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Experimental and numerical study of a small-scale and low-velocity indoor diffuser for displacement ventilation: Isothermal floor
Applied Thermal Engineering 87 (2015) 79e88
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Experimental and numerical study of a small-scale and low-velocity indoor diffuser for displacement ventilation: Isothermal floor ndez-Gutie rrez a, Ignacio Gonza lez-Prieto a, Luis Parras a, Alberto Ferna Manuel Cejudo-Lo pez a, Carlos del Pino a, * rrez-Castillo b, Jose Paloma Gutie a b
laga, Andalucía Tech, Escuela T laga, Spain Universidad de Ma ecnica Superior de Ingeniería Industrial, Campus de Teatinos s/n, 29071, Ma School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85202, USA
h i g h l i g h t s Boundary conditions well posed in a small-scale experimental setup. Airflow patterns were recorded, finding good agreement with numerical simulations. Numerics allowed us to know convective heat transfer in the vicinity of the floor. Accurate correlations given: heat flux strongly depends on temperature difference. Accurate correlations given: heat flux weakly depends on flow rate.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 July 2014 Accepted 1 December 2014 Available online 8 May 2015
The accurate knowledge of the dynamics and heat transfer of an attached cool radial jet in a laminar regime, discharging onto an isothermal floor and undisturbed atmosphere, is crucial to better control the parameters and reach a comfortable environment. In the present study, we first analyze a simple geometry of a small-scale, indoor displacement ventilation diffuser, by means of flow visualizations, and the PIV technique. In addition, we have carried out axisymmetric numerical simulations that show excellent agreement with qualitative and quantitative experimental data. Finally, we provide correlations of the heat flux as a function of non-dimensional parameters, finding out that the Nusselt number has a strong dependence on the Grashof number rather than the Reynolds number in this configuration. This simple model is a relevantly simple tool for practical engineering purposes such as the conditioning of large public areas with displacement ventilation systems. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Displacement flow diffuser CFD Thermal modeling
1. Introduction Displacement ventilation consists of flowing large amount of air with a temperature slightly lower than the comfort one, and using the heat of people and equipments in the treated zone to convect the air to the ceiling. Displacement ventilation offers the utmost in occupant comfort because the temperature and ventilation are near the optimal working point (see Refs. [1,2], and the references therein for more information about displacement ventilation). The study of theoretical and typical cold air structures in displacement ventilation are of great interest to define optimal design procedures [3]. Displacement ventilation creates both a high temperature and
* Corresponding author. Tel.: þ34 951952429; fax: þ34 951952605. E-mail address: [email protected] (C. Pino). http://dx.doi.org/10.1016/j.applthermaleng.2014.12.078 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
ventilation effectiveness [4], but it is difficult to find a simple approach of this system [5]. The dynamic of this flow is connected to the so-called wall jet, that is, the resulting attached plane jet impacting and spreading radially onto a surface. This problem has been studied in a laminar and turbulent regime theoretically [6], and it is in agreement with experimental observations [7]. The flow behaves as a radial boundary layer far away from the axis. The wall jet has been compared experimentally in the turbulent regime in the past [8,9], and in a room taking into account bouyancy [10]. Recently, a new similarity structure for the boundary layer of a turbulent wall jet was presented [11]. The same configuration can be analyzed using the pressure distribution and the wall stresses on the surface, and considering the ideal flow and boundary layer equations [12,13]. This problem has also been widely studied in the literature adding the energy equation due to its industrial
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applications. These applications include drying of paper, cooling of electronic equipments [14], or cooling turbomachinery elements [15], among others. The main work on the subject focused on the turbulent regime and the onset of primary or secondary vortices [16], together with the heat transfer, wall pressure distribution, and pressure loss [17]. These studies also focus on compressibility effects [18,19]. Generically, there are many difficulties in finding accurate turbulent models since the details of the experimental data are not well known. In addition, the thermal flow conditions may not be well posed in the geometry (see Ref. [20] and references therein). Furthermore, less information is given at moderate Reynolds numbers, beyond which the flow becomes unstable and even turbulent. A bidimensional flat jet has been also studied [21,22] with temperature variation and moderate Reynolds numbers, but no experiments were reported. This lack of knowledge in the laminar regime is the motivation for our work. The interest in this regime is to find out a first approximation of a low-velocity indoor diffuser for displacement ventilation. On the other hand, there are many studies in the literature concerning heating, ventilation and air conditioning (HVAC) systems for large public buildings. The research in this field has been motivated by the desire to reduce energy consumption [23], while fulfilling the demand, together with the increment of comfort requirements [24], even for systems with radiant ceilings [25]. In these works, there is a natural stratification from the floor to the ceiling, and only the area near the inhabited floor (mixing zone) must be treated. To achieve this objective, the study of the flow in displacement ventilation systems by means of diffusers combined with refrigerated floors initiated [24]. Most of the thermal load should be removed by the refrigerated floor, while the diffusers will provide the ventilation load. Thus, the air is pushed by displacement ventilation at low velocities and set-point temperatures for cooling near the comfort in the treated zones are imposed. The air creates a cold bed near the floor. Once the air finds a thermal load (person or equipment), its temperature increases, so the air rises to the non treated zone. Recently, this process has been applied in large public areas [26,27]. In our work we shed some new light for small-scale experiments and thermal models in a laminar regime. We analyze a reduced-scale cylindrical diffuser to better understand the heat transfer process from an experimental and numerical points of view. Thus, the correlations for full-scale rooms with displacement ventilation systems [28], the non-uniformity of the indoor temperature field, the effect of adiabatic and non-adiabatic obstructions in the conditioned zone, or the presence of air perturbations among others [29], are outside the scope of this study. Therefore, this work is only focused on the exact solution for the simplest (laminar) case of a reduced-scale cylindrical diffuser with no external sources and non-occupancy loads and well posed boundary conditions. Consequently, the turbulent regime is neglected due to the fact that the laminar cool round jet does not impact with any heat source, hence the convection (mixed) zone is not created in the experimental setup. One of the main challenges designing a HVAC system with displacement ventilation diffusers is to decide where to locate them, and this issue depends on the estimated load, among other factors. The exact position of these diffusers will improve the behavior of the whole system. For the sake of simplicity, the diffusers are installed equispaced, and little attention has been given to the general air distribution. Though this problem is very complex due to the nature of the flow (non steady, turbulent state with relatively high vertical temperature gradient in large areas), there are different turbulent models to achieve three-dimensional simulations successfully in a whole building [30]. However, these simulations are too time consuming, as the different boundaries and scales must be solved accurately [31], even by using nonlinear
RANS models [32]. To avoid this problem, different authors have been trying to simplify the calculations to obtain a first approach for the air distribution. This will allow us the accurate modeling of convective heat transfer correlations in rooms in terms of dimensional or non-dimensional parameters (see Refs. [29] or [1] and the references therein). For this reason, the problem of finding an optimal spatial distribution of the diffuser on the floor based on simple models will be interesting for real and practical applications to evaluate and control indoor systems. This is the main objective of this paper. To that end, we simplify the problem, taking into account the following constraints: laminar regime, non-refrigerated floor and non-occupancy loads. Thus, the result obtained will be a first order approximation for practical design applications. In this study, we only pay attention to the isothermal case, being the effect of the radiant floor analyzed in other manuscript. This paper is organized as follows. Firstly, a general description of the experimental setup and flow visualizations are included in section 2. The presentation of the laminar, steady and axisymmetric numerical results is given in section 3. The comparison between numerics and experimental data is shown in section 4. A simple laminar heat transfer model and its heat flux correlations are given in section 5. Finally, a summary of the main results is presented in section 6. 2. Experimental setup. flow visualization We used a small-scale diffuser depicted in Fig. 1 (a) in order to obtain a simple model, as well as a set of well-posed boundary conditions. The cylinder was precisely manufactured in perspex with a outer diameter 2Ro ¼ D ¼ 100 ± 0.06 mm, thickness of 2.5 mm ± 0.05 mm, total height H0 ¼ 212 ± 0.05 mm, and axial distance of the round slot h ¼ 10 ± 0.05 mm. This cylinder was inside a square cross-section chamber of one square meter (L1 L1 ¼ 10002 mm2), and a floor to ceiling distance of FCD ¼ 500 mm. The cylinder was also isolated from the surrounding air fluctuations outside. Special care was taken at this point to avoid disturbances and keep the flow axisymmetric and steady. There was a gap around the perimeter of L2 ¼ 50 mm between the chamber and the floor to avoid the movement of the cool air. As a result, the outflow air moved towards to a low level of the conditioned room. To ensure the same temperature conditions in each test, both the qualitative and the quantitative measurements were taken simultaneously. The whole test section was made of perspex to allow optical techniques such as Mie-scattering [33] [see, for instance, Fig. 1 (b)], Laser Doppler Anemometry (LDA) or Particle Image Velocimetry (PIV). Flow visualizations were acquired using a standard one Megapixel digital video camera at 25 frames per second (fps). On the other hand, for the LDA measurements we used Dantec 1D equipment and for PIV measurements we used two 500 mW continuous laser and a high speed camera up to 60,000 fps, though only 250 fps were required. An aluminum structure was used to hold the experimental setup. The different sections (floor, cylinder, ceiling and walls) were accurately aligned with a digital inclinometer to within ±0.1. The inlet flow rate, Q, and the inlet (Tin), ambient (Ta), floor (Tf) and ceiling (Tc) temperatures were controlled accurately by different equipments. To that end, we used digital and analogical flowmeters and two controlled heat exchangers for the inlet and floor temperatures. Typical values of the flow rate Q were inside the range 10e19 l/ min. The velocity inside the cylinder was uniform thanks to different layers of 6 mm diameter balls, stainless steel meshes of 0.25 mm2 free space, and one honeycomb of 6 mm hexagons and 50 mm length. Taking into account the values of the geometry, the final velocity at the exit of the round slot is close to 0.1 m/s. This
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Fig. 1. Sketch of the experimental setup (a), and instantaneous picture for Q ¼ 12 l/min and DT ¼ 2 K (b) for 106 mm z 239 mm and 100 mm r 100 mm.
value is the threshold that corresponds to the so-called diffuser near zone [34]. The flow rate of the piping system, Q, was measured upstream from the particle seeder. The flowmeter was also previously calibrated using LDA and PIV techniques. One can compute experimentally the flow rate with the following expression HZ0 þh
Q ¼ 2pRo
Vr ðzÞdz;
(1)
H0
where Vr is the dimensional radial component of the velocity field. The results confirmed minimal residual oscillations, which were ±200 ml/min in all the cases (±2.0% in the worst one). These experimental errors take into account the fluctuations of the measured radial velocity during the LDA measurements by means of 500 particles standard deviation, whilst the error using the PIV technique is based on the standard deviation of 500 instantaneous velocity fields.
Two silicone rubber heaters were fixed to an aluminum plate in the ceiling to control its set-temperature at Ta. This ambient temperature was also monitored by temperature sensors in three points of the chamber. The floor was also set to the same temperature Ta, to ensure no vertical temperature gradient at zero flow rate. The temperature variations in each test were set within ±0.1 K. Moreover, temperature differences between the inlet and the ambient were DT x TinTa from 0 K to 6 K. These values are close to those recommended in order to reach comfort conditions [1]. To guarantee the reliability of the experiments, different tests were repeated in several days with the same flow rate and DT conditions. At least a set of three different experiments were conducted with the same thermal conditions. The same results were obtained, confirming a good experimental procedure. Finally, it should be noticed that we carried out experiments with a 1.2 air changes per hour (ACH), approximately, taking into account the flow rates (Q) considered and the test chamber volume. This ACH value is in agreement with those used for practical displacement ventilation designing.
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The air in the whole pneumatic system was driven by a compressor to an external pressurized reservoir, that enforced a constant upstream pressure (p0 ¼ 3 bar) by means of a pressure reducer. The air temperature from the deposit was reduced downstream thanks to a heat exchanger 1 [see Fig. 1 (a)]. The cool air was pointed towards a particle seeder filled with olive oil upstream from the cylindrical diffuser. This setup allowed flow visualizations in the test section when a continuous laser beam of 500 mW and 532 nm wavelength was placed in a (r,z)-plane. An example of an instant picture for Q ¼ 12 l/min and DT ¼ 2 K near the outflow region D(r(D and 12 H0 (z(1:13H0 is shown in Fig. 1 (b). One can also observe that the flow was axisymmetric. Different images corresponding to different (r,z)-planes showed the same results within the experimental tolerances. Finally, a second heat exchanger was required to keep the floor temperature constant. 3. Numerical scheme We considered an incompressible air flow of density r(T) discharging from a diffuser of radius Ro at a certain temperature Tin into a cylindrical container with constant Ta, Tc and Tf temperatures. We used a cylindrical coordinate system (r, q, z) and we have assumed an axisymmetric flow, though the experimental setup had a square cross-section chamber. This small variation in the geometry will not affect the agreement between the numerical and the experimental results for the jet up to r / 450 mm. The computational domain for the inlet flow was equal to the one depicted in Fig. 1 (a), and the radial coordinate will be extended numerically to better appreciate the influence of the correlation of the heat flux in terms of the radial coordinate and the nondimensional parameters (see section 5). The Reynolds number is defined as
Um Ro Q ; ¼ Re ¼ 2phn n
Ta Tin DT ¼ ; Ta Tf
(3)
being Tf ¼ Ta. On the other hand, a normalized dimensionless temperature qT is useful to link the temperature of one cell (T) with the floor (Tf), the cool air supply (Tin) and the ambient (Ta) temperatures as follows:
T Tf T Ta : qT ¼ ¼ Tin Ta DT
(4)
The Prandtl number is the relation between the viscous and the thermal diffusivity, that is to say,
Pr ¼
n ; a
Gr ¼
bgDTR3o ; n2
(6)
where g is the gravity acceleration and b is the thermal expansion coefficient. Finally, the governing equations are the mass, momentum and energy conservation for an incompressible, steady state, axisymmetric, and laminar model using Boussinesq approximation (see, e.g. Refs. [35] or [36]) are written as follows:
! V$ V ¼ 0;
(7)
! ! ! 1 ! Gr g V $V V ¼ Vp þ V2 V 2 qT ; Re jgj Re
(8)
! V $VqT ¼
(9)
1 V2 qT ; Pr Re
! ! where V ¼ V =Um is the non-dimensional velocity field, corre! sponding to the cylindrical coordinate system, V ¼ ðVr ; Vq ; Vz Þ, and p ¼ rUp2 is the non-dimensional pressure. The software ANSYSm
Fluent solved the non linear NaviereStokes equations in the integral form using a finite-volume-method. The steady state solution was obtained using the SIMPLE coupling method between pressure and velocity. The above governing equations (7)e(9) are solved under the following boundary conditions:
!* chamber ceiling; 0 r 18 and z ¼ 10 : V ¼ 0; qT ¼ 0;
(2)
where Um is the bulk velocity, h the height of the round slot, and n(T) is the kinematic viscosity at a certain air supply temperature (Tin). To set the experimental Reynolds number, the fluctuations in the measured inlet temperature (Tin) were taken into account through the kinematic viscosity. We looked for a constant DT in the test, so that Reynolds numbers in the range of 170e320 were obtained. The non-dimensional radial and axial coordinates are r* ¼ r/Ro and z* ¼ z/Ro, where Ro is the characteristic length (outer radius of the cylindrical diffuser). The dimensionless parameter q is defined as
q¼
at constant pressure. Nevertheless, the value of the Prandtl number is constant, and approximately 0.7 in all cases considered in this study. Finally, the Grashof number is the relation between the buoyant and viscous forces, given by the equation:
(5)
being a the thermal diffusivity at the supply air temperature, a ¼ rcKp , where K is the thermal conductivity, and cp the specific heat
(10) !* chamber wall; r ¼ 18 and 1 z 10 : V ¼ 0; vqT =vn ¼ 0; (11) !* diffuser ceiling; 0 r 1 and z ¼ 4:45 : V ¼ 0; vqT =vn ¼ 0; (12) !* diffuser wall; r ¼ 1 and 0 z 4:25 V ¼ 0; vqT =vn ¼ 0; (13) chamber floor; 0 r 18 and z ¼ 0 :
!* V ¼ 0; qT ¼ 0; (14)
diffuser inflow; 0 r 1 and z ¼ 0 : Vz* ¼ 2h=Ro ; qT ¼ 1; (15) !* . chamberoutflow; r ¼18 and 0z 1: v V vn ¼0; vqT =vn¼0; (16) where n is the normal coordinate over the surface. The definitive axisymmetric grid consists in a mesh of 154,000 nodes. Boundary layers and gradient areas were refined to obtain accurate solutions. A grid convergence study was carried out, finding a relative error at different sample points lower than 0.1% in
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Fig. 2. Comparison of the experimental velocity measured by PIV for r* ¼ ±1 (left and right) and 4.25 z* 4.45, together with radial numerical velocity profile for the case Re z 210 and q ¼ 6.8 103.
comparison to a grid of twice the number of nodes used in this study. 4. Experimental and numerical results Experiments were conducted for a range of Reynolds number between 170 and 320, and several values of q. The results are analyzed below in two different regions: radial jet outflow area and boundary layer near the vicinity of the floor. 4.1. Difusser radial jet outflow area This region was significant to compare the experimental and numerical results. Apart from the calibration of the flowmeter described in section 2, PIV measurements were also carried out to validate the axisymmetric simulations in this region. This nonintrusive technique allows us to measure the velocity field in a (r*,z*)-plane near the round slot. We depict the velocity results in Fig. 2 for the case q ¼ 6.8 103, Re z 210 and r* ¼ ±1 and 4.25 z* 4.45. Two different aspects are shown. Firstly, it can be seen that there is a reasonable good agreement between the experimental and the axisymmetric numerical results, confirming the reliability of the computations. The top-hat-shape of the velocity profile at the exit of the cylindrical diffuser does not influence the dynamics of the flow near the floor since the cool air round jet was always controlled by buoyancy. Secondly, there is confirmation
83
that the flow is almost axisymmetric. The experimental data corresponding to two different planes, r* ¼ ±1 [see the flow visualization in Fig. 1 (b)] are equal within the experimental errors. One can also obtain quantitative information to compare with numerical data analyzing flow visualizations. To that end, the frames in each test were extracted from the video recorded. An average image was taken from post-processing 4000 frames. Fig. 3 shows an overview of the behavior presented by the radial outflow in the diffuser model. In the first series of averaged images, the temperature difference was constant (q ¼ 10.2 103) and the Reynolds number was increased from Re ¼ 211 (a) to 246 (b). Hence, flow visualizations changed slightly so the dynamics behavior of the radial cool jet was similar, resulting an increment in the maximum radial distance achieved with the Reynolds number. Only half a cylinder is shown in Fig. 3 to better appreciate the detail of the reattachment area, though the flow was axisymmetric as shown in Fig. 1 (b). The cool air jet flow from the round slot of the cylindrical diffuser fell down bonded to it due to bouyancy forces. One can observe in Fig. 3 that the air flow pattern increases its reattachment length with Reynolds number. In addition, small differences were found making the flow patterns visible for a constant Reynolds number and different values of q. As expected, the cool air jet decreases slightly and smoothly its maximum radial distance as the value of the temperature difference between the inlet and the undisturbed atmosphere (DT) increases for a constant Re (see below). Beyond a Reynolds number equals to 320, we observed the appearance of Kelvin Helmholtz instabilities in the shear layer, so the flow became unstable. Fig. 4 shows a qualitative comparison between the experimental and the numerical results for the case Re ¼ 246 and q ¼ 6.8 103. Fig. 4 (a) represents the averaged image and, after processing this image, the light intensity field is depicted in Fig. 4 (b). Finally, Fig. 4 (c) shows the numerical velocity field. The last two images agree very well, confirming a qualitative agreement. It is interesting at this point to make a quantitative comparison. To that end, radial Dr* ¼ Dr/Ro, and axial, Dz* ¼ Dz/Ro, non-dimensional distances were computed by means of the average image. This pair of values (Dr*, Dz*) correspond to the maximum length from (r*, z*) ¼ (1, 4, 45) to the reattachment area [as shown in Fig. 4 (b)]. We show a comparison between numerical results and experimental data for more pairs of values (Re, q). Thus, we represent Dr* and Dz* as function of the nondimensional temperature q, and several values of the Reynolds number Re in Figs. 5 (a) and b), respectively. The numerics reproduce the experimental results accurately. As expected, Dr* has lower values as q increases or Re decreases. On the other hand, Dz* was less affected by changing Re, and only small downward variations were observed as q increases.
Fig. 3. Average image for a constant q ¼ 10.2 103 and different values of the Reynolds number: Re ¼ 211 (a) and 246 (b). The region depicted corresponds to 0 r* 3.35 and 3 z* 5.
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Fig. 4. Comparison of the experimental particle concentration (a), post-processed image (b) and numerical velocity field (c) in a (r,z)-plane near the outflow from the diffuser for Re ¼ 246 and q ¼ 6.8 103. The region shown corresponds to 0 r* 2 and 2 z* 4.5.
We double checked experimentally that the values of Dr* and Dz* were equal for the left and right hand side of the diffuser (r* ¼ ±1), confirming again the symmetry of the flow. Finally, we compare in this section PIV results with the numerical velocity field data in a (r*, z*)-plane near the outflow of the diffuser in Fig. 6, showing excellent agreement. This result corresponds to the same case presented in Fig. 4. Once we checked that the axisymmetric and steady state simulations reproduced the experimental tests accurately, we focussed on the velocity and temperature fields in the vicinity of the floor to better understand the heat transfer process and the possible correlations in terms of dimensionless parameters. This will be described in the next section.
4.2. Boundary layer near the floor
Fig. 5. Comparison of dimensionless experimental [dashed lines with circles for Dr* (a) and squares for Dz* (b)] and numerical (solid line, diamonds) data, as function of the q for Re ¼ 176, 211 and 246.
Numerical simulations (a) and PIV measurements (b) of the cool jet impining onto the floor show the results depicted in Fig. 7, for the nondimensional velocity field in the vicinity of the floor in a (r*, z*)-plane and Re ¼ 246 and q ¼ 6.8 103. It is observed a good agreement and a change in the flow direction: the falling round jet bonded to the cylindrical difusser, parallel to the z* coordinate and analyzed in the previous section, became a spreading cool jet following the radial coordinate due to the presence of the floor. To compare both cases, we represent in Fig. 8 the nondimensional radial velocity Vr* as a function of the axial coordinate z* for r* ¼ 2.4 and the same values of Re and q depicted in Fig. 7. We observe again a reasonable good agreement between the numerical and experimental data, where both results correspond to those given for a typical boundary layer wall jet velocity profile. No PIV particles were found for z 0.5, so the velocity was almost zero. Numerical results and experimental data have been compared with a theoretical model. The spreading cool air bed flow over the floor is briefly analyzed in this section. Recall that the floor temperature is equal to the undisturbed ambient temperature for this theoretical model, so we compare our results with those given in Ref. [6] for the case of a non-heated flat jet. To that end, we adimensionalize the velocity field using the maximum velocity for each r * ðVr; max Þ, finding the dimensionless axial coordinate, z1=2 , at which the radial velocity profile reaches the value Vr;max =2. The
result of this computation is the function Vr ¼ V
Vr
r;maxðr * Þ
¼ f(h), where
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Fig. 6. Comparison of dimensionless experimental (a) and numerical (b) velocity magnitude near the outflow for the same case presented in Fig. 4.
h ¼ zz . We show in Fig. 9 Glauert's solution Vr ¼ f(h) in compar1=2
ison to the numerical simulations (a) and the experimental (PIV) results (b) for the case Re z 246 and q ¼ 6.8 103. We notice that the numerical solution obtained present a region of similarity near the floor. However, a new scaling is necessary in the far field (r* [ 1). In Fig. 9 (c), we show the maximum velocity, Vr;max , for the same numerical and experimental results presented in (a) and (b). One can obtain the tendency of the velocity in the far field, following its evolution with the non-dimensional radii. In the case of Glauert's laminar solution, both variables behave as Vr;max zr 3=2 and z1=2 zr 5=4 . In our case, Vr; max zr 1:17 and z1=2 zr 0:6941 showing a small disagreement in the values of the exponent due to the nature of the jet. In the case of Glauert's solution is a isothermal jet impining onto a wall, while in our study the temperature of the jet,
Tin, is lower than the wall (or floor), Ta (see, e.g. Ref. [37] for a boundary layer approximation of this case of study). 5. Laminar heat transfer model We will face the main goal of this research, obtaining heat flux correlations in terms of non-dimensional parameters. The heat transfer of the convective flow enhanced by the cool air can be calculated using numerical data and the Fourier's equation, as follows:
q ¼ k
vT : vz
(17)
The integrated heat flux, more interesting from an application point of view, is defined as
Z2p Z r qI ðrÞ ¼ 0
Ro
vT k r 0 dr 0 dq ¼ 2p vz
Zr k Ro
vT 0 0 r dr : vz
(18)
This measures the total heat exchanged between the floor and the cool air stream for a radial distance r. Instead of working with dimensional variables we can define an integral Nusselt number as
Fig. 7. Velocity magnitude for numerical simulations (a), together with the PIV results (b) for the case Re ¼ 246 and q ¼ 6.8 103.
Fig. 8. Comparison of the PIV radial velocity results (circles) and numerical simulations (squares) for the case Re ¼ 246 and q ¼ 6.8 103 at r* ¼ 2.4.
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NuI ðr Þ ¼
qI ðrÞ : kDTRo
(19)
Numerical results computed from this equation are depicted in solid lines in Fig. 10 for two different cases: a constant value of q and several Reynolds numbers (a), and vice versa (b). After processing numerical evolutions of NuI against r* shown in Fig. 10 (a) and (b), one can propose a fitting function NuI (r*) by means of the following expression
NuI ¼ A lnðr Þ þ B ; *
(20)
*
A and B being two variables which mainly depend on the flow rate, Q, and the temperature difference DT. Thus, we computed a set of values for A* and B* in (20) varying Re and q, using the slope in the far field of the function NuI (r*). To that end, we conducted several numerical simulations. In fact, our fittings correspond to those which make A* and B* depend on the Reynolds number (2), the Prandtl number (5), and the Grashof number (6). The final correlation for A* and B* with controlled parameters, Re and Gr, were computed, resulting the following expressions
A ¼ 0:243Re1:109 Gr 0:031 ; B ¼ 3:005 103 Re2:558 Gr0:698 :
(21)
Equations (20) and (21), are also represented in Fig. 10 with dashed lines. It is interesting to point out that buoyancy dominated flow dynamics near the vicinity of the floor. The cummulative heat is reasonably well represented for a wide set of simulations by equation (20), especially for r* 4. As expect, it is observed that NuI increases slightly with Reynolds number (or flow rate), but it changes dramatically to a strong dependence on the Grashof number, and therefore to the temperature difference, DT. Thus, one can notice that the parameter q has a significant effect on heat transfer in comparison to Reynolds number. As a result of this, notice the increment in the slope as this parameter was changed. This is also the case of the numerical velocity field in the vicinity of the floor (not shown): the maximum dimensional radial velocity, Vr, increases its value significantly with the temperature difference (DT), but it depends weakly on the flow rate. Finally, to obtain an estimation of the reliability of the model presented, we depict in Fig. 11 the computed integrated Nusselt value up to r* ¼ 10 (19) versus the estimated value obtained by the expressions (20) and (21), for all Reynolds and Grashof numbers studied. It is shown how accurately we can represent the real heat transfer exchanged between a diffuser and the floor for the isothermal case, so that 5% error is limited by dashed lines. 6. Conclusions
Fig. 9. Comparison between Glauert's solution Vr (black line) and numerical data (blue lines) (a) and those taken from PIV measurements (blue lines) (b) as a function of h. In (c) we compare numerical data and PIV measurements (green circles) for Vr;max against the nondimensional radial coordinate, r*. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
We have designed and built a test chamber to conduct experiments with a small scale diffuser for displacement ventilation that presents a stable, steady state and axisymmetric flow as a first approximation to evaluate and control this kind of indoor thermal systems. We have characterized a displacement diffuser in this setup by means of qualitative (flow visualizations) and quantitative (velocity field) measurements. The experimental data reproduced the shape of a cool radial air jet that emerges from a round slot onto an quiescent ambient that has the same temperature as the floor. Velocity and temperature fields obtained numerically show good agreement with the experimental data near the round slot and over the surface where a wall jet is formed. This jet behaves close to the theoretical velocity profile given by Glauert's solution (isothermal case), the discrepancy being justified by the thermal gradient of our
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case of study. Near the vicinity of the floor, we also observed a great influence of the temperature difference between the inlet air flow and the undisturbed ambient temperatures computing the heat flux. However, this latter is less sensitive to flow rate variations. The main result given is a precise correlation for the integral Nusselt number as a function of Reynolds and Grashof numbers taking into account a logarithmic function of the nondimensional radial distance. This model provides a useful description of convective heat transfer between the cool air and the floor in laminar flows and it is also useful for practical engineering purposes as first approximation, e.g. to design or to locate diffusers for displacement ventilation in buildings. A subsequent analysis will be carried out to introduce the radiant floor cooling into this model as a second part of this two-part study. References
Fig. 10. Radial profile of the integrated Nusselt numbers NuI as a function of the Reynolds numbers Re ¼ 176, 211, 246 and 281 for q ¼ 6.8 103 (a) and as a function of the normalized temperatures q ¼ 6.8 103, 10.2 103 and 12.4 103 for Re ¼ 246 (b). Solid lines represent numerical data from (17)e(19) and dashed lines correspond to (20) and (21).
Fig. 11. Integrated Nusselt number obtained from the model of equation (20) versus the integrated Nusselt number obtained by computational fluid dynamics (19).
[1] K.J. Loudermilk, Underfloor air distribution solutions for open office applications, ASHRAE Trans. 105 (1999) 605e613. [2] P.V. Nielsen, Displacement Ventilation e Theory and Design, Department of Building Technology and Structural Engineering. Aalborg University, Denmark, 1993. [3] Q. Chen, L. Glicksman, X. Yuan, S. Hu, Y. Hu, X. Yang, Performance Evaluation and Development of Design Guidelines for Displacement Ventilation, ASHRAE Research Project -RP-949, Report to ASHRAE TC 5. 3 -Room Air Distribution, 1999. [4] A. Awad, R.K. Calay, O.O. Badran, A.E. Holdo, An experimental study of stratified flow in enclosures, Appl. Therm. Eng. 28 (17e18) (2008) 2150e2158. [5] T. Zhang, K. Lee, Q. Chen, A simplified approach to describe complex diffusers in displacement ventilation for cfd simulations, Indoor Air 19 (3) (2009) 255e267. [6] M.N. Glauert, The wall jet, J. Fluid Mech. 1 (1956) 625e643. [7] R.A. Bajura, A.A. Szewczyk, Experimental investigation of a laminar twodimensional attached plane jet, Phys. Fluids 13 (7) (1970) 1653e1660. [8] P. Bakke, An experimental study of a wall jet, J. Fluid Mech. 2 (1957) 467e472. [9] K. Jambunathan, E. Lai, M. Moss, B. Button, A review of heat transfer data for single circular jet impingement, Int. J. Heat Fluid Flow 13 (1992) 106e115. [10] G. Cao, J. Kurnitski, M. Ruponen, O. Sepp€ anen, Experimental investigation and modelling of a buoyant attached plane jet in a room, Appl. Therm. Eng. 29 (14e15) (2009) 2790e2798. [11] G.I. Barenblatt, A.J. Chorin, V.M. Prostokishin, The turbulent wall-jet: a triplelayered structure and incomplete similarity, Proc. Natl. Acad. Sci. U S A 102 (25) (2005). [12] D.J. Phares, G.T. Smedley, R.C. Flagan, The wall shear stress produced by the normal impingement of a jet on a flat surface, J. Fluid Mech. 418 (2000) 351e375. [13] D.J. Phares, G.T. Smedley, R.C. Flagan, The inviscid impingement of a jet with arbitrary velocity profile, Phys. Fluids 12 (8) (2000) 2046e2055. [14] Y.H. Liu, S.Y. Tsai, C.C. Wang, Effect of driven frequency on flow and heat transfer of an impinging synthetic air jet, Appl. Therm. Eng. (2014), http:// dx.doi.org/10.1016/j.applthermaleng.2014.09.086. [15] K. Wang, H. Li, J. Zhu, Experimental study of heat transfer characteristic on jet impingement cooling with film extraction flow, Appl. Therm. Eng. 70 (2014) 620e629. [16] N. Didden, C. Ho, Unsteady separation in a boundary layer produced by an impinging jet, J. Fluid Mech. 160 (1985) 235e256. [17] F. Peper, W. Leiner, M. Fiebig, Impinging radial and inline jets: a comparison with regard to heat transfer, wall pressure distribution, and pressure loss, Exp. Therm. Fluid Sci. 14 (2) (1997) 194e204. [18] C. Donaldson, R.S. Snedeker, A study of free jet impingement. Part 2. Free jet turbulent structure and impingement heat transfer, J. Fluid Mech. 45 (1971) 477e512. [19] C. Donaldson, R.S. Snedeker, D. Margolis, A study of free jet impingement. Part 1. Mean properties of free and impinging jets, J. Fluid Mech. 45 (1971) 281e319. [20] M. Behnia, A. Parneix, P. Durbin, Accurate modeling of impinging jet heat transfer, Center for Turbulence Research, Annu. Res. Briefs 1 (1997) 149e164. [21] P.R. Kanna, M.K. Das, Conjugate heat transfer study of two-dimensional laminar incompressible offset jet flows, Numer. Heat. Transf. Part A Appl. 48 (7) (2005) 671e691. [22] P.R. Kanna, M.K. Das, Heat transfer study of two-dimensional laminar incompressible offset jet flows, Int. J. Therm. Sci. 47 (12) (2008) 1620e1629. [23] N.A. Tuan, K.D. Huang, Airflow circulation cell study of an air-conditioning energy-saving mechanism, Appl. Therm. Eng. 31 (17e18) (2011) 3956e3962. [24] P. Simmonds, W. Gaw, S. Holst, S. Reuss, Using radiant cooled floors to condition large spaces and maintain comfort conditions, ASHRAE Trans. 106 (2000) 695e701. [25] N. Fonseca, C. Cuevas, V. Lemort, Radiant ceiling systems coupled to its environment part 1: experimental analysis, Appl. Therm. Eng. 30 (14e15) (2011) 2187e2195.
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[32] A. Jouvray, P. Tucker, Y. Liu, On nonlinear RANS models when predicting more complex geometry room air flows, Int. J. Heat Fluid Flow 28 (2007) 275e288. [33] I.K. Toh, D. Honnery, J. Soria, Axial plus tangential entry swirling jet, Exp. Fluids 48 (2) (2010) 309e325. [34] A.K. Melikov, Personalized ventilation, Indoor Air 14 (2004) 157e167. nier, G. Lauriat, Effect of surface radiation on the breakdown of [35] H. Sun, E. Che steady natural convection flows in a square, air-filled cavity containing a centered inner body, Appl. Therm. Eng. 31 (2011) 1251e1262. [36] G.P. Bai, G.C. Gong, F.Y. Zhao, Z.X. Lin, Multiple thermal and moisture removals from the moving plate opposite to the impinging slot jet, Appl. Therm. Eng. 66 (2014) 252e265. ndez-Feria, C. del Pino, A. Fern rrez, Separation in the [37] R. Ferna andez-Gutie mixed convection boundary-layer radial flow over a constant temperature horizontal plate, Phys. Fluids 26 (2014) 103603.
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Experimental and numerical study of a small-scale and low-velocity indoor diffuser for displacement ventilation: Isothermal floor ndez-Gutie rrez a, Ignacio Gonza lez-Prieto a, Luis Parras a, Alberto Ferna Manuel Cejudo-Lo pez a, Carlos del Pino a, * rrez-Castillo b, Jose Paloma Gutie a b
laga, Andalucía Tech, Escuela T laga, Spain Universidad de Ma ecnica Superior de Ingeniería Industrial, Campus de Teatinos s/n, 29071, Ma School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85202, USA
h i g h l i g h t s Boundary conditions well posed in a small-scale experimental setup. Airflow patterns were recorded, finding good agreement with numerical simulations. Numerics allowed us to know convective heat transfer in the vicinity of the floor. Accurate correlations given: heat flux strongly depends on temperature difference. Accurate correlations given: heat flux weakly depends on flow rate.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 July 2014 Accepted 1 December 2014 Available online 8 May 2015
The accurate knowledge of the dynamics and heat transfer of an attached cool radial jet in a laminar regime, discharging onto an isothermal floor and undisturbed atmosphere, is crucial to better control the parameters and reach a comfortable environment. In the present study, we first analyze a simple geometry of a small-scale, indoor displacement ventilation diffuser, by means of flow visualizations, and the PIV technique. In addition, we have carried out axisymmetric numerical simulations that show excellent agreement with qualitative and quantitative experimental data. Finally, we provide correlations of the heat flux as a function of non-dimensional parameters, finding out that the Nusselt number has a strong dependence on the Grashof number rather than the Reynolds number in this configuration. This simple model is a relevantly simple tool for practical engineering purposes such as the conditioning of large public areas with displacement ventilation systems. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Displacement flow diffuser CFD Thermal modeling
1. Introduction Displacement ventilation consists of flowing large amount of air with a temperature slightly lower than the comfort one, and using the heat of people and equipments in the treated zone to convect the air to the ceiling. Displacement ventilation offers the utmost in occupant comfort because the temperature and ventilation are near the optimal working point (see Refs. [1,2], and the references therein for more information about displacement ventilation). The study of theoretical and typical cold air structures in displacement ventilation are of great interest to define optimal design procedures [3]. Displacement ventilation creates both a high temperature and
* Corresponding author. Tel.: þ34 951952429; fax: þ34 951952605. E-mail address: [email protected] (C. Pino). http://dx.doi.org/10.1016/j.applthermaleng.2014.12.078 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
ventilation effectiveness [4], but it is difficult to find a simple approach of this system [5]. The dynamic of this flow is connected to the so-called wall jet, that is, the resulting attached plane jet impacting and spreading radially onto a surface. This problem has been studied in a laminar and turbulent regime theoretically [6], and it is in agreement with experimental observations [7]. The flow behaves as a radial boundary layer far away from the axis. The wall jet has been compared experimentally in the turbulent regime in the past [8,9], and in a room taking into account bouyancy [10]. Recently, a new similarity structure for the boundary layer of a turbulent wall jet was presented [11]. The same configuration can be analyzed using the pressure distribution and the wall stresses on the surface, and considering the ideal flow and boundary layer equations [12,13]. This problem has also been widely studied in the literature adding the energy equation due to its industrial
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applications. These applications include drying of paper, cooling of electronic equipments [14], or cooling turbomachinery elements [15], among others. The main work on the subject focused on the turbulent regime and the onset of primary or secondary vortices [16], together with the heat transfer, wall pressure distribution, and pressure loss [17]. These studies also focus on compressibility effects [18,19]. Generically, there are many difficulties in finding accurate turbulent models since the details of the experimental data are not well known. In addition, the thermal flow conditions may not be well posed in the geometry (see Ref. [20] and references therein). Furthermore, less information is given at moderate Reynolds numbers, beyond which the flow becomes unstable and even turbulent. A bidimensional flat jet has been also studied [21,22] with temperature variation and moderate Reynolds numbers, but no experiments were reported. This lack of knowledge in the laminar regime is the motivation for our work. The interest in this regime is to find out a first approximation of a low-velocity indoor diffuser for displacement ventilation. On the other hand, there are many studies in the literature concerning heating, ventilation and air conditioning (HVAC) systems for large public buildings. The research in this field has been motivated by the desire to reduce energy consumption [23], while fulfilling the demand, together with the increment of comfort requirements [24], even for systems with radiant ceilings [25]. In these works, there is a natural stratification from the floor to the ceiling, and only the area near the inhabited floor (mixing zone) must be treated. To achieve this objective, the study of the flow in displacement ventilation systems by means of diffusers combined with refrigerated floors initiated [24]. Most of the thermal load should be removed by the refrigerated floor, while the diffusers will provide the ventilation load. Thus, the air is pushed by displacement ventilation at low velocities and set-point temperatures for cooling near the comfort in the treated zones are imposed. The air creates a cold bed near the floor. Once the air finds a thermal load (person or equipment), its temperature increases, so the air rises to the non treated zone. Recently, this process has been applied in large public areas [26,27]. In our work we shed some new light for small-scale experiments and thermal models in a laminar regime. We analyze a reduced-scale cylindrical diffuser to better understand the heat transfer process from an experimental and numerical points of view. Thus, the correlations for full-scale rooms with displacement ventilation systems [28], the non-uniformity of the indoor temperature field, the effect of adiabatic and non-adiabatic obstructions in the conditioned zone, or the presence of air perturbations among others [29], are outside the scope of this study. Therefore, this work is only focused on the exact solution for the simplest (laminar) case of a reduced-scale cylindrical diffuser with no external sources and non-occupancy loads and well posed boundary conditions. Consequently, the turbulent regime is neglected due to the fact that the laminar cool round jet does not impact with any heat source, hence the convection (mixed) zone is not created in the experimental setup. One of the main challenges designing a HVAC system with displacement ventilation diffusers is to decide where to locate them, and this issue depends on the estimated load, among other factors. The exact position of these diffusers will improve the behavior of the whole system. For the sake of simplicity, the diffusers are installed equispaced, and little attention has been given to the general air distribution. Though this problem is very complex due to the nature of the flow (non steady, turbulent state with relatively high vertical temperature gradient in large areas), there are different turbulent models to achieve three-dimensional simulations successfully in a whole building [30]. However, these simulations are too time consuming, as the different boundaries and scales must be solved accurately [31], even by using nonlinear
RANS models [32]. To avoid this problem, different authors have been trying to simplify the calculations to obtain a first approach for the air distribution. This will allow us the accurate modeling of convective heat transfer correlations in rooms in terms of dimensional or non-dimensional parameters (see Refs. [29] or [1] and the references therein). For this reason, the problem of finding an optimal spatial distribution of the diffuser on the floor based on simple models will be interesting for real and practical applications to evaluate and control indoor systems. This is the main objective of this paper. To that end, we simplify the problem, taking into account the following constraints: laminar regime, non-refrigerated floor and non-occupancy loads. Thus, the result obtained will be a first order approximation for practical design applications. In this study, we only pay attention to the isothermal case, being the effect of the radiant floor analyzed in other manuscript. This paper is organized as follows. Firstly, a general description of the experimental setup and flow visualizations are included in section 2. The presentation of the laminar, steady and axisymmetric numerical results is given in section 3. The comparison between numerics and experimental data is shown in section 4. A simple laminar heat transfer model and its heat flux correlations are given in section 5. Finally, a summary of the main results is presented in section 6. 2. Experimental setup. flow visualization We used a small-scale diffuser depicted in Fig. 1 (a) in order to obtain a simple model, as well as a set of well-posed boundary conditions. The cylinder was precisely manufactured in perspex with a outer diameter 2Ro ¼ D ¼ 100 ± 0.06 mm, thickness of 2.5 mm ± 0.05 mm, total height H0 ¼ 212 ± 0.05 mm, and axial distance of the round slot h ¼ 10 ± 0.05 mm. This cylinder was inside a square cross-section chamber of one square meter (L1 L1 ¼ 10002 mm2), and a floor to ceiling distance of FCD ¼ 500 mm. The cylinder was also isolated from the surrounding air fluctuations outside. Special care was taken at this point to avoid disturbances and keep the flow axisymmetric and steady. There was a gap around the perimeter of L2 ¼ 50 mm between the chamber and the floor to avoid the movement of the cool air. As a result, the outflow air moved towards to a low level of the conditioned room. To ensure the same temperature conditions in each test, both the qualitative and the quantitative measurements were taken simultaneously. The whole test section was made of perspex to allow optical techniques such as Mie-scattering [33] [see, for instance, Fig. 1 (b)], Laser Doppler Anemometry (LDA) or Particle Image Velocimetry (PIV). Flow visualizations were acquired using a standard one Megapixel digital video camera at 25 frames per second (fps). On the other hand, for the LDA measurements we used Dantec 1D equipment and for PIV measurements we used two 500 mW continuous laser and a high speed camera up to 60,000 fps, though only 250 fps were required. An aluminum structure was used to hold the experimental setup. The different sections (floor, cylinder, ceiling and walls) were accurately aligned with a digital inclinometer to within ±0.1. The inlet flow rate, Q, and the inlet (Tin), ambient (Ta), floor (Tf) and ceiling (Tc) temperatures were controlled accurately by different equipments. To that end, we used digital and analogical flowmeters and two controlled heat exchangers for the inlet and floor temperatures. Typical values of the flow rate Q were inside the range 10e19 l/ min. The velocity inside the cylinder was uniform thanks to different layers of 6 mm diameter balls, stainless steel meshes of 0.25 mm2 free space, and one honeycomb of 6 mm hexagons and 50 mm length. Taking into account the values of the geometry, the final velocity at the exit of the round slot is close to 0.1 m/s. This
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Fig. 1. Sketch of the experimental setup (a), and instantaneous picture for Q ¼ 12 l/min and DT ¼ 2 K (b) for 106 mm z 239 mm and 100 mm r 100 mm.
value is the threshold that corresponds to the so-called diffuser near zone [34]. The flow rate of the piping system, Q, was measured upstream from the particle seeder. The flowmeter was also previously calibrated using LDA and PIV techniques. One can compute experimentally the flow rate with the following expression HZ0 þh
Q ¼ 2pRo
Vr ðzÞdz;
(1)
H0
where Vr is the dimensional radial component of the velocity field. The results confirmed minimal residual oscillations, which were ±200 ml/min in all the cases (±2.0% in the worst one). These experimental errors take into account the fluctuations of the measured radial velocity during the LDA measurements by means of 500 particles standard deviation, whilst the error using the PIV technique is based on the standard deviation of 500 instantaneous velocity fields.
Two silicone rubber heaters were fixed to an aluminum plate in the ceiling to control its set-temperature at Ta. This ambient temperature was also monitored by temperature sensors in three points of the chamber. The floor was also set to the same temperature Ta, to ensure no vertical temperature gradient at zero flow rate. The temperature variations in each test were set within ±0.1 K. Moreover, temperature differences between the inlet and the ambient were DT x TinTa from 0 K to 6 K. These values are close to those recommended in order to reach comfort conditions [1]. To guarantee the reliability of the experiments, different tests were repeated in several days with the same flow rate and DT conditions. At least a set of three different experiments were conducted with the same thermal conditions. The same results were obtained, confirming a good experimental procedure. Finally, it should be noticed that we carried out experiments with a 1.2 air changes per hour (ACH), approximately, taking into account the flow rates (Q) considered and the test chamber volume. This ACH value is in agreement with those used for practical displacement ventilation designing.
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The air in the whole pneumatic system was driven by a compressor to an external pressurized reservoir, that enforced a constant upstream pressure (p0 ¼ 3 bar) by means of a pressure reducer. The air temperature from the deposit was reduced downstream thanks to a heat exchanger 1 [see Fig. 1 (a)]. The cool air was pointed towards a particle seeder filled with olive oil upstream from the cylindrical diffuser. This setup allowed flow visualizations in the test section when a continuous laser beam of 500 mW and 532 nm wavelength was placed in a (r,z)-plane. An example of an instant picture for Q ¼ 12 l/min and DT ¼ 2 K near the outflow region D(r(D and 12 H0 (z(1:13H0 is shown in Fig. 1 (b). One can also observe that the flow was axisymmetric. Different images corresponding to different (r,z)-planes showed the same results within the experimental tolerances. Finally, a second heat exchanger was required to keep the floor temperature constant. 3. Numerical scheme We considered an incompressible air flow of density r(T) discharging from a diffuser of radius Ro at a certain temperature Tin into a cylindrical container with constant Ta, Tc and Tf temperatures. We used a cylindrical coordinate system (r, q, z) and we have assumed an axisymmetric flow, though the experimental setup had a square cross-section chamber. This small variation in the geometry will not affect the agreement between the numerical and the experimental results for the jet up to r / 450 mm. The computational domain for the inlet flow was equal to the one depicted in Fig. 1 (a), and the radial coordinate will be extended numerically to better appreciate the influence of the correlation of the heat flux in terms of the radial coordinate and the nondimensional parameters (see section 5). The Reynolds number is defined as
Um Ro Q ; ¼ Re ¼ 2phn n
Ta Tin DT ¼ ; Ta Tf
(3)
being Tf ¼ Ta. On the other hand, a normalized dimensionless temperature qT is useful to link the temperature of one cell (T) with the floor (Tf), the cool air supply (Tin) and the ambient (Ta) temperatures as follows:
T Tf T Ta : qT ¼ ¼ Tin Ta DT
(4)
The Prandtl number is the relation between the viscous and the thermal diffusivity, that is to say,
Pr ¼
n ; a
Gr ¼
bgDTR3o ; n2
(6)
where g is the gravity acceleration and b is the thermal expansion coefficient. Finally, the governing equations are the mass, momentum and energy conservation for an incompressible, steady state, axisymmetric, and laminar model using Boussinesq approximation (see, e.g. Refs. [35] or [36]) are written as follows:
! V$ V ¼ 0;
(7)
! ! ! 1 ! Gr g V $V V ¼ Vp þ V2 V 2 qT ; Re jgj Re
(8)
! V $VqT ¼
(9)
1 V2 qT ; Pr Re
! ! where V ¼ V =Um is the non-dimensional velocity field, corre! sponding to the cylindrical coordinate system, V ¼ ðVr ; Vq ; Vz Þ, and p ¼ rUp2 is the non-dimensional pressure. The software ANSYSm
Fluent solved the non linear NaviereStokes equations in the integral form using a finite-volume-method. The steady state solution was obtained using the SIMPLE coupling method between pressure and velocity. The above governing equations (7)e(9) are solved under the following boundary conditions:
!* chamber ceiling; 0 r 18 and z ¼ 10 : V ¼ 0; qT ¼ 0;
(2)
where Um is the bulk velocity, h the height of the round slot, and n(T) is the kinematic viscosity at a certain air supply temperature (Tin). To set the experimental Reynolds number, the fluctuations in the measured inlet temperature (Tin) were taken into account through the kinematic viscosity. We looked for a constant DT in the test, so that Reynolds numbers in the range of 170e320 were obtained. The non-dimensional radial and axial coordinates are r* ¼ r/Ro and z* ¼ z/Ro, where Ro is the characteristic length (outer radius of the cylindrical diffuser). The dimensionless parameter q is defined as
q¼
at constant pressure. Nevertheless, the value of the Prandtl number is constant, and approximately 0.7 in all cases considered in this study. Finally, the Grashof number is the relation between the buoyant and viscous forces, given by the equation:
(5)
being a the thermal diffusivity at the supply air temperature, a ¼ rcKp , where K is the thermal conductivity, and cp the specific heat
(10) !* chamber wall; r ¼ 18 and 1 z 10 : V ¼ 0; vqT =vn ¼ 0; (11) !* diffuser ceiling; 0 r 1 and z ¼ 4:45 : V ¼ 0; vqT =vn ¼ 0; (12) !* diffuser wall; r ¼ 1 and 0 z 4:25 V ¼ 0; vqT =vn ¼ 0; (13) chamber floor; 0 r 18 and z ¼ 0 :
!* V ¼ 0; qT ¼ 0; (14)
diffuser inflow; 0 r 1 and z ¼ 0 : Vz* ¼ 2h=Ro ; qT ¼ 1; (15) !* . chamberoutflow; r ¼18 and 0z 1: v V vn ¼0; vqT =vn¼0; (16) where n is the normal coordinate over the surface. The definitive axisymmetric grid consists in a mesh of 154,000 nodes. Boundary layers and gradient areas were refined to obtain accurate solutions. A grid convergence study was carried out, finding a relative error at different sample points lower than 0.1% in
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Fig. 2. Comparison of the experimental velocity measured by PIV for r* ¼ ±1 (left and right) and 4.25 z* 4.45, together with radial numerical velocity profile for the case Re z 210 and q ¼ 6.8 103.
comparison to a grid of twice the number of nodes used in this study. 4. Experimental and numerical results Experiments were conducted for a range of Reynolds number between 170 and 320, and several values of q. The results are analyzed below in two different regions: radial jet outflow area and boundary layer near the vicinity of the floor. 4.1. Difusser radial jet outflow area This region was significant to compare the experimental and numerical results. Apart from the calibration of the flowmeter described in section 2, PIV measurements were also carried out to validate the axisymmetric simulations in this region. This nonintrusive technique allows us to measure the velocity field in a (r*,z*)-plane near the round slot. We depict the velocity results in Fig. 2 for the case q ¼ 6.8 103, Re z 210 and r* ¼ ±1 and 4.25 z* 4.45. Two different aspects are shown. Firstly, it can be seen that there is a reasonable good agreement between the experimental and the axisymmetric numerical results, confirming the reliability of the computations. The top-hat-shape of the velocity profile at the exit of the cylindrical diffuser does not influence the dynamics of the flow near the floor since the cool air round jet was always controlled by buoyancy. Secondly, there is confirmation
83
that the flow is almost axisymmetric. The experimental data corresponding to two different planes, r* ¼ ±1 [see the flow visualization in Fig. 1 (b)] are equal within the experimental errors. One can also obtain quantitative information to compare with numerical data analyzing flow visualizations. To that end, the frames in each test were extracted from the video recorded. An average image was taken from post-processing 4000 frames. Fig. 3 shows an overview of the behavior presented by the radial outflow in the diffuser model. In the first series of averaged images, the temperature difference was constant (q ¼ 10.2 103) and the Reynolds number was increased from Re ¼ 211 (a) to 246 (b). Hence, flow visualizations changed slightly so the dynamics behavior of the radial cool jet was similar, resulting an increment in the maximum radial distance achieved with the Reynolds number. Only half a cylinder is shown in Fig. 3 to better appreciate the detail of the reattachment area, though the flow was axisymmetric as shown in Fig. 1 (b). The cool air jet flow from the round slot of the cylindrical diffuser fell down bonded to it due to bouyancy forces. One can observe in Fig. 3 that the air flow pattern increases its reattachment length with Reynolds number. In addition, small differences were found making the flow patterns visible for a constant Reynolds number and different values of q. As expected, the cool air jet decreases slightly and smoothly its maximum radial distance as the value of the temperature difference between the inlet and the undisturbed atmosphere (DT) increases for a constant Re (see below). Beyond a Reynolds number equals to 320, we observed the appearance of Kelvin Helmholtz instabilities in the shear layer, so the flow became unstable. Fig. 4 shows a qualitative comparison between the experimental and the numerical results for the case Re ¼ 246 and q ¼ 6.8 103. Fig. 4 (a) represents the averaged image and, after processing this image, the light intensity field is depicted in Fig. 4 (b). Finally, Fig. 4 (c) shows the numerical velocity field. The last two images agree very well, confirming a qualitative agreement. It is interesting at this point to make a quantitative comparison. To that end, radial Dr* ¼ Dr/Ro, and axial, Dz* ¼ Dz/Ro, non-dimensional distances were computed by means of the average image. This pair of values (Dr*, Dz*) correspond to the maximum length from (r*, z*) ¼ (1, 4, 45) to the reattachment area [as shown in Fig. 4 (b)]. We show a comparison between numerical results and experimental data for more pairs of values (Re, q). Thus, we represent Dr* and Dz* as function of the nondimensional temperature q, and several values of the Reynolds number Re in Figs. 5 (a) and b), respectively. The numerics reproduce the experimental results accurately. As expected, Dr* has lower values as q increases or Re decreases. On the other hand, Dz* was less affected by changing Re, and only small downward variations were observed as q increases.
Fig. 3. Average image for a constant q ¼ 10.2 103 and different values of the Reynolds number: Re ¼ 211 (a) and 246 (b). The region depicted corresponds to 0 r* 3.35 and 3 z* 5.
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Fig. 4. Comparison of the experimental particle concentration (a), post-processed image (b) and numerical velocity field (c) in a (r,z)-plane near the outflow from the diffuser for Re ¼ 246 and q ¼ 6.8 103. The region shown corresponds to 0 r* 2 and 2 z* 4.5.
We double checked experimentally that the values of Dr* and Dz* were equal for the left and right hand side of the diffuser (r* ¼ ±1), confirming again the symmetry of the flow. Finally, we compare in this section PIV results with the numerical velocity field data in a (r*, z*)-plane near the outflow of the diffuser in Fig. 6, showing excellent agreement. This result corresponds to the same case presented in Fig. 4. Once we checked that the axisymmetric and steady state simulations reproduced the experimental tests accurately, we focussed on the velocity and temperature fields in the vicinity of the floor to better understand the heat transfer process and the possible correlations in terms of dimensionless parameters. This will be described in the next section.
4.2. Boundary layer near the floor
Fig. 5. Comparison of dimensionless experimental [dashed lines with circles for Dr* (a) and squares for Dz* (b)] and numerical (solid line, diamonds) data, as function of the q for Re ¼ 176, 211 and 246.
Numerical simulations (a) and PIV measurements (b) of the cool jet impining onto the floor show the results depicted in Fig. 7, for the nondimensional velocity field in the vicinity of the floor in a (r*, z*)-plane and Re ¼ 246 and q ¼ 6.8 103. It is observed a good agreement and a change in the flow direction: the falling round jet bonded to the cylindrical difusser, parallel to the z* coordinate and analyzed in the previous section, became a spreading cool jet following the radial coordinate due to the presence of the floor. To compare both cases, we represent in Fig. 8 the nondimensional radial velocity Vr* as a function of the axial coordinate z* for r* ¼ 2.4 and the same values of Re and q depicted in Fig. 7. We observe again a reasonable good agreement between the numerical and experimental data, where both results correspond to those given for a typical boundary layer wall jet velocity profile. No PIV particles were found for z 0.5, so the velocity was almost zero. Numerical results and experimental data have been compared with a theoretical model. The spreading cool air bed flow over the floor is briefly analyzed in this section. Recall that the floor temperature is equal to the undisturbed ambient temperature for this theoretical model, so we compare our results with those given in Ref. [6] for the case of a non-heated flat jet. To that end, we adimensionalize the velocity field using the maximum velocity for each r * ðVr; max Þ, finding the dimensionless axial coordinate, z1=2 , at which the radial velocity profile reaches the value Vr;max =2. The
result of this computation is the function Vr ¼ V
Vr
r;maxðr * Þ
¼ f(h), where
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Fig. 6. Comparison of dimensionless experimental (a) and numerical (b) velocity magnitude near the outflow for the same case presented in Fig. 4.
h ¼ zz . We show in Fig. 9 Glauert's solution Vr ¼ f(h) in compar1=2
ison to the numerical simulations (a) and the experimental (PIV) results (b) for the case Re z 246 and q ¼ 6.8 103. We notice that the numerical solution obtained present a region of similarity near the floor. However, a new scaling is necessary in the far field (r* [ 1). In Fig. 9 (c), we show the maximum velocity, Vr;max , for the same numerical and experimental results presented in (a) and (b). One can obtain the tendency of the velocity in the far field, following its evolution with the non-dimensional radii. In the case of Glauert's laminar solution, both variables behave as Vr;max zr 3=2 and z1=2 zr 5=4 . In our case, Vr; max zr 1:17 and z1=2 zr 0:6941 showing a small disagreement in the values of the exponent due to the nature of the jet. In the case of Glauert's solution is a isothermal jet impining onto a wall, while in our study the temperature of the jet,
Tin, is lower than the wall (or floor), Ta (see, e.g. Ref. [37] for a boundary layer approximation of this case of study). 5. Laminar heat transfer model We will face the main goal of this research, obtaining heat flux correlations in terms of non-dimensional parameters. The heat transfer of the convective flow enhanced by the cool air can be calculated using numerical data and the Fourier's equation, as follows:
q ¼ k
vT : vz
(17)
The integrated heat flux, more interesting from an application point of view, is defined as
Z2p Z r qI ðrÞ ¼ 0
Ro
vT k r 0 dr 0 dq ¼ 2p vz
Zr k Ro
vT 0 0 r dr : vz
(18)
This measures the total heat exchanged between the floor and the cool air stream for a radial distance r. Instead of working with dimensional variables we can define an integral Nusselt number as
Fig. 7. Velocity magnitude for numerical simulations (a), together with the PIV results (b) for the case Re ¼ 246 and q ¼ 6.8 103.
Fig. 8. Comparison of the PIV radial velocity results (circles) and numerical simulations (squares) for the case Re ¼ 246 and q ¼ 6.8 103 at r* ¼ 2.4.
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NuI ðr Þ ¼
qI ðrÞ : kDTRo
(19)
Numerical results computed from this equation are depicted in solid lines in Fig. 10 for two different cases: a constant value of q and several Reynolds numbers (a), and vice versa (b). After processing numerical evolutions of NuI against r* shown in Fig. 10 (a) and (b), one can propose a fitting function NuI (r*) by means of the following expression
NuI ¼ A lnðr Þ þ B ; *
(20)
*
A and B being two variables which mainly depend on the flow rate, Q, and the temperature difference DT. Thus, we computed a set of values for A* and B* in (20) varying Re and q, using the slope in the far field of the function NuI (r*). To that end, we conducted several numerical simulations. In fact, our fittings correspond to those which make A* and B* depend on the Reynolds number (2), the Prandtl number (5), and the Grashof number (6). The final correlation for A* and B* with controlled parameters, Re and Gr, were computed, resulting the following expressions
A ¼ 0:243Re1:109 Gr 0:031 ; B ¼ 3:005 103 Re2:558 Gr0:698 :
(21)
Equations (20) and (21), are also represented in Fig. 10 with dashed lines. It is interesting to point out that buoyancy dominated flow dynamics near the vicinity of the floor. The cummulative heat is reasonably well represented for a wide set of simulations by equation (20), especially for r* 4. As expect, it is observed that NuI increases slightly with Reynolds number (or flow rate), but it changes dramatically to a strong dependence on the Grashof number, and therefore to the temperature difference, DT. Thus, one can notice that the parameter q has a significant effect on heat transfer in comparison to Reynolds number. As a result of this, notice the increment in the slope as this parameter was changed. This is also the case of the numerical velocity field in the vicinity of the floor (not shown): the maximum dimensional radial velocity, Vr, increases its value significantly with the temperature difference (DT), but it depends weakly on the flow rate. Finally, to obtain an estimation of the reliability of the model presented, we depict in Fig. 11 the computed integrated Nusselt value up to r* ¼ 10 (19) versus the estimated value obtained by the expressions (20) and (21), for all Reynolds and Grashof numbers studied. It is shown how accurately we can represent the real heat transfer exchanged between a diffuser and the floor for the isothermal case, so that 5% error is limited by dashed lines. 6. Conclusions
Fig. 9. Comparison between Glauert's solution Vr (black line) and numerical data (blue lines) (a) and those taken from PIV measurements (blue lines) (b) as a function of h. In (c) we compare numerical data and PIV measurements (green circles) for Vr;max against the nondimensional radial coordinate, r*. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
We have designed and built a test chamber to conduct experiments with a small scale diffuser for displacement ventilation that presents a stable, steady state and axisymmetric flow as a first approximation to evaluate and control this kind of indoor thermal systems. We have characterized a displacement diffuser in this setup by means of qualitative (flow visualizations) and quantitative (velocity field) measurements. The experimental data reproduced the shape of a cool radial air jet that emerges from a round slot onto an quiescent ambient that has the same temperature as the floor. Velocity and temperature fields obtained numerically show good agreement with the experimental data near the round slot and over the surface where a wall jet is formed. This jet behaves close to the theoretical velocity profile given by Glauert's solution (isothermal case), the discrepancy being justified by the thermal gradient of our
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case of study. Near the vicinity of the floor, we also observed a great influence of the temperature difference between the inlet air flow and the undisturbed ambient temperatures computing the heat flux. However, this latter is less sensitive to flow rate variations. The main result given is a precise correlation for the integral Nusselt number as a function of Reynolds and Grashof numbers taking into account a logarithmic function of the nondimensional radial distance. This model provides a useful description of convective heat transfer between the cool air and the floor in laminar flows and it is also useful for practical engineering purposes as first approximation, e.g. to design or to locate diffusers for displacement ventilation in buildings. A subsequent analysis will be carried out to introduce the radiant floor cooling into this model as a second part of this two-part study. References
Fig. 10. Radial profile of the integrated Nusselt numbers NuI as a function of the Reynolds numbers Re ¼ 176, 211, 246 and 281 for q ¼ 6.8 103 (a) and as a function of the normalized temperatures q ¼ 6.8 103, 10.2 103 and 12.4 103 for Re ¼ 246 (b). Solid lines represent numerical data from (17)e(19) and dashed lines correspond to (20) and (21).
Fig. 11. Integrated Nusselt number obtained from the model of equation (20) versus the integrated Nusselt number obtained by computational fluid dynamics (19).
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