Prandtl Number Effect on Assisted Convective Heat Transfer through a Solar Collector

Transcription

1 Available at Appl. Appl. Math. ISSN: Applications and Applied Mathematics: An International Journal (AAM) Special Issue No. 2 (May 2016), pp th International Mathematics Conerence, March 20 22, 2014, IUB Campus, Bashundhara Dhaka, Bangladesh Prandtl Number Eect on Assisted Convective Heat Transer through a Solar Collector Rehena Nasrin *, Salma Parvin and M.A. Alim Department o Mathematics Bangladesh University o Engineering & Technology Dhaka-1000, Bangladesh * rehena@math.buet.ac.bd ABSTRACT Numerical study o the inluence o Prandtl number on orced convective heat transer through a riser pipe o a lat plate solar collector is done. The working luid is Al 2 O 3 /water nanoluid. By Finite Element Method the governing partial dierential equations are solved. The eect o the Prandtl number on the temperature and velocity ield has been depicted. Comprehensive average Nusselt number, average bulk temperature, mean velocity, mid-height temperature inside the pipe, mean output temperature and collector eiciency are presented or the governing parameter mentioned above. Nu increases by 16% with the variation o Pr rom 4.6 to 6.6 using nanoluid. Due to rising Pr heat transer rate increases but collector eiciency devalues. KEYWORDS: Assisted convection, inite element method, water-al 2 O 3 nanoluid, thermal eiciency MSC 2010 No.: 65, INTRODUCTION Solar collectors are key elements in many applications, such as building heating systems, solar drying devices, etc. Solar energy has the greatest potential o all the sources o renewable energy especially when other sources in the country have depleted. The luids with solid-sized nanoparticles suspended in them are called nanoluids. The suspended metallic or nonmetallic 22

2 AAM: Inter n. J., Special Issue, No. 2 (May 2016) 23 nanoparticles change the transport properties and heat transer characteristics o the base luid. Nanoluids are expected to exhibit superior heat transer properties compared with conventional heat transer luids. Forced convection occurs when a luid low is induced by an external orce. In other words, it is a mechanism in which luid motion is generated by an external source (like a pump, an or mixer, suction device etc.). Signiicant amounts o heat energy can be transported very eiciently by this system and it is ound very commonly in everyday lie, including central heating, air conditioning, steam turbines and in many other machines. Struckmann (2008) analyzed lat-plate solar collector where eorts had been made to combine a number o the most important actors into a single equation and thus ormulate a mathematical model which would describe the thermal perormance o the collector in a computationally eicient manner. Zambolin (2011) theoretically and experimentally perormed solar thermal collector systems and components. Sandhu (2013) experimentally studied temperature ield in lat-plate collector and heat transer enhancement with the use o insert devices. Various new conigurations o the conventional insert devices were tested over a wide range o Reynolds number ( ). Chabane et al. (2013) studied thermal perormance optimization o a lat plate solar air heater. The received energy and useul energy rates o the solar air heaters were evaluated or various air low rates were (0.0108, , , and kg.s -1 ) are investigated. Optimum values o air mass low rates were suggested maximizing the perormance o the solar collector. Mahian et al. (2013) perormed a review o the applications o nanoluids in solar energy. The eects o nanoluids on the perormance o solar collectors and solar water heaters rom the eiciency, economic and environmental consideration viewpoints and the challenges o using nanoluids in solar energy devices were discussed. Natarajan and Sathish (2009) studied role o nanoluids in solar water heater. Heat transer enhancement in solar devices is one o the key issues o energy saving and compact designs. The aim o this paper was to analyze and compare the heat transer properties o the nanoluids with the conventional luids. Eismann and Prasser (2013) investigated correction or the absorber edge eect in analytical models o lat plate solar collectors. They derived a new correlation or the in eiciency. Their correlation was easy to use or engineering purposes. The correlation allowed or more accurate eiciency/cost optimization. One o the major uncertainties o analytical models was eliminated. Wei et al. (2013) studied lat-plate solar heat collector with an integrated heat pipe. An improved structure o lat-plate solar heat collector applied in construction o the solar water heater system was proposed in this paper where the collector used one large integrated wickless heat pipe instead o side-by-side separate heat pipes. High stability and leak avoidance between the water cooling side and the solar heating side were the main advantages in their system. Nasrin and Alim (2012, 2013) studied ree and orced convective heat and mass transer with entropy generation o nanoluid having single as well as double nanoparticles considering dierent geometry namely complicated cavity, direct absorption solar collector, prism shaped

3 24 R. Nasrin et al. solar collector, solar collector having wavy absorber, lat plate solar collector etc. They analyzed nanoluid low with double nanoparticles using necessary modiication o established nanoparticles eective properties. They showed how to enhance thermal perormance o dierent types o solar collectors in their analyses. In literature review, it is seen that there has been a good number o works in the ield o heat transer system through a lat plate solar collector. But there is some scope to work with temperature, heat transer and enhancement o collector eiciency using nanoluid. In this paper, the orced convection low through the riser pipe o a lat plate solar collector is studied numerically. The objective o this paper is to present temperature and velocity proiles as well as heat transer system or the variation o Prandtl number. 2. PROBLEM FORMULATION I I be the intensity o solar radiation, incident on the aperture plane o the solar collector having a collector surace area o A, then the amount o solar radiation received by the collector is: Qi IA. (1) Basically, it is the product o the rate o transmission o the cover (λ) and the absorption rate o the absorber (к). Thus, Q recv I A. (2) As the collector absorbs heat its temperature is getting higher than that o the surrounding and heat is lost to the atmosphere by convection and radiation. The rate o heat loss (Q loss ) depends on the collector overall heat transer coeicient (U l ), the collector temperature (T col ) and ambient temperature (T amb ). So, loss l col amb Q U A T T. (3) Thus, the rate o useul energy extracted by the collector (Q usl ), expressed as a rate o extraction under steady state conditions, proportional to the rate o useul energy absorbed by the collector, less the amount lost by the collector to its surroundings. This is expressed as: Q Q Q I AU A T T. (4) usl recv loss l col amb It is also known that the rate o extraction o heat rom the collector may be measured by means o the amount o heat carried away in the luid passed through it. Thus, Q usl p out in mc T T, (5)

4 AAM: Inter n. J., Special Issue, No. 2 (May 2016) 25 where m, C p, T in and T out mass low rate, speciic heat at constant pressure, inlet and outlet luid temperature respectively. Equation (4) may be inconvenient because o the diiculty in deining the collector average temperature. It is convenient to deine a quantity that relates the actual useul energy gain o a collector to the useul gain i the whole collector surace were at the luid inlet temperature. This quantity is known as the collector heat removal actor (F R ) and is expressed as: F R mc p Tout Tin AI U T T l in amb. (6) The maximum possible useul energy gain in a solar collector occurs when the whole collector is at the inlet luid temperature. The actual useul energy gain (Q usl ), is ound by multiplying the collector heat removal actor (F R ) by the maximum possible useul energy gain. This allows the rewriting o equation (4) as: Qu F sl RAI Ul Tin Tamb. (7) The heat lux per unit area q is now denoted as Q usl A l in amb q I U T T. (8) Equation (7) is a widely used relationship or measuring collector energy gain and is generally known as the Hottel-Whillier-Bliss equation. The instantaneous thermal eiciency o the collector is: Q F usl R A I Ul Tin Tamb Tin Tamb FR FR Ul. (9) AI AI I A schematic diagram o the system and its cross sectional views are shown in Figures 1 (i)-(ii). The numerical computation is carried on taking single riser pipe o FPSC. FPSC with single riser pipe gives the average heat transer and luid low phenomena. Glass cover is at the top o the FPSC. It is made up o borosilicate which has thermal conductivity o 1.14 W/mK and reractive index o 1.47, speciic heat o 750 J/kgK and coeicient o sunlight transmission o 95%. The wavelength o visible light is roughly 700 nm. Thickness o glass cover is 0.005m. There is an air gap o 0.005m between glass cover and absorber plate. Air density = Kg/m 3, speciic heat = J/kgK and thermal conductivity = W/mK. All these properties o air domain represent air o temperature at 298K. A dark colored copper absorber plate is under the air gap. Length, width and thickness o the absorber plate are 1m, 0.15m and m

5 26 R. Nasrin et al. respectively. Coeicients o heat absorption and emission o copper absorber plate are 95% and 5% respectively. Solar radiance Absorber plate Out low Riser pipe Bonding conductance Inlow (i) Air gap Solar irradiation Glass cover Absorber plate Inlow Out low y // x Insulation Housing (ii) Figure 1. Computational domain o the FPSC (i) 3D and (ii) 2D longitudinal views The riser pipe has inner diameter 0.01 m and thickness m. The riser tube is also made in copper metal. A trapezium shaped bonding conductance o copper metal is attached to the absorber and riser pipe. The luid through the copper riser pipe is water-based nanoluid containing Al 2 O 3 nanoparticles whose thermo-physical properties are shown in Table 1. The two dimensional governing equations are:

6 AAM: Inter n. J., Special Issue, No. 2 (May 2016) 27 where u v 0, x y (10) 2 2 n u u v p u u n 2 2, x y x x y (11) 2 2 n u v v p v v n 2 2 x y y x y, (12) 2 2 u v T T n 2 2, x y x y (13) T x T y 2 2 a a k C n n p n is the thermal diusivity, 1 n s is the density,, (14) Cp 1 Cp Cp n s is the heat capacitance, n is the viscosity o Brinkman model (1952), k n k k 2k 2 k k s s k 2k k k s s is the thermal conductivity o Maxwell Garnett (MG) model (1904), ν Pr α ia the Prandtl number and

7 28 R. Nasrin et al. uinl Re ia the Reynolds number. The boundary conditions o the riser pipe are: at all solid boundaries: u = v = 0, at the solid-luid interace: k n T T k a solid y y n solid, at the inlet boundary: T T in, u = u in, at the outlet boundary: convective boundary condition p = 0, at the top surace o absorber: heat lux T k q I U T T y at outer surace o riser pipe: a a l in amb, and T y 0. Table 1. Thermo physical properties o luid and nanoparticles at 300K Physical Properties Fluid phase (Water) Al 2 O 3 C p (J/kgK) (kg/m 3 ) k (W/mK) (m 2 /s) For water based nanoluid low the expression o local Nusselt number is as ollows: Nu T kn y L kn k T k k q y ql k. (15)

8 AAM: Inter n. J., Special Issue, No. 2 (May 2016) 29 The non-dimensional orm o local heat transer rate at the riser pipe solid surace is k Nu k n Y. The above equations are non-dimensionalized by using the ollowing dimensionless, quantities X x y T Tin k, Y,. L L ql By integrating the local Nusselt number over the riser pipe, the average heat transer rate o the collector can be written as 1 L Nu Nu dx L. 0 The mean bulk temperature and average sub domain velocity o the luid inside the collector may be written as T av A A Td A da A Td A HL and Vd A A A Vav A da Vd A HL, (16) where A, L, H and V are the area, length and height o the absorber tube, magnitude o subdomain velocity, respectively. The overall heat transer loss rom the collector is the summation o three separate components, the top loss coeicient, the bottom loss coeicient and the edge loss coeicient. The empirical relations or these coeicients are mentioned by Duie and Beckman (1991) and Al-Ajlan et al. (2003) as ollows: Ul Ut Ub Ue, (17) where U t is the heat loss coeicient rom the top, U b is the heat loss coeicient rom the bottom, and U e is the heat loss coeicient rom the edges o collector. These can be calculated rom the ollowing relations:

9 30 R. Nasrin et al. U U t b N 1 Ta Tamb Ta Tamb c C T h 1 2N amb a a T amb N T a Nhamb c a N kb Ue Ae k 2L W e H, Ue, xb A xe LW, C 2 520* , and h h * N. a amb a 3. NUMERICAL IMPLEMENTATION The Galerkin inite element method Taylor and Hood (1973) and Dechaumphai (1999) is used to solve the non-dimensional governing equations along with boundary conditions. The equation o continuity has been used as a constraint due to mass conservation and this restriction may be used to ind the pressure distribution. The inite element method o Reddy (1994) is used to solve the equations (10) - (14), where the pressure P is eliminated by a constraint. The continuity equation is automatically ulilled or large values o this constraint. Then the velocity components (u, v) and temperature (T) are expanded using a basis set. The Galerkin inite element technique yields the subsequent nonlinear residual equations. Three points Gaussian quadrature is used to evaluate the integrals in these equations. The non-linear residual equations are solved using Newton Raphson method to determine the coeicients o the expansions Mesh Generation In the inite element method, the mesh generation is the technique to subdivide a domain into a set o sub-domains, called inite elements, control volume, etc. The discrete locations are deined by the numerical grid, at which the variables are to be calculated. It is basically a discrete representation o the geometric domain on which the problem is to be solved. The computational domains with irregular geometries by a collection o inite elements make the method a valuable practical tool or the solution o boundary value problems arising in various ields o engineering. Figure 2 displays the inite element mesh o the present physical domain. Figure 2. Mesh generation o the computational domain

10 AAM: Inter n. J., Special Issue, No. 2 (May 2016) Grid Independent Test An extensive mesh testing procedure is conducted to guarantee a grid-independent solution or Re = 480 and Pr = 5.8 in a solar collector. In the present work, we examine ive dierent nonuniorm grid systems with the ollowing number o elements within the resolution ield: 42,010, 99,832, 1,40,472, 1,68,040 and 1,92,548. The numerical scheme is carried out or highly precise key in the average Nusselt number or water-alumina nanoluid ( = 2%) as well as base luid ( = 0%) or the aoresaid elements to develop an understanding o the grid ineness as shown in Table 2. The scale o the average Nusselt numbers or nanoluid and clear water or 1,68,040 elements shows a little dierence with the results obtained or the other elements. Hence, considering the non-uniorm grid system o 1,68,040 elements is preerred or the computation. Table 2. Grid test at Pr = 5.8, = 2%, I = 215W/m 2 and Re = 480 Elements 42,010 99,832 1,40,472 1,68,040 1,92,548 Nu (Nanoluid) Nu (Base luid) Time (s) RESULTS AND DISCUSSION Finite element simulation is applied to perorm the analysis o laminar orced convection temperature and luid low through a riser pipe o a lat plate solar collector illed with water/alumina nanoluid. Eect o the Prandtl number (Pr) on heat transer and collector eiciency has been studied. The range o Pr or this investigation vary rom 4.6 to 6.6 where solar irradiation (I), Reynolds number (Re), and solid volume raction nanoluid () remain ixed at 215 W/m 2, 480 and 2% respectively. The mass low rate per unit area (m) is Kg/s, overall heat transer coeicient (U l ) is 8 W/m 2 K, aperature area o the collector (A) is 1.8 m 2, number o glass (N) is 1, the tilt angle o collector is 0 0 and nanoparticle size is 10 nm. Eect o Pr is shown in the Figures 3 (i)-(iii). The streamlines in the Figure 3(i) occupying the whole riser pipe is ound at the Prandtl number (Pr = 5.8). Color o streamlines at the solid suraces o riser pipe is deep blue and at the middle is red. This is due to the act that no-slip condition is maintained at the solid suraces o riser pipe. Thus maximum velocity is obtained at the middle o the pipe. Also the boundary plot represents that luid passes through the pipe and takes heat rom top surace o riser pipe. That s why boundary temperature becomes low near the inlet and gradually becomes high near the exit boundary. The temperature o top surace o riser pipe is higher than the bottom surace. Arrow plot indicates direction o luid low. From Figure

11 32 R. Nasrin et al. 3(iii) it is observed that nanoluid low enters at the let inlet opening and exits rom the right outlet opening. The eect o Prandtl number (Pr) on average Nusselt number (Nu) at the top o riser pipe, average temperature (T av ), magnitude o average velocity (V av ), mid-height temperature (T) o water-copper nanoluid, mean outlet temperature (T out ) and percentage o collector eiciency with Re = 480, I = 215W/m 2, = 2% are shown in Figures 4(i)-(vi). The values o Pr are chosen as 4.6, 5.2, 5.8 and 6.6. (i) (ii) (iii) Figure 3: Eect o Pr at 5.8 on (i) streamlines, (ii) boundary plot and (iii) arrow plot All these values o Prandtl number represent water at dierent temperatures. From the Figure 4(i) it is clearly observed that as Pr increases, heat transer rate enhances or both luids. Nu increases by 16% and 12% with the variation o Pr rom 4.6 to 6.6 or water-copper nanoluid and clear water respectively. As Pr increases average bulk temperature (T av ) o the luids decreases gradually or both luids. It is well known that increasing Prandtl number devalues luid temperature. Nanoluid has higher temperature than base luid.

12 AAM: Inter n. J., Special Issue, No. 2 (May 2016) 33 Mean velocity (V av ) devalues or higher Prandtl number because luid with higher viscosity can t move reely like lower viscous luid. Water-Al 2 O 3 nanoluid has lower mean velocity than clear water. Temperature (T) o water-cu nanoluid at the middle o riser pipe is seen in Figure 4(iv). Nanoluid temperature increases when it passes through the riser pipe. But temperature o nanoluid diminishes with growing Pr. (i) (ii) (iii) (iv) (v) (vi) Figure 4. Eect o Pr on (i) mean Nusselt number, (ii) mean bulk temperature, (iii) mean velocity, (iv) mid-height temperature, (v) mean output temperature and (vi) collector eiciency Figure 4(v) shows that mean outlet temperature (T out ) decreases with increasing Prandtl number. This describes in this way that increasing Prandtl number decreases luid temperature. As a result lower tempered luid can t become so hot at outlet exit like higher tempered luid. Average outlet temperature becomes 318K, 316K, 314K, 313K or nanoluid and 313K, 312K, 310K, 309K or water respectively with the variation o Pr = 4.6, 5.2, 5.8 and 6.6. Using higher values o Prandtl number the collector eiciency decreases. In this scheme water/aluminium oxide nanoluid ( = 2%) perorms better than clear water ( = 0%). Thermal

13 34 R. Nasrin et al. eiciency devalues rom 66%-45% or nanoluid and 52%-36% or water due to the variation o Prandtl number Pr rom 4.6 to CONCLUSION The results o the numerical analysis lead to the ollowing conclusions: The structure o the luid streamline, boundary plot and arrow plot through the solar collector is ound to signiicantly depend upon the Prandtl number. The Al 2 O 3 nanoparticle with the highest Pr is established to be most eective in enhancing perormance o heat transer rate than base luid. Collector eiciency is obtained lower or growing Pr. Mean bulk temperature diminishes or both luids with rising Pr. Mean velocity is obtained higher or base luid with alling Pr. ACKNOWLEDGEMENT The present numerical work is done in the Department o Mathematics, Bangladesh University o Engineering & Technology, Dhaka-1000, Bangladesh. Research Support & Publication Division, University Grants Commission, Agargaon, Bangladesh helps inancially or this work. REFERENCES Brinkman, H.C. (1952). The viscosity o concentrated suspensions and solution, Journal o Chemical Physics, Vol. 20, pp Chabane, F., Moummi, N., Benramache, S., Bensahal, D., Belahssen, O., Lemmadi, F.Z., (2013). Thermal perormance optimization o a lat plate solar air heater, Int. J. o Energy & Tech., Vol. 5, No. 8, pp Dechaumphai, P. (1999). Finite Element Method in Engineering, 2nd ed., Chulalongkorn University Press, Bangkok. Eismann, R., Prasser, H.-M. (2013). Correction or the absorber edge eect in analytical models o lat plate solar collectors, Solar Energy, Vol. 95, pp Mahian, O., Kianiar, A., Kalogirou, S.A., Pop, I., Wongwises, S. (2013). A review o the applications o nanoluids in solar energy, Int. J. o Heat and Mass Trans. Vol. 57, pp Maxwell-Garnett, J.C., (1904). Colours in metal glasses and in metallic ilms. Philos. Trans. Roy. Soc. A, Vol. 203, pp Nasrin, R. and Alim, M.A. (2012). Eect o radiation on convective low in a tilted solar collector illed with water-alumina nanoluid, Int. J. o Engg., Sci. & Tech., Vol. 4, No. 4, pp Nasrin, R. and Alim, M.A. (2012). Prandtl number eect on ree convective low in a solar collector utilizing nanoluid, Engg. e Transac., Vol. 7, No. 2, pp Nasrin, R. and Alim, M.A. (2013). Free convective low o nanoluid having two nanoparticles inside a complicated cavity, Int. J. o Heat and Mass Trans., Vol. 63, pp Nasrin, R. and Alim, M.A. (2013). Perormance o nanoluids on heat transer in a wavy solar collector, Int. J. o Engg., Sci. & Tech., Vol. 5, No. 3, pp

14 AAM: Inter n. J., Special Issue, No. 2 (May 2016) 35 Nasrin, R., Parvin, S. and Alim, M.A. (2013). Buoyant low o nanoluid or heat-mass transer through a thin layer, Mech. Engg. Res. J., Vol. 9, pp Natarajan, E. & Sathish, R. (2009). Role o nanoluids in solar water heater, Int. J. Adv. Manu. Techn., Vol. 45, 5 pages. DOI /s Reddy, J.N. and Gartling, D.K. (1994). The Finite Element Method in Heat Transer and Fluid Dynamics, CRC Press, Inc., Boca Raton, Florida. Sandhu, G. (2013). Experimental study o temperature ield in lat-plate collector and heat transer enhancement with the use o insert devices, M. o Engg. Sci. theseis, The School o Graduate and Postdoctoral Studies, The University o Western Ontario London, Ontario, Canada. Struckmann, F. (2008). Analysis o a lat-plate solar collector, Project Report, MVK160 Heat and Mass Transport, Lund, Sweden. Taylor, C., Hood, P. (1973). A numerical solution o the Navier-Stokes equations using inite element technique, Computer and Fluids, Vol. 1, pp Wei, L., Yuan, D., Tang, D., Wu, B. (2013). A study on a lat-plate type o solar heat collector with an integrated heat pipe, Solar Energy, Vol. 97, pp Zambolin, E. (2011). Theoretical and experimental study o solar thermal collector systems and components, Scuola di Dottorato di Ricerca in Ingegneria Industriale, Indirizzo Fisica Tecnica. NOMENCLATURES A aperature area o solar collector (m 2 ) A e area base on the perimeter o collector (m 2 ) A area o 2D cross section o riser pipe (m 2 ) C constant deined in section 2 C p speciic heat at constant pressure (Jkg -1 K -1 ) riction actor h convective heat transer coeicient (Wm -2 K -1 ) h a convective heat transer coeicient between glass and ambient air (Wm -2 K -1 ) H height o the collector (m) I solar irradiation (Wm -2 ) k thermal conductivity o luid (Wm -1 K -1 ) k b back insulation conductivity (Wm -1 K -1 ) k e edge insulation conductivity (Wm -1 K -1 ) L length o the solar collector (m) N number o glass Nu average Nusselt number p pressure (kgms -2 ) Pr Prandtl number q heat lux (Wm -2 ) Re Reynolds number T luid temperature (K) u, v, w velocity components along x, y, z direction (ms -1 )

15 36 R. Nasrin et al. U l overall heat transer coeicient (Wm -2 K -1 ) V magnitude o velocity (ms -1 ) V volume o riser pipe (m 3 ) x b back insulation thickness (m) x e edge insulation thickness (m) x, y Cartesian coordinates (m) X, Y dimensionless Cartesian coordinates W width o collector (m) Greek symbols thermal diusivity (m 2 s -1 ) tilt angle ( 0 ) λ transmitivity ε emissivity η thermal eiciency nanoparticles volume raction θ dimensionless luid temperature μ dynamic viscosity o the luid (m 2 s -1 ) ν kinematic viscosity o the luid (m 2 s -1 ) density o the luid (kgm -3 ) κ absorption coeicient Subscripts a absorber amb ambient av average b back c cover col collector e edge luid in input loss lost n nanoluid out output recv received s solid particle t top usl useul