Applied Energy 86 (2009) 762–771
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Heat and mass transfer modeling in rotary desiccant dehumidifiers Pascal Stabat, Dominique Marchio * Ecole des Mines de Paris – Centre Energétique et Procédés, 60 boulevard Saint Michel, F 75272 Paris cedex 06, France
a r t i c l e
i n f o
Available online 26 September 2008 Keywords: Desiccant Adsorption Heat and mass transfer Modeling Air conditioning
a b s t r a c t A desiccant wheel model has been developed in the aim to be adapted to building simulation tools. This model fulfils several criteria such as simplicity of parameterization, accuracy, possibility to characterize the equipment under all operation conditions and low computation time. The method of characteristics has been applied to the heat and mass transfer partial differential equations. This transformation provides new equations which are similar to those of a rotary heat exchanger. Then, the model is described by the Effectiveness-NTU method and it is identified from only one nominal rating point. The model has been compared to experimental and manufacturers’ data for a broad range of operating conditions. A good agreement has been found. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Desiccant evaporative cooling systems appear as an alternative to classical air-conditioners. The principle consists in drying the air in order to get a high potential of evaporative cooling of air. This technique is refrigerant free and uses few of electricity. In the other hand, as it is necessary to regenerate the desiccant wheel, thermal energy is required to heat up the wheel at temperatures in the range of 50–100 °C. The use of solar energy or waste heat can make this technique interesting. HVAC designers do not have any model to size and estimate the energy consumption of such systems. So, a model for desiccant wheels has been developed in order to be implemented in building dynamic simulation tools. This model aims to answer the requirements of such tools, that are: – accuracy; – low computation time; – simplicity of parameterization (the data available are often limited to those of manufacturers’ catalogues); – polyvalence (the model should characterize the equipment in all operating conditions such as variable air flow rate or variable regeneration temperature). 2. Fundamental heat and mass rate equations 2.1. Description of the desiccant wheel to be modeled In desiccant cooling systems, the rotary wheel is generally separated in two equal parts, one for the process, the other one for the * Corresponding author. Tel.: +33 1 40 51 91 80; fax: +33 1 46 34 24 91. E-mail address:
[email protected] (D. Marchio). 0306-2619/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.06.018
regeneration. Among the desiccants, the silica gel and the lithium chloride are the most used since they have a high capacity of adsorption on all the range of relative humidity of ambient air compared to molecular sieve, for example (Fig. 1). The evolution of vapor pressure at the desiccant surface in function of the moisture content in the wheel is shown in the Fig. 2. This Figure has been obtained based on a numerical model [2] solving the equations in Section 2.2 based on a second order Runge Kutta method. The curve shows the evolution of vapor pressure at the desiccant surface averaged on all the width of the wheel for each angular position. The isotherms are shown for temperatures of 30, 50, 70 and 90 °C. One can note a heating zone (from C to D) at constant adsorbed humidity corresponding to the hot air introduced in the wheel. The increase of the partial vapor pressure implies a transfer of moisture in the desiccant towards the air (from D to A). In the process side, one can first note a cooling zone (from A to B) which allows to decrease the temperature and water vapor pressure at desiccant surface. The cooling due to fresh air introduction is almost at constant adsorbed moisture. Then, the vapor pressure increases up to a level for which the desiccant can not adsorb anymore the water vapor contained in the air (from B to C). As a consequence, a fraction of process air flow rate is used for cooling of the wheel. This fraction is hot and almost no dehumidified. For this reason, it is common to install a purge section (Fig. 3) on desiccant wheels which allow to recover this hot air for regeneration and so reduce the average temperature and humidity ratio of outlet process air. The fraction of process air which passes through the purge section represents 2–4% of air flow rate in desiccant wheels. 2.2. Governing equations The desiccant wheel is modeled based on the thermal and mass balance equations in a small volume element of the wheel (Fig. 4)
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Nomenclature surface (m2) fluid capacitance rate (kgd.a. s1) ratio of fluid capacitance rate between regeneration and process (–) ratio of desiccant capacitance rate to minimum air capacitance rate (–) specific heat (J kg1 K1) wheel fraction used for regeneration (–) potential functions ([arbitrary dimension]) Fi (T,w)
A C C* Cr* cp f Fi H
matrix enthalpy J kg1 d air enthalpy J kg1
h
d:a:
1
2
convective heat transfer coefficient (W K m ) convective mass transfer coefficient (kgd.a. m2 s1) water vaporiation heat at 0°C (J kg1) thickness of the wheel (m) sorption heat (J kg1) differential heat of wetting (negative par convention) (J kg1) air mass flow rate (kgd.a. s1) mass of desiccant (kg) rotation speed of the wheel (tr s1) number of transfer units (–) partial pressure of water vapor (Pa) partial pressure of water vapor at saturation (Pa) Reynolds number (–) time (s) temperature (K) velocity (m s1) overall mass transfer coefficient (kgd.a. s1) 1 desiccant moisture content (kgv kgd )
hc hm hfg L Ls Lw _a m Md N NTU pv pvsat Re T T U UA W
in steady state conditions. The following equations are expressed _ a t and position x ¼ Lz. from dimensionless variables of time s ¼ mM d The assumptions are as follows:
Moisture content (g of water/ 100 g of desiccant)
1. the axial heat conduction and water vapor diffusion in the air are neglected; 2. the diffusion and capillarity in the desiccant matrix are assumed to be negligible in the axial direction (z);
60 isotherm at T = 25 ˚C
W X Z
1
humidity ratio (kgv kgd:a: ) axial position divided by the wheel thick (z/L) (–) axial position in the wheel (M)
Greek symbols a angular position e effectiveness
ah
aW
oT a ow a h ooT-aa
-
ðoH=o-Þ
k
1 ðoha =owa ÞT
r
ðoH=oT a Þðoha =oT a Þwa a oT o- wa _ at m Md
m s q /
C
T
air density (kg m3) relative humidity (%) ratio of air mass to desiccant mass (–)
Indices/exponents A air Eq equilibrium M matrix Min minimum value pi, po process inlet, outlet Pn ideal process outlet (infinite heat and mass transfer coefficients) T total V vapor ri, ro regeneration inlet, outlet
3. the temperature and humidity ratio gradients in the radial direction (r) are neglected, otherwise T = T(z, a) and w = w(z, a) are independent of the wheel radius; 4. the sorption is supposed without hysteresis; 5. the pressure drop in the wheel is neglected; 6. the inlet air temperature and humidity ratio are uniform; 7. the channels have all the same geometry; 8. the thermo physical properties of the matrix, the air and water vapor are constant in all the operating conditions; 9. the conditions are in steady state. The mass conservation equation is expressed as follows:
50
40
LiCl
oW owa owa þC þ ¼0 os os ox
Silicagel Molecular sieve 4A
The mass transfer equation can be written as:
30
oW hm A ¼ ðwa weq Þ _a os m
20
ð2Þ
The energy conservation equation is expressed as follows:
oH oha oha ¼0 þC þ os os ox
10
0
ð1Þ
ð3Þ
The heat transfer equation can be written as:
0
10
20
30 40 50 60 70 Relative humidity (%)
Fig. 1. Isotherms of different materials [1].
80
90
100
oH hm A hc A ¼ ðwa weq Þðhfg þ cpv T a Þ þ ðT T m Þ _a _a a os m m
ð4Þ
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3500
α=270°
90°C
Vapor pressure at desiccant surface (Pa)
reg
3000
α=0°
α=200°
70°C
D
α=180° process
2500
α=90°
2000 α=0°
50°C
A
1500 α=180°
1000 C
500
0 0.00
B
30°C
α=20°
0.04
0.08 Desiccant moisture content (kg/kg)
0.12
Fig. 2. Vapor pressure evolution at desiccant surface.
/ðT eq ; weq Þ ¼ C 1 W þ C 2 W 2
Dehumidified air
ð6Þ
2.3.2. Sorption heat The matrix enthalpy is expressed as follows:
H ¼ cpm T m þ cpv WT m þ
Z
W
Lw W
ð7Þ
0
The analysis of silica gel data [4] shows that the heat of wetting can be expressed as:
Lw ¼ hfg Dn expðkWÞ
Purge section
2.3.3. Convective heat and mass transfer coefficients The convective heat and mass transfer coefficient correlations from Bullock and Threlkeld [5] have been determined for water vapor adsorption on a silica gel. These correlations will be used afterwards:
Air of regeneration Fig. 3. Schematic of a desiccant wheel with a purge section.
Assuming a Lewis number, Le, equals to unity, the Eq. (4) becomes:
oH hm A ¼ ðh heq Þ _a a os m
ð8Þ
ð5Þ
The desiccant moisture content, W, and the matrix enthalpy, H, in the Eqs. (1)–(5) are determined based on the properties of the adsorbent. 2.3. Adsorbent properties 2.3.1. Sorption isotherm The adsorption is characterized by the moisture content of the adsorbent, W, which varies with the equilibrium temperature and the vapor pressure at the adsorbent surface. The adsorption isotherm depends on the adsorbent (Fig. 1). Moreover, one can find sometimes a hysteresis phenomenon. The isotherms are defined experimentally. Afterwards the isotherm proposed by Mathiprakasam and Lavan [3] for silica gels is used:
hc ¼ 0:671qua cpa Re0:51 hm ¼ 0:704qua Re0:51
ð9Þ ð10Þ
One can notice that Eqs. (9) and (10) allows to determine the Lewis number, hmhccpa , which is close to 1. 3. Solving method of fundamental equations Several models have been developed based on different approaches. 3.1. Empirical approach Some correlations [6–8] have been developed for desiccant wheels based on performance curves of manufacturers or experimental data. These correlations allow to assess the performance of a particular wheel in different conditions of temperature and humidity ratio at process and regeneration inlets. However, good results are obtained only for the desiccant
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α
Air of regeneration
α+dα
heater
z Air Dehumidified air
Process air One channel
α
α+dα
W
z
wa Process air
Dehumidified air
z=0
z=L Fig. 4. System model.
wheel which has been correlated and only for the domain of temperature, humidity ratio and air flow rate where the correlation has been established. 3.2. Theoretical approaches Finite difference models such as [9] are often used however they requires a very long computation time and the input data of these models are often unknown by building simulation tool users. Mathiprakasam and Lavan [3] have developed an analytical model based on a Laplace transformation of linearized heat and mass equations. The main drawback of this model is its difficulty to be parameterized. The method of characteristics [10–12] has been used for transforming the coupled non linear partial differential equations in a set of uncoupled differential equations. The new equations describe new independent variables which are called characteristic potentials. The characteristic potentials are associated with new parameters, ci, which can be assimilated to specific heat ratios. Thus, the new set of equations is analogous to the heat transfer equations in a non-hygroscopic rotary heat exchanger. The problem is then solved by analogy by using the solution of Kays and London [13] for rotary heat exchanger. The advantages of this method are: a good understanding of operation, in particular by an easy representation in psychrometric charts; a facilitated design by a performance prediction based on parameters and relationships similar to those used for heat exchangers (effectiveness and Number of Transfer Units); a rapid computation time. However, the parameterization of the model is still a problem. Howe [14] has simplified the model based on the analogy method. In his model, the effectiveness of the two characteristic potentials
are assumed to be constant and the potentials are described by the analytical formulas for silica gel defined by Jurinak [15]. The only drawback of this simplified model is that default values for the effectiveness should be used since they are unknown by the users of the model. Since no model fulfils completely the requirements defined above, a new model has been developed based on the method of characteristics. The model has been adapted in order to be parameterized only with the data provided by desiccant wheel manufacturers. 4. Solutions of combined heat and mass equations The equation system (1-2-3-5) forms a set of coupled partial differential hyperbolic equations. Two common solving techniques of hyperbolic equations exist: finite difference and characteristic method. The second technique will be used hereafter in order to get equations similar to those of rotary heat transfer equations. In order to make the analogy, further assumptions should be done: the Lewis number is assumed to be equal to 1; the derivatives of potential functions by enthalpy and humidity ratio are supposed to be constant; each ci is assumed to be a function of its associated Fi only. 4.1. Transformed equations The energy and mass conservation equations can be transformed in a set of uncoupled wave equations by introducing potential functions instead of enthalpy and humidity ratios. In the following, one first considers an ideal case where heat and mass transfer coefficients are infinite. Then, the method is extended to a non ideal case.
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In an ideal case, the matrix and the air are in a thermodynamic equilibrium, otherwise Tm = Ta = T and weq = wa = w. The governing equations may be transformed Fi functions [10] which satisfy the following equation:
oF i oF i ¼ ai ow T oT w
with i ¼ 1; 2
ð11Þ
and
ai kah oF i oF i ¼ 0 with i ¼ 1; 2 þrþC þ m os ox
ð12Þ
In which Fi are termed characteristic potentials and the ai are the solutions of the characteristic polynomial as follows:
a2i ai ðkah þ rm þ aW Þ þ ah kaW ¼ 0
ð13Þ
The Eqs. (11) and (12) form the characteristic equations equivalent of the Eqs. (1) and (3). Detailed calculations are presented in Appendix A. 4.2. Analogy with heat exchanger To get the analogy with a rotary sensible heat exchanger, the following assumptions are done:
oF eq oF a i ¼ i oha wa oha wa oF eq oF ai i ¼ ow ow a ha
with i ¼ 1; 2
ð14Þ
with i ¼ 1; 2
ð15Þ
a ha
This assumption is satisfied in the case where the air and the matrix are in a thermodynamic equilibrium (Tm = Ta and weq = wa) or if (oFi/ owa)ha and (o Fi/oha)wa are constant. The equilibrium condition would be achieved if the heat and mass transfer coefficients are infinite. This condition is not completely fulfilled since the heat and mass transfer coefficients are large but not infinite. The condition on the partial derivatives implies that the Fi functions are linear
combinations of the enthalpy and humidity ratio. Otherwise, the Fi functions should be parallel in a (h, w) chart. Fig. 5 shows that the second condition is not completely fulfilled since the Fi functions are not absolutely straight in a (h, w) chart. Since both of the conditions are almost satisfied, the assumption will be taken into account in the following. eq oF oF a With these assumptions, one can thus write that oTia jwa ¼ oT ia jwa . oF i By multiplying Eq. (A-6) by oT jw , one obtains:
ci
oF eq oF a oF a i þ C i þ i ¼ 0 with i ¼ 1; 2 os os ox
By multiplying equations (2) and (5) by and by adding them, one can find:
ð16Þ oF i j ow h
and
oF i j , oh w
respectively
oF eq i with i ¼ 1; 2 ¼ NUTw F ai F eq i os ai kah hm A with ci ¼ r and NUTw ¼ _a m m
ci
ð17Þ ð18Þ
The pair of Eqs. (16) and (17) forms now a system similar to those of a rotary heat exchanger. The ratio of specific heat is replaced by the ci. These ci depend on the two characteristic functions, Fi. This implies that the Eqs. (16) and (17) are not completely uncoupled. One should assume that the ci depend only of their associated potential function even if this condition is not perfectly fulfilled as shown on Fig. 6. The ci and the Fi of Fig. 6 have been calculated for a silica gel with a sorption isotherm according to Eq. (6) with coefficients C1 = 124 and C2 = 137 and a wetting heat according to Eq. (8) with coefficients Dn = 0.28 and k = 10.28. The ci should be parallel to their associated Fi on a psychrometric chart, but it is not the case. In the following, average ci will be considered in the effectiveness calculations. With these assumptions, the equations of the characteristic potentials can be solved for each Fi by analogy with rotary heat exchanger. The analogy method is summed up in the Table 1. The effectiveness of the two potential functions can be written as for a heat exchanger:
0.03
Humidity ratio (kg/kg)
0.025
0.02
F1 F2 0.015
0.01
F1 0.005
F2 0 40000
50000
60000
70000
80000
90000
100000
110000
enthalpy (J/kg) Fig. 5. Potential functions for a silica gel in a (h, w) diagram.
120000
130000
140000
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0.030
F1
F2
γ2=50
γ2=30
γ1=0.2
γ2=20
humidity ratio (kg/kg)
γ2=10
0.020 γ2=5 γ1=0.3
γ1=0.4
0.010 γ1=0.5 γ1=0.6 γ1=0.8
0.000 20
γ2=2
γ1=0.7
30
40
50
60
70
80
90
100
temperature (°C) Fig. 6. Potential functions and ci for a silica gel in a psychrometric chart.
Table 1 Analogy between the desiccant wheel and a rotary non-hygroscopic heat exchanger Sensible heat exchanger [13]
Heat exchanger in Fi
i: analogous potential j: wheel side (process or regeneration) _ ja ðcpa þ w cpv Þ Cj ¼ m
_ ja C ij ¼ m
C r
N i C ri ¼ Mm_dmin cmin a ij or C ri ¼ min C irj ¼ M_d jN c
¼
Cr C min
¼
M cm N C min
min C ¼ CCmax
1 NTUT0 ¼ C min
h
1 ðhc AÞj
þ ðhc
e ¼ ecf ðNTUT0 ; C Þ 1
1 AÞ3j
C i ¼
i1
1 1:93 9ðC r Þ
NTUm 0 ¼
ma
C imin C imax 1 C imin
h
1 ðhm AÞj
þ ðhm
ei ¼ ecf ðNTUm 0 ; Ci Þ 1
1 AÞ3j
1 1:93 9ðC ri Þ
i1
ecf is the effectiveness of a counter flow heat exchanger,
ecf is the effectiveness of a counter flow heat exchanger,
Or at low rotation speed (C r 6 0:4Þ,
Or at low rotation speed (C r 6 0:4Þ,
e ¼ C r
e ¼ C r
e1 ¼
C 1p F 1po F 1pi C 1min F 1ri F 1pi
ð19Þ
e2 ¼
C 2p F 2po F 2pi C 2min F 2ri F 2pi
ð20Þ
The intersection point between the two potential functions, F1pi and F2pi, termed pn on the Fig. 7, corresponds to an ideal case where the heat and mass transfer coefficients are infinite. The outlet state of process is located at the intersection of the potential functions F1px and F2px calculated by the effectiveness relations (19) and (20) which accounts for the real case. One can notice on Fig. 7 that the first potential function (corresponding to the positive root of Eq. (13)) is close to a constant enthalpy line and the second one (corresponding to the negative root of Eq. (13)) is close to a constant relative humidity line.
5. Analogy model adaptation for building simulation tools The users of building simulation tools are often lacking in technical data on heating, ventilation and air conditioning systems. So, the developed model of desiccant wheels should be easy to parameterize through operating data provided in manufacturers’ catalogues. For the moment, the model requires parameters such as the rotation speed of the wheel, the mass of desiccant, the heat exchange surface, the heat transfer coefficient, which are generally unknown. So, the analogy model should be adapted in order to define these parameters thanks to operating data of the desiccant wheel. In this aim, default values are provided for the potential functions, Fi, and specific heat ratios, ci. Then, the model is completely parameterized thanks to an operating point.
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humidity ratio (kg/kg)
0.030
0.020
F2pi
0.010
F1ri
ri
pi F2po
F1po F1pi
po
F2ri pn
0.000 20
30
40
50
60
70
80
90
100
Temperature (°C) Fig. 7. Potential functions in a psychrometric chart for the determination of the outlet process air state.
5.1. Selection of Fi and ci
5.2. Model parameterization
Banks [16] has proposed analytical expressions of characteristic potentials. Here, default potential functions have been set based on the study of manufacturers’ catalogue data [17,18] and experimental data [19]. Four types of desiccant wheel have been studied. Three wheels include a section purge. Three wheels are covered with silica gel, the last one with lithium chloride. The analogy with a heat exchanger assumes that the balance on the Fi functions is respected, thus:
In order to determine the outlet states of a given desiccant wheel, the effectiveness related to the potential functions should be assessed. As shown on Table 1, the effectiveness depend on the product of desiccant mass by rotation speed Md N, and the product of the mass transfer coefficient by the exchange surface, hm S. These data are generally unknown. The aim is now to determine these unknown parameters thanks to one operating point of the desiccant wheel. With an operating point and the default potential functions, one can calculate the two effectiveness based on Eqs. (19) and (20). Concerning the first effectiveness relation, Maclaine-Cross [10] has shown by numerical calculations that if NTUo P 4.5, C* 6 1, Cr* 6 0.4 and the ratio of mass transfer coefficients between process side and regeneration side is equal to 1, the first effectiveness can be reduced to:
C 1p ðF 1po F 1pi Þ ¼ C 1r ðF 1ro F 1ri Þ
ð21Þ
C 2p ðF 2po
ð22Þ
F 2pi Þ ¼
C 2r ðF 2ro
F 2ri Þ
The default potential functions have been set in order to comply with conditions (21) and (22) and to be as much as possible representative of the average characteristics of desiccant wheels. Only the second potential analytical function of Banks [15] answers to the conditions (21) and (22). Furthermore, this function fits quite well with experimental data, so it has been selected. Concerning the first potential function, the enthalpy function has been chosen since it fulfils the conditions (21) and (22) and it fits quite well with experimental data.
F1 ¼ h
ð23Þ
ð273:15 þ TÞ1:5 1:1w0:08 F2 ¼ 6360
ð24Þ
Moreover, Maclaine-Cross [10] has set a technique for the determination of the average ci. However, since the model turns out to be weakly sensitive to the choice of the ci, default average values on the operating range of desiccant wheels are taken as follows:
c1 ¼ 0:3 c2 ¼ 30
M N
e1 ¼ C r1 ¼ _dmin c1min ma
ð27Þ
With only one operating point, the product MdN can be assessed since a default value is used for c1min . The Eq. (27) can be used since the required conditions for its use are generally fulfilled. Then, the Number of Transfer Units or more precisely the overall mass transfer coefficient should be determined. In this aim, the effectiveness related to the second potential function at one operating point is used. Contrary to the first effectiveness which is close to 0.1, the second one is near 0.8 such as the Kays and London effectiveness relation [12] should be used (see Table 1). For each side of the wheel, the overall mass transfer coefficient can be expressed as follows:
ð25Þ
UAm p ¼ hm AT ð1 f Þ
ð28Þ
ð26Þ
UAm r ¼ h m AT f
ð29Þ
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By considering the Eq. (10) and assuming the dynamic viscosity is constant, one can rewrite the overall mass transfer coefficients such as: 0
0:51 _ 0:49 UAm p ¼ hm AT map ð1 f Þ 0
UAm 0 ¼
#
C 2min 1 C 2 _ 0:49 m ap ð1 f Þ 0 1 1 1 C 2 C B1 e2 1 9ðC Þ1:93 B C r2 ln B 1 C @ A 1 1 e2 1 1:93 9ðC r2 Þ 1
0:51
"
Temperature of inlet process air
Humidity ratio of inlet process air (wpi)
Humidity ratio of regeneration air
Regeneration temperature
[17]
100%/80%
22–36 °C
[18]
(100%–80%– 60%)/(100%– 80%–60%) 100%/100%
20–40 °C
7–15 g/kg 7–20 g/kg
wpi + 0.8 g/kg 7–20 g/kg
55 °C, 75 °C, 95 °C 40 °C, 55 °C, 70 °C
12.5– 26.3 g/kg
wpi
88 °C
C 2
þ
1
0:51 _ 0:49 m ar f
[19]
35 °C
ð32Þ tials is determined and so the outlet conditions of the wheel are found.
If C 2 ¼ 1, Eq. (32) becomes:
UAm 0
Process/ regeneration air flow rate (in % of nominal process air flow rate)
ð31Þ
0 hm AT
By defining ¼ and if < 1, and by using the effectiveness relation in Table 1, one can write:
"
Data
ð30Þ
_ 0:49 0:51 UAm r ¼ hm AT mar f UAm 0
Table 2 Validation domain
# 1 ¼ þ 0:49 0:51 C 2min 0:51 _ ar f m _ 0:49 m ap ð1 f Þ 1 e2 1 1 1:93 9ðC r2 Þ 1 1 e2 1 1 1:93 9ðC r2 Þ
6. Validation of the model
1
The model has been validated by comparison with both manufacturers’ data [17,18] and experimental data [19] on a large range of temperatures and humidity ratios (Table 2). A study of error propagation in a desiccant evaporative cooling system [2] has shown that errors on the desiccant wheel outlet state of ±2 °C and ±1 g/kg cause a maximum error of 1.1 °C and 0.7 g/kg on the supply air state of a desiccant evaporative cooling system. This leads to a maximum error on water consumption of humidifiers of 13.3% and a maximum error on the regeneration power of 14.2%. The tolerance on the model is so fixed to ±2 °C for temperature and ±1 g/kg for humidity ratio. The Fig. 8 shows that the model gives good results in all the studied cases. The deviation of the model compared to experimental and manufacturers’ data is below the imposed tolerance except for few points. One notices that the maximum deviation in enthal-
ð33Þ
So, the parameter UAm 0 can be calculated by using the relation (32) or (33). Only one operating point is thus necessary to parameterize the model. To calculate different operating states of the wheel, first, the overall mass transfer conductance should be calculated by using relations (30) and (31) and then the effectiveness can be assessed. By using Eqs. (19) and (20), the characteristic potentials at the outlet of the wheel are deducted. By an iterative calculation, the crossing point between these two characteristic poten-
1.5
1
Δw (g/kg)
0.5
0
-0.5
LiCl - constant air flow rate [17] -1
Silicagel - constant air flow rate [16] Silicagel - constant air flow rate [18] iso wet-bulb temperature
-1.5 -3
-2
-1
0
1
2
ΔT (°C) Fig. 8. Comparison between the model and manufacturers’ data [17,18] and experimental data [19].
3
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P. Stabat, D. Marchio / Applied Energy 86 (2009) 762–771
Fig. 9. Comparison between the model and manufacturers’ data [18] for variable regeneration and process airflow rates.
py is of 2.5% for silica gel wheels and 4.3% for the lithium chloride wheel. The results are better for silica gel. The enthalpy line probably fits better the first potential function of silica gel than those of lithium chloride. The model has been also validated for process and regeneration airflow rates of 80% and 60% of the nominal values. The model has been parameterized from one operating point with process and regeneration air flow rates of 10,000 m3/h. Fig. 9 shows that the model can determine operating points of a desiccant wheel even for conditions where the process and the regeneration airflow rates are changed. Three points are presented on the Figure for each case of airflow rate, representing three temperature and humidity ratio conditions at the process inlet. 7. Conclusions The proposed model of dehumidification wheels for desiccant evaporative cooling applications answers to the aims of integration into building simulation tools. The model is: easy to parameterize (the knowledge of only one operating point of the wheel is required); rapidly computed (the model includes only one loop which implies at maximum 18 iterations. The simulation of one operating point requires less than one thousandth of second in a Matlab environment with a Pentium IV 1.9 GHz); able to assess all common operating conditions with satisfying accuracy. The use of the analogy method by introducing the characteristic equations turns out to be a good solution to reach these objectives. In order to get a model which is easy to parameterize, some simplification have been added to the analogy method: The thermal capacity ratios ci are assumed to be constant; The effectiveness of the first potential function is supposed to be lower than 0.4;
The potential functions are imposed whatever the adsorbent; The purge section which is often used in desiccant wheels is not taken into account. Indeed, the ci are relatively constant and the effectiveness of the first potential is generally close to 0.1 on the operating domain of desiccant wheels. However, the two last assumptions are more difficult to justify. Despite these two simplifications, the model provides good results compared to experimental and manufacturers’ data.
Appendix A. Intermediate calculations Eqs. (1) and (3) can be rewritten as:
oW oT oW ow ow þ þ C þ ¼0 oT w os ow T os ox oH oh oT oH oh ow oh oT þ C þ þ C þ oT w oT os ow T ow T os oT w ox w oh ow þ ¼0 owT ox
ðA- 1Þ
ðA-2Þ
One notes:
oT ah ¼ ow h T k ¼ 1 ðoH=oWÞ ðoh=owÞ T
oT oT aW ¼ ow m ¼ oW W w ðoH=oTÞW aW r ¼ ðoh=oTÞw b ¼ m ¼ oW ow T
oh By dividing the Eq. (A-2) by oT jw and considering oT oh jh ¼ ow jT oT j , the Eqs. (A-1) and (A-2) become: ow oh w
1 oT ow ow þ ð b þ CÞ þ ¼0 os ox m os ow 1 oT aW r þ ð1 kÞ ah þ C þ ah ðk 1Þ þ ah C os os m m oT ow þ þ ah ¼0 ox ox
that
ðA-3Þ
ðA-4Þ
P. Stabat, D. Marchio / Applied Energy 86 (2009) 762–771
One introduces the parameter ai which represents the roots of the characteristic equation. By multiplying equation (A-3) by (ai ah) and by adding the Eq. (A-4), one finds:
a a oT ah i h þ r ð1 kÞÞ þ C os m m a ow oT aW ow W þ ðai ah Þ þ C ah ð1 kÞ þ ah C þ þ ai ¼0 os ox ox m m ðA-5Þ One searches then the ai which verify:
ji
oT ow oT ow þ ai þ ai ¼0 þ os os ox ox
ai kah with ai ka ji ¼ m þ r þ Ca ¼ ci þ C,It is so h m þ r þ C ai ¼ ai mW þ C þ ah k amW . By simplifying, this condition becomes:
ðA-6Þ necessary
a2i ai ðkah þ rm þ aW Þ þ ah kaW ¼ 0
that
ðA-7Þ
For each ai, there is a Fi function which satisfies the following equation:
oF i oF i ¼ a i owT oT w
with i ¼ 1; 2
ðA-8Þ
The Fi are termed characteristic potentials. By multiplying the Eq. (A-6) by (oFi/oT)w, one obtains:
ji
oF i oF i þ ¼ 0 pour i ¼ 1; 2 os ox
ðA-9Þ
The Eqs. (A-8) and (A-9) define the characteristics equations equivalent to Eqs. (1) and (3). The Eq. (A-8), similar to a wave equation, gives:
ai ¼
oT owF i
pour i ¼ 1; 2
References [1] Desiccant Rotors. MUNTERS Dehumidification, Amesbury, USA: 1992.
ðA-10Þ
771
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