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Combined effect of bottom reflectivity and water turbidity on steady state thermal efficiency of salt gradient solar pond
Energy Conversion and Management 45 (2004) 73–81 www.elsevier.com/locate/enconman
Combined effect of bottom reflectivity and water turbidity on steady state thermal efficiency of salt gradient solar pond M. Husain, P.S. Patil, S.R. Patil, S.K. Samdarshi
*
North Maharshtra University, School of Physical Sciences, # 80, Jalgaon (MS) 425001, India Received 9 December 2002; accepted 24 May 2003
Abstract In salt gradient solar ponds, the clarity of water and absorptivity of the bottom are important concerns. However, both are practically difficult to maintain beyond a certain limit. The reflectivity of the bottom causes the loss of a fraction of the incident radiation flux, resulting in lower absorption of flux in the pond. Turbidity hinders the propagation of radiation. Thereby it decreases the flux reaching the storage zone. Both these factors lower the efficiency of the pond significantly. However, the same turbidity also prevents the loss of radiation reflected from the bottom. Hence, the combined effect is compensatory to some extent. The present work is an analysis of the combined effect of the bottomÕs reflectivity and water turbidity on the steady state efficiency of solar ponds. It is found that in the case of a reflective bottom, turbidity, within certain limits, improves the efficiency of pond. This is apparently contradictory to the conventional beliefs about the pond. Nevertheless, this conclusion is of practical importance for design and maintenance of solar ponds. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Salt gradient solar pond; Reflectivity of bottom; Turbidity of water
1. Introduction The salt gradient solar pond (SGSP) is a viable system for solar thermal energy conversion and long term storage. Typically, a SGSP is a large body of saline water, exposed to the ambient, characterized by three zones, the upper convective zone (UCZ), the non-convective zone (NCZ) and the storage zone (STZ), as shown in Fig. 1. The radiation flux with direct and diffuse components, incident at the surface of pond, is partly reflected with the remainder penetrating the *
Corresponding author. E-mail address: drsksamdarshi@rediffmail.com (S.K. Samdarshi).
0196-8904/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0196-8904(03)00123-7
74
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
Nomenclature d h Ix Is IT K L NCZ R DT STZ UCZ x xUCZ as q h g
depth from surface to interface of NCZ–STZ (m) non dimensional transmission function incoming radiation flux at depth x (W/m2 ) radiation flux at pondÕs surface (W/m2 ) net radiation flux at depth (W/m2 ) conductivity of pondÕs liquid (W/m K) depth of pond (m) non-convective zone normalized transmission function temperature difference between STZ and UCZ of pond (K) storage zone upper convective zone depth (m) depth from surface to interface of UCZ–NCZ (m) absorptivity–transmissivity product reflectivity turbidity (nephelometric turbidity unit) efficiency
surface. The penetrated part attenuates and gets absorbed within the liquid mass of the pond. A part of the radiation flux reaching the bottom may get reflected back, depending upon the reflectivity of the bottom. This part travels towards the surface. At the surface, a part of it crosses the water–air interface and is lost, and the remainder reflects back inside. This phenomenon of reflection between the surface and bottom continues until the radiation energy vanishes by getting absorbed and lost to the atmosphere. Kooi [1] did a steady state analysis of a reflective bottom SGSP considering the reflected radiation flux coming from the bottom. He concluded that the reflectivity significantly lowers the steady state thermal efficiency of the pond. He used the empirical formulation due to Bryant and Colbeck [2] and extended it to account for reflected radiation also. Hull [3] analyzed the phenomenon of multiple reflections of radiation between the bottom and surface and developed universal functions to account for it. Using them, he also concluded that the reflectivity lowers the steady state efficiency of the pond. Srinivasan and Guha [4] and Sezai and Tasdemiroglu [5] used HullÕs model to analyze the long term thermal performance of a solar pond. Their conclusions are also similar to that in Refs. [1,3]. It is always desirable to keep the bottom of the pond as a perfect absorber, but in the course of operation, it becomes reflective due to several unavoidable reasons [4,6]. Earlier models considered the transmission of radiation flux in pure or saline water. Wang and Yagoobi [7] presented a review of existing models. Hawlader [8] investigated the effect of the clarity of water on the extinction coefficient and, consequently, on the effectiveness of solar ponds. He reported that the extremely clear water of Lake Tahoe had an extinction coefficient one fifth
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
75
Fig. 1. Schematic diagram of a salt gradient solar pond.
that of the turbid water in Lake Castle and one tenth that of a solar pond. He found that typical values of extinction coefficient could vary from 0.05 to 0.5 m1 . However, Hawlader [8] did not give a parameter to quantify the clarity of water. Wang and Yagoobi [9] first used turbidity as a parameter in a context with a pond to quantify the clarity of a pondÕs water. They used Nephelometric turbidity units (NTU) to express the turbidity. Their experimental investigations confirmed that the salt content of water does not have a significant impact on the phenomenon of propagation/attenuation of radiation flux in water, but the turbidity of water greatly reduces the forward propagation of radiation flux, causing higher attenuation. Using experimental data, they developed an empirical formulation for estimating the radiation flux at a depth in turbid water. Later, Wang and Yagoobi [10] used this formulation to analyze the effect of turbidity on the thermal performance of a solar pond. They considered the bottom of the solar pond as a black body, i.e. a perfect absorber of radiation flux. They found the impact of turbidity on attenuation of radiation flux is of such magnitude that no realistic analysis can be done while ignoring it. There is a considerable drop in thermal performance of a pond due to turbidity. Solar ponds, being exposed systems, cannot have particle free (non-turbid) water, even if they are regularly treated with chemical coagulants. Wind continuously dumps dust loads into the pond. In tropical countries, it is a major problem. Coagulants do not yield nil turbidity water [11]. Generally the turbidity records of ponds are not available. Wang and Yagoobi [10] compared the actual temperature data of the El Paso pond with their simulated temperature results. The actual data corresponded to the simulated values of temperature for 0.3 NTU turbidity level, while it deviated strongly from the values for 0.0 NTU. This indicates the El Paso pond had a turbidity level around 0.3 NTU. It is clear from the foregoing discussion that water turbidity and bottom reflectivity both reduce the thermal efficiency of a solar pond. Though unwanted, both are realities. While the earlier works considered both the parameters separately, the present work aims to couple both. In the present work, the formulation of Wang and Yagoobi [9], which originally accounts only for
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M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
incoming radiation flux, is empirically extended to account for the reflected radiation flux also. Using it, the combined effect of both the parameters on the steady state thermal efficiency of the pond is analyzed. It is found that the combined effect is of compensatory nature to some extent. In the case of a reflective bottom of the solar pond, an optimum level of water turbidity improves the steady state thermal efficiency of the pond. This is in extreme contrast to the established beliefs about the pond. It suggests radically reconsidering the maintenance of the pond.
2. Theory 2.1. Estimation of radiation flux in turbid water Turbidity refers to the presence of colloidal sized particles (103 to 106 mm size) in water [12]. Because of their size, they reflect the visible radiation flux at right angles, thereby hindering the forward propagation of it. This causes higher attenuation of the radiation flux in its path. The following model, based upon their experimental work, may be used to calculate the incident radiation at a depth x in turbid water [9]. Ix ðh; xÞ ¼ Is hðh; xÞ
ð1Þ
where Is , h and x are the radiation flux incident at the surface of the pond, turbidity of water in NTU and depth in meters, respectively. The non-dimensional transmission function h is defined as hðh; xÞ ¼ hð0:3; xÞRðh; xÞ
ð2Þ
where hð0:3; xÞ ¼ 0:58 0:076 lnð100xÞ
ð3Þ
Rðh; xÞ ¼ 1:0 0:1975xðh 0:3Þ þ 0:0144xðh 0:3Þ2
ð4Þ
and
During their experimentation, Wang and Yagoobi [9] varied the depth of water and turbidity in the range 0 < x < 1:34 m and 0:3 < h < 5 NTU, respectively. However, Wang and Yagoobi [10] used the formulation up to a depth of 2.15 m for modeling of a solar pond, which was 3.15 m deep. Here, 2.15 m referred to the depth of the interface of the NCZ and STZ. This is believed to be justified because expression (2) is a well defined function. The expressions (1)–(4) are used for a uniform turbidity level in water. For non-uniform turbidity levels, the above mentioned expressions are extended as [9] ! ! j j i i1 X X X X ð5Þ h hj ; lk h hjþ1 ; lk hðf ðhi Þ; xÞ ¼ j¼1
where x¼
i X j¼1
lj
k¼1
j¼1
k¼1
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
77
Here, i is the number of sub-layers of water in which the turbidity is varied, and lj is the thickness of the jTH sub-layer. Eq. (5) considers the incoming radiation flux only. It does not consider the bottom reflected part. In the present work, Eq. (5) is extended to incorporate the reflected part also in the following form ! ! ( ! j j i i1 n m X X X X X X h hj ; lk h hjþ1 ; lk q h hm ; lk hT ðf ðhi Þ; xÞ ¼ j¼1
j¼1
k¼1 n1 X
h hmþ1 ;
m¼1
m X
!)
k¼1
m¼1
lk
k¼1
ð6Þ
k¼1
where n is the total number of sub-layers that the radiation flux travels in its path from the surface to the bottom and back to the k TH sub-layer (refer Fig. 1). The reflectivity of the bottom is denoted by q. The multiple reflections between the bottom and surface are ignored because their magnitude is very small. Eq. (1) now becomes IT ðh; xÞ ¼ Is hT ðh; xÞ
ð7Þ
where IT is the net radiation flux at depth x resulting from the incoming and bottom reflected parts. 2.2. The steady state thermal efficiency of SGSP The steady state thermal efficiency of a solar pond is expressed as [13] g ¼ as KDT =fIs ðd xUCZ Þg
ð8Þ
where K and DT are, respectively, the conductivity of the pondÕs liquid and the temperature difference between the STZ and the UCZ of the pond. The NCZ thickness is denoted by d xUCZ . Keeping other parameters constant, the thermal efficiency is directly proportional to as, the transmissivity–absorptivitty product. It is defined as [1,13] Z d IT ðh; xÞ dx ð9Þ as ¼ f1=ðd xUCZ Þg xUCZ
Kooi [13] evaluated expression (9) using the empirical formulation of Bryant and Colbeck [2] to estimate IT . Bryant and Colbeck considered only incoming radiation flux. Kooi [1] further evaluated Eq. (9) considering the bottom reflected radiation flux also. For this, he empirically extended the formulation of Bryant and Colbeck to account for the reflected radiation flux. In the present work, the radiation flux in water is estimated by expression (7), i.e. by considering the effect of water turbidity also. The product as is calculated by solving expression (9) numerically. As the thermal efficiency is directly proportional to as, the product as is taken as a parameter to analyze the efficiency of a solar pond. For the sake of comparison, s is also calculated for turbidity-free water using the expression due to Hull [3], as given in Appendix A. Three cases of turbidity profile are considered: case (i) uniform turbidity throughout the depth, case (ii) linearly increasing turbidity from the UCZ–NCZ interface to the NCZ–STZ interface and case (iii) linearly decreasing turbidity from the UCZ–NCZ interface to the NCZ–STZ interface.
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M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
In the UCZ and STZ, the turbidity is always considered to be uniform because these are mixed zones.
3. Results and discussion (a) Fig. 2 shows the variation of the as the product as a function of reflectivity ðqÞ for a few selected values of the uniform water turbidity profile, i.e. case (i). The NCZ thickness is taken as 0.5 m. It is evident that zero reflectivity yields the highest value of the as product, but this is a hypothetical case. A real pond cannot be totally devoid of reflectivity as well as turbidity. In the very low range of reflectivity also, the as value is inversely related to the value of turbidity. However, a trend opposite to that discussed above is observed at moderate and high reflectivity values. That is, a higher as value is obtained with higher turbidity. This is so because at higher reflectivity, the reflected part of the radiation flux is also significant. The turbidity that hinders the incoming radiation flux does the same with the reflected part coming from the bottom. Thus, the loss of the reflected part is prevented due to turbidity, and the as value is increased. In Fig. 3, the same parameters are taken with the NCZ thickness as 1.5 m. Similar conclusions could be drawn from this also. (b) The phenomenon described in (a) has a larger impact at shallow pond depths in which a higher amount of radiation flux reaches the bottom, and hence, the reflected part is also large. (c) Similar results are obtained for the downward increasing turbidity profile, i.e. case (ii), also, as shown in Figs. 4 and 5, that is, a higher as product value with higher turbidity in the case of a reflective bottom. However, with case (iii), i.e. downward decreasing turbidity profile, the as product value is lower with the higher turbidity value for the practical range of reflectivity of the bottom, Fig. 5. A higher as product with increased turbidity is obtained only at very high
Fig. 2. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (i).
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
79
Fig. 3. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (i).
Fig. 4. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (ii).
reflectivity values, which is an unrealistic case. Hence, the turbidity profile of case (iii) is the most undesirable for the pondÕs performance. The results imply that there exists an optimum value of turbidity in the water for a reflective bottom solar pond, which provides a higher efficiency than turbidity free water. This value depends upon the type of turbidity profile, the pondÕs dimensions and the bottomÕs reflectivity. These results have practical significance. A perfectly non-reflective bottom is difficult to obtain in reality. The bottom becomes reflective due to several reasons. Normal soil has a reflectivity of 0.14 [6]. Generally, the bottom is lined black in the beginning, but the pond, being an exposed
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M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
Fig. 5. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (iii).
system, accumulates wind borne silt and litter. The reflectivity of silt and litter is around 0.3 [4,6]. The precipitation of salt at the bottom takes place due to temperature fluctuations within the STZ. The reflectivity of salt is in the range of 0.5–0.7 [3,4]. Hull [14] has suggested a membrane stratified pond. Such a pond will not accumulate wind borne silt and litter. Even in such a pond, the precipitation of salt is unavoidable. Under such conditions, i.e. a reflective bottom, a selected level of turbidity can improve the efficiency of the pond. Reflectivity values around 0.14, 0.3 and 0.5–0.7 are of practical importance. However, in Figs. 2–5, the reflectivity value is varied from 0.0 to 1.0 for the sake of completeness of analysis. 4. Conclusions The combined effect of water turbidity and bottom reflectivity on the thermal performance of a salt gradient solar pond is analyzed. The combined effect is compensatory to certain limits. In the case of a reflective bottom, an optimum value of turbidity in the water, depending upon the pondÕs dimension, yields a better thermal efficiency than turbidity free water. After the commissioning of a pond, there is no control on the reflectivity of the bottom. However, the water turbidity is a parameter, that which can be controlled to a certain extent by regulating the coagulant dose applied to the pond or even by artificially adding turbidity into it. This turbidity can partially compensate for the losses due to reflectivity, thereby improving the thermal efficiency of pond. Acknowledgements The resources provided for research by the All India Council of Technical Education and University Grants Commission, India, is gratefully acknowledged by the authors.
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Appendix A Hull [3] has given the following expression to calculate the as product Z d as ¼ uT ðxÞ dx
ð10Þ
xUCZ
where uT ; the net radiation flux at a depth x is estimated as uT ¼ u1 ðxÞ þ qu1 ðLÞff ðbðxÞÞ gðaðxÞÞg=f1 qf ðbðLÞÞg where gðaðxÞÞ and f bðxÞ are functions defined in Ref. [3].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Kooi CF. Solar Energy 1981;31:113. Bryant HC, Colbeck I. Solar Energy 1977;19:321. Hull JR. Solar Energy 1982;29(5):385. Srinivasan J, Guha A. Solar Energy 1987;39(4):361. Sezai I, Tasdemiroglu E. Solar Energy 1995;55(4):311. Hunn BD, Calafeel DO. Solar Energy 1977;19:87. Wang J, Yagoobi S. Proceedings of the second ASME–JSES–JSME International Solar Energy Conference, Reno. NE. 59, 1991. Hawlader MNA. Solar Energy 1980;25:461. Wang J, Yagoobi S. Solar Energy 1994;52(5):429. Wang J, Yagoobi S. Solar Energy 1995;54(5):301. Weber WJ. Physicochemical processes for water quality control. New York: Wiley, Interscience Publication; 1972. p. 89. Sawyer CN, McCarty PL. Chemistry for environmental engineering 3e. International Students Ed. Singapore: McGraw-Hill Publishing Company; 1978. p. 289. Kooi CF. Solar Energy 1979;25:37. Hull JR. Solar Energy 1980;25:317.
Combined effect of bottom reflectivity and water turbidity on steady state thermal efficiency of salt gradient solar pond M. Husain, P.S. Patil, S.R. Patil, S.K. Samdarshi
*
North Maharshtra University, School of Physical Sciences, # 80, Jalgaon (MS) 425001, India Received 9 December 2002; accepted 24 May 2003
Abstract In salt gradient solar ponds, the clarity of water and absorptivity of the bottom are important concerns. However, both are practically difficult to maintain beyond a certain limit. The reflectivity of the bottom causes the loss of a fraction of the incident radiation flux, resulting in lower absorption of flux in the pond. Turbidity hinders the propagation of radiation. Thereby it decreases the flux reaching the storage zone. Both these factors lower the efficiency of the pond significantly. However, the same turbidity also prevents the loss of radiation reflected from the bottom. Hence, the combined effect is compensatory to some extent. The present work is an analysis of the combined effect of the bottomÕs reflectivity and water turbidity on the steady state efficiency of solar ponds. It is found that in the case of a reflective bottom, turbidity, within certain limits, improves the efficiency of pond. This is apparently contradictory to the conventional beliefs about the pond. Nevertheless, this conclusion is of practical importance for design and maintenance of solar ponds. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Salt gradient solar pond; Reflectivity of bottom; Turbidity of water
1. Introduction The salt gradient solar pond (SGSP) is a viable system for solar thermal energy conversion and long term storage. Typically, a SGSP is a large body of saline water, exposed to the ambient, characterized by three zones, the upper convective zone (UCZ), the non-convective zone (NCZ) and the storage zone (STZ), as shown in Fig. 1. The radiation flux with direct and diffuse components, incident at the surface of pond, is partly reflected with the remainder penetrating the *
Corresponding author. E-mail address: drsksamdarshi@rediffmail.com (S.K. Samdarshi).
0196-8904/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0196-8904(03)00123-7
74
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
Nomenclature d h Ix Is IT K L NCZ R DT STZ UCZ x xUCZ as q h g
depth from surface to interface of NCZ–STZ (m) non dimensional transmission function incoming radiation flux at depth x (W/m2 ) radiation flux at pondÕs surface (W/m2 ) net radiation flux at depth (W/m2 ) conductivity of pondÕs liquid (W/m K) depth of pond (m) non-convective zone normalized transmission function temperature difference between STZ and UCZ of pond (K) storage zone upper convective zone depth (m) depth from surface to interface of UCZ–NCZ (m) absorptivity–transmissivity product reflectivity turbidity (nephelometric turbidity unit) efficiency
surface. The penetrated part attenuates and gets absorbed within the liquid mass of the pond. A part of the radiation flux reaching the bottom may get reflected back, depending upon the reflectivity of the bottom. This part travels towards the surface. At the surface, a part of it crosses the water–air interface and is lost, and the remainder reflects back inside. This phenomenon of reflection between the surface and bottom continues until the radiation energy vanishes by getting absorbed and lost to the atmosphere. Kooi [1] did a steady state analysis of a reflective bottom SGSP considering the reflected radiation flux coming from the bottom. He concluded that the reflectivity significantly lowers the steady state thermal efficiency of the pond. He used the empirical formulation due to Bryant and Colbeck [2] and extended it to account for reflected radiation also. Hull [3] analyzed the phenomenon of multiple reflections of radiation between the bottom and surface and developed universal functions to account for it. Using them, he also concluded that the reflectivity lowers the steady state efficiency of the pond. Srinivasan and Guha [4] and Sezai and Tasdemiroglu [5] used HullÕs model to analyze the long term thermal performance of a solar pond. Their conclusions are also similar to that in Refs. [1,3]. It is always desirable to keep the bottom of the pond as a perfect absorber, but in the course of operation, it becomes reflective due to several unavoidable reasons [4,6]. Earlier models considered the transmission of radiation flux in pure or saline water. Wang and Yagoobi [7] presented a review of existing models. Hawlader [8] investigated the effect of the clarity of water on the extinction coefficient and, consequently, on the effectiveness of solar ponds. He reported that the extremely clear water of Lake Tahoe had an extinction coefficient one fifth
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
75
Fig. 1. Schematic diagram of a salt gradient solar pond.
that of the turbid water in Lake Castle and one tenth that of a solar pond. He found that typical values of extinction coefficient could vary from 0.05 to 0.5 m1 . However, Hawlader [8] did not give a parameter to quantify the clarity of water. Wang and Yagoobi [9] first used turbidity as a parameter in a context with a pond to quantify the clarity of a pondÕs water. They used Nephelometric turbidity units (NTU) to express the turbidity. Their experimental investigations confirmed that the salt content of water does not have a significant impact on the phenomenon of propagation/attenuation of radiation flux in water, but the turbidity of water greatly reduces the forward propagation of radiation flux, causing higher attenuation. Using experimental data, they developed an empirical formulation for estimating the radiation flux at a depth in turbid water. Later, Wang and Yagoobi [10] used this formulation to analyze the effect of turbidity on the thermal performance of a solar pond. They considered the bottom of the solar pond as a black body, i.e. a perfect absorber of radiation flux. They found the impact of turbidity on attenuation of radiation flux is of such magnitude that no realistic analysis can be done while ignoring it. There is a considerable drop in thermal performance of a pond due to turbidity. Solar ponds, being exposed systems, cannot have particle free (non-turbid) water, even if they are regularly treated with chemical coagulants. Wind continuously dumps dust loads into the pond. In tropical countries, it is a major problem. Coagulants do not yield nil turbidity water [11]. Generally the turbidity records of ponds are not available. Wang and Yagoobi [10] compared the actual temperature data of the El Paso pond with their simulated temperature results. The actual data corresponded to the simulated values of temperature for 0.3 NTU turbidity level, while it deviated strongly from the values for 0.0 NTU. This indicates the El Paso pond had a turbidity level around 0.3 NTU. It is clear from the foregoing discussion that water turbidity and bottom reflectivity both reduce the thermal efficiency of a solar pond. Though unwanted, both are realities. While the earlier works considered both the parameters separately, the present work aims to couple both. In the present work, the formulation of Wang and Yagoobi [9], which originally accounts only for
76
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
incoming radiation flux, is empirically extended to account for the reflected radiation flux also. Using it, the combined effect of both the parameters on the steady state thermal efficiency of the pond is analyzed. It is found that the combined effect is of compensatory nature to some extent. In the case of a reflective bottom of the solar pond, an optimum level of water turbidity improves the steady state thermal efficiency of the pond. This is in extreme contrast to the established beliefs about the pond. It suggests radically reconsidering the maintenance of the pond.
2. Theory 2.1. Estimation of radiation flux in turbid water Turbidity refers to the presence of colloidal sized particles (103 to 106 mm size) in water [12]. Because of their size, they reflect the visible radiation flux at right angles, thereby hindering the forward propagation of it. This causes higher attenuation of the radiation flux in its path. The following model, based upon their experimental work, may be used to calculate the incident radiation at a depth x in turbid water [9]. Ix ðh; xÞ ¼ Is hðh; xÞ
ð1Þ
where Is , h and x are the radiation flux incident at the surface of the pond, turbidity of water in NTU and depth in meters, respectively. The non-dimensional transmission function h is defined as hðh; xÞ ¼ hð0:3; xÞRðh; xÞ
ð2Þ
where hð0:3; xÞ ¼ 0:58 0:076 lnð100xÞ
ð3Þ
Rðh; xÞ ¼ 1:0 0:1975xðh 0:3Þ þ 0:0144xðh 0:3Þ2
ð4Þ
and
During their experimentation, Wang and Yagoobi [9] varied the depth of water and turbidity in the range 0 < x < 1:34 m and 0:3 < h < 5 NTU, respectively. However, Wang and Yagoobi [10] used the formulation up to a depth of 2.15 m for modeling of a solar pond, which was 3.15 m deep. Here, 2.15 m referred to the depth of the interface of the NCZ and STZ. This is believed to be justified because expression (2) is a well defined function. The expressions (1)–(4) are used for a uniform turbidity level in water. For non-uniform turbidity levels, the above mentioned expressions are extended as [9] ! ! j j i i1 X X X X ð5Þ h hj ; lk h hjþ1 ; lk hðf ðhi Þ; xÞ ¼ j¼1
where x¼
i X j¼1
lj
k¼1
j¼1
k¼1
M. Husain et al. / Energy Conversion and Management 45 (2004) 73–81
77
Here, i is the number of sub-layers of water in which the turbidity is varied, and lj is the thickness of the jTH sub-layer. Eq. (5) considers the incoming radiation flux only. It does not consider the bottom reflected part. In the present work, Eq. (5) is extended to incorporate the reflected part also in the following form ! ! ( ! j j i i1 n m X X X X X X h hj ; lk h hjþ1 ; lk q h hm ; lk hT ðf ðhi Þ; xÞ ¼ j¼1
j¼1
k¼1 n1 X
h hmþ1 ;
m¼1
m X
!)
k¼1
m¼1
lk
k¼1
ð6Þ
k¼1
where n is the total number of sub-layers that the radiation flux travels in its path from the surface to the bottom and back to the k TH sub-layer (refer Fig. 1). The reflectivity of the bottom is denoted by q. The multiple reflections between the bottom and surface are ignored because their magnitude is very small. Eq. (1) now becomes IT ðh; xÞ ¼ Is hT ðh; xÞ
ð7Þ
where IT is the net radiation flux at depth x resulting from the incoming and bottom reflected parts. 2.2. The steady state thermal efficiency of SGSP The steady state thermal efficiency of a solar pond is expressed as [13] g ¼ as KDT =fIs ðd xUCZ Þg
ð8Þ
where K and DT are, respectively, the conductivity of the pondÕs liquid and the temperature difference between the STZ and the UCZ of the pond. The NCZ thickness is denoted by d xUCZ . Keeping other parameters constant, the thermal efficiency is directly proportional to as, the transmissivity–absorptivitty product. It is defined as [1,13] Z d IT ðh; xÞ dx ð9Þ as ¼ f1=ðd xUCZ Þg xUCZ
Kooi [13] evaluated expression (9) using the empirical formulation of Bryant and Colbeck [2] to estimate IT . Bryant and Colbeck considered only incoming radiation flux. Kooi [1] further evaluated Eq. (9) considering the bottom reflected radiation flux also. For this, he empirically extended the formulation of Bryant and Colbeck to account for the reflected radiation flux. In the present work, the radiation flux in water is estimated by expression (7), i.e. by considering the effect of water turbidity also. The product as is calculated by solving expression (9) numerically. As the thermal efficiency is directly proportional to as, the product as is taken as a parameter to analyze the efficiency of a solar pond. For the sake of comparison, s is also calculated for turbidity-free water using the expression due to Hull [3], as given in Appendix A. Three cases of turbidity profile are considered: case (i) uniform turbidity throughout the depth, case (ii) linearly increasing turbidity from the UCZ–NCZ interface to the NCZ–STZ interface and case (iii) linearly decreasing turbidity from the UCZ–NCZ interface to the NCZ–STZ interface.
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In the UCZ and STZ, the turbidity is always considered to be uniform because these are mixed zones.
3. Results and discussion (a) Fig. 2 shows the variation of the as the product as a function of reflectivity ðqÞ for a few selected values of the uniform water turbidity profile, i.e. case (i). The NCZ thickness is taken as 0.5 m. It is evident that zero reflectivity yields the highest value of the as product, but this is a hypothetical case. A real pond cannot be totally devoid of reflectivity as well as turbidity. In the very low range of reflectivity also, the as value is inversely related to the value of turbidity. However, a trend opposite to that discussed above is observed at moderate and high reflectivity values. That is, a higher as value is obtained with higher turbidity. This is so because at higher reflectivity, the reflected part of the radiation flux is also significant. The turbidity that hinders the incoming radiation flux does the same with the reflected part coming from the bottom. Thus, the loss of the reflected part is prevented due to turbidity, and the as value is increased. In Fig. 3, the same parameters are taken with the NCZ thickness as 1.5 m. Similar conclusions could be drawn from this also. (b) The phenomenon described in (a) has a larger impact at shallow pond depths in which a higher amount of radiation flux reaches the bottom, and hence, the reflected part is also large. (c) Similar results are obtained for the downward increasing turbidity profile, i.e. case (ii), also, as shown in Figs. 4 and 5, that is, a higher as product value with higher turbidity in the case of a reflective bottom. However, with case (iii), i.e. downward decreasing turbidity profile, the as product value is lower with the higher turbidity value for the practical range of reflectivity of the bottom, Fig. 5. A higher as product with increased turbidity is obtained only at very high
Fig. 2. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (i).
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Fig. 3. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (i).
Fig. 4. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (ii).
reflectivity values, which is an unrealistic case. Hence, the turbidity profile of case (iii) is the most undesirable for the pondÕs performance. The results imply that there exists an optimum value of turbidity in the water for a reflective bottom solar pond, which provides a higher efficiency than turbidity free water. This value depends upon the type of turbidity profile, the pondÕs dimensions and the bottomÕs reflectivity. These results have practical significance. A perfectly non-reflective bottom is difficult to obtain in reality. The bottom becomes reflective due to several reasons. Normal soil has a reflectivity of 0.14 [6]. Generally, the bottom is lined black in the beginning, but the pond, being an exposed
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Fig. 5. Absorptivity–transmissivity product as a function of reflectivity for a few turbidity profiles of case (iii).
system, accumulates wind borne silt and litter. The reflectivity of silt and litter is around 0.3 [4,6]. The precipitation of salt at the bottom takes place due to temperature fluctuations within the STZ. The reflectivity of salt is in the range of 0.5–0.7 [3,4]. Hull [14] has suggested a membrane stratified pond. Such a pond will not accumulate wind borne silt and litter. Even in such a pond, the precipitation of salt is unavoidable. Under such conditions, i.e. a reflective bottom, a selected level of turbidity can improve the efficiency of the pond. Reflectivity values around 0.14, 0.3 and 0.5–0.7 are of practical importance. However, in Figs. 2–5, the reflectivity value is varied from 0.0 to 1.0 for the sake of completeness of analysis. 4. Conclusions The combined effect of water turbidity and bottom reflectivity on the thermal performance of a salt gradient solar pond is analyzed. The combined effect is compensatory to certain limits. In the case of a reflective bottom, an optimum value of turbidity in the water, depending upon the pondÕs dimension, yields a better thermal efficiency than turbidity free water. After the commissioning of a pond, there is no control on the reflectivity of the bottom. However, the water turbidity is a parameter, that which can be controlled to a certain extent by regulating the coagulant dose applied to the pond or even by artificially adding turbidity into it. This turbidity can partially compensate for the losses due to reflectivity, thereby improving the thermal efficiency of pond. Acknowledgements The resources provided for research by the All India Council of Technical Education and University Grants Commission, India, is gratefully acknowledged by the authors.
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Appendix A Hull [3] has given the following expression to calculate the as product Z d as ¼ uT ðxÞ dx
ð10Þ
xUCZ
where uT ; the net radiation flux at a depth x is estimated as uT ¼ u1 ðxÞ þ qu1 ðLÞff ðbðxÞÞ gðaðxÞÞg=f1 qf ðbðLÞÞg where gðaðxÞÞ and f bðxÞ are functions defined in Ref. [3].
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