{"id":136718,"date":"2025-06-12T09:08:14","date_gmt":"2025-06-12T07:08:14","guid":{"rendered":"https:\/\/www.velasolaris.com\/?post_type=handbuch&#038;p=136718"},"modified":"2025-10-20T16:37:51","modified_gmt":"2025-10-20T14:37:51","slug":"collector-model-according-to-iso-9806","status":"publish","type":"handbuch","link":"https:\/\/www.velasolaris.com\/en\/handbuch\/polysun-designer\/producers\/solar-thermal-collectors\/collector-model-according-to-iso-9806\/","title":{"rendered":"Collector Model according to ISO 9806"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\" id=\"h-solar-thermal-collectors-calculation-per-iso-9806\">Solar Thermal Collectors Calculation per ISO 9806<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">From Polysun version 2025.5, solar thermal collectors are calculated according to the current ISO standard 9806.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In this article, you will find out:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Which parameters the ISO 9806 model uses in Polysun (e.g. Eta0,b, a1, a2, etc.)<\/li>\n\n\n\n<li>How irradiation and losses are calculated<\/li>\n\n\n\n<li>What the Incident Angle Modifier (IAM) is and how it is implemented depending on the collector type<\/li>\n\n\n\n<li>How the calculated temperature of the collector is determined over time (dynamic heat capacity, etc.)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table is-style-regular\"><table class=\"has-fixed-layout\"><tbody><tr><td>Parameters<\/td><td>Unit<\/td><td>Symbol<\/td><\/tr><tr><td>Number of collectors<\/td><td>[-]<\/td><td>\\(N_{module}\\)<\/td><\/tr><tr><td>Total gross area<\/td><td>\\(m^{2}\\)<\/td><td>\\(A_{gross}\\)<\/td><\/tr><tr><td>Wind speed at the collector array<\/td><td>[-]<\/td><td>\\(f_{wind}\\)<\/td><\/tr><tr><td>Orientation<\/td><td>\u00b0<\/td><td>\\(\\gamma\\)<\/td><\/tr><tr><td>Tilt angle<\/td><td>\u00b0<\/td><td>\\(\\beta\\)<\/td><\/tr><tr><td>Rotation<\/td><td>\u00b0<\/td><td>\\(\\phi\\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The catalogue structure depends on the previously selected norm.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\"><table class=\"has-fixed-layout\"><tbody><tr><td>Parameters<\/td><td>Unit<\/td><td>Symbol<\/td><\/tr><tr><td>Eta0,b: Efficiency<\/td><td>[-]<\/td><td>\\(\\eta_{0,b}\\)<\/td><\/tr><tr><td>a1: Heat loss coefficient<\/td><td>\\(\\frac{W}{m^{2}K}\\)<\/td><td>\\(a_{1}\\)<\/td><\/tr><tr><td>a2: Temperature dependence of the heat loss coefficient<\/td><td>\\(\\frac{W}{m^{2}K^{2}}\\)<\/td><td>\\(a_{2}\\)<\/td><\/tr><tr><td>a3: Wind speed dependence of the heat loss coefficient<\/td><td>\\(\\frac{J}{m^{3}K}\\)<\/td><td>\\(a_{3}\\)<\/td><\/tr><tr><td>a4: Sky temperature dependence of the heat loss coefficient<\/td><td>[-]<\/td><td>\\(a_{4}\\)<\/td><\/tr><tr><td>a5: Effective thermal capacity<\/td><td>\\(\\frac{J}{m^{3}K}\\)<\/td><td>\\(a_{5}\\)<\/td><\/tr><tr><td>a6: Wind speed dependence of the zero loss efficiency<\/td><td>\\(\\frac{s}{m}\\)<\/td><td><br>\\(a_{6}\\)<\/td><\/tr><tr><td>a7: Wind speed dependence of IR radiation exchange<\/td><td>\\(\\frac{W}{m^{2}K^{4}}\\)<\/td><td>\\(a_{7}\\)<\/td><\/tr><tr><td>a8: Radiation loss<\/td><td>\\(\\frac{W}{m^{2}K^{4}}\\)<\/td><td>\\(a_{8}\\)<\/td><\/tr><tr><td>Kd: Incidence angle modifier for diffuse solar radiation<\/td><td>[-]<\/td><td>\\(K_{d}\\)<\/td><\/tr><tr><td>kt 10 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,10}\\)<\/td><\/tr><tr><td>kt 20 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,20}\\)<\/td><\/tr><tr><td>kt 30 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,30}\\)<\/td><\/tr><tr><td>kt 40 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,40}\\)<\/td><\/tr><tr><td>kt 50 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,50}\\)<\/td><\/tr><tr><td>kt 60 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,60}\\)<\/td><\/tr><tr><td>kt 70 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,70}\\)<\/td><\/tr><tr><td>kt 80 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,80}\\)<\/td><\/tr><tr><td>kt 90 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{t,90}\\)<\/td><\/tr><tr><td>kl 10 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,10}\\)<\/td><\/tr><tr><td>kl 20 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,20}\\)<\/td><\/tr><tr><td>kl 30 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,30}\\)<\/td><\/tr><tr><td>kl 40 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,40}\\)<\/td><\/tr><tr><td>kl 50 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,50}\\)<\/td><\/tr><tr><td>kl 60 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,60}\\)<\/td><\/tr><tr><td>kl 70 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,70}\\)<\/td><\/tr><tr><td>kl 80 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,80}\\)<\/td><\/tr><tr><td>kl 90 \u00b0<\/td><td>\u00b0<\/td><td>\\(k_{l,90}\\)<\/td><\/tr><tr><td>Collector axis orientation<\/td><td>[-]<\/td><td>\\(N_{axis}\\)<\/td><\/tr><tr><td>Volume<\/td><td>\\(l\\)<\/td><td>\\(V\\)<\/td><\/tr><tr><td>Max temperature<\/td><td>\u00b0 C<\/td><td>\\(T_{max}\\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The \\(k_{t}\\) and \\(k_{l}\\) parameters modify the angle of incidence for direct solar irradiation for transverse \\(k_{t}\\) and longitudinal \\(k_{l}\\) angles of incidence. The relationship is described in more detail in the section explaining the IAM values.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-calculation-of-latex-e-sol-global-solar-irradiation\">Calculation of \\(E_{sol}\\) \u2013 global solar irradiation<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The available irradiation on the collector field is described as \\(E_{sol}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\dot{E}_{sol} = (I_{b}+I_{H}+I_{d})\\cdot A_{gross} = I_{sol}\\cdot A_{gross}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The following definitions apply:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\(I_{b}\\) : direct (beam) tilted solar irradiation with \\(G_{b}\\) (direct solar radiation) and \\(\\theta_{I}\\) (angle of incidence)&nbsp;<br><br> \\(I_{b}=G_{b}\\cdot \\cos(\\theta_{I})\\)<br><\/li>\n\n\n\n<li>\\(I_{H}\\) : reduced global irradiance taking into account reflection and albedo effects of the surroundings. Here \\(G_{H}\\) stands for the reduced solar radiation depending on the horizon, \\(\\alpha\\) for the albedo factor and \\(F_{p}\\) for the reflected view factor&nbsp;<br><br> \\(I_{H}=G_{H}\\cdot \\alpha\\cdot F_{\\rho}\\)<br><\/li>\n\n\n\n<li>\\(I_{d}\\) : Diffuse irradiance with \\(G_{d}\\) (diffuse global irradiance) and \\(F_{d}\\) (proportion of diffuse irradiance into the collector surface)&nbsp;<strong> <\/strong><br><br> \\(I_{d}=G_{d}\\cdot F_{d}\\)<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-calculation-of-latex-q-sol-the-solar-thermal-yield\">Calculation of \\(Q_{sol}\\) &#8211; the solar thermal yield<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The solar thermal yield is calculated using the following equations:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\dot{Q}_{sol}=A_{gross}\\cdot (\\eta_{0,b}\\cdot I_{b}\\cdot K_{b}(\\theta_{t},\\theta_{l})+\\eta_{0,b}\\cdot I_{d}\\cdot K_{d}+\\eta_{0,b}\\cdot I_{H}-\\dot{q}_{loss})\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The term \\(\\dot{q}_{loss}\\) describes the sum of all losses that result as follows:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\dot{q}_{loss} = \\dot{q}_{loss,1}+\\dot{q}_{loss,2}+\\dot{q}_{loss,wind}+\\dot{q}_{loss,sky}+\\dot{q}_{loss,wind,0}+\\dot{q}_{loss,wind,IR}+\\dot{q}_{loss,irr}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In detail, the loss terms are defined as:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\"><table class=\"has-fixed-layout\"><tbody><tr><td>Description<\/td><td>Formula<\/td><\/tr><tr><td>First order heat loss<\/td><td>\\(\\dot{q}_{loss,1}=a_{1}\\cdot (T_{m}-T_{a})\\)<\/td><\/tr><tr><td>Second order heat loss<\/td><td> \\(\\dot{q}_{loss,2}=a_{2}\\cdot(T_{m}-T_{a})^{2}\\)<\/td><\/tr><tr><td>Wind effect on heat loss<\/td><td>\\(\\dot{q}_{loss,wind}=a_{3}\\cdot f_{wind}\\cdot u'(T_{m}-T_{a})\\)<\/td><\/tr><tr><td>Sky temperature on heat loss<\/td><td>\\(\\dot{q}_{loss,sky}=a_{4}\\cdot(\\sigma\\cdot T_{a}^{4}-I_{L})\\)<\/td><\/tr><tr><td>Wind effect on zero loss efficiency<\/td><td>\\(\\dot{q}_{loss,wind,0}=a_{6}\\cdot f_{wind}\\cdot u&#8217;\\cdot I_{sol}\\)<\/td><\/tr><tr><td>Wind effect on infrared loss<\/td><td>\\(\\dot{q}_{loss,wind,IR}=a_{7}\\cdot f_{wind}\\cdot u&#8217;\\cdot (I_{L}-\\sigma\\cdot T_{a}^{4})\\)<\/td><\/tr><tr><td>Irradiance loss<\/td><td>\\(\\dot{q}_{loss,irr}=a_{8}\\cdot(T_{m}-T_{a})^{4}\\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">With the following variables:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\(T_{m}\\) mean collector temperature<\/li>\n\n\n\n<li>\\(T_{a}\\) ambient temperature<\/li>\n\n\n\n<li>\\(u\u2019\\) reduced wind speed according to \\(u&#8217;=u-3\\frac{m}{s}\\)<\/li>\n\n\n\n<li>\\(I_{L}\\) long-wave irradiance \\(\\sigma\\) (Stefan-Boltzmann constant)<br> \\(I_{L}=G_{long-wave}\\cdot \\frac{1+\\cos(\\beta)}{2}+\\sigma\\cdot T_{a}^{4}\\cdot \\frac{1-\\cos(\\beta)}{2}\\)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">A state of equilibrium is assumed for the second-order heat loss and radiation loss terms if the ambient temperature is greater than the collector temperature:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(T_{a}&gt;T_{m}\\to \\dot{q}_{loss,2}=0, \\dot{q}_{loss,irr}=0\\)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-incident-angle-modifier-iam-latex-k-b-calculation\">Incident angle modifier (IAM) \u2013 \\(K_{b}\\) calculation<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The calculation of the IAM factor \\(K_{b}(\\theta_{t},\\theta_{l})\\) depends on the collector type. The following relationship applies for unglazed and flat plate collectors:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(K_{b}(\\theta_{t},\\theta_{l})=f_{axis}(\\gamma_{l},\\theta_{I})\\cdot \\cos^{2}(\\Phi)+f_{axis}(\\gamma_{t},\\theta_{I})\\cdot \\sin^{2}(\\Phi)\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For all other collector types: \\(K_{b}(\\theta_{t},\\theta_{l})=f_{axis}(\\gamma_{l},\\theta_{l})\\cdot f_{axis}(\\gamma_{t},\\theta_{t})\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\theta_{l}=\\arctan(\\cos(\\Phi)\\cdot \\left| \\tan(\\theta_{I}) \\right|)\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\theta_{t}=\\arctan(\\sin(\\Phi)\\cdot \\left| \\tan(\\theta_{I}) \\right|)\\)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\(\\theta_{l}\\) : angle of incidence on the collector surface<\/li>\n\n\n\n<li>\\(\\Phi\\) : angle as a function of rotation \\(\\Phi\\) (input in user interface) \\(\\Phi=rad(\\phi+90\\cdot (1-N_{axis}))+f_{\\theta_{I}}\\)<\/li>\n\n\n\n<li>The following applies to \\(f_{\\theta_{I}}\\) :\n<ul class=\"wp-block-list\">\n<li>\\(\\alpha_{s}\\) : sun elevation angle<\/li>\n\n\n\n<li>\\(\\beta\\) : inclination angle<\/li>\n\n\n\n<li>\\(\\gamma\\) : azimuth angle, depending on the tracking system<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">If \\(\\cos(\\beta)=0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f_{\\theta_{I}}=\\arctan(\\cos(\\alpha_{s})\\cdot \\frac{\\cos(\\gamma&#8217;)}{\\sin(\\alpha_{s})})\\) , if \\(\\sin(\\alpha_{s})=0\\) \\(f_{\\theta_{I}=0}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">else if \\(\\cos(\\beta)=\\sin(\\alpha_{s})\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f_{\\theta_{I}}=\\arctan(\\cos(\\alpha_{s})\\cdot \\frac{\\sin(\\gamma&#8217;)}{\\cos(\\beta)\\cdot (\\sin(\\beta)-\\cos(\\alpha_{s})\\cdot \\cos(\\gamma&#8217;))})\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">else <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f_{\\theta_{I}}=\\arccos(1-\\cos(\\theta_{I})\\cdot \\cos(\\beta) \\cdot \\frac{\\sin(\\beta)-f}{\\sin(\\theta_{I})\\cdot \\left[ 1-\\sin(\\beta)\\cdot (\\sin(\\beta)-f) \\right]})\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f=\\cos(\\beta)\\cdot \\frac{\\sin(\\beta)-\\cos(\\gamma&#8217;)\\cdot \\cos(\\alpha_{s})}{\\cos(\\beta)-\\sin(\\alpha_{s})}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The function \\(f_{axis}(\\gamma,\\theta)\\) is calculated using the IAM angle values contained in the collector database:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f_{axis}(\\gamma,\\theta)=k_{\\gamma,i}+(\\theta_{\\gamma}-\\theta_{i})\\cdot \\frac{k_{\\gamma,i+1}-k_{\\gamma,i}}{\\theta_{i+1}-\\theta_{i}}\\)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-calculation-of-the-collector-temperature\">Calculation of the collector temperature<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The temperature in the collector system is determined according to the first law of thermodynamics for open systems. The following diagram illustrates the principle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"767\" height=\"556\" src=\"https:\/\/www.velasolaris.com\/wp-content\/uploads\/2025\/06\/Screenshot-2025-06-05-115525.jpg\" alt=\"\" class=\"wp-image-136593\" style=\"width:614px;height:auto\" srcset=\"https:\/\/www.velasolaris.com\/wp-content\/uploads\/2025\/06\/Screenshot-2025-06-05-115525.jpg 767w, https:\/\/www.velasolaris.com\/wp-content\/uploads\/2025\/06\/Screenshot-2025-06-05-115525-300x217.jpg 300w, https:\/\/www.velasolaris.com\/wp-content\/uploads\/2025\/06\/Screenshot-2025-06-05-115525-200x145.jpg 200w, https:\/\/www.velasolaris.com\/wp-content\/uploads\/2025\/06\/Screenshot-2025-06-05-115525-400x290.jpg 400w, https:\/\/www.velasolaris.com\/wp-content\/uploads\/2025\/06\/Screenshot-2025-06-05-115525-600x435.jpg 600w\" sizes=\"auto, (max-width: 767px) 100vw, 767px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">As described above, the solar thermal yield is determined according to the following equation:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\dot{Q}_{sol}=A_{gross}\\cdot (\\eta_{0,b}\\cdot I_{b}\\cdot K_{b}(\\theta_{t},\\theta_{l})+\\eta_{0,b}\\cdot I_{d}\\cdot K_{d}+\\eta_{0,b}\\cdot I_{H}-\\dot{q}_{loss})=\\dot{E}_{sol,net}-\\dot{Q}_{loss}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the energy balance of the first law of thermodynamics, this results in:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\frac{dE_{coll}}{dt}=\\dot{E}_{sol,net}-\\dot{Q}_{loss}+P_{in}-P_{out}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or more specifically as a derivative of the collector temperature over time:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(m_{coll}\\cdot c_{p}\\frac{dT_{m}}{dt}=\\dot{E}_{sol,net}-\\dot{Q}_{loss}+P_{in}-P_{out}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From the catalogue entry, the dynamic heat capacity per square metre is given by the value \\(a_{5}\\). From this, the total heat capacity of the collector field can be reformulated as follows:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(A_{gross}\\cdot a_{5}\\frac{dT_{m}}{dt}=\\dot{E}_{sol,net}-\\dot{Q}_{loss}+P_{in}-P_{out}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rearranging the equation allows the collector temperature in the respective time step to be derived.<\/p>\n","protected":false},"author":135679,"featured_media":0,"parent":117670,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-136718","handbuch","type-handbuch","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v28.0 (Yoast SEO v28.0) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Solar Thermal Collectors Calculation per ISO 9806<\/title>\n<meta name=\"description\" content=\"Calculate solar thermal collectors in Polysun per ISO 9806. 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