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Experimental and simulation studies on a single pass, double duct solar air heater
Energy Conversion and Management 44 (2003) 1209–1227 www.elsevier.com/locate/enconman
Experimental and simulation studies on a single pass, double duct solar air heater F.K. Forson a, M.A.A. Nazha b, H. Rajakaruna a
b,*
Department of Mechanical Engineering, Kwame Nkrumah University of Science & Technology, Kumasi, Ghana b School of Engineering & Technology, De Montfort University, Queens Building, Leicester LE1 9BH, UK Received 15 March 2002; accepted 10 June 2002
Abstract A mathematical model of a single pass, double duct solar air heater (SPDDSAH) is described. The model provides a design tool capable of predicting: incident solar radiation, heat transfer coefficients, mean air flow rates, mean air temperature and relative humidity at the exit. Results from the simulation are presented and compared with experimental ones obtained on a full scale air heater and a small scale laboratory one. Reasonable agreement between the predicted and measured values is demonstrated. Predicted results from a parametric study are also presented. It is shown that significant improvement in the SPDDSAH performance can be obtained with an appropriate choice of the collector parameters and the top to bottom channel depth ratio of the two ducts. The air mass flow rate is shown to be the dominant factor in determining the overall efficiency of the heater. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Solar air heater; Mathematical modelling; Heat transfer
1. Introduction Solar air heaters are employed in many applications requiring low to moderate temperatures (below 60 °C), such as crop drying and space heating. The principal types of these heaters are: the single pass with front duct (SPFDSAH), rear duct (SPRDSAH), double duct (SPDDSAH) and double-pass (DPSAH). In any of these four different arrangements, either a non-porous or a porous/infiltrating absorber can be used. It has been observed [1,2] that the double pass solar air
*
Corresponding author. Tel.: +44-116-2577097; fax: +44-116-2577692. E-mail addresses: [email protected] (F.K. Forson), [email protected] (M.A.A. Nazha), [email protected] (H. Rajakaruna). 0196-8904/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 2 ) 0 0 1 3 9 - 5
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Nomenclature A CP f Gr Gsc hr;pc hr;pb hw hr;cs h1 h2 h3 h4 Hd Ho H HT Ic k L m_ Nu P Ps Pvs Pr Q Ra Re s S T Tf U W
area specific heat capacity of air (J/kg K) friction factor Grashof number solar constant (W/m2 ) radiation heat transfer coefficient from air heater absorber plate to cover (W/m2 K) radiation heat transfer coefficient from air heater absorber plate to base plate (W/m2 K) wind heat transfer coefficient (W/m2 K) radiant heat transfer from collector cover to sky (W/m2 K) heat transfer coefficient between heated air in top channel and glass cover (W/m2 K) heat transfer coefficient between upper surface of absorber plate and heated air in top channel (W/m2 K) heat transfer coefficient between lower surface of absorber plate and heated air in bottom channel (W/m2 K) heat transfer coefficient between base plate and heated air in bottom channel (W/m2 K) monthly average daily diffuse radiation on horizontal plane (J/m2 ) extraterrestrial irradiance on horizontal plane (W/m2 ) monthly average daily radiation on horizontal plane (J/m2 ) monthly average daily radiation on tilted plane (J/m2 ) energy incident on collector cover per unit area per unit time (W/m2 ) thermal conductivity (kW/m K) length of primary collector (m) air mass flow rate (kg/s) Nusselt number air heater top to bottom channel depth ratio vapour pressure of water at surface of wet material (N/m2 ) water vapour saturation pressure (N/m2 ) Prandtl number air heater length to width ratio Rayleigh Number Reynolds number depth of solar air heater channel (m) monthly average absorbed solar radiation per unit area (W/m2 ) temperature or average temperature (K) mean temperature (K) heat transfer coefficient (W/m2 K) width of air heater (m)
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Greek b e q r
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symbols air heater angle of tilt (°) emittance density of air (kg/m3 ) Stefan–Boltzman constant (kW/m2 K4 )
Subscripts a, amb ambient b base plate c collector cover e edge p absorber plate s sky in inlet out outlet 1 top (upper) channel 2 bottom (lower) channel
heaters perform better than the conventional single pass systems. However, their application in natural convective flow is limited, since air needs to be forced through the two flow channels for efficient utilisation of the system. Studies on the three types of single pass solar air heaters employing a solid absorber have shown that the performance of the SPDDSAH is superior to the other two [3–5]. Several studies have been conducted on the SPDDSAH, but issues regarding the best geometrical proportions are not conclusive. Nevertheless, it is accepted that the determination of the optimum collector length will greatly aid the design of future air heaters [4]. For a SPDDSAH in which the flow is by forced convection, Pawar et al. [6] recommend that for higher efficiencies, the collector length should be 1.5–2.5 m. This was based on a fixed collector width of 1.0 m, suggesting that air heaters having length to width ratios in the range 1.5–2.5 are ideal. However, for a SPDDSAH in which the flow is purely by natural convection, recommendations for optimal designs for specific applications are lacking. Such recommendations require proper understanding of the effects of the design variables and would involve studying the implications of a number of different configurations under varied operating conditions. Accomplishing such a task on full scale prototypes can be time consuming and expensive. Small scale models provide useful information but cannot simulate all the features of the full scale system. Alternatively, a mathematical model of the physical system can be developed and used for investigating the effect of changes in the design and operating conditions on the systemÕs performance at a relatively low cost. Such a model, however, must be validated against experimental results before it can be used with confidence as a design aid for future systems. Mathematical models for predicting the useful energy gained by a flat plate solar collector are generally based on the equation: qu ¼ F 0 ½ðsaÞe Ic UL ðTf Ta Þ
ð1Þ
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where F 0 and UL are the collector efficiency factor and the collector heat loss coefficient, respectively, Tf is the local time average fluid temperature and ðsaÞe is the effective transmittance– absorptance product. An examination of Eq. (1) suggests that the collectorÕs efficiency (the amount of useful energy collected for a given solar radiation input) can be increased by increasing F 0 and ðsaÞe and reducing UL . Parker [7] developed expressions for F 0 and UL for a SPDDSAH, but in arriving at the solution, he made simplifying assumptions regarding the mass flow rate and the heat transfer coefficients. Boindi et al. [8] also derived expressions for F 0 and UL by assuming equal temperatures of the air in the two channels. It is noted from these expressions that the values of F 0 and UL are related to the heat transfer parameters, which are also dependent on the characteristics of the absorber plate, the transparent cover and the base plate of the system. Pawar et al. [6] developed a steady state model aimed at predicting the thermal behaviour of air heaters. Their model provides no means of estimating the air flow rates but uses assumed values, and it over predicts the collector efficiency by 10–15%. Ong [9,10] manipulated the basic governing equations and employed the matrix inversion technique to solve the resulting constitutive equations. In OngÕs model, the mass flow is assumed. The model over predicts the mean temperature of the heated air near the inlet of the collector and under predicts it near the outlet. The mass flow rate affects the performance of a collector significantly, and a proper method of estimating its value is, therefore, necessary. Hegazy [11,12] did modelling work of a SPDDSAH. The work showed that the channel depth-to-length ratio is an important parameter in determining the useful heat gain. His study suggests that for variable flow operation, the optimum depth-to-length ratio should be 0.0025 for both natural [11] and forced convection [12]. Using the above optimum value, he conducted another theoretical study on the effect of collector shape. The study showed that for a constant area, variable width collectors exhibit performance behaviour similar to constant width ones, but with a marginal decrease in collector efficiency [13]. These studies did not address the effect of top-to-bottom channel depth ratio of the two ducts. The work presented in this paper aims at providing a mathematical model capable of predicting the performance of a SPDDSAH, including the properties of the heated air at the exit. It also aims at using the developed model to investigate the effects of the design and operating variables on the performance of a SPDDSAH, perform sensitivity analysis and outline recommendations for an optimal design configuration for a given air heater aperture area that maximises the air temperature and mass flow rate and minimises the relative humidity. The model is validated by comparing the simulation results with those obtained using a small scale laboratory model and a full scale prototype attached to a commercial crop dryer.
2. Description of the system The schematic configuration of the SPDDSAH is illustrated in Fig. 1. Basically, it is an air heater that utilises a non-porous absorber. There are two rectangular flow channels, one above and one below the absorber plate. The collector has a single glazing, and the bottom plywood board serves as both the base or bottom plate and the insulation. The absorber plate is suspended between the glazing and the bottom plate, and the system is also thermally insulated from the sides. Air flow through the air heater is by natural convection. Ambient air enters the two channels at x ¼ 0, and the hot air emerges at x ¼ L. The two hot air streams mix in the plenum.
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Fig. 1. Schematic diagram of a SPDDSAH.
3. Formulation of the mathematical model The equations governing the performance of the system are formulated by coupling the energy balance equations of the components of the air heater with those for the useful heat extracted in the two channels, making the following assumptions: 1. The air heater operates under steady state conditions. 2. The capacitance of the absorber plate is negligible. 3. The operating temperatures of the air heater components are constant. 4. The area of the absorber plate is equal to the aperture area of the collector. 5. The temperature of the air varies only in the direction of the flow (x direction). 6. The sky is considered a blackbody for long wavelength radiation. 7. The heat flow through the glazing and the base plate is one dimensional and in the y direction. 8. The air heater is treated as an inclined chimney. 9. The air heater is considered to be made of sections of equal length in the direction of airflow with each section receiving heated air from the previous section and delivering heated air to the following one. 10. The physical properties of air are assumed to vary linearly with temperature in the range 293 K < T < 333 K. 11. The ambient temperature and wind speed are constant and equal to the long term monthly average day time values. 3.1. Energy balance equations The energy balance equations for the various components of the system and an expression that relates the useful energy absorbed by the heated air to the incident energy (solar radiation) on the system are given below:
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cover plate: S c þ hr;pc ðTp Tc Þ þ h1 ðTf;1 Tc Þ ¼ hw ðTc Ta Þ þ hr;cs ðTc Ts Þ
ð2Þ
air stream 1 (air in top channel): dTf;1 þ Ue;1 s1 ðTf;1 Ta Þ dx
ð3Þ
S p ¼ h3 ðTp Tf;2 Þ þ h2 ðTp Tf;1 Þ þ hr;pb ðTp Tb Þ þ hr;pc ðTp Tc Þ
ð4Þ
h2 W ðTp Tf;1 Þ ¼ h1 W ðTf;1 Tc Þ þ m_ 1 CP a absorber plate:
air stream 2 (air in bottom channel): h3 W ðTp Tf;2 Þ ¼ h4 W ðTf;2 Tb Þ þ m_ 2 CP a
dTf;2 þ Ue;2 s2 ðTf;2 Ta Þ dx
ð5Þ
base plate: h4 ðTf;2 Tb Þ þ hr;pb ðTp Tb Þ ¼ Ub ðTb Ta Þ
ð6Þ
efficiency of the system: gc ¼
m_ 1 CP ;1 ðTf;1out Tf;1in Þ þ m_ 2 CP ;2 ðTf;2out Tf;2in Þ I c Ac
ð7Þ
3.2. Modelling of airflow rate The air flow rate through a chimney is caused by a natural draft resulting from the temperature difference between the hot air at the inlet and the cold air outside, and the pressure drop Dt (N/m2 ) is given by [14]: Tf;m Ta ð8Þ Dt ¼ 233:9qa h Tf;m where Tf;m is the mean temperature of the heated air inside the chimney (K), and h is the vertical height of the stack (m). To account for heat losses in a chimney, the Tf;m , valid for chimneys less than 9 m tall, is given by [15]: Tf;m ¼ 5=9fTa 0:65ðTa Tf;out Þg
ð9Þ
where Ta and Tf;out are the temperatures at the inlet and outlet of the chimney, respectively. Considering the air heater as an inclined chimney and neglecting inertia forces, the thermal pressure difference inside the collector is balanced by the pressure drop due to friction across the collector, and hence, the mass flow rate of air in each of the two channels is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)ffi u ( u Tf;out Ta ð10Þ m_ i ¼ qi Wsi CV ;i Fv;i t2 233:9qa h Tf;out þ 137 Ta pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where h ¼ L sin b, Fv;i ¼ dh;i =fLqi , dh;i ¼ hydraulic mean depth of the duct and L is the collector length (m), Cv;i the coefficient of velocity and W the width of the collector (m).
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3.3. Estimation of heat transfer coefficients The radiative heat transfer coefficients from the absorber to the glazing, hr;pc , and from the absorber to the bottom plate, hr;pb , are given by an expression of the general form:
1 1 2 2 þ 1 ð11Þ hr;pi ¼ rðTp þ Ti ÞðTp þ Ti Þ ep ei where i represents c, b. The heat transfer by radiation from the glazing of a solar collector to the sky is given by hr;cs ¼ rec ðTc2 þ Ts2 ÞðTc þ Ts Þ
ð12Þ
where the sky temperature, Ts is evaluated from an equation given by Swinbank [16] and Tc is the temperature of the air heater top cover or glazing. Hollands et al. [17] give the relationship between Nusselt number and Rayleigh number for free convective heat exchange through a tilted cavity with tilt angle, b, 0 6 b 6 60° as: ! # " 1=3 1708ðsin 1:8bÞ1:6 1708 Raðcos bÞ 1 1 þ ð13Þ Nux ¼ 1 þ 1:44 1 Raðcos bÞ Raðcos bÞ 5830 where ½X is defined by: ½X ¼ ðjX j þ X Þ=2 The average value of the Nusselt number is related to the local value by the relation Nu ¼ 43Nux
ð14Þ
Eqs. (13) and (14) are used to evaluate the heat transfer coefficients, h2 and h4 , between the hot air and the heated absorber plate for large aspect ratios ðL=s > 12Þ. For small aspect ratios, ðL=sÞ 6 12, it is suggested [18] that reasonable results may be obtained for rectangular cavities as follows: b=bc NuV ðb=4bc Þ NuL ¼ NuH ðsin bc Þ ð15Þ NuH Values of bc are provided by Incropera and DeWitt [18]. The convection heat transfer coefficient for a horizontal cavity, NuH , with heated plate below is obtained from Eq. (13) by making b ¼ 0. For a vertical rectangular cavity, with aspect ratio from 2 to 10, Pr < 105 and 103 < Ra < 1010 , the following correlation has been suggested [18]: 0:28 1=4 Pr L Nuv ¼ 0:22 ð16Þ Ra 0:2 þ Pr s The characteristic length in the expression for Nusselt number for Eqs. (13)–(16) is the depth of the corresponding air channel. In using these equations, the properties of air are evaluated at the mean temperature of the plates enclosing the cavity. The heat transfer coefficient, h1 , between the hot air and the inner surface of the relatively cold tilted glass cover is estimated using the following correlation [19]:
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Nu ¼
h i h1 X ¼ 0:27 ðGrPr cos bÞ1=3 k
ð17Þ
where X is the characteristic length taken as the surface area divided by the perimeter that encompasses the area. The heat transfer coefficient, h3 , is calculated using the correlation for calculating the heat transfer coefficients between two horizontal plates (with the hot plate uppermost) and accounting for the tilt as follows [20]: Nu ¼ 0:135ðGrPr cos bÞ1=3
ð18Þ
The wind heat transfer coefficient is evaluated using the correlation proposed by Othieno [21] for angles of tilt in the range 0–20° hw ¼ kð75 þ 0:42Re0:6 Þ=X
ð19Þ
where the characteristic length, X, is four times the plate area divided by the plate perimeter. The back loss coefficient Ub is given by Klein [22]: Ub ¼ 1=ðx=k þ 1=hb Þ
ð20Þ
where k and x are the thermal conductivity and thickness of the back insulation, respectively, and hb is the convection coefficient between the bottom of the insulation and the environment. The values of hb range from 12.5 to 25 W/m2 K. The edge heat losses are estimated by assuming one dimensional sideways heat flow around the perimeter of the collector system. The edge loss heat transfer coefficient is referred to the collector area Ac and given by: Uei ¼ ðUAÞedge =Ac ¼ fðk=xÞsi g=W
ð21Þ
3.4. Radiation on the collector surface The monthly average daily extraterrestrial radiation on a horizontal surface, H o , is [23]: ! i h 24 3600 360 n pxs Gsc 1 þ 0:033 cos cos / cos d sin xs þ sin / sin d ð22Þ Ho ¼ p 365 180 ¼ mean day (based on solar radiation) of the month, / ¼ latitude of where Gsc ¼ 1367 W/m2 , n the site, d ¼ declination and xs ¼ sunset hour angle. Eq. (22) is valid for evaluating H o for latitudes in the range 60° to þ60° [23]. The Angstrom–Page regression equation, as developed for Kumasi, Ghana [24], was used to relate the monthly average daily radiation to the extraterrestrial radiation on a horizontal surface. The mean monthly daily total radiation on a tilted surface, H T , is related to H and R as: H T ¼ RH
ð23Þ
where H is the monthly average radiation on a horizontal surface, and the factor R is an estimate of the sum of three terms corresponding to beam, diffuse and ground-reflected radiation on the inclined surface.
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Klein and Theilacker [25] proposed an algorithm for estimating R valid for any surface ori for south facing collectors situated in entation and for all latitudes. The simplified expression of R the northern hemisphere was used. This algorithm requires an estimate of H d =H , the monthly average diffuse radiation fraction. H d =H was related to K T (¼ H=H o ), the ratio of the average daily total to extraterrestrial radiation or the monthly average daily clearance index. The correlation of Erbs et al. [26], independent of the season, was used for estimating H d =H . The monthly average transmittance–absorptance product, ðsaÞav , is related to the monthly average radiation incident on the collector and the monthly average solar radiation absorbed per unit absorber plate area, S ph 0
S ph ¼ H T ðsaÞav ¼ R H
ð24Þ
The radiation absorbed by the absorber per unit area (S ph ) is evaluated by introducing a factor to account for the effects of dust and shading on the collector performance. Hence, the effective monthly average daily absorbed radiation per unit area (by the absorber) is given by ! n 0 00 00 S ph ¼ fR a þ b Ho ð25Þ N where H o is the monthly average daily extraterrestrial radiation on a horizontal surface, n is the monthly average daily hours of bright sunshine, a00 and b00 are empirical constants dependent on the location and N is the monthly average of the maximum possible hours of bright sunshine (i.e. the day length of the average day of the month). In this model, it is assumed that solar radiation is collected over a time interval of Nds , corresponding to the day length for drying, on each day of the month. Hence, the steady state intensity of the monthly mean absorbed solar radiation (by the absorber), S p , is given as Ho 0 00 00 n S p ¼ fR a þ b ð26Þ N 3600 N ds where f is a factor to account for the combined effects of shading and dust and n is the number of bright sunshine hours on the mean day of the month. The intensity of solar radiation absorbed by the collector cover per unit area is also given by S c ¼ ac RK T
Ho Nds 3600
ð27Þ
3.5. Thermo-physical properties of air in the plenum The air inside the plenum is a combination of the air streams exiting the two channels. In estimating the temperature of the air in the plenum, we assume adiabatic mixing of the two air streams from the two channels to achieve a conservative estimate of the plenum air temperature. This assumption is made on the premise that once the air is inside the plenum, there is no radiative or conductive heat transfer to the air. In real terms, the effect of heat losses is taken into account by the factors n and c based on experimentation. The expression for the average temperature of the air in the plenum, Tf , is given by:
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Tf ¼ n
m_ 1 CP ;1 Tf;1 þ m_ CP ;2 Tf;2 þc ðm_ 1 þ m_ 2 Þ
ð28Þ
where n and c are constants determined experimentally (a linear regression, using Eq. (28) and 208 data points, obtained on both the full scale and the small scale models, produced n ¼ 0:94 and c ¼ 0:56). The water vapour saturation pressure, Pvs , for the temperature range of 0–473 K is given by: 2 3 lnðpvs Þ ¼ C1 =Tdb þ C2 þ C3 Tdb þ C4 Tdb þ C5 Tdb þ C6 lnðTdb Þ
ð29Þ
where C1 ¼ 5:8002206 10 , C2 ¼ 5:516256, C3 ¼ 4:8640239 10 , C4 ¼ 4:1764768 105 , C5 ¼ 1:4452093 108 , C6 ¼ 6:5459763 and Tdb is the dry bulb temperature. The humidity ratio, H, (kg water vapour/kg dry air) in the temperature range of 225–533 K is given by: 3
H¼
0:6219Pv Patm Pv
2
ð30Þ
The partial pressure of water vapour, Pv , is evaluated using Eq. (29) by replacing the dry bulb temperature with its corresponding dew point temperature, Td , of a given sample of moist air. The dew point temperature can be calculated from the following equation: Td ¼ a þ ba þ ca2 þ da3 þ eðPv Þ0:1984
ð31Þ
where Td ¼ dew point temperature, °C, a ¼ lnðPv Þ, Pv ¼ water vapour partial pressure, kPa, a ¼ 6:54, b ¼ 14:526, c ¼ 0:7389, d ¼ 0:09486 and e ¼ 0:4569. The relative humidity, RH, is, by definition, given by Pv ð32Þ RH ¼ Pvs The physical properties of air are assumed to vary linearly with temperature. 4. Solution procedure A FORTRAN computer program was written and used to solve the model equations. Two input variables of the model, namely the solar intensity, H T , and the ambient temperature, Ta , are time dependent, and hence, the values of the absorber plate temperature, Tp , fluid temperatures in the top and bottom channels, Tf;1 , Tf;2 , respectively, which are dependent on H T and ambient temperature, Ta , are also time dependent. In order to simplify the solution procedure, it is assumed that the air heater is subject to a constant insulation, and hence, the plate temperature is assumed constant. Eqs. (2), (4) and (6) are used to write implicit expressions for Tf;1 , Tf;2 , Tb and Tc in terms of the defined parameters. These are then substituted into Eqs. (3) and (5) resulting in the following first order differential equations: dTf;1 ¼ /1 Tf;1 þ /2 dx
ð33Þ
dTf;2 ¼ /3 Tf;2 þ /4 dx
ð34Þ
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The finite difference technique is employed to solve Eqs. (33) and (34). An explicit forward differencing method is used with an iterative technique where Tf;i ¼ ðTf;iþ1 þ Tf;i Þ=2
and
dTi Tiþ1 Ti ¼ dx Dx
ð35Þ
5. Experiments Two sets of experiments were conducted, one set using a small scale laboratory model of the SPDDSAH and the other using a full scale commercial type SPDDSAH. The small scale air heater had an aperture measuring 440 mm wide and 740 mm long with an overall duct depth of 75 mm. Ten, 100 W each, infrared heat lamps were used to provide the incident energy of average intensity of 464 W/m2 on the plane of the collector. The full scale field model air heater is attached to a commercial size drying chamber in Ghana. It measures 6.75 m wide and 12.6 m long with an overall duct depth of 280 mm. The detailed description of the design of this heater and dryer combination has been given previously [27]. 5.1. Laboratory or indoor experiments Three tests were conducted for three different locations of the absorber plate between the cover plate and the base plate. The purpose of this was to assess the effect of the top to bottom channel depth ratio on the performance of the SPDDSAH. This ratio was set at 0.364:1, 1.08:1 and 3.42:1 in the three tests, respectively. Each test was conducted over a period of at least 3 days with the infrared lamps switched on for between 5 and 7 h each day. 5.1.1. Instrumentation of the small scale model The laboratory model was instrumented with appropriate thermocouples for measuring the glazing, absorber plate, air in the flow channels and base plate temperatures along the flow direction of the collector. The top and bottom values of the glazing, absorber and base plate temperatures were measured at the middle and the outlet section of the collector. The temperatures of the air were, however, taken along the centre line of the two channels at the inlet and outlet sections. These measurements were used to compute the weighted mean averages of the temperatures. A pyranometer (Kipp and Zonen type CA 1 No. 754379) with microvagalvanometer (type AL 4) readout was employed to measure the global radiation at several points over the field of the collector and averaged to give the average ‘‘solar’’ irradiance. Two hot wire anemometers, for measuring velocities in the range 0–2 m/s, were used to measure the air velocities along the centre line of the air channels at the inlet and outlet of the collector and the wind speeds inside the enclosed test room. These velocity readings were taken at hourly intervals. A hand held hygrometer was used to measure the relative humidity of air at the collectorÕs inlet and outlet. 5.2. Field or outdoor experiment The outdoor experiment was conducted on a SPDDSAH with a fixed top to bottom channel depth ratio of 1:1.
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5.2.1. Instrumentation of the large scale model An LI200S LI-COR PYRANOMETER located at the middle of the collector was used to measure the incident global solar radiation. The collector was instrumented with k-type thermocouples for measuring the temperatures of the components of the air heater, i.e. the absorber plate, the glazing and the base plate, at three locations along the length of the air heater. At each of these three locations, the top and the bottom temperatures of the components were taken. The temperatures of the air in the two flow channels of the air heater were measured at the inlet and outlet sections. The ambient temperature and the temperature of the air in the plenum were also measured with thermocouples. The ambient wind speed, the outlet air velocity and relative humidity at the inlet and outlet of the collector were also obtained.
6. Results and discussion A comparison of the daily hourly mean values of the measured parameters and their corresponding predicted values for three top to bottom channel depth ratios of the laboratory SPDDSAH is presented in Table 1. The values presented are based on a steady state incident radiation of 464 W/m2 . The daily hourly average values of the day time values of the measured parameters and their corresponding predicted values for the full scale air heater are summarised in Table 2.
7. Parametric studies on the small scale SPDDSAH In order to optimise the design of the SPDDSAH, parametric studies were conducted on the laboratory air heater using the simulation model. The study was designed to examine the factors that are within the control of a solar air heater designer and that can be optimised in order to improve its performance. The focus of the study reported in this paper, therefore, was concerned with determining the best air heater design configuration without compromising on cost. Table 1 Comparison of predicted and measured average values for the laboratory air heater Air heater model: performance indicators Tc (°C) Tp (°C)
Tb (°C) Tf;1 (°C) Tf;2 (°C) Tf (°C)
m_ (kg/s)
RH (%)
gc (%)
35 28 )20
49.6 56.5 þ13.9
Test 1 Measured 46.8 (P ¼ 0:364:1) Predicted 43.3 Difference (%) )7.1
56.1 57.0 þ1.6
27.5 25.0 )9.1
46.1 42.0 )8.9
24.9 26.2 þ5.2
27.5 29.5 þ7.3
0.0164 0.0144 )12.2
Test 2 Measured (P ¼ 1:08:1) Predicted Difference (%)
44.8 44.8 0
54.5 56.5 þ3.7
29 29 0
38.9 36.8 )5.4
33.2 31.8 )4.2
31.9 34.3 þ7.5
0.0123 0.0130 þ5.7
35.8 29.7 )17.0
63.5 64.0 þ0.8
Test 3 Measured 45.3 (P ¼ 3:42:1) Predicted 43.4 Difference (%) )4.4
52.6 55.4 þ5.3
33.1 31.6 )4.5
34.5 34.1 )1.1
38.9 36.1 )7.2
37.1 34.4 )7.3
0.0139 0.0145 þ4.3
39.8 31.7 )20.4
71.3 71.0 )1.0
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Table 2 Comparison of predicted and measured average values for the full scale air heater Parameter
Measured Predicted Difference (%)
Tc (°C)
Tp (°C)
Tb (°C)
Tf;1 (°C)
Tf;2 (°C)
Tf (°C)
m_ (kg/s) RH2 (%)
RAD (W/m2 )
41.4 42.1 þ1.7
42.4 47.5 þ12
36.5 37.3 þ2.2
40.2 41.9 þ4.2
35.4 39.2 þ10.7
37.6 40.3 þ7.2
0.369 0.350 )5.1
341 479 þ40.5
43.1 34.8 )19.7
The investigation assessed the effect of the following four parameters on the performance of the air heater: (i) (ii) (iii) (iv)
the air flow channel top to bottom channel depth ratio, P (¼ s1 =s2 ), the length to width ratio, Q (¼ L=W ), the overall collector channel depth, s (¼ s1 þ s2 ), the area of the collector, Ac .
Using the same base line design parameters of the laboratory air heater (P ¼ 0:364, Q ¼ 1:586, s ¼ 75 mm and Ac ¼ 0:317 m2 ), the value of only one parameter was varied for a given set and the effect of this change on the collectorÕs predicted performance was obtained. Fig. 2 shows that as the top to bottom channel depth ratio is increased for a given overall depth, there is an increase in the temperature and a reduction in the mass flow rate of the heated air in the bottom channel; and the reverse occurs in the upper channel. A better way of looking at Fig. 2 is to consider an increasing P (i.e. P > 1), and a decreasing P (i.e. P < 1). These two cases signify an increasing greater depth of the top channel relative to the bottom one and an increasing greater depth of the bottom channel relative to the top one,
Fig. 2. Predicted variation of air temperature and mass flow rate with P.
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respectively. In both cases, a decrease in the depth of a channel results in an increase in the flow resistance, leading to a reduction in the mass flow and an increase in the temperature. However, the temperature of the heated air in the upper channel appears to increase more rapidly when P < 1 and reducing than the case for the lower channel with P > 1 and increasing. This prediction is supported by the experimental findings presented in Table 1. The efficiency of the air heater also increases with increasing P, while the relative humidity (normalised to the inlet condition) decreases as shown in Fig. 3. Fig. 4 shows that an increase in the value of S causes an increase in the mass flow rate and a decrease in the temperature of the heated air in both channels. An increase in the channel flow area results in a lower flow resistance and, hence, an increase in the mass flow rate, accompanied by a reduction in the air temperature. The efficiency of the heater also increases with S as shown in Fig. 5. A similar trend of results has been reported in the literature [10]. The useful heat transferred to the air is given by Qu ¼ m_ CP DT . Since DT decreases as the overall depth (S) increases, the increase in efficiency suggests that the increase in mass flow rate outweighs the effect of the decrease in the corresponding temperature of the heated air. Thus, the mass flow becomes the dominant determining factor governing the performance of the air heater with an increase in the value of the overall collector depth. However, the extent to which S can be increased to achieve an increase in collector efficiency is limited, since for collectors with a tilt angle up to 60°, the aspect ratio (L=s) of the collector should be between 20 and 200 [28]. This restriction places a limit on the maximum value of S for a collector of given area. It is also significant to note that with this option, the increase in efficiency is at the expense of the relative humidity, which increases with S as shown in Fig. 5. Thus, increasing the air heater channel depth may result in heated air conditions that are unfavourable for certain applications (such as drying purposes). On the other hand, it can be inferred from the results that in order to achieve a higher air stream temperature, a
Fig. 3. Predicted variation of efficiency and relative humidity with P.
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Fig. 4. Predicted variation of air temperature and mass flow rate with S.
Fig. 5. Predicted variation of efficiency and relative humidity with S.
small flow channel depth is required. However, this may cause large frictional losses over the length of the air heater, resulting in a reduced mass flow rate. A compromise is needed that takes into account the specific application of the air heater. Fig. 6 shows that as Q is increased, the temperature of the air in each channel increases, while the mass flow rate decreases. An increase in the value of Q for a given collector area implies a reduction in the value of the width of the collector and an increase in its length. Thus, for a given
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Fig. 6. Predicted variation of air temperature and mass flow rate with Q.
configuration of the collector with S, P and Ac maintained constant, there is a reduction in the cross-sectional area of each channel as Q increases. This leads to an increase in turbulence, resulting in a decrease in the mass flow accompanied by an increase in the heated air temperature. Fig. 7 indicates that increasing Q leads to reductions in the efficiency and the normalised relative humidity of the heated air. The temperature of the heated air, however, increases with increasing Q. This suggests that the mass flow rate is again the dominant factor governing the performance of the air heater. Reducing the length to width ratio for a given collector is an
Fig. 7. Predicted variation of efficiency and relative humidity with Q.
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Fig. 8. Predicted variation of air temperature and mass flow rate with Ac .
alternative proposition for increasing its efficiency without compromising on the cost, but it is at the expense of decreasing temperatures and increasing relative humidity of the heated air. Fig. 8 clearly illustrates that an increase in the collector area will lead to an increase in both the temperature and the mass flow rate. This option does not necessarily alter the flow channel configuration, but the total solar radiation absorbed by the collector is increased and, consequently, the available energy increases. However, the efficiency of the collector decreases, and the relative humidity of the heated air increases, as shown in Fig. 9.
Fig. 9. Predicted variation of efficiency and relative humidity with Ac .
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It is evident from the study, that for an existing SPDDSAH, the most effective modification that can be made to enhance its performance appears to be the re-positioning of the absorber plate to obtain higher values of P. As for new designs, a number of parameters have to be chosen carefully with the ultimate purpose of the heater in mind to achieve the most effective performance.
8. Conclusion The following conclusions may be drawn from the foregoing case and parametric studies: 1. The developed mathematical model provides reasonable predictions of the performance of a SPDDSAH and can be a useful design tool for future development of a SPDDSAH to satisfy specific applications. 2. Significant improvement in the SPDDSAH performance may be obtained through a careful choice of a number of collector parameters. 3. A compromise between higher values of Q, Ac and P is needed to achieve the required temperature, mass flow rates and relative humidity of the heated air. 4. Increasing the top to bottom channel ratio is a cost effective way of optimising the thermal performance of a SPDDSAH. 5. The mass flow rate is the dominant factor in determining the efficiency of a natural convection air heater, hence a careful estimation of the mass flow rate is essential.
Acknowledgements The authors wish to acknowledge the Commonwealth Scholarship Commission, the British Council and the Ghana Government for sponsoring the Ph.D. studies of the first author at De Montfort University, Leicester, England. The technical support of the Technicians of the Department of Mechanical Engineering, UST-Ghana and the School of Engineering and Technology, DMU, England, during the construction and instrumentation of the set ups are also acknowledged.
References [1] Mohamad A. A high efficiency solar air heater. Sol Energy 1997;60(2):71–6. [2] Wijeysundera NE, Lee AH, Tjioe LE. Thermal performance study of two-pass solar air heaters. Sol Energy 1982;28(5):363–70. [3] Macedo IC, Altemani CAC. Experimental evaluation of natural convection solar air heaters. Sol Energy 1978;20(5):367–9. [4] Ong KS. Results of investigation into forced and natural convection solar air heater and crop dryers. Reg J Energy Heat Mass Transfer 1982;4(1):29–45. [5] Close DJ. Solar air heaters for low and moderate temperature applications. Sol Energy 1963;7(3):117–24. [6] Pawar RS, Takwale MG, Bhide VG. Evaluation of the performance of the solar air heater. Energy Convers Manage 1994;35(1994):699–708.
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[7] Parker BF. Design equations for solar air heaters. Trans ASAE 1980;23:1494–9, 1504. [8] Boindi P, Cicala L, Farina G. Performance analysis of solar air heaters of conventional design. Sol Energy 1988;41(1):101–7. [9] Ong KS. Thermal performance of solar air heaters: mathematical model and solution procedure. Sol Energy 1995;55(2):93–109. [10] Ong KS. Thermal performance of solar air heaters: experimental correlation. Sol Energy 1995;55(3):209–20. [11] Hegazy AA. Performance of flat plate solar air heaters with optimum channel geometry for constant/variable flow operation. Energy Convers Manage 2000;41:401–17. [12] Hegazy AA. Comparative study of the performances of four photovoltaic/thermal solar air collectors. Energy Convers Manage 2000;41:861–81. [13] Hegazy AA. Thermohydraulic performance of air heating solar collectors with variable width flat absorber plates. Energy Convers Manage 2000;41:1361–78. [14] Achenbach PR, Cole SD. Performance of fourteen masonry chimneys under steady state conditions. Trans ASHVE 1949;55:129–54. [15] Colborne WG, Moffat WC. Fundamental analysis of chimney performance. ASHRAE J 1959;1(3):55–60. [16] Swinbank WC. Long-wave radiation from clear skies. Quart J Roy Meteor Soc 1963;89:339–48. [17] Hollands KGT, Unny TE, Raithby GD, Konicek L. Free convective heat transfer across inclined air layers. Trans ASME, J Heat Transfer 1976;98:189–93. [18] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. 4th ed New York: John Wiley & Sons; 1996, p. 509–12. [19] El Sayed MM. Mathematical modelling of a thin layer solar kiln. Trans ASME, J Sol Energy Eng 1990;12:196–203. [20] Wong HY. Handbook of essential formulae and data on heat transfer for engineers. New York: Longman Inc.; 1977, p. 54–5. [21] Othieno H. Natural convection solar crop dryers in Kenya-Theory and practical application. CSC Technical Publication Series No. 24, 1987. [22] Klein SA. Calculation of the monthly-average transmittance–absorptance product. Sol Energy 1979;23(6):547–51. [23] Duffie JA, Beckman WA. Solar engineering of thermal processes. 2nd ed. New York: Wiley Interscience; 1991. [24] Jackson EA, Akuffo FO. Correlation between monthly average global irradiation and relative duration of sunshine at Kumasi. Energy Convers Manage 1992;33(1):13–22. [25] Klein SA, Theilacker JC. An algorithm for calculating monthly-average radiation on inclined surfaces. Trans ASME, J Solar Energy Eng 1981;103:29–33. [26] Erbs DGS, Klein SA, Dufie JA. Estimation of the diffuse radiation fraction for hourly daily and monthly-average global radiation. Sol Energy 1982;28:293–302. [27] Forson FK, Akuffo FO, Nazha MAA. Natural convection solar crop-dryers of commercial scale in Ghana: design construction and performance. Ambient Energy 1996;17(3):123–30. [28] Buchberg H, Catton I, Edwards DK. Natural convection in enclosed spaces: review of applications to solar energy collection. ASME Paper 74-WA/HT-12 1974.
Experimental and simulation studies on a single pass, double duct solar air heater F.K. Forson a, M.A.A. Nazha b, H. Rajakaruna a
b,*
Department of Mechanical Engineering, Kwame Nkrumah University of Science & Technology, Kumasi, Ghana b School of Engineering & Technology, De Montfort University, Queens Building, Leicester LE1 9BH, UK Received 15 March 2002; accepted 10 June 2002
Abstract A mathematical model of a single pass, double duct solar air heater (SPDDSAH) is described. The model provides a design tool capable of predicting: incident solar radiation, heat transfer coefficients, mean air flow rates, mean air temperature and relative humidity at the exit. Results from the simulation are presented and compared with experimental ones obtained on a full scale air heater and a small scale laboratory one. Reasonable agreement between the predicted and measured values is demonstrated. Predicted results from a parametric study are also presented. It is shown that significant improvement in the SPDDSAH performance can be obtained with an appropriate choice of the collector parameters and the top to bottom channel depth ratio of the two ducts. The air mass flow rate is shown to be the dominant factor in determining the overall efficiency of the heater. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Solar air heater; Mathematical modelling; Heat transfer
1. Introduction Solar air heaters are employed in many applications requiring low to moderate temperatures (below 60 °C), such as crop drying and space heating. The principal types of these heaters are: the single pass with front duct (SPFDSAH), rear duct (SPRDSAH), double duct (SPDDSAH) and double-pass (DPSAH). In any of these four different arrangements, either a non-porous or a porous/infiltrating absorber can be used. It has been observed [1,2] that the double pass solar air
*
Corresponding author. Tel.: +44-116-2577097; fax: +44-116-2577692. E-mail addresses: [email protected] (F.K. Forson), [email protected] (M.A.A. Nazha), [email protected] (H. Rajakaruna). 0196-8904/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 2 ) 0 0 1 3 9 - 5
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Nomenclature A CP f Gr Gsc hr;pc hr;pb hw hr;cs h1 h2 h3 h4 Hd Ho H HT Ic k L m_ Nu P Ps Pvs Pr Q Ra Re s S T Tf U W
area specific heat capacity of air (J/kg K) friction factor Grashof number solar constant (W/m2 ) radiation heat transfer coefficient from air heater absorber plate to cover (W/m2 K) radiation heat transfer coefficient from air heater absorber plate to base plate (W/m2 K) wind heat transfer coefficient (W/m2 K) radiant heat transfer from collector cover to sky (W/m2 K) heat transfer coefficient between heated air in top channel and glass cover (W/m2 K) heat transfer coefficient between upper surface of absorber plate and heated air in top channel (W/m2 K) heat transfer coefficient between lower surface of absorber plate and heated air in bottom channel (W/m2 K) heat transfer coefficient between base plate and heated air in bottom channel (W/m2 K) monthly average daily diffuse radiation on horizontal plane (J/m2 ) extraterrestrial irradiance on horizontal plane (W/m2 ) monthly average daily radiation on horizontal plane (J/m2 ) monthly average daily radiation on tilted plane (J/m2 ) energy incident on collector cover per unit area per unit time (W/m2 ) thermal conductivity (kW/m K) length of primary collector (m) air mass flow rate (kg/s) Nusselt number air heater top to bottom channel depth ratio vapour pressure of water at surface of wet material (N/m2 ) water vapour saturation pressure (N/m2 ) Prandtl number air heater length to width ratio Rayleigh Number Reynolds number depth of solar air heater channel (m) monthly average absorbed solar radiation per unit area (W/m2 ) temperature or average temperature (K) mean temperature (K) heat transfer coefficient (W/m2 K) width of air heater (m)
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Greek b e q r
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symbols air heater angle of tilt (°) emittance density of air (kg/m3 ) Stefan–Boltzman constant (kW/m2 K4 )
Subscripts a, amb ambient b base plate c collector cover e edge p absorber plate s sky in inlet out outlet 1 top (upper) channel 2 bottom (lower) channel
heaters perform better than the conventional single pass systems. However, their application in natural convective flow is limited, since air needs to be forced through the two flow channels for efficient utilisation of the system. Studies on the three types of single pass solar air heaters employing a solid absorber have shown that the performance of the SPDDSAH is superior to the other two [3–5]. Several studies have been conducted on the SPDDSAH, but issues regarding the best geometrical proportions are not conclusive. Nevertheless, it is accepted that the determination of the optimum collector length will greatly aid the design of future air heaters [4]. For a SPDDSAH in which the flow is by forced convection, Pawar et al. [6] recommend that for higher efficiencies, the collector length should be 1.5–2.5 m. This was based on a fixed collector width of 1.0 m, suggesting that air heaters having length to width ratios in the range 1.5–2.5 are ideal. However, for a SPDDSAH in which the flow is purely by natural convection, recommendations for optimal designs for specific applications are lacking. Such recommendations require proper understanding of the effects of the design variables and would involve studying the implications of a number of different configurations under varied operating conditions. Accomplishing such a task on full scale prototypes can be time consuming and expensive. Small scale models provide useful information but cannot simulate all the features of the full scale system. Alternatively, a mathematical model of the physical system can be developed and used for investigating the effect of changes in the design and operating conditions on the systemÕs performance at a relatively low cost. Such a model, however, must be validated against experimental results before it can be used with confidence as a design aid for future systems. Mathematical models for predicting the useful energy gained by a flat plate solar collector are generally based on the equation: qu ¼ F 0 ½ðsaÞe Ic UL ðTf Ta Þ
ð1Þ
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where F 0 and UL are the collector efficiency factor and the collector heat loss coefficient, respectively, Tf is the local time average fluid temperature and ðsaÞe is the effective transmittance– absorptance product. An examination of Eq. (1) suggests that the collectorÕs efficiency (the amount of useful energy collected for a given solar radiation input) can be increased by increasing F 0 and ðsaÞe and reducing UL . Parker [7] developed expressions for F 0 and UL for a SPDDSAH, but in arriving at the solution, he made simplifying assumptions regarding the mass flow rate and the heat transfer coefficients. Boindi et al. [8] also derived expressions for F 0 and UL by assuming equal temperatures of the air in the two channels. It is noted from these expressions that the values of F 0 and UL are related to the heat transfer parameters, which are also dependent on the characteristics of the absorber plate, the transparent cover and the base plate of the system. Pawar et al. [6] developed a steady state model aimed at predicting the thermal behaviour of air heaters. Their model provides no means of estimating the air flow rates but uses assumed values, and it over predicts the collector efficiency by 10–15%. Ong [9,10] manipulated the basic governing equations and employed the matrix inversion technique to solve the resulting constitutive equations. In OngÕs model, the mass flow is assumed. The model over predicts the mean temperature of the heated air near the inlet of the collector and under predicts it near the outlet. The mass flow rate affects the performance of a collector significantly, and a proper method of estimating its value is, therefore, necessary. Hegazy [11,12] did modelling work of a SPDDSAH. The work showed that the channel depth-to-length ratio is an important parameter in determining the useful heat gain. His study suggests that for variable flow operation, the optimum depth-to-length ratio should be 0.0025 for both natural [11] and forced convection [12]. Using the above optimum value, he conducted another theoretical study on the effect of collector shape. The study showed that for a constant area, variable width collectors exhibit performance behaviour similar to constant width ones, but with a marginal decrease in collector efficiency [13]. These studies did not address the effect of top-to-bottom channel depth ratio of the two ducts. The work presented in this paper aims at providing a mathematical model capable of predicting the performance of a SPDDSAH, including the properties of the heated air at the exit. It also aims at using the developed model to investigate the effects of the design and operating variables on the performance of a SPDDSAH, perform sensitivity analysis and outline recommendations for an optimal design configuration for a given air heater aperture area that maximises the air temperature and mass flow rate and minimises the relative humidity. The model is validated by comparing the simulation results with those obtained using a small scale laboratory model and a full scale prototype attached to a commercial crop dryer.
2. Description of the system The schematic configuration of the SPDDSAH is illustrated in Fig. 1. Basically, it is an air heater that utilises a non-porous absorber. There are two rectangular flow channels, one above and one below the absorber plate. The collector has a single glazing, and the bottom plywood board serves as both the base or bottom plate and the insulation. The absorber plate is suspended between the glazing and the bottom plate, and the system is also thermally insulated from the sides. Air flow through the air heater is by natural convection. Ambient air enters the two channels at x ¼ 0, and the hot air emerges at x ¼ L. The two hot air streams mix in the plenum.
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Fig. 1. Schematic diagram of a SPDDSAH.
3. Formulation of the mathematical model The equations governing the performance of the system are formulated by coupling the energy balance equations of the components of the air heater with those for the useful heat extracted in the two channels, making the following assumptions: 1. The air heater operates under steady state conditions. 2. The capacitance of the absorber plate is negligible. 3. The operating temperatures of the air heater components are constant. 4. The area of the absorber plate is equal to the aperture area of the collector. 5. The temperature of the air varies only in the direction of the flow (x direction). 6. The sky is considered a blackbody for long wavelength radiation. 7. The heat flow through the glazing and the base plate is one dimensional and in the y direction. 8. The air heater is treated as an inclined chimney. 9. The air heater is considered to be made of sections of equal length in the direction of airflow with each section receiving heated air from the previous section and delivering heated air to the following one. 10. The physical properties of air are assumed to vary linearly with temperature in the range 293 K < T < 333 K. 11. The ambient temperature and wind speed are constant and equal to the long term monthly average day time values. 3.1. Energy balance equations The energy balance equations for the various components of the system and an expression that relates the useful energy absorbed by the heated air to the incident energy (solar radiation) on the system are given below:
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cover plate: S c þ hr;pc ðTp Tc Þ þ h1 ðTf;1 Tc Þ ¼ hw ðTc Ta Þ þ hr;cs ðTc Ts Þ
ð2Þ
air stream 1 (air in top channel): dTf;1 þ Ue;1 s1 ðTf;1 Ta Þ dx
ð3Þ
S p ¼ h3 ðTp Tf;2 Þ þ h2 ðTp Tf;1 Þ þ hr;pb ðTp Tb Þ þ hr;pc ðTp Tc Þ
ð4Þ
h2 W ðTp Tf;1 Þ ¼ h1 W ðTf;1 Tc Þ þ m_ 1 CP a absorber plate:
air stream 2 (air in bottom channel): h3 W ðTp Tf;2 Þ ¼ h4 W ðTf;2 Tb Þ þ m_ 2 CP a
dTf;2 þ Ue;2 s2 ðTf;2 Ta Þ dx
ð5Þ
base plate: h4 ðTf;2 Tb Þ þ hr;pb ðTp Tb Þ ¼ Ub ðTb Ta Þ
ð6Þ
efficiency of the system: gc ¼
m_ 1 CP ;1 ðTf;1out Tf;1in Þ þ m_ 2 CP ;2 ðTf;2out Tf;2in Þ I c Ac
ð7Þ
3.2. Modelling of airflow rate The air flow rate through a chimney is caused by a natural draft resulting from the temperature difference between the hot air at the inlet and the cold air outside, and the pressure drop Dt (N/m2 ) is given by [14]: Tf;m Ta ð8Þ Dt ¼ 233:9qa h Tf;m where Tf;m is the mean temperature of the heated air inside the chimney (K), and h is the vertical height of the stack (m). To account for heat losses in a chimney, the Tf;m , valid for chimneys less than 9 m tall, is given by [15]: Tf;m ¼ 5=9fTa 0:65ðTa Tf;out Þg
ð9Þ
where Ta and Tf;out are the temperatures at the inlet and outlet of the chimney, respectively. Considering the air heater as an inclined chimney and neglecting inertia forces, the thermal pressure difference inside the collector is balanced by the pressure drop due to friction across the collector, and hence, the mass flow rate of air in each of the two channels is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)ffi u ( u Tf;out Ta ð10Þ m_ i ¼ qi Wsi CV ;i Fv;i t2 233:9qa h Tf;out þ 137 Ta pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where h ¼ L sin b, Fv;i ¼ dh;i =fLqi , dh;i ¼ hydraulic mean depth of the duct and L is the collector length (m), Cv;i the coefficient of velocity and W the width of the collector (m).
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3.3. Estimation of heat transfer coefficients The radiative heat transfer coefficients from the absorber to the glazing, hr;pc , and from the absorber to the bottom plate, hr;pb , are given by an expression of the general form:
1 1 2 2 þ 1 ð11Þ hr;pi ¼ rðTp þ Ti ÞðTp þ Ti Þ ep ei where i represents c, b. The heat transfer by radiation from the glazing of a solar collector to the sky is given by hr;cs ¼ rec ðTc2 þ Ts2 ÞðTc þ Ts Þ
ð12Þ
where the sky temperature, Ts is evaluated from an equation given by Swinbank [16] and Tc is the temperature of the air heater top cover or glazing. Hollands et al. [17] give the relationship between Nusselt number and Rayleigh number for free convective heat exchange through a tilted cavity with tilt angle, b, 0 6 b 6 60° as: ! # " 1=3 1708ðsin 1:8bÞ1:6 1708 Raðcos bÞ 1 1 þ ð13Þ Nux ¼ 1 þ 1:44 1 Raðcos bÞ Raðcos bÞ 5830 where ½X is defined by: ½X ¼ ðjX j þ X Þ=2 The average value of the Nusselt number is related to the local value by the relation Nu ¼ 43Nux
ð14Þ
Eqs. (13) and (14) are used to evaluate the heat transfer coefficients, h2 and h4 , between the hot air and the heated absorber plate for large aspect ratios ðL=s > 12Þ. For small aspect ratios, ðL=sÞ 6 12, it is suggested [18] that reasonable results may be obtained for rectangular cavities as follows: b=bc NuV ðb=4bc Þ NuL ¼ NuH ðsin bc Þ ð15Þ NuH Values of bc are provided by Incropera and DeWitt [18]. The convection heat transfer coefficient for a horizontal cavity, NuH , with heated plate below is obtained from Eq. (13) by making b ¼ 0. For a vertical rectangular cavity, with aspect ratio from 2 to 10, Pr < 105 and 103 < Ra < 1010 , the following correlation has been suggested [18]: 0:28 1=4 Pr L Nuv ¼ 0:22 ð16Þ Ra 0:2 þ Pr s The characteristic length in the expression for Nusselt number for Eqs. (13)–(16) is the depth of the corresponding air channel. In using these equations, the properties of air are evaluated at the mean temperature of the plates enclosing the cavity. The heat transfer coefficient, h1 , between the hot air and the inner surface of the relatively cold tilted glass cover is estimated using the following correlation [19]:
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Nu ¼
h i h1 X ¼ 0:27 ðGrPr cos bÞ1=3 k
ð17Þ
where X is the characteristic length taken as the surface area divided by the perimeter that encompasses the area. The heat transfer coefficient, h3 , is calculated using the correlation for calculating the heat transfer coefficients between two horizontal plates (with the hot plate uppermost) and accounting for the tilt as follows [20]: Nu ¼ 0:135ðGrPr cos bÞ1=3
ð18Þ
The wind heat transfer coefficient is evaluated using the correlation proposed by Othieno [21] for angles of tilt in the range 0–20° hw ¼ kð75 þ 0:42Re0:6 Þ=X
ð19Þ
where the characteristic length, X, is four times the plate area divided by the plate perimeter. The back loss coefficient Ub is given by Klein [22]: Ub ¼ 1=ðx=k þ 1=hb Þ
ð20Þ
where k and x are the thermal conductivity and thickness of the back insulation, respectively, and hb is the convection coefficient between the bottom of the insulation and the environment. The values of hb range from 12.5 to 25 W/m2 K. The edge heat losses are estimated by assuming one dimensional sideways heat flow around the perimeter of the collector system. The edge loss heat transfer coefficient is referred to the collector area Ac and given by: Uei ¼ ðUAÞedge =Ac ¼ fðk=xÞsi g=W
ð21Þ
3.4. Radiation on the collector surface The monthly average daily extraterrestrial radiation on a horizontal surface, H o , is [23]: ! i h 24 3600 360 n pxs Gsc 1 þ 0:033 cos cos / cos d sin xs þ sin / sin d ð22Þ Ho ¼ p 365 180 ¼ mean day (based on solar radiation) of the month, / ¼ latitude of where Gsc ¼ 1367 W/m2 , n the site, d ¼ declination and xs ¼ sunset hour angle. Eq. (22) is valid for evaluating H o for latitudes in the range 60° to þ60° [23]. The Angstrom–Page regression equation, as developed for Kumasi, Ghana [24], was used to relate the monthly average daily radiation to the extraterrestrial radiation on a horizontal surface. The mean monthly daily total radiation on a tilted surface, H T , is related to H and R as: H T ¼ RH
ð23Þ
where H is the monthly average radiation on a horizontal surface, and the factor R is an estimate of the sum of three terms corresponding to beam, diffuse and ground-reflected radiation on the inclined surface.
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Klein and Theilacker [25] proposed an algorithm for estimating R valid for any surface ori for south facing collectors situated in entation and for all latitudes. The simplified expression of R the northern hemisphere was used. This algorithm requires an estimate of H d =H , the monthly average diffuse radiation fraction. H d =H was related to K T (¼ H=H o ), the ratio of the average daily total to extraterrestrial radiation or the monthly average daily clearance index. The correlation of Erbs et al. [26], independent of the season, was used for estimating H d =H . The monthly average transmittance–absorptance product, ðsaÞav , is related to the monthly average radiation incident on the collector and the monthly average solar radiation absorbed per unit absorber plate area, S ph 0
S ph ¼ H T ðsaÞav ¼ R H
ð24Þ
The radiation absorbed by the absorber per unit area (S ph ) is evaluated by introducing a factor to account for the effects of dust and shading on the collector performance. Hence, the effective monthly average daily absorbed radiation per unit area (by the absorber) is given by ! n 0 00 00 S ph ¼ fR a þ b Ho ð25Þ N where H o is the monthly average daily extraterrestrial radiation on a horizontal surface, n is the monthly average daily hours of bright sunshine, a00 and b00 are empirical constants dependent on the location and N is the monthly average of the maximum possible hours of bright sunshine (i.e. the day length of the average day of the month). In this model, it is assumed that solar radiation is collected over a time interval of Nds , corresponding to the day length for drying, on each day of the month. Hence, the steady state intensity of the monthly mean absorbed solar radiation (by the absorber), S p , is given as Ho 0 00 00 n S p ¼ fR a þ b ð26Þ N 3600 N ds where f is a factor to account for the combined effects of shading and dust and n is the number of bright sunshine hours on the mean day of the month. The intensity of solar radiation absorbed by the collector cover per unit area is also given by S c ¼ ac RK T
Ho Nds 3600
ð27Þ
3.5. Thermo-physical properties of air in the plenum The air inside the plenum is a combination of the air streams exiting the two channels. In estimating the temperature of the air in the plenum, we assume adiabatic mixing of the two air streams from the two channels to achieve a conservative estimate of the plenum air temperature. This assumption is made on the premise that once the air is inside the plenum, there is no radiative or conductive heat transfer to the air. In real terms, the effect of heat losses is taken into account by the factors n and c based on experimentation. The expression for the average temperature of the air in the plenum, Tf , is given by:
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Tf ¼ n
m_ 1 CP ;1 Tf;1 þ m_ CP ;2 Tf;2 þc ðm_ 1 þ m_ 2 Þ
ð28Þ
where n and c are constants determined experimentally (a linear regression, using Eq. (28) and 208 data points, obtained on both the full scale and the small scale models, produced n ¼ 0:94 and c ¼ 0:56). The water vapour saturation pressure, Pvs , for the temperature range of 0–473 K is given by: 2 3 lnðpvs Þ ¼ C1 =Tdb þ C2 þ C3 Tdb þ C4 Tdb þ C5 Tdb þ C6 lnðTdb Þ
ð29Þ
where C1 ¼ 5:8002206 10 , C2 ¼ 5:516256, C3 ¼ 4:8640239 10 , C4 ¼ 4:1764768 105 , C5 ¼ 1:4452093 108 , C6 ¼ 6:5459763 and Tdb is the dry bulb temperature. The humidity ratio, H, (kg water vapour/kg dry air) in the temperature range of 225–533 K is given by: 3
H¼
0:6219Pv Patm Pv
2
ð30Þ
The partial pressure of water vapour, Pv , is evaluated using Eq. (29) by replacing the dry bulb temperature with its corresponding dew point temperature, Td , of a given sample of moist air. The dew point temperature can be calculated from the following equation: Td ¼ a þ ba þ ca2 þ da3 þ eðPv Þ0:1984
ð31Þ
where Td ¼ dew point temperature, °C, a ¼ lnðPv Þ, Pv ¼ water vapour partial pressure, kPa, a ¼ 6:54, b ¼ 14:526, c ¼ 0:7389, d ¼ 0:09486 and e ¼ 0:4569. The relative humidity, RH, is, by definition, given by Pv ð32Þ RH ¼ Pvs The physical properties of air are assumed to vary linearly with temperature. 4. Solution procedure A FORTRAN computer program was written and used to solve the model equations. Two input variables of the model, namely the solar intensity, H T , and the ambient temperature, Ta , are time dependent, and hence, the values of the absorber plate temperature, Tp , fluid temperatures in the top and bottom channels, Tf;1 , Tf;2 , respectively, which are dependent on H T and ambient temperature, Ta , are also time dependent. In order to simplify the solution procedure, it is assumed that the air heater is subject to a constant insulation, and hence, the plate temperature is assumed constant. Eqs. (2), (4) and (6) are used to write implicit expressions for Tf;1 , Tf;2 , Tb and Tc in terms of the defined parameters. These are then substituted into Eqs. (3) and (5) resulting in the following first order differential equations: dTf;1 ¼ /1 Tf;1 þ /2 dx
ð33Þ
dTf;2 ¼ /3 Tf;2 þ /4 dx
ð34Þ
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The finite difference technique is employed to solve Eqs. (33) and (34). An explicit forward differencing method is used with an iterative technique where Tf;i ¼ ðTf;iþ1 þ Tf;i Þ=2
and
dTi Tiþ1 Ti ¼ dx Dx
ð35Þ
5. Experiments Two sets of experiments were conducted, one set using a small scale laboratory model of the SPDDSAH and the other using a full scale commercial type SPDDSAH. The small scale air heater had an aperture measuring 440 mm wide and 740 mm long with an overall duct depth of 75 mm. Ten, 100 W each, infrared heat lamps were used to provide the incident energy of average intensity of 464 W/m2 on the plane of the collector. The full scale field model air heater is attached to a commercial size drying chamber in Ghana. It measures 6.75 m wide and 12.6 m long with an overall duct depth of 280 mm. The detailed description of the design of this heater and dryer combination has been given previously [27]. 5.1. Laboratory or indoor experiments Three tests were conducted for three different locations of the absorber plate between the cover plate and the base plate. The purpose of this was to assess the effect of the top to bottom channel depth ratio on the performance of the SPDDSAH. This ratio was set at 0.364:1, 1.08:1 and 3.42:1 in the three tests, respectively. Each test was conducted over a period of at least 3 days with the infrared lamps switched on for between 5 and 7 h each day. 5.1.1. Instrumentation of the small scale model The laboratory model was instrumented with appropriate thermocouples for measuring the glazing, absorber plate, air in the flow channels and base plate temperatures along the flow direction of the collector. The top and bottom values of the glazing, absorber and base plate temperatures were measured at the middle and the outlet section of the collector. The temperatures of the air were, however, taken along the centre line of the two channels at the inlet and outlet sections. These measurements were used to compute the weighted mean averages of the temperatures. A pyranometer (Kipp and Zonen type CA 1 No. 754379) with microvagalvanometer (type AL 4) readout was employed to measure the global radiation at several points over the field of the collector and averaged to give the average ‘‘solar’’ irradiance. Two hot wire anemometers, for measuring velocities in the range 0–2 m/s, were used to measure the air velocities along the centre line of the air channels at the inlet and outlet of the collector and the wind speeds inside the enclosed test room. These velocity readings were taken at hourly intervals. A hand held hygrometer was used to measure the relative humidity of air at the collectorÕs inlet and outlet. 5.2. Field or outdoor experiment The outdoor experiment was conducted on a SPDDSAH with a fixed top to bottom channel depth ratio of 1:1.
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5.2.1. Instrumentation of the large scale model An LI200S LI-COR PYRANOMETER located at the middle of the collector was used to measure the incident global solar radiation. The collector was instrumented with k-type thermocouples for measuring the temperatures of the components of the air heater, i.e. the absorber plate, the glazing and the base plate, at three locations along the length of the air heater. At each of these three locations, the top and the bottom temperatures of the components were taken. The temperatures of the air in the two flow channels of the air heater were measured at the inlet and outlet sections. The ambient temperature and the temperature of the air in the plenum were also measured with thermocouples. The ambient wind speed, the outlet air velocity and relative humidity at the inlet and outlet of the collector were also obtained.
6. Results and discussion A comparison of the daily hourly mean values of the measured parameters and their corresponding predicted values for three top to bottom channel depth ratios of the laboratory SPDDSAH is presented in Table 1. The values presented are based on a steady state incident radiation of 464 W/m2 . The daily hourly average values of the day time values of the measured parameters and their corresponding predicted values for the full scale air heater are summarised in Table 2.
7. Parametric studies on the small scale SPDDSAH In order to optimise the design of the SPDDSAH, parametric studies were conducted on the laboratory air heater using the simulation model. The study was designed to examine the factors that are within the control of a solar air heater designer and that can be optimised in order to improve its performance. The focus of the study reported in this paper, therefore, was concerned with determining the best air heater design configuration without compromising on cost. Table 1 Comparison of predicted and measured average values for the laboratory air heater Air heater model: performance indicators Tc (°C) Tp (°C)
Tb (°C) Tf;1 (°C) Tf;2 (°C) Tf (°C)
m_ (kg/s)
RH (%)
gc (%)
35 28 )20
49.6 56.5 þ13.9
Test 1 Measured 46.8 (P ¼ 0:364:1) Predicted 43.3 Difference (%) )7.1
56.1 57.0 þ1.6
27.5 25.0 )9.1
46.1 42.0 )8.9
24.9 26.2 þ5.2
27.5 29.5 þ7.3
0.0164 0.0144 )12.2
Test 2 Measured (P ¼ 1:08:1) Predicted Difference (%)
44.8 44.8 0
54.5 56.5 þ3.7
29 29 0
38.9 36.8 )5.4
33.2 31.8 )4.2
31.9 34.3 þ7.5
0.0123 0.0130 þ5.7
35.8 29.7 )17.0
63.5 64.0 þ0.8
Test 3 Measured 45.3 (P ¼ 3:42:1) Predicted 43.4 Difference (%) )4.4
52.6 55.4 þ5.3
33.1 31.6 )4.5
34.5 34.1 )1.1
38.9 36.1 )7.2
37.1 34.4 )7.3
0.0139 0.0145 þ4.3
39.8 31.7 )20.4
71.3 71.0 )1.0
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Table 2 Comparison of predicted and measured average values for the full scale air heater Parameter
Measured Predicted Difference (%)
Tc (°C)
Tp (°C)
Tb (°C)
Tf;1 (°C)
Tf;2 (°C)
Tf (°C)
m_ (kg/s) RH2 (%)
RAD (W/m2 )
41.4 42.1 þ1.7
42.4 47.5 þ12
36.5 37.3 þ2.2
40.2 41.9 þ4.2
35.4 39.2 þ10.7
37.6 40.3 þ7.2
0.369 0.350 )5.1
341 479 þ40.5
43.1 34.8 )19.7
The investigation assessed the effect of the following four parameters on the performance of the air heater: (i) (ii) (iii) (iv)
the air flow channel top to bottom channel depth ratio, P (¼ s1 =s2 ), the length to width ratio, Q (¼ L=W ), the overall collector channel depth, s (¼ s1 þ s2 ), the area of the collector, Ac .
Using the same base line design parameters of the laboratory air heater (P ¼ 0:364, Q ¼ 1:586, s ¼ 75 mm and Ac ¼ 0:317 m2 ), the value of only one parameter was varied for a given set and the effect of this change on the collectorÕs predicted performance was obtained. Fig. 2 shows that as the top to bottom channel depth ratio is increased for a given overall depth, there is an increase in the temperature and a reduction in the mass flow rate of the heated air in the bottom channel; and the reverse occurs in the upper channel. A better way of looking at Fig. 2 is to consider an increasing P (i.e. P > 1), and a decreasing P (i.e. P < 1). These two cases signify an increasing greater depth of the top channel relative to the bottom one and an increasing greater depth of the bottom channel relative to the top one,
Fig. 2. Predicted variation of air temperature and mass flow rate with P.
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respectively. In both cases, a decrease in the depth of a channel results in an increase in the flow resistance, leading to a reduction in the mass flow and an increase in the temperature. However, the temperature of the heated air in the upper channel appears to increase more rapidly when P < 1 and reducing than the case for the lower channel with P > 1 and increasing. This prediction is supported by the experimental findings presented in Table 1. The efficiency of the air heater also increases with increasing P, while the relative humidity (normalised to the inlet condition) decreases as shown in Fig. 3. Fig. 4 shows that an increase in the value of S causes an increase in the mass flow rate and a decrease in the temperature of the heated air in both channels. An increase in the channel flow area results in a lower flow resistance and, hence, an increase in the mass flow rate, accompanied by a reduction in the air temperature. The efficiency of the heater also increases with S as shown in Fig. 5. A similar trend of results has been reported in the literature [10]. The useful heat transferred to the air is given by Qu ¼ m_ CP DT . Since DT decreases as the overall depth (S) increases, the increase in efficiency suggests that the increase in mass flow rate outweighs the effect of the decrease in the corresponding temperature of the heated air. Thus, the mass flow becomes the dominant determining factor governing the performance of the air heater with an increase in the value of the overall collector depth. However, the extent to which S can be increased to achieve an increase in collector efficiency is limited, since for collectors with a tilt angle up to 60°, the aspect ratio (L=s) of the collector should be between 20 and 200 [28]. This restriction places a limit on the maximum value of S for a collector of given area. It is also significant to note that with this option, the increase in efficiency is at the expense of the relative humidity, which increases with S as shown in Fig. 5. Thus, increasing the air heater channel depth may result in heated air conditions that are unfavourable for certain applications (such as drying purposes). On the other hand, it can be inferred from the results that in order to achieve a higher air stream temperature, a
Fig. 3. Predicted variation of efficiency and relative humidity with P.
F.K. Forson et al. / Energy Conversion and Management 44 (2003) 1209–1227
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Fig. 4. Predicted variation of air temperature and mass flow rate with S.
Fig. 5. Predicted variation of efficiency and relative humidity with S.
small flow channel depth is required. However, this may cause large frictional losses over the length of the air heater, resulting in a reduced mass flow rate. A compromise is needed that takes into account the specific application of the air heater. Fig. 6 shows that as Q is increased, the temperature of the air in each channel increases, while the mass flow rate decreases. An increase in the value of Q for a given collector area implies a reduction in the value of the width of the collector and an increase in its length. Thus, for a given
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Fig. 6. Predicted variation of air temperature and mass flow rate with Q.
configuration of the collector with S, P and Ac maintained constant, there is a reduction in the cross-sectional area of each channel as Q increases. This leads to an increase in turbulence, resulting in a decrease in the mass flow accompanied by an increase in the heated air temperature. Fig. 7 indicates that increasing Q leads to reductions in the efficiency and the normalised relative humidity of the heated air. The temperature of the heated air, however, increases with increasing Q. This suggests that the mass flow rate is again the dominant factor governing the performance of the air heater. Reducing the length to width ratio for a given collector is an
Fig. 7. Predicted variation of efficiency and relative humidity with Q.
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Fig. 8. Predicted variation of air temperature and mass flow rate with Ac .
alternative proposition for increasing its efficiency without compromising on the cost, but it is at the expense of decreasing temperatures and increasing relative humidity of the heated air. Fig. 8 clearly illustrates that an increase in the collector area will lead to an increase in both the temperature and the mass flow rate. This option does not necessarily alter the flow channel configuration, but the total solar radiation absorbed by the collector is increased and, consequently, the available energy increases. However, the efficiency of the collector decreases, and the relative humidity of the heated air increases, as shown in Fig. 9.
Fig. 9. Predicted variation of efficiency and relative humidity with Ac .
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It is evident from the study, that for an existing SPDDSAH, the most effective modification that can be made to enhance its performance appears to be the re-positioning of the absorber plate to obtain higher values of P. As for new designs, a number of parameters have to be chosen carefully with the ultimate purpose of the heater in mind to achieve the most effective performance.
8. Conclusion The following conclusions may be drawn from the foregoing case and parametric studies: 1. The developed mathematical model provides reasonable predictions of the performance of a SPDDSAH and can be a useful design tool for future development of a SPDDSAH to satisfy specific applications. 2. Significant improvement in the SPDDSAH performance may be obtained through a careful choice of a number of collector parameters. 3. A compromise between higher values of Q, Ac and P is needed to achieve the required temperature, mass flow rates and relative humidity of the heated air. 4. Increasing the top to bottom channel ratio is a cost effective way of optimising the thermal performance of a SPDDSAH. 5. The mass flow rate is the dominant factor in determining the efficiency of a natural convection air heater, hence a careful estimation of the mass flow rate is essential.
Acknowledgements The authors wish to acknowledge the Commonwealth Scholarship Commission, the British Council and the Ghana Government for sponsoring the Ph.D. studies of the first author at De Montfort University, Leicester, England. The technical support of the Technicians of the Department of Mechanical Engineering, UST-Ghana and the School of Engineering and Technology, DMU, England, during the construction and instrumentation of the set ups are also acknowledged.
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