the adsorption mode (s) sads duration time of adsorption/desorption cycle (s) scyc duration time of the desorption mode (s) sdes j relative pressure, i.e., ratio of the partial to the saturation vapour pressure
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Fig. 1 e Schematic representation of the half-channel domain.
method to be adopted. In fact, most of the existing laboratories are not prepared to vary the ambient pressure, and validation of the proposed methods still remains incomplete. The modelling of the performance of desiccant wheels is of great relevance for product optimization by the manufacturer, as well as to perform parametric studies and yearly simulations of the performance of global systems integrating desiccant wheels, towards a more rational energy use. Different methods of numerical solution have been used by many researchers with different simplified treatments of the fluid and solid domains to predict the behaviour of air dehumidifying systems. The solution of the coupled heat and mass transfer was formerly presented by Maclaine-Cross and Banks (1972) by using the analogy with the heat transfer process. Later on Banks (1985a,b) presented a more accurate solution based on a nonlinear analogy method. Most physical models used in numerical studies of desiccant wheels have been analysed through 1-D mathematical formulation based on the hypotheses of negligible cross-direction resistances to heat and mass transfer inside the desiccant wall (Dai et al., 2001; Cejudo et al., 2002; Zhang et al., 2003; Golubovic and Worek, 2004; Harshe et al., 2005; Gao et al., 2005; Elsayed et al., 2006; Chung and Lee, 2009; Chung et al., 2009). The validity of such one-dimensional models is acceptable to relatively thin
desiccant walls of the hygroscopic matrix and was investigated by some authors (Ruivo, 2005; Sphaier and Worek, 2006; Ruivo et al., 2006, 2008a). Other studies were performed using 2-D mathematical formulations taking into account the resistance to the diffusion of mass inside the desiccant porous wall (Ruivo, 2005; Niu and Zhang, 2002; Zhang and Niu, 2002; Sphaier and Worek, 2004; Ruivo et al., 2007a,b). Several parametric studies were performed in the above referenced works, but only the study of Golubovic and Worek (2004) is focussed on the effect of pressure on sorption in desiccant wheels employed to dry pressurized airflows. They emphasize the possibility and the implications of the occurrence of condensation in the regeneration portion of the wheel, affecting the global performance. Only one study was found in the literature specifically addressed to the effect of altitude on the performance of a desiccant cooling system (Pesaran and Heiden, 1994), namely the behaviour of the different system components for decreasing atmospheric pressure, from 101,325 to 81,060 Pa. The objective of the present work is to analyze the influence of decreasing atmospheric pressure on the performance of desiccant wheels, a subject with practical relevance when these equipments operate at significantly high altitudes, as recognized by the ASHRAE Technical Committee TC 8.12 (call for proposals for the research project 1339-TRP, “Selection of Desiccant Equipment at Altitude”, 2009).
Table 1 e Meanings of f, rf , uf , Gf and Sf in the conservation equation Eq (1). Gf
f
rf
uf
Desiccant domain Mass of adsorbed water Energy
X[ Td
rsd rsd
0 0
rsd Ds;eff lwpm cpwpm
Airflow domain Mass of gaseous mixture
1
rf
uf
0
Mass of water vapour
4v,f
rf
uf
0
Energy
Tf
rf
uf
0
Momentum
uf
rf
uf
0
Conservation equation
Sf Sv[;d Se,d
jv;gs Hc
jv;gs Hc jv;gs ðhv hfg0 Þ jh;gs ðTf Td Þ Hc cp f Hc cp f 0
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In the present work, a numerical model is used accounting for the heat and mass transfer phenomena occurring simultaneously with the sorption processes in the hygroscopic matrix of a desiccant wheel. The study is based on the simulation of a representative channel, using a simplified model to predict its transient behaviour, starting from an imposed set of initial conditions until the stationary regime of adsorption/ desorption cycles is achieved. This version of the model was derived from a detailed numerical model (Ruivo, 2005; Ruivo et al., 2007a,b) and it is valid for the cases with thin desiccant walls, where the resistances to heat and mass transfer in the cross-direction inside the porous medium are negligible (Ruivo, 2005; Ruivo et al., 2006, 2008a). Four parametric studies were performed to derive the correlations for the correction factor of the mass transfer rate when a desiccant wheel is operating at non-standard atmospheric conditions. The first set of parametric studies (PS1, PS2 and PS3) concerns to three alternative forms of specifying the inlet conditions of both airflows. In a fourth parametric study (PS4), a correlation for the correction factor taking into account the influence of the rotation speed of the wheel was also derived. The numerical model was not yet validated. A serious comparison of numerical results with experimental data would require that all the properties of the real desiccant layer and of the desiccant matrix were well known, but there is a lack of information for the complete characterization of the desiccant media. In some published experimental works, some property values or correlations are missing. Moreover, examples of exhaustive experimental research on the behaviour of a desiccant wheel (Cejudo et al., 2002) show significant mass and energy imbalances between the regeneration and the process airflows.
2.
Mathematical and numerical modelling
2.1.
Physical model and simplifications
Only a representative channel of the hygroscopic matrix of the desiccant wheel is modelled, the adsorption/desorption cycle being imposed by suddenly changing the inlet conditions and the direction of the airflow. The problem consists of a parallelplate channel which is cyclically submitted to alternating airflow inlet conditions and stream direction, producing a sequence of adsorption and desorption processes in the desiccant walls of the channel. The physical domain can be reduced to a half-channel as schematically represented in Fig. 1. The airflow is modelled as a piston-like flow and the coefficients of convective heat and mass transfer with the wall are assigned typical values for laminar flow regime, as described in detail in Ruivo (2005) and in Ruivo et al. (2007a). The desiccant wall of the channel is a relatively thin porous layer of silica-gel RD, and it is modelled by a simplified model where the resistances to heat and mass transfer in the y direction inside the porous medium are assumed negligible. The thickness of the represented desiccant wall Hp corresponds to the half-thickness of the real wall in the hygroscopic matrix. This simplified model was derived from a detailed model described by Ruivo (2005) and Ruivo et al. (2007a,b), who concluded that its use is valid when the thickness of the
desiccant layer is lower than approximately 0.1 mm (Ruivo, 2005; Ruivo et al., 2006, 2008a). The resistances to heat and mass transfer in the streamwise direction inside the desiccant porous medium are relatively high and they are taken into account in the simplified model. The adsorption/desorption process is also considered taking into account that water vapour and adsorbed water co-exist in equilibrium inside the desiccant porous medium, this equilibrium being characterised by sorption isotherms without hysteresis. The properties of the desiccant layer are those of silica-gel RD (Pesaran, 1983). The ordinary diffusion of water vapour is neglected due to the small dimension of the pores (Pesaran, 1983). However the real resistances to the surface diffusion of adsorbed water and to the Knudsen diffusion of water vapour are considered in the x direction. The upper and lateral boundaries of the wall domain are considered impermeable and adiabatic. The domain of the airflow is also simulated by a simplified model assuming bulk flow with negligible pressure losses. The convective heat and mass transfer at the solidegas interface is modelled by using convective transfer coefficients suitable for laminar channel flow. The lower boundary corresponds to a symmetry plane in the airflow domain and therefore it is considered also as impermeable and adiabatic.
2.2.
Governing equations
The complete set of conservation equations to be solved by the numerical model in both domains can be reduced to the generic form (cf. Nomenclature for further symbol identification): v v vf (1) rf f þ rf uf f Gf Sf ¼ 0; vt vx vx where the density rf , the velocity uf , the diffusion coefficient Gf and the source-term Sf assume the different meanings listed in Table 1, depending on the nature of the generic variable f considered. A common and useful simplification consists of assuming that the vapour and the adsorbed water inside the porous medium are locally in thermodynamic equilibrium, thus dispensing the solution of one of the mass conservation equations. So, the mass fraction of vapour is specified through the local equilibrium condition. The source-term Sv[;d can be evaluated as
Fig. 2 e Effective coefficients of surface and of Knudsen mass diffusion.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
Sv[;d
v jv;x jv;gs v rg$v 4v;d ¼ þ ; Hc vx vt
(2)
where the longitudinal Knudsen diffusion mass flux of water vapour is given by DK;eff v rg$v 4v;d ; (3) jv;x ¼ 3g$v vx and the convective mass flux of water vapour at the interface is 4v;f 4v;d jv;gs ¼ hm rf : (4) 1 4v;d The source-term in the energy conservation equation for the desiccant layer is v v Se;d ¼ rsd X[ h$lig rg$v 4v;p hfg0 jv;x hv þ j[;x h$[ vt vx jv;gs hv hfg0 þ jh;gs ; (5) þ Hp cpwpm where the longitudinal mass flux of adsorbed water by surface diffusion is given by j[;x ¼ Ds;eff rsd
vX[ ; vx
and the convective heat flux at the interface is jh;gs ¼ hh Tf Td :
(6)
(7)
In the evaluation of the convective fluxes at the interface at a particular position x, the state of the interface was assumed to be equal to the internal state of the desiccant due to the hypothesis of null gradients in the cross-direction inside the desiccant porous medium. Moreover, according to the low mass transfer rate theory (Mills, 1985), the influence of the mass transfer at the interface on the coefficient hh is assumed to be negligible. The heat convection coefficient hh is estimated after the Nusselt number for developed laminar channel flow, considering constant and uniform temperature at the interface, and corrected to account for the real shape of the channels of the hygroscopic matrix (Ruivo et al., 2007b). Regarding the mass convection coefficient, hm, the Sherwood number is related to Nu according to the ChiltoneColburn analogy Sh ¼ Nu Le1=3 ;
(8)
the Lewis number being defined by Le ¼ lf = rf cpf Df :
711
that the convective heat transfer coefficient hh is practically independent of the atmospheric pressure. Like the thermal conductivity, the specific heat cpf is also assumed to be independent of the atmospheric pressure, ¸ engel, 1998; Pesaran and Heiden, contrarily to Df and rf (e.g. C 1994). However, these two properties have inverse variations with pressure, therefore the product rf Df is independent of pressure, and likewise the Lewis and Sherwood numbers (see Equations (8) and (9)). From Equation (11) it may be concluded that the product rf hm used in Equation (4) to evaluate the local mass flux remains also independent from pressure variations. The Sherwood number is also constant, as it varies with the properties defining the Lewis number (Eq. (9)). Like the thermal conductivity, the specific heat cpf is also assumed to be independent of the atmospheric pressure, contrarily to Df and rf (e.g. C ¸ engel, 1998; Pesaran and Heiden, 1994). These considerations lead to the conclusion that the atmospheric pressure influences the mass convection coefficient hm, this effect being taken into account in the numerical model. The density of the gaseous mixture inside the pores and in the airflow depends considerably on pressure, and was estimated assuming ideal gas behaviour for the airevapour mixture. It was assumed in this simplified model that the pressure field in the desiccant porous medium and in the airflow channel is uniform and equal to the local atmospheric pressure, which value is strongly dependent on altitude. The water vapour content in the airflow is related with the mass fraction by wv ¼ 4v;f =ð1 4v;f Þ and the mass fraction of water vapour in the air-mixture inside the pores of the desiccant is related by: 4v;d ¼
Rg jpvs ; Rg jpvs þ Rv p jpvs
(12)
where pvs is the saturation pressure and j is the ratio of vapour partial pressures pv/pvs.
2.3.
Initial and boundary conditions
The cyclic process with a duration scyc is divided into the adsorption and the desorption modes, with durations sads and sdes. From the point of view of the modelled channel, the adsorption and the desorption processes occur, respectively, when 0 < s sads and sads < s scyc. At the stationary cyclic regime, the amounts of mass and heat transferred during the
(9)
Thus, the convection coefficients are estimated as hh ¼ Nu lf =dhyd ;
(10)
and hm ¼ Sh Df =dhyd ;
(11)
the hydraulic diameter being dhyd ¼ 4Hc, or dhyd ¼ 43m/as. The Nusselt number is considered constant due to the assumption of fully developed laminar channel flow. The thermal conductivity of the moist air is assumed to be independent of the atmospheric pressure, an assumption that is commonly adopted for gases and indicated in classical literature of heat transfer (e.g., C¸engel, 1998), as well as in Pesaran and Heiden (1994). Consequently, it is reasonable to assume
Fig. 3 e Curves of the sorption equilibrium of silica-gel with moist air.
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adsorption process are equal to those occurring during the desorption process. In fact, the modelled channel that is representative of the matrix is submitted to an initial transient process that must be started at a certain condition. The transition of mode, from adsorption to desorption, or from desorption to adsorption, is done by suddenly changing the inlet airflow conditions and reversing the airflow direction in the channel. After a certain number of adsorption/desorption cycles, the differences between two consecutive cycles are negligible, meaning that the stationary cyclic regime was achieved. The initial condition for the sequence of the cycles corresponds to the beginning of one of the modes of the cycle (adsorption or desorption), imposing uniform distributions for temperature and adsorbed water content in the desiccant and assuming that the airflows are initially in thermodynamic equilibrium with the desiccant medium. The inlet airflow conditions are chosen according to the starting mode considered, by imposing constant and uniform values of uf ¼ uin (or the corresponding mass velocity, Fm ¼ Fm,in), Tf ¼ Tin and 4v;f ¼ 4v;in . Lateral and the upper boundaries of the desiccant portion of calculation domain are considered adiabatic and impermeable, which is recognized by zero normal-gradient conditions for temperature and mass concentrations.
a
Numerical solution procedure
The differential equations are discretized via the finite-volume method adopting a fully-implicit scheme for the integration of the transient terms (Patankar, 1980). The physical domain is divided into a set of control-volumes, whose representative nodes form a non-uniform grid that is strongly refined near the solidegas interface, and towards the inlet and outlet sections of the channel. The discretization equations for the primary variables are formally equivalent. The tri-diagonal matrix algorithm (TDMA) is used to iteratively solve the set of algebraic equations for each variable field in each time-step. The values of the diffusion coefficients at the control-volume interfaces are estimated by the harmonic mean (Patankar, 1980). The nonlinearities implied by the changing properties, as well as the strong coupling between the different fields and the local constraint of the phase change require a careful underelaxation of the source-terms of the equations along the iterative calculation. Within the desiccant sub-domain, the equilibrium value of the vapour mass fraction is locally specified through Eq. (12), according to the procedure suggested in Patankar (1980). The simulation of a transient process consists of a stepby-step calculation procedure, each time-step being automatically adapted along the simulation, weighted by the time variation of the primary variables. This technique ensures a suitable representation of the transient evolutions and reduces the global time consumption of the simulations. Preliminary grid-independence tests were carried out to choose a numerical grid with a suitable refinement near the interface (Ruivo, 2005).
20 p =101325 Pa
0 ×106 (kg s -1 m-2 ) Jm
2.4.
part of the relations for the dry air, water vapour and liquid water was derived from thermodynamics tables (C¸engel, 1998) in the form of polynomial expressions (Ruivo, 2005). The properties of the air-mixture such as the specific heat and the thermal conductivity are weighted averages based on the dry air and water vapour mass fractions. Similarly, the specific heat and the thermal conductivity of the wet desiccant medium are weighted averages based on the mass fraction of each component (dry air, water vapour, adsorbed water and dry desiccant). The properties of silica-gel RD, the relations for the equilibrium condition, the heat of wetting, the adsorbed water enthalpy and the adsorption heat are indicated in Ruivo et al. (2007a). The dependences of the coefficients of surface and Knudsen mass diffusion on the temperature and on the adsorbed water content were derived after the expressions in Pesaran (1983) and are shown in Fig. 2. The pressure p does not affect these diffusion coefficients, as well as the equilibrium condition when it is written in the usual form of j ¼ f ðX[;eq Þ. Some curves of the sorption equilibrium for temperatures of 30 and 100 C at 60,000 and 101,325 Pa are shown in Fig. 3 in the form X[;eq ¼ gðT; wv Þ, being observed that, in this representation, the influence of the atmospheric pressure at high temperature is quite negligible.
L c=0.1 m
15
0.3 10 0.5
.
5
0 1.0
1.5
2.0
2.5 -1
b
20
15
p =101325 Pa
10
.
5
0 0
5
10
15
20 -1
2.5.
Evaluation of properties and coefficients
The model takes into account the changes occurring in the airflow properties and in the diffusion coefficients. The major
3.0
-2
F m,in (kg s m )
0 Jm ×106 (kg s -1 m-2 )
712
25
30
-3
F m,in /L c (kg s m )
Fig. 4 e Predicted mass transfer rate at sea-level conditions (study PS1) as a function of (a) Fm,in and Lc and (b) of the ratio Fm,in/Lc.
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2.6.
Evaluation of the mass transfer rate
is regenerated. So, at steady state conditions, the mass transfer rate occurring in the adsorption zone is equal to that occurring in the desorption zone. Therefore, considering the adsorption mode, the following expression can be deduced for the mass transfer rate between both airflows, per unit of transfer area of the matrix (Ruivo et al., 2007b): Zsads _ Jv;gs ds: (14) J_m ¼ scyc
The mass transfer rate per unit of transfer area occurring in the modelled channel is evaluated by Ruivo et al. (2007a): 1 J_v;gs ¼ xc
Zxc jv;gs dx:
(13)
0
The behaviour of the modelled channel enables the prediction of the global behaviour of the hygroscopic rotor crossed by two airflows at steady state conditions, working as a desiccant wheel. The adsorption mode corresponds to the adsorption zone of the rotor matrix, where the dehumidification of the process air occurs, while the desorption mode corresponds to the desorption zone, where the rotor matrix
0
3.
Study cases and results
Four parametric studies, hereafter designated PS1, PS2, PS3 and PS4, were performed considering a desiccant wheel
a 1.05 p =101325 Pa 1.00 90,000 0.95 80,000
kp
0.90 70,000
0.85 0.80
60,000 0.75 0.70 0
5
10
15
20 -1
25
30
-3
F m,in /L c (kg s m )
b
1.05 p =101325 Pa 1.00
log(10 × k p)
90,000 0.95 80,000 70,000
linear regression
0.90
60,000 0.85 0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
log(100×L c/F m,in) Fig. 5 e Correction factor kp of the mass transfer rate for non-standard atmospheric pressure (study PS1): (a) in a common representation; (b) in logarithmic scales.
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0.14
1.10
0.12
1.05
0.10
1.00
0.08
0.95
0.06
0.90
0.04
0.85
0.02
0.80
0.00
0.75
-0.02
b kp
a kp
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0.70
60,000
70,000
80,000
90,000
100,000
110,000
p (Pa) Fig. 6 e Linear regressions for the coefficient akp and for the term bkp (study PS1). divided into two equal parts (sads ¼ sdes), where the adsorption and desorption zones are equal and crossed by countercurrent airflows. The compact corrugated matrix is composed by sinusoidal cross section channels, having a specific transfer area as ¼ 3198 m2 m3 and a porosity 3m ¼ 0.84. According to Ruivo et al. (2007b), the chosen matrix corresponds to cell B3 (Hcell ¼ 1.5 mm, Pcell ¼ Psin ¼ 3 mm, Ep ¼ 0.1 mm, Hsin ¼ 1.27 mm), the representative channel of the matrix being modelled with Hp ¼ 0.05 mm, Hc ¼ 0.263 mm and with a Nusselt number of 2.09. For the highest value of inlet mass velocity considered in the present work (Fm,in ¼ 3 kg s1 m2), the Reynolds number is 237, which confirms the validity of the assumption of laminar regime airflow. In a first parametric study (PS1), the inlet states of both airflows are characterised by Tin,ads ¼ 30 C, wv,in,ads ¼ 0.01 kg kg1, and Tin,des ¼ 100 C, wv,in,des ¼ 0.01 kg kg1. The rotor speed was fixed to 7.2 rotations per hour, that corresponds to scyc ¼ 500 s. Concerning the main goal of this case 1.1
k*p
1.0
0.9
study, a wide range of values of the atmospheric pressure p with practical interest was considered (60,000, 70,000, 80,000, 90,000 and 101,325 Pa). Three values for the channel length Lc were chosen, namely 0.1, 0.3 and 0.5 m, the lower value being representative of a matrix with a short channel length and the last one with a long one. The mass airflow rates per unit of frontal area (mass velocity) entering in both zones of the desiccant wheel were considered equal to Fm,in and three different values were considered in the parametric study (1, 2 and 3 kg s1 m2). The simulation started with the adsorption mode, departing from an equilibrium condition for the desiccant layer defined by the initial uniform distributions of temperature and of adsorbed water content, respectively, with the values of 100 C and 0.0121 kg kg1. The numerically predicted results for the cyclic behaviour of the modelled channel were used to derive the behaviour of a hygroscopic matrix rotating through the adsorption and the desorption zones and continuously exchanging heat and mass with the two air streams, similarly to what occurs in a hygroscopic wheel at steady state conditions. The results of this parametric study are plotted in Figs. 4e7. Fig. 4(a) refers to the predicted mass transfer rate per unit of 0 transfer area J_m for sea-level conditions ( p ¼ 101,325 Pa). As 0 expected, it is seen that J_m becomes larger by increasing the mass velocity Fm,in and decreasing the channel length Lc.
0.8
Table 2 e Parameters of the linear regression akp, bkp, R2kp for the different lines of Fig. 5(b). p (Pa)
0.7 0.7
0.8
0.9
1.0
1.1
kp
Fig. 7 e Correlation between the kp values estimated by Eq. (18) and the kp values obtained with the numerical results (study PS1).
60,000 70,000 80,000 90,000 101,325
akp
bkp
R2kp
0.1357 0.094 0.0589 0.0291 0
0.7877 0.8553 0.9106 0.9564 1
0.9949 0.9915 0.9884 0.9851 1
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Table 3 e Parameters of the linear regressions of akp and bkp. aa
ba
R2a
ab
bb
R2b
3.2693 106 3.1777 106 1.2611 106
0.32595 0.32118 0.12664
0.9903 0.9994 0.9934
5.109 106 10.44 106 4.104 106
0.4919 0.8956 0.5908
0.987 0.9844 0.9912
Parametric study PS1 PS2 PS3
0 Comparing the curves J_m w Fm,in, it is observed that the same 0 J_m value is achieved with Fm,in ¼ 1 kg s1 m2, for Lc ¼ 0.1 m, and with Fm,in ¼ 3 kg s1 m2, for Lc ¼ 0.3 m, i.e., points with the same value of the ratio Fm,in/Lc. This behaviour corroborates the theory for rotary heat exchangers where the nondimensional NTU parameter is a function of the ratio between the transfer area and the mass flow rate. Consequently, the predicted influences of the mass velocity and of the channel length on the mass transfer rate of the desiccant matrix at standard atmospheric pressure are alternatively condensed in a single curve in Fig. 4(b). The effect of a pressure decrease on the mass transfer rate 0 can be expressed as a correction factor kp ¼ J_m =J_m , which is represented in Fig. 5(a). Fig. 5(b) is an alternative representation of Fig. 5(a) using transformed logarithmic scales. These new curves evidence a linear tendency, suggesting the use of the linear regression which is represented by the continuous
lines in Fig. 5(b). So, the correction factor can be estimated by the following expression: Lc þ bkp : (15) log 10kp ¼ akp log 100 Fm;in The values of the coefficient akp, the term bkp and the correlation factor R2kp for each line in Fig. 5(b) are listed in Table 2. The calculated values of the correlation factor indicate a good fitting of the linear regression in all cases. Using the data of Table 2, the dependences of akp and bkp on pressure are represented in Fig. 6, and they are well represented by, respectively: akp ¼ aa p þ ba
(16)
and bkp ¼ ab p þ bb :
(17)
a 1.2 p =60,000 Pa 1.0
kp
τ cyc=100 s 250
0.8
500 750
1000
0.6 0
5
10
15
20 -1
25
30
-3
F m,in /L c (kg s m )
b
1.1 p =80,000 Pa
1.0
kp
τ cyc=100 s 250
500
0.9 750
1000
0.8 0
5
10
15
20 -1
25
30
-3
F m,in /L c (kg s m ) Fig. 8 e Correction factor of the mass transfer rate (study PS4) for different cycle durations at a non-standard atmospheric pressure of: (a) 60,000 Pa and (b) 80,000 Pa.
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The values of the coefficients aa and ab, terms ba and bb and correlation factors R2a and R2b are listed in Table 3. Manipulating Eq. (15) and taking into account Eqs. (16) and (17), a first expression is obtained to estimate the correction factor: kp ¼ 0:1 10½ðaa pþba Þ logð100 Lc =Fm;in Þþab pþbb :
(18)
Fig. 7 shows the correlation between the estimated values kp with Eq. (18) and the kp values obtained after the results of the numerical modelling, i.e., kp ¼ 0:9971kp þ 0:0026, with a root mean square deviation RMSD ¼ 0.0092 and a correlation factor R2 ¼ 0.9837. According to Fig. 5(a), it may be concluded that the effect of the atmospheric pressure on the mass transfer rate is practically negligible for values of the ratio Fm,in/Lc between 2 and 3, approximately. For Fm,in/Lc > 3, the correction factor kp becomes smaller with the decrease of atmospheric pressure. The lower values of kp, meaning a reduction of over 25% in ‘, are observed for the case with higher ratio Fm,in/Lc when p ¼ 60,000 Pa. In a second study (PS2), the values of the correction factor are determined for cases with fixed values of j, instead of a fixed value of the water vapour content as in the preceding parametric study. It is important to stress that the parameter (j ¼ pv/pvs) corresponds exactly to the relative humidity, for states of the moist air when the temperature is equal or lower than the particular value of the water saturation temperature Tsat at the atmospheric pressure p. Moreover, the relative humidity concept is not applicable when the moist air temperature is greater than Tsat, a situation that is commonly found in regeneration airflow inlet conditions. The inlet states of both airflows in the present parametric study are characterised by Tin,ads ¼ 30 C, jin,ads ¼ 0.377, and Tin,des ¼ 100 C, jin,ads ¼ 0.016, the water vapour content and the vapour mass fraction increasing with the atmospheric pressure (simulated values: 60,000, 70,000, 80,000, 90,000 and 101,325 Pa).
Following a procedure similar to that of the study PS1, the 0 results obtained for the correction factor (kp ¼ J_m =J_m ) were analysed. The achieved values of the coefficients aa and ab, terms ba and bb and correlation factors R2a and R2b are listed in Table 3. The correlation between the values kp estimated with Eq. (18) and the kp values is acceptable. The determined values of the root mean square deviation and the correlation factor were RMSD ¼ 0.006773773 and R2 ¼ 0.9972, respectively. In the previous parametric studies the mass velocity Fm,in was used as inlet airflow condition. Now, the correction factor values are determined for a set of cases (SC3) with a fixed and equal value of the airflow velocity, uin, in both inlet process and regeneration zones. All the other inlet conditions are those of the first study (PS1). The purposed expression for the calculation of kp is: kp ¼ 0:1 10½ðaa pþba Þ logð100 Lc =uin Þþab pþbb :
(19)
The calculated values of the coefficients aa and ab, and terms ba and bb are listed in Table 3, as well as the correlation factors R2a and R2b . In the cases of the study PS3, the correlation factor and the root mean square deviation of the linear correlation between kp and kp were R2 ¼ 0.9922 and RMSD ¼ 0.0131, respectively. In a fourth parametric study (PS4), the influence of the rotor speed on the correction factor kp was investigated. In all previous parametric studies, the duration of the cycle was fixed at 500 s. A wide range of values of scyc was considered (100, 250, 500, 750 and 1000 s) in the numerical simulations, as well as three values of the pressure (60,000, 80,000 and 101,325 Pa). The same values of the ratio Fm,in/Lc and of inlet state conditions of the study PS1 were considered in the present parametric study. The values of kp were determined and the results plotted in Fig. 8(a) and (b), respectively, for 60,000 Pa and 80,000 Pa. It is observed that at high values of scyc (low rotation speeds of the wheel), the correction factor is smaller than for the highest wheel rotation speed, and also
1.1
1.0
k p,
cyc
0.9
0.8
0.7
0.6 0.6
0.7
0.8
0.9
1.0
1.1
kp Fig. 9 e Correlation between the correction factor values estimated by Eq. (20) and those obtained with the numerical results (study PS4).
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
that this difference increases with the decreasing in the atmospheric pressure. The shape of the curves in Fig. 8(a) and (b) suggests that it is easy to derive a correlation for kp. Taking as a reference the correction factor for scycref ¼ 500 s, kp,cycref and manipulating the obtained data, the following correction expression was derived: kp;cyc ¼ kp;cycref þ 0:5077 5 105 p 9 107 s2cyc 2:06 104 scyc þ 0:7998 :
(20)
Fig. 9 shows the correlation between the values kp,cyc estimated with Eq. (20) and the kp values obtained after the results of the numerical modelling of the cases of this fourth parametric study. The corresponding values of the root mean square deviation and of the correlation factor were RMSD ¼ 0.00878 and R2 ¼ 0.9904, respectively. The processing and analysis of results of the different set of parametric studies led to the conclusion that it is relatively easy to develop correlations to estimate the pressure correction factor kp for different operating conditions characterised by a wide ranges of the atmospheric pressure, of the rotation speed and of the ratio Fm,in/Lc.
4.
Conclusions
In the present work, a numerical model that predicts the heat and mass transfer occurring in a channel of a desiccant wheel was used to evaluate the influence of the atmospheric pressure on the mass transfer rate. Parametric studies were performed for different values of the channel length and of both airflow rates in a range of values with practical interest, considering also a large range of values of the rotation speed of the wheel. The results of the parametric study PS1 indicate that the decreasing effect of increasing altitude on the mass transfer rate can be well expressed by a correction factor kp expressed by a simple correlation that may be a useful estimator for the selection of desiccant wheels for non-standard atmospheric operating conditions. The derived correlations of kp for the conditions of the others parametric studies (PS2, PS3 and PS4) were also tested, a good agreement being achieved between the estimated correction factor and the one calculated after the numerical results. There are many other parameters than those considered in the present study that influence the mass transfer rate occurring in desiccant wheels (e.g., Ruivo, 2005; Chung and Lee, 2009; Ruivo et al., 2007b, 2008b). Therefore, additional effort is still needed to verify the validity of this corrective procedure for different inlet conditions, partitions of the desiccant wheel, as well as matrix specific transfer area and porosity. Furthermore, the channel walls can also be of different thicknesses and composed of different desiccant materials.
Appendix. Supplementary data Supplementary data related to this article can be found online at doi:10.1016/j.ijrefrig.2010.11.013.
717
references
Banks, P.J., 1985a. Prediction of heat and mass regenerator performance using nonlinear analogy method: Part 1 e basis. ASME J. Heat Transfer 107, 222e229. Banks, P.J., 1985b. Prediction of heat and mass regenerator performance using nonlinear analogy method: Part 2 e comparison of methods. ASME J. Heat Transfer 107, 230e238. C¸engel, Y., 1998. Heat Transfer e A Practical Approach. McGrawHill, Boston. Cejudo, J.M., Moreno, R., Carrillo, A., 2002. Physical and neural network models of a silica gel desiccant wheel. Energy Build. 34, 837e844. Chung, J.D., Lee, D.Y., 2009. Effect of desiccant isotherm on the performance of desiccant wheel. Int. J. Refrigeration 32, 720e726. Chung, J.D., Lee, D.Y., Yoon, S.M., 2009. Optimization of desiccant wheel speed and area ratio of regeneration to dehumidification as a function of regeneration temperature. Solar Energy 83, 625e635. Dai, Y.J., Wang, R.Z., Zhang, H.F., 2001. Parameter analysis to improve rotary desiccant dehumidification using a mathematical model. Int. J. Therm. Sci. 40, 400e408. Elsayed, S.S., Hamamoto, Y., Akisawa, A., Kashiwagi, T., 2006. Analysis of an air cycle refrigerator driving air conditioning system integrated desiccant system. Int. J. Refrigeration 29, 219e228. Gao, Z., Mei, V.C., Tomlinson, J.J., 2005. Theoretical analysis of dehumidification process in a desiccant wheel. Heat Mass Transfer 41, 1033e1042. Golubovic, M.N., Worek, W.M., 2004. Influence of elevated pressure on sorption in desiccant wheels. Numer. Heat Transfer A 45, 869e886. Harshe, Y.M., Utikar, R.P., Ranade, V.V., Pahwa, D., 2005. Modeling of rotary desiccant wheels. Chem. Eng. Technol. 28, 1473e1479. Maclaine-Cross, I.L., Banks, P.J., 1972. Coupled heat and mass transfer in regenerators-prediction using an analogy with heat transfer. Int. J. Heat Mass Transfer 15, 1225e1242. Mills, A., 1985. Heat and Mass Transfer. Irwin, London. Niu, J.L., Zhang, L.Z., 2002. Effects of wall thickness on the heat and moisture transfer in desiccant wheels for air dehumidification and enthalpy recovery. Int. Commun. Heat Mass Transfer 29, 255e268. Patankar, S., 1980. Numerical Heat Transfer and Fluid Flow. In: Hemisphere. McGraw-Hill, Washington, DC. Pesaran, A., 1983. Moisture transport in silica gel particle beds. PhD Thesis, University of California, Los Angeles. Pesaran, A.A., Heiden, R., 1994. The influence of altitude on the performance of desiccant-cooling systems. Energy 19, 1165e1179. Ruivo, C.R., 2005. Modelac¸a˜o nume´rica dos feno´menos de transfereˆncia de calor e de massa em rodas higrosco´picas. PhD Thesis, University of Coimbra, Coimbra, Portugal (in Portuguese). Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2006. Analysis of simplifying assumptions for the numerical modeling of the heat and mass transfer in a porous desiccant medium. Numer. Heat Transfer A 49, 851e872. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2007a. On the behaviour of hygroscopic wheels: part I e channel modeling. Int. J. Heat Mass Transfer 50, 4812e4822. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2007b. On the behaviour of hygroscopic wheels: part II e rotor performance. Int. J. Heat Mass Transfer 50, 4823e4832. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2008a. On the validity of lumped capacitance approaches for the numerical prediction
718
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
of heat and mass transfer in desiccant airflow systems. Int. J. Therm. Sci. 47, 282e292. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2008b. Numerical study of the behavior of an elementary desiccant layer of a hygroscopic rotor. Numer. Heat Transfer A 53, 1037e1053. Sphaier, L.A., Worek, W.M., 2004. Analysis of heat and mass transfer in porous sorbents used in rotary regenerators. Int. J. Heat Mass Transfer 47, 3415e3430.
Sphaier, L.A., Worek, W.M., 2006. Comparisons between 2-D and 1-D formulations of heat and mass transfer in rotary regenerators. Numer. Heat Transfer B 49, 223e237. Zhang, L.Z., Niu, J.L., 2002. Performance comparisons of desiccant wheels for air dehumidification and enthalpy recovery. Appl. Therm. Eng. 22, 1347e1367. Zhang, X.J., Dai, Y.J., Wang, R.Z., 2003. A simulation study of heat and mass transfer in a honeycomb rotary desiccant dehumidifier. Appl. Therm. Eng. 23, 989e1003.
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Influence of the atmospheric pressure on the mass transfer rate of desiccant wheels C.R. Ruivo a,b,*, J.J. Costa b, A.R. Figueiredo b a b
Dep. Eng. Mec., Instituto Superior de Engenharia, University of Algarve, Campus da Penha, 8005-139 Faro, Portugal ADAI, Department of Mechanical Engineering, University of Coimbra, Rua Luı´s Reis Santos, 3030-788 Coimbra, Portugal
article info
abstract
Article history:
The performance of a desiccant wheel is evaluated by modelling a representative channel.
Received 17 December 2009
The hypothesis of negligible resistances to heat and mass transfer in the cross-direction is
Received in revised form
assumed in the thin porous desiccant wall of the channels and the airflow is treated as
16 October 2010
a bulk flow. Parametric studies were conducted to investigate the influence of the atmo-
Accepted 26 November 2010
spheric pressure decrease from 101,325 Pa to 60,000 Pa (0e4217 m of altitude) on the mass
Available online 3 December 2010
transfer rate of desiccant wheels considering distinct channel lengths and different inlet airflow rates, a large range of values of the rotation speed, as well as three alternative ways
Keywords:
to specify the inlet conditions of the regeneration and of the process airflows. A procedure
Adsorption
to derive correlations based on the numerical results is presented for the correction factor
Desiccant wheel
of the mass transfer rate when a desiccant wheel is operating at non-standard atmospheric
Modelling
conditions. Four parametric studies were performed, the derived correlations were tested
Steady state
and a good agreement was found between the estimated correction factor and the
Pressure
correction factor calculated after the numerical results. ª 2010 Elsevier Ltd and IIR. All rights reserved.
Influence de la pression atmosphe´rique sur la vitesse de transfert de masse des roues de´shydratantes Mots cle´s : Adsorption ; Roue deshydratante ; Modelisation ; Regime permanent ; Pression
1.
Introduction
Usually the available performance data of dehumidification and cooling systems integrating desiccant wheels are based on atmospheric conditions at sea level. The use of specific manufacturer software permits to select or to rate the equipment, but an arduous task still remains: to perform the
energy dynamic simulation of the complete system and building or of the process during a year or a season. The existing standards and guidelines do not address the methods that should be used to predict the performance of a desiccant wheel at high altitudes. Manufacturers of desiccant wheels have devised methods for restating performance, but there is no consensus on the better
* Corresponding author. Dep. Eng. Mec., Instituto Superior de Engenharia, University of Algarve, Campus da Penha, 8005-139 Faro, Portugal. Tel.: þ351 289 800100; fax: þ351 289 888405. E-mail address: [email protected] (C.R. Ruivo). 0140-7007/$ e see front matter ª 2010 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2010.11.013
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Nomenclature a as b cp;f
cp;f cpwpm Df DK,eff Ds,eff dhyd Fm Fm,in Hc Hp hads hfg0 hh h[ hlig hm hv J_m 0 J_m
J_v;gs jh,gs jh,x j[;x jv,gs jv,sx kp kp kp,cyc kp,cycref Lc Le Nu p pvs
coefficient of the linear regression specific transfer area of the hygroscopic matrix (m2 m3) term of the linear regression specific heat of the moist air referred to the unit of mass of mixture (J kg1 K1) mean specific heat of the moist air referred to the unit of mass of mixture (J kg1 K1) mean specific heat of the wet porous medium (J kg1 K1) diffusion coefficient of water vapour in the airflow (m2 s1) effective coefficient of Knudsen diffusion (m2 s1) effective coefficient of surface diffusion of adsorbed water (m2 s1) hydraulic diameter (m) airflow mass velocity (kg s1 m2) airflow mass velocity at the inlet of the channel (kg s1 m2) channel half-height (cf. Fig. 1) (m) half-thickness of the desiccant wall (cf. Fig. 1) (m) heat of adsorption (J kg1) latent heat of vaporization (J kg1) convection heat transfer coefficient (W m2 K1) enthalpy of adsorbed water (J kg1) heat of wetting (J kg1) convection mass transfer coefficient (m s1) enthalpy of vapour in the desiccant wall (J kg1) global mass transfer rate per unit of transfer area of matrix (kg s1 m2) global mass transfer rate per unit of transfer area of matrix at sea-level atmospheric pressure (kg s1 m2) gas side mass transfer rate at the interface per unit of transfer area (kg s1 m2) gas side convective heat flux at the interface (W m2) axial heat conduction flux in the desiccant (W m2) axial surface diffusion flux in the desiccant (kg s1 m2) gas side convective mass flux at the interface (kg s1 m2) axial Knudsen diffusion flux in the desiccant (kg s1 m2) correction factor for the effect of atmospheric 0 pressure on the mass transfer rate (J_m =J_m ) correction factor kp predicted by a correlation expressed by Eq. (15) correction factor kp predicted by the Eq. (19) reference correction factor kp used in Eq. (19) channel length (cf. Fig. 1) (m) Lewis number Nusselt number pressure of the airevapour mixture (Pa) saturation pressure (Pa)
Rg Rg$v Rv R2 RMSD Sh Se,d Sv[;d Sf Td Tf Tin t uf uin uf wv wv,in
X[ x y
gas constant of the air (J kg1 K1) gas constant of the airevapour mixture (J kg1 K1) gas constant of the water vapour (J kg1 K1) coefficient of the correlation root mean square deviation Sherwood number source-term in the energy conservation equation source-term in the water mass conservation equation generic source-term in Eq. (1) temperature of the desiccant ( C) temperature of the airflow ( C) bulk airflow temperature at the inlet of the channel ( C) time (s) bulk airflow velocity (m s1) bulk airflow velocity at the channel inlet (m s1) velocity (m s1) water vapour content (kg kg1, i.e., kg of water vapour per kg of dry air) bulk airflow water vapour content at the channel inlet (kg kg1, i.e., kg of water vapour per kg of dry air) adsorbed water content in the desiccant (kg kg1, i.e., kg of adsorbed water per kg of dry desiccant) spatial coordinate (m) spatial coordinate (m)
Greek symbols volume fraction of the gaseous mixture within the 3g$v porous medium porosity of the hygroscopic matrix 3m f generic variable generic diffusion coefficient in Eq. (1) Gf vapour mass fraction in the desiccant (kg kg1, i.e., 4v;d kg of water vapour per kg of moist air) vapour mass fraction in the airflow (kg kg1, i.e., 4v;f kg of water vapour per kg of moist air) bulk airflow vapour mass fraction at the inlet of 4v;in the channel (kg kg1, i.e., kg of water vapour per kg of moist air) thermal conductivity of the airflow (W m1 K1) lf thermal conductivity of wet porous medium lwpm (W m1 K1) density of the airflow (kg m3) rf density of the gas mixture (kg m3) r[ apparent density of the gas mixture inside the rg$v porous medium (kg m3) apparent density of the dry desiccant (kg m3) rsd generic term in Eq. (1) rf s time coordinate in an adsorption/desorption cycle (s) duration time of the adsorption mode (s) sads duration time of adsorption/desorption cycle (s) scyc duration time of the desorption mode (s) sdes j relative pressure, i.e., ratio of the partial to the saturation vapour pressure
709
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Fig. 1 e Schematic representation of the half-channel domain.
method to be adopted. In fact, most of the existing laboratories are not prepared to vary the ambient pressure, and validation of the proposed methods still remains incomplete. The modelling of the performance of desiccant wheels is of great relevance for product optimization by the manufacturer, as well as to perform parametric studies and yearly simulations of the performance of global systems integrating desiccant wheels, towards a more rational energy use. Different methods of numerical solution have been used by many researchers with different simplified treatments of the fluid and solid domains to predict the behaviour of air dehumidifying systems. The solution of the coupled heat and mass transfer was formerly presented by Maclaine-Cross and Banks (1972) by using the analogy with the heat transfer process. Later on Banks (1985a,b) presented a more accurate solution based on a nonlinear analogy method. Most physical models used in numerical studies of desiccant wheels have been analysed through 1-D mathematical formulation based on the hypotheses of negligible cross-direction resistances to heat and mass transfer inside the desiccant wall (Dai et al., 2001; Cejudo et al., 2002; Zhang et al., 2003; Golubovic and Worek, 2004; Harshe et al., 2005; Gao et al., 2005; Elsayed et al., 2006; Chung and Lee, 2009; Chung et al., 2009). The validity of such one-dimensional models is acceptable to relatively thin
desiccant walls of the hygroscopic matrix and was investigated by some authors (Ruivo, 2005; Sphaier and Worek, 2006; Ruivo et al., 2006, 2008a). Other studies were performed using 2-D mathematical formulations taking into account the resistance to the diffusion of mass inside the desiccant porous wall (Ruivo, 2005; Niu and Zhang, 2002; Zhang and Niu, 2002; Sphaier and Worek, 2004; Ruivo et al., 2007a,b). Several parametric studies were performed in the above referenced works, but only the study of Golubovic and Worek (2004) is focussed on the effect of pressure on sorption in desiccant wheels employed to dry pressurized airflows. They emphasize the possibility and the implications of the occurrence of condensation in the regeneration portion of the wheel, affecting the global performance. Only one study was found in the literature specifically addressed to the effect of altitude on the performance of a desiccant cooling system (Pesaran and Heiden, 1994), namely the behaviour of the different system components for decreasing atmospheric pressure, from 101,325 to 81,060 Pa. The objective of the present work is to analyze the influence of decreasing atmospheric pressure on the performance of desiccant wheels, a subject with practical relevance when these equipments operate at significantly high altitudes, as recognized by the ASHRAE Technical Committee TC 8.12 (call for proposals for the research project 1339-TRP, “Selection of Desiccant Equipment at Altitude”, 2009).
Table 1 e Meanings of f, rf , uf , Gf and Sf in the conservation equation Eq (1). Gf
f
rf
uf
Desiccant domain Mass of adsorbed water Energy
X[ Td
rsd rsd
0 0
rsd Ds;eff lwpm cpwpm
Airflow domain Mass of gaseous mixture
1
rf
uf
0
Mass of water vapour
4v,f
rf
uf
0
Energy
Tf
rf
uf
0
Momentum
uf
rf
uf
0
Conservation equation
Sf Sv[;d Se,d
jv;gs Hc
jv;gs Hc jv;gs ðhv hfg0 Þ jh;gs ðTf Td Þ Hc cp f Hc cp f 0
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In the present work, a numerical model is used accounting for the heat and mass transfer phenomena occurring simultaneously with the sorption processes in the hygroscopic matrix of a desiccant wheel. The study is based on the simulation of a representative channel, using a simplified model to predict its transient behaviour, starting from an imposed set of initial conditions until the stationary regime of adsorption/ desorption cycles is achieved. This version of the model was derived from a detailed numerical model (Ruivo, 2005; Ruivo et al., 2007a,b) and it is valid for the cases with thin desiccant walls, where the resistances to heat and mass transfer in the cross-direction inside the porous medium are negligible (Ruivo, 2005; Ruivo et al., 2006, 2008a). Four parametric studies were performed to derive the correlations for the correction factor of the mass transfer rate when a desiccant wheel is operating at non-standard atmospheric conditions. The first set of parametric studies (PS1, PS2 and PS3) concerns to three alternative forms of specifying the inlet conditions of both airflows. In a fourth parametric study (PS4), a correlation for the correction factor taking into account the influence of the rotation speed of the wheel was also derived. The numerical model was not yet validated. A serious comparison of numerical results with experimental data would require that all the properties of the real desiccant layer and of the desiccant matrix were well known, but there is a lack of information for the complete characterization of the desiccant media. In some published experimental works, some property values or correlations are missing. Moreover, examples of exhaustive experimental research on the behaviour of a desiccant wheel (Cejudo et al., 2002) show significant mass and energy imbalances between the regeneration and the process airflows.
2.
Mathematical and numerical modelling
2.1.
Physical model and simplifications
Only a representative channel of the hygroscopic matrix of the desiccant wheel is modelled, the adsorption/desorption cycle being imposed by suddenly changing the inlet conditions and the direction of the airflow. The problem consists of a parallelplate channel which is cyclically submitted to alternating airflow inlet conditions and stream direction, producing a sequence of adsorption and desorption processes in the desiccant walls of the channel. The physical domain can be reduced to a half-channel as schematically represented in Fig. 1. The airflow is modelled as a piston-like flow and the coefficients of convective heat and mass transfer with the wall are assigned typical values for laminar flow regime, as described in detail in Ruivo (2005) and in Ruivo et al. (2007a). The desiccant wall of the channel is a relatively thin porous layer of silica-gel RD, and it is modelled by a simplified model where the resistances to heat and mass transfer in the y direction inside the porous medium are assumed negligible. The thickness of the represented desiccant wall Hp corresponds to the half-thickness of the real wall in the hygroscopic matrix. This simplified model was derived from a detailed model described by Ruivo (2005) and Ruivo et al. (2007a,b), who concluded that its use is valid when the thickness of the
desiccant layer is lower than approximately 0.1 mm (Ruivo, 2005; Ruivo et al., 2006, 2008a). The resistances to heat and mass transfer in the streamwise direction inside the desiccant porous medium are relatively high and they are taken into account in the simplified model. The adsorption/desorption process is also considered taking into account that water vapour and adsorbed water co-exist in equilibrium inside the desiccant porous medium, this equilibrium being characterised by sorption isotherms without hysteresis. The properties of the desiccant layer are those of silica-gel RD (Pesaran, 1983). The ordinary diffusion of water vapour is neglected due to the small dimension of the pores (Pesaran, 1983). However the real resistances to the surface diffusion of adsorbed water and to the Knudsen diffusion of water vapour are considered in the x direction. The upper and lateral boundaries of the wall domain are considered impermeable and adiabatic. The domain of the airflow is also simulated by a simplified model assuming bulk flow with negligible pressure losses. The convective heat and mass transfer at the solidegas interface is modelled by using convective transfer coefficients suitable for laminar channel flow. The lower boundary corresponds to a symmetry plane in the airflow domain and therefore it is considered also as impermeable and adiabatic.
2.2.
Governing equations
The complete set of conservation equations to be solved by the numerical model in both domains can be reduced to the generic form (cf. Nomenclature for further symbol identification): v v vf (1) rf f þ rf uf f Gf Sf ¼ 0; vt vx vx where the density rf , the velocity uf , the diffusion coefficient Gf and the source-term Sf assume the different meanings listed in Table 1, depending on the nature of the generic variable f considered. A common and useful simplification consists of assuming that the vapour and the adsorbed water inside the porous medium are locally in thermodynamic equilibrium, thus dispensing the solution of one of the mass conservation equations. So, the mass fraction of vapour is specified through the local equilibrium condition. The source-term Sv[;d can be evaluated as
Fig. 2 e Effective coefficients of surface and of Knudsen mass diffusion.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
Sv[;d
v jv;x jv;gs v rg$v 4v;d ¼ þ ; Hc vx vt
(2)
where the longitudinal Knudsen diffusion mass flux of water vapour is given by DK;eff v rg$v 4v;d ; (3) jv;x ¼ 3g$v vx and the convective mass flux of water vapour at the interface is 4v;f 4v;d jv;gs ¼ hm rf : (4) 1 4v;d The source-term in the energy conservation equation for the desiccant layer is v v Se;d ¼ rsd X[ h$lig rg$v 4v;p hfg0 jv;x hv þ j[;x h$[ vt vx jv;gs hv hfg0 þ jh;gs ; (5) þ Hp cpwpm where the longitudinal mass flux of adsorbed water by surface diffusion is given by j[;x ¼ Ds;eff rsd
vX[ ; vx
and the convective heat flux at the interface is jh;gs ¼ hh Tf Td :
(6)
(7)
In the evaluation of the convective fluxes at the interface at a particular position x, the state of the interface was assumed to be equal to the internal state of the desiccant due to the hypothesis of null gradients in the cross-direction inside the desiccant porous medium. Moreover, according to the low mass transfer rate theory (Mills, 1985), the influence of the mass transfer at the interface on the coefficient hh is assumed to be negligible. The heat convection coefficient hh is estimated after the Nusselt number for developed laminar channel flow, considering constant and uniform temperature at the interface, and corrected to account for the real shape of the channels of the hygroscopic matrix (Ruivo et al., 2007b). Regarding the mass convection coefficient, hm, the Sherwood number is related to Nu according to the ChiltoneColburn analogy Sh ¼ Nu Le1=3 ;
(8)
the Lewis number being defined by Le ¼ lf = rf cpf Df :
711
that the convective heat transfer coefficient hh is practically independent of the atmospheric pressure. Like the thermal conductivity, the specific heat cpf is also assumed to be independent of the atmospheric pressure, ¸ engel, 1998; Pesaran and Heiden, contrarily to Df and rf (e.g. C 1994). However, these two properties have inverse variations with pressure, therefore the product rf Df is independent of pressure, and likewise the Lewis and Sherwood numbers (see Equations (8) and (9)). From Equation (11) it may be concluded that the product rf hm used in Equation (4) to evaluate the local mass flux remains also independent from pressure variations. The Sherwood number is also constant, as it varies with the properties defining the Lewis number (Eq. (9)). Like the thermal conductivity, the specific heat cpf is also assumed to be independent of the atmospheric pressure, contrarily to Df and rf (e.g. C ¸ engel, 1998; Pesaran and Heiden, 1994). These considerations lead to the conclusion that the atmospheric pressure influences the mass convection coefficient hm, this effect being taken into account in the numerical model. The density of the gaseous mixture inside the pores and in the airflow depends considerably on pressure, and was estimated assuming ideal gas behaviour for the airevapour mixture. It was assumed in this simplified model that the pressure field in the desiccant porous medium and in the airflow channel is uniform and equal to the local atmospheric pressure, which value is strongly dependent on altitude. The water vapour content in the airflow is related with the mass fraction by wv ¼ 4v;f =ð1 4v;f Þ and the mass fraction of water vapour in the air-mixture inside the pores of the desiccant is related by: 4v;d ¼
Rg jpvs ; Rg jpvs þ Rv p jpvs
(12)
where pvs is the saturation pressure and j is the ratio of vapour partial pressures pv/pvs.
2.3.
Initial and boundary conditions
The cyclic process with a duration scyc is divided into the adsorption and the desorption modes, with durations sads and sdes. From the point of view of the modelled channel, the adsorption and the desorption processes occur, respectively, when 0 < s sads and sads < s scyc. At the stationary cyclic regime, the amounts of mass and heat transferred during the
(9)
Thus, the convection coefficients are estimated as hh ¼ Nu lf =dhyd ;
(10)
and hm ¼ Sh Df =dhyd ;
(11)
the hydraulic diameter being dhyd ¼ 4Hc, or dhyd ¼ 43m/as. The Nusselt number is considered constant due to the assumption of fully developed laminar channel flow. The thermal conductivity of the moist air is assumed to be independent of the atmospheric pressure, an assumption that is commonly adopted for gases and indicated in classical literature of heat transfer (e.g., C¸engel, 1998), as well as in Pesaran and Heiden (1994). Consequently, it is reasonable to assume
Fig. 3 e Curves of the sorption equilibrium of silica-gel with moist air.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
adsorption process are equal to those occurring during the desorption process. In fact, the modelled channel that is representative of the matrix is submitted to an initial transient process that must be started at a certain condition. The transition of mode, from adsorption to desorption, or from desorption to adsorption, is done by suddenly changing the inlet airflow conditions and reversing the airflow direction in the channel. After a certain number of adsorption/desorption cycles, the differences between two consecutive cycles are negligible, meaning that the stationary cyclic regime was achieved. The initial condition for the sequence of the cycles corresponds to the beginning of one of the modes of the cycle (adsorption or desorption), imposing uniform distributions for temperature and adsorbed water content in the desiccant and assuming that the airflows are initially in thermodynamic equilibrium with the desiccant medium. The inlet airflow conditions are chosen according to the starting mode considered, by imposing constant and uniform values of uf ¼ uin (or the corresponding mass velocity, Fm ¼ Fm,in), Tf ¼ Tin and 4v;f ¼ 4v;in . Lateral and the upper boundaries of the desiccant portion of calculation domain are considered adiabatic and impermeable, which is recognized by zero normal-gradient conditions for temperature and mass concentrations.
a
Numerical solution procedure
The differential equations are discretized via the finite-volume method adopting a fully-implicit scheme for the integration of the transient terms (Patankar, 1980). The physical domain is divided into a set of control-volumes, whose representative nodes form a non-uniform grid that is strongly refined near the solidegas interface, and towards the inlet and outlet sections of the channel. The discretization equations for the primary variables are formally equivalent. The tri-diagonal matrix algorithm (TDMA) is used to iteratively solve the set of algebraic equations for each variable field in each time-step. The values of the diffusion coefficients at the control-volume interfaces are estimated by the harmonic mean (Patankar, 1980). The nonlinearities implied by the changing properties, as well as the strong coupling between the different fields and the local constraint of the phase change require a careful underelaxation of the source-terms of the equations along the iterative calculation. Within the desiccant sub-domain, the equilibrium value of the vapour mass fraction is locally specified through Eq. (12), according to the procedure suggested in Patankar (1980). The simulation of a transient process consists of a stepby-step calculation procedure, each time-step being automatically adapted along the simulation, weighted by the time variation of the primary variables. This technique ensures a suitable representation of the transient evolutions and reduces the global time consumption of the simulations. Preliminary grid-independence tests were carried out to choose a numerical grid with a suitable refinement near the interface (Ruivo, 2005).
20 p =101325 Pa
0 ×106 (kg s -1 m-2 ) Jm
2.4.
part of the relations for the dry air, water vapour and liquid water was derived from thermodynamics tables (C¸engel, 1998) in the form of polynomial expressions (Ruivo, 2005). The properties of the air-mixture such as the specific heat and the thermal conductivity are weighted averages based on the dry air and water vapour mass fractions. Similarly, the specific heat and the thermal conductivity of the wet desiccant medium are weighted averages based on the mass fraction of each component (dry air, water vapour, adsorbed water and dry desiccant). The properties of silica-gel RD, the relations for the equilibrium condition, the heat of wetting, the adsorbed water enthalpy and the adsorption heat are indicated in Ruivo et al. (2007a). The dependences of the coefficients of surface and Knudsen mass diffusion on the temperature and on the adsorbed water content were derived after the expressions in Pesaran (1983) and are shown in Fig. 2. The pressure p does not affect these diffusion coefficients, as well as the equilibrium condition when it is written in the usual form of j ¼ f ðX[;eq Þ. Some curves of the sorption equilibrium for temperatures of 30 and 100 C at 60,000 and 101,325 Pa are shown in Fig. 3 in the form X[;eq ¼ gðT; wv Þ, being observed that, in this representation, the influence of the atmospheric pressure at high temperature is quite negligible.
L c=0.1 m
15
0.3 10 0.5
.
5
0 1.0
1.5
2.0
2.5 -1
b
20
15
p =101325 Pa
10
.
5
0 0
5
10
15
20 -1
2.5.
Evaluation of properties and coefficients
The model takes into account the changes occurring in the airflow properties and in the diffusion coefficients. The major
3.0
-2
F m,in (kg s m )
0 Jm ×106 (kg s -1 m-2 )
712
25
30
-3
F m,in /L c (kg s m )
Fig. 4 e Predicted mass transfer rate at sea-level conditions (study PS1) as a function of (a) Fm,in and Lc and (b) of the ratio Fm,in/Lc.
713
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
2.6.
Evaluation of the mass transfer rate
is regenerated. So, at steady state conditions, the mass transfer rate occurring in the adsorption zone is equal to that occurring in the desorption zone. Therefore, considering the adsorption mode, the following expression can be deduced for the mass transfer rate between both airflows, per unit of transfer area of the matrix (Ruivo et al., 2007b): Zsads _ Jv;gs ds: (14) J_m ¼ scyc
The mass transfer rate per unit of transfer area occurring in the modelled channel is evaluated by Ruivo et al. (2007a): 1 J_v;gs ¼ xc
Zxc jv;gs dx:
(13)
0
The behaviour of the modelled channel enables the prediction of the global behaviour of the hygroscopic rotor crossed by two airflows at steady state conditions, working as a desiccant wheel. The adsorption mode corresponds to the adsorption zone of the rotor matrix, where the dehumidification of the process air occurs, while the desorption mode corresponds to the desorption zone, where the rotor matrix
0
3.
Study cases and results
Four parametric studies, hereafter designated PS1, PS2, PS3 and PS4, were performed considering a desiccant wheel
a 1.05 p =101325 Pa 1.00 90,000 0.95 80,000
kp
0.90 70,000
0.85 0.80
60,000 0.75 0.70 0
5
10
15
20 -1
25
30
-3
F m,in /L c (kg s m )
b
1.05 p =101325 Pa 1.00
log(10 × k p)
90,000 0.95 80,000 70,000
linear regression
0.90
60,000 0.85 0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
log(100×L c/F m,in) Fig. 5 e Correction factor kp of the mass transfer rate for non-standard atmospheric pressure (study PS1): (a) in a common representation; (b) in logarithmic scales.
714
0.14
1.10
0.12
1.05
0.10
1.00
0.08
0.95
0.06
0.90
0.04
0.85
0.02
0.80
0.00
0.75
-0.02
b kp
a kp
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
0.70
60,000
70,000
80,000
90,000
100,000
110,000
p (Pa) Fig. 6 e Linear regressions for the coefficient akp and for the term bkp (study PS1). divided into two equal parts (sads ¼ sdes), where the adsorption and desorption zones are equal and crossed by countercurrent airflows. The compact corrugated matrix is composed by sinusoidal cross section channels, having a specific transfer area as ¼ 3198 m2 m3 and a porosity 3m ¼ 0.84. According to Ruivo et al. (2007b), the chosen matrix corresponds to cell B3 (Hcell ¼ 1.5 mm, Pcell ¼ Psin ¼ 3 mm, Ep ¼ 0.1 mm, Hsin ¼ 1.27 mm), the representative channel of the matrix being modelled with Hp ¼ 0.05 mm, Hc ¼ 0.263 mm and with a Nusselt number of 2.09. For the highest value of inlet mass velocity considered in the present work (Fm,in ¼ 3 kg s1 m2), the Reynolds number is 237, which confirms the validity of the assumption of laminar regime airflow. In a first parametric study (PS1), the inlet states of both airflows are characterised by Tin,ads ¼ 30 C, wv,in,ads ¼ 0.01 kg kg1, and Tin,des ¼ 100 C, wv,in,des ¼ 0.01 kg kg1. The rotor speed was fixed to 7.2 rotations per hour, that corresponds to scyc ¼ 500 s. Concerning the main goal of this case 1.1
k*p
1.0
0.9
study, a wide range of values of the atmospheric pressure p with practical interest was considered (60,000, 70,000, 80,000, 90,000 and 101,325 Pa). Three values for the channel length Lc were chosen, namely 0.1, 0.3 and 0.5 m, the lower value being representative of a matrix with a short channel length and the last one with a long one. The mass airflow rates per unit of frontal area (mass velocity) entering in both zones of the desiccant wheel were considered equal to Fm,in and three different values were considered in the parametric study (1, 2 and 3 kg s1 m2). The simulation started with the adsorption mode, departing from an equilibrium condition for the desiccant layer defined by the initial uniform distributions of temperature and of adsorbed water content, respectively, with the values of 100 C and 0.0121 kg kg1. The numerically predicted results for the cyclic behaviour of the modelled channel were used to derive the behaviour of a hygroscopic matrix rotating through the adsorption and the desorption zones and continuously exchanging heat and mass with the two air streams, similarly to what occurs in a hygroscopic wheel at steady state conditions. The results of this parametric study are plotted in Figs. 4e7. Fig. 4(a) refers to the predicted mass transfer rate per unit of 0 transfer area J_m for sea-level conditions ( p ¼ 101,325 Pa). As 0 expected, it is seen that J_m becomes larger by increasing the mass velocity Fm,in and decreasing the channel length Lc.
0.8
Table 2 e Parameters of the linear regression akp, bkp, R2kp for the different lines of Fig. 5(b). p (Pa)
0.7 0.7
0.8
0.9
1.0
1.1
kp
Fig. 7 e Correlation between the kp values estimated by Eq. (18) and the kp values obtained with the numerical results (study PS1).
60,000 70,000 80,000 90,000 101,325
akp
bkp
R2kp
0.1357 0.094 0.0589 0.0291 0
0.7877 0.8553 0.9106 0.9564 1
0.9949 0.9915 0.9884 0.9851 1
715
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
Table 3 e Parameters of the linear regressions of akp and bkp. aa
ba
R2a
ab
bb
R2b
3.2693 106 3.1777 106 1.2611 106
0.32595 0.32118 0.12664
0.9903 0.9994 0.9934
5.109 106 10.44 106 4.104 106
0.4919 0.8956 0.5908
0.987 0.9844 0.9912
Parametric study PS1 PS2 PS3
0 Comparing the curves J_m w Fm,in, it is observed that the same 0 J_m value is achieved with Fm,in ¼ 1 kg s1 m2, for Lc ¼ 0.1 m, and with Fm,in ¼ 3 kg s1 m2, for Lc ¼ 0.3 m, i.e., points with the same value of the ratio Fm,in/Lc. This behaviour corroborates the theory for rotary heat exchangers where the nondimensional NTU parameter is a function of the ratio between the transfer area and the mass flow rate. Consequently, the predicted influences of the mass velocity and of the channel length on the mass transfer rate of the desiccant matrix at standard atmospheric pressure are alternatively condensed in a single curve in Fig. 4(b). The effect of a pressure decrease on the mass transfer rate 0 can be expressed as a correction factor kp ¼ J_m =J_m , which is represented in Fig. 5(a). Fig. 5(b) is an alternative representation of Fig. 5(a) using transformed logarithmic scales. These new curves evidence a linear tendency, suggesting the use of the linear regression which is represented by the continuous
lines in Fig. 5(b). So, the correction factor can be estimated by the following expression: Lc þ bkp : (15) log 10kp ¼ akp log 100 Fm;in The values of the coefficient akp, the term bkp and the correlation factor R2kp for each line in Fig. 5(b) are listed in Table 2. The calculated values of the correlation factor indicate a good fitting of the linear regression in all cases. Using the data of Table 2, the dependences of akp and bkp on pressure are represented in Fig. 6, and they are well represented by, respectively: akp ¼ aa p þ ba
(16)
and bkp ¼ ab p þ bb :
(17)
a 1.2 p =60,000 Pa 1.0
kp
τ cyc=100 s 250
0.8
500 750
1000
0.6 0
5
10
15
20 -1
25
30
-3
F m,in /L c (kg s m )
b
1.1 p =80,000 Pa
1.0
kp
τ cyc=100 s 250
500
0.9 750
1000
0.8 0
5
10
15
20 -1
25
30
-3
F m,in /L c (kg s m ) Fig. 8 e Correction factor of the mass transfer rate (study PS4) for different cycle durations at a non-standard atmospheric pressure of: (a) 60,000 Pa and (b) 80,000 Pa.
716
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
The values of the coefficients aa and ab, terms ba and bb and correlation factors R2a and R2b are listed in Table 3. Manipulating Eq. (15) and taking into account Eqs. (16) and (17), a first expression is obtained to estimate the correction factor: kp ¼ 0:1 10½ðaa pþba Þ logð100 Lc =Fm;in Þþab pþbb :
(18)
Fig. 7 shows the correlation between the estimated values kp with Eq. (18) and the kp values obtained after the results of the numerical modelling, i.e., kp ¼ 0:9971kp þ 0:0026, with a root mean square deviation RMSD ¼ 0.0092 and a correlation factor R2 ¼ 0.9837. According to Fig. 5(a), it may be concluded that the effect of the atmospheric pressure on the mass transfer rate is practically negligible for values of the ratio Fm,in/Lc between 2 and 3, approximately. For Fm,in/Lc > 3, the correction factor kp becomes smaller with the decrease of atmospheric pressure. The lower values of kp, meaning a reduction of over 25% in ‘, are observed for the case with higher ratio Fm,in/Lc when p ¼ 60,000 Pa. In a second study (PS2), the values of the correction factor are determined for cases with fixed values of j, instead of a fixed value of the water vapour content as in the preceding parametric study. It is important to stress that the parameter (j ¼ pv/pvs) corresponds exactly to the relative humidity, for states of the moist air when the temperature is equal or lower than the particular value of the water saturation temperature Tsat at the atmospheric pressure p. Moreover, the relative humidity concept is not applicable when the moist air temperature is greater than Tsat, a situation that is commonly found in regeneration airflow inlet conditions. The inlet states of both airflows in the present parametric study are characterised by Tin,ads ¼ 30 C, jin,ads ¼ 0.377, and Tin,des ¼ 100 C, jin,ads ¼ 0.016, the water vapour content and the vapour mass fraction increasing with the atmospheric pressure (simulated values: 60,000, 70,000, 80,000, 90,000 and 101,325 Pa).
Following a procedure similar to that of the study PS1, the 0 results obtained for the correction factor (kp ¼ J_m =J_m ) were analysed. The achieved values of the coefficients aa and ab, terms ba and bb and correlation factors R2a and R2b are listed in Table 3. The correlation between the values kp estimated with Eq. (18) and the kp values is acceptable. The determined values of the root mean square deviation and the correlation factor were RMSD ¼ 0.006773773 and R2 ¼ 0.9972, respectively. In the previous parametric studies the mass velocity Fm,in was used as inlet airflow condition. Now, the correction factor values are determined for a set of cases (SC3) with a fixed and equal value of the airflow velocity, uin, in both inlet process and regeneration zones. All the other inlet conditions are those of the first study (PS1). The purposed expression for the calculation of kp is: kp ¼ 0:1 10½ðaa pþba Þ logð100 Lc =uin Þþab pþbb :
(19)
The calculated values of the coefficients aa and ab, and terms ba and bb are listed in Table 3, as well as the correlation factors R2a and R2b . In the cases of the study PS3, the correlation factor and the root mean square deviation of the linear correlation between kp and kp were R2 ¼ 0.9922 and RMSD ¼ 0.0131, respectively. In a fourth parametric study (PS4), the influence of the rotor speed on the correction factor kp was investigated. In all previous parametric studies, the duration of the cycle was fixed at 500 s. A wide range of values of scyc was considered (100, 250, 500, 750 and 1000 s) in the numerical simulations, as well as three values of the pressure (60,000, 80,000 and 101,325 Pa). The same values of the ratio Fm,in/Lc and of inlet state conditions of the study PS1 were considered in the present parametric study. The values of kp were determined and the results plotted in Fig. 8(a) and (b), respectively, for 60,000 Pa and 80,000 Pa. It is observed that at high values of scyc (low rotation speeds of the wheel), the correction factor is smaller than for the highest wheel rotation speed, and also
1.1
1.0
k p,
cyc
0.9
0.8
0.7
0.6 0.6
0.7
0.8
0.9
1.0
1.1
kp Fig. 9 e Correlation between the correction factor values estimated by Eq. (20) and those obtained with the numerical results (study PS4).
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
that this difference increases with the decreasing in the atmospheric pressure. The shape of the curves in Fig. 8(a) and (b) suggests that it is easy to derive a correlation for kp. Taking as a reference the correction factor for scycref ¼ 500 s, kp,cycref and manipulating the obtained data, the following correction expression was derived: kp;cyc ¼ kp;cycref þ 0:5077 5 105 p 9 107 s2cyc 2:06 104 scyc þ 0:7998 :
(20)
Fig. 9 shows the correlation between the values kp,cyc estimated with Eq. (20) and the kp values obtained after the results of the numerical modelling of the cases of this fourth parametric study. The corresponding values of the root mean square deviation and of the correlation factor were RMSD ¼ 0.00878 and R2 ¼ 0.9904, respectively. The processing and analysis of results of the different set of parametric studies led to the conclusion that it is relatively easy to develop correlations to estimate the pressure correction factor kp for different operating conditions characterised by a wide ranges of the atmospheric pressure, of the rotation speed and of the ratio Fm,in/Lc.
4.
Conclusions
In the present work, a numerical model that predicts the heat and mass transfer occurring in a channel of a desiccant wheel was used to evaluate the influence of the atmospheric pressure on the mass transfer rate. Parametric studies were performed for different values of the channel length and of both airflow rates in a range of values with practical interest, considering also a large range of values of the rotation speed of the wheel. The results of the parametric study PS1 indicate that the decreasing effect of increasing altitude on the mass transfer rate can be well expressed by a correction factor kp expressed by a simple correlation that may be a useful estimator for the selection of desiccant wheels for non-standard atmospheric operating conditions. The derived correlations of kp for the conditions of the others parametric studies (PS2, PS3 and PS4) were also tested, a good agreement being achieved between the estimated correction factor and the one calculated after the numerical results. There are many other parameters than those considered in the present study that influence the mass transfer rate occurring in desiccant wheels (e.g., Ruivo, 2005; Chung and Lee, 2009; Ruivo et al., 2007b, 2008b). Therefore, additional effort is still needed to verify the validity of this corrective procedure for different inlet conditions, partitions of the desiccant wheel, as well as matrix specific transfer area and porosity. Furthermore, the channel walls can also be of different thicknesses and composed of different desiccant materials.
Appendix. Supplementary data Supplementary data related to this article can be found online at doi:10.1016/j.ijrefrig.2010.11.013.
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references
Banks, P.J., 1985a. Prediction of heat and mass regenerator performance using nonlinear analogy method: Part 1 e basis. ASME J. Heat Transfer 107, 222e229. Banks, P.J., 1985b. Prediction of heat and mass regenerator performance using nonlinear analogy method: Part 2 e comparison of methods. ASME J. Heat Transfer 107, 230e238. C¸engel, Y., 1998. Heat Transfer e A Practical Approach. McGrawHill, Boston. Cejudo, J.M., Moreno, R., Carrillo, A., 2002. Physical and neural network models of a silica gel desiccant wheel. Energy Build. 34, 837e844. Chung, J.D., Lee, D.Y., 2009. Effect of desiccant isotherm on the performance of desiccant wheel. Int. J. Refrigeration 32, 720e726. Chung, J.D., Lee, D.Y., Yoon, S.M., 2009. Optimization of desiccant wheel speed and area ratio of regeneration to dehumidification as a function of regeneration temperature. Solar Energy 83, 625e635. Dai, Y.J., Wang, R.Z., Zhang, H.F., 2001. Parameter analysis to improve rotary desiccant dehumidification using a mathematical model. Int. J. Therm. Sci. 40, 400e408. Elsayed, S.S., Hamamoto, Y., Akisawa, A., Kashiwagi, T., 2006. Analysis of an air cycle refrigerator driving air conditioning system integrated desiccant system. Int. J. Refrigeration 29, 219e228. Gao, Z., Mei, V.C., Tomlinson, J.J., 2005. Theoretical analysis of dehumidification process in a desiccant wheel. Heat Mass Transfer 41, 1033e1042. Golubovic, M.N., Worek, W.M., 2004. Influence of elevated pressure on sorption in desiccant wheels. Numer. Heat Transfer A 45, 869e886. Harshe, Y.M., Utikar, R.P., Ranade, V.V., Pahwa, D., 2005. Modeling of rotary desiccant wheels. Chem. Eng. Technol. 28, 1473e1479. Maclaine-Cross, I.L., Banks, P.J., 1972. Coupled heat and mass transfer in regenerators-prediction using an analogy with heat transfer. Int. J. Heat Mass Transfer 15, 1225e1242. Mills, A., 1985. Heat and Mass Transfer. Irwin, London. Niu, J.L., Zhang, L.Z., 2002. Effects of wall thickness on the heat and moisture transfer in desiccant wheels for air dehumidification and enthalpy recovery. Int. Commun. Heat Mass Transfer 29, 255e268. Patankar, S., 1980. Numerical Heat Transfer and Fluid Flow. In: Hemisphere. McGraw-Hill, Washington, DC. Pesaran, A., 1983. Moisture transport in silica gel particle beds. PhD Thesis, University of California, Los Angeles. Pesaran, A.A., Heiden, R., 1994. The influence of altitude on the performance of desiccant-cooling systems. Energy 19, 1165e1179. Ruivo, C.R., 2005. Modelac¸a˜o nume´rica dos feno´menos de transfereˆncia de calor e de massa em rodas higrosco´picas. PhD Thesis, University of Coimbra, Coimbra, Portugal (in Portuguese). Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2006. Analysis of simplifying assumptions for the numerical modeling of the heat and mass transfer in a porous desiccant medium. Numer. Heat Transfer A 49, 851e872. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2007a. On the behaviour of hygroscopic wheels: part I e channel modeling. Int. J. Heat Mass Transfer 50, 4812e4822. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2007b. On the behaviour of hygroscopic wheels: part II e rotor performance. Int. J. Heat Mass Transfer 50, 4823e4832. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2008a. On the validity of lumped capacitance approaches for the numerical prediction
718
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 7 0 7 e7 1 8
of heat and mass transfer in desiccant airflow systems. Int. J. Therm. Sci. 47, 282e292. Ruivo, C.R., Costa, J.J., Figueiredo, A.R., 2008b. Numerical study of the behavior of an elementary desiccant layer of a hygroscopic rotor. Numer. Heat Transfer A 53, 1037e1053. Sphaier, L.A., Worek, W.M., 2004. Analysis of heat and mass transfer in porous sorbents used in rotary regenerators. Int. J. Heat Mass Transfer 47, 3415e3430.
Sphaier, L.A., Worek, W.M., 2006. Comparisons between 2-D and 1-D formulations of heat and mass transfer in rotary regenerators. Numer. Heat Transfer B 49, 223e237. Zhang, L.Z., Niu, J.L., 2002. Performance comparisons of desiccant wheels for air dehumidification and enthalpy recovery. Appl. Therm. Eng. 22, 1347e1367. Zhang, X.J., Dai, Y.J., Wang, R.Z., 2003. A simulation study of heat and mass transfer in a honeycomb rotary desiccant dehumidifier. Appl. Therm. Eng. 23, 989e1003.