Investigation of free convection in a vertical water channel

Experimental Thermal and Fluid Science xxx (2014) xxx–xxx

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Investigation of free convection in a vertical water channel Mario Misale, Marco Fossa, Giovanni Tanda ⇑ DIME, Università degli Studi di Genova, Genova, Italy

a r t i c l e

i n f o

Article history: Available online xxxx Keywords: Free convection Vertical channel Schlieren Water flow

a b s t r a c t Natural convection in an asymmetrically heated, vertical channel was studied both experimentally and computationally. The experiments were performed in water for an aspect ratio of the vertical channel (ratio between spacing and height) equal to 0.1. The schlieren technique was used to obtain the local heat transfer coefficient; this work seemingly represents the first attempt to apply this optical method to measure the local heat transfer coefficient using water as convective fluid. Numerical simulations demonstrated that thermal and fluid flow fields can be regarded as two-dimensional inside the channel and that the heated wall of the channel is practically isothermal. The good agreement between experimental and numerical distributions of the heat transfer coefficient encourages the use of the schlieren technique for the study of free-convection heat transfer in water flows. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Natural convection in open-ended vertical channels is encountered in many practical applications and has been extensively investigated in the past by many authors (e.g., [1–5]). Literature papers, however, are typically restricted to natural convection with air as convective fluid while experiments performed in water are relatively rare. Sparrow and co-workers [6–8] investigated the natural convection heat transfer in water vertical channels with one or two walls heated, taking into account the effects of the channel aspect ratio and channel inclination. A standard heating-foils/thermocouples technique was used to deduce the average heat transfer coefficient from the readings of the electrical power dissipated and the thermocouple array deployed into the wall material and in the fluid. More recent investigations of the water free convection in vertical channels are presented in [9,10] for asymmetric and symmetric heating conditions, respectively; again, heat transfer characteristics were provided on an average basis. This paper presents measurements of the local heat transfer coefficient along the heated wall of a vertical channel using water as working fluid. The channel is formed by an electrically heated plate bounded by two unheated walls to form two identical, oneside heated, vertical channels open at the top and bottom to permit the natural circulation of the convective fluid (water). A schlieren technique has been employed to reconstruct the distribution of ⇑ Corresponding author. Address: DIME/MASET, Università degli Studi di Genova, via Montallegro 1, I-16145 Genova, Italy. Tel.: +39 0103532557; fax: +39 0103532566. E-mail address: [email protected] (G. Tanda).

the heat transfer coefficient. This optical technique has been extensively used for the measurement of heat transfer coefficients in air (e.g., [11–14]), whereas its use in water was restricted in the past to qualitative observations like flow visualization (see, e.g., [15]). 2. The experiment 2.1. The test section The schematic view of the test section utilized in the experiments is shown in Fig. 1a. It basically consisted in two adjacent, identical, asymmetrically heated, vertical channels placed inside a tank filled with water. The heated wall of the channels was made of two thin sheets of chrome-plated copper with an electric foil heater sandwiched between them. The two copper sheets were sealed with a waterproof cement to prevent any contact between the heater and the fluid. When electrical power was delivered to the resistance, owing to the high thermal conductivity of copper, the heated plate attained a uniform surface temperature at steady state. The dimensions of the heated plate were the following: height H = 87 mm, length L = 48 mm, overall thickness t = 8 mm. The heated plate was bounded by two 5 mm-thick shrouding walls, smooth and unheated, made of bakelite. The spacing S between each unheated wall and the heated plate, set equal on both sides, was 8.7 mm, thus corresponding to a channel aspect ratio S/H equal to 0.1. The symmetrical arrangement of the heated plate/shrouding walls assembly allows the optical measurements to be repeated on both sides and, owing to the symmetry, averaged at the same elevation, thus reducing the experimental error. The heated plate/shrouding walls assembly

http://dx.doi.org/10.1016/j.expthermflusci.2014.01.022 0894-1777/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: M. Misale et al., Investigation of free convection in a vertical water channel, Exp. Therm. Fluid Sci. (2014), http:// dx.doi.org/10.1016/j.expthermflusci.2014.01.022

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Nomenclature f2 H h k L n nair n0 S T t x, y, z

focal length of the schlieren head (m) channel height (m) local convective heat transfer coefficient (W/m2 K) fluid thermal conductivity (W/m K) plate length (m) refractive index of water refractive index of ambient air refractive index of water at a standard condition channel spacing (m) temperature (°C) or (K) heated plate thickness (m) spatial Cartesian coordinates (m)

Greek symbols a, a0 light ray angular deflection (rad) D light ray displacement (m) Subscripts f refers to (inlet) fluid conditions w refers to conditions at the wall y refers to the y direction

six thermocouples deployed in the plate material. The fluid temperature Tf was measured by a thermocouple located at the inlet section of one of the two identical channels; an additional thermocouple, able to travel vertically within the tank and outside the channels, was used to check the presence of undesirable water temperature stratification.

2.2. The optical arrangement

Fig. 1. Schematic drawing of (a) the test section (frontal view) and of (b) the water tank (3D view). Dimensions, in (mm), not to scale.

was suspended, by using a supporting frame and thin nylon wires, inside a chamber with inner dimensions 180  65  390 mm (width  length  height). The water chamber and the coordinate system adopted for the vertical channels are schematically reported in Fig. 1b. The chamber was filled with distilled water up to a 352 mm level from the tank inner floor and was open at the top side to provide ambient pressure conditions at the water/ air interface. The exit of the channels was situated 75 mm below the water surface and the entrance at 190 mm above the tank inner floor. The vertical sides of the chamber normal to the light beam were made of 6 mm-thick, high quality glasses so as to permit the schlieren measurements. The remaining sides and the bottom of the tank were made of 10 mm-thick chrome-plated copper, finned on the ambient air side to facilitate the dissipation of the input power to the laboratory ambient air. The heated plate and the water tank were instrumented with 0.5-mm-dia sheathed thermocouples, calibrated to ±0.05 K. Six thermocouples were embedded in the wall of the heated plate at different locations through 0.5-mm-dia holes drilled into the material as close to the exposed surfaces as possible. The wall temperature Tw was obtained by averaging the readings of the

A Z-type schlieren arrangement, schematically shown in Fig. 2, was used for the visualization of thermal field and to infer the local heat transfer coefficient at the heated wall/fluid interface. The optical principles are here briefly summarized. A non-coherent light beam from a vertical slit source, collimated by the concave mirror M1, passes through the test section. Here, the heated plate and the two unheated vertical walls are deployed with the length L aligned to the travelling light beam (i.e. along the z-coordinate of Figs. 1b and 2). A second concave mirror M2, is then used to project a real image of the slit source in the focal plane and a real image of the test section onto a screen or camera. The occurrence of thermal gradients in the test section leads to inhomogeneities of the water refractive index; as a consequence the light rays undergo angular deflections. Fig. 2 shows a light ray deflected, within the test section, by an angle ay, whose extent is related to thermal gradients, along the y direction, in the water. As the light ray emerges into the surrounding air, the deflection angle is modified, according to Snell’s law, and it becomes a0y ¼ ay ðn0 =nair Þ, where n0 (=1.335) and nair (=1.0003) are the refractive indices of water and air at ambient conditions (18–26 °C, atmospheric pressure), respectively. Regions of the optical field characterized by the same light deflection in the y–z plane can be identified by shifting an opaque vertical filament in the focal plane of mirror M2, as shown in Fig. 3 (focal filament method). When a deflected light ray is stopped by the focal filament, the image of the corresponding region of fluid will appear dark on the screen, while the remaining field will be bright. A typical example of a photograph taken by the schlieren apparatus using the focal filament method is reported in Fig. 4. The amount of the angular deflection of a disturbed ray can be deduced by measuring, in the focal plane of mirror M2, the distance Dy between the middle of the undisturbed image of the slit source and the centerline of the filament, i.e. the distance Dy between filament positions 1 and 2 displayed in Fig. 3. This distance corresponds to the displacement impressed to the filament, directly measured by a micrometer, and it is related to the local angular deflection a0y by the simple formula Dy ¼ f2 a0y where f2 is the focal length of the mirror M2 (also called the schlieren head). It is worth noting that Dy represents the light ray displacement, in the focal

Please cite this article in press as: M. Misale et al., Investigation of free convection in a vertical water channel, Exp. Therm. Fluid Sci. (2014), http:// dx.doi.org/10.1016/j.expthermflusci.2014.01.022

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Fig. 2. Schematic layout of the Z-shaped schlieren system (top view); M1 and M2 are 380 mm diameter concave mirrors (with focal lengths f1 and f2, respectively), ay is the ycomponent of the light ray angular deflection within the test section, a0y is the angular deflection of the light ray emerging into the ambient air, and Dy is the related displacement at the focal plane of mirror M2 (f1 = f2 = 1.9 m, c = 9°, distance between mirrors about 8 m).

Fig. 3. Measurement of light displacement Dy in the focal plane of schlieren mirror M2 by means of the focal filament technique.

plane of the schlieren head, associated with the angular deflection of the light ray after it has passed through the test section. Then, moving the focal filament in the focal plane, parallel to itself, allows one to measure the values of the light ray deflections (and the related light ray displacements) in the whole optical field. If the thermal field in the test section is assumed to be twodimensional (i.e. temperature is independent of the z-coordinate), it can be easily demonstrated that the local heat transfer coefficient is related to the light ray displacement by the following equation:

h¼

  kw nair Dw ðT w  T f Þ f2 Lðdn=dTÞw

ð1Þ

where T is the temperature, n is the water refractive index, L is the test section length (in the z direction), k is the water thermal conductivity, nair is the ambient air refractive index and subscripts w and f denote wall and (inlet) fluid conditions, respectively. In particular, Dw is the displacement of the light ray passing in the vicinity of the wall at the desired x-location. Details on the derivation of Eq. (1) are provided in a companion paper [16]. The derivative of n versus T has been obtained by processing data reported in [17]:

dn=dT ¼ 1:1869  105  4:4407  106 T þ 2:5  108 T 2 where the water temperature is expressed in °C.

ð2Þ

Fig. 4. A typical schlieren image recorded with the focal filament method: the filament shadow (visible in the right-hand side channel) identifies the images of points of the optical field deflecting the light, in the y direction, by the same (known) amount.

2.3. Experimental procedure Experiments were performed by delivering a given electrical power to the heater, in order to achieve the desired uniform temperature over the copper block (the heated plate). At the steady state, measurements of wall and water temperatures were obtained by the readings from the thermocouple deployed in the

Please cite this article in press as: M. Misale et al., Investigation of free convection in a vertical water channel, Exp. Therm. Fluid Sci. (2014), http:// dx.doi.org/10.1016/j.expthermflusci.2014.01.022

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material and in the fluid; then, for each run, the optical measurement of the light deflections at the vertical walls was performed in order to obtain the local heat transfer coefficients at several locations along the vertical surfaces of the heated plate. The experiments were performed by keeping the input power, delivered to the heated plate, in the 3.5–4 W range, giving a wall-to-fluid temperature difference of 1.35–1.65 K. 2.4. Experimental uncertainty The experimental uncertainty in the results was evaluated by using a root-square combination of the contributions made by the uncertainties in each of the individual measurements. The uncertainty in the local heat transfer coefficient h as measured by the schlieren method (Eq. (1)) is generally sensitive to errors in the readings of light displacement and wall and fluid temperatures. Further sources of errors are due to optics (aberrations, misalignments, etc.) and to uncertainty in optical properties of water. Taking into account all these parameters gives an uncertainty (at the 95% confidence level), in the measured h values, in the 15–18% range.

and 25 °C for test No. 2 (the different room air temperature was expected to affect the steady-state plate and water temperatures). These parameters were considered as input data for the respective numerical simulations. The same table reports measured and calculated wall-to-fluid temperature difference for each case. For the investigated range of the parameters, the values of Tw and Tf resulted, at the steady-state, between 20 and 27 °C. The selected values of the input power gave rise to small wall-to-fluid temperature differences. This circumstance was addressed by preliminary experiments that indicated values of TwTf in the 1–2 K range to facilitate the schlieren measurements with the present optical apparatus. Computed velocity and temperature fields, for a power input of 3.5 W delivered to the heater and room air temperature set at 18 °C (test conditions No. 1), are presented in Figs. 5–7. Fig. 5 illustrates the velocity module and the velocity vector in the x–y plane at half length L of the heated plate (z = L/2): as expected, an up-moving buoyant water flow is symmetrically induced inside the channels, while a down-moving flow occurs very close to the side walls of the tank. The map of velocity vector reveals a weak recirculating flow in the channel. This recirculation is situated adjacent to the unheated wall in the upper half of the channel, as shown in Fig. 6; the same phenomenon was experimentally and computationally

3. Results and discussion A numerical study of this problem was conducted to supplement the experimental work. Numerical simulations were performed by means of FloTHERMÒ [18], a commercially available, finite volume based, CFD software package. Numerical results provided also further insights concerning the flow and thermal field in the whole test assembly (channels and water tank) that cannot be directly deduced by schlieren or thermocouple measurements. The numerical solution of the present problem is relatively routine and can be tackled by a number of different approaches available in the literature. The analysis was based on a model considering the laminar, buoyancy-induced water flow in the tank coupled with the conduction in the walls (heated plate, channel unheated walls, tank walls) and with the air free convection on the external side of the tank. For this purpose, the 3D computational domain encompassed the water tank and a sufficiently large external region including the surrounding laboratory room air. A non-uniform Cartesian grid featured by more than 6  106 computational cells was adopted. Preliminary tests were conducted on a grid with a double number of cells without finding significant differences in the calculation results; the lower grid density was thus selected for all runs since it required less computing resources. For each simulation, the input variables were the heat flux uniformly dissipated by the foil heater and the air temperature of the laboratory room (where the heat flux is discharged from the water tank). The thermophysical properties of water were assumed to be constant and evaluated at the film temperature (Tw + Tf)/2 provided by the respective experiment. Moreover, thermophysical properties of materials (copper, bakelite, glass) were provided. The operating conditions for the experimental runs are given in Table 1. The input power delivered to the heated plate was set equal to 3.5 W for tests Nos. 1 and 2, and to 4 W for test No. 3. Laboratory room air temperature was 18 °C for tests Nos. 1 and 3

Table 1 Characteristics of the investigated cases. Test no.

Input power, W

Room air temp., °C

TwTf measured, K

TwTf computed, K

1 2 3

3.5 3.5 4

18 25 18

1.45 1.35 1.65

1.35 1.3 1.5

Fig. 5. Map of the velocity (top) and velocity vectors (bottom), in the x–y plane at z = L/2, for test conditions No. 1. Only data for the upper part of the tank are presented.

Please cite this article in press as: M. Misale et al., Investigation of free convection in a vertical water channel, Exp. Therm. Fluid Sci. (2014), http:// dx.doi.org/10.1016/j.expthermflusci.2014.01.022

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observed in [6] as the Rayleigh number exceeded a threshold value. A large mass of water between the exterior sides of the channels and the tank sides is featured by no motion; in this region, the isotherm contours on the same x–y plane (at z = L/2), plotted in Fig. 7, show a not-negligible thermal stratification. Indeed, the calculated temperature difference between two points, taken in the stagnant fluid and aligned to the inlet and outlet sections of the channels, respectively (i.e. at a vertical distance equal to the channel height), is 0.3 K, i.e. about 20% of the mean wall-to-fluid temperature difference. This computed result is in excellent agreement with the experimental measurements performed inside the tank by the thermocouple travelling through the fluid. The calculated distribution of wall temperature over the heat plate surface exposed to the water flow, not indicated in the figure, justified the assumption of a nearly-uniform wall temperature employed to deduce the heat transfer coefficient h from the experiment by applying Eq. (1). Moreover, thermal gradients along the direction normal to the

Fig. 6. Detail of the map of the velocity vectors, for test conditions No. 1, showing the flow reversal close to the unheated walls.

Fig. 7. Map of the fluid temperature, in the x–y plane at z = L/2, for test conditions No. 1. Only data for the upper part of the tank are presented.

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plate, and calculated at the wall-to-fluid interface, were practically insensitive (within 3%) to the z-coordinate, thus supporting the assumption of h values independent of z, as required by the experimental technique. Computed and measured heat transfer coefficients along the vertical side of the heated plate, for test No. 1, are reported in Fig. 8, where symbols refer to experimental values, obtained by applying Eq. (1), and lines refer to numerical and theoretical values. More precisely, the solid line shows the computed h values for the real experimental conditions (case A), the dashed–dotted line shows the h values computed by considering a test section of indefinite length (in the z direction) submerged in a tank of very large size in the y direction (case B), while the dotted line represents the h values given by the theoretical solution for an isolated isothermal vertical plate with Prandtl number equal to 7 (case C), reported in [19]. Comparison between case A and case B shows the effect of the thermal stratification (not occurring in simulation B), which negatively affects the heat transfer performance inside the channel, as documented in [20] where, for a stratification level equal to that for the present experiment, a 10% reduction in the mean heat transfer coefficient over an isolated, isothermal vertical surface was calculated. Comparison between case B and case C shows the effect of the presence of the frontal wall opposite to the heated plate. It is apparent that this effect is almost negligible, despite the low aspect ratio (S/H = 0.1) of the channel. In practice, h in the channel is slightly enhanced (with respect to the isolated plate) near the entrance due to the draught effect, but it is slightly reduced towards the channel outlet due to the interaction of the growing thermal boundary layer with the opposite unheated wall. In the vicinity of the exit section, the local h values for case B show a sudden marked increase, not registered by the theoretical solution (case C), which neglects the second derivative of temperature in the streamwise direction. On an average basis, the mean h value for case B is slightly higher than that for case C, in agreement with [7], where the mean h value for the channel with S/H = 0.1 was experimentally found to be higher (by 3%) than that for the isolated vertical surface (S/H ? 1). Differences between the experimental results and the numerical simulations for the real experimental conditions (case A) are well within the experimental uncertainty. Numerical and experimental h-distributions for all the three test conditions (Nos. 1–3) are plotted in Fig. 9. Minor differences in both computed and measured h profiles can be observed between test Nos. 1 and 2, where the same input power is dissipated. This finding can be ascribed to the different laboratory room

Fig. 8. Measured and numerical/theoretical distributions of the heat transfer coefficient h along the vertical side of the heated plate for test No. 1 (local h values are averaged along the z direction).

Please cite this article in press as: M. Misale et al., Investigation of free convection in a vertical water channel, Exp. Therm. Fluid Sci. (2014), http:// dx.doi.org/10.1016/j.expthermflusci.2014.01.022

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Experiments displayed a good agreement with the numerical results. References

Fig. 9. Measured and computed distributions of the heat transfer coefficient h along the vertical side of the heated plate for all test conditions (local h values are averaged along the z direction).

air temperature, which in turn affects the steady-state temperature values of water in the tank and induces slight variations of the thermophysical properties (in particular of dn/dT for water). Again, measured data show a good agreement with the numerical simulations. It is worth noting that, for a parallel investigation performed in air, the similar level of agreement was encountered between the experimental results, obtained by using the same schlieren apparatus and technique [12], and numerical predictions [21]. 4. Conclusions An experimental and numerical study of water natural convection in an open-ended, asymmetrically heated, vertical channel has been conducted. Experiments were performed for two identical vertical channels (formed by a heated plate and two unheated walls) placed inside a large tank filled with water. Care was taken to ensure a nearly two-dimensional flow and thermal field inside the channels and a uniform wall temperature condition at the heated plate surface. Numerical simulations confirmed these assumptions. Local heat transfer coefficients have been measured by using a schlieren optical technique, typically employed for experimental investigations in gases and here successfully adopted for a liquid.

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Please cite this article in press as: M. Misale et al., Investigation of free convection in a vertical water channel, Exp. Therm. Fluid Sci. (2014), http:// dx.doi.org/10.1016/j.expthermflusci.2014.01.022