Desalination 209 (2007) 144–150

Numerical study of the natural convection process in the parabolic–cylindrical solar collector Talal Kassem F.M.E.E. P.O. Box 86, Damascus, Syria email: [email protected]

Abstract This paper deals with numerical prediction of natural convection heat transfer in the annular space between the circular receiver tube and the glass envelope in the so parabolic–cylindrical solar collector, for the case of an isothermal outer cylinder with a sinusoidal local heat flux distribution on the inner cylinder. The dimensional Darcy–Boussinesq’s equations are formulated in bicylindrical coordinates and solved numerically using finitedifference based on successive-overrelaxation iteration. The parametric effect on the heat transfer characteristics is discussed with respect to eccentricity and azimuthal angular location of the inner cylinder for eccentric annulus. The two-dimensional structure of the fluid flow and temperature distributions as well as local and average Nusselt numbers are obtained. It is demonstrated that the heat transfer in the annulus can indeed be optimized by a proper choice of the eccentricity. Keywords: Heat transfer; Natural convection; Eccentric annulus

1. Introduction Natural convection heat transfer in horizontal enclosures of concentric and eccentric cylindrical annular form has received increased attention due to the fundamental importance in practical applications and the interesting feature of the specific heat transport phenomena. Kuehn et al. [1] compiled a comprehensive review of the available experimental results for natural convection heat transfer between horizontal concentric and eccentric cylinders and proposed correlating equations using a

conduction boundary-layer model. Prusa et al. [2] constructed a finite difference numerical simulation using a special coordinate transformation. Their computations included relative eccentricities between –0.652 and 0.623 and Rayleigh number Ra =1.2 × 104. Cho et al. [3], and Shin [4–7] solved the problem using bicylindrical coordinates. The resulting equations are solved using the Gauss–Seidel and successive over-relaxation procedures. Ratzel et al. [8] examined the effect of nonuniform temperature distribution and eccentricity on natural convection heat transfer.

The Ninth Arab International Conference on Solar Energy (AICSE-9), Kingdom of Bahrain 0011-9164/06/$– See front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.desal.2007.04.023

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In this paper, to extend the existing knowledge on the natural convection heat transfer in a horizontal cylindrical annuli, a numerical analysis is made using finite difference method based on successive over-relaxation iteration. The angle of inclination is varied to three distinct values of a = 0°, 45° and 90°. The temperature of the outer cylinder is held constant at 293 K and the inner cylinder is exposed to a nonuniform heat flux q. Prandtl is fixed with that of air, 0.706, throughout the study while the eccentricity is changed from 0.01 ≤ e ≤ 0.4.

h1 > 0 and 0 < h2 < h1. The corresponding radii ratio r2/r1 = sinh h2/sinh h1, a = r1 sinh h1 = r2 sinh h2, and the eccentricity e = e/(r2–r1). we assume that the fluid motion is two dimensional. the 2-D governing equations of the dimensionless stream function, vorticity, and temperature can be written respectively in bicylindrical coordinates as follows:

2. Numerical study

∂T ⎫ ⎧ ⎡⎣ F (η ,θ ) sin α + G (η ,θ ) cos α ⎤⎦ ⎪ 1 ⎪ ∂η ⎪⎪ ⎨ ⎬ H ⎪ ∂T ⎪ + ⎡ F (η ,θ ) cos α − G (η ,θ ) sin α ⎤⎦ ⎪⎩ ⎣ ∂θ ⎪⎭ 2 2 1 ⎛∂ ω ∂ ω ⎞ + 2⎜ 2 + 2 ⎟ H ⎝∂η ∂θ⎠

2.1. Derivation of governing equations Consider a Newtonian fluid confined between two eccentric cylinders or radii, r1 and r2 > r1, the line connecting the centres of the two cylinders forms angle α with the gravity vector g. Our appropriate model is that of steady, laminar, thermal convection of a fluid due to nonuniformities of density in a gravitational field. In accordance with the usual Bossinesq approximation, density changes are neglected except insofar as they affect the body force term in the equations of motion. The other properties of fluid are independent of temperature. The problem is formulated using bicylindrical coordinates [9] that are natural for this problem since the physical boundaries are identified with constant value coordinates. The transformation from the cartesian coordinates (x, y) into the bicylindrical coordinates (h,q) is achieved through the formula:

η − iθ x + iy = a coth 2 where constant h lines are the circles: a2 ( x − a coth η ) 2 + y 2 = sinh 2 η

(1)

ω =−

1 ⎛ ∂ 2ψ ∂ 2ψ ⎞ + 2 ⎟ H 2 ⎜⎝ ∂ 2η ∂θ⎠

(3)

vη ∂ω vθ ∂ω + = H ∂η H ∂θ

vη ∂ω vθ ∂ω 1 ⎛ ∂ 2T ∂ 2T ⎞ + = + H ∂η H ∂θ PrH 2 ⎜⎝ ∂ 2η ∂ 2θ ⎟⎠ where h=

(4)

(5)

1 ∂ψ 1 ∂ψ a ; vη = ; vθ = cosh η − cos θ H ∂θ H ∂η

(6)

and

1 − cos θ cosh η , cosh η − cos θ sin θ sinh η G (η ,θ ) = cosh η − cos θ F (η ,θ ) =

(7)

The boundary conditions imposed in the present problem are On the inner cylinder:

vη = vθ = ψ = 0 (2)

Consequently, the two eccentric cylinders can be described by two nonzero h coordinates

The heat flux on this wall is q = − q0 (1 + b cos φ )

g β Dh4

λν T 2

=

1 ⎛ ∂T ⎞ H T ⎜⎝ ∂η ⎟⎠ η =η

(8) 1

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and the temperature of the inner cylinder allowed to vary according to ⎫ ⎧18Tw+1 − 9Tw+2 + 2Tw+3 ⎪ 1⎪ 4 ⎡ Tw = ⎨ g β Dh ⎤ ⎬ 11 ⎪+6 Δη ⎢ H q0 (1 + b cos φ ) ⎥ λ ν 2 ⎥⎦ ⎪⎭ ⎢⎣ ⎩

(9)

On the outer cylinder:

vη = vθ = ψ = 0 and Tw = 0 The periodic condition at branch cut (q = –p or q = + p) is

T−π = T+ π ω −π = ω + π and ψ −π = ψ + π The local Nusselt number, Nul, is defined as

qw Dh (10) λ ΔT where qw is the heat flux at either of the cylinders. In accord with Eq. (10), the local Nusselt number at the outer cylinder is N ul =

N ul =

1 ⎛ ∂T ⎞ HT ⎜⎝ ∂η ⎟⎠ η =η

(11) 2

∂T/∂h can be obtained from a 2nd-order Taylor series approximation:

−3Tw + 4 Tw −1 − Tw − 2 ⎛ ∂T ⎞ ⎜⎝ ∂η ⎟⎠ = 2 Δη w

(12)

where Δh is the size of the interval in h direction. – The averaged Nusselt number Nul is obtained by integrating (11) over the surface of the outer cylinder:

N ul =

1 2π

Fig. 1. The coordinates system.

+π

∫ N ul dθ

(13)

−π

The integral in this equation is approximated with the aid of Simpson’s method. 3. Computational procedure The partial differential equations which describe the natural convection process are

represented discretely through use of the familiar control-volume method of finite-difference and Patankar’s power law technique [10] is used to simulate the nonlinar terms in the energy and motion equations. The grid system in the physical domine is described in Fig. 1. Through the use of bicylinderical coordinates we effectively transform the computational domain into rectangular h 2 < h < h1; –p < q < +p. Uniform meshes in h and q direction have been used in the computation where the grid spacing in the h direction is Δh = (hi – he/2h – 1) and in the q direction Δq = 4Δh. The resultant discertized equations are put in the form convenient for iteration in the successive over-relaxation method (SOR) [11]. The relaxation factors used are 0.70, 0.08, and 0.80 for T, w, and y, respectively, for Rayleigh numbers nearly 106. The number of grid points are (22 × 42), (22 × 45), and (22 × 50) for e = 0.01, e = 0.2, and e = 0.4, respectively. The

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Fig. 2. Isotherma for annuli in the same angle of inclination (e= 0) but of diferent eccentricity: a = 0.01(a), 0.2(b), 0.4(c), and (Ra = 106, Pr = 0.702, r2/r1 = 2).

iterative procedure was terminated when the following relative-error convergence criterion was satisfied:

Ω n+1 − Ω n (14) ≤ 10 −3 Ω n+1 where Ω indicates one of the values T or y at the nth step of the iterations. Max

4. Results and discussion For the verification of our mathematical model we compared our numerical results obtained for the Nusselt number at the inner cylinder wall (Nul) in the case of natural convection heat transfer between concentric cylinders subject to constant temperature boundray with those of an experimental investigation were presented by Liu et al. [12] at Pr = 0.7, Ra ≈ 106, Ti – Te =

58.33°C and (De – Di)/2Di = 0.75. We find a good agreement between the experimental and our numerical results. 4.1. The flow and temperature fields We examine first the effect of the eccentricity on the flow and temperature fields. In Figs. 2 (a–c) and 3 (a–c) the isotherms and the streamlines are presented for the particular case where a = 0° with Ra ≈ 106, Pr = 0.702 and the diameter ratio is uniform as r2 / r1 = 2, but the eccentricity is varied to three distinct values of ε+ = 0.01, 0.2 and 0.4. In this case the flow and temperature fields are symmetric which respect to the line interconnecting the cylinders centers. The flow consists of two counter rotating convective cells. At the bottom portion of the annulus (between q = 75°, 270°) the flow is inert and

Fig. 3. Streamlines for annuli in the same angle of inclination (e = 0) but of different eccentricities: a = 0.01(a), 0.2(b), 0.4(c) and (Ra = 106, Pr = 0.702, r2/r1 = 2).

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Fig. 4. Isotherma for annuli in the same eccentricity (ε = 0) but of diferent angles of inclination: a = 0°(a), 45°(b), 90°(c), and (Ra ≈ 106, Pr = 0.702, r2/r1 = 2).

stably stratified. The cross indicate the location of the maximum value of the stream function, which be the centre of rotation. In Fig. 3(a) the maximum is located near the 145° position. As the eccentricity increases, the centre of rotation moves upwards and the maximum is about 20° from the top as shown in Fig. 3(c) with the lower portion of the annulus practically stagnant. It is observed that Fig. 3(b) shows a multicellular flow pattern in which the fluid in the secondary cells at top of the annulus has the opposite sense of rotation from that of the primary cell. Next, we examine the effect of the angle of inclination (a) on the flow and temperature fields. We depict in Figs. 4(a–c) and 5(a–c) the isotherms and streamlines for Ra ≈ 106, Pr = 0.702, r2/r1 = 2, and for a given eccentricity ε+ =0.2. The effect of the various angles of inclination a = 0°, 45°, and 90° can be seen clearly by the displacement of the maximum value of the

temperature from the position θ = 180°, in the case of symmetry a = 0°, to anther position θ = 115°, in the case of (a = 90°). The variation of the local Nusselt numbers at the inner cylinder wall (Nul)i and the outer one (Nul)e with the angle q for (a = 0°), Ra ≈ 106, Pr = 0.702, r2/r1 = 2, and for different eccentricities ε+ = 0.01, 0.02 and 0.4 are shown in Figs.(6) and (7). The local Nusselt numbers (Nul)i, (Nul)e are observed to increase to a maximum value at q = 180°. The influence of the eccentricity upon the Nusselt number is directly indicated by the decrease of the maximum values of the local Nusselt numbers at the tow cylinders as ε+ is increased to 0.4. 5. Conclusions Free connective heat transfer in a Newtonian fluid bounded by tow horizontal eccentric cylinders has been analytically examined. The

Fig. 5. Streamlines for annuli in the same eccentricity (ε = 0.2) but of different angles of inclination: a = 0°(a), 45°(b), 90°(c) and (Ra ≈ 106, Pr = 0.702, r2/r1 = 2).

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Fig. 6. Variation of the local Nusselt rulers at the inner cylinder are (Nul)i and outer one (Nul)e with the angle q for (a=0°) but of different eccentricities (ε° = 0.001. 0.2, and 0.4) and for (Ra ≈ 106, Pr = 0.702, r2/r1 = 2).

resulting finite difference equations were successfully solved by the successive-over relaxation iteration. The effects of the eccentricity and the angle of inclination on the flow and temperature fields have been quantitatively studied. The conclusions are summarized as follows: In the limited case of a = 0°, Ra ≈ 106, Pr = 0.702, and r2/r1 = 2 themaximum local Nusselt numbers (Nul)i at the inner cylinder and (Nul)e at the outer cylinder have a maxima at q = 180°, and they show a fairly large dependence on the eccentricity. The heat flow from the inner cylinder farward the outer cylinder can be reduced by using. Where the heat transport is minimized with the eccentricity of 0.04. Nomenclature a b Cp Dh e g F, G h

scale factor in bicylindrical coordinates (m) constant, Eq. (8) specific heat of the fluid (J/kg k) hydraulic diameter = 2[r2 – r1] (m) distance between centres of circular cylinders in eccentric annuli (m) acceleration of gravity (m/s2) functions, defined in Eq. (7) metric coefficient, defined in Eq. (6)

dimensionless h, [h/Dh] Prandtl number = nrcp/l density of heat flux (w/m2) Rayleigh number inner and outer radii of annulus, respectively (m) T dimensionless temperature of the fluid = g b Dh3 (Tf – T2)/n2 Tf temperatureof the fluid (K) T2 temperature at outer cylinder (K) ΔT temperature difference between the fluid and the outer cylinder (K) vh, vq dimensionless velocity components in h,q direction x, y cartesian coordinates Greek symbols a inclination angle b thermal expansion coefficient (1/k) e eccentricity = e/ r2–r1 h, q bicylindrical coordinates l thermal conductivity of the fluid (w/mk) n kinematic viscosity (m2/s) r density of the fluid (kg/m3) y dimensionless stream function w dimensionless vorticity function Subscripts w wall H Pr qv Ra r1, r2

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