{"id":6731,"date":"2024-11-29T12:28:27","date_gmt":"2024-11-29T12:28:27","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=6731"},"modified":"2024-11-29T12:28:28","modified_gmt":"2024-11-29T12:28:28","slug":"second-law-analysis","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/11\/29\/second-law-analysis\/","title":{"rendered":"Second law analysis"},"content":{"rendered":"\n

The analysis presented here is based on Bejan\u2019s work (Bejan et al., 1981;\u00a0Bejan, 1995). The analysis, however, is adapted to imaging collectors, because entropy generation minimization is more important to high-temperature systems. Consider that the collector has an aperture area (or total heliostat area),\u00a0A<\/em>a<\/sub>, and receives solar radiation at the rate\u00a0Q<\/em>\u2217<\/sup>\u00a0from the sun, as shown in\u00a0Figure 3.48. The net solar heat transfer,\u00a0Q<\/em>\u2217<\/sup>, is proportional to the collector area,\u00a0A<\/em>a<\/sub>, and the proportionality factor,\u00a0q<\/em>\u2217<\/sup>\u00a0(W\/m2<\/sup>), which varies with geographical position on the earth, the orientation of the collector, meteorological\u00a0conditions, and the time of day. In the present analysis,\u00a0q<\/em>\u2217<\/sup>\u00a0is assumed to be constant and the system is in a steady state; that is,<\/p>\n\n\n\n

\"image\"(3.144)<\/strong><\/p>\n\n\n\n

For concentrating systems, q<\/em>\u2217<\/sup> is the solar energy falling on the reflector. To obtain the energy falling on the collector receiver, the tracking mechanism accuracy, the optical errors of the mirror, including its reflectance, and the optical properties of the receiver glazing must be considered.<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

FIGURE 3.48<\/strong> <\/a>Imaging concentrating collector model.<\/p>\n\n\n\n

Therefore, the radiation falling on the receiver,\u00a0\"image\", is a function of the optical efficiency, which accounts for all these errors. For the concentrating collectors, Eq.\u00a0(3.116)\u00a0can be used. The radiation falling on the receiver is (Kalogirou, 2004):<\/p>\n\n\n\n

\"image\"(3.145)<\/strong><\/p>\n\n\n\n

The incident solar radiation is partly delivered to a power cycle (or user) as heat transfer Q<\/em> at the receiver temperature, T<\/em>r<\/sub>. The remaining fraction, Q<\/em>o<\/sub>, represents the collector-ambient heat loss:<\/p>\n\n\n\n

\"image\"(3.146)<\/strong><\/p>\n\n\n\n

For imaging concentrating collectors, Q<\/em>o<\/sub> is proportional to the receiver-ambient temperature difference and to the receiver area as:<\/p>\n\n\n\n

\"image\"(3.147)<\/strong><\/p>\n\n\n\n

where U<\/em>r<\/sub> is the overall heat transfer coefficient based on A<\/em>r<\/sub>. It should be noted that U<\/em>r<\/sub> is a characteristic constant of the collector.<\/p>\n\n\n\n

Combining Eqs\u00a0(3.146)\u00a0and\u00a0(3.147), it is apparent that the maximum receiver temperature occurs when\u00a0Q<\/em>\u00a0=\u00a00, that is, when the entire solar heat transfer\u00a0Q<\/em>\u2217<\/sup>\u00a0is lost to the ambient. The maximum collector temperature is given in dimensionless form by:<\/p>\n\n\n\n

\"image\"(3.148)<\/strong><\/a><\/p>\n\n\n\n

Combining Eqs\u00a0(3.145)\u00a0and\u00a0(3.148),<\/p>\n\n\n\n

\"image\"(3.149)<\/strong><\/p>\n\n\n\n

Considering that C<\/em> = A<\/em>a<\/sub>\/A<\/em>r<\/sub>, then:<\/p>\n\n\n\n

\"image\"(3.150)<\/strong><\/p>\n\n\n\n

As can be seen from Eq.\u00a0(3.150),\u00a0\u03b8<\/em>max<\/sub>\u00a0is proportional to\u00a0C<\/em>, that is, the higher the concentration ratio of the collector, the higher are\u00a0\u03b8<\/em>max<\/sub>\u00a0and\u00a0T<\/em>r,max<\/sub>. The term\u00a0T<\/em>r,max<\/sub>\u00a0in Eq.\u00a0(3.148)\u00a0is also known as the\u00a0stagnation temperature of the collector<\/em>, that is, the temperature that can be obtained at a no flow condition. In dimensionless form, the collector temperature,\u00a0\u03b8<\/em>\u00a0=\u00a0T<\/em>r<\/sub>\/T<\/em>o<\/sub>, varies between 1 and\u00a0\u03b8<\/em>max<\/sub>, depending on the heat delivery rate,\u00a0Q<\/em>. The stagnation temperature,\u00a0\u03b8<\/em>max<\/sub>, is the parameter that describes the performance of the collector with regard to collector-ambient heat loss, since there is no flow through the collector and all the energy collected is used to raise the temperature of the working fluid to the stagnation temperature, which is fixed at a value corresponding to the energy collected equal to energy loss to ambient. Hence, the collector efficiency is given by:<\/p>\n\n\n\n

\"image\"(3.151)<\/strong><\/p>\n\n\n\n

Therefore \u03b7<\/em>c<\/sub> is a linear function of collector temperature. At the stagnation point, the heat transfer, Q<\/em>, carries zero exergy<\/em>, or zero potential for producing useful work.<\/p>\n\n\n\n

3.7.1 Minimum entropy generation rate<\/h3>\n\n\n\n

The minimization of the entropy generation rate is the same as the maximization of the power output. The process of solar energy collection is accompanied by the generation of entropy upstream of the collector, downstream of the collector, and inside the collector, as shown in\u00a0Figure 3.49.<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

FIGURE 3.49<\/strong> <\/a>Exergy flow diagram.<\/p>\n\n\n\n

The exergy inflow coming from the solar radiation falling on the collector surface is:<\/p>\n\n\n\n

\"image\"(3.152)<\/strong><\/p>\n\n\n\n

where\u00a0T<\/em>\u2217<\/sub>\u00a0is the apparent sun temperature as an exergy source. In this analysis, the value suggested by\u00a0Petela (1964)\u00a0is adopted, that is,\u00a0T<\/em>\u2217<\/sub>\u00a0is approximately equal to \u00beT<\/em>s<\/sub>, where\u00a0T<\/em>s<\/sub>\u00a0is the apparent blackbody temperature of the sun, which is about 5770\u00a0K. Therefore, the\u00a0T<\/em>\u2217<\/sub>\u00a0considered here is 4330\u00a0K. It should be noted that, in this analysis,\u00a0T<\/em>\u2217<\/sub>\u00a0is also considered constant; and because its value is much greater than\u00a0T<\/em>o<\/sub>,\u00a0E<\/em>in<\/sub>\u00a0is very near\u00a0Q<\/em>\u2217<\/sup>. The output exergy from the collector is given by:<\/p>\n\n\n\n

\"image\"(3.153)<\/strong><\/p>\n\n\n\n

whereas the difference between\u00a0E<\/em>in<\/sub>\u00a0and\u00a0E<\/em>out<\/sub>\u00a0represents the destroyed exergy. From\u00a0Figure 3.49, the entropy generation rate can be written as:<\/p>\n\n\n\n

\"image\"(3.154)<\/strong><\/a><\/p>\n\n\n\n

This equation can be written with the help of Eq.\u00a0(3.146)\u00a0as:<\/p>\n\n\n\n

\"image\"(3.155)<\/strong><\/p>\n\n\n\n

By using Eqs\u00a0(3.152)\u00a0and\u00a0(3.153), Eq.\u00a0(3.155)\u00a0can be written as:<\/p>\n\n\n\n

\"image\"(3.156)<\/strong><\/p>\n\n\n\n

or<\/p>\n\n\n\n

\"image\"(3.157)<\/strong><\/p>\n\n\n\n

Therefore, if we consider E<\/em>in<\/sub> constant, the maximization of the exergy output (E<\/em>out<\/sub>) is the same as the minimization of the total entropy generation, S<\/em>gen<\/sub>.<\/p>\n\n\n\n

3.7.2 Optimum collector temperature<\/h3>\n\n\n\n

By substituting Eqs\u00a0(3.146)\u00a0and\u00a0(3.147)\u00a0into Eq.\u00a0(3.155), the rate of entropy generation can be written as:<\/p>\n\n\n\n

\"image\"(3.158)<\/strong><\/p>\n\n\n\n

By applying Eq.\u00a0(3.150)\u00a0in Eq.\u00a0(3.158)\u00a0and performing various manipulations,<\/p>\n\n\n\n

\"image\"(3.159)<\/strong><\/a><\/p>\n\n\n\n

The dimensionless term, S<\/em>gen<\/sub>\/U<\/em>r<\/sub>A<\/em>r<\/sub>, accounts for the fact that the entropy generation rate scales with the finite size of the system, which is described by A<\/em>r<\/sub> = A<\/em>a<\/sub>\/C<\/em>.<\/p>\n\n\n\n

By differentiating Eq.\u00a0(3.159)\u00a0with respect to\u00a0\u03b8<\/em>\u00a0and setting it to 0, the optimum collector temperature (\u03b8<\/em>opt<\/sub>) for minimum entropy generation is obtained:<\/p>\n\n\n\n

\"image\"(3.160)<\/strong><\/p>\n\n\n\n

By substituting\u00a0\u03b8<\/em>max<\/sub>\u00a0with\u00a0T<\/em>r,max<\/sub>\/T<\/em>o<\/sub>\u00a0and\u00a0\u03b8<\/em>opt<\/sub>\u00a0with\u00a0T<\/em>r,opt<\/sub>\/T<\/em>o<\/sub>, Eq.\u00a0(3.160)\u00a0can be written as:<\/p>\n\n\n\n

\"image\"(3.161)<\/strong><\/p>\n\n\n\n

This equation states that the optimal collector temperature is the geometric average of the maximum collector (stagnation) temperature and the ambient temperature. Typical stagnation temperatures and the resulting optimum operating temperatures for various types of concentrating collectors are shown in\u00a0Table 3.4. The stagnation temperatures shown in\u00a0Table 3.4\u00a0are estimated by considering mainly the collector radiation losses.<\/p>\n\n\n\n

Table 3.4<\/p>\n\n\n\n

<\/a>Optimum Collector Temperatures for Various Types of Concentrating Collectors<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

<\/a>Note: Ambient temperature considered = 25 \u00b0C.<\/p>\n\n\n\n

As can be seen from the data presented in\u00a0Table 3.4\u00a0for high-performance collectors such as the central receiver, it is better to operate the system at high flow rates to lower the temperature around the value shown instead of operating at very high temperature to obtain higher thermodynamic efficiency from the collector system.<\/p>\n\n\n\n

By applying Eq.\u00a0(3.160)\u00a0to Eq.\u00a0(3.159), the corresponding minimum entropy generation rate is:<\/p>\n\n\n\n

\"image\"(3.162)<\/strong><\/p>\n\n\n\n

where\u00a0\u03b8<\/em>\u2217<\/sub>\u00a0=\u00a0T<\/em>\u2217<\/sub>\/T<\/em>o<\/sub>. It should be noted that, for flat-plate and low-concentration ratio collectors, the last term of Eq.\u00a0(3.162)\u00a0is negligible, since\u00a0\u03b8<\/em>\u2217<\/sub>\u00a0is much bigger than\u00a0\u03b8<\/em>max<\/sub>\u00a0\u2212\u00a01; but it is not for higher concentration collectors such as the central receiver and the parabolic dish ones, which have stagnation temperatures of several hundreds of degrees.<\/p>\n\n\n\n

By applying the stagnation temperatures shown in\u00a0Table 3.4\u00a0to Eq.\u00a0(3.162), the dimensionless entropy generated against the collector concentration ratios considered here, as shown in\u00a0Figure 3.50, is obtained.<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

FIGURE 3.50<\/strong> <\/a>Entropy generated and optimum temperatures against collector concentration ratio.<\/p>\n\n\n\n

3.7.3 Non-isothermal collector<\/h3>\n\n\n\n

So far, the analysis was carried out considering an isothermal collector. For a non-isothermal one, which is a more realistic model, particularly for long PTCs, and by applying the principle of energy conservation,<\/p>\n\n\n\n

\"image\"(3.163)<\/strong><\/p>\n\n\n\n

where x<\/em> is from 0 to L<\/em> (the collector length). The generated entropy can be obtained from:<\/p>\n\n\n\n

\"image\"(3.164)<\/strong><\/p>\n\n\n\n

From an overall energy balance, the total heat loss is:<\/p>\n\n\n\n

\"image\"(3.165)<\/strong><\/p>\n\n\n\n

Substituting Eq.\u00a0(3.165)\u00a0into Eq.\u00a0(3.164)\u00a0and performing the necessary manipulations, the following relation is obtained:<\/p>\n\n\n\n

\"image\"(3.166)<\/strong><\/p>\n\n\n\n

where \u03b8<\/em>out<\/sub> = T<\/em>out<\/sub>\/T<\/em>o<\/sub>, \u03b8<\/em>in<\/sub> = T<\/em>in<\/sub>\/T<\/em>o<\/sub>, N<\/em>s<\/sub> is the entropy generation number, and M<\/em> is the mass flow number given by:<\/p>\n\n\n\n

\"image\"(3.167)<\/strong><\/p>\n\n\n\n

<\/a>and<\/p>\n\n\n\n

\"image\"(3.168)<\/strong><\/p>\n\n\n\n

If the inlet temperature is fixed, \u03b8<\/em>in<\/sub> = 1, then the entropy generation rate is a function of only M<\/em> and \u03b8<\/em>out<\/sub>. These parameters are interdependent because the collector outlet temperature depends on the mass flow rate.<\/p>\n","protected":false},"excerpt":{"rendered":"

The analysis presented here is based on Bejan\u2019s work (Bejan et al., 1981;\u00a0Bejan, 1995). The analysis, however, is adapted to imaging collectors, because entropy generation minimization is more important to high-temperature systems. Consider that the collector has an aperture area (or total heliostat area),\u00a0Aa, and receives solar radiation at the rate\u00a0Q\u2217\u00a0from the sun, as shown […]<\/p>\n","protected":false},"author":1,"featured_media":6688,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[699],"tags":[],"class_list":["post-6731","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-3-solar-energy-collectors"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/11\/battery_1976451.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/6731","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=6731"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/6731\/revisions"}],"predecessor-version":[{"id":6732,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/6731\/revisions\/6732"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/6688"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=6731"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=6731"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=6731"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}