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Evaluation of thermal field in buoyancy-induced flows by a Schlieren method
Evaluation of Thermal Field in BuoyancyInduced Flows by a Schlieren Method F. Devia
G. Milano G. Tanda Dipartimento di Ingegneria Energetica, Universith di Genova, Genova, Italy
• The aim of this work is to perform a quantitative experimental investigation of two-dimensional buoyancy-induced flows using a schlieren method. The schlieren technique uses the refraction of light rays to display a pattern related to the temperature field. The employed schlieren system makes use of an opaque filament to identify regions of fluid deflecting light rays by the same amount. An iterative inverse technique is developed to reconstruct the temperature field from optical data. After a description of the experimental technique and the data processing features, results concerning the temperature profile reconstruction for natural convective flows are presented. Attention is devoted to air buoyancy flows induced by a single heated vertical plate and by an array of four heated vertical plates. Reconstructed temperature data are compared with theoretical predictions and with direct temperature measurements performed by thermocouples. Keywords: schlieren method, natural convection, thermal field, inverse solution
INTRODUCTION In recent years the development of optical techniques has made it easier to measure the temperature distribution in flow fields. The basic concept of the schlieren technique is to record the deflection undergone by a light ray as it passes through a fluid region characterized by refractive index inhomogeneities. Since the light deflections are related to refractive index gradients and thus to the thermal field, the measurements can be processed to reconstruct the temperature profiles in the visualized region of fluid. A large variety of schlieren methods for quantitative evaluations have been described in the literature (knife-edge [1], Ronchi [1], moir6 [2], specklegram [3], color-coding systems [4], etc). In this work, light deflection measurements are performed by using an opaque filament that can be shifted in the focal plane of the schlieren head. A broad description of the focal filament method for establishing curves of equal angles of light deviation is given in Ref. 5. The schlieren system employed here is relatively simple (neither laser source nor light intensity measurements are required), straightforward to align, easy to operate, and inexpensive. The distribution of the light deviations obtained for a given experimental run is then processed by an iterative inverse technique to reconstruct the thermal field. The solution of the nonlinear inverse problem is obtained by minimization of the squared error function of light deflection residuals, using the OLS (ordinary least squares) estimation method coupled to the Gauss minimization algorithm. This technique is applied to the study of two-dimensional laminar natural convection of air flowing on heated
vertical plates. In the first case, the thermal field around a single, isolated, isothermal vertical plate was analyzed, and the reconstructed temperature profiles were compared with Ostrach's theoretical predictions [6]. In a second case, attention was turned to a more complicated plate configuration, consisting of a staggered array of four vertical plates. The latter configuration can simulate either heat exchangers with staggered fins or electronic devices cooled by natural convection of air. Temperature profiles were obtained when the plates were heated equally and arranged at different interplate spacings. APPARATUS The schlieren system is graphically shown in Fig. la. A noncoherent light beam coming from a slit source is collimated by the concave mirror M 1 to form parallel light rays. Owing to the inhomogeneities of the fluid refractive index in the test section, the light rays are deflected and focused by the concave mirror M 2 onto different positions in its focal plane. All rays are then projected on the screen or on a camera. Regions of the test section having the same light angular deflection a (or the same light displacement A) can be identified by using an opaque filament (a thin dark strip) in the focal plane of mirror M 2 (focal filament method [5]). When a disturbed light ray is stopped by the focal filament, the image of the corresponding region of the test section will appear dark on the screen and the remaining field will be bright. The shift A of the disturbed ray in the focal plane of mirror M 2 can be recorded by measuring the distance between the middle of the undisturbed image of the slit source and the centerline of the filament, as shown in Fig. lb. Therefore,
Address correspondence to Dr. Ing. Giovanni Tanda, Dipartimento Ingegneria Energetica, Universith di Genova, via all'Opera Pia 15/a, 1-16145 Genova (Italy).
Experimental Thermal and Fluid Science 1994; 8:1-9 © 1994 by Elsevier Science Inc., 655 Avenue of the Americas, New York, NY 10010
0894-1777/94/$7.00
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F. l)cvia ct al. focal filament
f-J / ~
""-,.~
mirror M 2
/ ~
,~ ~/~ndisturbed ray
///~---J~\
~
test section
disturbed raYment
~
"'1~
light source a
b
Figure 1. (a) Schlieren system (top view). M 1 and M 2 a r e 0.38 m diameter concave mirrors (with focal lengths fl and J), respectively), a is the angular deflection of the light ray, and A is the related displacement at the focal plane of mirror M2 (fl = f2 = 1.9 m, 3' = 9°, distance between mirrors about 8 m). (b) Measurement of light displacement A at the focal plane of mirror M2; the focal filament is shifted from position 1 to position 2. moving the filament in the direction of deflections allows one to define the locus of the images of points that deflect light by the same amount. If the thermal field is assumed to be two-dimensional (i.e., temperature is independent of z coordinate), the relation between the light ray shift A and the local temperature gradient can be easily obtained from the equation [1] A = ~(3T/Oy)/T
2
(1)
where T is the absolute temperature and y is the direction in which the light deflection is recorded. The constant 1~ = - K P L f z / R n o was equal to -0.0456 m 2. K in the present experiment. (K is the Gladstone-Dale constant, P the pressure, L the length of the disturbed optical path, f2 the focal length of the schlieren head, R the ideal gas constant, and n 0 the fluid refractive index at reference conditions.) A schematic view of the test section is presented in Fig. 2. Four vertical plates, suspended by nylon lines, were affixed at three different levels in the vertical coordinate. The symbols A, B, and C are assigned to denote plates located at the lower, middle, and upper stages, respectively. Each plate was made of two aluminum layers with a plane electrical heater sandwiched between them. The dimensions of each plate are: height H = 65 mm, thickness D = 8 mm, and length L = 300 mm (in the direction of the undisturbed light rays). Dimensions were chosen to favor a two-dimensional thermal field around the heated plates. Surface temperature was measured by means of 10 fine-gauge thermocouples (type K, calibrated to _+0.1 K) embedded in the material. Due to the high thermal conductivity of aluminum, each plate, heated by a given amount of power input, reached a uniform temperature over its entire surface. At the steady state, a plate temperature uniformity generally within + 1% of the mean wallto-fluid temperature difference was attained for the spatially distributed thermocouples. Due to the low thermal emittance of polished aluminum, the heat transfer by radiation was reduced to a small value and the thermal power was mainly exchanged by natural convection be-
X
tdy
SO
Figure 2. Schematic view of the test section (H, D, and L denote the height, thickness, and length of each plate, respectively; S is the interplate spacing, and g is the gravity vector).
tween the exposed surfaces of the plates and the surrounding air. It is worth noting that measurements of radiation heat transfer are not required for the thermal field reconstruction by the schlieren method. The ambient air temperature was measured by five shielded thermocouples deployed below the plate assembly. In some experimental runs a number of fine-gauge, type K thermocoupies were located in different positions near the heated plate assembly to enable a comparison of temperature measurements with the reconstructed temperature profiles in the fluid. Two different arrangements of plates were considered. In the first case, experiments were conducted for a single heated vertical plate. A second set of experiments were performe d in the presence of all four vertical heated plates. In the latter case, each plate of the assembly dissipated the same thermal power. The interplate spacing S (see Fig. 2) was parametrically varied to obtain values of
Thermal Field in Buoyancy-Induced Flow 3 the spacing-to-height ratio ( S / H ) ranging between 0.3 and 0.1. For both of the arrangements considered (single-plate and four-plate assembly), experimental runs consisted in (1) heating the single plate or the plate assembly (12-24 h were allowed for the attainment of steady-state conditions), (2) measuring wall and ambient air temperatures, (3) visualizing the regions at equal light deviation, and (4) processing the schlieren images to reconstruct temperature profiles. Schlieren photographs were obtained with the focal filament set in the vertical direction and at different distances from the undisturbed image of the slit source. The location of the curves at equal light deflection (called isodeflection lines) was obtained by identifying, for a set of photographs, the coordinates of the filament shadow centerlines. The filament was moved only in the y direction (orthogonal to the vertical surfaces of the plates) along which the largest temperature gradients were expected. It should be noted that temperature gradients in the x direction, even though not directly measured, are not neglected; their effect is included in the horizontal light deflection measured values. The technique employed to reconstruct the thermal field from optical data is outlined in the following section.
assuming that the unknown temperature distribution T(y) along a line at constant x is represented by a proper class of functions. Two kinds of parameterization were developed and tested: a local second-order, three-points polynomial approximation (moving parabola) and the cubic B-splines approximation. The results of several simulated experiments showed that both kinds of parameterization give a good approximation of temperature distribution in buoyancy-induced flows but that better accuracy and smoothness can generally be obtained by the use of the cubic B-splines approximation. This latter parameterization allows us to approximate the temperature profile by the transformations T = T1 + (T2 - T1)u3/6
for Yl < y < (5y 1 + y 2 ) / 6
(7a) T= [Tl(5-3u-
3u z + 2 u 3)
+ T2(1 + 3u + 3u z - 3u 3) + T3u3]/6 for (5y 1 + y 2 ) / 6 _
T = [Ti_3(1 - 3u + 3u 2 - u 3) + T/_2(4 - 6u 2 + 3u 3) + T/_I(I + 3u + 3u 2 - 3U3) + T/u3]/6
DATA PROCESSING
for ( Y i - 3 + 4yi-2 + Yi-1)/6
If the measured and calculated displacements of light rays are denoted by Am and A c, for the solution of the inverse problem in the OLS formulation it is necessary to find a vector of unknown temperature T = {T/}, i = 1, N, that minimizes the sum
< Y < (Yi 2 + 4yi-I + Yi)/6
S
=
[A m
--
Ac(Z)]t[Am
-- Ac(t)] = min
(2)
where N is the total number of measured light deflections and [ ]t denotes the transposition operator. The calculated displacement is given by At(T) = 1 2 ( O T / a y ) / T z
(3)
and the unknown vector of parameters T is subjected to some constraints by using appropriate thermal boundary conditions, which depend on the kind of each convective experiment. As the calculated function Ac is a nonlinear function of T, the minimization process has been constructed with successive approximations, recursively using the linearized Gauss algorithm. In vectorial form the iterative process at the generic iteration k + 1 can be written T k + 1 = T k __ 1 p k grad{st,} (4) and stops when the minimum values of the target functional s is reached. In Eq. (4), the gradient of s and the var/covar matrix P at the generic iteration k can be determined as follows: grad{s} =
-2X t. [A m -
P = [X'. X ] - '
Ac]
(5)
(7"o)
(7C)
T = [TN_2(1 -- 3u + 3u 2 -- u 3) + TN_ 1(4 -- 6u 2 + 3u 3) + TN(1 + 3u + 3u 2 - 2u3)]/6
(7d)
for (YN-2 + 4YN-1 + Y N ) / 6
(7e)
f°r (YN-1 + 5YN)/6
(6)
where X is the sensitivity matrix of A c with respect to the parameters {T,.}at the measuring points Yi, i = 1, N. It can be seen from Eq. (3) that in order to calculate the function Ac, temperature and temperature gradients of the fluid must be evaluated. This can be accomplished by
-1/T+l/Ty=o=(foYA(y)dy)/12
(8)
If the function A(y) is known with high precision and at several points close to each other, Eq. (8) in principle provides the solution with the best accuracy, because no
4
F. Devia et al.
assumption at all is made on the function T ( y ) . The only (numerical) error, due to the kind of discrete formula a d o p t e d for integrating Eq. (8), can be reduced to a negligible quantity. If, on the other hand, the values of A(y) are p e r t u r b e d by noise, as always occurs in experimental tests, a n d / o r data are available at a limited number of points, the accuracy of the direct integration technique decreases and tends to be comparable to or worse than that given by the iterative solution of Eq. (4). In general, the latter should be p r e f e r r e d for real applications because it gives not only the unknown t e m p e r a t u r e but also the confidence-bound estimate of each temperature value calculated. This information may be vital in processing noisy measurements. Owing to the importance of the influence of measurement errors on the reconstructed t e m p e r a t u r e field, a detailed analysis on this topic has been performed, and several results of simulated experiments are r e p o r t e d here. In Table 1 the values of light shift m e a r e calculated from the t e m p e r a t u r e distribution Te given by the exact Ostrach solution for a single vertical plate at uniform wall temperature T w = 354.6 K (81.45°C) cooled by laminar air free convection at the t e m p e r a t u r e T~ = 293.6 K (20.45°C), at the vertical elevation of 5 mm from the leading edge of the plate. The input h e values are processed by using two methods: (1) the iterative inverse technique {Eqs. (2)-(7)] and (2) the direct integration technique [Eq. (8)]. The t e m p e r a t u r e values reconstructed by means of the two above methods are designated T and T*, respectively. The
2~r confidence bounds ( 2 0 / 1 odds) for T and 7"* values are r e p o r t e d in the case of artificially perturbed A input data. The perturbation, which simulates random measurement errors, was obtained by superposing upon the exact values of A e a white, zero-mean, Gaussian noise with a standard deviation o:,± such that 2~a = X A .... ax- Three values of X were tested: X = 0.05, 0.10, and 0.20: it can be noted that the intermediate value X = 0.10 can be considered representative of the effective experimental errors associated with the light shift measurement and estimated according to the procedure outlined by Moffat [7]. The exact (Ostrach) t e m p e r a t u r e values 7 I, are compared at first with those reconstructed by the iterative Gauss technique ( T ) and by direct integration (T*) in the case of exact, unperturbed input data (2O'a = 0). As can be seen, both the inverse solution based on the cubic B-splines approximation and the direct integration give excellent results. The analysis of the solution with p e r t u r b e d input data (2~ra = X A . . . . . ) was p e r f o r m e d by the Monte Carlo technique (2000 runs for each set of data). A n ensemble of estimation errors e~ = T - T e was generated, where T e is the vector of exact (simulated) values and T (equal to T or T*) is the vector of estimated values. The variance and the corresponding standard deviation of the vector e T for the two above-mentioned reconstruction techniques were calculated (by a sample statistics analysis) and compared. As shown in Table 1, the 2trT. confidence bound of the direct integration values is slightly better for the half of
Table 1. Results of a Simulated Experiment: Comparison Between Exact (T e) and Reconstructed (T and T*) Temperature Distributions Using the Theoretical Solution of Ostrach [6] as Input Data a
y
(m)
(m)
T,, (°C)
2 ~ra = 0 T (°C)
0.00 0.27 0.54 0.82 1.09 1.36 1.63 1.91 2.18 2.45 2.72 2.99 3.27 3.54 3.81 4.08 4.36 4.63 5.17 5.72 6.26 6.81 7.35 8.43 9.94
8.20 8.48 8.68 8.74 8.62 7.76 7.08 6.31 5.49 4.69 3.93 3.25 2.65 2.14 1.71 1.36 1.07 0.66 0.40 0.24 0.14 0.09 0.03 0.01 0.00
81.45 75.30 69.18 63.18 57.41 51.98 46.99 42.50 38.57 35.18 32.32 29.94 27.98 26.40 25.13 24.12 23.31 22.68 21.79 21.26 20.93 20.73 20.62 20.51 20.45
81.45 75.30 69.18 63.18 57.41 51.97 46.99 42.50 38.57 35.17 32.32 29.93 27.99 26.39 25.13 24.11 23.31 22.68 21.77 21.26 20.91 20.74 20.61 20.52 20.45
×
10 3
A e ×
10 3
T* (°C)
20-a = 0.05A e, max 20 r 2~rr, (°C) (°C)
20-a = 0.10A e. max 2tr r 2trr, (°C) (°C)
20;a = 0.20A e, max 2~rr 2tr r, (°C) (°C)
81.45 75.30 69.18 63.18 57.41 51.98 46.98 42.50 38.56 35.18 32.32 29.94 27.98 26.40 25.13 24.11 23.31 22.68 21.79 21.25 20.93 20.73 20.61 20.50 20.45
0.00 0.23 0.43 0.49 0.57 0.61 0.66 0.67 0.70 0.72 0.74 0.76 0.78 0.80 0.81 0.83 0.84 0.85 1.09 0.92 1.13 0.95 1.05 0.78 0.00
0.00 0.46 0.85 0.96 1.14 1.20 1.30 1.33 1.39 1.43 1.47 1.51 1.54 1.59 1.60 1.65 1.65 1.67 2.16 1.81 2.24 1.89 2.08 1,54 0.00
0.00 0.92 1.69 1.93 2.27 2.40 2.61 2.67 2.79 2.85 2.94 3.01 3.07 3.17 3.21 3.30 3.31 3.35 4.32 3.63 4.48 3.78 4.15 3.07 0.00
0.00 0.22 0.37 0.47 0.54 0.59 0.64 0.67 0.70 0.73 0.75 0.77 0.80 0.83 0.86 0.89 0.91 0.94 1.02 1.11 1.20 1.27 1.34 1.57 1.96
0.00 0.44 0.74 0.92 1.06 1.17 1.25 1.31 1.37 1.43 1.48 1.53 1.58 1.64 1.70 1.75 1.80 1.87 2.01 2.20 2.37 2.52 2.65 3.11 3.88
aThe 20" confidence bounds are reported for both temperature reconstruction methods, using artificially perturbed A input data.
0.00 0.87 1.48 1.85 2.12 2.33 2.49 2.62 2.73 2.85 2.95 3.05 3.16 3.27 3.39 3.50 3.60 3.73 4.02 4.40 4.74 5.04 5.31 6.21 7.76
Thermal Field in Buoyancy-Induced Flow 5 the points closer to the heated wall. Conversely, the 2(r r confidence bound of the inverse technique values is better for the other half of the points, owing to the second boundary condition (T = Too far from the heated wall). In this regard it can be observed that the error affecting the temperature evaluated by the direct integration technique can be very large for the points closest to the fluid at rest. Another important conclusion that can be drawn from Table 1 is the proportionality between o-a and trr (or crr.)--doubling the standard deviation of the noise affecting the measured data doubles the standard deviation of reconstructed temperature values. This behavior remains valid for values of tra larger than those reported in Table 1. This means that the inverse method here implemented gives "stable" solutions and no "regularization" technique is needed. The results of Table 1 refer to the temperature reconstruction field in the case of a relatively large set of data points (25). In Table 2 the same thermal problem is solved for a set of only seven points perturbed by a noise with 2tra = 0.10 Ae, max. The results reported in Table 2 show that when the measured input data are few and are affected by noise, the inverse technique should be preferred because the probable errors affecting the reconstructed temperature values are less everywhere (2~rr < 20"r.). It should be noted that the error predicted by the inverse technique at the first and last points is zero, because at these two points the temperature values are known (boundary conditions) and therefore the associated sensitivity coefficients and variances are zero too. On the other hand, the effect of errors in the wall and ambient fluid temperature measurements can be easily taken into account. In conclusion, when T~ and T~ measurement errors are also considered and their interaction with other errors (introduced by the mathematical model and by light-shift A measurements) is taken into account, the overall experimental uncertainty (at the 95% confidence level) in T values, reconstructed from a set of 25 input data points, varies between 2% and 4% of T~ - T~.
voted at first to the case of a single vertical plate, for which the development of a temperature profile in laminar regime is theoretically known and has been experimentally investigated in several papers [e.g., 3, 8, 9]. Figure 3 shows a typical example of a schlieren photograph corresponding to a given focal filament displacement. From a set of such photographs, the contours of a number of isodeflection lines were obtained as reported in Fig. 4. It should be observed that during the reconstruction process a larger number of lines were used than are plotted in the figure. Some results are presented in Fig. 5 in terms of the dimensionless temperature 0 = ( T - T o o ) / ( T ~ - T ~ ) as a function of the dimensionless similarity variable 77 = (Grx/4)l/4y/x. Measured data refer to two values of T w - T~ (20.9 and 61 K) and to three x elevations (25, 35, and 45 mm from the leading edge). The reference temperature T r = T w - 0.38(T~ - To~) suggested in Ref. 10 was used to evaluate the kinematic viscosity of air, while the thermal expansion coefficient was evaluated at T~. The solid line refers to the theoretical prediction of Ostrach [6] for a single isothermal vertical plate in the laminar regime (Pr = 0.72). Also presented is a set of data obtained by Kastell et al [3] by a specklegram technique for T~ - T~o = 61.2 K and x = 30.8 and 50 mm. Inspection of the figure shows good agreement of all the measured data with the theoretical profile. A comparison of present data with those reported in Ref. 3 reveals slight discrepancies, limited to the regions far from the vertical wall, probably due to the uncertainties associated with the measurement methods and with the algorithms used to obtain temperature values. In the second phase, the thermal field near a four-vertical-plate assembly was investigated. This configuration (schematically depicted in Fig. 2) may simulate either
TEMPERATURE FIELD RECONSTRUCTION F O R LAMINAR NATURAL C O N V E C T I O N FLOWS In order to assess the accuracy of the optical method and that of the data processing procedure, attention was de-
AE
Q
m
Q
?
2. Results of a Simulated Experiment--Comparison Between 2 (r Confidence Bounds for Temperature Reconstructed by the Inverse Technique (T) and Direct Integration (T*) a
Table
y × 103 (m)
Ae X 103 (m)
Te (°C)
2tr T (°C)
2o-,r, (°C)
0.00 1.09 2.18 3.27 4.36 6.26 9.94
8.20 8.62 5.49 2.65 1.07 0.14 0.00
81.45 57.41 38.57 27.98 23.31 20.93 20.45
0.00 1.79 2.76 2.84 3.16 3.35 0.00
0.00 1.90 3.09 3.34 3.62 4.45 6.95
~A set of seven input data (with 2tra uncertainty equal to 0.10Ae,max) is considered.
®
Figure 3. Photograph of a schlieren image. Natural convection on a single heated vertical plate. 1, Heated plate; 2, filament shadow; 3, check pins; 4, nylon lines supporting the plate; 5, electric wires; 6, thermocouple wires; 7, two-dimensional calibrated grid.
6
F. Dcvia et al. heaters embedded in the plates. It can be noted that the isodeflection profiles (Fig. 6) are more complicated than in the single-plate case, and they may present change of sign, minimum, maximum, and inflection points. The reconstructed fluid temperature profiles are displayed for three experiments in Figs. 7a, 7b, and 7c. The power input to each plate was 6.5 W, 90% of which was expected to be exchanged by natural convection at the exposed surfaces, while the remaining 10% is transferred to the surroundings by thermal radiation. The spacing parameter S/H was set equal to 0.3 (Fig. 7a), 0.2 (Fig. 7b), and 0.1 (Fig. 7c). In order to favor a compact presentation of results, only a limited region of the visualized thermal field is plotted. Continuous lines represent reconstructed T - T.~ profiles at different elevations, namely at 5 mm from the trailing edge of plate A; at 5, 15, 25, 35, 45,
A.103 ,m
6177 1
1011'11
I 2 3 4 5
0.3 0.8 1.2 1.9 2.5
6 7 8 9 10
-0.3 -0.8 -I .2 -I .9 -2.5
I °.211
1~III115
Figure 4. Curves of equal light displacement A for natural convection on a single heated vertical plate (Tw - T~ = 20.9 K). compact heat exchangers with staggered fins or natural convection air-cooled electronic devices. For the sake of brevity, only a limited number of experiments are shown. A more detailed analysis for this natural convection system is presented in Ref. 11. Experiments were conducted by delivering an equal amount of electric power to the
1.0
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i
i
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Tw-Too= \ ,_8 08 ' k
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200K x: 25 mm • 30.8 35 • 45 •
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(Pr=0.72]
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~, _
I ~ ' ~ - - ~ T ~
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tlx "~
01.2K[3]
I 2 3 4
0.3 0.8 1.9 2.5
5 6 7 8
-0.3 -0.8 -I .9 -2.5
rl = (Orx/4)'/'y/X
Figure 5. Dimensionless temperature O profiles as a function of similarity variable -q, for free convection on a vertical plate. Comparison between present data ( 0 , I , v , O, D, v), measured data from Ref. 3 (*, "k), and the predicted values by Ostrach [6] (solid lines).
Figure 6. Curves at equal light displacement A for natural convection on a vertical plane assembly. Each plate dissipates the same thermal power (6.5 W). S/H = 0.1.
Thermal Field in Buoyancy-Induced Flow 7 and 55 mm from the leading edge of plate B (both sides); and at 5 and 20 mm from the leading edge of plate C. In addition, direct air temperature measurements obtained by thermocouples placed in several locations are reported (using blackened circles) for S / H = 0.1 and 0.3. Near each temperature profile, the corresponding heat transfer coefficient h at the wall is indicated. This quantity was easily computed by the relationship ~T) k~ ~ wT,~- T~
h = -
(9)
where (OT/On),, is the temperature gradient along the normal to plate surfaces (evaluated at the wall), Tw is the uniform wall temperature, and k w is the air thermal conductivity (evaluated at the corresponding wall temperature). Temperature drops Tw - Too between walls and air ranged from 21.5 to 27 K depending on S / H value and on the specific location of each plate. As expected, the highest temperature gradients are present near the leading edge of the walls. Downstream of each plate, the air temperature is increased owing to the contact of the buoyant flow with the heated walls. In these
regions (called thermal plumes or wakes) the flow showed small instabilities as pointed out by thermocouple measurements. This phenomenon introduced additional uncertainties in the optical measurements, and it partially explains differences between reconstructed and thermocouple data (3 K at the most, ie, about 14% of wall-to-ambient mean temperature drop). Further interesting features can be inferred from the results plotted in Figs. 7a, 7b, and 7c. It should be observed that in the four-plate configuration each plate induces a buoyant upflow that may impinge on the plates situated downstream or interact with their thermal boundary layers. The extent to which heat transfer from each plate is affected by these convective interactions can be deduced by local heat transfer coefficients reported in the figure. The outer sides of plates B are likely to be free of interactions, and heat transfer coefficient distributions are practically independent of S / H values. Conversely, the preheating of the upmoving airflow induced by plate A tends to degrade heat transfer from the inner sides of plates B and from both sides of plate C. When the interplate spacing is relatively large (namely, for S / H = 0.3, Fig. 7a), the heat transfer coefficients along the inner
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a
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b
c
Figure 7. Profiles of T - T= for natural convection from vertical plate arrays having different interplate distances. Each plate dissipates the same thermal power (6.5 W). Solid lines, reconstructed data; blackened circles, thermocouple measurements. (a) S / H = 0.3; (b) S / H = 0.2; (c) S / H = 0.1. The value of the local heat transfer coefficient h [W/(m2K)] is indicated at different locations. The symbol CL denotes the symmetry axis of each array configuration; the vertical bars denote the temperature scale (10 K).
8 F. Devia et al. side of plates B, compared with those on the outer side at the same elevation, are reduced by over 25% at the leading edge and by 10% to 6% as one moves toward the trailing edge. Comparison of heat transfer coefficients at the leading edge of plate C with those of plates B (outer side) at similar locations shows reductions (25-30%) due to the mentioned interaction with the thermal plume from plate A. At the lowest interplate distances ( S / H = 0.2, Fig. 7b and S / H = 0.1, Fig. 7c), convective interactions among the plates become stronger. Two conflicting factors are present when the interplate spacing is reduced: (1) the preheating of the airstream flowing in the channel formed by the two facing plates B, which tends to reduce the heat transfer from the impinged surfaces, and (2) the acceleration of flow near the leading edge of plates B (inner side) and C, due to the contraction of flow passage area, which locally tends to enhance the heat transfer. The flow acceleration is responsible for the heat transfer enhancement at the leading edge of plates B (inner side) when S / H is reduced. Conversely, far from the leading edge, the preheated airflow leads to a decrease in heat transfer coefficients as S / H varies from 0.3 to 0.1. In average, the heat transfer coefficient on the inner face of plates B for S / H = 0.1 is 18% lower than that for S / H = 0.3. Considerations on heat transfer from plate C are limited to the two measurement locations (situated near the leading edge) that are plotted in the figure. Local heat transfer coefficients at the same level increase and then decrease by progressively reducing S / H from 0.3 to 0.2 to 0.1. This trend is related to the competition between the two above-mentioned conflicting factors. P R A C T I C A L S I G N I F I C A N C E AND CONCLUSION A schlieren system is used for the study of two-dimensional, laminar, natural convection thermal fields. Temperature profiles were obtained from light ray deflection measurements, properly processed by an iterative inverse technique. The main advantages of the system employed are its relative simplicity, the ease of application, the low cost, and the satisfactory accuracy of results. First, thermal fields reconstructed for a single isothermal vertical plate were found to be in good agreement with predictions of Ostrach. Then the thermal field around four staggered vertical plates was obtained. Convective interactions among the plates were identified by examining the local heat transfer coefficients, determined from temperature distributions near the heated walls. Temperature results obtained for the single-plate case and for the four-plate configuration prove the validity of the method and its applicability to the study of thermal fields in complicated buoyancy-induced flows.
NOMENCLATURE D
fl,~ Gr~ g H
plate thickness, m focal lengths, m Grashof number [ = ~ g ( T w - T~)x3/ u2], dimensionless acceleration of gravity, m / s z plate height, m
h K k L no
local heat transfer coefficient, W / ( m : . K) Gladstone-Dale constant, m3/kg thermal conductivity of the fluid, W / ( m • K) plate length, m fluid refractive index at standard temperature and pressure, dimensionless H normal coordinate, m P pressure, Pa Pr Prandtl number, dimensionless R ideal gas constant, J / ( k g • K) S distance between plates (see Fig. 2), m T absolute temperature, K T* temperature obtained by direct integration, K X, y, 2" spatial Cartesian coordinates, m Greek Symbols light ray angular deflection, rad thermal expansion coefficient of the fluid, K - l light ray displacement, m similarity variable [= (Grx/4)l/4(y/x)], dimensionless O temperature function [= ( T - T~)/(Tw - T~)], dimensionless u kinematic viscosity of the fluid, m2/s ~r standard deviation (generic) constant in Eqs. (1), (3), (8) (= - K P L f 2 / R n o ) , m 2. K ct /~ A 7/
Subscripts calculated exact measured maximum reference refers to reconstructed temperature values refers to wall conditions refers to ambient conditions A refers to light ray shift measurements
c e m max r T w
REFERENCES 1. Eckert, E. R. G., and Goldstein, R. J., Measurements in Heat Transfer, 2nd ed., McGraw-Hill, New York, 1976. 2. Tabei, K., and Shirai, H., Temperature and/or Density Measurements of Asymmetrical Flow Fields by Means of the Moir6Schlieren Method (Method of Transformation of Moir6 Data and Its Application to the Temperature Measurement of a Combustion Flame from a Rectangular Burner), JSME Int. J,, Ser. 11, 33, 249-255, 1990. 3. Kastell, D., Kihm, K. D., and Flechter, L. S., Study of Laminar Thermal Boundary Layers Occurring Around the Leading Edge of a Vertical Isothermal Wall Using a Specklegram Technique, Exp. Fluids, 13, 249-256, 1992. 4. Settles, G. S., Colour-Coding Schlieren Techniques for the Optical Study of Heat and Fluid Flow, Int. J. Heat Fluid Flow, 6, 3-15, 1985. 5. Vasil'ev, L. A., Schlieren Methods, Keter, New York, 1971. 6. Ostraeh, S., An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force, NACA Rep. 1111, 1953. 7. Moffat, J. R., Describing the Uncertainties in Experimental Results, Exp. Thermal Fluid Sci., 1, 3-17, 1988.
Thermal Field in Buoyancy-Induced Flow 8. Schmidt, E., and Beckmann, W., Das Temperatur- und Geschwindigkeitsfeld von einer W~irme abgebenden senkrechten Platte bei natiirlicher Konvektion, Forsch-lng.-Wes., 1, 391-406, 1930. 9. Eckert, E. R. G., and Soehngen, E. E., Studies on Heat Transfer in Laminar Free Convection with the Zehnder-Mach Interferometer, USAF Rep. 5747, 1948.
9
10. Sparrow, E. M., and Gregg, J. L., The Variable Fluid-Property Problem in Free Convection, Trans. ASME, 80, 879-886, 1958. 11. Tanda, G., Natural Convection Heat Transfer From a Staggered Vertical Plate Array, ASME J. Heat Transfer (in press).
Received March 22, 1993; revised August 2, 1993
G. Milano G. Tanda Dipartimento di Ingegneria Energetica, Universith di Genova, Genova, Italy
• The aim of this work is to perform a quantitative experimental investigation of two-dimensional buoyancy-induced flows using a schlieren method. The schlieren technique uses the refraction of light rays to display a pattern related to the temperature field. The employed schlieren system makes use of an opaque filament to identify regions of fluid deflecting light rays by the same amount. An iterative inverse technique is developed to reconstruct the temperature field from optical data. After a description of the experimental technique and the data processing features, results concerning the temperature profile reconstruction for natural convective flows are presented. Attention is devoted to air buoyancy flows induced by a single heated vertical plate and by an array of four heated vertical plates. Reconstructed temperature data are compared with theoretical predictions and with direct temperature measurements performed by thermocouples. Keywords: schlieren method, natural convection, thermal field, inverse solution
INTRODUCTION In recent years the development of optical techniques has made it easier to measure the temperature distribution in flow fields. The basic concept of the schlieren technique is to record the deflection undergone by a light ray as it passes through a fluid region characterized by refractive index inhomogeneities. Since the light deflections are related to refractive index gradients and thus to the thermal field, the measurements can be processed to reconstruct the temperature profiles in the visualized region of fluid. A large variety of schlieren methods for quantitative evaluations have been described in the literature (knife-edge [1], Ronchi [1], moir6 [2], specklegram [3], color-coding systems [4], etc). In this work, light deflection measurements are performed by using an opaque filament that can be shifted in the focal plane of the schlieren head. A broad description of the focal filament method for establishing curves of equal angles of light deviation is given in Ref. 5. The schlieren system employed here is relatively simple (neither laser source nor light intensity measurements are required), straightforward to align, easy to operate, and inexpensive. The distribution of the light deviations obtained for a given experimental run is then processed by an iterative inverse technique to reconstruct the thermal field. The solution of the nonlinear inverse problem is obtained by minimization of the squared error function of light deflection residuals, using the OLS (ordinary least squares) estimation method coupled to the Gauss minimization algorithm. This technique is applied to the study of two-dimensional laminar natural convection of air flowing on heated
vertical plates. In the first case, the thermal field around a single, isolated, isothermal vertical plate was analyzed, and the reconstructed temperature profiles were compared with Ostrach's theoretical predictions [6]. In a second case, attention was turned to a more complicated plate configuration, consisting of a staggered array of four vertical plates. The latter configuration can simulate either heat exchangers with staggered fins or electronic devices cooled by natural convection of air. Temperature profiles were obtained when the plates were heated equally and arranged at different interplate spacings. APPARATUS The schlieren system is graphically shown in Fig. la. A noncoherent light beam coming from a slit source is collimated by the concave mirror M 1 to form parallel light rays. Owing to the inhomogeneities of the fluid refractive index in the test section, the light rays are deflected and focused by the concave mirror M 2 onto different positions in its focal plane. All rays are then projected on the screen or on a camera. Regions of the test section having the same light angular deflection a (or the same light displacement A) can be identified by using an opaque filament (a thin dark strip) in the focal plane of mirror M 2 (focal filament method [5]). When a disturbed light ray is stopped by the focal filament, the image of the corresponding region of the test section will appear dark on the screen and the remaining field will be bright. The shift A of the disturbed ray in the focal plane of mirror M 2 can be recorded by measuring the distance between the middle of the undisturbed image of the slit source and the centerline of the filament, as shown in Fig. lb. Therefore,
Address correspondence to Dr. Ing. Giovanni Tanda, Dipartimento Ingegneria Energetica, Universith di Genova, via all'Opera Pia 15/a, 1-16145 Genova (Italy).
Experimental Thermal and Fluid Science 1994; 8:1-9 © 1994 by Elsevier Science Inc., 655 Avenue of the Americas, New York, NY 10010
0894-1777/94/$7.00
2
F. l)cvia ct al. focal filament
f-J / ~
""-,.~
mirror M 2
/ ~
,~ ~/~ndisturbed ray
///~---J~\
~
test section
disturbed raYment
~
"'1~
light source a
b
Figure 1. (a) Schlieren system (top view). M 1 and M 2 a r e 0.38 m diameter concave mirrors (with focal lengths fl and J), respectively), a is the angular deflection of the light ray, and A is the related displacement at the focal plane of mirror M2 (fl = f2 = 1.9 m, 3' = 9°, distance between mirrors about 8 m). (b) Measurement of light displacement A at the focal plane of mirror M2; the focal filament is shifted from position 1 to position 2. moving the filament in the direction of deflections allows one to define the locus of the images of points that deflect light by the same amount. If the thermal field is assumed to be two-dimensional (i.e., temperature is independent of z coordinate), the relation between the light ray shift A and the local temperature gradient can be easily obtained from the equation [1] A = ~(3T/Oy)/T
2
(1)
where T is the absolute temperature and y is the direction in which the light deflection is recorded. The constant 1~ = - K P L f z / R n o was equal to -0.0456 m 2. K in the present experiment. (K is the Gladstone-Dale constant, P the pressure, L the length of the disturbed optical path, f2 the focal length of the schlieren head, R the ideal gas constant, and n 0 the fluid refractive index at reference conditions.) A schematic view of the test section is presented in Fig. 2. Four vertical plates, suspended by nylon lines, were affixed at three different levels in the vertical coordinate. The symbols A, B, and C are assigned to denote plates located at the lower, middle, and upper stages, respectively. Each plate was made of two aluminum layers with a plane electrical heater sandwiched between them. The dimensions of each plate are: height H = 65 mm, thickness D = 8 mm, and length L = 300 mm (in the direction of the undisturbed light rays). Dimensions were chosen to favor a two-dimensional thermal field around the heated plates. Surface temperature was measured by means of 10 fine-gauge thermocouples (type K, calibrated to _+0.1 K) embedded in the material. Due to the high thermal conductivity of aluminum, each plate, heated by a given amount of power input, reached a uniform temperature over its entire surface. At the steady state, a plate temperature uniformity generally within + 1% of the mean wallto-fluid temperature difference was attained for the spatially distributed thermocouples. Due to the low thermal emittance of polished aluminum, the heat transfer by radiation was reduced to a small value and the thermal power was mainly exchanged by natural convection be-
X
tdy
SO
Figure 2. Schematic view of the test section (H, D, and L denote the height, thickness, and length of each plate, respectively; S is the interplate spacing, and g is the gravity vector).
tween the exposed surfaces of the plates and the surrounding air. It is worth noting that measurements of radiation heat transfer are not required for the thermal field reconstruction by the schlieren method. The ambient air temperature was measured by five shielded thermocouples deployed below the plate assembly. In some experimental runs a number of fine-gauge, type K thermocoupies were located in different positions near the heated plate assembly to enable a comparison of temperature measurements with the reconstructed temperature profiles in the fluid. Two different arrangements of plates were considered. In the first case, experiments were conducted for a single heated vertical plate. A second set of experiments were performe d in the presence of all four vertical heated plates. In the latter case, each plate of the assembly dissipated the same thermal power. The interplate spacing S (see Fig. 2) was parametrically varied to obtain values of
Thermal Field in Buoyancy-Induced Flow 3 the spacing-to-height ratio ( S / H ) ranging between 0.3 and 0.1. For both of the arrangements considered (single-plate and four-plate assembly), experimental runs consisted in (1) heating the single plate or the plate assembly (12-24 h were allowed for the attainment of steady-state conditions), (2) measuring wall and ambient air temperatures, (3) visualizing the regions at equal light deviation, and (4) processing the schlieren images to reconstruct temperature profiles. Schlieren photographs were obtained with the focal filament set in the vertical direction and at different distances from the undisturbed image of the slit source. The location of the curves at equal light deflection (called isodeflection lines) was obtained by identifying, for a set of photographs, the coordinates of the filament shadow centerlines. The filament was moved only in the y direction (orthogonal to the vertical surfaces of the plates) along which the largest temperature gradients were expected. It should be noted that temperature gradients in the x direction, even though not directly measured, are not neglected; their effect is included in the horizontal light deflection measured values. The technique employed to reconstruct the thermal field from optical data is outlined in the following section.
assuming that the unknown temperature distribution T(y) along a line at constant x is represented by a proper class of functions. Two kinds of parameterization were developed and tested: a local second-order, three-points polynomial approximation (moving parabola) and the cubic B-splines approximation. The results of several simulated experiments showed that both kinds of parameterization give a good approximation of temperature distribution in buoyancy-induced flows but that better accuracy and smoothness can generally be obtained by the use of the cubic B-splines approximation. This latter parameterization allows us to approximate the temperature profile by the transformations T = T1 + (T2 - T1)u3/6
for Yl < y < (5y 1 + y 2 ) / 6
(7a) T= [Tl(5-3u-
3u z + 2 u 3)
+ T2(1 + 3u + 3u z - 3u 3) + T3u3]/6 for (5y 1 + y 2 ) / 6 _
T = [Ti_3(1 - 3u + 3u 2 - u 3) + T/_2(4 - 6u 2 + 3u 3) + T/_I(I + 3u + 3u 2 - 3U3) + T/u3]/6
DATA PROCESSING
for ( Y i - 3 + 4yi-2 + Yi-1)/6
If the measured and calculated displacements of light rays are denoted by Am and A c, for the solution of the inverse problem in the OLS formulation it is necessary to find a vector of unknown temperature T = {T/}, i = 1, N, that minimizes the sum
< Y < (Yi 2 + 4yi-I + Yi)/6
S
=
[A m
--
Ac(Z)]t[Am
-- Ac(t)] = min
(2)
where N is the total number of measured light deflections and [ ]t denotes the transposition operator. The calculated displacement is given by At(T) = 1 2 ( O T / a y ) / T z
(3)
and the unknown vector of parameters T is subjected to some constraints by using appropriate thermal boundary conditions, which depend on the kind of each convective experiment. As the calculated function Ac is a nonlinear function of T, the minimization process has been constructed with successive approximations, recursively using the linearized Gauss algorithm. In vectorial form the iterative process at the generic iteration k + 1 can be written T k + 1 = T k __ 1 p k grad{st,} (4) and stops when the minimum values of the target functional s is reached. In Eq. (4), the gradient of s and the var/covar matrix P at the generic iteration k can be determined as follows: grad{s} =
-2X t. [A m -
P = [X'. X ] - '
Ac]
(5)
(7"o)
(7C)
T = [TN_2(1 -- 3u + 3u 2 -- u 3) + TN_ 1(4 -- 6u 2 + 3u 3) + TN(1 + 3u + 3u 2 - 2u3)]/6
(7d)
for (YN-2 + 4YN-1 + Y N ) / 6
(7e)
f°r (YN-1 + 5YN)/6
(6)
where X is the sensitivity matrix of A c with respect to the parameters {T,.}at the measuring points Yi, i = 1, N. It can be seen from Eq. (3) that in order to calculate the function Ac, temperature and temperature gradients of the fluid must be evaluated. This can be accomplished by
-1/T+l/Ty=o=(foYA(y)dy)/12
(8)
If the function A(y) is known with high precision and at several points close to each other, Eq. (8) in principle provides the solution with the best accuracy, because no
4
F. Devia et al.
assumption at all is made on the function T ( y ) . The only (numerical) error, due to the kind of discrete formula a d o p t e d for integrating Eq. (8), can be reduced to a negligible quantity. If, on the other hand, the values of A(y) are p e r t u r b e d by noise, as always occurs in experimental tests, a n d / o r data are available at a limited number of points, the accuracy of the direct integration technique decreases and tends to be comparable to or worse than that given by the iterative solution of Eq. (4). In general, the latter should be p r e f e r r e d for real applications because it gives not only the unknown t e m p e r a t u r e but also the confidence-bound estimate of each temperature value calculated. This information may be vital in processing noisy measurements. Owing to the importance of the influence of measurement errors on the reconstructed t e m p e r a t u r e field, a detailed analysis on this topic has been performed, and several results of simulated experiments are r e p o r t e d here. In Table 1 the values of light shift m e a r e calculated from the t e m p e r a t u r e distribution Te given by the exact Ostrach solution for a single vertical plate at uniform wall temperature T w = 354.6 K (81.45°C) cooled by laminar air free convection at the t e m p e r a t u r e T~ = 293.6 K (20.45°C), at the vertical elevation of 5 mm from the leading edge of the plate. The input h e values are processed by using two methods: (1) the iterative inverse technique {Eqs. (2)-(7)] and (2) the direct integration technique [Eq. (8)]. The t e m p e r a t u r e values reconstructed by means of the two above methods are designated T and T*, respectively. The
2~r confidence bounds ( 2 0 / 1 odds) for T and 7"* values are r e p o r t e d in the case of artificially perturbed A input data. The perturbation, which simulates random measurement errors, was obtained by superposing upon the exact values of A e a white, zero-mean, Gaussian noise with a standard deviation o:,± such that 2~a = X A .... ax- Three values of X were tested: X = 0.05, 0.10, and 0.20: it can be noted that the intermediate value X = 0.10 can be considered representative of the effective experimental errors associated with the light shift measurement and estimated according to the procedure outlined by Moffat [7]. The exact (Ostrach) t e m p e r a t u r e values 7 I, are compared at first with those reconstructed by the iterative Gauss technique ( T ) and by direct integration (T*) in the case of exact, unperturbed input data (2O'a = 0). As can be seen, both the inverse solution based on the cubic B-splines approximation and the direct integration give excellent results. The analysis of the solution with p e r t u r b e d input data (2~ra = X A . . . . . ) was p e r f o r m e d by the Monte Carlo technique (2000 runs for each set of data). A n ensemble of estimation errors e~ = T - T e was generated, where T e is the vector of exact (simulated) values and T (equal to T or T*) is the vector of estimated values. The variance and the corresponding standard deviation of the vector e T for the two above-mentioned reconstruction techniques were calculated (by a sample statistics analysis) and compared. As shown in Table 1, the 2trT. confidence bound of the direct integration values is slightly better for the half of
Table 1. Results of a Simulated Experiment: Comparison Between Exact (T e) and Reconstructed (T and T*) Temperature Distributions Using the Theoretical Solution of Ostrach [6] as Input Data a
y
(m)
(m)
T,, (°C)
2 ~ra = 0 T (°C)
0.00 0.27 0.54 0.82 1.09 1.36 1.63 1.91 2.18 2.45 2.72 2.99 3.27 3.54 3.81 4.08 4.36 4.63 5.17 5.72 6.26 6.81 7.35 8.43 9.94
8.20 8.48 8.68 8.74 8.62 7.76 7.08 6.31 5.49 4.69 3.93 3.25 2.65 2.14 1.71 1.36 1.07 0.66 0.40 0.24 0.14 0.09 0.03 0.01 0.00
81.45 75.30 69.18 63.18 57.41 51.98 46.99 42.50 38.57 35.18 32.32 29.94 27.98 26.40 25.13 24.12 23.31 22.68 21.79 21.26 20.93 20.73 20.62 20.51 20.45
81.45 75.30 69.18 63.18 57.41 51.97 46.99 42.50 38.57 35.17 32.32 29.93 27.99 26.39 25.13 24.11 23.31 22.68 21.77 21.26 20.91 20.74 20.61 20.52 20.45
×
10 3
A e ×
10 3
T* (°C)
20-a = 0.05A e, max 20 r 2~rr, (°C) (°C)
20-a = 0.10A e. max 2tr r 2trr, (°C) (°C)
20;a = 0.20A e, max 2~rr 2tr r, (°C) (°C)
81.45 75.30 69.18 63.18 57.41 51.98 46.98 42.50 38.56 35.18 32.32 29.94 27.98 26.40 25.13 24.11 23.31 22.68 21.79 21.25 20.93 20.73 20.61 20.50 20.45
0.00 0.23 0.43 0.49 0.57 0.61 0.66 0.67 0.70 0.72 0.74 0.76 0.78 0.80 0.81 0.83 0.84 0.85 1.09 0.92 1.13 0.95 1.05 0.78 0.00
0.00 0.46 0.85 0.96 1.14 1.20 1.30 1.33 1.39 1.43 1.47 1.51 1.54 1.59 1.60 1.65 1.65 1.67 2.16 1.81 2.24 1.89 2.08 1,54 0.00
0.00 0.92 1.69 1.93 2.27 2.40 2.61 2.67 2.79 2.85 2.94 3.01 3.07 3.17 3.21 3.30 3.31 3.35 4.32 3.63 4.48 3.78 4.15 3.07 0.00
0.00 0.22 0.37 0.47 0.54 0.59 0.64 0.67 0.70 0.73 0.75 0.77 0.80 0.83 0.86 0.89 0.91 0.94 1.02 1.11 1.20 1.27 1.34 1.57 1.96
0.00 0.44 0.74 0.92 1.06 1.17 1.25 1.31 1.37 1.43 1.48 1.53 1.58 1.64 1.70 1.75 1.80 1.87 2.01 2.20 2.37 2.52 2.65 3.11 3.88
aThe 20" confidence bounds are reported for both temperature reconstruction methods, using artificially perturbed A input data.
0.00 0.87 1.48 1.85 2.12 2.33 2.49 2.62 2.73 2.85 2.95 3.05 3.16 3.27 3.39 3.50 3.60 3.73 4.02 4.40 4.74 5.04 5.31 6.21 7.76
Thermal Field in Buoyancy-Induced Flow 5 the points closer to the heated wall. Conversely, the 2(r r confidence bound of the inverse technique values is better for the other half of the points, owing to the second boundary condition (T = Too far from the heated wall). In this regard it can be observed that the error affecting the temperature evaluated by the direct integration technique can be very large for the points closest to the fluid at rest. Another important conclusion that can be drawn from Table 1 is the proportionality between o-a and trr (or crr.)--doubling the standard deviation of the noise affecting the measured data doubles the standard deviation of reconstructed temperature values. This behavior remains valid for values of tra larger than those reported in Table 1. This means that the inverse method here implemented gives "stable" solutions and no "regularization" technique is needed. The results of Table 1 refer to the temperature reconstruction field in the case of a relatively large set of data points (25). In Table 2 the same thermal problem is solved for a set of only seven points perturbed by a noise with 2tra = 0.10 Ae, max. The results reported in Table 2 show that when the measured input data are few and are affected by noise, the inverse technique should be preferred because the probable errors affecting the reconstructed temperature values are less everywhere (2~rr < 20"r.). It should be noted that the error predicted by the inverse technique at the first and last points is zero, because at these two points the temperature values are known (boundary conditions) and therefore the associated sensitivity coefficients and variances are zero too. On the other hand, the effect of errors in the wall and ambient fluid temperature measurements can be easily taken into account. In conclusion, when T~ and T~ measurement errors are also considered and their interaction with other errors (introduced by the mathematical model and by light-shift A measurements) is taken into account, the overall experimental uncertainty (at the 95% confidence level) in T values, reconstructed from a set of 25 input data points, varies between 2% and 4% of T~ - T~.
voted at first to the case of a single vertical plate, for which the development of a temperature profile in laminar regime is theoretically known and has been experimentally investigated in several papers [e.g., 3, 8, 9]. Figure 3 shows a typical example of a schlieren photograph corresponding to a given focal filament displacement. From a set of such photographs, the contours of a number of isodeflection lines were obtained as reported in Fig. 4. It should be observed that during the reconstruction process a larger number of lines were used than are plotted in the figure. Some results are presented in Fig. 5 in terms of the dimensionless temperature 0 = ( T - T o o ) / ( T ~ - T ~ ) as a function of the dimensionless similarity variable 77 = (Grx/4)l/4y/x. Measured data refer to two values of T w - T~ (20.9 and 61 K) and to three x elevations (25, 35, and 45 mm from the leading edge). The reference temperature T r = T w - 0.38(T~ - To~) suggested in Ref. 10 was used to evaluate the kinematic viscosity of air, while the thermal expansion coefficient was evaluated at T~. The solid line refers to the theoretical prediction of Ostrach [6] for a single isothermal vertical plate in the laminar regime (Pr = 0.72). Also presented is a set of data obtained by Kastell et al [3] by a specklegram technique for T~ - T~o = 61.2 K and x = 30.8 and 50 mm. Inspection of the figure shows good agreement of all the measured data with the theoretical profile. A comparison of present data with those reported in Ref. 3 reveals slight discrepancies, limited to the regions far from the vertical wall, probably due to the uncertainties associated with the measurement methods and with the algorithms used to obtain temperature values. In the second phase, the thermal field near a four-vertical-plate assembly was investigated. This configuration (schematically depicted in Fig. 2) may simulate either
TEMPERATURE FIELD RECONSTRUCTION F O R LAMINAR NATURAL C O N V E C T I O N FLOWS In order to assess the accuracy of the optical method and that of the data processing procedure, attention was de-
AE
Q
m
Q
?
2. Results of a Simulated Experiment--Comparison Between 2 (r Confidence Bounds for Temperature Reconstructed by the Inverse Technique (T) and Direct Integration (T*) a
Table
y × 103 (m)
Ae X 103 (m)
Te (°C)
2tr T (°C)
2o-,r, (°C)
0.00 1.09 2.18 3.27 4.36 6.26 9.94
8.20 8.62 5.49 2.65 1.07 0.14 0.00
81.45 57.41 38.57 27.98 23.31 20.93 20.45
0.00 1.79 2.76 2.84 3.16 3.35 0.00
0.00 1.90 3.09 3.34 3.62 4.45 6.95
~A set of seven input data (with 2tra uncertainty equal to 0.10Ae,max) is considered.
®
Figure 3. Photograph of a schlieren image. Natural convection on a single heated vertical plate. 1, Heated plate; 2, filament shadow; 3, check pins; 4, nylon lines supporting the plate; 5, electric wires; 6, thermocouple wires; 7, two-dimensional calibrated grid.
6
F. Dcvia et al. heaters embedded in the plates. It can be noted that the isodeflection profiles (Fig. 6) are more complicated than in the single-plate case, and they may present change of sign, minimum, maximum, and inflection points. The reconstructed fluid temperature profiles are displayed for three experiments in Figs. 7a, 7b, and 7c. The power input to each plate was 6.5 W, 90% of which was expected to be exchanged by natural convection at the exposed surfaces, while the remaining 10% is transferred to the surroundings by thermal radiation. The spacing parameter S/H was set equal to 0.3 (Fig. 7a), 0.2 (Fig. 7b), and 0.1 (Fig. 7c). In order to favor a compact presentation of results, only a limited region of the visualized thermal field is plotted. Continuous lines represent reconstructed T - T.~ profiles at different elevations, namely at 5 mm from the trailing edge of plate A; at 5, 15, 25, 35, 45,
A.103 ,m
6177 1
1011'11
I 2 3 4 5
0.3 0.8 1.2 1.9 2.5
6 7 8 9 10
-0.3 -0.8 -I .2 -I .9 -2.5
I °.211
1~III115
Figure 4. Curves of equal light displacement A for natural convection on a single heated vertical plate (Tw - T~ = 20.9 K). compact heat exchangers with staggered fins or natural convection air-cooled electronic devices. For the sake of brevity, only a limited number of experiments are shown. A more detailed analysis for this natural convection system is presented in Ref. 11. Experiments were conducted by delivering an equal amount of electric power to the
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Figure 5. Dimensionless temperature O profiles as a function of similarity variable -q, for free convection on a vertical plate. Comparison between present data ( 0 , I , v , O, D, v), measured data from Ref. 3 (*, "k), and the predicted values by Ostrach [6] (solid lines).
Figure 6. Curves at equal light displacement A for natural convection on a vertical plane assembly. Each plate dissipates the same thermal power (6.5 W). S/H = 0.1.
Thermal Field in Buoyancy-Induced Flow 7 and 55 mm from the leading edge of plate B (both sides); and at 5 and 20 mm from the leading edge of plate C. In addition, direct air temperature measurements obtained by thermocouples placed in several locations are reported (using blackened circles) for S / H = 0.1 and 0.3. Near each temperature profile, the corresponding heat transfer coefficient h at the wall is indicated. This quantity was easily computed by the relationship ~T) k~ ~ wT,~- T~
h = -
(9)
where (OT/On),, is the temperature gradient along the normal to plate surfaces (evaluated at the wall), Tw is the uniform wall temperature, and k w is the air thermal conductivity (evaluated at the corresponding wall temperature). Temperature drops Tw - Too between walls and air ranged from 21.5 to 27 K depending on S / H value and on the specific location of each plate. As expected, the highest temperature gradients are present near the leading edge of the walls. Downstream of each plate, the air temperature is increased owing to the contact of the buoyant flow with the heated walls. In these
regions (called thermal plumes or wakes) the flow showed small instabilities as pointed out by thermocouple measurements. This phenomenon introduced additional uncertainties in the optical measurements, and it partially explains differences between reconstructed and thermocouple data (3 K at the most, ie, about 14% of wall-to-ambient mean temperature drop). Further interesting features can be inferred from the results plotted in Figs. 7a, 7b, and 7c. It should be observed that in the four-plate configuration each plate induces a buoyant upflow that may impinge on the plates situated downstream or interact with their thermal boundary layers. The extent to which heat transfer from each plate is affected by these convective interactions can be deduced by local heat transfer coefficients reported in the figure. The outer sides of plates B are likely to be free of interactions, and heat transfer coefficient distributions are practically independent of S / H values. Conversely, the preheating of the upmoving airflow induced by plate A tends to degrade heat transfer from the inner sides of plates B and from both sides of plate C. When the interplate spacing is relatively large (namely, for S / H = 0.3, Fig. 7a), the heat transfer coefficients along the inner
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Figure 7. Profiles of T - T= for natural convection from vertical plate arrays having different interplate distances. Each plate dissipates the same thermal power (6.5 W). Solid lines, reconstructed data; blackened circles, thermocouple measurements. (a) S / H = 0.3; (b) S / H = 0.2; (c) S / H = 0.1. The value of the local heat transfer coefficient h [W/(m2K)] is indicated at different locations. The symbol CL denotes the symmetry axis of each array configuration; the vertical bars denote the temperature scale (10 K).
8 F. Devia et al. side of plates B, compared with those on the outer side at the same elevation, are reduced by over 25% at the leading edge and by 10% to 6% as one moves toward the trailing edge. Comparison of heat transfer coefficients at the leading edge of plate C with those of plates B (outer side) at similar locations shows reductions (25-30%) due to the mentioned interaction with the thermal plume from plate A. At the lowest interplate distances ( S / H = 0.2, Fig. 7b and S / H = 0.1, Fig. 7c), convective interactions among the plates become stronger. Two conflicting factors are present when the interplate spacing is reduced: (1) the preheating of the airstream flowing in the channel formed by the two facing plates B, which tends to reduce the heat transfer from the impinged surfaces, and (2) the acceleration of flow near the leading edge of plates B (inner side) and C, due to the contraction of flow passage area, which locally tends to enhance the heat transfer. The flow acceleration is responsible for the heat transfer enhancement at the leading edge of plates B (inner side) when S / H is reduced. Conversely, far from the leading edge, the preheated airflow leads to a decrease in heat transfer coefficients as S / H varies from 0.3 to 0.1. In average, the heat transfer coefficient on the inner face of plates B for S / H = 0.1 is 18% lower than that for S / H = 0.3. Considerations on heat transfer from plate C are limited to the two measurement locations (situated near the leading edge) that are plotted in the figure. Local heat transfer coefficients at the same level increase and then decrease by progressively reducing S / H from 0.3 to 0.2 to 0.1. This trend is related to the competition between the two above-mentioned conflicting factors. P R A C T I C A L S I G N I F I C A N C E AND CONCLUSION A schlieren system is used for the study of two-dimensional, laminar, natural convection thermal fields. Temperature profiles were obtained from light ray deflection measurements, properly processed by an iterative inverse technique. The main advantages of the system employed are its relative simplicity, the ease of application, the low cost, and the satisfactory accuracy of results. First, thermal fields reconstructed for a single isothermal vertical plate were found to be in good agreement with predictions of Ostrach. Then the thermal field around four staggered vertical plates was obtained. Convective interactions among the plates were identified by examining the local heat transfer coefficients, determined from temperature distributions near the heated walls. Temperature results obtained for the single-plate case and for the four-plate configuration prove the validity of the method and its applicability to the study of thermal fields in complicated buoyancy-induced flows.
NOMENCLATURE D
fl,~ Gr~ g H
plate thickness, m focal lengths, m Grashof number [ = ~ g ( T w - T~)x3/ u2], dimensionless acceleration of gravity, m / s z plate height, m
h K k L no
local heat transfer coefficient, W / ( m : . K) Gladstone-Dale constant, m3/kg thermal conductivity of the fluid, W / ( m • K) plate length, m fluid refractive index at standard temperature and pressure, dimensionless H normal coordinate, m P pressure, Pa Pr Prandtl number, dimensionless R ideal gas constant, J / ( k g • K) S distance between plates (see Fig. 2), m T absolute temperature, K T* temperature obtained by direct integration, K X, y, 2" spatial Cartesian coordinates, m Greek Symbols light ray angular deflection, rad thermal expansion coefficient of the fluid, K - l light ray displacement, m similarity variable [= (Grx/4)l/4(y/x)], dimensionless O temperature function [= ( T - T~)/(Tw - T~)], dimensionless u kinematic viscosity of the fluid, m2/s ~r standard deviation (generic) constant in Eqs. (1), (3), (8) (= - K P L f 2 / R n o ) , m 2. K ct /~ A 7/
Subscripts calculated exact measured maximum reference refers to reconstructed temperature values refers to wall conditions refers to ambient conditions A refers to light ray shift measurements
c e m max r T w
REFERENCES 1. Eckert, E. R. G., and Goldstein, R. J., Measurements in Heat Transfer, 2nd ed., McGraw-Hill, New York, 1976. 2. Tabei, K., and Shirai, H., Temperature and/or Density Measurements of Asymmetrical Flow Fields by Means of the Moir6Schlieren Method (Method of Transformation of Moir6 Data and Its Application to the Temperature Measurement of a Combustion Flame from a Rectangular Burner), JSME Int. J,, Ser. 11, 33, 249-255, 1990. 3. Kastell, D., Kihm, K. D., and Flechter, L. S., Study of Laminar Thermal Boundary Layers Occurring Around the Leading Edge of a Vertical Isothermal Wall Using a Specklegram Technique, Exp. Fluids, 13, 249-256, 1992. 4. Settles, G. S., Colour-Coding Schlieren Techniques for the Optical Study of Heat and Fluid Flow, Int. J. Heat Fluid Flow, 6, 3-15, 1985. 5. Vasil'ev, L. A., Schlieren Methods, Keter, New York, 1971. 6. Ostraeh, S., An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force, NACA Rep. 1111, 1953. 7. Moffat, J. R., Describing the Uncertainties in Experimental Results, Exp. Thermal Fluid Sci., 1, 3-17, 1988.
Thermal Field in Buoyancy-Induced Flow 8. Schmidt, E., and Beckmann, W., Das Temperatur- und Geschwindigkeitsfeld von einer W~irme abgebenden senkrechten Platte bei natiirlicher Konvektion, Forsch-lng.-Wes., 1, 391-406, 1930. 9. Eckert, E. R. G., and Soehngen, E. E., Studies on Heat Transfer in Laminar Free Convection with the Zehnder-Mach Interferometer, USAF Rep. 5747, 1948.
9
10. Sparrow, E. M., and Gregg, J. L., The Variable Fluid-Property Problem in Free Convection, Trans. ASME, 80, 879-886, 1958. 11. Tanda, G., Natural Convection Heat Transfer From a Staggered Vertical Plate Array, ASME J. Heat Transfer (in press).
Received March 22, 1993; revised August 2, 1993