performance model for a horizontal tube falling film evaporator

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Performance Model for a Horizontal TubeFalling Film EvaporatorRohit Sharma a & Sushanta K. Mitra aa Department of Mechanical Engineering , Indian Institute ofTechnology , Bombay, Mumbai, IndiaPublished online: 06 Feb 2007.

To cite this article: Rohit Sharma & Sushanta K. Mitra (2005) Performance Model for a Horizontal TubeFalling Film Evaporator, International Journal of Green Energy, 2:1, 109-127

To link to this article: http://dx.doi.org/10.1081/GE-200051315

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International Journal of Green Energy, 2: 109–127, 2005Copyright © Taylor & Francis Inc.ISSN: 0197-1522 print / 1543-5083 onlineDOI: 10.1081/GE-200051315

109

PERFORMANCE MODEL FOR A HORIZONTAL TUBE FALLING FILM EVAPORATOR

Rohit Sharma and Sushanta K. Mitra*Department of Mechanical Engineering, Indian Institute of Technology, Bombay,Mumbai, India

A model is developed to predict the evaporative heat transfer coefficient in a horizontaltube-falling evaporator and has been applied to evaluate overall performance of a desalina-tion unit. Performance variation with different parameters like operation temperature, typeof water distribution system, mass flow rate of distilled water inside the exchanger areanalyzed. It has been observed that the model is able to predict the trends of heat transfercharacteristics of the evaporator reasonably well. However, at low liquid film flow rateconditions, the model overpredicts the heat transfer characteristics marginally. In order toimprove the evaporative exchanger performance, it is observed that preheating of the liquidfilm before injection into the evaporator is desirable. Calculations are also performed toestimate the value of overall heat transfer coefficient for a typical desalination unit.

Keywords: Desalination; Falling-film; Horizontal-tube; Evaporative heat transfer

INTRODUCTION

The desalination of sea water is a process vital to many countries in the arid regionsof the world. Desalination plants can take different forms, with the most popular being themulti-stage flashing design (Ansari & Owen, 1999), where the brine is heated by passingat relatively high pressure through the inside of the tubes while steam is condensing on theoutside. Other types include the multi-effect boiling evaporator design, in which thestream-flowing tubes are submerged inside a pool of water, but this has a relatively lowheat transfer coefficient (Ansari & Owen, 1999). Recently the concept of falling filmevaporator has been introduced and has become popular in the industries due to its highheat transfer coefficient (Ansari & Owen, 1999).

Literature reviews reveal that limited modeling for heat and mass transfers is avail-able for a generalized desalination unit. Analytical study for the heat transfer of evaporatingliquid films on a vertical surface, in laminar regime, is conducted by Nusselt (Hewitt &Taylor, 1970 ). An expression for the film thickness was developed. A simple model forcombined evaporation and boiling of liquid films was developed by Lorenz and YungLorenz & Yung (1979). The developing length was estimated by using Nusselts’s expres-sion. Chyu Chyu & Bergles (1987) presented two models for the saturated, non-boiling,falling film evaporation on a horizontal tube. Both models are based on three defined heat

*Correspondence: Sushanta K. Mitra, Department of Mechanical Engineering, Indian Institute ofTechnology, Bombay, Mumbai, 400076, India. E-mail:[email protected]

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110 R. SHARMA AND S. K. MITRA

transfer regions: the jet impingement region, the thermal developing region, and the thermaldeveloped region. The only difference between the two models is in the manner withwhich the thermal developed region is being considered. In one of the models, the thermaldeveloped region is based on the Nusselt equations (Hewitt & Taylor, 1970 ), whereas theother one is based on experiments conducted by Chun and Seben Chun & Seban (1999).

A generalized model for a desalination unit working on a horizontal tube-fallingfilm evaporator is considered here. This model incorporates property variation with tem-perature. Various flow regimes are also taken into account here. The numerical resultspresented in the model have been compared with the experimental results of BourouniBourouni et al. (1997).

PROBLEM STATEMENT

The schematic of the desalination unit has been shown in the Fig. 1 (Bourouni et al.1997). The steam flows through the tubes with an inlet temperature of about 80°C. Thecooling air moves up in the space between the tubes. The liquid film is dripped from thedistributor and it flows downward from one tube to another. The feed water distributionsystem is a perforated tube, with a row of holes (2 × 10–3 m in diameter) placed at 5 × 10–3 mseparation. A fraction of water is evaporated and carried away by ascending air flow,which is maintained by the blower. At the top of the exchanger, the hot humid air is drivento the condenser, where distilled water is recuperated. Water drips down and hot steamflows inside the tubes in cross-counter current direction.

A typical flow configuration over the tube bundle is shown in Fig. 2, where thecoordinates are chosen in such a way that positive z is in the direction of flow of the film.The evaporator has been divided into number of cells where the ith cell represents a singletube. For the ith cell, steam enters at temperature Tin,hl,i, flows with Reynolds numberRein,hl,i and leaves at Tout,hl,i. In the ith cell, film temperature is Tin,f,i and the mass flow rate

Figure 1 Schematic of a desalination unit.

Humid Air

Condenser

Evaporator

Distilled WaterSalt Water

Cold Water

Blower

Pump

Hot Water

Air

Pre heated salt water

Distributor

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PERFORMANCE MODEL FOR FILM EVAPORATOR 111

is Γin,f,i. The film falls on the next tube with temperature Tout,j,i and has a mass flow rate ofΓout,f,i, which is lower than Γin,f,i. Expressions for heat transfer coefficient for evaporationand condensation for a single tube are used for evaluating performance of the evaporator.

The mathematical model is based on the following assumptions:

• The entire surface of the tube is covered with the liquid film, i.e., there is a perfect wetting.• The flow is steady with out any interruption.• The film thickness is small as compared to the tube diameter.• There is no nucleate boiling within the film.• The surface tension and pressure gradient effects are negligible.• The temperature of the tube wall is uniform.• The evaporation takes place at liquid-vapour interface.• The humid air is considered as a perfect gas.• The liquid-gas interface is at thermodynamic equilibrium and there is no gas dissolution.• The heat transfer by radiation is not considered.

ANALYSIS

Energy balance is performed for the fluid flows in the ith cell, where energy andmass transfer interactions take place due to the condensation of steam, evaporation of thefalling film and diffusion of water into air. In the computational cells, the energy balanceequation for the hot liquid can be written as,

Figure 2 Flow over a tube bundleSharma & Mitra (2004).

S

Water

Air

XO

Z

B

ith

cell C.V.

Tin,f,i

Tout,fi,i out,f,i

Steam

Re in,f,i in,f,i

Re out,f,i

Γ

Γ

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where in Eq. (1) is the mass flow rate of hot steam or water flowing through the tube,Tw,int is the temperature of the inner surface of the wall and dint is the internal diameter ofthe tube. Average steam or hot water temperature has been considered for the calcula-tion purpose. For the liquid film energy balance, it is assumed that the evaporation takesplace at the interface only and the film thickness is negligible as compared to the tuberadius and can be written as,

where is the mass flow rate of the film falling on the outer surface of the tube ofdiameter . The heat transfer between the liquid film and air can occur with or withoutevaporation at the interface. If the vapour pressure at the interface is equal to the saturationpressure then no evaporation takes place. The evaporation of a film on a heated surfacecan take place by bubble formation or by evaporation at the interface. In particular, thelater mechanism prevails when the temperature differences are low. As the temperaturedifferences are less than 10°C, the evaporation at the interface has been considered. In thiscase, the energy balance is given by:

where in Eq. (3) is the mass flow rate of air in the cell. If the vapour pressure at theinterface is less than the saturation pressure, evaporation occurs at the interface betweenthe air and the liquid film. In addition to the energy convected away from the film surface,heat is transferred from the liquid to air stream by the evaporation of water. This energydepends on the latent heat of vaporization and the mass of water evaporated, which can bewritten as:

The first term on the left hand side of Eq. (4) represents the heat transferred into air due toevaporation taking place at the interface, where is the mass of water evaporated andHlat is the latent heat of vapourization of water. For the case with evaporation, the amountof water evaporated is given by Fick’s law as (Colburn, 1933):

�m CpdT

dzd h T Thl hl

hlhl hl w= −( )π int int, (1)

�mhl

Thl

π d h T T m CpdT

dm Cp

dT

dext f w ext I out f ff

hl hlhl

, ,−( )+ =� �z z

(2)

�mout f,dext

�m CpdT

dzd h T Tin g g

gext g I g, = −( )π

(3)

�min g,

� �m H m CpdT

dzd h T Tev lat in g g

gext g I g+ = −( ), π (4)

�mev

�m DM

R T

dP

dzev vv

o g

v

I

= −

(5)

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where RO represents perfect gas constant, Pv is the partial pressure at the interface. The coef-ficient Dv represents the diffusivity of vapour in the air and is given by (Colburn, 1933 ):

where Pt is the total pressure and T0 = 273K. If q is the fraction of mass evaporated, onecan write the following mass conservation equations for the film and the air:

From Eqs. (1)-(4), it is observed that the values of various heat transfer coeffi-cients are needed to solve the energy balance equations. The heat transfer , used inEq. (1), represents the convective heat transfer coefficient for steam flowing inside thetubes. The steam flow is the turbulent region, and hence Colburn’s correlation is used(Colburn, 1933 ):

The convective heat transfer coefficient for gas flow around a tube bundle, in Eqs.(3)-(4), for Reynolds number from 2000 to 32000 is given by (Colburn, 1933 ):

Evaporative heat transfer coefficient, in Eq. (2), which is a strong function of theflow regime, has been correlated for different flow patterns, as written here (Owens,1978):

DP

T

Tvt

g= +

2 261

1 81.

.

o

(6)

� �m qmev in f= , (7)

� �m q mout f in f, ,= −( )1 (8)

� � �m m mout g in g ev, ,= + (9)

hhl

hk

dhlhl

hl hl= 0 023 0 8 13.

int

.Re Pr (10)

hg

h d

k

Cp

k

d ug ext

g

g

g

ext g g

g

=

0 33.

,µ ρµ

13

max (11)

hf

hv

gk

L

dRef

f

f extf

2

3

0 11

31

32 2

=

−. ;

.

for laminar regiime (12)

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SOLUTION METHOD

To take into account the variation of fluid properties as a function of temperature, allphysical properties are evaluated at the average temperatures of the fluids in the cell. Asthe fluid properties are evaluated at mean fluid temperatures in the cell, which areunknown, and the relations between them are not linear functions, the proposed equationsare not linear. To express the energy balance in a cell, the energy balance equations pre-sented before have been transformed to the following equation:

where the temperature differences are defined as:

The temperature gap can be deducted from the other variables. This justifies thatthere are only four independent variables. The problem is thus transformed to a non-linear multi-variable function minimization. In each cell of the heat exchanger, solvingfor energy equations is equivalent to minimizing the function fi. A downhill simplexalgorithm has been used here. An initial guess for is used and value of fi iscalculated. After each iteration j, the variable incrementation giving a maximum

hv

gk

L

d

Cp

kff

f ext

f f

f

2

3

0 1 0 513

0 185

=

. ;

. .µ

for turbulent regime (13)

f T T T T m Cp T d L h Ti i i i i hl hl i i hl i∆ ∆ ∆ ∆ ∆ ∆1 2 3 4 1, , , , , , , , , ,|( )= −� π int 22

1 5 3

,

, , , , ,

|i

hl hl i out f f i ext i f i i

hl

m Cp T m Cp T d L h T

m C

+ − −

+

� �

�

∆ ∆ ∆π

pp T m Cp T d L h T

m Cp T

hl i out f f i ext i g i i

hl hl i

∆ ∆ ∆

∆

1 5 4

1

, , , , ,

,

− −

+ −

�

� �

π

mm Cp T m Cp T m Hout f f i in g i g i ev i lat i, , , , , , , |∆ ∆5 4 0− − =� �

(14)

∆T T Ti in hl i out hl i1, , , , ,= − (15)

∆T T Ti hl i w i2, , , ,= − int (16)

∆T T Ti w ext i i3 1, , , ,= − (17)

∆T T Ti l i g i4, , ,= − (18)

∆T T Ti in f i out f i5, , , , ,= − (19)

∆T i5,

∆Ti i, , ,0 1 4=ti i, ,=( )1 4

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decrease in the function is determined. The solution is obtained when the function resi-due is less than 10–4. The solution procedure can be summarized in the following steps:

1. Temperature, velocity and mass fraction profiles are given at the top of the evaporator(first stage i = 1). For other cells, the variable profiles are deducted from calculation inthe previous cell.

2. Inputs are initial variables values , maximum iteration number, and incrementsteps .

3. Calculation of the minimum value of the function at the iteration j; Min(fi, j).4. Calculation of the new variable value, giving a maximum of the function decrease

.5. Convergence test: If then repeat steps 3 and 4.6. Calculation of air partial vapour pressure Pv,i and saturation pressure at liquid-air inter-

face Psat,i.7. If then steps 2-5 are repeated with .8. Calculation of the parameters corresponding

to the (i + 1)th cell.9. Redo the steps 2-8 until all the cells of the evaporator are being considered.

RESULTS AND DISCUSSIONS

Calculations were performed with the initial guess °C and the incrementvalue ti = 0.2sec and the maximum number of iterations = 20. The evaporative heat transfercoefficient and mass of water evaporated have been plotted against various parameters likesteam inlet temperature, position of the tube in the evaporator, and liquid film Reynoldsnumber. Figure 3 shows the variation of evaporative heat transfer coefficient with the filmReynolds number. Average evaporative film heat transfer coefficient is a decreasing func-tion of film Reynolds number as the Eq. (12) has a negative exponent on film Reynoldsnumber. Physically, with an increase in film Reynolds number, the film thicknessincreases and hence the heat transfer coefficient decreases. For a typical desalination unit,film Reynolds number is of the order of 300, which is laminar regime. The expression forcondensation heat transfer coefficient is given Eq. (10), which has a positive exponentterm over Reynolds number and hence it is an increasing function of steam flow Reynoldsnumber.

Figure 4 depicts a linear relationship between mass of water evaporated and steaminlet temperature. The experimental and the theoretical models are in good agreement. It isobserved that the mass of water evaporated increases with an increase in the inlet temper-ature. This is because with the increase in hot liquid inlet temperature, the amount of heatused for sensible heating is reduced whereas the portion absorbed for latent heatingincreases. Fig. 5 shows the variation of mass of water evaporated with Z. The distance isbeing given in terms of dimensionless number Z defined by where B is thevertical height difference between the consecutive tubes in a column. It is found that themass of water evaporated decreases with the distance in the downward direction, as nearthe lowest most rows of tubes, the temperature of air is lowest. Hence, part of the energy isbeing used for air heating rather than latent heating of water.

Figure 6 shows the variation of mass of water evaporated with film Reynolds num-ber. It is observed that the mass of water evaporated first rapidly increases with increase infilm Reynolds number and then becomes asymptotically constant. Such a behaviour is due

∆Ti i, , ,0 1 4=ti i, ,=1 4

∆ ∆T T tij ij i= +−1

| ( , ) ( , ) |Min f j Min f ji i− − ≥ −1 10 4

P Pv i sat i, ,> �mev i, = 0

T T X m mout hl i out f i in i out f i in g i, , , , , , , , ,, , , ,� � and

∆T = 4

Z z B= /

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Figure 3 Variation of heat transfer coefficient with film Reynolds number.

0 500 1000 1500 2000 2500 30000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Ref

Hea

t tra

nsfe

r co

effic

ient

(W

/m2 K

)

Figure 4 Variation of mass of water evaporated with steam inlet temperature for Tin,f = 50°C, Tin,g = 30°C,Rein,hl = 25,000 and Rein,f = 400.

40 45 50 55 60 65 70 75 80 85 900

1

2

3

4

5

6

7

8

9

10

Tin,hl

mev

10–

4 kg/s

o Experiment[6] Model

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Figure 5 Variation of mass of water evaporated with the down-ward direction for Tin,hl = 80°C, Tin,f = 76°C,Tin,g = 30°C Rein,hl = 25,000, and Rein,f = 400.

0 5 10 15 20 25 30 350.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Z

mev

10–

4 kg/s

Figure 6 Variation of mass of water evaporated with film Reynolds number for Tin,hl= 80°C, Tin,f= 50°C,Rein,hl= 25,000, and Tin,g= 30°C.

0 200 400 600 800 1000 12004

5

6

7

8

9

10

11

Rein,f

o Experiment[6] Model

mev

10–

3 kg/s

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to the fact the heat transfer coefficient decreases due to an increase in film thickness. Withthe increase in Reynolds number there is an increase in mass flow rate, which results in anincrease in mass of water evaporated. However, it gradually becomes constant as theevaporative heat transfer coefficient also becomes constant for higher Reynolds number,as shown in Fig. 3. For lower Reynolds numbers, the experimental and the theoreticaltrends are similar, but at higher Reynolds numbers, there is an opposite trend. The differ-ence is due to the fact that at higher Reynolds numbers there is splashing of water wherenozzles are used as the water distribution system (Bourouni & Tadrist, 1999). For experi-ments performed with perforated slot type water distribution system, the theoretical andthe experimental values are similar (Bourouni et al., 1997), as less pulverization of wateroccurs for perforated slot type water distribution system.

The mass of water evaporated increases with the increase in hot water Reynoldsnumber because in turbulent regime of flow, heat transfer coefficient increases with theReynolds number. For high values of the Reynolds number, there is less decrease in massof water evaporated with Z. As shown in Fig. 7, at low values of hot liquid flow rate, evap-oration does not take place in the lower part of the exchanger. With increase in ,evaporation becomes more uniform in the exchanger and there is less variation in the massof water evaporated with Z.

Figure 8 shows that the mass of water evaporated decreases in an asymptotic wayfrom higher to lower cells, for inlet condition corresponding to Tin,f = 75°C. For Tin,f ≤70°C, the mass of water evaporated increases in the first few cells and reaches maximumand then decreases continuously in an asymptotic way. This trend is due to liquid film pre-heating in the first few cells of the exchanger. As shown in Fig. 8, a decrease in Tin,f leadsto the reduction in the evaporation in higher cells of the exchanger. Overall the total

Figure 7 Variation of mass of water evaporated with Rein,hlfor Tin,hl = 78°C, Tin, f= 76°C, Tin,g = 42°C, Rein,f=350, Rein,g = 7000.

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Z

mev

(10–

4 kg/s

)

+ Rein,hl

=45,000* Re

in,hl=27,000

o Rein,hl

=9,000

mev

10–

4 kg/s

Rein,hl

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amount of water evaporated shows significant reduction when Tin,f decreases. The heattransfered from the steam flowing through the tubes is used for sensible heating of air,film and latent heating of the film. In the top part of the exchanger, the air temperature ismaximum, hence the sensible heat requirement of air is less than at the bottom part of theexchanger. As a result, heat transfered from steam is used either for sensible heating ofliquid film or latent heating of the film. The fraction of heat used for sensible heatingdepends on the temperature difference between the film and the steam. To improve theexchanger performance, this temperature difference should be minimcal, and hence pre-heating of the film before injection is needed. Figure 9 shows the variation of water evap-orated with Reynolds number. It is found that for the first few cells, there is an increase inthe mass of water evaporated.

The experimental investigation presented by Bourouni Bourouni et al. (1997) doesnot allow the evaluation of different parameters, like mass of water evaporated, averagetemperature, and local air humidity. Sensitivity analysis has been carried out for variousoperational conditions to investigate their influence on the local and global parametervariations. An increase in hot water inlet temperature improves the evaporation in thehigher part of the exchanger. In fact an increase in Tin,hl causes an increase in liquid filmtemperature in the first few cells of the evaporator. For lower mass flow rates, evaporationdoes not take place in the lower parts of the evaporator. With the increase in hot waterinlet Reynolds number, evaporation becomes uniform in the exchanger, and the mass ofwater evaporated profile becomes linear, as shown in Fig. 7. The increase in mass of waterevaporated with inlet parameters can be explained on the basis of an increase in the enthalpicflux . As shown in Fig. 8, the amount of evaporated water decreases in anasymptotic way from top to bottom cells, for an inlet temperature of 75°C. For operations

Figure 8 Variation of mass of water evaporated with Tin,f for Tin,g = 42°C, Rein,hl = 25000, Rein,f = 350, Rein,g =6000.

0 5 10 15 20 25 30 35 401

2

3

4

5

6

7

Z

mev

(10–

4kg

/s)

+ Tin,f

=75°C

* Tin,f

=70°C

o Tin,f

=60°C

( )�m Cp Thl hl hl∆

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corresponding to Tin,f ≤ 70°C, mass of water evaporated increases in first few cells andthen decrease. This is caused by liquid film preheating in the first few cells. An increasein air humidity leads to an increase in partial vapour pressure in the air in the cells. Onthe other hand, the saturation pressure is independent of air humidity and depends onlyon the saturation temperature. Therefore, the difference between the partial vapour pres-sure in air and the interface decreases when the air humidity increases. This causes adecrease in the quantity of water diffused from liquid interface to air, i.e., the mass ofwater evaporated.

HEAT TRANSFER CALCULATIONS FOR A TYPICAL PLANT

A detailed calculation is performed for a desalination unit used in the cement plant.The maximum value of desalinated water is around 2000 tonnes per day. An estimate ofheat transfer coefficient helps in calculating the mass of steam required for a given amountof desalinated water and heat transfer surface, which can be varied according to futurerequirements. The plant has a rectangular layout of tubes with 120 tubes in a row and alarge number of tube rows. It can be assumed that after four to five tube rows, the film willhave water at saturated state and is uniformly distributed over all the 120 tubes in a row.At maximum load of 2000 tonnes per day, the mass flow rate of water per tube is

per tube. Length of tube is L = 4.8m,so the mass flow rate

per unit length of the tube on one side of the tube is kg/sec m. The tubes

Figure 9 Variation of mass of water evaporated with Rein,f for Tin,hl = 80°C, Tin,f = 50°C, Tin,g = 42°C, Rein,hl =25000, Rein,g = 6000.

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

Z

mev

(10–4

kg/s

)+ Re

in,f=400

* Rein,f

=300 o Re

in,f=200

2000 10

24 3600 1200 193

3×× ×

= . kg/sec

Γ =×

=0 193

4 8 20 02

.

..

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have a triangular pitch of 29 mm and a outer diameter of 22 mm. The liquid film can beassumed to be falling from one tube to next tube from a height B = 47 mm with a free fall

velocity given by m/sec. Steam is flowing inside inside the tubes at 80°C

and the liquid film is at saturation temperature corresponding to the stage operation pres-sure. It has been assumed that the film losses all its momentum after it strikes a tube andthe tubes have an inner wall temperature uniformly maintained at 80°C.

When the liquid film falls on the tube bundles, the heat transfer coefficient ishighest at the top-most stagnation point due to high turbulence and high velocity gradi-ent. As the film thickness is much smaller than the tube radius and the impingement jetflow is expected to be influential only in a short region, the flow at the top of the hori-zontal tube may be modeled as a liquid jet impinging on a flat plate. The flow field canbe divided into three zones: stagnation zone, impingement flow, and uniform parallelflow, as shown in Fig. 10.

The stagnation zone is characterized by a velocity just outside the hydrodynamicboundary layer of the jet flow umax, linearly proportional to the distance x* measured fromstagnation point along the periphery of the tube. The local heat transfer coefficient in thestagnation flow zone can be written as (Chyu & Bergles, 1987 ):

The film Reynolds number is given and all the properties are calculated at

80°C, where Γ is the mass flow one one side of the tube and µ = 355 × 10-6 Pas. The

value of for the stagnation region is given by Fig. 10, which is 5/6 in this

case. The jet velocity uj for the case of falling film evaporator is that of a free body falling

from a height B, given be . The jet width is calculated from conservation of mass

and is given by . As shown in Fig. 10, the stagnation region is valid for 0 < x*/w*

< 0.6 for which the velocity gradient is constant. The angular position of stagnation region

is given by , where R is the radius of the tube. For the falling film evapo-

rator under consideration, the values are calculated at saturation temperature of 80°C and

jet width w* is 42.88 × 10–6m. It is found that the heat transfer coefficient hstg and angular

position Φstg are 2.04 × 105W/m2K and 1.225–3rad, respectively.The impingement flow zone covers the region between the stagnation zone and

the uniform parallel flow zone. As shown in Fig. 10, the impingement flow zone isvalid for 0.5 < umax/uj < 1.0. This corresponds to 0.6 < x*/w* < 2.0 and up to

u gB= =2 0 96.

h kd u u

d x w

u

wstgj j= ⋅

1 03 1 3

0 5

.( )

( * *) */

.

Pr max

ν(20)

Ref = 4Γµ

d u u

d x wj( )

( * *)max

2gB

wu j

* = 2Γρ

φstgw

R=

0 6.*

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. For laminar flow regime, the local heat transfer coefficient can be

written as:

where the local Reynolds number Rex is of the order of 105.By the end of the impingement region, the flow is considered to be hydro-dynami-

cally developed from which the velocity distribution and film thickness can be calculated.The film thickness δ is given by (Chyu & Bergles, 1987 ):

As there is less variation in mass flow rate of film in a thermal developing regionΓ(Φ), which can be assumed to be constant, the most part of heat flux is used for thermalboundary layer development. The variation of film thickness with respect to angular position

Figure 10 Distribution of velocity jet outside the hydrodynamic boundary layer

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

x^*/w^*

u max

/uj

Stagnation Flow region

Impingement Flow region

Uniform Parallel Flow Region

φimpw

R=

2 0.*

Nuh x

k

xu ximp

impx x= = =0 037 1 3 0 8. ;

( )/ .Pr Re Re max

ν (21)

δµ

ρ ρ ρ φφφ

( )( )

/

( )sin=

−

31 3

Γg f f g (22)

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measured from top-most stagnation point is given by Fig. 11. This plot is valid forconstant values of Γ(Φ) and it leads to almost constant values of film thickness. Hence, thevalue of film thickness at Φ = 90° is used in the entire thermal developing region calcu-lation. With respect to a coordinate system moving at speeds the same as that of the film,the change of temperature profile in the developing region can be obtained by solving theone-dimensional transient heat conduction equation, which can be written as (Chyu &Bergles, 1987 ):

The solution is composed of the steady state term and the transient term in form of aninfinite series. The temperature profile can be assumed to be steady and stabilized at

. The average heat flux from top of the tube to the angular position φ is given by

(Chyu & Bergles, 1987):

Figure 11 Variation of film thickness with respect to angular position in rad.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Angular position, Radian

Film

thic

knes

s

T T

T T

y n y

nn

s

w s n

( , ) sinτ δδ π

π δπ ατδ

−−

= − −

=

∞

∑12

2 2

21

( / )

exp

(23)

τδπαd =

2

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where

For thermal developing region from Φi to Φd, the average heat transfer coefficient can becalculated from the following relation:

In case of this desalination unit, the average heat transfer coefficient in thermal developingregion up to 27° is 9000W/m2K.

For the thermal developed region, one may consider heat transfer in the film by con-duction only, which is accurate for low mass flow rates when convection term is not muchimportant. Correlations based on experiments for falling film evaporating along a verticalwall are more accurate than the theoretical model based on conduction (Chyu & Bergles,1987 ). For such correlations, the average heat transfer coefficient for laminar range is canbe written as (Chun & Seban, 1999):

and for laminar-wavy region it is given by (Chun & Seban, 1999):

and finally for the turbulent region, it is given by (Chun & Seban, 1999):

The desalination unit under study is in the laminar-wavy flow regime and the correspond-ing heat transfer coefficient is 7064W/m2K. The average heat transfer coefficient can thenbe calculated from the following expression:

′′ = −−

−qd w s

f f gFk T T

g,( )

/

( )( )

0

1 3

3φρ ρ ρ

µ Γ (24)

FP n

n P Pn

= + − − ==

∞

∑12 1

12

2

12π

ππατδ

[ exp( )]; (25)

hq q

T Tdd d i d

d i w sdi d

d i,( )

,( ) ,( )

( )( );φ φ

φ φφ φ

φ φφ

π−− −=

′′ − ′′

− −=0 0 1

ααµρR g

3 4

5

1 3Γ

/

(26)

h

k g

gfd νµ

µρσ

2 1 3

1 34

3

1 11

1 10

= ≤

−−/

/

/

. Re forΓ

(27)

h

k g

gfd ν µρσ µ

2 1 3

0 223

1 11

0 822 1

=

≤ ≤−

−/

.

/

. Re for4 Γ

4450 1 06Pr− .(28)

h fd

k

v

g

2 1 33 0 65 0 4 1 063 8 10 1450

= × ≥− −

/. . .. Pr Re Prfor

Γµ

(29)

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and it comes out to be 7644W/m2K. It is observed that the value of is largely influ-enced by the thermal developed region heat transfer coefficient, which covers 85% of theheat transfer surface of the tube.

CONCLUSION

An analytical model to predict heat and mass transfer characteristics of a horizon-tal-tube-falling film evaporator has been presented here. The evaporator is designed tobe used in brackish water desalination, functioning on aero-evapo-condensation pro-cess. The model is valid for film Reynolds numbers between 400 to 630, as in this rangethe experimental and the theoretical results for mass of water evaporated are almost thesame. For film Reynolds number less than 400, the mass flow rate reduces, and hencethere is an improper wetting of the tube surface. The experimental and the theoreticalresults show opposite trends for film Reynolds numbers greater than 630, which can beattributed to the splashing of water at the water distribution system. It is also observedthat for every 10°C temperature difference between the liquid film and hot water at theexchanger inlet the evaporation is not maximum in the first cell for film temperaturesbelow 70°C. This causes a non-negligible loss of mass of distilled water quantity. Toimprove the evaporative exchanger performances, it is important to preheat the liquidfilm before injection into the evaporator so that there is a temperature difference on theorder of 5°C between the film and the steam flowing through the tubes. The influence ofinlet parameters on the heat and mass transfer in the exchanger has been studied.Numerical results show an increase in exchanger performance in terms of mass of waterevaporated when both the inlet temperature and mass flow rates of hot water and liquidfilm are increased. These trends are in good agreement with known experimental resultsin the literature. Typical plant data is also used to calculate the overall heat transfercoefficient for a desalination unit.

ACKNOWLEDGMENT

Financial assistance from Larsen & Toubro, India Ltd., in form of Built India schol-arship is acknowledged.

REFERENCES

Ansari, A.D., & Owen, I. (1999). Thermal and hydrodynamic analysis of the condensation and evap-oration processes in the horizontal tube desalination plant. International Journal of Heat andMass Transfer, 42:1633–1644.

Bourouni, K., & Tadrist, L. (1999). Experimentation and modeling of an innovative geothermaldesalination unit. Desalination, 125:47–153.

Bourouni, K., Martin, R., Tadrsit, L., & Tadrist, H. (1997). Experimental investigation of evaporationperformance of a desalination prototype using aero-evapo-condensation process. Desalination,114:111–128.

h h h hstgstg

impimp stg

dd imp=

+

−

+

−

φ

π

φ φ

π

φ φ

π +

−

h fd

dπ φπ

(30)

h

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Chun, K.R., & Seban, R.A. (1999). Heat transfer to evaporating liquid film. International Journal ofHeat and Mass Transfer, 42:1633–1644.

Chyu, M.C., & Bergles, A.E. (1987). An Analytical and Experimental Study of Falling-Film Evapo-rator on a Horizontal Tube. Journal of Heat Transfer, 109:983–990.

Colburn, A.P. (1933). A method of correlating forced convection heat transfer and comparison withfluid friction. Transactions of AIChE, 29:74

Hewitt, G.F., & Taylor, H. (1970). Annular Two-phase flow. Pergamon.Lorenz, J. L., & Yung, D. (1979). A note on combined boiling evaporation of liquid films on hori-

zontal tubes. ASME Journal of Heat transfer 2: 391–396.Owens, W.L. (1978). Correlation of thin film evaporation heat transfer coefficients for horizontal

tubes. Conference of Ocean Thermal Energy conversion, 1: 20–22.Sharma, R., & Mitra, S. K. (2004). Heat transfer analysis of horizontal tube falling film evapo-

rator. In: Proceeding of CSME. University of Western Ontario, London, Canada: CSMEForum 2004.

NOMENCLATURE

B distance between two consecutive tube rowCp specific heatDu diffusion coefficientd tube diameterHtat latent heat of evaporationh heat transfer coefficient

average heat transfer coefficientk thermal conductivityL tube lengthMv molecular weightNu Nusselt numberPr Prandtl numberPt total pressurePv partial pressureq evaporated mass fraction

heat fluxR tube diameterRe Reynolds numberRo perfect gas constantT local Temperature∆T temperature difference

average temperatureu velocityw* jet widthx* distance measured from stagnation pointy distance measured perpendicular to tube surfaceZ dimensionless coordinatesz direction along the film flowα thermal diffusivityδ film thicknessΓ mass flow rate/lengthµ dynamic viscosityv kinematic viscosityρ densityτ time

h

′′q

T

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Φ angular positionσ surface tension

Subscriptsd thermal developing regionext externalg airhl hot liquidf filmfd fully developed regionI Interfacei ithcellin inletint internalimp impingement regionout outletsat saturationstg stagnation regionw wall

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performance model for a horizontal tube falling film evaporator

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