# Simplified Collector Performance Model

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1 Simplified Collector Performance Model Prediction of the thermal output of various solar collectors: The quantity of thermal energy produced by any solar collector can be described by the energy balance equation where Q out = Q opt Q loss Q out Q opt Q loss = rate of thermal energy output = rate of optical energy absorbed by receiver = rate of thermal energy lost from receiver. Source: Solar energy Fundamentals and Design, William B. Stine & Raymond W. Harrigan, John Wiley 7Sons, 1985

2 Optical Energy Absorbed by the Receiver Q opt = A a ρ s,m τ g α r RSI a A a = specular reflectance of concentrating mirror, if any (1.0 for nonconcentrating flat-plate collector) ρ s,m = transmittance of any glass envelope covering the receiver (e.g. glass cover plate in a flat-plate collector) A a = aperture area of the collector S = receiver shading factor (fraction of collector aperture not shadowed by the receiver; 1.0 for a flat-plate collector) R = receiver intercept factor (fraction of reflected beam intercepted by receiver; 1.0 for a flat-plate collector) I a = isolation (irradiance) incident on collector aperture (W/m 2 ) α r = absorbance of the receiver S, R, α r, ρ s,m and τ g are constants (in a more detailed model, the dependence of the angle of the incident insolation can be considered) dependent only on the materials used and the structure accuracy of the collector - they can be lumped into a single constant term η opt, the optical efficiency of the collector. For a flat-plate collector utilizing no reflectors, S, R and ρ s,m are set equal to 1.0.

3 Thermal Energy Lost from the Receiver Q loss = A r U l ( T r T a ) A r T r T a = surface area of the receiver (m 2 ) = averaged receiver temperature ( o C) = ambient temperature ( o C) U l = overall heat loss coefficient (W/m 2 o C) T r = T out + T in 2 T out is the temperature in degrees C of the fluid leaving the collector while T in is the temperature of the fluid entering the collector. The heat loss coefficient U l is not a simple constant but instead may vary as heat-loss mechanisms change with temperature. For example as the temperature increases, radiant heat loss from the receiver increases.

4 Q out = A a η opt I a A r U l η col = Q out A a I a = η opt A r A a U l η col = η opt U l C g Collector Efficiency ( ) T r T a I a ( T r T a ) ( T r T a ) I a Where η col is the aperture efficiency of the collector and A r /A a is the geometric concentration ratio (C g ) C g = A a A r T s Aperture Receiver

5 Reflectance of Surfaces θ θ Specular reflection Diffuse reflection General reflection In general, the magnitude of the reflected intensity in a particular direction for a given surface is a function of the wave length and the spatial distribution of the incident radiation. The transmittance, reflectance and absorption are functions of the incoming radiation, thickness, refractive index and extinction coefficient of the material. The refractive index n, and extinction coefficient of the material K of the cover material are functions of the wavelength of the radiation. However, for our purposes here, all properties will be assumed to be independent of the wavelength, which is the case for glass.

6 Radiation Transmission and Absorption Reflection of unpolarized Radiation: r perpendicular = r 1 = sin2 θ 2 θ 1 sin 2 θ 2 + θ 1 For radiation at normal incidences: If one medium is air ( ) ( ) ( ) ( ) r parallel = r 2 = tan2 θ 2 θ 1 tan 2 θ 2 + θ 1 r = I r = 1 ( I i 2 r 1 + r 2 ) ( ) 2 ( ) 2 r = n n 1 2 n 1 + n 2 ( ) 2 ( ) 2 r = n 1 1 n 1 +1 n is the refractive index of the medium n 1 n 2 = sinθ 2 sinθ 1 Snell s law θ 1 and θ 2 are the angles of incidence and refraction

7 Nonabsorbing Cover Transmission Neglecting absorption in the cover material and considering only the perpendicular or parallel component of the polarization of the incoming radiation, the transmittance can be written as: τ 1 = τ 2 = 1 r 1 1+ r 1 τ 1 = 1 r 1 1+ r 1

8 Nonabsorbing Cover Transmission Since the components of r 1 and r 2 are not equal (except at normal incidence), the transmittance of initially unpolarized radiation is the average transmission of two components, τ r = 1 1 r r r 1 1+ r 2 where the subscript r is a reminder that only reflection losses are considered. For a system of N covers of the same material, τ rn = 1 1 r 1 1 r ( 2N 1)r 1 1+ ( 2N 1)r 2

9 Example The average index of refraction of glass for the solar spectrum is Calculate the reflectance of one surface of glass at normal incidence and at 60 degrees. Also calculate the transmittance of two covers of nonabsorbing glass at normal incidence and at 60 degrees. At normal incidence: At an incidence angle of 60 o ( ) 2 ( ) = r = n n 1 2 n 1 + n 2 2 = sin60 θ 2 = sin 1 = r = 1 ( 2 r + r 1 2)= 1 sin ( ) sin ( ) + tann 2 ( 25.42) tann 2 ( 94.58) = τ r ()= (0.0434) = 0.85 τ r ( 60)= = (0.185) 1+ 3(0.001)

10 Absorption by Glazing The absorption of radiation in a partially transparent medium is described by Bouguer s law, which is based on the assumption that the absorbed radiation is proportional to the local intensity in the medium and the distance x the radiation has traveled in the medium: di = IKdx Where K is the extinction coefficient (4m -1 ~ 32m -1 ) which is assumed to be constant in the solar spectrum. Integrating along the actual path length in the medium from 0 to L/cos θ 2 yields: τ a = I transmitted I incident = e KL cosθ 2

11 Optical Properties of Cover Systems 1 r 2 τ perpendicular = τ perpendicular 1 r perpendicular a 1+ r perpendicular 1 (r perpendicular τ a ) 2 ρ perpendicular = r perpendicular (1+ τ a τ perpendicular ) 1 r α perpendicular = (1 τ a ) perpendicular 1 r perpendicular τ a Transmittance Reflectance Absorptance For practical collector covers: τ a > 0.9 and r = 0.1 The transmittance of a single cover τ a τ r The absorptance of the collector cover, α 1 τ a The reflectance of a single cover, ρ τ a -τ

12 Calculate the transmittance, reflectance and absorptance of a single glass cover 2.3 mm thick at an angle of 60 degrees. The extinction coefficient of the glass is 32 m -1. At an incidence angle of 60 o, the optical path length is Example KL cosθ 2 = cos34.58 = τ a = e = Refraction angle calculated in an earlier example. The transmittance is found by averaging the transmittance of the parallel and perpendicular components of polarization or simply: τ = = The reflectance is found: α = = The absorptance is found to be: ρ = = The above results are also valid to identical multiple covers. The length L equal to the total cover system thickness.

13 Transmission - Absorption Transmittance-absorption Product (τα): Transmittance of the cover system at the desired angle = τ The angular absorption of the absorber plate = α Of the incident energy, τα is absorbed by the absorber plate and (1- α) τ is reflected back to the cover system. The reflection from the absorber plate is diffusive. ρ d refers to the reflectance of the cover system for diffuse radiation incident from the bottom side and can be estimated to be the difference between τ a and τ at an angle 60 o. The fraction of the incident energy ultimately absorbed is: τα τα = 1 ( 1 α)ρ d

14 Transmittance-absorption Product

15 Basic Flat-Plate Collector

16 Flat-Plate Collectors In the solar collector, the energy transfer is from source of radiant energy to a fluid. The flux of incident radiation ~ 1000 W/m 2 ( it is variable) The wavelength range: μm Energy delivery at moderate temperatures ~ 400K Use both beam and diffusive solar radiation - do not require tracking the Sun. Major applications: solar water heating, building heating, air conditioning and industrial process heat. Maximum temperature ~ 400K Source: Solar Engineering of Thermal processes by Duffie & Beckman, chapter 6.

17 Absorbed Radiation The solar radiation absorbed by a collector per unit area of absorber S is equal to the difference between the incident solar radiation and the optical losses and is given by the expression: S = I b R b ( τα) b + I d τα 1+ cosβ ( ) d Where (1+cos b)/2 and (1-cos b)/2 are the view factors from the collector to the sky and from the collector to the ground respectively. The subscripts b, d, and g represent beam, diffuse and ground. τα = transmittance -absorption product - a property of a cover-absorber combination. The geometric factor R b is the ratio of beam radiation on the tilted surface to that on a horizontal surface. I is the solar radiation J/m 2 and r g is the ground reflectance 2 + ρ g ( I b + I d )τα 1 cosβ ( ) g 2

18 The thermal energy lost from the collector to the surroundings by conduction, convection and infrared radiation = U L (T pm -T a ) Where, U L is the heat transfer coefficient, T pm and T a are mean absorber plate temperature and ambient temperature respectively. In steady state the useful energy out put of a collector of area A c : T pm is difficult to calculate or measure since it is a function of collector design, the incident solar radiation and the entering fluid conditions. Collector performance: Q u = A c [ S U L ( T pm T a )] Energy Balance η = A c Q u dt G T dt Incident total solar energy

19 Energy Balance Collector heat removal factor : Relates the actual useful energy gain of a collector to the useful gain if the whole collector surface were at the fluid inlet temperature. F R = m Ý C p A c U L 1 exp A U F ' C L m Ý C p Q u = A c F R [ S U L ( T i T a )] where U o is the heat transfer resistance from the fluid to the ambient air. Collector flow factor: F ' = U o U L F '' = F R F '

20 Collector Efficiency Factor

21 Collector Flow Factor

22 Energy Balance Q u = A c F [ R S U L ( T i T a )] Where T i is the inlet fluid temperature and F R is the heat removal factor. Instantaneous efficiency: η i = Q u = F R G T τα A c G T [ ( ) av U L (T i T a )] G T G T is given in W/m 2

23 Homework Calculate the daily useful gain and efficiency of 10 solar collector modules installed optimally (first determine the optimum orientation) in parallel at the college of engineering. The hourly radiation on the plane of the collector I T, the hourly radiation absorbed by the absorber plate S, and the hourly ambient temperature T a are given in the table. For the collector assume the overall loss coefficient U L to be 8 W/m 2 C and the plate efficiency factor F to be The flow rate through each 1x2m collector panel is o.03 kg/s and the inlet fluid temperature remains constant at 40 o C. Reference: Chapters 5 & 6 of Solar engineering of thermal processes, Duffie & Beckman, 2nd edition, Wiley Interscience, 1991.

24 Solar Collector Parameters

25 Liquid Heating Collector System

26 Determination of instantaneous h with beam radiation nearly normal to the absorbing surface. Determination of effects of angle of incidence of the solar radiation. Determination of collector time constant - a measure of effective heat capacity. The useful gain: Collector Characterization Q u = m C p ( ) T o T i Measure the fluid inlet and outlet temperatures and the fluid flow rate.

27 Water Heater Configurations Natural circulation system One tank forced circulation system Antifreeze loop and internal heat exchanger Antifreeze loop and external heat exchanger

28 Design Problem Using TRNSYS A transient process simulation program for the study of solar processes and its applications Thermal performance of solar water heater Develop a two cover collector design for a hot water load of 3000kg water/day at a minimum temperature of 335K evenly distributed between the hours of Solar collector total effective area = 65 m 2. The collector is designed to operate in Tallahassee.

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