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Heat-and-mass transfers modelled for rotary desiccant dehumidifiers
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APPLIED ENERGY Applied Energy 85 (2008) 128–142 www.elsevier.com/locate/apenergy
Heat-and-mass transfers modelled for rotary desiccant dehumidifiers Pascal Stabat, Dominique Marchio
*
Ecole des Mines de Paris – Centre Energe´tique et Proce´de´s 60, Boulevard Saint Michel, F 75272 Paris cedex 06, France
Abstract A desiccant-wheel behavioural model has been developed. This model fulfils several criteria, such as simplicity of parameterization; accuracy; ability to characterize the equipment behaviour under all operation conditions; and short computation time. The method of characteristics has been applied to the heat-and-mass transfer partial-differential equations. This transformation provides equations which are similar to those of a rotary heat-exchanger. Then, the model is described by the effectiveness-NTU method and identified from only one nominal-rating point. The model predictions have been compared with experimental and manufacturers’ data for a broad range of operating conditions. Good agreement has been found. 2007 Elsevier Ltd. All rights reserved. Keywords: Desiccant; Adsorption; Heat-and-mass transfer; Modelling; Air conditioning
1. Introduction Desiccant evaporative cooling systems are an alternative to classical air-conditioners. The principle consists in drying the air in order to achieve a high potential rate of evaporative cooling of air. This technique is refrigerant free and uses little electricity. However, as it is necessary to regenerate the desiccant wheel, thermal energy is required to heat up the wheel to temperatures in the range of 50–100 C. The use of solar energy or waste heat can make this technique attractive. HVAC designers do not have any mathematical model to size and estimate the energy consumption of such systems. So, a model for desiccant wheels has been developed in order to be included in building dynamic-simulation tools. This model aims to answer the requirements of such tools, namely: – accuracy; – short computation time;
*
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 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.06.017
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
Nomenclature surface area (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 Æ kg1 K1) potential functions Fi(T,w) (arbitrary dimension) wheel fraction used for regeneration (–) matrix enthalpy (J Æ kg1 d ) air enthalpy (J Æ kg1 ) d:a: convective heat-transfer coefficient (W Æ K1 m2) water vaporization heat at 0 C (J Æ kg1) convective mass-transfer coefficient (kgd.a. Æ m2 s1) thickness of the wheel (m) sorption heat (J Æ kg1) differential heat of wetting (negative by convention) (J Æ kg1) mass of desiccant (kg) air-mass flow rate (kgd.a. Æ s1) rotational speed of the wheel (tr Æ s1) number of transfer units (–) partial pressure of water vapour (Pa) partial pressure of water vapour at saturation (Pa) Reynolds number (–) temperature (K) time (s) overall mass-transfer coefficient (kgd.a. s1) velocity (m Æ s1) desiccant’s moisture-content (kgv Æ kg1 d ) humidity ratio (kgv Æ kg1 ) d:a: axial position divided by the wheel thickness (z/L) (–) axial position in the wheel (m) angular position a ð@H =@xÞT @T a @T a ah ¼ @w a ¼ m ¼ @T w @xa @x wa k ¼ 1 ð@h@ wa Þ a
A C C* Cr* cp Fi f H h hc hfg hm L LS Lw Md m_ a N NTU pv Pvsat Re T t UA u W w x z a
h
x
T
e effectiveness q air density (kg Æ m3) ð@H =@T Þ r ¼ ð@ha =@T aaÞwxa s ¼ mM_ adt / relative humidity (%) C ratio of air-mass to desiccant mass indices/exponents (–) Subscripts a air eq equilibrium m matrix min minimum value pi, po process inlet and outlet, respectively pn ideal process outlet (infinite heat-and-mass transfer coefficients) ri, ro regeneration inlet, outlet T total v vapour
129
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– simplicity of parameterization (the data available are often limited to those in manufacturers’ catalogues); – polyvalence (the model should characterize the equipment in all operating conditions, such as variable airflow rate or variable regeneration-temperature). 2. Fundamental heat-and-mass rate equations 2.1. Description of the desiccant wheel whose behaviour is to be modelled In desiccant-cooling systems, the rotary wheel consists of two equal parts, one for the process, the other for the regeneration. Among the desiccants, silica gel and lithium chloride are most used since they have high capacities of adsorption throughout the range of relative humidity of ambient air compared with a molecular sieve, for example (Fig. 1). The evolution of vapour pressure at the desiccant surface is a function of the moisture content in the wheel as shown in Fig. 2. This figure is based on a numerical model [2] achieved by solving the equations in Section 2.2 based on a second-order Runge Kutta method. The curve shows the evolution of vapour pressure at the desiccant surface averaged for 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 to the wheel. The increase of the partial vapour-pressure implies a transfer of moisture in the desiccant towards the air (from D to A). For the process side, there is a cooling zone (from A to B) which facilitates decreases in the temperature and water–vapour pressure at the desiccant surface. The cooling, due to fresh-air introduction, occurs almost at constant adsorbed moisture. Then, the vapour pressure increases up to a level for which the desiccant cannot adsorb any more water vapour contained in the air (from B to C). As a consequence, a fraction of the process-air flow rate is used for cooling of the wheel. This fraction is hot and almost not dehumidified. For this reason, it is common to install a purge section (Fig. 3) on desiccant wheels: this allows one to recover this hot air for regeneration and so reduce the average temperature and humidity ratio of the outlet process-air. The fraction of process air which passes through the purge section represents 2–4% of the air-flow rate in desiccant wheels. 2.2. Governing equations
Moisture content (g of water/ 100 g of dessiccant)
The desiccant wheel is modelled, based on the thermal and mass balance equations in a small volume element of the wheel (Fig. 4) in steady-state conditions. The following equations express the dimensionless variables of time s ¼ mM_ adt and position x ¼ Lz .
60 isotherm at T = 25 °C
50
LiCl Silicagel Molecular sieve 4A
40
30
20
10
0 0
10
20
30
40
50
60
70
80
Relative humidity (%)
Fig. 1. Isotherms of different materials [1].
90
100
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142 3500
α=270°
α=200°
90°C
Vapour pressure at desiccant surface (Pa)
reg
3000
131
α=0°
70°C
D
α=180° process
2500
α=90°
2000 α=0°
A
50°C
1500 α=180°
1000 C 500
30°C
B α=20°
0 0.00
0.04
0.08
0.12
Desiccant moisture content (kg/kg)
Fig. 2. Vapour pressure evolution at desiccant surface.
The assumptions are as follows: • the axial heat conduction and water-vapour diffusion in the air are neglected; • the diffusion and capillarity in the desiccant matrix are assumed to be negligible in the axial direction (z); • the temperature and humidity ratio gradients in the radial direction (r) are neglected; i.e. T = T (z, a) and w = w (z, a) are independent of the wheel radius; • the sorption is supposed to occur without hysteresis; • the pressure drop in the wheel is neglected; • the inlet-air temperature and humidity ratio are uniform; • the channels have all the same geometry; • the thermo-physical properties of the matrix, the air and water vapour are constant in all the operating conditions; • a steady state applies.
Dehumidified air
Purge section Air of regeneration Fig. 3. Schematic of a desiccant wheel with a purge section.
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α+dα
Air of regeneration
α
heater
z A ir Dehumidified air
Process air One channel
α
α+dα
W z
wa Process air
Dehumidified air
z=0
z=L Fig. 4. System model.
The mass-conservation equation is expressed as follows: oW owa owa þC þ ¼0 os os ox
ð1Þ
The mass-transfer equation can be written as: oW hm A ¼ ðwa weq Þ os m_ a
ð2Þ
The energy-conservation equation is expressed as follows: oH oha oha þC þ ¼0 os os ox
ð3Þ
The heat-transfer equation can be written as: oH hm A hc A ¼ ðwa weq Þðhfg þ cpv T a Þ þ ðT a T m Þ os m_ a m_ a
ð4Þ
Assuming a Lewis number, Le, equal to unity, then Eq. (4) becomes: oH hm A ¼ ðha heq Þ os m_ a
ð5Þ
The desiccant moisture-content, W, and the matrix enthalpy, H, in Eqs. (1)–(5) are based on the properties of the adsorbent.
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133
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 vapour pressure at the adsorbent surface. The adsorption isotherm depends on the adsorbent material properties (Fig. 1). Moreover, one can sometimes find a hysteresis phenomenon. The isotherms are defined experimentally: the isotherm proposed by Mathiprakasam and Lavan [3] for silica gels is used: /ðT eq ; weq Þ ¼ C1 W þ C2 W 2
ð6Þ
2.3.2. Sorption heat The matrix enthalpy is expressed as follows: Z w H ¼ cpm T m þ cpv WT m þ 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
ð8Þ
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-vapour adsorption on a silica gel. These correlations will be used: hc ¼ 0:671qua cpa Re0:51 hm ¼ 0:704qua Re
ð9Þ
0:51
Eqs. (9) and (10) allow one to determine the Lewis number,
ð10Þ hc , hm cpa
which is close to unity.
3. Solving the 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 the performance curves of manufacturers or experimental data. These correlations allow one to assess the performance of a particular wheel under different conditions of temperature and humidity ratio at the process and regeneration inlets. However, good results are obtained only for the considered desiccant wheel and only for the domain of temperature, humidity ratio and air-flow rate where the correlation has been well established. 3.2. Theoretical approaches Finite-difference models such as those in reference [9] are often used. However they require prolonged computation times 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 lies in its difficulty to be parameterized. The method of characteristics [10–12] has been used for transforming the coupled non-linear partial-differential equations into a set of uncoupled differential-equations. The new equations describe new independentvariables which are called characteristic potentials. These are associated with new parameters, ci, which can be associated with specific-heat ratios. Thus, the new set of equations is analogous to the heat-transfer equations
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in a non-hygroscopic rotary heat-exchanger. The problem is then solved by analogy from the solution of Kays and London [13] for a rotary heat-exchanger. This method achieves: • A good understanding of the 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 (namely effectiveness and number of transfer units); and • A short 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 analytical formulae 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 the combined-heat-and mass equations The equation system (1)–(3), (5) forms a set of coupled partial-differential hyperbolic equations. Two common solving techniques for hyperbolic equations exist: normally the finite-difference and the characteristic method. The second technique will be used hereafter in order to obtain equations nominally similar to those for rotary heat-transfer equations. In order to achieve the analogy, further assumptions should be made: • The Lewis number is assumed to equal unity. • The derivatives of the potential functions by enthalpy and humidity ratio are supposed to be invariant. • Each ci is assumed to be a function only of its associated Fi. 4.1. Transformed equations The energy-and-mass conservation equations can be transformed to become 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. In an ideal case, the matrix and the air are in thermodynamic equilibrium, with Tm = Ta = T and weq ¼ wa ¼ w: The governing equations may be transformed by using Fi functions [10], which satisfy the following equation: oF i oF i ¼ a with i ¼ 1; 2 ð11Þ i ow oT T
and
w
ai kah oF i oF i þrþC þ ¼ 0 with i ¼ 1; 2 ð12Þ v os ox in which Fi are termed characteristic potentials and the ai are the solutions of the characteristic polynomial as follows: a2i ai ðkah þ rv þ aw Þ þ ah kaw ¼ 0 ð13Þ Eqs. (11) and (12) form the characteristic equations equivalent of Eqs. (1) and (3). Detailed calculations are presented in Appendix A.
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135
4.2. Analogy of dehumidifier with a heat exchanger To obtain the analogy with a rotary sensible heat-exchanger, the following assumptions are made. oF eq oF ai i ¼ with i ¼ 1; 2 oha wa oha wa oF eq oF ai i ¼ with i ¼ 1; 2 owa ha owa ha
ð14Þ ð15Þ
This assumption is satisfied in the case where the air and the matrix are in thermodynamic equilibrium (Tm = Ta and weq = wa) or if (oFi/owa)ha and (oFi/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 imposed 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 i With these assumptions, one can thus write that oTia ¼ oT ai . By multiplying Eq. (A-6) by oF , one oT w wa wa obtains: ci
oF eq oF a oF a i þC i þ i ¼0 os os ox
with i ¼ 1; 2
By multiplying Eqs. (2) and (5) by ci
oF eq i ¼ NUTw F ai F eq i os
oF i ow h
and
ð16Þ
oF i , oh w
respectively and by adding them, then:
with i ¼ 1; 2
ð17Þ
with ci ¼ r
ai kah v
NUTw ¼
and
hm A m_ a
ð18Þ
0.03
Humidity ratio (kg/kg)
0.025
0.02
F1 F2 0.0 15
0.01
F1 0.005
F2 0 40000
50000
60000
70000
80000
90000
100000
110000
120000
Enthalpy(J/kg)
Fig. 5. Potential functions for a silica gel in a (h, w) diagram.
130000
140000
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F2
F1
ι 2=50
ι 4=30
ι 1=0.2
ι 4=20
Humidity ratio (kg/kg)
ι 4=10
0.020 ι 4=5 ι 1=0.3
ι 1=0.4
0.010 ι 1=0.5 ι 1=0.6 ι 1=0.8 0.000 20
30
ι 4=2
ι 1=0.7
40
50
60
70
80
90
100
Temperature (°C)
Fig. 6. Potential functions and ci for a silica gel in a psychrometric chart.
The pair of Eqs. (16) and (17) forms a system similar to that for 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 Eqs. (16) and (17) are not completely uncoupled. One should assume that the ci depend only on 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 silica gel with a sorption isotherm described by 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 this is not the case. In the following, the 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 the procedure for the 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: 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
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) Cij ¼ m_ ja
Cj ¼ m_ ja ðcpa þ wcpv Þ
dN i c Cri ¼ M or C ri ¼ min C iri ¼ Mm_djN cij m_ min min
r mN Cr ¼ CCmin ¼ Mc C min min C ¼ CCmax
a
a
Ci
h
i1
Ci ¼ C imin
max
h
i1
1 1 NTUT0 ¼ Cmin þ 1 ðhc AÞj h ðhc AÞ3j1 i T e ¼ ecf NTU0 ; C 1 9ðC Þ1:93
1 1 NTUm þ 1 0 ¼ Ci min mðhm AÞ jh ðhm AÞ3j1 i ei ¼ ecf NTU0 ; C i 1 9ðC Þ1:93
ecf is the effectiveness of a counter-flow heat-exchanger, or, at low rotation-speed (Cr 6 0:4Þ; e ¼ C r
ecf is the effectiveness of a counter-flow heat-exchanger, or, at low rotation-speed ðCr 6 0:4Þ; e ¼ C r
r
r
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137
Humidity ratio (kg/kg)
0.030
0.020
F2pi
0.010
F1ri
ri
pi
F2po
F1po F1pi
0.000 20
po
F2ri pn
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.
The intersection point between the two potential functions, F1pi and F2pi, see Fig. 7, corresponds to an ideal case where the heat-and-mass transfer coefficients are infinite. The outlet process state is located at the intersection of the potential functions F1px and F2px, as calculated by the effectiveness relations (19) and (20) which account 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. Adaptation of model for a building-simulation tool The users of building-behaviour 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 rotational speed of the wheel, the mass of desiccant, the heat-exchange surface, and the heattransfer coefficient, which are generally unknown. So, the analogy model should be adapted in order to define these parameters via 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. 5.1. Selection of Fi and ci Banks [16] 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 C 1p ðF 1po F 1pi Þ ¼ C 1r ðF 1ro F 1ri Þ
ð21Þ
C 2p ðF 2po F 2pi Þ ¼ C 2r ðF 2ro F 2ri Þ
ð22Þ
and
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P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
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 [16] satisfies 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 conditions (21) and (22) and it fits quite well with experimental data. F1 ¼ h F2 ¼
ð23Þ
ð273:15 þ T Þ 6360
1;5
1:1w0:08
ð24Þ
Moreover, Maclaine-Cross [9] 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 ci, the default average values for the operating range of desiccant wheels are taken as follows: c1 ¼ 0:3
ð25Þ
c2 ¼ 30
ð26Þ
5.2. Model parameterization In order to determine the outlet states of a given desiccant wheel, the effectiveness related to the potential functions should be assessed. As shown in Table 1, the effectiveness depends on the product of desiccant mass by rotational speed, i.e. Md Æ N, and the product of the mass-transfer coefficient by the exchange surface, i.e. hm.S. These data are generally unknown. The aim is now to determine these unknown parameters via one operating point of the desiccant wheel. With an operating point and the default potential-functions, one can calculate the effectiveness based on Eqs. (19) and (20). Concerning the first effectiveness relation, Maclaine-Cross [9] has shown, by numerical calculations, that if NTUo P4.5, C* 6 1, Cr* 6 0.4 and the ratio of mass-transfer coefficients between process side and regeneration side is equal to unity, the first effectiveness can be reduced to: e1 ¼ C rl ¼
Md N 1 cmin _ min m a
ð27Þ
With only one operating point, the product MdN can be assessed since a default value is used for c1min . 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. To achieve 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 [13] should be used (see Table 1). For each side of the wheel, the overall mass-transfer coefficient can be expressed as follows: UAm p ¼ hm AT ð1 f Þ
ð28Þ
UAm r
ð29Þ
¼ hm A T f
By considering Eq. (10) and assuming the dynamic viscosity is constant, one can rewrite the overall masstransfer coefficients as: 0 _ 0:49 UAm p ¼ hm A T m ap ð1 f Þ 0 0:51 _ 0:49 UAm r ¼ hm A T m ar f
0:51
ð30Þ ð31Þ
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139
0 Defining UAm 0 ¼ hm AT and if C 2 < 1, and by using the effectiveness relation in Table 1, one can write:
" UAm 0 ¼
1 m_ 0:49 ap ð1
fÞ
0:51
þ
1
#
0:51 m_ 0:49 ar f
0
1 1 1 C 2 C B1 e2 1 9ð9C Þ1:93 C 2min r2 B C ln B 1 C A 1 C 2 @ 1 e2 1 9ðC1Þ1:93
ð32Þ
1 e2 1 9ðC1Þ1:93 r2 C 2min 1 1 e2 1 9ðC1Þ1:93
ð33Þ
r2
If C 2 ¼ 1, Eq. (32) becomes: " UAm 0 ¼
1 m_ 0:49 ap ð1
fÞ
0:51
þ
1 0:51 m_ 0:49 ar f
#
r2
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 the 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 deduced. By an iterative process, the crossing point between these two characteristic potentials is determined and so the outlet conditions of the wheel are found. 6. Validation of the model The model has been validated by comparison with both manufacturers’ data [17,18] and experimental measurements [19] for large ranges of temperatures and humidity ratios (Table 2). A study of error propagation in a desiccant evaporative-cooling system [2] has shown that errors for 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 with respect to water consumption of humidifiers of 13.3% and a maximum error on the regeneration power of 14.2%. The tolerance on the model is fixed to ± 2 C for temperature and ±1 g/kg for the humidity ratio. Fig. 8 shows that the model gives good results in all the studied cases. The deviation of the model’s predictions compared with experimental and manufacturers’ data is usually below the imposed tolerance. The maximum deviation in enthalpy is 2.5% for silica-gel wheels and 4.3% for the lithium-chloride wheels. 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 for one operating point with process and regeneration air-flow rate 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.
Table 2 Validation domain Reference Process/regeneration air-flow rate Temperature of Humidity radio of inlet Humidity ratio of Regeneration (as % of nominal process air-flow rate) inlet process air process air (Wpi) regeneration air temperature [17] [18] [19]
100%/80% (100%–80%–60%)/(100%–80%–60%) 100%/100%
22–36 C 20–40 C 35 C
7–15 g/kg 7–20 g/kg 12.5–26.3 g/kg
wpi + 0.8 g/kg 7–20 g/kg wpi
55 C, 75 C, 95 C 40 C, 55 C, 70 C 88 C
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P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142 1.5
1
Δ w (g/kg)
0.5
0
-0.5 LiCl - constant air flow rate [18]
-1
Silicagel - constant air flow rate [17] Silicagel - constant air flow rate [19] iso wet-bulb temperature
-1.5 -3
-2
-1
0
1
2
3
Δ T (°C)
Fig. 8. Comparison between the model predictions, manufacturers’ data [17,18] and experimental measurement [19].
1.5
1
Δ w (g/kg)
0.5
0
-0.5 3
-1
-1.5 -3
Process/Regeneration air flow rate: 10 000 / 10 000 m /h 3 Process/Regeneration air flow rate: 10 000 / 8 000 m /h 3 Process/Regeneration air flow rate: 10 000 / 6 000 m /h Process/Regeneration air flow rate: 8 000 / 8 000 m3/h 3 Process/Regeneration air flow rate: 6 000 / 6 000 m /h
-2
-1
0
1
2
3
ΔT (°C)
Fig. 9. Comparison between the model predictions and manufacturers’ data [18] for variable regeneration and process airflow-rates.
7. Conclusions The proposed model of dehumidification wheels for desiccant evaporative-cooling applications satisfies the aims of integration into building-simulation tools. The model is: • Easy to parameterize (needing only a knowledge of only one operating point of the wheel). • Rapidly computed (the model including only one loop which implies at maximum 18 iterations. The simulation of one operating point requires less than one thousandth of a second in a Matlab environment with a Pentium IV 1,9 GHz). • Able to assess all common operating conditions with satisfying accuracy.
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
141
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 devise a model which is easy to parameterize, some simplification have been added to the analogy method: • • • •
All values of the thermal capacity ratio ci are assumed to be constant. The effectiveness of the first potential-function is supposed to be less 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 values are relatively constant and the effectiveness of the first potential is generally close to 0.1 in 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 with experimental and manufacturers’ data.
Appendix A. Intermediate calculations Eqs. (1) and (3) can be rewritten as: oW oT oW ow ow þ þ ¼0 þC oT w os ow T os ox oH oh oT oH oh ow oh oT oh ow þ þ þ ¼0 þC þC oT w oT w os ow T ow T os oT w ox owT ox
ðA-1Þ ðA-2Þ
One notes:
oT oT oT ah ¼ a w ¼ v ¼ ow h ow w oW w ðoH =oW ÞT ðoH =oT Þw aw oW ¼ k¼1 r¼ b¼ ow T ðoh=owÞT ðoh=oT Þw v oh ¼ oh oT , Eqs. (A-1) and (A-2) become: By dividing Eq. (A-2) by oT and considering that oT ow h ow T oh w w 1 oT ow ow þ ðb þ CÞ þ ¼0 v os os ox ow oT 1 oT aW ow þ ah ðk 1Þ þ þ ah ¼0 r þ ð1 kÞ þ ah C ah þ C v os os ox ox v
ðA-3Þ ðA-4Þ
One introduces the parameter ai which represents the roots of the characteristic equation. By multiplying Eq. (A-3) by (ai ah) and by adding Eq. (A-4), one finds: a a oT a ow oT ah aW ow i h W þ ð ai a h Þ þ þ ai ¼0 þ r ð1 kÞ þ C þ C ah ð1 kÞ þ ah C os os ox ox v v v v ðA-5Þ One searches then for values of ai which verify: ji
oT ow oT ow þ ai þ ai þ ¼0 os os ox ox
ðA-6Þ
h h þ r þ C ¼ ci þ C, It is necessary that ai ka þ r þ C ai ¼ ai avw þ C þ ah k avw . with ji ¼ ai ka v v By simplifying, this condition becomes: a2i ai ðkah þ rv þ aw Þ þ ah kaw ¼ 0
ðA-7Þ
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For each ai there is a Fi function which satisfies the following equation: oF i oF i ¼ ai with i ¼ 1; 2 ow oT
ðA-8Þ
w
The Fi values are termed characteristic potentials. By multiplying Eq. (A-6) by (oFi/oT)w, one obtains: ji
oF i oF i þ ¼0 os ox
for i ¼ 1; 2
ðA-9Þ
Eq. (A-8) and (A-9) define characteristic equations which are equivalent to Eqs. (1) and (3), respectively. Eq. (A-8), which is similar to a wave equation, gives: oT ai ¼ for i ¼ 1; 2 ðA-10Þ oW Fi
References [1] Desiccant Rotors. MUNTERS Dehumidification; 1992. [2] Stabat P. Mode´lisation de composants de syste`mes de climatisation mettant en œuvre l’adsorption et l’e´vaporation d’eau. The`se de doctoral, Ecole des Mines de Paris, France; 2003. [3] Mathiprakasam B, Lavan Z. Performance predictions for adiabatic desiccant-dehumidifiers using linear solutions. J Solar Energy Eng 1980:102. [4] Brandemuehl MJ. Analysis of heat-and-mass transfer regenerators with time varying or spatially non-uniform inlet temperatures. Ph.D Thesis, University of Wisconsin; 1982. [5] Bullock CE, Threlkeld JL. Dehumidification of moist air by adiabatic adsorption. Trans ASHRAE 1966:72. [6] Behne M. Alternatives to compressive cooling in non-residential buildings to reduce primary-energy consumption. Final Report LBL 1997. [7] Lindholm T. Evaporative and desiccant-cooling techniques, feasibility when applied to air conditioning. Ph.D. Thesis, Chalmers University of Technology, Sweden; 2000. [8] Beccali M, Butera F, Guanella R, Adhikari RS. Performance evaluation of rotary desiccant-wheels using a simplified psychrometric model as the design tool. EPIC 2002 ATVC conference 2, Lyon 2002; 419–424. [9] Maclaine-Cross IL. A theory of combined heat-and-mass transfer in regenerators. Ph.D. Thesis, Department of Mechanical Engineering, Monash University, Australia; 1974. [10] Banks PJ. Coupled equilibrium heat and single adsorbate transfer in fluid flow through a porous medium – I Characteristic potentials and specific capacity ratios. Chem Eng Sci 1972;27:1143–55. [11] Maclaine-Cross IL, Banks PJ. Coupled heat-and-mass transfer in regenerators – prediction using an analogy with heat transfer. Int J Heat Mass Transfer 1972;15:1225–42. [12] Close DJ, Banks PJ. Coupled equilibrium heat and single-adsorbate transfer in fluid flow through a porous medium – II Predictions for a silica-gel air-drier using characteristic charts. Chem Eng Sci 1972;27:1157–69. [13] Kays WM, London AL. Compact Heat-Exchangers. third ed. New York: McGraw-Hill; 1984. [14] Howe R. Rotary Desiccant Dehumidifier. TRNLIB – Libraries of User-Written TRNSYS Components, Solar Energy Laboratory, http://sel.me.wisc.edu/trnsvs/trnlib/dessi.htm: 1983. [15] Jurinak JJ. Open-cycle solid desiccant cooling – component models and system simulation. Ph.D. Thesis, University of Wisconsin – Madison; 1982. [16] Banks PJ. Predictions of heat-and-mass regenerator performance using non-linear analogy method: Part 1. Basis, Part 2. Comparison of methods. ASME J Heat Transfer 1985;107:222–38. [17] MCC – Series Cooling Cassette, manufacturer’s catalogue, Munters; 1994. [18] SECO Desiccant/enthalpy/dehumidification rotors. Software SECO, Klingenburg; 2002. [19] Slayzak SJ, Pesaran AA, Hancock CE. Experimental evaluation of commercial-desiccant dehumidifier-wheels. http://www.nrel.gov/ desiccantcool/wheels.html; 2002.
APPLIED ENERGY Applied Energy 85 (2008) 128–142 www.elsevier.com/locate/apenergy
Heat-and-mass transfers modelled for rotary desiccant dehumidifiers Pascal Stabat, Dominique Marchio
*
Ecole des Mines de Paris – Centre Energe´tique et Proce´de´s 60, Boulevard Saint Michel, F 75272 Paris cedex 06, France
Abstract A desiccant-wheel behavioural model has been developed. This model fulfils several criteria, such as simplicity of parameterization; accuracy; ability to characterize the equipment behaviour under all operation conditions; and short computation time. The method of characteristics has been applied to the heat-and-mass transfer partial-differential equations. This transformation provides equations which are similar to those of a rotary heat-exchanger. Then, the model is described by the effectiveness-NTU method and identified from only one nominal-rating point. The model predictions have been compared with experimental and manufacturers’ data for a broad range of operating conditions. Good agreement has been found. 2007 Elsevier Ltd. All rights reserved. Keywords: Desiccant; Adsorption; Heat-and-mass transfer; Modelling; Air conditioning
1. Introduction Desiccant evaporative cooling systems are an alternative to classical air-conditioners. The principle consists in drying the air in order to achieve a high potential rate of evaporative cooling of air. This technique is refrigerant free and uses little electricity. However, as it is necessary to regenerate the desiccant wheel, thermal energy is required to heat up the wheel to temperatures in the range of 50–100 C. The use of solar energy or waste heat can make this technique attractive. HVAC designers do not have any mathematical model to size and estimate the energy consumption of such systems. So, a model for desiccant wheels has been developed in order to be included in building dynamic-simulation tools. This model aims to answer the requirements of such tools, namely: – accuracy; – short computation time;
*
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 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.06.017
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
Nomenclature surface area (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 Æ kg1 K1) potential functions Fi(T,w) (arbitrary dimension) wheel fraction used for regeneration (–) matrix enthalpy (J Æ kg1 d ) air enthalpy (J Æ kg1 ) d:a: convective heat-transfer coefficient (W Æ K1 m2) water vaporization heat at 0 C (J Æ kg1) convective mass-transfer coefficient (kgd.a. Æ m2 s1) thickness of the wheel (m) sorption heat (J Æ kg1) differential heat of wetting (negative by convention) (J Æ kg1) mass of desiccant (kg) air-mass flow rate (kgd.a. Æ s1) rotational speed of the wheel (tr Æ s1) number of transfer units (–) partial pressure of water vapour (Pa) partial pressure of water vapour at saturation (Pa) Reynolds number (–) temperature (K) time (s) overall mass-transfer coefficient (kgd.a. s1) velocity (m Æ s1) desiccant’s moisture-content (kgv Æ kg1 d ) humidity ratio (kgv Æ kg1 ) d:a: axial position divided by the wheel thickness (z/L) (–) axial position in the wheel (m) angular position a ð@H =@xÞT @T a @T a ah ¼ @w a ¼ m ¼ @T w @xa @x wa k ¼ 1 ð@h@ wa Þ a
A C C* Cr* cp Fi f H h hc hfg hm L LS Lw Md m_ a N NTU pv Pvsat Re T t UA u W w x z a
h
x
T
e effectiveness q air density (kg Æ m3) ð@H =@T Þ r ¼ ð@ha =@T aaÞwxa s ¼ mM_ adt / relative humidity (%) C ratio of air-mass to desiccant mass indices/exponents (–) Subscripts a air eq equilibrium m matrix min minimum value pi, po process inlet and outlet, respectively pn ideal process outlet (infinite heat-and-mass transfer coefficients) ri, ro regeneration inlet, outlet T total v vapour
129
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– simplicity of parameterization (the data available are often limited to those in manufacturers’ catalogues); – polyvalence (the model should characterize the equipment in all operating conditions, such as variable airflow rate or variable regeneration-temperature). 2. Fundamental heat-and-mass rate equations 2.1. Description of the desiccant wheel whose behaviour is to be modelled In desiccant-cooling systems, the rotary wheel consists of two equal parts, one for the process, the other for the regeneration. Among the desiccants, silica gel and lithium chloride are most used since they have high capacities of adsorption throughout the range of relative humidity of ambient air compared with a molecular sieve, for example (Fig. 1). The evolution of vapour pressure at the desiccant surface is a function of the moisture content in the wheel as shown in Fig. 2. This figure is based on a numerical model [2] achieved by solving the equations in Section 2.2 based on a second-order Runge Kutta method. The curve shows the evolution of vapour pressure at the desiccant surface averaged for 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 to the wheel. The increase of the partial vapour-pressure implies a transfer of moisture in the desiccant towards the air (from D to A). For the process side, there is a cooling zone (from A to B) which facilitates decreases in the temperature and water–vapour pressure at the desiccant surface. The cooling, due to fresh-air introduction, occurs almost at constant adsorbed moisture. Then, the vapour pressure increases up to a level for which the desiccant cannot adsorb any more water vapour contained in the air (from B to C). As a consequence, a fraction of the process-air flow rate is used for cooling of the wheel. This fraction is hot and almost not dehumidified. For this reason, it is common to install a purge section (Fig. 3) on desiccant wheels: this allows one to recover this hot air for regeneration and so reduce the average temperature and humidity ratio of the outlet process-air. The fraction of process air which passes through the purge section represents 2–4% of the air-flow rate in desiccant wheels. 2.2. Governing equations
Moisture content (g of water/ 100 g of dessiccant)
The desiccant wheel is modelled, based on the thermal and mass balance equations in a small volume element of the wheel (Fig. 4) in steady-state conditions. The following equations express the dimensionless variables of time s ¼ mM_ adt and position x ¼ Lz .
60 isotherm at T = 25 °C
50
LiCl Silicagel Molecular sieve 4A
40
30
20
10
0 0
10
20
30
40
50
60
70
80
Relative humidity (%)
Fig. 1. Isotherms of different materials [1].
90
100
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142 3500
α=270°
α=200°
90°C
Vapour pressure at desiccant surface (Pa)
reg
3000
131
α=0°
70°C
D
α=180° process
2500
α=90°
2000 α=0°
A
50°C
1500 α=180°
1000 C 500
30°C
B α=20°
0 0.00
0.04
0.08
0.12
Desiccant moisture content (kg/kg)
Fig. 2. Vapour pressure evolution at desiccant surface.
The assumptions are as follows: • the axial heat conduction and water-vapour diffusion in the air are neglected; • the diffusion and capillarity in the desiccant matrix are assumed to be negligible in the axial direction (z); • the temperature and humidity ratio gradients in the radial direction (r) are neglected; i.e. T = T (z, a) and w = w (z, a) are independent of the wheel radius; • the sorption is supposed to occur without hysteresis; • the pressure drop in the wheel is neglected; • the inlet-air temperature and humidity ratio are uniform; • the channels have all the same geometry; • the thermo-physical properties of the matrix, the air and water vapour are constant in all the operating conditions; • a steady state applies.
Dehumidified air
Purge section Air of regeneration Fig. 3. Schematic of a desiccant wheel with a purge section.
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α+dα
Air of regeneration
α
heater
z A ir Dehumidified air
Process air One channel
α
α+dα
W z
wa Process air
Dehumidified air
z=0
z=L Fig. 4. System model.
The mass-conservation equation is expressed as follows: oW owa owa þC þ ¼0 os os ox
ð1Þ
The mass-transfer equation can be written as: oW hm A ¼ ðwa weq Þ os m_ a
ð2Þ
The energy-conservation equation is expressed as follows: oH oha oha þC þ ¼0 os os ox
ð3Þ
The heat-transfer equation can be written as: oH hm A hc A ¼ ðwa weq Þðhfg þ cpv T a Þ þ ðT a T m Þ os m_ a m_ a
ð4Þ
Assuming a Lewis number, Le, equal to unity, then Eq. (4) becomes: oH hm A ¼ ðha heq Þ os m_ a
ð5Þ
The desiccant moisture-content, W, and the matrix enthalpy, H, in Eqs. (1)–(5) are based on the properties of the adsorbent.
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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 vapour pressure at the adsorbent surface. The adsorption isotherm depends on the adsorbent material properties (Fig. 1). Moreover, one can sometimes find a hysteresis phenomenon. The isotherms are defined experimentally: the isotherm proposed by Mathiprakasam and Lavan [3] for silica gels is used: /ðT eq ; weq Þ ¼ C1 W þ C2 W 2
ð6Þ
2.3.2. Sorption heat The matrix enthalpy is expressed as follows: Z w H ¼ cpm T m þ cpv WT m þ 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
ð8Þ
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-vapour adsorption on a silica gel. These correlations will be used: hc ¼ 0:671qua cpa Re0:51 hm ¼ 0:704qua Re
ð9Þ
0:51
Eqs. (9) and (10) allow one to determine the Lewis number,
ð10Þ hc , hm cpa
which is close to unity.
3. Solving the 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 the performance curves of manufacturers or experimental data. These correlations allow one to assess the performance of a particular wheel under different conditions of temperature and humidity ratio at the process and regeneration inlets. However, good results are obtained only for the considered desiccant wheel and only for the domain of temperature, humidity ratio and air-flow rate where the correlation has been well established. 3.2. Theoretical approaches Finite-difference models such as those in reference [9] are often used. However they require prolonged computation times 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 lies in its difficulty to be parameterized. The method of characteristics [10–12] has been used for transforming the coupled non-linear partial-differential equations into a set of uncoupled differential-equations. The new equations describe new independentvariables which are called characteristic potentials. These are associated with new parameters, ci, which can be associated with specific-heat ratios. Thus, the new set of equations is analogous to the heat-transfer equations
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in a non-hygroscopic rotary heat-exchanger. The problem is then solved by analogy from the solution of Kays and London [13] for a rotary heat-exchanger. This method achieves: • A good understanding of the 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 (namely effectiveness and number of transfer units); and • A short 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 analytical formulae 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 the combined-heat-and mass equations The equation system (1)–(3), (5) forms a set of coupled partial-differential hyperbolic equations. Two common solving techniques for hyperbolic equations exist: normally the finite-difference and the characteristic method. The second technique will be used hereafter in order to obtain equations nominally similar to those for rotary heat-transfer equations. In order to achieve the analogy, further assumptions should be made: • The Lewis number is assumed to equal unity. • The derivatives of the potential functions by enthalpy and humidity ratio are supposed to be invariant. • Each ci is assumed to be a function only of its associated Fi. 4.1. Transformed equations The energy-and-mass conservation equations can be transformed to become 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. In an ideal case, the matrix and the air are in thermodynamic equilibrium, with Tm = Ta = T and weq ¼ wa ¼ w: The governing equations may be transformed by using Fi functions [10], which satisfy the following equation: oF i oF i ¼ a with i ¼ 1; 2 ð11Þ i ow oT T
and
w
ai kah oF i oF i þrþC þ ¼ 0 with i ¼ 1; 2 ð12Þ v os ox in which Fi are termed characteristic potentials and the ai are the solutions of the characteristic polynomial as follows: a2i ai ðkah þ rv þ aw Þ þ ah kaw ¼ 0 ð13Þ Eqs. (11) and (12) form the characteristic equations equivalent of Eqs. (1) and (3). Detailed calculations are presented in Appendix A.
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135
4.2. Analogy of dehumidifier with a heat exchanger To obtain the analogy with a rotary sensible heat-exchanger, the following assumptions are made. oF eq oF ai i ¼ with i ¼ 1; 2 oha wa oha wa oF eq oF ai i ¼ with i ¼ 1; 2 owa ha owa ha
ð14Þ ð15Þ
This assumption is satisfied in the case where the air and the matrix are in thermodynamic equilibrium (Tm = Ta and weq = wa) or if (oFi/owa)ha and (oFi/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 imposed 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 i With these assumptions, one can thus write that oTia ¼ oT ai . By multiplying Eq. (A-6) by oF , one oT w wa wa obtains: ci
oF eq oF a oF a i þC i þ i ¼0 os os ox
with i ¼ 1; 2
By multiplying Eqs. (2) and (5) by ci
oF eq i ¼ NUTw F ai F eq i os
oF i ow h
and
ð16Þ
oF i , oh w
respectively and by adding them, then:
with i ¼ 1; 2
ð17Þ
with ci ¼ r
ai kah v
NUTw ¼
and
hm A m_ a
ð18Þ
0.03
Humidity ratio (kg/kg)
0.025
0.02
F1 F2 0.0 15
0.01
F1 0.005
F2 0 40000
50000
60000
70000
80000
90000
100000
110000
120000
Enthalpy(J/kg)
Fig. 5. Potential functions for a silica gel in a (h, w) diagram.
130000
140000
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P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142 0.030
F2
F1
ι 2=50
ι 4=30
ι 1=0.2
ι 4=20
Humidity ratio (kg/kg)
ι 4=10
0.020 ι 4=5 ι 1=0.3
ι 1=0.4
0.010 ι 1=0.5 ι 1=0.6 ι 1=0.8 0.000 20
30
ι 4=2
ι 1=0.7
40
50
60
70
80
90
100
Temperature (°C)
Fig. 6. Potential functions and ci for a silica gel in a psychrometric chart.
The pair of Eqs. (16) and (17) forms a system similar to that for 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 Eqs. (16) and (17) are not completely uncoupled. One should assume that the ci depend only on 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 silica gel with a sorption isotherm described by 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 this is not the case. In the following, the 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 the procedure for the 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: 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
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) Cij ¼ m_ ja
Cj ¼ m_ ja ðcpa þ wcpv Þ
dN i c Cri ¼ M or C ri ¼ min C iri ¼ Mm_djN cij m_ min min
r mN Cr ¼ CCmin ¼ Mc C min min C ¼ CCmax
a
a
Ci
h
i1
Ci ¼ C imin
max
h
i1
1 1 NTUT0 ¼ Cmin þ 1 ðhc AÞj h ðhc AÞ3j1 i T e ¼ ecf NTU0 ; C 1 9ðC Þ1:93
1 1 NTUm þ 1 0 ¼ Ci min mðhm AÞ jh ðhm AÞ3j1 i ei ¼ ecf NTU0 ; C i 1 9ðC Þ1:93
ecf is the effectiveness of a counter-flow heat-exchanger, or, at low rotation-speed (Cr 6 0:4Þ; e ¼ C r
ecf is the effectiveness of a counter-flow heat-exchanger, or, at low rotation-speed ðCr 6 0:4Þ; e ¼ C r
r
r
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
137
Humidity ratio (kg/kg)
0.030
0.020
F2pi
0.010
F1ri
ri
pi
F2po
F1po F1pi
0.000 20
po
F2ri pn
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.
The intersection point between the two potential functions, F1pi and F2pi, see Fig. 7, corresponds to an ideal case where the heat-and-mass transfer coefficients are infinite. The outlet process state is located at the intersection of the potential functions F1px and F2px, as calculated by the effectiveness relations (19) and (20) which account 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. Adaptation of model for a building-simulation tool The users of building-behaviour 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 rotational speed of the wheel, the mass of desiccant, the heat-exchange surface, and the heattransfer coefficient, which are generally unknown. So, the analogy model should be adapted in order to define these parameters via 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. 5.1. Selection of Fi and ci Banks [16] 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 C 1p ðF 1po F 1pi Þ ¼ C 1r ðF 1ro F 1ri Þ
ð21Þ
C 2p ðF 2po F 2pi Þ ¼ C 2r ðF 2ro F 2ri Þ
ð22Þ
and
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P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
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 [16] satisfies 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 conditions (21) and (22) and it fits quite well with experimental data. F1 ¼ h F2 ¼
ð23Þ
ð273:15 þ T Þ 6360
1;5
1:1w0:08
ð24Þ
Moreover, Maclaine-Cross [9] 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 ci, the default average values for the operating range of desiccant wheels are taken as follows: c1 ¼ 0:3
ð25Þ
c2 ¼ 30
ð26Þ
5.2. Model parameterization In order to determine the outlet states of a given desiccant wheel, the effectiveness related to the potential functions should be assessed. As shown in Table 1, the effectiveness depends on the product of desiccant mass by rotational speed, i.e. Md Æ N, and the product of the mass-transfer coefficient by the exchange surface, i.e. hm.S. These data are generally unknown. The aim is now to determine these unknown parameters via one operating point of the desiccant wheel. With an operating point and the default potential-functions, one can calculate the effectiveness based on Eqs. (19) and (20). Concerning the first effectiveness relation, Maclaine-Cross [9] has shown, by numerical calculations, that if NTUo P4.5, C* 6 1, Cr* 6 0.4 and the ratio of mass-transfer coefficients between process side and regeneration side is equal to unity, the first effectiveness can be reduced to: e1 ¼ C rl ¼
Md N 1 cmin _ min m a
ð27Þ
With only one operating point, the product MdN can be assessed since a default value is used for c1min . 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. To achieve 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 [13] should be used (see Table 1). For each side of the wheel, the overall mass-transfer coefficient can be expressed as follows: UAm p ¼ hm AT ð1 f Þ
ð28Þ
UAm r
ð29Þ
¼ hm A T f
By considering Eq. (10) and assuming the dynamic viscosity is constant, one can rewrite the overall masstransfer coefficients as: 0 _ 0:49 UAm p ¼ hm A T m ap ð1 f Þ 0 0:51 _ 0:49 UAm r ¼ hm A T m ar f
0:51
ð30Þ ð31Þ
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
139
0 Defining UAm 0 ¼ hm AT and if C 2 < 1, and by using the effectiveness relation in Table 1, one can write:
" UAm 0 ¼
1 m_ 0:49 ap ð1
fÞ
0:51
þ
1
#
0:51 m_ 0:49 ar f
0
1 1 1 C 2 C B1 e2 1 9ð9C Þ1:93 C 2min r2 B C ln B 1 C A 1 C 2 @ 1 e2 1 9ðC1Þ1:93
ð32Þ
1 e2 1 9ðC1Þ1:93 r2 C 2min 1 1 e2 1 9ðC1Þ1:93
ð33Þ
r2
If C 2 ¼ 1, Eq. (32) becomes: " UAm 0 ¼
1 m_ 0:49 ap ð1
fÞ
0:51
þ
1 0:51 m_ 0:49 ar f
#
r2
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 the 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 deduced. By an iterative process, the crossing point between these two characteristic potentials is determined and so the outlet conditions of the wheel are found. 6. Validation of the model The model has been validated by comparison with both manufacturers’ data [17,18] and experimental measurements [19] for large ranges of temperatures and humidity ratios (Table 2). A study of error propagation in a desiccant evaporative-cooling system [2] has shown that errors for 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 with respect to water consumption of humidifiers of 13.3% and a maximum error on the regeneration power of 14.2%. The tolerance on the model is fixed to ± 2 C for temperature and ±1 g/kg for the humidity ratio. Fig. 8 shows that the model gives good results in all the studied cases. The deviation of the model’s predictions compared with experimental and manufacturers’ data is usually below the imposed tolerance. The maximum deviation in enthalpy is 2.5% for silica-gel wheels and 4.3% for the lithium-chloride wheels. 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 for one operating point with process and regeneration air-flow rate 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.
Table 2 Validation domain Reference Process/regeneration air-flow rate Temperature of Humidity radio of inlet Humidity ratio of Regeneration (as % of nominal process air-flow rate) inlet process air process air (Wpi) regeneration air temperature [17] [18] [19]
100%/80% (100%–80%–60%)/(100%–80%–60%) 100%/100%
22–36 C 20–40 C 35 C
7–15 g/kg 7–20 g/kg 12.5–26.3 g/kg
wpi + 0.8 g/kg 7–20 g/kg wpi
55 C, 75 C, 95 C 40 C, 55 C, 70 C 88 C
140
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142 1.5
1
Δ w (g/kg)
0.5
0
-0.5 LiCl - constant air flow rate [18]
-1
Silicagel - constant air flow rate [17] Silicagel - constant air flow rate [19] iso wet-bulb temperature
-1.5 -3
-2
-1
0
1
2
3
Δ T (°C)
Fig. 8. Comparison between the model predictions, manufacturers’ data [17,18] and experimental measurement [19].
1.5
1
Δ w (g/kg)
0.5
0
-0.5 3
-1
-1.5 -3
Process/Regeneration air flow rate: 10 000 / 10 000 m /h 3 Process/Regeneration air flow rate: 10 000 / 8 000 m /h 3 Process/Regeneration air flow rate: 10 000 / 6 000 m /h Process/Regeneration air flow rate: 8 000 / 8 000 m3/h 3 Process/Regeneration air flow rate: 6 000 / 6 000 m /h
-2
-1
0
1
2
3
ΔT (°C)
Fig. 9. Comparison between the model predictions and manufacturers’ data [18] for variable regeneration and process airflow-rates.
7. Conclusions The proposed model of dehumidification wheels for desiccant evaporative-cooling applications satisfies the aims of integration into building-simulation tools. The model is: • Easy to parameterize (needing only a knowledge of only one operating point of the wheel). • Rapidly computed (the model including only one loop which implies at maximum 18 iterations. The simulation of one operating point requires less than one thousandth of a second in a Matlab environment with a Pentium IV 1,9 GHz). • Able to assess all common operating conditions with satisfying accuracy.
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
141
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 devise a model which is easy to parameterize, some simplification have been added to the analogy method: • • • •
All values of the thermal capacity ratio ci are assumed to be constant. The effectiveness of the first potential-function is supposed to be less 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 values are relatively constant and the effectiveness of the first potential is generally close to 0.1 in 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 with experimental and manufacturers’ data.
Appendix A. Intermediate calculations Eqs. (1) and (3) can be rewritten as: oW oT oW ow ow þ þ ¼0 þC oT w os ow T os ox oH oh oT oH oh ow oh oT oh ow þ þ þ ¼0 þC þC oT w oT w os ow T ow T os oT w ox owT ox
ðA-1Þ ðA-2Þ
One notes:
oT oT oT ah ¼ a w ¼ v ¼ ow h ow w oW w ðoH =oW ÞT ðoH =oT Þw aw oW ¼ k¼1 r¼ b¼ ow T ðoh=owÞT ðoh=oT Þw v oh ¼ oh oT , Eqs. (A-1) and (A-2) become: By dividing Eq. (A-2) by oT and considering that oT ow h ow T oh w w 1 oT ow ow þ ðb þ CÞ þ ¼0 v os os ox ow oT 1 oT aW ow þ ah ðk 1Þ þ þ ah ¼0 r þ ð1 kÞ þ ah C ah þ C v os os ox ox v
ðA-3Þ ðA-4Þ
One introduces the parameter ai which represents the roots of the characteristic equation. By multiplying Eq. (A-3) by (ai ah) and by adding Eq. (A-4), one finds: a a oT a ow oT ah aW ow i h W þ ð ai a h Þ þ þ ai ¼0 þ r ð1 kÞ þ C þ C ah ð1 kÞ þ ah C os os ox ox v v v v ðA-5Þ One searches then for values of ai which verify: ji
oT ow oT ow þ ai þ ai þ ¼0 os os ox ox
ðA-6Þ
h h þ r þ C ¼ ci þ C, It is necessary that ai ka þ r þ C ai ¼ ai avw þ C þ ah k avw . with ji ¼ ai ka v v By simplifying, this condition becomes: a2i ai ðkah þ rv þ aw Þ þ ah kaw ¼ 0
ðA-7Þ
142
P. Stabat, D. Marchio / Applied Energy 85 (2008) 128–142
For each ai there is a Fi function which satisfies the following equation: oF i oF i ¼ ai with i ¼ 1; 2 ow oT
ðA-8Þ
w
The Fi values are termed characteristic potentials. By multiplying Eq. (A-6) by (oFi/oT)w, one obtains: ji
oF i oF i þ ¼0 os ox
for i ¼ 1; 2
ðA-9Þ
Eq. (A-8) and (A-9) define characteristic equations which are equivalent to Eqs. (1) and (3), respectively. Eq. (A-8), which is similar to a wave equation, gives: oT ai ¼ for i ¼ 1; 2 ðA-10Þ oW Fi
References [1] Desiccant Rotors. MUNTERS Dehumidification; 1992. [2] Stabat P. Mode´lisation de composants de syste`mes de climatisation mettant en œuvre l’adsorption et l’e´vaporation d’eau. The`se de doctoral, Ecole des Mines de Paris, France; 2003. [3] Mathiprakasam B, Lavan Z. Performance predictions for adiabatic desiccant-dehumidifiers using linear solutions. J Solar Energy Eng 1980:102. [4] Brandemuehl MJ. Analysis of heat-and-mass transfer regenerators with time varying or spatially non-uniform inlet temperatures. Ph.D Thesis, University of Wisconsin; 1982. [5] Bullock CE, Threlkeld JL. Dehumidification of moist air by adiabatic adsorption. Trans ASHRAE 1966:72. [6] Behne M. Alternatives to compressive cooling in non-residential buildings to reduce primary-energy consumption. Final Report LBL 1997. [7] Lindholm T. Evaporative and desiccant-cooling techniques, feasibility when applied to air conditioning. Ph.D. Thesis, Chalmers University of Technology, Sweden; 2000. [8] Beccali M, Butera F, Guanella R, Adhikari RS. Performance evaluation of rotary desiccant-wheels using a simplified psychrometric model as the design tool. EPIC 2002 ATVC conference 2, Lyon 2002; 419–424. [9] Maclaine-Cross IL. A theory of combined heat-and-mass transfer in regenerators. Ph.D. Thesis, Department of Mechanical Engineering, Monash University, Australia; 1974. [10] Banks PJ. Coupled equilibrium heat and single adsorbate transfer in fluid flow through a porous medium – I Characteristic potentials and specific capacity ratios. Chem Eng Sci 1972;27:1143–55. [11] Maclaine-Cross IL, Banks PJ. Coupled heat-and-mass transfer in regenerators – prediction using an analogy with heat transfer. Int J Heat Mass Transfer 1972;15:1225–42. [12] Close DJ, Banks PJ. Coupled equilibrium heat and single-adsorbate transfer in fluid flow through a porous medium – II Predictions for a silica-gel air-drier using characteristic charts. Chem Eng Sci 1972;27:1157–69. [13] Kays WM, London AL. Compact Heat-Exchangers. third ed. New York: McGraw-Hill; 1984. [14] Howe R. Rotary Desiccant Dehumidifier. TRNLIB – Libraries of User-Written TRNSYS Components, Solar Energy Laboratory, http://sel.me.wisc.edu/trnsvs/trnlib/dessi.htm: 1983. [15] Jurinak JJ. Open-cycle solid desiccant cooling – component models and system simulation. Ph.D. Thesis, University of Wisconsin – Madison; 1982. [16] Banks PJ. Predictions of heat-and-mass regenerator performance using non-linear analogy method: Part 1. Basis, Part 2. Comparison of methods. ASME J Heat Transfer 1985;107:222–38. [17] MCC – Series Cooling Cassette, manufacturer’s catalogue, Munters; 1994. [18] SECO Desiccant/enthalpy/dehumidification rotors. Software SECO, Klingenburg; 2002. [19] Slayzak SJ, Pesaran AA, Hancock CE. Experimental evaluation of commercial-desiccant dehumidifier-wheels. http://www.nrel.gov/ desiccantcool/wheels.html; 2002.