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Heat transfer by free convection from an isothermal vertical round plate in unlimited space
Applied Energy 68 (2001) 187±201
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Heat transfer by free convection from an isothermal vertical round plate in unlimited space Witold M. Lewandowski *, Ewa Radziemska Technical University of Gdansk, Department of Apparatus and Chemical Machinery, ul.G.Narutowicza 11/12, 80-952 Gdansk, Poland Received 30 March 2000; received in revised form 22 August 2000; accepted 26 August 2000
Abstract A theoretical solution of natural convective heat transfer from isothermal round plates mounted vertically in unlimited space, is presented. With simplifying assumptions typical for natural heat transfer process, equations for the velocity pro®le in the boundary layer and the average velocity were obtained. Using this velocity, the energy ¯ow within the boundary layer was balanced and compared with the energy transferred from the surface of the vertical plate according to the Newton's law. The solution of the resulting dierential equation is presented in the form of a correlation between the dimensionless Nusselt and Rayleigh numbers. The theoretical result is compared with the correlation of numerical results obtained using fluent. Experimental measurements of heat transfer from a heated round vertical plate 0.07 m in diameter were performed in both water and air. The theoretical, numerical, and experimental results are all in good agreement. # 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Natural convective heat transfer from vertical plates is an important phenomenon occurring in many practical situations. A review of 25 publications describing work on heated plates of this con®guration was presented by Lewandowski and Kubski in 1983 [1]. An average of those results, in the form of a correlation between the Nusselt and Rayleigh numbers, is given in the ®rst row of Table 1. The range of deviations of the 25 results from the mean was approximately 25%, and constant over the whole range of Rayleigh numbers investigated. * Corresponding author. Tel.: +48-058-347-24-10; fax: +48-458-347-2658. E-mail address: [email protected] (W.M. Lewandowski). 0306-2619/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(00)00053-2
188
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
Nomenclature a A Ak C Cp D g
l Cp
Gr i K K1 Nu
g TD3 2
D l
thermal diusivity (m2/s) control surface, Fig. 2 (m2) control surface on the plate, Fig. 2 (m2) integration constant (-) speci®c heat at constant pressure (J/(kgK)) plate diameter (m) acceleration due to gravity (m/s2) Grashof number (-) enthalpy (J/kg) constant in Equation (13) (-) constant in thermal conductivity relations, Eqs. (30) and (31) (W/K) Nusselt number (-) pressure (N/m2) Prandtl number (-) heat ¯ow (W)
p Pr=/a Q g TD3 Ra a T Tw T1 T u Wx Wy x x y
temperature ( C or K) wall temperature ( C) bulk ¯uid temperature ( C) temperature dierence (K) function of the boundary layer thickness (m4) velocity component in x-direction (m/s) velocity component in y-direction (m/s) dimensionless distance parallel to the plate (-) coordinate parallel to the plate (m) coordinate perpendicular to the plate (m)
Greek symbols lam l llam
heat transfer coecient (W/(m2 K)) average volumetric thermal expansion coecient (1/K) dimensionless boundary layer thickness (-) boundary layer thickness (m) thickness of the laminate (m) nondimentional temperature (-) thermal conductivity of the ¯uid (W/(m K)) thermal conductivity of the laminate (W/(m K)) kinematic viscosity (m2/s) density (kg/m3)
Rayleigh number (-)
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
189
Results from more recent studies of convective heat transfer from isothermal vertical plates in unlimited space, published after 1983 [2±7], are summarised in Table 1 and shown on a plot of the Nusselt number versus the Rayleigh number in Fig. 1. In the review by Jorne [2] there are four correlations which we designate as R (by Ruckenstein), L (by Lorenz), P (by Pohlhausen), and S (by Squire) [2]. The average of the 11 results in Table 1 and Fig. 1, on free convection heat transfer from isothermal rectangular or square plates, gives the following relation between the Nusselt and Rayleigh numbers: Nu 0:634 Ra1=4
1
The discrepancy between the relation based on the pre-1983 [1] (Table 1, ®rst row) and post-1983 [Eq. (1)] data is probably caused by the inclusion in the 1983 review [1] Table 1 Comparison of Nusselt±Rayleigh relation for vertical plates Authors
Nusselt±Rayleigh relations
Lewandowski and Kubski [1] (1983) (review) Jorne [2] (1984) R (review) L P S
0.252
Nu=0.550 Ra
Nu=0.503Ra1/4 (Ruckenstein) Nu=0.548Ra1/4 (Lorenz) Nu=0.517Ra1/4 (Pohlhausen) Nu
0:677 Ra1=4 (Squire) 1 20=21Pr1=4
Notes The mean correlation from the results of 25 authors Theory and local values Theory Theory, Pr=0.72 Theory
Herwig [3] (1985)
Nu=0.9681Gr1/4 Nu=1.058Ra1/4 for air Nu=0.647Ra1/4 for water
Theory for a semi in®nite plate
Lewandowski and Kubski [4], (1984)
Nu=0.612Ra1/4 104
Experiment in water, glycerine for rectangular plate
Miyamoto et al. [5], (1985)
Nu=0.448+0.460Gr1/4 Nu=0.448+0.503Ra1/4 for air Nu=0.448+0.308Ra1/4 for water
Numerical solution of ®nite dierences method
Lewandowski [6], (1991)
Nuch=1.586(Rach F)1/4 Nu=0.631Ra1/4 for vertical F=0.025
Analytical solution for inclined plates
Yu and Lin [7], (1988)
Nu=(4/3)F(Pr)( Ra)1/4 for vertical plate, where:
Solution for a semi-in®nite isothermal inclined plate and for 0.01 4 Pr 4 1
F(Pr)=0.6[(1+Pr)/(1+2.006Pr1/2+ +2.034Pr)]1/4 and =Pr/(1+Pr) Al.-Arabi and Sakr [8], (1988)
Nu=0.54(GrPrcos )1/4 1.15 104< GrPrcos
Experimental studies for inclined plates (0< <80 ), =0-vertical
Present study (29)
Nu=0.667Ra1/4
Theoretical solution for round vertical plate
190
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
results of studies of surfaces with uniform heat ¯uxes, as well as results obtained in the turbulent regime of free convective heat transfer. By contrast, the later results summarised in Table 1, are all from studies of isothermal surfaces in the laminar regime. The shapes of all theoretically and experimentally investigated isothermal plates mentioned in Table 1 were rectangular or square. The constant width and height of
Fig. 1. Graphical presentation of the results of natural convective heat transfer from isothermal vertical plates collected in Table 1, recalculated for Pr=0.7 (top) and Pr=5±6 (bottom). The numbers in brackets are the literature sources (References and Table 1). The shaded Eq. (29) is our theoretical solution.
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
191
these plates leads to a two-dimensional model for the convective ¯uid ¯ow. For other vertical plate shapes, such as triangular or circular, the problem is more complicated, because in these cases the ¯uid ¯ow structure is three-dimensional and the value of the local coecient of convective heat transfer depends not only on the height of the plate, but also on its width. Vertical plates, whose width is not constant, require a dierent, three-dimensional, model of the phenomenon. In such cases, the length of the boundary layer and its thickness change with lateral position on the plate. The theoretical and experimental exploration of heat transfer from a circular plate is the subject of this paper. 2. Physical model of the phenomenon The cross-section of a round plate and its boundary layer are shown in Fig. 2a. Streamlines, assumed to be parallel and vertical, are shown in Fig. 2b. As one moves across the plate, the length of the boundary layer varies from zero at one edge, through the diameter of the plate, D, and back to zero. In addition to the assumption that the streamlines are parallel, other simplifying assumptions used in the model, typical for natural convection in ¯uids, are as follows: . the ¯uid is incompressible and the ¯ow is laminar,
Fig. 2. Physical model of convective heat transfer from a vertical, isothermal, round plate. (a) Side view, (b) front view.
192
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
. the magnitude of the component of velocity parallel to the surface is much greater than the magnitude of the normal component (Wx >>Wy), . the physical properties of the ¯uid in the boundary layer and in the undisturbed region are constant, and . the value of the Prandtl number is in the range, 0.6 < Pr < 6, so the thicknesses of the thermal and hydraulic boundary layers are similar (Th ).
3. Simpli®ed analytical solution The quasi-analytical solution of the Navier±Stokes equations in the presence of a temperature pro®le in the boundary layer is described by:
T ÿ T1 y 2 1ÿ Tw ÿ T1
2
The mean value of the velocity in the boundary layer was adapted for a vertical isothermal surface from reference [6]: 2
x g T : W 40 The change in mass ¯ow rate in a control volume of the boundary layer is: x : dm d AW where A is the cross-sectional area of the boundary layer: p A 2 x D ÿ x The amount of heat necessary to produce this change in the mass ¯ow is: ÿ x dQ idm Cp Tx ÿ T1 d AW
3
4
5
6
The mean value of the temperature in the boundary layer is: ÿ
1 Tx ÿ T1
0
y 2 T T 1 ÿ dy 3
7
and substitution of this relation into Eq. (6) gives:
dQ
x Cp Td AW 3
8
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
According to Newton's equation, the heat transferred is: @ TdAk dQ TdAk ÿl @y y0 where dAk is a control surface, shown in Fig. 2, on the heated round plate: p dAk 2 x D ÿ x dx From Eq. (2), the dimensionless temperature gradient at the wall is: @ 2 ÿ @y y0
193
9
10
11
By equating the heat ¯ows given by Eqs. (8) and (9) and substituting in Eqs. (3), (5), (10) and (11) one obtains: p d 3 x D ÿ x 240lCp p 12 g T dx x D ÿ x Introducing the plate diameter, D, as the characteristic linear dimension and de®ning:
x l ; x ; a ; Cp D D
RaD
g TD3 a
Eq. (12) can be rewritten: p d 3 x 1 ÿ x 240 p K Ra dx D x 1 ÿ x
13
14
Dierentiating Eq. (14) gives: 33 d 4 4 ÿ K 2x 2 1 ÿ x dx
15
d 4 1 ÿ 2x K dx 2x 1 ÿ x
16
or 33
Introducing u
4 K
and
du
43 d; K
17
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W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
the non-homogeneous dierential equation, Eq. (16) becomes: 3 du u 2x ÿ 1 ÿ 1 4 dx 2x 1 ÿ x
18
The homogeneous equation, 3 du u 2x ÿ 1 ÿ 0 4 dx 2x 1 ÿ x
19
has the solution, u
C0
20
2
2x 1 ÿ x3
Using the integration constant (C0 =C0 (x)) gives the relation: 2 4 2 C0 23 x 1 ÿ x3 dx 3
21
The expression inside the integral in Eq. (21), according to the Czebyshev criterion, cannot be expressed in the form of an elementary function, but it can be expanded in a Taylor series: 2 2 1 4 7 4 14 5 2 x ÿ x x 1 ÿ x3 x3 1 ÿ x ÿ x2 ÿ x3 ÿ 3 9 81 243 729
22
and the integral, Eq. (21), becomes: 4 2 3 5 1 8 1 11 2 14 7 7 17 20 x 3 ÿ x 3 ÿ x 3 C1 C0 23 x3 ÿ x3 ÿ x 3 ÿ 3 5 4 33 189 1377 2430
23
Evaluating the integration constant, C1, using the boundary condition, for x 0; 0; u 0; and C1 0
24
Eq. (20) becomes:
u
4 2 3 5 1 8 1 11 2 14 7 7 17 20 3 3 3 3 3 3 3 2 x ÿ x ÿ x ÿ x ÿ x ÿ x 3 5 4 33 189 1377 2430 2
2x 1 ÿ x3
25
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
195
Elimination of the variable, u, using its de®nition, Eq. (18), gives an expression describing the thickness and shape of the boundary layer on the plate: 1 3 5 1 8 1 11 2 14 7 17 7 20 4 3 3 3 3 3 3 1 1 x ÿ x ÿ x ÿ x ÿ x ÿ x 240 4 4 4 5 1 1 4 33 189 1377 2430 K4 u4 1 Ra 3 x 1 ÿ x6 26 The value of the heat transfer coecient may be determined from:
2l
27
The non-dimensional form of the local heat transfer coecient, expressed using the Nusselt number is: D 2 l 14 1 1 1 x 1 ÿ x6 4 RaD 1 20 3 5 1 8 1 11 2 14 7 17 7 20 4 3 3 3 3 3 3 x ÿ x ÿ x ÿ x ÿ x ÿ x 5 4 33 189 1377 2430
NuD
28
and the mean value of the Nusselt number is then: NuD
1 0
1
NuD dx 0:667Ra4D
29
4. The analysis of the theoretical solution Compared with the average of the theoretical and experimental convective heat transfer coecients on vertical square or rectangular plates [1], the present result, Eq. (29), for vertical round plates is larger by 17.4%. This augmentation of heat transfer from round plates seems, at ®rst, rather high, but may be explained by noting that the results collected in the 1983 review paper [1] included, in addition to isothermal cases, those having uniform heat ¯ux and results obtained in the turbulent regime of free convective heat transfer. The present study considers only the conditions of isothermal plates and the laminar regime of heat transfer, so is not directly comparable with the earlier correlation. The present theoretical result, Eq. (29), is also not directly comparable with all of the results collected in Table 1, because of the presence of the Prandtl number as a parameter in some of the correlations in Table 1. The Nusselt±Grashof relations in
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W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
Table 1 were recalculated into the form of Nusselt±Rayleigh relations for Pr=0.7 (air) and Pr=5-6 (water) in order to determine the average, Eq. (1). Comparison of Eq. (29) with Eq. (1) suggests that the Nusselt number for heat transfer from the round plate is 5.2% greater than those for rectangular and square plates. The most directly comparable results for vertical rectangular plates are those presented in reference [6], described by relation Nu=0.631Ra1/4 (Table 1), obtained by the same method and using the same assumptions as in the present work. Comparing this result with Eq. (29), we see that the average heat transfer coecient for round plates is approximately 6% higher than that for rectangular plates. We conclude that the shape of a vertical plate can have a signi®cant in¯uence on its heat transfer coecient. The explanation of the eect is that, for a rectangular plate, the thickness of the boundary layer and values of heat transfer coecient are functions of the plate height only and are constants over the width of the plate, but for a round plate, the boundary layer thickness and heat transfer coecient are functions, not only of distance in the direction of the ¯ow, but also of lateral distance from the centre of the plate. Because the length and thickness of the boundary layer go to zero at the edges of the round plate, the heat transfer coecient approaches in®nity there. In consequence, the average value of the coecient is greater for round than for square or rectangular vertical plates. 5. Experimental apparatus and procedures The experimental studies were made in a rectangular glass tank, 0.5 0.5 1.0 m, in water and air. The vertical round plate had a special layered construction, consisting of two circular copper plates of diameter, D=0.07 m, both cemented to a layer of glass laminate using epoxy-resin, as shown in Fig. 3. The plate was heated by warm water ¯owing through a chamber in contact with one of the copper plates. The thickness of the copper plates (5 mm each) ensured that the surface in contact with the surroundings was isothermal ( 0.1 C). The plate and heater were insulated on the side and bottom using polyurethane foam. The plate temperature was regulated using a thermostat on the water supply to the heater. Eleven copper-constantan thermocouples were soldered to the copper plates for measurement of the temperatures on the surface in contacted with the ¯uid (Tw,i) and the temperature dierences between the two copper plates separated by the glass laminate (Tlam=(Tw1,i-Tw,i)). The average temperature of the plate was the average of the temperatures measured on that plate, e.g. (Tw =(1/n)Tw,i), and the average temperature drop across the plate (copper-glass laminate-copper) was (Tlam,i= (1/n)( (Tw1,i-Tw,i)), where n is the number of thermocouples. The heat transferred from the warm water in the heater, through the plate, into the surrounding ¯uid was determined from the following relations: Q
D2 llam Tlam K1water Tlam 4 lam
for water
30
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
and Q where: Qrad
D2 llam Tlam ÿ Qrad K1air Tlam Tlam ÿ Qrad 4 lam D2 "5:86 4
" # Tw 273:14 4 T1 273:14 4 ÿ 100 100
for air
197
31
32
is the heat transferred by radiation from the vertical plate in air. For the polished copper plate, the emissivity and coecient of surface con®guration are "=0.6 and = 0.831, respectively.
Fig. 3. Schematic diagram of experimental apparatus with enlarged detail of the cross-section of the round vertical plate.
198
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
The relation describing the eective thermal conductivity of the layer of glass laminate and two copper plates as a function of the temperature (Tlam) was measured using a separate apparatus. The results were K1water=5.295 W/K for water and K1air=1.193+0.002Tlam W/K for air, with the average temperature given by Tlam=(Tw1,i+Tw,i)/(2n). The dierences between results obtained when calibrating the plate in water and in air were due to the in¯uence of radiation. The uncertainty in the determination of K1 was estimated to be 5.0% in water and 5.2% in air. Estimates of the uncertainties in the determination of the Rayleigh numbers are 4.5% in water, 1.1% in air and in the determination of the Nusselt numbers, 6.7% in water and 6.6% in air. Additional information about the experimental procedure, method of heating the plate, measurements of temperature, and heat losses may be found in previous publications [9±11]. 6. Experimental results The results of the experiments are shown on a plot of the Nusselt number versus the Rayleigh number in Fig. 4 for the measurements in water and in Fig. 5 for the measurements in air. Both dimensionless groups were evaluated using the diameter of the plate as the characteristic length. Least-squares ®t to the data produced the following relationships: Nu 0:587 Ra1=4
for water
33
Nu 0:655 Ra1=4
for air
34
in quite good agreement with the theoretical prediction, Eq. (29).
Fig. 4. Experimental data obtained for water (points) compared with the theoretical solution, Eq. (29) (solid line).
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
199
Fig. 5. Experimental data obtained for air (points) compared with the theoretical solution, Eq. (29) (solid line).
7. Numerical calculations The numerical results were obtained using the commercial program fluent/uns, with mesh generator GeoMesh, for modelling ¯uid ¯ow and heat transfer in complex geometries. The solution-adaptive grid capability is particularly useful for accurately predicting ¯ow ®elds in regions having large gradients, such as boundary layers. The geometry and mesh were created using Gambit, duplicating the actual geometry of the experimental apparatus, which is convenient for comparison of the results of the experiments with the numerical calculations. In order to obtain a converged solution, it was necessary to create the grid, import it to the solver, select the solver formulation, and choose the basic equations to be solved, in this case the laminar heat transfer model. After specifying material properties, the boundary conditions, and initialising the ¯ow, the converged solution was obtained in the form of velocity and temperature ®elds. The results were also obtained in the form of a Nusselt±Rayleigh relation. Due to fluent/uns capabilities, temperatures on both sides of the plate and in the ¯uid could be obtained in every converged solution. To be most useful, the results of the numerical calculations, for the assumed geometry and the initial and boundary conditions, e.g. Tw, T1, D, the Boussinesq approximation, steady state, properties of the ¯uid, etc. should be presented in graphical and numerical form. The relationship between the Nusselt and Rayleigh numbers, based on the results of the numerical calculations, is: Nu 0:699 Ra1=4
for air
36
This correlation, presented in graphical form in Fig. 6, diers from the theoretical prediction, Eq. (29), by only+4.5%. The numerically calculated temperature and velocity ®elds in the neighbourhood of the plate supported the proposed physical model and assumptions serving as the basis for the theoretical result, Eq. (29).
200
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
8. General comparison and conclusions In Fig. 7 is a comparison of the analytical solution, Eq. (29), with the experimental results obtained in water, Eq. (33), and air, Eq. (34), and with the Nusselt±Rayleigh relation obtained from the numerical calculations for air, Eq. (36). The divergence of the experimental and numerical results from the theoretical prediction is no more than 15% over the entire range of Rayleigh numbers investigated. In conclusion, the coecient for natural convection heat transfer from verticallymounted circular plates is approximately 6% greater than that from vertical rectangular or square plates having heights equal to the diameter of the circular plate. However, this conclusion does not apply to vertical quadrangular plates whose edges are not parallel and perpendicular to the direction of gravitation.
Fig. 6. Numerical results performed for air (points) compared with the theoretical solution, Eq. (29) (solid line).
Fig. 7. The comparison of experimental, theoretical, and numerical results.
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
201
Acknowledgements This research was supported by Scienti®c Research Grant of the Chemistry Faculty of Technical University of GdanÄsk, under Theses nr.DS 012233/0951 and BW 012233/098. References [1] Lewandowski WM, Kubski P. Methodical investigation of free convection from vertical and horizontal plates. WaÈrme- und StouÈbertagung 1983;17:147±54. [2] Jorne J. Mass or heat transfer coecients under assisting or non-assisting natural convection near a vertical wall. Chem Eng Sci 1984;39(12):1701. [3] Herwing H. An asymptotic approach to free convection ¯ow at maximum density. Chem Eng Sci 1985;40(9):1709. [4] Lewandowski W, Kubski P. Eect of the use of the balance and gradient methods as a result of experimental investigations of natural convection action with regard to the conception and construction of measuring apparatus. WaÈrme- und StouÈbertagung 1984;18:156±247. [5] Miyamoto M, et al. Free convection heat transfer from vertical and horizontal short plates. Int J Heat Mass Transfer 1985;28(9):1733. [6] Lewandowski WM. Natural convection heat transfer from plates of ®nite dimensions. Int J Heat and Mass Transfer 1991;34(3):875±85. [7] Yu W-S, Lin H-T. Free convection heat transfer from an isothermal plate with arbitrary inclination. WaÈrme- und StouÈbertagung 1988;23:203±11. [8] Al-Arabi M, Sakr B. Natural convection heat transfer from inclined isothermal plates. Int J Heat and Mass Transfer 1988;31:559±66. [9] Lewandowski WM. Radziemska E. Natural convection heat transfer from isothermal vertical round space, III Ð Baltic Heat Transfer Conference, 1999. [10] Radziemska E. Theoretical, experimental, visualizational and numerical study of the mechanism of convective heat losses from ¯at, round bottoms of tanks and other devices. Doctor's thesis, Technical University of Gdansk, 1999, [11] Radziemska E, Lewandowski WM. Heat transfer by natural convection from isothermal downwardfacing round plate in unlimited space, (submitted to Applied Energy).
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Heat transfer by free convection from an isothermal vertical round plate in unlimited space Witold M. Lewandowski *, Ewa Radziemska Technical University of Gdansk, Department of Apparatus and Chemical Machinery, ul.G.Narutowicza 11/12, 80-952 Gdansk, Poland Received 30 March 2000; received in revised form 22 August 2000; accepted 26 August 2000
Abstract A theoretical solution of natural convective heat transfer from isothermal round plates mounted vertically in unlimited space, is presented. With simplifying assumptions typical for natural heat transfer process, equations for the velocity pro®le in the boundary layer and the average velocity were obtained. Using this velocity, the energy ¯ow within the boundary layer was balanced and compared with the energy transferred from the surface of the vertical plate according to the Newton's law. The solution of the resulting dierential equation is presented in the form of a correlation between the dimensionless Nusselt and Rayleigh numbers. The theoretical result is compared with the correlation of numerical results obtained using fluent. Experimental measurements of heat transfer from a heated round vertical plate 0.07 m in diameter were performed in both water and air. The theoretical, numerical, and experimental results are all in good agreement. # 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Natural convective heat transfer from vertical plates is an important phenomenon occurring in many practical situations. A review of 25 publications describing work on heated plates of this con®guration was presented by Lewandowski and Kubski in 1983 [1]. An average of those results, in the form of a correlation between the Nusselt and Rayleigh numbers, is given in the ®rst row of Table 1. The range of deviations of the 25 results from the mean was approximately 25%, and constant over the whole range of Rayleigh numbers investigated. * Corresponding author. Tel.: +48-058-347-24-10; fax: +48-458-347-2658. E-mail address: [email protected] (W.M. Lewandowski). 0306-2619/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(00)00053-2
188
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
Nomenclature a A Ak C Cp D g
l Cp
Gr i K K1 Nu
g TD3 2
D l
thermal diusivity (m2/s) control surface, Fig. 2 (m2) control surface on the plate, Fig. 2 (m2) integration constant (-) speci®c heat at constant pressure (J/(kgK)) plate diameter (m) acceleration due to gravity (m/s2) Grashof number (-) enthalpy (J/kg) constant in Equation (13) (-) constant in thermal conductivity relations, Eqs. (30) and (31) (W/K) Nusselt number (-) pressure (N/m2) Prandtl number (-) heat ¯ow (W)
p Pr=/a Q g TD3 Ra a T Tw T1 T u Wx Wy x x y
temperature ( C or K) wall temperature ( C) bulk ¯uid temperature ( C) temperature dierence (K) function of the boundary layer thickness (m4) velocity component in x-direction (m/s) velocity component in y-direction (m/s) dimensionless distance parallel to the plate (-) coordinate parallel to the plate (m) coordinate perpendicular to the plate (m)
Greek symbols lam l llam
heat transfer coecient (W/(m2 K)) average volumetric thermal expansion coecient (1/K) dimensionless boundary layer thickness (-) boundary layer thickness (m) thickness of the laminate (m) nondimentional temperature (-) thermal conductivity of the ¯uid (W/(m K)) thermal conductivity of the laminate (W/(m K)) kinematic viscosity (m2/s) density (kg/m3)
Rayleigh number (-)
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
189
Results from more recent studies of convective heat transfer from isothermal vertical plates in unlimited space, published after 1983 [2±7], are summarised in Table 1 and shown on a plot of the Nusselt number versus the Rayleigh number in Fig. 1. In the review by Jorne [2] there are four correlations which we designate as R (by Ruckenstein), L (by Lorenz), P (by Pohlhausen), and S (by Squire) [2]. The average of the 11 results in Table 1 and Fig. 1, on free convection heat transfer from isothermal rectangular or square plates, gives the following relation between the Nusselt and Rayleigh numbers: Nu 0:634 Ra1=4
1
The discrepancy between the relation based on the pre-1983 [1] (Table 1, ®rst row) and post-1983 [Eq. (1)] data is probably caused by the inclusion in the 1983 review [1] Table 1 Comparison of Nusselt±Rayleigh relation for vertical plates Authors
Nusselt±Rayleigh relations
Lewandowski and Kubski [1] (1983) (review) Jorne [2] (1984) R (review) L P S
0.252
Nu=0.550 Ra
Nu=0.503Ra1/4 (Ruckenstein) Nu=0.548Ra1/4 (Lorenz) Nu=0.517Ra1/4 (Pohlhausen) Nu
0:677 Ra1=4 (Squire) 1 20=21Pr1=4
Notes The mean correlation from the results of 25 authors Theory and local values Theory Theory, Pr=0.72 Theory
Herwig [3] (1985)
Nu=0.9681Gr1/4 Nu=1.058Ra1/4 for air Nu=0.647Ra1/4 for water
Theory for a semi in®nite plate
Lewandowski and Kubski [4], (1984)
Nu=0.612Ra1/4 104
Experiment in water, glycerine for rectangular plate
Miyamoto et al. [5], (1985)
Nu=0.448+0.460Gr1/4 Nu=0.448+0.503Ra1/4 for air Nu=0.448+0.308Ra1/4 for water
Numerical solution of ®nite dierences method
Lewandowski [6], (1991)
Nuch=1.586(Rach F)1/4 Nu=0.631Ra1/4 for vertical F=0.025
Analytical solution for inclined plates
Yu and Lin [7], (1988)
Nu=(4/3)F(Pr)( Ra)1/4 for vertical plate, where:
Solution for a semi-in®nite isothermal inclined plate and for 0.01 4 Pr 4 1
F(Pr)=0.6[(1+Pr)/(1+2.006Pr1/2+ +2.034Pr)]1/4 and =Pr/(1+Pr) Al.-Arabi and Sakr [8], (1988)
Nu=0.54(GrPrcos )1/4 1.15 104< GrPrcos
Experimental studies for inclined plates (0< <80 ), =0-vertical
Present study (29)
Nu=0.667Ra1/4
Theoretical solution for round vertical plate
190
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results of studies of surfaces with uniform heat ¯uxes, as well as results obtained in the turbulent regime of free convective heat transfer. By contrast, the later results summarised in Table 1, are all from studies of isothermal surfaces in the laminar regime. The shapes of all theoretically and experimentally investigated isothermal plates mentioned in Table 1 were rectangular or square. The constant width and height of
Fig. 1. Graphical presentation of the results of natural convective heat transfer from isothermal vertical plates collected in Table 1, recalculated for Pr=0.7 (top) and Pr=5±6 (bottom). The numbers in brackets are the literature sources (References and Table 1). The shaded Eq. (29) is our theoretical solution.
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191
these plates leads to a two-dimensional model for the convective ¯uid ¯ow. For other vertical plate shapes, such as triangular or circular, the problem is more complicated, because in these cases the ¯uid ¯ow structure is three-dimensional and the value of the local coecient of convective heat transfer depends not only on the height of the plate, but also on its width. Vertical plates, whose width is not constant, require a dierent, three-dimensional, model of the phenomenon. In such cases, the length of the boundary layer and its thickness change with lateral position on the plate. The theoretical and experimental exploration of heat transfer from a circular plate is the subject of this paper. 2. Physical model of the phenomenon The cross-section of a round plate and its boundary layer are shown in Fig. 2a. Streamlines, assumed to be parallel and vertical, are shown in Fig. 2b. As one moves across the plate, the length of the boundary layer varies from zero at one edge, through the diameter of the plate, D, and back to zero. In addition to the assumption that the streamlines are parallel, other simplifying assumptions used in the model, typical for natural convection in ¯uids, are as follows: . the ¯uid is incompressible and the ¯ow is laminar,
Fig. 2. Physical model of convective heat transfer from a vertical, isothermal, round plate. (a) Side view, (b) front view.
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. the magnitude of the component of velocity parallel to the surface is much greater than the magnitude of the normal component (Wx >>Wy), . the physical properties of the ¯uid in the boundary layer and in the undisturbed region are constant, and . the value of the Prandtl number is in the range, 0.6 < Pr < 6, so the thicknesses of the thermal and hydraulic boundary layers are similar (Th ).
3. Simpli®ed analytical solution The quasi-analytical solution of the Navier±Stokes equations in the presence of a temperature pro®le in the boundary layer is described by:
T ÿ T1 y 2 1ÿ Tw ÿ T1
2
The mean value of the velocity in the boundary layer was adapted for a vertical isothermal surface from reference [6]: 2
x g T : W 40 The change in mass ¯ow rate in a control volume of the boundary layer is: x : dm d AW where A is the cross-sectional area of the boundary layer: p A 2 x D ÿ x The amount of heat necessary to produce this change in the mass ¯ow is: ÿ x dQ idm Cp Tx ÿ T1 d AW
3
4
5
6
The mean value of the temperature in the boundary layer is: ÿ
1 Tx ÿ T1
0
y 2 T T 1 ÿ dy 3
7
and substitution of this relation into Eq. (6) gives:
dQ
x Cp Td AW 3
8
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
According to Newton's equation, the heat transferred is: @ TdAk dQ TdAk ÿl @y y0 where dAk is a control surface, shown in Fig. 2, on the heated round plate: p dAk 2 x D ÿ x dx From Eq. (2), the dimensionless temperature gradient at the wall is: @ 2 ÿ @y y0
193
9
10
11
By equating the heat ¯ows given by Eqs. (8) and (9) and substituting in Eqs. (3), (5), (10) and (11) one obtains: p d 3 x D ÿ x 240lCp p 12 g T dx x D ÿ x Introducing the plate diameter, D, as the characteristic linear dimension and de®ning:
x l ; x ; a ; Cp D D
RaD
g TD3 a
Eq. (12) can be rewritten: p d 3 x 1 ÿ x 240 p K Ra dx D x 1 ÿ x
13
14
Dierentiating Eq. (14) gives: 33 d 4 4 ÿ K 2x 2 1 ÿ x dx
15
d 4 1 ÿ 2x K dx 2x 1 ÿ x
16
or 33
Introducing u
4 K
and
du
43 d; K
17
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the non-homogeneous dierential equation, Eq. (16) becomes: 3 du u 2x ÿ 1 ÿ 1 4 dx 2x 1 ÿ x
18
The homogeneous equation, 3 du u 2x ÿ 1 ÿ 0 4 dx 2x 1 ÿ x
19
has the solution, u
C0
20
2
2x 1 ÿ x3
Using the integration constant (C0 =C0 (x)) gives the relation: 2 4 2 C0 23 x 1 ÿ x3 dx 3
21
The expression inside the integral in Eq. (21), according to the Czebyshev criterion, cannot be expressed in the form of an elementary function, but it can be expanded in a Taylor series: 2 2 1 4 7 4 14 5 2 x ÿ x x 1 ÿ x3 x3 1 ÿ x ÿ x2 ÿ x3 ÿ 3 9 81 243 729
22
and the integral, Eq. (21), becomes: 4 2 3 5 1 8 1 11 2 14 7 7 17 20 x 3 ÿ x 3 ÿ x 3 C1 C0 23 x3 ÿ x3 ÿ x 3 ÿ 3 5 4 33 189 1377 2430
23
Evaluating the integration constant, C1, using the boundary condition, for x 0; 0; u 0; and C1 0
24
Eq. (20) becomes:
u
4 2 3 5 1 8 1 11 2 14 7 7 17 20 3 3 3 3 3 3 3 2 x ÿ x ÿ x ÿ x ÿ x ÿ x 3 5 4 33 189 1377 2430 2
2x 1 ÿ x3
25
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195
Elimination of the variable, u, using its de®nition, Eq. (18), gives an expression describing the thickness and shape of the boundary layer on the plate: 1 3 5 1 8 1 11 2 14 7 17 7 20 4 3 3 3 3 3 3 1 1 x ÿ x ÿ x ÿ x ÿ x ÿ x 240 4 4 4 5 1 1 4 33 189 1377 2430 K4 u4 1 Ra 3 x 1 ÿ x6 26 The value of the heat transfer coecient may be determined from:
2l
27
The non-dimensional form of the local heat transfer coecient, expressed using the Nusselt number is: D 2 l 14 1 1 1 x 1 ÿ x6 4 RaD 1 20 3 5 1 8 1 11 2 14 7 17 7 20 4 3 3 3 3 3 3 x ÿ x ÿ x ÿ x ÿ x ÿ x 5 4 33 189 1377 2430
NuD
28
and the mean value of the Nusselt number is then: NuD
1 0
1
NuD dx 0:667Ra4D
29
4. The analysis of the theoretical solution Compared with the average of the theoretical and experimental convective heat transfer coecients on vertical square or rectangular plates [1], the present result, Eq. (29), for vertical round plates is larger by 17.4%. This augmentation of heat transfer from round plates seems, at ®rst, rather high, but may be explained by noting that the results collected in the 1983 review paper [1] included, in addition to isothermal cases, those having uniform heat ¯ux and results obtained in the turbulent regime of free convective heat transfer. The present study considers only the conditions of isothermal plates and the laminar regime of heat transfer, so is not directly comparable with the earlier correlation. The present theoretical result, Eq. (29), is also not directly comparable with all of the results collected in Table 1, because of the presence of the Prandtl number as a parameter in some of the correlations in Table 1. The Nusselt±Grashof relations in
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Table 1 were recalculated into the form of Nusselt±Rayleigh relations for Pr=0.7 (air) and Pr=5-6 (water) in order to determine the average, Eq. (1). Comparison of Eq. (29) with Eq. (1) suggests that the Nusselt number for heat transfer from the round plate is 5.2% greater than those for rectangular and square plates. The most directly comparable results for vertical rectangular plates are those presented in reference [6], described by relation Nu=0.631Ra1/4 (Table 1), obtained by the same method and using the same assumptions as in the present work. Comparing this result with Eq. (29), we see that the average heat transfer coecient for round plates is approximately 6% higher than that for rectangular plates. We conclude that the shape of a vertical plate can have a signi®cant in¯uence on its heat transfer coecient. The explanation of the eect is that, for a rectangular plate, the thickness of the boundary layer and values of heat transfer coecient are functions of the plate height only and are constants over the width of the plate, but for a round plate, the boundary layer thickness and heat transfer coecient are functions, not only of distance in the direction of the ¯ow, but also of lateral distance from the centre of the plate. Because the length and thickness of the boundary layer go to zero at the edges of the round plate, the heat transfer coecient approaches in®nity there. In consequence, the average value of the coecient is greater for round than for square or rectangular vertical plates. 5. Experimental apparatus and procedures The experimental studies were made in a rectangular glass tank, 0.5 0.5 1.0 m, in water and air. The vertical round plate had a special layered construction, consisting of two circular copper plates of diameter, D=0.07 m, both cemented to a layer of glass laminate using epoxy-resin, as shown in Fig. 3. The plate was heated by warm water ¯owing through a chamber in contact with one of the copper plates. The thickness of the copper plates (5 mm each) ensured that the surface in contact with the surroundings was isothermal ( 0.1 C). The plate and heater were insulated on the side and bottom using polyurethane foam. The plate temperature was regulated using a thermostat on the water supply to the heater. Eleven copper-constantan thermocouples were soldered to the copper plates for measurement of the temperatures on the surface in contacted with the ¯uid (Tw,i) and the temperature dierences between the two copper plates separated by the glass laminate (Tlam=(Tw1,i-Tw,i)). The average temperature of the plate was the average of the temperatures measured on that plate, e.g. (Tw =(1/n)Tw,i), and the average temperature drop across the plate (copper-glass laminate-copper) was (Tlam,i= (1/n)( (Tw1,i-Tw,i)), where n is the number of thermocouples. The heat transferred from the warm water in the heater, through the plate, into the surrounding ¯uid was determined from the following relations: Q
D2 llam Tlam K1water Tlam 4 lam
for water
30
W.M. Lewandowski, E. Radziemska / Applied Energy 68 (2001) 187±201
and Q where: Qrad
D2 llam Tlam ÿ Qrad K1air Tlam Tlam ÿ Qrad 4 lam D2 "5:86 4
" # Tw 273:14 4 T1 273:14 4 ÿ 100 100
for air
197
31
32
is the heat transferred by radiation from the vertical plate in air. For the polished copper plate, the emissivity and coecient of surface con®guration are "=0.6 and = 0.831, respectively.
Fig. 3. Schematic diagram of experimental apparatus with enlarged detail of the cross-section of the round vertical plate.
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The relation describing the eective thermal conductivity of the layer of glass laminate and two copper plates as a function of the temperature (Tlam) was measured using a separate apparatus. The results were K1water=5.295 W/K for water and K1air=1.193+0.002Tlam W/K for air, with the average temperature given by Tlam=(Tw1,i+Tw,i)/(2n). The dierences between results obtained when calibrating the plate in water and in air were due to the in¯uence of radiation. The uncertainty in the determination of K1 was estimated to be 5.0% in water and 5.2% in air. Estimates of the uncertainties in the determination of the Rayleigh numbers are 4.5% in water, 1.1% in air and in the determination of the Nusselt numbers, 6.7% in water and 6.6% in air. Additional information about the experimental procedure, method of heating the plate, measurements of temperature, and heat losses may be found in previous publications [9±11]. 6. Experimental results The results of the experiments are shown on a plot of the Nusselt number versus the Rayleigh number in Fig. 4 for the measurements in water and in Fig. 5 for the measurements in air. Both dimensionless groups were evaluated using the diameter of the plate as the characteristic length. Least-squares ®t to the data produced the following relationships: Nu 0:587 Ra1=4
for water
33
Nu 0:655 Ra1=4
for air
34
in quite good agreement with the theoretical prediction, Eq. (29).
Fig. 4. Experimental data obtained for water (points) compared with the theoretical solution, Eq. (29) (solid line).
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199
Fig. 5. Experimental data obtained for air (points) compared with the theoretical solution, Eq. (29) (solid line).
7. Numerical calculations The numerical results were obtained using the commercial program fluent/uns, with mesh generator GeoMesh, for modelling ¯uid ¯ow and heat transfer in complex geometries. The solution-adaptive grid capability is particularly useful for accurately predicting ¯ow ®elds in regions having large gradients, such as boundary layers. The geometry and mesh were created using Gambit, duplicating the actual geometry of the experimental apparatus, which is convenient for comparison of the results of the experiments with the numerical calculations. In order to obtain a converged solution, it was necessary to create the grid, import it to the solver, select the solver formulation, and choose the basic equations to be solved, in this case the laminar heat transfer model. After specifying material properties, the boundary conditions, and initialising the ¯ow, the converged solution was obtained in the form of velocity and temperature ®elds. The results were also obtained in the form of a Nusselt±Rayleigh relation. Due to fluent/uns capabilities, temperatures on both sides of the plate and in the ¯uid could be obtained in every converged solution. To be most useful, the results of the numerical calculations, for the assumed geometry and the initial and boundary conditions, e.g. Tw, T1, D, the Boussinesq approximation, steady state, properties of the ¯uid, etc. should be presented in graphical and numerical form. The relationship between the Nusselt and Rayleigh numbers, based on the results of the numerical calculations, is: Nu 0:699 Ra1=4
for air
36
This correlation, presented in graphical form in Fig. 6, diers from the theoretical prediction, Eq. (29), by only+4.5%. The numerically calculated temperature and velocity ®elds in the neighbourhood of the plate supported the proposed physical model and assumptions serving as the basis for the theoretical result, Eq. (29).
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8. General comparison and conclusions In Fig. 7 is a comparison of the analytical solution, Eq. (29), with the experimental results obtained in water, Eq. (33), and air, Eq. (34), and with the Nusselt±Rayleigh relation obtained from the numerical calculations for air, Eq. (36). The divergence of the experimental and numerical results from the theoretical prediction is no more than 15% over the entire range of Rayleigh numbers investigated. In conclusion, the coecient for natural convection heat transfer from verticallymounted circular plates is approximately 6% greater than that from vertical rectangular or square plates having heights equal to the diameter of the circular plate. However, this conclusion does not apply to vertical quadrangular plates whose edges are not parallel and perpendicular to the direction of gravitation.
Fig. 6. Numerical results performed for air (points) compared with the theoretical solution, Eq. (29) (solid line).
Fig. 7. The comparison of experimental, theoretical, and numerical results.
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201
Acknowledgements This research was supported by Scienti®c Research Grant of the Chemistry Faculty of Technical University of GdanÄsk, under Theses nr.DS 012233/0951 and BW 012233/098. References [1] Lewandowski WM, Kubski P. Methodical investigation of free convection from vertical and horizontal plates. WaÈrme- und StouÈbertagung 1983;17:147±54. [2] Jorne J. Mass or heat transfer coecients under assisting or non-assisting natural convection near a vertical wall. Chem Eng Sci 1984;39(12):1701. [3] Herwing H. An asymptotic approach to free convection ¯ow at maximum density. Chem Eng Sci 1985;40(9):1709. [4] Lewandowski W, Kubski P. Eect of the use of the balance and gradient methods as a result of experimental investigations of natural convection action with regard to the conception and construction of measuring apparatus. WaÈrme- und StouÈbertagung 1984;18:156±247. [5] Miyamoto M, et al. Free convection heat transfer from vertical and horizontal short plates. Int J Heat Mass Transfer 1985;28(9):1733. [6] Lewandowski WM. Natural convection heat transfer from plates of ®nite dimensions. Int J Heat and Mass Transfer 1991;34(3):875±85. [7] Yu W-S, Lin H-T. Free convection heat transfer from an isothermal plate with arbitrary inclination. WaÈrme- und StouÈbertagung 1988;23:203±11. [8] Al-Arabi M, Sakr B. Natural convection heat transfer from inclined isothermal plates. Int J Heat and Mass Transfer 1988;31:559±66. [9] Lewandowski WM. Radziemska E. Natural convection heat transfer from isothermal vertical round space, III Ð Baltic Heat Transfer Conference, 1999. [10] Radziemska E. Theoretical, experimental, visualizational and numerical study of the mechanism of convective heat losses from ¯at, round bottoms of tanks and other devices. Doctor's thesis, Technical University of Gdansk, 1999, [11] Radziemska E, Lewandowski WM. Heat transfer by natural convection from isothermal downwardfacing round plate in unlimited space, (submitted to Applied Energy).