Convective effects in air layers bound by cellular honeycomb arrays *

Transcription

1 Journal o Scientiic & Industrial Research Vol. 64, August 5, pp Convective eects in air layers bound by cellular honeycomb arrays * Pawan Kumar 1 and N D Kaushika, * 1 Centre or Energy Studies, Indian Institute o Technology, Hauz Khas, New Delhi Bharati Vidyapeeth s College o Engineering, A-4, Paschim Vihar, New Delhi Received 14 June 4; revised 4 April 5; accepted 14 June 5 Rayleigh-Benard convection in a luid (air) layer in presence o transparent cellular structure (honeycomb) is reviewed or application to the engineering design o transparent insulation materials devices. The explicit numerical computations o Rayleigh numbers o the characteristic equation at higher wave numbers corresponding to the onset o instability are carried out. Results highlight a new acet o theory that it can indeed be used to estimate the critical Rayleigh number, which corresponds to the layer bound by vertical walls o cellular structure. A relationship between wave number and aspect ratio o honeycomb structure is evolved. Subsequently the stability o inclined luid layer is considered and base low existence and the need or its consideration in terms o Nusselt number is discussed. An outline o governing equations o the square celled air temperature, vorticity, stream unctions and velocity are presented and it is ound that the Nusselt number has a value in the range or a wide range o Rayleigh numbers in near critical Rayleigh regime. This range may be considered as a trade o region or the engineering design o honeycomb devices. Convection can be suppressed by varying the aspect ratio o honeycomb cell. Honeycomb structure o aspect ratio 1-15 seems suitable or suppression o convection in air layer o depth 5- cm or temperature dierence ( T ) o to 1. Keywords: Critical Rayleigh number, Stability theory, Convection suppression, Honeycomb, Transparent insulation material IPC Code: B1D 47/; F16 59/6 Introduction The air layers have been used as an insulating material or engineering applications such as nuclear reactors and building elements (double glazed windows, solar collectors, integrated collector storage systems). In such applications, it is important to suppress natural convection in the air layer because the thermal transer coeicient across the layer increases with convective mass transport. Honeycomb cellular structures have been suggested as a means to suppress the convection in air layers heated rom below. The air layer bound by transparent honeycomb are solar transparent; yet provide good thermal insulation and are oten reerred to as transparent insulation material in solar energy context. The luid mechanical treatment o this type o problem or square cells has been given by several researchers 1-6 and or rectangular cell by Koutsoheras & Charters 7. According to these investigations, when a cellular matrix (Fig. 1) is introduced in a luid layer o ininite horizontal extent and inite vertical depth () *Author or correspondence Tel: ; Teleax: E mail: pksharma197@yahoo.com, ndkaushika@yahoo.com contained between two isothermal bounding suraces and heated at the bottom, the vertical walls provide extra viscous resistance to the onset o convection and eectively raise the value o critical Rayleigh numbers (Rc), which depend upon physical shape and aspect ratio (A/d) o honeycomb cell. In this paper, an alternative approach or the evaluation o critical Rayleigh numbers o air layers in presence o honeycomb structures has been developed. The approach illustrates a new acet, mathematical and physical, o general theory o hydrodynamic stability o luid layer o horizontal ininite extent heated rom below (Rayleigh Benard problem). It involves numerical solution or higher wave numbers o transcendental characteristic equations o normal mode analysis 8. Furthermore, inclined air layer bound by cellular structures have many applications in advanced glazing o solar thermal system. In inclined layers, Hollands et al 9-1 have pointed out the inevitable existence o base low and need or consideration in terms o Nusselt number. In this paper, Nusselt number has been examined as a unction o Rayleigh number near the critical Rayleigh number in inclined

2 KUMAR & KAUSHIKA: CONVECTION IN AIR AYERS BOUND BY CEUAR HONEYCOMB ARRAYS 63 (a) Square honeycomb (c) Up slope slat Fig. 1 Schematics o cellular structures air layer bound by cellular matrix. The objective is to investigate trade os or the engineering design o honeycomb devices. Convective Eects in Horizontal Air ayer with Honeycomb Pellew & Southwell in 194 were irst to develop the numerical solution or critical Rayleigh regime stability problem or two rigid boundaries in a horizontal luid (air) layers heated rom below. In this context, two methods were used: 1) Energy method; and ) Normal mode technique. Chandrasekhar 8 gave a detailed analysis o Rayleigh Benard problem based on normal mode technique. The ollowing two assumptions were made during the ormulations o the governing equations: 1. Boussinesq Approximation Density o the luid is constant except in buoyancy term,. Fourier s aw o volume expansion The change in density is related with the change in temperature given by δρ αρ ( T T ), which gives ρ ρ[1 α( T T )] ρ[1 αγ y] ; where α is the coeicient o volume expansion, T is the initial temperature, ρ is the initial density and T T T γ is adverse temperature gradient. y y Applying these assumptions to the governing equations o horizontal air layer heated rom below bound by suraces y± /, the time dependent governing equations ater neglecting the product o perturbations and their powers (higher than one) reduce to the orm: u v w + +. (1) x y z u 1 p + u t ρ x υ. ()

3 64 J SCI IND RES VO. 64 AUGUST 5 v 1 p α Tg + υ v. (3) t ρ y w 1 p + υ w. (4) t ρ z DT Dt κ T. (5) T and u v w at y and y. (6) Following normal mode technique, assuming perturbations, are given by u U ( y)exp[ i( ax+ bz) + ct], v V ( y)exp[ i( ax+ bz) + ct], w W ( y)exp[ i( ax+ bz) + ct], T θ ( y)exp[ i( ax+ bz) + ct], p ρ P( y) exp[ i( ax + bz) + ct], (7) where k( a,, b ) is the wave number and c is the complex wave velocity. When these relations are substituted into Eqs (1) to (5), and all variables except V and θ are eliminated, one get [( D k )( D k ) c/ ] V gk / υ α θ υ, (8) [( D k ) c / k ] θ γ / kv. (9) Using transormations y c υ h σ r y,, P, k, υ κ κθ gα γ θ, D D and R γ κυ 4 (1) the non-dimensional orm o Eqs (8 & 9), on removing bars rom y, D and θ and substituting 1 and σ (marginal state o system) gives: ( D h ) V h Rθ (11) ( D h ) θ V (1) On eliminating θ between Eqs (11) and (1), we get 3 ( D h ) V + h RV. (13) For rigid suraces ( y± 1/ ); the boundary conditions may be taken as V DV D h V at y ( ) ± 1/. (14) Assuming V Acosq y+ Bcoshqy + Bcoshqy as even solution and V Asinq y + Bsinhqy+ Bsinhqy as odd solution or Eq. (13), where A, B are arbitrary 3 constants, q is a root o the Eq. ( q h ) Rh and is given by qq 1 +iq such that q h{ (1 + n+ n ) n}, q h{ (1 + n+ n ) (1 + n)} 1 1 q h n and ( 1) (15) The characteristic Eqs or even and odd solutions o Eq. (13) obtained by using Eq. (15) are as: Even characteristic equation: 1 1 im{( 3 + i) qtanh q} + q tan q, (16) 1 1 Odd characteristic equation: 1 q cot q Im[( 3 + i) sinhq isinq, (17) coshq1 cosq 1 q The transcendental Eqs (16) and (17) are solved iteratively to obtain characteristic values o Rayleigh number as a unction o wave number h. Chandrasekhar 8 reported results o such solutions or the values o wave numbers (-8) and used the criteria that minimum Rayleigh number would correspond to critical Rayleigh number at the wave number o minima. From the plot o Rayleigh numbers, at which the instability sets in, the critical Rayleigh numbers o R c at wave number h3.117 and Rc when h5.365 were obtained. In above analysis or Rayleigh number, the even perturbations are more adverse to the stability than the odd perturbations (Fig. ). The wave number h is a measure o nondimensional width o the luid cell. Holland 11 correlated wave number with aspect ratio o cellular

4 KUMAR & KAUSHIKA: CONVECTION IN AIR AYERS BOUND BY CEUAR HONEYCOMB ARRAYS 65 Table 1 Rayleigh number versus wave numbers corresponding to the onset o instability h Ra rom even solution Ra rom odd solution h Ra rom even solution Ra rom odd solution Fig. Rayleigh numbers corresponding to the onset o instability as a unction o wave numbers honeycomb array used or convection suppression as ollows: h π 5A π 5( / d). (18) The above argument suggests that i one can draw curves between Rayleigh number and higher values o h corresponding to even and odd solutions o characteristic equation, the points on lower curve would correspond to critical Rayleigh number (the onset o instability) at the corresponding wave numbers. Authors, thereore, solved the transcendental Eqs (16) and (17) or the wave number up to 4 or minimum Ra (Table 1). Following Edwards & Catton 1, expressed as: R ( a +3.9) 3 c ( a +7.97) Rc may be where, a mπ 5 A and.75 m 1 (19) Fig. 3 Comparative study o critical Rayleigh numbers as a unction o aspect ratio. The computed values o R c corresponding to m exhibit close agreement with experimentally measured values o Heitz & Westwater 14. Thereore, critical Rayleigh number curve o present analysis corresponding to the onset o instability as a unction o wave number is compared with the analysis o Edward & Catton 1 (Fig. 3). Results o Edward & Catton 1 belong to the luid mechanical treatment o luid layer with honeycomb structure. The similarity in representation (Fig. 3) concludes that Table1 or odd solution may be used or critical Rayleigh number as a unction o wave number. The critical Rayleigh number or wave numbers, not given in Table 1, may be calculated iteratively by solving Eq. (17) or required h. The values o critical Rayleigh number may be used or the engineering design o honeycomb devices. The Rayleigh number o a luid layer heated rom below may be expressed as:

5 66 J SCI IND RES VO. 64 AUGUST 5 3 gα T Ra () κν So, or convection suppression (Ra<Rc) Rcκν (1) gα ( T ) max 3 For air, one has 5 g 9.8 m/ s, κ.67 1 m / s, υ m / s, α (.97 1 ) C () Thereore κυ ξ (say) (3) gα ξr Hence c ( T) max (4) 3 where ξ is number, which has dierent values or dierent materials. The computed ( T) max values o air layer obtained rom Eq. (4) are illustrated in Figs 4a and 4b respectively. It shows that cell width (8-1 mm) is required to suppress convection or temperature dierence (-1 C) or 1-15 cm depth o air layer with honeycomb. Convective Eects in Inclined Air ayer with Honeycomb I an air layer is inclined rom horizontal and heated rom below, the luid can never be stationary. Since the unbalanced buoyancy orce always creates some motion, the low, or which Ra< 178/cosβ, is the base low. The motion consists o a single cell with luid rising rom the hot surace and alling along the cold one and turning at upper and lower extremes o layer. The low o convective heat across the layer will be normal to the plates and in middle region the heat transer across the layer is only by conduction. This base low luid movement must be considered in the study o convective heat transer through the inclined luid layer bound by honeycomb cellular panel. Nusselt number (the dimensionless parameter characteristic o the ratio o convective heat low to the conduction) must thereore, be considered in addition to critical Rayleigh number. Mathematical model o convective stability o inclined luid (air) layer bound by square honeycomb was given by Koutsoheras & Charters 7, Arulanantham et al 1 etc. The critical Rayleigh as well as post-critical Rayleigh Regimes are examined through Nusselt number. Following assumptions have been made to study the convective eects in inclined air layer bound by square honeycomb (Fig. 5). i) The density o the luid remains constant except in buoyancy terms; θ ii) Zero heat lux through the side walls i.e. ; x iii) inear variation in temperature, Y i.e. θ θh + ( θh θc ) ; iv) The thickness and conductivity o the walls are inite. Non-dimensional governing equation o continuity, momentum and energy equations are given by u v + x y u u u + u + v τ x y sin p u u Ra PT r β + Pr + x x y v + u v + v v Ra PT r cosβ p τ x y y v v + Pr + x y + u + v + τ x y x y T T T T T (5) (6) (7). (8) The two dimensional conduction Eq or the wall is given by θw θw θw κ w + t X Y (9) κ w is the thermal diusivity o the wall material. The total convective and conductive heat transer (Q y ) crossing a plane at constant y is given by

6 KUMAR & KAUSHIKA: CONVECTION IN AIR AYERS BOUND BY CEUAR HONEYCOMB ARRAYS 67 Fig. 4a Variation o ( T ) max or honeycomb with its depth () and cell size (d) Fig. 4b Variation o ( T ) with aspect ratio(a) or honeycomb o depth () max

7 68 J SCI IND RES VO. 64 AUGUST 5 D D T Qy ρ Cp vt dx K dx y (3) The average Nusselt number is given by Nu y Qy K T (31) Ater non-dimensionalisation o ( Q y ) and substituting it in Nu y, Nusselt number is given by Fig. 5 Schematic o square honeycomb Fig. 6 Eect o tilt angle on Nusselt number: Ra1,, A1 1/ A 1 T Nuy A v T dx y (3) Arulanantham et al 1 developed an algorithm or the calculation o Nusselt number using Eq. (3). It has been used here to investigate eect on Nusselt number o tilt angle and Rayleigh number (Figs 6 & 7). For a wide range o Rayleigh number, Nusselt number was rom 1 to 1.5, which implies that the convection is very weak and almost suppressed. This range o Nu can be regarded as the trade o range or engineering design o honeycomb with nonconvective air cell. It can also be considered as theoretical basis o Nu 1. used by Holland 15 in experiments on convection suppression. The relations based on correlations or the Nusselt number o a square celled honeycomb given by Smart et al 16 and Cane et al 1 are as ollows: sin β π cos β Ra Nu A (33) The range o validity is 4 o o ( Ra/ A ) < 6, 3 < β < 9 and A 4. Hollands 1 considered the base low and recommended the engineering design o honeycomb with Nusselt number o 1.. The minimum aspect ratio A that is required just to suppress the convection is given by: Fig. 7 Eect o Ra on Nusselt number: A1, RK, A1, β4 1/ 4 Ra F( ) 4 A β (34)

8 KUMAR & KAUSHIKA: CONVECTION IN AIR AYERS BOUND BY CEUAR HONEYCOMB ARRAYS 69 where ( ) 1/( sin β ) o o F( β ) 4.45cos ( β 6), 3 β 9 (35) And the value o Ra used here is deined by Eq. 4 3 Ra 737(1 + 1 ) β1 T (1 ) p β (36) where β 1 1/ Tm, Tm( ink) ( Th+ Tc)/, T Th Tc, and p is the atmospheric pressure or mean temperature, 8K Tm 5K. By analytical observations, above expression is also valid or β, i F (β) is taken as 1.7 and linear interpolation o values ( - 3 ) is also permitted. For air, at atmospheric pressure and moderate temperature (8 K Tm 37 K), minimum required cell width is to just suppress the convection is 1/4 (1 ) d 1 c( β )(1+ β ) β ( T ) 1/ 1/ /4 (1 ) m c( β )(1+ β ) β ( T ) 1/ 1/4 1 1 cm (37) 1 where is in m, β 1 is in K -1, T is in K. The Tm unction c ( β ) is given by o o c( β ) 1.3 F( β ) or 3 β sin β, or o β 3 o (38) The results are computed or the minimum cell width required to suppress the convection or the angle o inclination o 1, 4 and 7 based on the above correlations or honeycomb cell depth (-16 cm) and T (-1 C), which is the range o interest in solar thermal applications (Figs 8a - 8c). It is ound that cell width (7-19 mm) is suicient to suppress the convection. Conclusions The Rayleigh-Benard convection in a luid (air) layer bound by transparent cellular structure (honeycomb) is examined or application to the engineering design o transparent insulation materials devices. The derivation o characteristic equation o thermo hydrodynamic instability o horizontal layer Fig. 8a Honeycomb cell width required or convection suppression (tilt angle β1 )

9 61 J SCI IND RES VO. 64 AUGUST 5 Fig. 8b Honeycomb cell width required or convection suppression (tilt angle β4 ) Fig. 8c Honeycomb cell width required or convection suppression (tilt angle β7 )

10 KUMAR & KAUSHIKA: CONVECTION IN AIR AYERS BOUND BY CEUAR HONEYCOMB ARRAYS 611 heated rom below, based on normal mode technique, is presented. The explicit numerical computations o Rayleigh numbers o the characteristic equation at higher wave numbers corresponding to the onset o instability are carried out. The results highlight a new acet that the resultant Rayleigh numbers can be used to estimate the critical Rayleigh number or luid layer with honeycomb structure. Furthermore, convection can be suppressed by a suitable choice o the aspect ratio o honeycomb cell. The honeycomb structure o aspect ratio 1-15 seems suitable or suppression o convection in air layer o depth 5-15 cm or the temperature dierence ( T ) in the range o to 1. Subsequently, critical and post critical Rayleigh regime convective stability o inclined air layer bounded by cellular structures were investigated. The consideration o base low is essential in inclined layers. The base low can be taken into account through the consideration o Nusselt number in addition to the critical Rayleigh numbers. The governing equations o square celled air temperature, vorticity, stream unctions and velocity are presented. The eect was studied o tilt angle and Rayleigh number on Nusselt number(1-1.5), which was o practical interest in solar energy application. The range o Nusselt number corresponds to the almost nonconvective state o luid. This range may be considered as the trade o region or the engineering design o convection suppression devices. It may, thereore, be used or the engineering design o non convective honeycomb cell. Reerences 1 Arulanantham M, Singh T P & Kaushika N D, Convective heat transer across transparent honeycomb insulation materials, Energy Convers Mgmt, 35 (1994) Kaiser A S, Zamora B & Viedma A, Correlation or Nusselt number in natural convection in vertical convergent channels at uniorm wall temperature by a numerical investigation, Int J Heat Fluid Flow, 5 (4) Kaushika N D, Design and development o honeycomb and slatted convection suppression devices or solar energy applications, Final Technical Report, CSIR Research project No. 3 (156)/85/ EMR-II (1989). 4 Kaushika N D, Priya Padma & Singh T P, Convection theory o honeycomb and slat devices or solar energy applications, Energy Resource Technol, (199) Hideo I, Chauanshan D A I & Akihiko H, The convective instability in a microemulsion phase- change material slurry layer, JSME Int J, 47B (4) Kaushika N D & Sumathy K, Solar transparent insulation materials: A review, Renewable Sustainable Energy Rev, 7 (3) Koutsoheras W & Charters W W S, Natural convection phenomena in inclined cells with inite side wall, Solar Energy, 19 (1977) Chandrasekhar S, Hydrodynamic and Hydromagnetic Stability (Oxord Clarandan Press, ondon) 1961, Hollands K G T, Konicek, Unny T E & Raithby G D, Free convection heat transer across inclined air layers. Trans ASME J Heat Mass Transer, 98 (1976) Cane R D, Hollands K G T, Raithby G D & Unny T E, Free convection heat transer across inclined honeycomb panels, Trans ASME J Heat Mass Transer, 99 (1977) Holland K G T, Honeycomb devices in lat- plate solar collectors, Solar Energy, 9 (1965) Edwards D K & Catton I, Prediction o heat transer by natural convection in closed cylinders heated rom below, Int J Heat Mass Transer, 1 (1969) Sharma M S & Kaushika N D, Design and perormance characteristics o honeycomb solar pond, Energy Convers Mgmt, 7 (1986) Heitz W & Westwater J W, Critical Rayleigh numbers o natural convection o water conined in square cells with /D rom.5 to 8, Trans ASME J Heat Mass Transer, 93 (1971) Holland K G T, Advanced non-concentrating solar collectors in solar energy conversion: An introductory course, edited by A E Dixon and J D aslie (Pergamon Press, Oxord) 1979, Smart D R, Hollands K G T & Raithby G D, Free convection heat transer across rectangular celled diathermous honeycombs, Trans ASME J Heat Mass Transer, 1 (198) Nomenclature A Aspect Ratio o cell A w C p d g K w K p P r R R c RK Aspect Ratio o wall Speciic heat d δ Honeycomb cell width Acceleration due to gravity Thermal conductivity o wall Thermal conductivity o luid Depth o honeycomb cell P Dimensionless pressure ρκ Prandtl Number υ κ 4 gαγ Rayleigh Number κυ 3 gα θ κυ Critical Rayleigh Number wall conductivity K luid conductivity K w

11 61 J SCI IND RES VO. 64 AUGUST 5 T θ Dimensionless temperature θc θ θ T u v h c Perturbation in temperature U Dimensionless velocity κ V Dimensionless velocity κ X x Dimensionless distance Distance romletsideo wall Depth o Honeycombcell y Y Dimensionless distance Distance rom bottomo Honeycomb Depth o Honeycombcell α γ δ ρ ρ υ κ Coeicient o volume expansion β Tilt angle Adverse temperature gradient T T T y y Wall width Density o the wall at time t The initial density Kinematic viscosity µ ρ Thermal diusivity o the luid κ τ Dimensionless time t Time t