Journal of Thermal Science Vol.19, No.3 (2010) 253260

Received: November 2009 Roy J. Issa: Assistant Professor

www.springerlink.com

DOI: 10.1007/s11630-010-0253-8 Article ID: 1003-2169(2010)03-0253-08

Heat Transfer Optimization for Air-Mist Cooling between a Stack of Parallel Plates

Roy J. Issa Mechanical Engineering Division, Department of Engineering and Computer Science West Texas A&M University Canyon, Texas 79016, USA E-mail: rissa@mail.wtamu.edu, Tel: (806) 651-5261

Science Press and Institute of Engineering Thermophysics, CAS and Springer-Verlag Berlin Heidelberg 2010

A theoretical model is developed to predict the upper limit heat transfer between a stack of parallel plates subject to multiphase cooling by air-mist flow. The model predicts the optimal separation dis-tance between the plates based on the development of the boundary layers for small and large separa-tion distances, and for dilute mist conditions. Simulation results show the optimal separation distance to be strongly dependent on the liquid-to-air mass flow rate loading ratio, and reach a limit for a criti-cal loading. For these dilute spray conditions, complete evaporation of the droplets takes place. Simu-lation results also show the optimal separation distance decreases with the increase in the mist flow rate. The proposed theoretical model shall lead to a better understanding of the design of fins spacing in heat exchangers where multiphase spray cooling is used.

Keywords: Air-mist, Multiphase, Heat transfer, Liquid-to-air loading

Introduction

In the external cooling of cross-flow finned tubular heat exchangers, a single-phase fluid is traditionally used to cool a bundle of tubes. Substantial amount of research has also been done during the last few decades on heat exchangers using multiphase fluids, particularly air and water sprays. Yang and Clark [1] conducted experimental studies on tubular heat exchangers with plain, louvered, and perforated fins using air-water sprays. Their results showed the friction losses were not affected by the water sprays, and improvements in the multiphase heat transfer coefficient, as compared to cooling by air alone, were in the range between 40 to 45% for air Reynolds numbers ranging from 500 to 1000. Bhatti and Savery [2] intro-

duced fine size water droplets into the thermal boundary layer formed around the fins for a heat exchanger which resulted in a great improvement in the heat transfer proc-ess. Trela [3] developed a numerical model to calculate the heat transfer coefficient for a laminar air-mist flow over a flat plate. Walczyk [4] conducted studies on an air-fin roof-type condenser sprayed with water (up to 10% by weight). His experiments showed heat transfer enhancement close to 125 %.

Song et al. [5] developed a theoretical model for the cooling of a finned channel in a heat exchanger. The model examined the heat transfer enhancement of the air-cooled fins by the evaporative cooling of a thin water film. Heat transfer enhancement was shown to depend greatly on the fin thickness, and for fins that are not suf-

254 J. Therm. Sci., Vol.19, No.3, 2010

Nomenclature A Area (m2) Viscous energy source a, b Constants Liquid-to-air mass flow rate loading ratiocp Specific heat at constant pressure (J/kg.K) g Gas thermal diffusivity (m2/s) D Plate-to-plate distance (m) Density (kg/m3) H Plate height (m) Dynamic viscosity (N.s/m2) h Enthalpy (J/kg) Shear stress (N/m2) hD Mass transfer coefficient (m/s) V Kinematic viscosity (m2/s) hfg Enthalpy of vaporization (J/kg) Finite difference hg Gas heat transfer coefficient (W/m2.K) Flow boundary layer thickness (m) k Conductivity (W/m.K) Partial derivative notation L Length of the entire package of fins (m) Subscripts Le Lewis number 1 Channel entry side m Mass flow rate (kg/s) 2 Channel exit side n Number of channels a Dry air Nu Nusselt number c Conduction/convection at the wall P Pressure, total pressure (N/m2) ch Per channel Pr Prandtl number d Droplets

Q Total heat transfer from n channels (W) e Evaporation chQ Heat transfer rate per channel (W) g Gas (air and water vapor) "q Heat flux per plate (W/m2) i at y equal to 0 dq Droplets sensible heating/volume (W/m3) l Liquid

Re Reynolds number opt Optimal T Temperature (K) pl Plate u x-component for velocity (m/s) t Total v y-component for velocity (m/s) v Vapor

U Average velocity for developed flow (m/s) w Wall W Channel width (m) x Channel cross-section Mist quality Free stream x, y Rectangular coordinates (m) Superscripts Greek letters " Flux Absolute humidity Overbar Relative humidity _ Average

ficiently thick, heat transfer enhancement by evaporative cooling was shown to decrease. Esterhuyse and Kroger [6] studied the effect of ionization of the water droplets in air on the surface wetting of fins in a heat exchanger device. Droplets were electrostatically charged to the same polarity as the heat exchanger surface to prevent them from impacting and wetting the surface; therefore, reducing corrosion. Their test results showed that drop-lets deposition decreased with the increase in the electric charge, and for a sufficient high charge, no deposition was found to occur on the fins. Novak et al. [7] experi-mentally investigated mist cooling in a vertical channel.

Water mass fractions between 10 to 19% were applied in a circular channel with droplets size distribution from 8 to 100 microns. Their results showed mist cooling can increase the heat transfer coefficient by several orders of magnitude compared to air alone. Novak et al. [8] also developed a numerical model for the air-mist cooling inside vertical channels. Good agreement was obtained between their test results and the model. The research conducted by all these researchers did not consider what the best fins arrangement should be in the confined space of a heat exchanger. None of the above researchers con-ducted studies on the optimization of fins spacing for

Roy J. Issa. Heat Transfer Optimization for Air-Mist Cooling between a Stack of Parallel Plates 255

maximum possible heat transfer when air-mist cooling is used. The only analysis reported in the review of litera-ture on the optimization of fins spacing was that by Bejan and Sciubba [9] for cooling by forced air.

Theoretical Approach

Optimizing the plate-to-plate spacing, D, in a heat ex-changer system where the cooling medium is an air-mist spray consists of considering two extreme cases for the boundary layers developing at the plate walls: a) small-D limit approach where the two boundary layers merge quickly together in the channel, and b) large-D limit ap-proach where the two wall boundary layers remain sepa-rate in the channel. The optimal separation distance be-tween the plates resulting in maximum heat transfer oc-curs where the solution by both approaches intersect each other.

The heat transfer from an air-mist spray is very much dependent on the spray droplets size and the liquid-to-air loading ratio. The smaller the droplets size, the lower is the liquid-to-air loading, the easier it is for the droplets to evaporate in the confined channel between the parallel plates; thus leading to higher heat transfer enhancement. Also, smaller size droplets will better cool the thermal boundary layer near the plate surface by increasing the near surface local vapor convection. However, high liq-uid-to-air loadings or large size droplets will have a det-rimental effect on the surface cooling because of the sur-face flooding that can occur. The analysis presented in this paper is for mist size droplets and dilute sprays (low liquid-to-air loadings).

The objective of this research is to obtain a relation-ship for the optimal distance between a stack of parallel plates (Fig. 1) for maximum internal cooling as function of the air-mist flow operating conditions. Simulation re-sults will reveal the influence of the liquid-to-air loading ratio on this optimal distance.

Fig. 1 Internal cooling of a stack of parallel plates by air-mist.

Small-D limit (fully developed boundary layers) In the small-D limit case, the separation distance be-

tween the plates is assumed to be very small such that the

boundary layers forming on the opposing plates quickly merge together near the entrance of the channel. Forced air from a fan or blower mixes with fine-size water drop-lets from a spray nozzle before entering the channel as shown in Fig. 2. The flow is considered to be laminar in this case. The percentage by weight of the water droplets in the liquid/air mixture is assumed to be very small; therefore, the thickness of the water film layers develop-ing at the walls can be considered negligible. Further-more, perfect mixing between the air and water droplets is assumed to take place that results in homogeneous cooling across the channel. In this analysis, the plates are held at a temperature that is higher than both the air and water droplets incoming temperatures.

Fig. 2 Air-mist flow between parallel plates (small-D limit). Applying the first law of thermodynamics to the

air-mist flow between the entry and exit regions of the parallel plates (1 channel):

( ) ( )( )( )

,2 ,2 ,2 ,2 ,1 ,1 ,1 ,1

,2 ,2 ,2 ,2 ,2 ,2

,1 ,1 ,1 ,1 ,1 ,1

ch a a d d a a d d

a a v v d d

a a v v d d

Q m h m h m h m h

m h m h m h

m h m h m h

= + += + + + +

(1)

The conservation of mass flow rate can be applied separately to both dry air and water. dry air:

,1 ,2a a am m m= = (2-a) water:

,1 ,1 ,2 ,2v d v dm m m m+ = + (2-b) Define 1 and 2 to be the absolute humidity at the

entry and exit side of the channel, respectively: 1 ,1,1

1,1 1 1 ,1

0.622 gva g

Pmm P P

= = (3-a)

2 ,2,22

,2 2 2 ,2

0.622 gva g

Pmm P P

= = (3-b)

Substituting Eq. (2-a), (3-a) and (3-b) into Eq. (2-b):

( )( )

,2 ,1 ,2 ,1

,1 2 1

d d v v

d a

m m m m

m m = =

(4)

Also, by substituting Eq. (2-a) and (4) into Eq. (1), and re-arranging, the equation for heat transfer in the channel becomes:

256 J. Therm. Sci., Vol.19, No.3, 2010

, 2 1 2 ,2 1 ,1

,1 ,2 2 1 ,2 ,1 ,1

, 2 1 2 ,2 1 ,1

2 1 ,2 ,1 ,2 1

( )

( )

( ) ( )

( ) ( , )

ch a p a a v a v

d d a d d d

a p a a v v

a d d d d

Q m c T T m h m h

m h m h m hm c T T m h h

m h m h h

= + +

= + +

(5)

Let be the liquid-to-air mass flow rate loading ratio: ,1d

a

mm

= (6) Substituting Eq. (6) into (5), chQ can be written as:

, 2 1 2 ,2 1 ,1

2 1 ,2 ,2 ,1

[ ( ) ( )

( ) ( )]ch a p a v v

d d d

Q m c T T h h

h h h

= + +

(7)

Where the air mass flow is:

a a x am A U HDU = = (8) The pressure drop by the air-mist flow in the channel,

P, can be approximated by:

212gWUPD

= (9) From Eq. (9), an expression for the air velocity is ob-

tained: 2

12 g

D PUW

= (10) The air mass flow rate can now be expressed as a

function of the pressure drop by substituting Eq. (10) into (8):

3112

aa

g

Hm PDW

= (11)

By substituting Eq. (11) into (7), the equation for chQ can be re-written as:

32 1

2 1 2 1

2 1

1= [ ( )12

+( ) ( )+ ( )]

ach p,a

g

v,2 v,1 d,2

d, d,

HQ PD c T T W

h h h h h

(12)

The total heat transfer from n channels formed be-tween (n+1) parallel plates is:

chQ = nQ (13) where,

LnD

= (14) Therefore, the total heat transfer for the air-mist cool-

ing inside n channels can now be calculated: 2

, 2 1

2 ,2 1 ,1 2 1 ,2

,2 ,1

1 [ ( )12

( ) ( )

( )]

ap a

g

v v d

d d

HLQ PD c T TW

h h hh h

= + +

(15)

Large-D limit (separate boundary layers) In the large-D limit case, the separation distance

between the plates is assumed to be quite large such that the two boundary layers on the opposing plates do not meet (Fig. 3). In this case, dilute water flow conditions are assumed to prevail, and as a result, the water film boundary layer thickness is considered to be negligible. Since the thermal and hydrodynamic boundary layers do not meet, the temperature and velocity of the mixture will vary along the cross section of the channel. The formula-tion that is presented here is for a laminar flow.

Fig. 3 Air-mist flow between parallel plates (large-D limit). For an incompressible flow, the continuity equation

for the gas mixture is:

0g gu vx y

+ = (16) The velocity profile inside the boundary layer can be

represented by the following cubic function: 33 1

2 2gu y y

u = (17)

where, 4.64 g x/u = (18)

By substituting Eq. (18) into (17), the gas velocity profile can be expressed as:

1/ 2 3/ 21/ 2 3/ 2 30.323 0.005g

g g

u u ux y x yu

=

(19) where,vg is calculated by integrating Eq. (16) between the limits:

,0

yg

g g iu

v v yx

= (20) and the average value of gv inside the gas boundary layer

can then be obtained: 1/ 2 1/ 2 1/ 2

, 0.6091g g iv v w u = + (21) ,g iv is the average velocity of the water vapor blowing

into the gas boundary layer at the interface of the water

Roy J. Issa. Heat Transfer Optimization for Air-Mist Cooling between a Stack of Parallel Plates 257

film and gas boundary layers. ,g iv is approximated

from: "

, ,/g i e v wv m = (22) In this application, dilute spray conditions prevail.

Therefore, the liquid film deposition on the wall is much less than the gas boundary layer thickness ( gl

258 J. Therm. Sci., Vol.19, No.3, 2010

" " "t c eq q q= + (38)

For a laminar gas flow over a flat plate, the average shear stress at the wall is evaluated from [10]:

2 1/ 20.664 Rew g g wu = (39) where,

Re gwg

u W= (40)

Substituting Eq. (40) into (39), w becomes: 3/ 2 1/ 2 1/ 20.664w g gu W = (41)

The pressure drop in the gas flow due to the wall shear stress is:

3/ 2 1/ 2 1/ 21.3282 g gw

u WWPD D

= = (42) From Eq. (42), the gas velocity can be calculated as a

function of the pressure drop: 2/3

1/ 2 1/ 21.328g g

D PuW

= (43)

For a laminar gas flow over a plate, Nusselt number is approximated from the following empirical correlation [10] for Pr 0.6:

1/3 1/ 20.664 Pr Reg wg

h WNu

k= = (44)

Substituting Eq. (40) and (43) into Eq. (44), the aver-age heat transfer coefficient of the gas, gh , is then ob-

tained: 2 /3 1/3 1/3 1/3 2 /3 1/30.6041 Prg g g g gh k P W D = (45)

The total heat transfer for the internal cooling of n channels (formed between n+1 parallel plates) consists of the heat transfer from 2n separate boundary layers. The total heat transfer is:

" "2 2pl t tLQ nA q WHqD

= = (46) Substituting Eq. (34), (37), (38) and (45) into Eq. (46),

the equation for the total heat transfer becomes:

( )

2/ 3 1/ 3 1/ 3 1/3 1/3 2/ 3

,2

,

, ,2 / 3

,

1.2082 Pr *

( )

exp( / ) 1

( )

1 exp( / )

g g g g

w g g

g g g g

w p l g g g

gg g p g g g g g

v w v fg

g p g

Q k P W LHD

T T v

v

T T c k kvc v

h

c Le

=

+ + +

(47) The optimal separation distance between the plates for

maximum heat transfer, Dopt, occurs when:

arglim limsmall D l e D

it itQ Q = (48)

And optD can be shown as:

( ){3 / 8

3/8 1/ 4 1/8 1/ 2 3/8 1/ 2 1/ 4

, 2 ,2 1 ,1 2 1 ,2

,2 ,1

,2

,

2.726 Pr

( ) ( ) ( )

( )( ) *

exp( / ) 1

( )

1 exp( / )

opt g g g g g

p g w v v d

w g gd d

g g g

w p l g g g

gg g p g g g g g

D k W P

c T T h h h

T T vh h

v

T T c k kvc v

= +

+ + +

( ) 3/8, ,

2 /3,

v w v fg

g p g

h

c Le

+

Special case (cooling by air only) When the cooling fluid is air only, Eq. (15) will reduce

to: , 21 ( )

12a p a

wa

c HLQ P T T DW

=

(50)

Also, Eq. (47) will reduce to:

[ ]

2/ 3 1/ 3 1/ 3 1/3 1/3 2/ 31.2082 Pr *

( )exp( / ) 1

a a a a

a aw

a a a a

Q k P W LHDv

T Tv

=

(51)

By equating the above two equations, the optimal separation distance between the plates becomes:

[ ]3/8 3/8 1/8 1/ 2 3/8 3/8

3/81/ 2 3/8 1/ 4 1/ 4 3/8,

2.726 Pr

exp( / 1)a a a a a

opt

a p a a a a a a

k W vD

c P v

=

(52)

Numerical Simulations

Simulation results presented in this paper attempt to portray the sensitivity of the heat transfer rate occurring inside n channels (formed between a stack of n+1 parallel plates) on the separation distance between the plates. Equation (15) for the small-D limit is plotted together with Eq. (47) as function of the separation distance D. Figures 4 through 6 show the results of three cases: cooling by air, and cooling by air-mist with liquid-to-air loadings of 1% and 5%. The thermal properties of the gas mixture (air and vapor) are calculated using the equation

(49)

Roy J. Issa. Heat Transfer Optimization for Air-Mist Cooling between a Stack of Parallel Plates 259

proposed by Brokaw [11]. Simulations are presented for the case where a stack of parallel plates that are 1 m wide (W), 0.1 m high (H), and are packaged in a space that is 0.3 m long (L). The plates are held at a constant tem-perature of 80. The air temperature is assumed to be 20 , and the air inlet velocity is 1 m/s. Simulations show the optimal distance between the plates to be strongly dependent on the liquid-to-air loading, and decreases as the liquid mass flow rate increases. For these low liquid loadings and operating conditions, water is assumed to completely evaporate by the time it exits the channels formed by the parallel plates.

Figure 7 shows the relationship formed between the optimal separation distance, Dopt, and the liquid-to-air loading ratio, The optimal separation distance is shown to reach a saturation level as the liquid-to-air loading

Fig. 4 Cooling of parallel plates by air alone. (= 0, am = 0.0323 kg/s, ,1dm = 0 kg/s, L = 0.3 m, W = 1 m, H = 0.1 m, T = 20, Tw = 80, U = 1 m/s)

Fig. 5 Cooling of parallel plates by air-mist. ( = 1/100, am = 0.0323 kg/s, ,1dm = 0.000323 kg/s, L = 0.3 m,

W = 1 m, H = 0.1 m, T = 20 oC, Tw = 80 oC, U = 1 m/s)

Fig. 6 Cooling of parallel plates by air-mist. ( = 5/100,

am = 0.0323 kg/s, ,1dm = 0.00162 kg/s, L = 0.3 m, W = 1 m, H = 0.1 m, T = 20, Tw = 80, U = 1 m/s)

Fig. 7 Optimal plate-to-plate distance versus liquid-to-air

loading ratio ( am = 0.0323 kg/s, L = 0.3 m, W = 1 m, H = 0.1 m, T = 20, Tw = 80, U = 1 m/s)

approaches 10%. As the loading ratio increases above 10%, water as a thin liquid film start forming at the plates.

Conclusions

A mathematical model is developed to maximize the internal cooling between a stack of parallel plates where the cooling fluid is a mixture of air and mist. The model predicts the optimal separation distance between the plates for dilute water spray conditions (low liquid-to-air loading ratios). Formulation for the optimal separation distance is based on the assumption of laminar flow con-ditions. Heat transfer between the plates is shown to be strongly dependent on the amount of water droplets in the liquid/air flow mixture. The optimal separation dis-tance between the plates is shown to decrease with the increase in the water content in air, but reaches a satura-

260 J. Therm. Sci., Vol.19, No.3, 2010

tion level at a relatively low liquid-to-air loading ratio of around 10% for the considered cases.

References

[1] Yang, W.J., and Clark, D.W.: Spray Cooling of Air- Cooled Compact Heat Exchangers, International Journal of Heat and Mass Transfer, vol.18, no.2, pp. 311317, (1975).

[2] Bhatti, M., and Savery, C.: Augmentation of Heat Trans-fer in a Laminar External Boundary Layer by the Vapori-zation of Suspended Droplets, Journal of Heat Transfer, Ser. C, vol.97, no.2, pp. 179184, (1975).

[3] Trela, M.: An Approximate Calculation of Heat Transfer during Flow of an Air-Water Mist along a Heated Flat Plate, International Journal of Heat and Mass Transfer, vol.24, no.4, pp. 749755, (1981).

[4] Walczyk, H.: Enhancement of Heat Transfer from Air-Fin Coolers with Water Spray, Chemical Engineering and Processing, vol.32, no.2, pp. 131138, (1993).

[5] Song, C.H., Lee, D.Y., and Ro, S.T.: Cooling Enhance-ment in an Air-Cooled Finned Heat Exchanger by Thin Water Film Evaporation, International Journal of Heat and Mass Transfer, vol.46, no.7, pp. 12411249, (2003).

[6] Esterhuyse, B.D., and Kroger, D.G.: The Effect of Ioniza-

tion of Spray in Cooling Air on the Wetting Characteris-tics of Finned Tube Heat Exchanger, Applied Thermal Engineering, vol.25, no.17-18, pp. 31293137, (2005).

[7] Novak, V., Sadowski, D.L., Schoonover, K.G., Abdel- Khalik, S.I., and Ghiaasiaan, S.M.: Heat Transfer in Two- Component Internal Mist Cooling: Part I. Experimental Investigation, Nuclear Engineering and Design, vol.238, no.9, pp. 23412350, (2008).

[8] Novak, V., Sadowski, D.L., Schoonover, K.G., Abdel- Khalik, S.I., and Ghiaasiaan, S.M.: Heat Transfer in Two- Component Internal Mist Cooling: Part II. Mechanistic Modeling, Nuclear Engineering and Design, vol.238, no.9, pp. 23512358, (2008).

[9] Bejan, A., and Sciubba, E.: The Optimal Spacing of Par-allel Plates Cooled by Forced Convection, International Journal of Heat and Mass Transfer, vol.35, no.12, pp. 32593264, (1992).

[10] Howarth, L.: On the Solution of Laminar Boundary Layer Equations, Proceedings of the Royal Society of London, Series A, vol.164, no.919, pp. 547579, (1938).

[11] Brokaw, R.S.: Approximate Formulas for Viscosity and Thermal Conductivity of Gas Mixtures, NASA Technical Note D-2502, Lewis Research Center, Cleveland, Ohio, (1964).

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