Effects of spray modeling on heat and mass transfer in air–water spray systems in parallel flow

International Communications in Heat and Mass Transfer 34 (2007) 878 – 886 www.elsevier.com/locate/ichmt

Effects of spray modeling on heat and mass transfer in air–water spray systems in parallel flow☆ R. Sureshkumar, P.L. Dhar, S.R. Kale ⁎ Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi 110016, India Available online 19 April 2007

Abstract The heat and mass transfer equations for a co-flow water spray in air were solved for different combinations of drop diameter category and velocity sub-class, and compared with experimental data. For uniform drop velocities, the number of categories was increased to 10, 50, 100 and 200, as against 5 in an earlier study. Best predictions were obtained with 100 categories. For 5 categories, 10, 20 and 40 velocity sub-classes within each category were introduced, and predictions were slightly better than with a single velocity. Results with 100 categories–1 velocity and 10 categories–10 velocity sub-classes were similar; the latter matched experiments mostly within ± 15% as against ± 30% in an earlier study. © 2007 Elsevier Ltd. All rights reserved. Keywords: Spray; Evaporative cooling; Modeling; Drop categories; Velocity sub-class

1. Introduction In evaporative cooling systems, water sprays are employed to cool incoming hot air. The sprays consist of relatively large drops (typically 200 μm diameter) and diameter changes due to evaporation are marginal (1–2%) (Kachhwaha et al. [1], Sureshkumar [2]). To predict the performance of such coolers, changes in air temperature (i.e. dry bulb temperature, DBT) and humidity (i.e. specific humidity, ω) need to be calculated. For this purpose, it is important to accurately model the drop diameter–velocity distribution resulting from jet break-up, and typical results are reported in Kachhwaha et al. [1]. The structure of a spray can be described by macroscopic parameters, e.g., spray angle and spray penetration, and by microscopic parameters, such as, drop size and velocity distributions, and various mean drop diameters. These parameters, the latter in particular, are required for heat and mass transfer simulations. Kachhwaha et al. [1] modeled the spray as uniform velocity drops that were classified into five diameter (size) categories, and then solved the conservation equations for air flowing in a 0.58 × 0.58 × 2 m long duct for a steady 1-dimensional co-flow (parallel) configuration. The duct length was divided into equal slices within which rapid transverse mixing was assumed. Heat and mass transfer between drops and air were modeled as source terms in the conservation equations for air. ☆

Communicated by A.R. Balakrishnan and S. Jayanti. ⁎ Corresponding author. E-mail address: [email protected] (S.R. Kale).

0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.03.001

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

Nomenclature As a0..3 B Cd Cpd Cp,av D D30 ¯ D f g ˜h ha hd hfg hfg,0 hfg,wall hfg h˜ N n n¯ P S¯e S¯m S¯mv T U UN U¯ u W We x, y Z

surface area of drop Lagrangian constants 12/Weber number drag coefficient specific heat of drop average specific heat of air drop diameter volume mean diameter normalized drop diameter joint drop diameter–velocity distribution function constant for gravity heat transfer coefficient specific enthalpy of air specific enthalpy of drop specific enthalpy of evaporation of water specific enthalpy of evaporation of water at 0 °C specific enthalpy of evaporation of water at wall temperature specific enthalpy of evaporation of water mass transfer coefficient number distribution of drops number of drops normalized number distribution duct perimeter dimensionless source term for energy dimensionless source term for mass dimensionless source term for momentum air temperature drop velocity water velocity at the nozzle non-dimensional drop velocity air velocity drop velocity relative to air Weber number coordinate axes parameter in MEP formalism

Greek α ΔD Δn ω ρ σ

half mean cone angle drop diameter interval number interval mass fraction of water vapor in air density water surface tension

Subscripts 0 freestream value a air i drop diameter category l liquid max maximum min minimum x, y coordinate directions

879

880

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

2. Limitations of previous work Prior experiments by Kachhwaha et al. [1] had large uncertainties in air and water temperature, ± 0.5 and ± 1.8 °C, respectively. In the experiments by Sureshkumar [2], these were reduced to ± 0.3 and ± 0.1 °C, respectively. Both the number of drop categories and their velocities are better resolved here. Details of these and other improvements are given below. 2.1. Nozzle design and spray photography Instead of commercially available nozzles used in the earlier study, here, precision machined nozzles were produced, Fig. 1. Four nozzle outlet diameters of 3, 4, 5 and 5.5 mm. were used with water pressures of 1, 2 and 3 bar(g). The spray was photographed with a high-speed SLR film camera and a typical photo is shown in Fig. 2. Here, the rectangular strip with marker circles was placed at the focal plane for scaling/calibration giving a magnification of 7. Separate calibrations were performed with the strip placed at known distances away from the focal plane. The pictures were projected on a screen where the diameters were measured and then scaled to obtain the actual drop diameter. Sharply focused drops were scaled using the focal plane calibration. The haziness of some out-of-focus drops was the criteria for selecting the appropriate scaling factor for these drops; not all out-of-focus drops were measured. With this technique both measurement accuracy and number of drops measured were greater than the earlier study, and consequently, the drop size distribution was better resolved. The resolution of drop diameter measurements was limited by the marker circle diameter that was 0.5 mm., and consequently, the drop diameter was resolved in intervals of 74 μm. A typical plot of the normalized drop numbers counted in each

Fig. 1. Cross-section view of the nozzle.

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

881

Fig. 2. Photograph of a typical spray.

range is shown in Fig. 3. This sizing technique was used because qffiffiffiffiffiffiffiffiffiffiffi image processing software was not 3 Dn available. Using this data, the volume-mean diameter, D30 ¼ 3 RD was calculated. These values were RDn identical for all flow conditions and nozzle diameters, however, studies (Aigal et al. [3]) show that variations in D30 do occur with both increase in air velocity and nozzle pressure. In the present study, this variation could be within the resolution range (74 μm) and, hence, not adequately resolved. The uncertainty due to the resolution limit for a mean drop diameter of 330 μm is ± 22%. The spray angle was measured from still photographs and it increased with pressure and nozzle outlet diameter. Under still conditions, the maximum half-cone angle was 39.5° for 5.5 mm nozzle at 3 bar(g). The maximum variation in half-cone angle was 4 to 5° for pressures of 1 to 3 bar(g). The increase in this angle with nozzle diameter is gradual but for a steep increase (9.5°) between 3 and 4 mm diameters for 1 bar(g). In parallel airflow, the spray is deflected inwards, reducing the half-cone angle by up to 9° for the 4 mm nozzle. When impinged on a flat sheet normal to the spray axis, the drops formed an annular pattern which indicated that they move between two concentric cones that have a common tip at the nozzle and whose diameter ratio at any axial location is between 0.45 and 0.5; an exception was the 3 mm diameter nozzle at 1 bar(g) where it was 0.9 that can be attributed to the very small flow rate. In the simulations, the drops were “injected” along two vectors in a vertical plane, each inclined to the nozzle centerline at half the mean cone angle (α) — one oriented above and the other below the nozzle axis.

Fig. 3. Typical normalized size distribution from image analysis. Nozzle diameter 4 mm, pressure 3 bar(g).

882

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

2.2. Drop size and velocity distribution function The drop diameters and velocities in a spray cannot be computed analytically from first principles, hence, the maximum entropy formalism (MEP) employed by Kachhwaha et al. [1] was used. From the photographs, the number distribution function, fn(D), was obtained as: fn ð DÞ ¼

Dn¯ Dni =DDi ¼ N P DDi Dni

ð1Þ

i

If the drop diameter D is treated as a continuous variable in the range 0 to ∞, or between finite limits, Dmin to Dmax, Eq. (1) can be written as below for use in the MEP. fn ð DÞ ¼

d n¯ dn=dD ¼ dD Z∞ ðdn=dDÞdD

ð2Þ

0

The function fn(D) was obtained from the Rosin–Rammler distribution using the volume mean diameter, D30. The maximum entropy principle (Li and Tankin [4] and Li et al. [5]) was employed; Eqs. (3a)–(3c) below are, respectively, the non-dimensional mass, momentum and energy conservation equations with their source terms, and Eq. (3d) is for normalization. ZD¯ max ZU¯ max ¯ min D

ð3aÞ

¯ dU ¯ dD ¯ ¼ 1 þ S¯mv ¯3 U f D

ð3bÞ

U¯ min

ZD¯ max ZU¯ max ¯ min D

¯3 d U ¯ dD ¯ ¼ 1 þ S¯m f D

U¯ min

ZD¯ max ZU¯ max   ¯ þ BD ¯2 d U ¯3 U ¯ dD ¯ ¼ 1 þ S¯e f D ¯ min D

ð3cÞ

U¯ min

U¯ max ZD¯ max Z

¯ dD ¯¼1 f dU ¯ min D

ð3dÞ

U¯ min

Here, f is the joint drop diameter–velocity distribution function which maximizes Shannon's entropy subject to the constraints of Eqs. (3a)–(3d) and has the form: n  o ¯ 3  a2 D ¯3 U ¯  a3 D ¯3 U ¯ þ BD ¯2 ¯2 exp a0  a1 D f ¼ 3D ð4Þ ¯ = D/D30 and non-dimensional velocity U¯ = U/U0, Weber number where the non-dimensional diameter D 2 We = ρlUN D30/σ, and B = 12/We. The number distribution function obtained by integrating Eq. (4) from minimum to maximum velocities becomes: dN ¼ ¯ dD

U¯ max Z

¯¼ fd U

 1=2     ¯ 3 D a2 ¯3 ¯ 3  a1  2 D ½erf ðZmax Þ  erf ðZmin Þexp a0  a3 BD 2 a3 4a3

U¯ min

      a22 3 1=2 a22 3 1=2 ¯ ¯ ¯ max þ 2a ¯ D D and Z ¼ U þ . where Zmax ¼ U a a 3 min min 3 2a3 3

ð5Þ

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

883

Likewise, the number-based velocity distribution is obtained by integrating Eq. (4) over minimum to maximum R D¯ drop diameters as dN¯ ¼ D¯ fd D¯. The distinct features of the nozzles and the atomization are not explicitly modeled but dU appear via the various measured parameters. An analytical solution of these equations is not possible, necessitating a numerical solution. max

min

2.3. Heat and mass transfer simulations The drop trajectory and heat and mass transfer with air were calculated by solving the conservation equations as a system of ODEs (Kachhwaha et al. [1], Sureshkumar [2]). These are listed below. (a) Conservation of mass Accounting for moisture addition, the conservation of mass for air is: qa ð1  xÞuA ¼ Constant

ð6aÞ

The increase in water vapor mass fraction in air occurs due to evaporation from drops, tunnel surfaces and drift eliminator, and is given by:

dx X ni h˜ i As;i xs;i  x h˜x;wall Pðxwall  xÞ x du x dqa ¼   þ Ux;i u dx Au u dx qa dx i The drop diameter change for category ‘i’ due to evaporation is:

dDi h˜ i qa xs;i  x ¼ dx ql Ux;i

ð6bÞ

ð6cÞ

(b) Conservation of momentum In the experiments, the wind tunnel was at ambient pressure and momentum exchange between drops and air was negligible. Therefore, the air momentum equation was not used in the simulations. The drop x- and y-momentum equations are, respectively:

0:75CD;i qa Wi Ux;i  u dUx;i 3Ux;i dDi ¼ ð7aÞ  dx Di dx Ux;i ql Di dUy;i gðql  qa Þ 0:75CD;i qa Wi Uy;i 3Uy;i dDi ¼   Ux;i ql dx Ux;i ql Di Di dx

ð7bÞ

The drop trajectory equation, for ‘i’-category drops is given by: dyi Uy;i ¼ dx Ux;i

ð7cÞ

(c) Conservation of energy The conservation of energy for air with heat transfer from drops and walls is: dTa ¼ dx

h i

X ni hfg;i pD2i h˜i qa xs;i  x  ni h˜ i pD2i ðTa  Td Þ Cp;av qa uUx;i h i þ hfg;wall h˜ x;wall qa ðxwall  xÞ  h˜ wall ðTa  Twall Þ i



hfg;0 dx Cp;av dx

   P ha 1 du 1 dqa þ  Cp;av qa uA Cp;av u dx qa dx ð8aÞ

884

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

The conservation of energy for drops includes drop temperature change for ‘i’ category due to heat and mass transfer with air: h



i ˜ i Ta  Td;i  hfg h˜i q xs;i  x 6 h a dTd;i 3hd;i dDi ¼ ð8bÞ  dx Cpd;i Di dx Di ql Ux;i Cpd;i (d) Source terms The mass, x-momentum (y-momentum is similar) and energy source terms, respectively are given by Eqs. (9a), (9b) and (9c): 3

Dout  D3in Sm;d ¼ pqd N ð9aÞ 6

ud;out D3out  ud;in D3in Smu;d ¼ pqd N ð9bÞ 6 h i



hv  hfg;o d;out D3out  hv  hfg;o d;in D3in Þ Se;d ¼ pqd N ð9cÞ 6 Various property correlations detailed in Kachhwaha et al. [1] were used. By specifying air inlet and spray conditions, the outlet air DBT and specific humidity were computed. For ease of distinction, the term parcel is used, defined as a unique combination of initial diameter, initial velocity magnitude and initial velocity direction (vector). After setting the initial drop diameter and velocity, a Lagrangian marching technique using a 4th-order Runge–Kutta method is employed. The contribution of each parcel was obtained by multiplying the individual drop contribution by the number of similar drops. Subsequent summing up of contributions from each parcel gives the total cooling and moisture addition. These simulations differ from those by Kachhwaha et al. [1] in the number of drop conservation equations that formed the set of ODEs. Kachhwaha et al. [1] employed 5 categories with a single velocity for 2 directions that resulted in 50 ODEs for drops, and a total of 52 ODEs. Here, for 5 categories, there were 10 velocity subclasses for 2 directions resulting in 500 ODEs for drops for a total of 502 ODEs. In both cases, the system of ODEs was solved iteratively for air and drops until changes in air properties were less than a specified value. 3. Spray modeling Kachhwaha et al. [1] employed five categories with a uniform velocity resulting in 10 parcels. Here, first, the number of drop categories was successively increased from 5 to 50, 100 and 400 by suitably discretizing the joint drop diameter–velocity distribution function. Next, 10 different velocity magnitudes were assigned (velocity sub-classes) and they were also obtained from the joint distribution. Based on the joint distribution function, a fraction of the drops within each category were assigned one of the ten velocities. These parcels are more realistic of drop initial conditions and were inputs to the heat and mass transfer simulation. 4. Simulation results Results from these simulations are compared with the experimental data of Sureshkumar [2] that were obtained during two climatic conditions, viz., hot–dry and hot–humid, for each nozzle size, at three water pressures and three air velocities (1, 2 and 3 m s− 1). Predictions using the code of Kachhwaha et al. [1] were in agreement with experimental data within ± 30%. On increasing the number of drop categories from 5 to 50, 100 and 400 while assigning the same velocity to each drop, the predictions were almost identical. The best agreement with experiments was obtained with 100 categories (200 parcels). The results with 5 diameter categories and 10 velocity sub-classes showed better agreement with experiments than with a uniform velocity. An increase in diameter categories from 5 to 20 or 40 while employing 10

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

885

Fig. 4. Comparison of DBT residuals from predictions of the three spray models. Model 1: 5 diameters × 1 velocity; Model 2: 100 diameters × 1 velocity; Model 3: 10 diameters × 10 velocities.

velocity sub-classes did not have any effect on the simulation outcomes. Thus, 10 drop diameter categories and 10 velocity sub-classes appear to be optimum for simulation purposes. The effects of the above are illustrated in the whisker plots of Fig. 4. Here, the DBT residual, defined as the difference between predicted and experimental DBT, is plotted for the three methods. Method 1 is for 5 categories with uniform velocity (Kachhwaha et al. [1]), Method 2 is for 100 categories with uniform velocity, and Method 3 is for 10 categories with 10 velocity sub-classes. The statistics shows that the mean predictions from Methods 2 and 3 are better than that of Method 1. Further, there are no significant differences between the variances of Methods 2 and 3 indicating that both perform equally well. Identical conclusions are seen in the analysis of the specific humidity residual. In subsequent simulations, therefore, 10 categories with 10 velocity sub-classes were taken as inputs to the simulations. Comparisons between simulations and experiments are presented in Fig. 5 for hot–dry and in Fig. 6 for hot–humid climates. In the former, evaporative cooling is significantly greater than in the latter and uncertainties relatively lower. In hot–dry climate, the simulations predict less DBT change than in experiments, Fig. 5(a), and the opposite is seen for specific humidity change, Fig. 5(b). The spread, however, is less than that of Kachhwaha et al. [1]. For hot–humid conditions, there is an under-prediction of DBT changes, Fig. 6(a), as well as in specific humidity change, Fig. 6(b).

Fig. 5. Comparison of predictions with experimental data in hot and dry conditions for four nozzle diameters. (a) DBT, (b) Specific humidity.

886

R. Sureshkumar et al. / International Communications in Heat and Mass Transfer 34 (2007) 878–886

Fig. 6. Comparison of predictions with experimental data in hot and humid conditions for four nozzle diameters. (a) DBT, (b) Specific humidity.

Most agreements are within ± 15% and the remaining few within ± 30%. Considering the inherent uncertainties in the empirical correlations for heat and mass transfer, the overall uncertainty in predictions is reasonable. There could be other reasons for the mismatch. The 1-D model cannot adequately capture the 3-D phenomena resulting from the conical spray interacting with air flow in the square duct. Span-wise variations would be significant and, hence, rapid span-wise mixing assumed in the simulations would be inadequate. Spray photographs showed that the inter-drop spacing in their flow direction is about 5–10 drop diameters and, therefore, drop–drop interactions would have a significant effect on heat and mass transfer. 5. Conclusions The DBT and specific humidity changes in an air-water spray system in parallel flow were obtained from a 1-D heat and mass transfer simulation. The number of drop diameter categories and velocity sub-classes were varied. A larger number of categories and different velocities within a category improved the predictions up to a certain level of discretization beyond which there was no change in the simulation results. Thus, by using an optimum number of categories and velocity sub-classes, reasonably accurate predictions are obtainable with savings in computation time. References [1] S.S. Kachhwaha, P.L. Dhar, S.R. Kale, Experimental studies and numerical simulation of evaporative cooling of air with a water spray — I. Horizontal parallel flow, Int. J. Heat Mass Transfer 41 (1998) 447–464. [2] R. Sureshkumar, Heat and mass transfer studies on air and water-spray interactions, Ph.D. thesis, Mechanical Engineering Department, IIT Delhi, 2000. [3] A.K. Aigal, S.K. Singhal, B.P. Pundir, Analysis of measured droplet size distribution of air deflected diesel spray, Proc. IMechE J Automobi. Eng. 208 (1995) 33–43. [4] X. Li, R.S. Tankin, Derivation of droplet size distribution in sprays by using information theory, Combust. Sci. Technol. 60 (1988) 345–357. [5] X. Li, L.P. Chin, R.S. Tankin, T. Jackson, J. Stutrud, G. Switzer, Comparison between experiments and predictions based on maximum entropy for sprays from pressure atomizer, Combust. Flame 86 (1990) 73–89.