Natural convection in a vertical strip immersed in a porous medium

Transcription

1 European Journal o Mechanics B/Fluids 22 (2003) Natural convection in a vertical strip immersed in a porous medium L. Martínez-Suástegui a,c.treviño b,,f.méndez a a Facultad de Ingeniería, UNAM, Mexico D.F., Mexico b Facultad de Ciencias, UNAM, Mexico D.F., Mexico Received 17 February 2003; received in revised orm 18 July 2003; accepted 18 August 2003 Abstract In this work, the conjugated heat transer characteristics o a thin vertical strip o inite length, placed in a porous medium has been studied using numerical and asymptotic techniques. The nondimensional temperature distribution in the strip and the reduced Nusselt number at the top o the strip are obtained as a unction o the thermal penetration parameter s, which measures the thermal region where the temperature o the strip decays to the ambient temperature o the surrounding luid. The numerical values o this nondimensional parameter permits to classiy the dierent physical regimes, showing dierent solutions: a thermally long behaviour, an intermediate transition and a short strip limit Éditions scientiiques et médicales Elsevier SAS. All rights reserved. Keywords: Porous medium; Natural convection; Heat transer 1. Introduction In this work we deal with the problem o the coupled conjugate conduction-natural convective heat transer in a vertical solid strip totally embedded in a porous medium. The book by Pop and Ingham [1] presents abundant theoretical evidence, showing the importance o the thermal interaction with dierent heat transer mechanisms in practical systems. In the same direction, the book o Sundén and Heggs [2] shows that this simple geometrical coniguration is ound in a broad range o scientiic and engineering problems associated with dierent industrial applications. Additional examples on this and other related topics can be obtained in the books by Ingham and Pop [3], Nield and Bejan [4] and Vaai [5]. Lock and Gunn [6] did the irst theoretical study dealing with the conjugate conduction-ree convection problem o a long vertical thin strip or in embedded in a porous medium. They obtained sel-similar solutions or the vertical in geometry. Cheng and Minkowycz [7], Kuehn et al. [8] and Sparrow and Acharya [9] developed equivalent analyses or the same type o problems. Based on these studies, Pop et al. [10] obtained a set o similarity solutions or a long vertical plate projecting downward rom a heated horizontal plane base at uniorm temperature, or the case o the thermal conductivity-in thickness product varying as a power o distance rom a certain speciied origin. Later, Pop et al. [11], improved the above analysis by developing a inite-dierence numerical scheme or the case o uniorm thickness and thermal conductivity o the in. Pop and Nakayama [12] reviewed the problem o conjugate convective heat transer rom a vertical in embedded in a luid-saturated porous media. The above mentioned authors mainly consider an ininitely long in in order to ormulate the thermally coupled governing equations. For instance, Pop and Nakayama [12] introduced an unknown characteristic length x b (Eq. (14) in [12]) in order to nondimensionalize the governing equations and selected an appropriate origin o coordinates. However, the physical interpretation to choose this length scale was not enough clariied. One o the objectives o the present work is to show that this length scale, called in this paper L, can be easily estimated using an order o magnitude analysis o the governing equations. Furthermore, we avoid the unnecessary condition to assume an ininitely long in by considering a vertical strip or in o inite * Corresponding author. address: ctrev@servidor.unam.mx (C. Treviño) /$ see ront matter 2003 Éditions scientiiques et médicales Elsevier SAS. All rights reserved. doi: /j.euromechlu

2 546 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) length L. Thereore, we introduce a nondimensional parameter L /L, which identiies all possible physical regimes or this conjugate heat transer process. Using order o magnitude estimates o the coupled governing equations, we identiy the characteristic thermal penetration length L, where the temperature o an ininitely long in would decay to the ambient temperature o the surrounding porous medium. As was previously mentioned, the ratio o L to the actual length L o the in is a undamental parameter that serves to classiy the possible thermal regimes. Additional nondimensional parameters in the problem are the Rayleigh number (to be deined below) and the aspect ratio o the strip. Perturbation and numerical methods are employed, together with the Darcy and boundary layer approximations or the ree convection low, to analyze the transition rom a thermally short in (s = L /L 1) to a thermally long in (s 1) and to ascertain the inluence o the thermal properties o the strip and porous medium on the overall heat transer rate. Finally, the analytical solutions obtained using perturbation techniques are compared with numerical results. 2. Basic equations The physical model and a suitable coordinate system are given in Fig. 1. A vertical heated conducting strip o length L and thickness 2h L, is totally embedded in a luid-saturated porous medium with a temperature T. The upper surace o the strip is assumed to have a uniorm temperature T 0 >T, whilst the lower surace is assumed to be adiabatic. Heat is transerred rom the top o the strip to the luid-saturated porous medium through the strip. Consider irst an ininitely long strip. Owing to the heat loss to the surrounding luid-saturated porous medium, the temperature o the strip decreases downward rom its top, decaying towards the ambient temperature o the luid in a thermal penetration region, with a characteristic length L which can be estimated rom the balance o heat transer to the luid-saturated porous medium and heat conduction along the strip. Assuming that the Rayleigh number Ra = gkβ(t 0 T )L /α m ν is very large compared with unity (where g, K, β, α m and ν are the gravity acceleration, the speciic permeability o the porous medium, the thermal expansion coeicient, the thermal diusivity o the luid-saturated porous medium and the kinematic viscosity, respectively), the low around the strip is conined to a natural convection boundary layer o characteristic thickness δ = L / Ra 1/2. The total heat lost by conduction to the luid-saturated porous medium per unit time and unit width o the strip is o the order o L k T/δ, while the heat conducted along the strip is o order hk s T /L,wherek and k s are the thermal conductivity o the porous medium and the strip material, respectively (k = φk + (1 φ)k m [4], where φ is the porosity, k and k m are the thermal conductivities o the luid and the porous matrix, respectively).the balance o these two luxes yields L = h (k s/k) 2/3 Ra 1/3, h (1) Fig. 1. Sketch o the studied problem.

3 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) where Ra h = gkβ(t 0 T )h/α m ν is the Rayleigh number based on the hal-thickness o the strip. The boundary layer approximation is then valid or values o the ratio o thermal conductivities k s /k L /h 1. For values o L L,the actual length o the strip is irrelevant and L is the appropriate characteristic length in the heat transer process. This regime corresponds to the thermally long strip. A parameter relating both characteristic lengths is deined by s = L /L. The thermally long strip corresponds to values o s 1. Using the estimate o the thermal penetration length, the total heat lux transerred at the top o the strip per unit time and unit width o the strip is o order, Q hk s (T 0 T )/L, or in nondimensional orm, Nu Long = Q/k(T 0 T ) Ra 1/2. The order o magnitude o the heat lux going rom the solid to the luid in the porous medium is k s T sh /h k T Ra 1/2 /L,where T sh denotes the variation o the solid temperature in the transverse direction. From this relationship we obtain T sh / T (k/k s )(h/l ) Ra 1/2 (h/l ) 2. Thereore, or values o h L, the temperature variation in the transverse direction o the strip is very small compared with the overall temperature dierence (thermally thin solid). This approximation is valid or values o k s /k Ra 1/2 h. In the opposite case o short strips with h L L,thatiss 1, the longitudinal temperature variation rom the top to the bottom o the strip, T sl say, can be estimated rom the energy balance Q hk s T sl /L Lk T /δ, whereδ is the characteristic thermal boundary layer thickness, δ L/ Ra 1/2 L. This balance gives T sl T /s 3/2.Thismeansthat the temperature o the strip is almost uniorm and equal to its top temperature or values o s large compared with unity. In this regime, k s T sh /h k T Ra 1/2 L /L, andthen T sh/ T (h/l ) 3/2 (h/l) 1/2 (h/l ) 2 s 1/2. The thermally thin approximation is valid or values o s (L /h) 4, as long as Ra L 1. In the limit o short strips, the overall Nusselt number is then Nu short Ra 1/2 /s 1/2, or values o s 1. The advantages o the long strip regime is clearly seen when the overall Nusselt numbers or both regimes are compared. Using the Darcy Boussinesq and boundary-layer approximations, the natural convection low in the luid-saturated porous medium is described by the ollowing governing equations [4] u x + v y = 0, u = gkβ ν (T T ), u T x + v T y = α 2 T y 2, (2) (3) (4) or the mass conservation, momentum and energy, respectively. The heat equation or the strip is given by 2 T s x T s y 2 = 0. (5) In the above equations, u, v are the velocity components along the x,y axes, T and T s are the temperatures o the luid-saturated porous medium and the solid plate, respectively. Eqs. (2) (5) are to be solved with the ollowing boundary conditions: v = 0, T = T s, k T y = k T s s on y = 0, (6) y T s y = 0, T s (L, y) = 1, (7) y= h T s x = 0, (8) x=0 u 0, T T as y and x = 0 with y 0. (9) I the aspect ratio o the strip h/l is assumed to be very small compared with unity, Eq. (4) can be integrated in the transverse direction, resulting h d2 T s dx 2 + k k s T y = 0. 0 In this case, the temperature at the strip is assumed to depend only on the longitudinal coordinate alone. In the ollowing two sections we present the asymptotic solutions or long and short strips, respectively. (10)

4 548 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) Long strips For thermally long strips, L L (s 1), the appropriate characteristic length is L. The ollowing nondimensional variables are introduced or this regime σ = L 0 x L, Z = y L Ra 1/2, U = u, u c V = v u c Ra 1/2, θ = T T T 0 T, θ s = T s T T 0 T, where L 0 is a length to be computed later and u c is the characteristic low velocity, deined by u c = gkβ(t 0 T )/ν. The nondimensional orm o the governing Eqs. (2) (5), are now or this regime U σ + V Z = 0, (12) U = θ, (13) U θ σ + V θ Z = 2 θ Z 2, d 2 θ s dσ 2 + θ Z = 0, (15) 0 with the corresponding nondimensional boundary conditions V = 0, θ = θ s at Z = 0, U,θ 0 orz,σ, θ s = 1 at σ = σ 0, θ s 0 orσ. (16) Here, σ 0 = (L 0 L)/L. This problem, as shown by [7,10], has a sel-similar solution o the orm θ = θ s (σ ) dh dξ, 3/2 [ V = σ 0 σ 2 h + 2ξ dh ], ξ = σ 3/2 Z dξ 0 σ 2, (17) where θ s is ound to be θ s = (σ 0 /σ ) 3. The unction h(ξ) satisies the ordinary dierential equation d 3 ( ) h dh 2 dξ h d2 h dξ dξ 2 = 0 (18) with the boundary conditions h(0) = dh dξ 1 = dh ξ=0 dξ = 0. ξ (19) (11) Fig. 2. Numerical solution o unction h(ξ), obtained by solving Eqs. (18) and (19) in the long strip regime.

5 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) The value o σ 0 can be obtained by introducing Eqs. (17) into the energy equation or the strip (15), resulting σ 0 = (12/d 2 h/dξ 2 η=0 ) 2/ The length L 0 is then L 0 = L L. Fig. 2 shows the unction h(ξ), obtained ater solving numerically Eqs. (18), (19), using a central dierence scheme with the aid o a quasi-linearization technique. The nondimensional temperature o the strip then is given by ( θ s = σ 0 s σ 0 s + 1 x/l ) 3 or s 0. (20) Finally, the nondimensional heat lux or Nusselt number at the bottom o the strip is given by Nu Ra 1/2 = dθ s dσ = , or s 0. (21) σ =σ0 σ 0 4. Short strips For large values o s compared with unity (L L ), the appropriate characteristic length is L. The heat transer problem in this regime can be studied using the nondimensional variables and χ = x L, θ = T (x,y) T T 0 T θ s = T s(x) T, η= Ra 1/2 L y T 0 T L 1/2, (22) x1/2 = u u c = η, = vx1/2 [ V Ra L L 1/2 χ u c χ + 2 η ], (23) 2 η where Ra L is the Rayleigh number based on the length o the strip. In nondimensional variables, Eq. (10) becomes s 3/2 d2 θ s dχ 2 = 1 2 χ 1/2 η 2, (24) η=0 with the boundary conditions at the top and bottom o the strip given by dθ s θ s = 1 atχ = 1 and = 0 atχ = 0. (25) dχ The nondimensional governing equations or the luid immersed in the porous medium reduces to 3 η [ 2 η 2 = χ 2 η χ η 2 ] χ η 2. (26) The associated nondimensional boundary conditions are then = η θ s at η = 0 and 0 η or η, (27) plus conditions o regularity at the origin o the boundary layer χ = 0. An asymptotic solution o problem (24) (27) or large values o s can be sought as a regular expansion in powers o s 3/2, which is the small parameter dictated by Eq. (24). The solution can then be written as { { θs θs0 { θsj (χ) } = 0 (η) } + 1 s j=1 3j/2 j (χ, η) }. (28) The leading term or the nondimensional temperature in the strip is clearly θ s0 = 1. Carrying expansions given in Eqs. (28) into the nondimensional Eqs. (24) and (26) with the associated boundary conditions, and keeping terms up to order s 3,the ollowing set o equations are obtained. For the solid: d 2 θ sj dχ 2 = 1 2 j 1 χ 1/2 η 2 η=0 or j 1, (29) with the boundary conditions dθ sj dχ = θ sj (1) = 0, χ=0 or j 1. (30)

6 550 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) For the luid immersed in the porous medium: d 3 0 dη d dη 2 = 0, (31) 3 1 η ( 2 ) [ η 2 + d2 0 d0 dη 2 2 ] 1 χ 1 dη χ η d2 0 1 dη 2 = 0, (32) χ etc., with the boundary conditions valid or j 0 and j = j η θ sj = 0 at η = 0 (33) j η = 0 or η. Eq. (31), with the boundary conditions (33) and (34) or j = 0, corresponds to the classical problem o convection in a porous medium adjacent to a heated isothermal vertical plate [7], giving d 2 0 /dη 2 0 = G 0 = The irst order correction or the nondimensional temperature o the strip given by Eq. (29), can be obtained ater integrating it twice to yield θ s1 = n=0,3/2 a n χ n = 4G 0 ( 1 χ 3/2 ). (35) 3 Here a 0 = a 3/2 = 4G 0 /3. The solution to the linear Eq. (32) with the corresponding boundary conditions must be o the orm 1 (χ, η) = n=0,3/2 a nχ n g n (η),whereg n (η) satisies the ordinary dierential equations d 3 g n dη d 2 g n 0 dη 2 n d ( ) 0 dg n 1 d 2 dη dη + 2 n 0 dη 2 g n = 0, or n = 0, 3/2. (36) The associated boundary conditions or g n are g n (0) = dg n dη 1 = dg n η=0 dη = 0. (37) η The solution o Eqs. (36) and (37), gives the values G 1 (n) = d 2 g n /dη 2 0. It can be easily shown, using the invariance properties o the boundary layer equations, that G 1 (0) = 3/2G 0 = The obtained numerical value o G 1 (3/2) = Following the same procedure, the second order correction o the nondimensional temperature in the strip is given by θ s2 = a n G 1 (n) ( χ n+1/2 ) 1. (38) (n + 1/2)(n + 3/2) n=0,3/2 Summarizing, the nondimensional temperature o the plate or large values o the parameter s, up to terms o order s 3, can be written as θ s = 1 4 G 0 ( 1 χ 3/2 ) 3 s 3/ The reduced Nusselt number, Nu / Ra 1/2 = s dθ s /dχ 1 is then given by [ 1 6G 0 G 1 (3/2) 3s 3/2 Nu Ra = 2G 0 1/2 s 1/2 G 0 s 3 [ 12G0 ( 1 χ 3/2 ) G 1 (3/2) ( 1 χ 3)] + O ( s 9/2). (39) (34) ] + O ( s 7/2), or s. (40) The thermal properties o the material, appearing in the deinition o L play no role in the heat transer at the leading order, because Ra 1/2 /s 1/2 = Ra 1/2 L and is independent o L. 5. Numerical results Eqs. (24) to (27) have been numerically solved or dierent values o parameter s in the thermally short strip regime. The boundary value problem, represented by Eq. (24) has been transormed to an initial value problem by introducing the new nondimensional longitudinal coordinate ζ = χ/s. Thereore, Eqs. (24) and (26), take the orm

7 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) Fig. 3. Numerical results or the nondimensional temperature in the strip as a unction o the normalized nondimensional longitudinal coordinate χ, or dierent values o s. For a value o s = 0.05, the asymptotic solution or long strips, given by Eq. (20), is also plotted. d 2 θ s dζ 2 = 1 2 ζ 1/2 η 2, η=0 (41) 3 η [ 2 η 2 = ζ 2 η ζ η 2 ] ζ η 2, (42) with the boundary conditions at the top and bottom o the strip given by θ s = 1 atζ = ζ and dθ s = 0 atζ = 0, (43) dζ where ζ = 1/s. Close to the bottom o the strip, ζ 0, θ s behaves like θ s = a a3/2 G 0 ζ 3/2 + O ( ζ 7/2), (44) where a 1 is a constant to be given a priori. For a given value o a, avalueoζ and s is obtained. For values o s>1, Eq. (39) is used in order to have a preliminary nondimensional temperature proile at the strip. The nondimensional heat lux 2 / η 2 0 as a unction o ζ is obtained by integrating Eq. (42) and the corresponding boundary conditions. This equation has been decomposed in one irst order and one second order equations and solved using central dierence discretization. Once the nondimensional heat lux at each ζ position has been obtained, the new nondimensional temperature proile in the strip is obtained ater solving Eq. (41), with the starting behavior (44), using a ourth-order Runge Kutta technique. The process is repeated until a convergence criterion is ulilled. Fig. 3 shows the numerical results o the nondimensional temperature proiles or the strip as a unction o the nondimensional longitudinal coordinate, or dierent values o s. In this igure, the nondimensional temperature o the strip has been plotted by using the long strip approximation or a value o s = The reduced Nusselt number Nu / Ra 1/2 is plotted in Fig. 4 as a unction o parameter s, covering the ull transition rom the short to long strips. The numerical results are plotted with open circles. The one and two terms asymptotic solutions given by Eq. (40), obtained or large values o s (short strips) are plotted, together with the asymptotic solution or long strips (s 0), given by Eq. (21). This igure shows that the short strip approximation is practically valid or values o s>4, whereas the long strip approximation holds or values o s< Conclusions In this paper, we have extended previous results or the conjugated heat transer in a vertical strip with a prescribed temperature at its top and immersed in a luid-saturated porous medium, using analytical and numerical techniques. For large values o the Rayleigh number, the problem depends on one nondimensional parameter: s = L /L. We show that the existing

8 552 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) Fig. 4. Reduced Nusselt number, Nu / Ra 1/2 as a unction o s. The open circles correspond to the numerical solution and the broken lines correspond to asymptotic solutions or long and short strips. regimes are well classiied by the assumed values o the nondimensional parameter s, ranging rom the limit o a thermally long strip to the limit o a thermally short strip. In addition, the temperature o the strip, which depends on the thermal properties o the strip and the porous medium as well as on the thickness o the strip, decreases to the temperature o the porous medium in a thermal penetration length L. This is deined as the length needed to reach the ambient temperature o the luid-saturated porous medium. Due to the inite thermal conductivity o the strip material, the heat transer by conduction along the strip is a relevant mechanism that modiies the previous estimations o the reduced Nusselt number with prescribed boundary conditions. In particular, we identiy that the limit o s 1, corresponds to the case o an ininitely long strip [12]. In this case, the problem has a sel-similarclosed orm solution or small values o s, giving an overall Nusselt number strongly dependent on the thermal conductivity o the strip material and its thickness. For long strips, the actual length o the strip plays no role on the heat transer process, resulting heat lux at the top o the strip given by Q = k 2/3 k 1/3 s [ ] (T 0 T ) 4/3 gkβh 1/3. (45) α m ν In the opposite case o large values o s (thermally short strip), the reduced Nusselt number depends weakly on the thermal properties o the strip material, as given by the irst term in Eq. (40). The leading order behavior is in act independent on the thermal conductivity o the strip. A two-term asymptotic solution is derived or the limit o s. We have shown that the short strip approximation is valid or values o s>4, whereas the long strip approximation [7,10] holds or values o s<0.4. In this work, the ull transition rom short strip (low heat transer rates) to long strip (large heat transer rates) has been treated using numerical and asymptotic techniques. Acknowledgements CT and LM acknowledges the support o CONACyT and DGAPA-UNAM (IN104301) o México. Reerences [1] I. Pop, D.B. Ingham, Convective Heat Transer: Mathematical and Computational Modelling o Viscous Fluids in Porous Media, Pergamon, Oxord, [2] B. Sundén, P.J. Heggs (Eds.), Recent Advances in Analysis o Heat Transer or Fin Type Suraces, WIT Press, Southampton, Boston, [3] D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Pergamon, Oxord, [4] D.A. Nield, A. Bejan, Convection in Porous Media, 2nd Edition, Springer, New York, [5] K. Vaai (Ed.), Handbook o Porous Media, Marcel Dekker, New York, 2000.

9 L. Martínez-Suástegui et al. / European Journal o Mechanics B/Fluids 22 (2003) [6] G.S. Lock, J.C. Gunn, Laminar ree convection rom a downward-projecting in, J. Heat Transer 90 (1968) [7] P. Cheng, W.J. Minkowycz, Free convection about a vertical lat plate embedded in a porous medium with application to heat transer rom a dike, J. Geophys. Res. 82 (1977) [8] T.H. Kuehn, S.S. Kwon, A.K. Tolpadi, Similarity solution or conjugate natural convection heat transer rom a long vertical in, Int. J. Heat Mass Transer 26 (1983) [9] E.M. Sparrow, S. Acharya, A natural convection in with a solution-determined nonmonotonically varying heat transer coeicient, J. Heat Transer 103 (1981) [10] I. Pop, J.K. Sunada, P. Cheng, W.J. Minkowycz, Conjugate ree convection rom long vertical plate ins embbeded in a porous medium at high Rayleigh numbers, Int. J. Heat Mass Transer 28 (1985) [11] I. Pop, D.B. Ingham, P.J. Heggs, D. Gardner, Conjugate heat transer rom a downward projecting in immersed in a porous medium, in: Proc. 8th Int. Heat Transer Conerence, San Francisco, CA, [12] I. Pop, A. Nakayama, Conjugate ree and mixed convection heat transer rom a vertical in embbeded in a porous medium, in: B. Sundén, P.J. Heggs (Eds.), Recent Advances in Analysis o Heat Transer or Fin Type Suraces, WIT Press, Southampton, Boston, 2000, Chapter 4.