Irreversibility analysis of falling film absorption over a cooled horizontal tube

International Journal of Heat and Mass Transfer 88 (2015) 755–765

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Irreversibility analysis of falling film absorption over a cooled horizontal tube Niccolò Giannetti a,⇑, Andrea Rocchetti b, Kiyoshi Saito a, Seiichi Yamaguchi a a b

Department of Applied Mechanics and Aerospace Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan DIEF – Department of Industrial Engineering of Florence, Via Santa Marta, 3, 50139 Firenze, Italy

a r t i c l e

i n f o

Article history: Received 20 November 2014 Accepted 5 May 2015 Available online 22 May 2015 Keywords: Irreversibility Entropy generation Absorption cycle Horizontal tube

a b s t r a c t Based on a numerical study of the water vapour absorption process in LiBr–H2O solution, for a laminar, gravity driven, viscous, incompressible liquid film, flowing over a horizontal cooled tube, irreversibilities related to fluid friction, heat transfer, mass transfer and their coupling effects have been locally and globally examined. The hydrodynamic description is based on Nusselt boundary layer assumptions. The tangential and normal velocity components, respectively obtained from momentum and continuity equations, have been used for the numerical solution of mass and energy transport equations in the two-dimensional domain defined by the film thickness and the position along the tube surface. Local entropy generation calculation can be performed referring to the calculated velocity, temperature and concentration fields. Results have been explored in different operative conditions, in order to examine comprehensively the impact of the various irreversibility sources and to identify the least irreversible solution mass flow-rate for the absorber. As a parallel, a refined understanding of the absorption process can be obtained. Considering absorption at the film interface and cooling effect at the tube wall, the analysis thermodynamically characterises the absorption process which occurs inside actual falling film heat exchangers and establishes a criterion for their thermodynamic optimisation. Results suggest the importance to operate at reduced mass flow rates with a thin uniform film. Meanwhile, tension-active additives are required to realise this condition. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction All the real processes occurring in an energy conversion system are associated to an unavoidable degradation of the original amount of energy. The second law of thermodynamics provides a qualitative description of physical processes and is critical to identify their limitations. Namely, according to this general design issue, thermal design and basic thermodynamics are to be employed together with the purpose of identifying the optimum size or operating regime of a certain engineering system, where by ‘‘optimum’’ the least exergy destroying condition, which can still assure the fundamental engineering function, is intended. Devices in which simultaneous heat and mass exchanges occur are commonly used in the power and refrigeration industries, as well as air conditioning where both temperature and humidity might be simultaneously controlled. This devices are also part of absorption machines as generator and absorber. The use of the vapour absorption cycle for heat driven energy systems was among the first popular and widely used methods of refrigeration. Even

⇑ Corresponding author. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.022 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

though the development of vapour compression cycles has limited the implementation field of vapour absorption systems, the main benefits of absorption cycle are still evident: since a negligible amount of electricity is needed, waste heat can be used as the main energy source, and higher reliability can be ascribed to the absence of moving parts. In addition, typically used refrigerants (water or ammonia) are not responsible of ozone depletion effect. The fundamental heat and mass transfer processes constituting the absorption cycle are realised inside specific heat exchangers, whose characteristics have decisive effects on the overall system efficiency, on its dimensions and its cost. In the conventional case of falling film heat exchangers high transfer coefficient and low pressure drop can be obtained. However, the attempt to experimentally and theoretically describe the complex heat and mass transfer mechanism occurring inside these devices is still incomplete and has not led to conclusive approaches. In terms of modelling efforts, [1–5] presented simplified models for falling film absorption of water vapour over a horizontal tube. Similarly, they solved the problem with a finite difference method and studied the effect of different parameters on the coupled heat and mass transfer processes. Among the possible scenarios, entropy generation minimisation has been widely accepted as a method for heat

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Nomenclature a C cp D E g G h H j k M N P q r R Re S s T u v V x y

thermal diffusivity [m2s1] molar concentration [molm3] isobaric molar heat [Jmol1K1] mass diffusivity [m2s1] Entropy generation rate per tube unit length [Wm1K1] gravity [ms2] mass flux per unit surface [kgm2s1] molar enthalpy [Jmol1] number of nodes in radial direction molar flux [molm2s1] thermal conductivity [Wm1K1] molar weight [kgmol1] number of nodes in tangential direction pressure [kPa] heat flux per tube unit length [kWm1] outer tube radius [m] grid ratio in normal direction Reynolds Number volumetric entropy generation rate [Wm3K1] molar entropy [Jmol1K1] temperature [K] streamwise velocity [ms1] radial velocity [ms1] total velocity [ms1] local tangential position [m] local normal position [m]

u g s C d

l q c x

general parameter identification dimensionless normal position shear stress tensor [Pa] mass flow rate per unit length [kgs1m1] film Thickness [m] viscosity [Pas] density [kgm3] chemical potential [Jmol1] LiBr mass concentration

Subscripts 0 standard abs absorption c convection d diffusion e equilibrium f friction G global H2O water i, j node indexes if interface in inlet min minimum S solution sat phases equilibrium t thermal v vapour w wall

Greek symbols b streamwise Angle [rad] e dimensionless tangential position

exchangers’ design. Entropy can be used to evaluate the irreversibility introduced, characterise the quality of energy-conversion, and eventually, develop consistent criteria for the optimisation and control of a component or the system. The existence of thermal, velocity and concentration gradients in the computational field representing the absorptive film yields a non-equilibrium state, responsible of entropy generation (better defined as entropy variation due to irreversibility). To the authors’ knowledge, few researchers [6–9] have previously carried out second law analyses of heat and mass exchange devices. In particular, [10,11] report a second law-based analytical study for gas absorption into a laminar, falling, viscous, incompressible, liquid film. The main conclusion states that entropy generation is mainly ruled by the coupling effects between heat and mass transfer near the gas–liquid interface and by the viscous irreversibility when approaching the solid wall. However, heat transfer at the wall has not been included in the problem. Simultaneous cooling and absorption allow the process to be maintained far from the thermodynamic equilibrium at which absorption will not occur. The main purpose of this work is to perform a numerical and parametric analysis of the volumetric entropy generation rate inside the computational domain representing real LiBr–H2O absorptive films, where energy and species transport equations are solved numerically. Results can be used to reduce irreversibility in a falling film heat exchanger, in order to optimise the absorption process both locally and globally. 2. Model description and numerical solution The system in question is showed schematically in Fig. 1. A single horizontal tube is considered and the LiBr–H2O solution flows

viscously down over it driven by gravity as a laminar incompressible liquid, while vapour mass transfer process occurs at the interface of the flowing film. The heat released by the absorption is rejected to the cooling water flowing inside the tube. Heat and mass transfer characteristics have been studied by solving numerically transport of mass and energy equations, under the following main assumptions: 1. The flow is steady, laminar and without interfacial waves. 2. Thermodynamic equilibrium exists at the film interface with the vapour. 3. There is no shear force between the liquid film and the vapour. 4. Disturbance at the edges of the system are neglected assuming that both the tube circumference and length are large comparing to the film thickness. 5. Physical properties are function of the inlet concentration and temperature, but remain constant while flowing on the single tube. 6. Heat transfer to vapour phase is negligible. 7. Outside tube surface temperature is constant and equal to the coolant temperature. 8. Body fitted coordinates (x along the tube surface and y normal to it at any point) are used because the film thickness is small if compared to the tube circumference [5]. The hydrodynamic description is based on Nusselt boundary layer assumptions. The tangential and normal velocity components (respectively, Eqs. (1) and (2)), from continuity and momentum equations, have been used for solving numerically the transport of mass and energy (respectively, Eqs. (4) and (3)).

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For the boundary conditions at the interface second order backward difference is used for the first derivative of diffusion terms. While for the wall condition second order forward difference approximates the first derivative of concentration. The calculation is performed between b = p/N and b = p(N1)/N because of the definition domain of the velocity field (Eqs. (1) and (2)). 3. Entropy generation The local irreversibility analysis of water vapour absorption through a laminar, gravity driven, viscous, incompressible liquid film, flowing over a horizontal cooled tube, by the identification of different entropy sources is performed from the obtained temperature, velocity and concentration fields. Owing to the simultaneous heat and mass transfer, four sources of irreversibility can be recognised: irreversibility related to heat transfer, fluid friction, coupled effects between heat and mass transfer by convection and coupled effects between heat and mass transfer by diffusion. Further assumptions required for the entropy generation mathematical formulation are: 1. Physical absorption (no chemical reactions). 2. Steady state regime of the two dimensional flow. 3. Gravity driven-laminar flow of a Newtonian, incompressible, viscous, liquid film (LiBr–H2O solution). 4. Water vapour absorbed is considered as a perfect gas. 5. The absorption process takes place at constant pressure.

Fig. 1. Local coordinate system of the flowing film.



u¼

qg 1 sin b dy  y2 2 l

v ¼



 ð1Þ 

qgy2 dd 1 y sin b þ d cos b r 3 2l dx

ð2Þ

u

@T @T @2T þv ¼a 2 @x @y @y

ð3Þ

u

@x @x @2x þv ¼D 2 @x @y @y

ð4Þ

In order to obtain temperature and concentration fields, the solution-method introduces a dimensionless coordinate transformation (Eqs. (5) and (6), respectively in the circumferential and radial directions) and is parallel to that of [12].

e¼ g¼

x

pr y d

¼

b

p

ð5Þ

ð6Þ

The grid generation and finite difference approximations of the partial derivatives match [13] to reduce numerical instabilities. A cosine type grid is employed in the g direction to make the grid finer where steeper gradients are expected, i.e. near the tube wall surface and at the vapour interface. For both energy and species transport equations, first order backward difference is applied for the convective term in the e direction. In the g direction second order central difference is used both for the first and the second derivative [13], respectively of the convective and diffusive terms.

The existence of thermal, velocity and concentration gradients in the computational field representing the absorbing falling film yields a non-equilibrium state, which is responsible of entropy variation (entropy generation). According to the problem formulation and to the introduced assumptions, the volumetric rate of entropy generation due to friction, heat and mass transfer is given by,

  ! s : rð V Þ 1 ! ! ! ! 1 þ SG ¼ q  r  jv  r ðcv Þ T T T

ð7Þ

These three terms can be, respectively, related to heat transfer, fluid friction and mass transfer irreversibilities. The heat flux q includes the heat flux given by the Fourier Law and the enthalpy flux due to species diffusion.

! ! ! q ¼ kS r ðTÞ  hv jv

ð8Þ

As a result the rearrangement of the expression gives,

SG ¼

! ! 2 1 ! ! s : rð V Þ 1 ! ! ðTÞ ðTÞ þ k r þ h j r  jv  rðcv Þ  S v v T T T2 T2 1

ð9Þ

Considering the absorbed water vapour as an ideal gas and the only species diffusing through the liquid solution of LiBr–H2O, molar concentration of the diffusing species Cv, can be derived directly from the mass concentration field of LiBr (x) resulting from the solution of energy and species transport equations.

C v ;ij ¼

qS M H2 0

ð1  xij Þ 

qS M H2 0

ð1  xin Þ

ð10Þ

The molar chemical potential of the water vapour can be calculated as [14],

cv ðT; P0 Þ ¼ cp;v ðT  T 0 Þ  Tcp;v ln  Tsv ;0 ðT 0 ; P0 Þ



 T þ hv ;0 ðT 0 ; P0 Þ T0 ð11Þ

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N. Giannetti et al. / International Journal of Heat and Mass Transfer 88 (2015) 755–765

10000

300 250

1000 100

Sd St

200 150 100

10

0.82 1.00

0.47

0.14

0.95 0.00

0.75

(a)

0.85

ε

0.65

0.45

ε

0.55

0.25

0.35

0.07

0.18

0.30

0.42

0.53

0.65

0.77

0.88

0.82

1.00

0.47

0.00

0.14 η

0

0.05 0.15

50 1

η

(b)

2000

0.25

-4000

0.20

-7000

0.15

-10000

0.10

Sf

0.03 0.93 0.69 0.37 0.10 0.00

0.23

(c)

0.13

η

0.33

0.43

0.63

0.53

0.05 0.93 0.83 0.73

0.82 1.00

0.47

0.14

0.87 0.00

0.77

0.67

ε

0.57

0.47

0.27

ε 0.37

-13000

0.07 0.17

Sc

-1000

0.00

η

(d) 5000 1500

SG

-2000

(e) 3 1

Fig. 2. Local entropy generation rate [kWm

K

1 1

]; C = 0.045 kgm

s

 sv ¼ cp;v ln

ð12Þ



T þ sv ;0 T0

ð13Þ

The shear stress is simply given by,

  @u @ v s ¼ lS þ @y @x

η

, xin = 60%, Tin = 46.6 °C, P = 1 kPa, Tw = 32 °C, flowing over a tube with outer radius r = 9 mm.

where P0 and T0 are the standard values of pressure and temperature. The expression of vapour molar enthalpy and entropy are given respectively by,

hv ¼ cp;v ðT  T 0 Þ þ hv ;0

0.82 0.00 0.05 0.18 0.37 0.58 0.78 0.93 1.00

0.67

ε

0.52

0.37

0.22

-9000

0.07

-5500

ð14Þ

Thus, the local volumetric entropy generation rate can be expressed as,

"    # ( "     #      ) 2 2 2 2 2 @T @T lS @u @v @u @v þ þ þ SG ¼ 2 þ 2 þ @x @y @x @y T @y @x T           hv @T @T 1 @ cv @ cv þ 2 jv ;x þ jv ;y  jv ;x þ jv ;y ð15Þ @x @y @x @y T T kS

The transverse and axial molar fluxes are given respectively by,

jv ;x ¼ C v u  Dv S

@C v @x

ð16Þ

jv ;y ¼ C v v  Dv S

@C v @y

ð17Þ

By substituting the fluxes and enthalpy expressions and by calculating the derivatives of the molar chemical potential, the expression can be finally reorganized for representing the case of gas absorption into a falling liquid film.

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8.E-02

800

1100

0.09

SG

SG

St

600

0.08

6.E-02

800

Sf

400

0.07

4.E-02

0.06 2.E-02

200

St

500

Sc

0.05

Sd

0

S

0.E+00

Sf

S

Sf Sd

-2.E-02

-200

0.04

200 Sf

Sc

0.03

-4.E-02

-400

0.02

-100 -6.E-02

-600

0.01 -8.E-02

-800 0

0.2

0.4

ε

0.6

0.8

1

-400

Fig. 3. Distribution of different entropy generation groups [kWm3K1] in the stream-wise direction. C = 0.045 kgm1s1, xin = 60%, Tin = 46.6 °C, P = 1 kPa, Tw = 32 °C, flowing over a tube with outer radius r = 9 mm.

"  ( "   2 #  2 # 2 2 @T @T lS @u @v þ þ 2 þ @x @y @x T @y T2    2 )  @u @v þ cp;v ðT  T 0 Þ þ hv ;0 þ þ @y @x       T Cv @T @T þ sv ;0 þT cp;v ln v þ u T0 @y @x T2   Dv S @C v @T @C v @T þ  2 @y @y @x @x T

SG ¼

200 Re

Fig. 4. Different groups of global volumetric entropy generation rate [kWm3K1] as a function of film Reynolds number (abscissa in logarithmic scale); xin = 60%, Tin = 46.6 °C, P = 1 kPa, Tw = 32 °C, flowing over a tube with outer radius r = 9 mm.

Regarding the thermodynamic formulation of the present problem, the solution properties are calculated for the inlet values of temperature and concentration at the absorber pressure, with reference to [15]. 4. Results

ð18Þ

"   2 # 2 @T @T þ @x @y T2 kS

ð19Þ

The second term is due to fluid friction.

Sf ¼

20

kS

Different terms, related to different entropy variation sources, can be distinguished. The first term of the right-hand side of Eq. (18) stands for the irreversibility due to heat transfer St.

St ¼

0.00 2

( "   2 #    2 ) 2 @u @v @u @v þ þ 2 þ @x @y T @y @x

lS

ð20Þ

The third and the fourth terms are related to the coupling effects between heat and mass transfer, by convection and diffusion, respectively.

       T Cv @T @T þ sv ;0 Sc ¼ cp;v ðT  T 0 Þ þ hv ;0 þ T cp;v ln v þu 2 T0 @y @x T ð21Þ      T þ sv ;0 Sd ¼  cp;v ðT  T 0 Þ þ hv ;0 þ T cp;v ln T0    Dv S @C v @T @C v @T þ  @x @x T 2 @y @y

ð22Þ

4.1. Local entropy generation analysis A general analysis has been carried out for typical conditions of a single absorber tube, working with inlet coolant temperature of 32 °C and external pressure of 1.0 kPa. The initial conditions of the LiBr–H2O solution film have been set at the equilibrium for a 60% concentration solution. Furthermore, the variation of mass flow rate due to absorption of water vapour is considered negligible. This assumption is valid for mass-flow rates per unit of tube length higher than 0.001 kgm1s1 and, accordingly, this analysis has been performed in a consistent mass flow-rate range [13]. Since Nusselt integral solution for velocity distribution is not defined at the inlet and outlet positions, respectively e = 0 and e = 1, the total entropy generation SG and each entropy generation groups (St, Sc, Sd and Sf) have been evaluated between e = 1/N and e = N-1/N, inside the whole film thickness. Thermal entropy generation (Fig. 2(a), logarithmic scale) makes evidence of the effect of both heat transfers at the wall and at the liquid–vapour interface. At the latter position a relative maximum can be highlighted, corresponding to the position at which the absorbed flux maximises the temperature gradient. In the condition considered, the impact of heat transfer at the wall on thermal irreversibility is much higher than that of absorption. Furthermore, the temperature gradient component normal to the wall and to the free surface is also dependent on the film thickness distribution. A

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(a)

(b)

750

200 180

700

160 r 7.0 mm

650

11.0

140

9.0

120

600

St

r 7.0 mm

Sd 100

9.0 550

80

500

60

11.0

40 450

20 0

400 4

(c)

Re

40

4

(d)

100

Re

40

0.06

50 0.05

0 11.0

0.04

-50

9.0

Sc -100

Sf 0.03

r 7.0 mm

11.0

-150

0.02

9.0

-200 r 7.0 mm

0.01

-250 -300

0

4

Re

40

4

40

Re

3 1

Fig. 5. Effect of different radii on the single entropy generation groups [kWm

800 750 700 650 600 r 7.0 mm

SG 550 500

9.0

450 11.0

400 350 300 4

40

Re

Fig. 6. Volumetric entropy generation [kWm3K1] as a function of Reynolds number for different values of the tube radius (abscissa in logarithmic scale); Tw = 32 °C, xin = 60%, Tin = 46.6 °C, P = 1 kPa.

reduction of d increases both heat transfer and the rate of absorption. The thickening of the film brings about the opposite effect [5][13]. Owing to the equilibrium hypothesis at the film interface, the solution of the coupled heat and mass transfer process occurring inside the domain is strongly related to the solution inlet conditions as well as to the operative pressure and boundary conditions. Since the concentration of 60% is the equilibrium concentration at the inlet temperature of 46.6 °C, the absorption starts after the thermal boundary layer reaches the interface, and the position at which this occurs depends primarily on the solution mass flow-rate, the tube radius and the coolant temperature (Fig. 2(b)). Near the vapour-solution interface, where absorption takes place, the entropy generation group due to the coupled effect between heat and mass transfer by convection (Fig. 2(c)) is at its maximum (absolute value). A local maximum can be identified

K

] (abscissa in logarithmic scale); Tw = 32 °C, xin = 60%, Tin = 46.6 °C, P = 1 kPa.

and can be related to the conflicting effects of absorption mass-flux and velocity field. The boundary condition for the concentration gradient at the tube wall requires this entropy generation group to be constantly zero at that position. In the second half of the tube surface this group assumes negative values: this behaviour can be explained considering that the local temperature decreases in the stream-wise direction and vapour concentration Cv increases in that region (Eq. (21), where the term Cvu/T2_@T/@x is dominant). Friction related irreversibility (Fig. 2(d)) decreases regularly from the wall to the interface and reaches its maximum value in the vertical part of the tube, where the velocity field gradient is maximum, since the tangential velocity assumes its highest value and the film thickness is at its lowest. The friction related irreversibility shows generally a magnitude range lower than the other entropy generation groups. The total entropy generation rate SG is illustrated in Fig. 2(e) as the superimposition of the different groups previously identified. It can be highlighted that the total entropy generation rate SG of an absorptive LiBr–H2O solution, flowing on a cooled horizontal tube, always shows a local minimum in the radial direction except in the first and the last parts of the tube surface, where the total entropy generation rate is relentlessly decreasing. The minimum is determined by the contemporaneity of wall heat transfer, friction and coupled heat and mass transfer at the interface, and it is positioned at the penetration distance of the diffusing vapour inside the film. Fig. 3 shows the distribution of the various entropy generation groups in the stream-wise direction and the way they are combined together to obtain the global entropy generation SG. The friction related entropy generation group refers to the secondary axes. The value associated at each position e is obtained from the integral in the dimensionless radial direction g, and accordingly corresponds to the average value over the film thickness at that specific position. It can be highlighted that each entropy generation group has a local maximum value along the tube surface and the global

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N. Giannetti et al. / International Journal of Heat and Mass Transfer 88 (2015) 755–765

(a)

(b)

700

200 180

600

Tw 32 C

Tw 32 C

160 140

500

34 C

120 34 C

St 400

Sd 100

36 C

80 36 C

300

60 40

200

20 0

100 4

(c)

Re

40

4

(d)

50

40

Re

0.1 Tw 32 C

0

34 C -50

36 C 36 C

-100

34 C

Sc

Sf 0.01

-150 Tw 32 C -200 -250 0.001

-300 4

Re

40

4

40

Re

Fig. 7. Effect of different wall temperatures on the single entropy generation groups [kWm3K1] (abscissa in logarithmic scale); xin = 60%, Tin = 46.6 °C, r = 9 mm, P = 1 kPa.

900 800 700 600 Tw 32 C

SG 500 400

34 C

300

36 C

200 100 4

40

Re

Fig. 8. Volumetric entropy generation [kWm3K1] as a function of Reynolds number for different values of the tube wall temperature (abscissa in logarithmic scale); xin = 60%, Tin = 46.6 °C, r = 9 mm, P = 1 kPa.

irreversibility is maximum at a position close to the vertical part of the tube, where, due to the combined effects of temperature, concentration and velocity fields, gradients are at the highest value. Furthermore, for the typical conditions of falling film heat exchangers the friction related-entropy generation group appear to have the smallest impact on the global irreversibility.

4.2. Global entropy generation analysis The fundamental design approach adopted in the followings is to thermodynamically determine the optimum operating regime of a system. However, the ‘‘optimal’’ condition for a system or a component can be defined in different ways depending on their

main purpose. As previously stated, in this work by ‘‘optimum’’ the least irreversible operating condition for a specified objective is meant, or otherwise, the most desirable tradeoff between two or more competing irreversibilities [6]. Entropy generation minimisation had been also applied to design counter-flow heat exchangers [16–18] or desiccant systems [19–22]. The present main purpose is to identify second law optima for a particular heat and mass transfer system, where an increased number of competing irreversibilities must be considered. As a consequence, the basic problem of heat transfer can be reduced to a particular case of this general analysis. The parametric analysis makes evidence that a minimum entropy generation, establishing the optimal thermodynamic condition, can always be identified in terms of solution Reynolds number.

Re ¼

4C

l

ð23Þ

Firstly, the general trend of the different groups has been examined (Fig. 4). The value associated at each Reynolds number is obtained from the double integral in the dimensionless radial g and tangential e directions, and accordingly corresponds to the average value over the whole transversal section of the film. Increasing Reynolds determine increasing friction and decreasing absorption rates, consequently, their respective entropy generation groups show consistent trends. The thermal related irreversibilities show a maximum value, established by the conflicting effects of increasing extension of the entrance region, increasing film thickness and decreasing absorption heat release. Finally, the entropy generation group related to the coupled effects of mass convection and heat transfer shows a minimum value, which can be explained considering that, increasing Reynolds number, convection is amplified while absorption heat release is reduced. As a result, the global

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(a)

(b)

210

Ti 46°C 180 Ti 46 C

400

150 120

97°C

St

S

97 C

d

90 60 177 C

177°C 30

40

0

4

(c)

40

400

Re

4

(d)

50

40

400

Re

0.1

0 -50 -100 Sc

Ti 46°C

177°C 97°C

Sf 0.01

97°C

-150

177°C -200 Ti 46°C -250

0.001

-300 4

40

400

Re

4

40

400

Re

Fig. 9. Effect of different inlet solution temperatures on the single entropy generation groups [kWm3K1] (b), (c) (abscissa in logarithmic scale) (a), (d) (logarithmic axes); r = 9 mm lines labelled as 46 °C (win = 60%, Tw = 32 °C, P = 1 kPa), lines labelled as 97 °C (win = 60%, Tw = 83 °C, P = 12.5 kPa) and lines labelled as 177 °C (win = 63%, Tw = 163 °C, P = 149 kPa).

Ti 46 °C 400

SG 97 °C

177 °C 40 4

40

400

Re

3 1

Fig. 10. Volumetric entropy generation [kWm K ] as a function of Reynolds number for different operative conditions (logarithmic axes); r = 9 mm lines labelled as 46 °C (win = 60%, Tw = 32 °C, P = 1 kPa), lines labelled as 97 °C (win = 60%, Tw = 83 °C, P = 12.5 kPa) and lines labelled as 177 °C (win = 63%, Tw = 163 °C, P = 149 kPa).

entropy generation rate, obtained by the superimposition of all these phenomena, shows both a local minimum and a local maximum in the considered range of operative condition. After the local maximum SG starts decreasing again because the solution at the film interface flows too fast and absorption process does not have enough time to occur, while the effect of friction is still small. If an extended Reynolds range is considered, after a certain value of this parameter, friction related irreversibilities would have a relative importance strong enough to cause the global entropy generation to increase relentlessly. Nonetheless, due to the assumption of a laminar film and with reference to the operative

conditions-range of interest, the following analysis is extended to a range of solution mass flow-rates compatible with absorption applications. Fig. 5(a)–(d) show the effect of different tube radii on each entropy generation group. The effect of radius can be primarily related to the change in intensity of the radial velocity field (Eq. (2)), which directly influences Sc, Sd, Sf (Eqs. (20)–(22)) and indirectly St through the released heat of absorption. A smaller radius increases thermal irreversibility St (Fig. 5(a)) and moves the position of its maximum to a lower Reynolds. A longer flowing time of the solution over a tube with bigger radius can explain higher values of the entropy generation group related to vapour diffusion inside the film thickness Sd (Fig. 5(b)). When the tube radius is reduced, curves representing Sc (Fig. 5(c)) are shifted to lower entropy generation rates and the general trend corresponds to lower Reynolds numbers. Finally, friction related irreversibility Sf (Fig. 5(d)) is slightly affected by the tube radius. In Fig. 6, the influence of the tube radius on the global entropy generation for fixed inlet and boundary conditions is presented. As a rule, the lower the tube radius the higher the volumetric entropy generation rate. The trend of the average volumetric entropy generation rate SG shows a decreasing behaviour in the low Reynolds region, where the effect of the film thickness is preponderant, and a subsequent increasing one at high Reynolds, where the effect of the velocity field dominates the heat and mass transfer process, increasing local gradients and related irreversibilities. The compromise between these conflicting effects establishes the position of the minimum value of the volumetric entropy generation rate, which occurs at lower Reynolds number when the tube radius decreases. Fig. 7(a)–(d) describe the effect of different tube wall temperatures on each entropy generation group. In general, they make

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(a)

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Fig. 11. Effects of different inlet solution concentration (left side) and absorber pressure (right side) on the single entropy generation groups [kWm3K1] (a), (b), (c) (abscissa in logarithmic scale) (d) (logarithmic axes); Tw = 32 °C, Tin = 46.6 °C, r = 9 mm.

evidence of the fact that a lower wall temperature increases temperature gradients and, once the temperature gradient reach the interface, also concentration gradients (Fig. 7(b)). The trend of the thermal related entropy generation group is shifted to lower values and the maximum slightly moves to higher Reynolds when

the temperature of the coolant is increased (Fig. 7(a)). Similar behaviour is shown by the absolute value of the entropy generation group Sc (Fig. 7(c)). Friction related irreversibility (Fig. 7(d), logarithmic scale) is increased by a lower value of the solution temperature T inside the film thickness (Eq. (20)).

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850 in

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750 61 %

700 650

SG

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600 550 500 450 400 4

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40

Fig. 12. Volumetric entropy generation [kWm3K1] as a function of Reynolds number for different inlet concentrations (abscissa in logarithmic scale); Tw = 32 °C, Tin = 46.6 °C, r = 9 mm, P = 1 kPa.

In general, a lower tube wall temperature increases both heat transfer and, increasing the driving force for vapour absorption, mass transfer at the interface. Accordingly, Fig. 8 makes evidence of a higher global entropy generation when tube wall temperature is decreased, while the optimal Reynolds is weakly dependent on this parameter. Furthermore, in order to extend the analysis to a wide range of operative conditions, including those typical of different applications, the volumetric entropy generation rate has been studied when different inlet temperatures, and, due to the inlet equilibrium hypothesis, different concentration and absorber pressure, are used. Fig. 9(a)–(d) describe the effect of the inlet solution temperature on each entropy generation group. This parameter has a big influence on the properties of the solution and, in general, when its value is increased, the trend of each irreversibility group is maintained, but moved to a higher Reynolds range. Thermal irreversibility St (Fig. 9(a)) decreases because of higher values of the solution temperature inside the calculation domain, even if the thermal conductance is also increased (Eq. (19)). Similarly, the group related to the coupled effects of mass convection and heat transfer Sc (Fig. 9(c)) is mainly scaled by the value of the solution temperature T (Eq. (21)). Friction irreversibility Sf (Fig. 9(d)) decreases with higher inlet temperatures also because of a lower viscosity (Eq. (20)). Contrarily, the entropy generation group related to vapour diffusion inside the film thickness Sd (Fig. 9(b)) is dominated by an increased diffusivity of water vapour at higher solution temperatures.

2000 1800 1600 1400

SG

1200 1000 P 2.0 kPa

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200 4

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Fig. 13. Volumetric entropy generation [kWm3K1] as a function of Reynolds number for different absorber pressure (abscissa in logarithmic scale); Tw = 32 °C, Tin = 46.6 °C, r = 9 mm, xin = 60%.

In general, higher temperature applications (i.e. representing absorbers of single and multiple lift heat transformers) have lower global entropy generation and cope with higher solution Reynolds (Fig. 10). The thermodynamic optimum Reynolds, for fixed operative conditions, increases for higher temperature applications, such as heat transformer absorbers. Optimal Reynolds numbers of 22, 73 and 127 are obtained, respectively, for inlet solution temperatures of 46, 97 and 177 °C. These values have been obtained without considering partial wetting of the surface, but usually, they correspond to solution mass flow rates which are not able to assure complete wetting, unless tension-active surfactants are added to the LiBr–H2O solution. When the inlet concentration of the solution is increased the equilibrium hypothesis at the inlet is relaxed and the solution enters the calculation domain as a sub-cooled film. Fig. 11(a)–(d) describe the similar effects of higher inlet solution concentration and higher absorber pressure on each entropy generation group. As a rule, increasing the absorber pressure or the inlet solution concentration, for the same value of the others parameters, increases the absorbed vapour mass flux. Under the same point of view, since in the following graphs the effect of an increased absorber pressure is stronger than a higher solution concentration, the irreversibility related to friction Sf decreases for increasing absorption pressures due to the higher heat released for the absorption of water vapour, while increases for higher concentration because the effect of an increased viscosity is higher than that of heat of absorption on temperature distribution. The higher the concentration the higher the optimal Reynolds number (Fig. 12). Finally, Fig. 13 highlights that increasing the absorber pressure also entropy generation increases. This behaviour can be explained considering that a higher vapour pressure directly determines higher absorption rate at the interface and, indirectly, higher heat transfer at the wall. 5. Conclusions The LiBr–H2O concentration and temperature distributions inside the laminar falling film have been obtained from the numerical solution of the coupled species and energy transport equations. Velocity, temperature and concentration fields, in turn, allow estimating gradients and fluxes of these variables and, eventually, the local volumetric entropy generation of the absorptive film flowing over a cooled horizontal tube. Various entropy generation groups, distinguished with regard to different entropy variations sources, have been discussed and analysed both locally and globally. The parametric analysis performed makes evidence of a minimum entropy generation which can be always identified in terms of solution Reynolds number. Furthermore, the importance of the entropy generation group related to the coupled effect of heat and convective mass transfer on this advantageous thermodynamic condition has been highlighted and the behaviour of each entropy generation group has been described. As a rule, lower tube radius, inlet temperature, inlet concentration and absorber pressure correspond to lower values of the optimal Reynolds number for the absorber, while the tube wall temperature shows a weak influence on that condition. This analysis characterises the irreversibility of the process occurring in real absorbers and has been used to identify the least irreversible value of the solution mass flow-rate for various operating conditions. These results make evidence of the importance to work at reduced mass flow rates with a thin uniform film. As a consequence, tension-active additives are critical to realise this condition. Also, it can be observed that changes in parameters’ values (such as lower tube radii, lower coolant temperature or lower mass flow-rates), which, in general, bring about an enhancement in the

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absorber performance, are associated to higher irreversibilities. As a consequence, the performance improvement related to the effect of these parameters should be compared to the related increase of irreversibility to identify the optimal thermodynamic condition for the absorber and entropy can been used to define a criterion for the thermodynamic optimisation of the absorber. Conflict of interest None declared. References [1] A. Yig˘it, A numerical study of heat and mass transfer in falling film absorber, Int. Commun. Heat Mass Transfer 26 (2) (1999) 269–278. [2] J.W. Andberg, G.C. Vliet, A simplified model for absorption of vapors into liquid films flowing over cooled horizontal tubes, ASHRAE Trans. 93 (1987) 2454– 2466. [3] V.D. Papaefthimiou, I.P. Koronaki, D.C. Karampinos, E.D. Rogdakis, A novel approach for modelling LiBr–H2O falling film absorption on cooled horizontal bundle of tubes, Int. J. Refrig. 35 (4) (2012) 1115–1122. [4] K. Banasiak, J. Kozioł, Mathematical modelling of a LiBr–H2O absorption chiller including two-dimensional distributions of temperature and concentration fields for heat and mass exchangers, Int. J. Therm. Sci. 48 (9) (2009) 1755– 1764. [5] F. Babadi, B. Farhanieh, Characteristics of heat and mass transfer in vapor absorption of falling film flow on a horizontal tube, Int. Commun. Heat Mass Transfer 32 (9) (2005) 1253–1265. [6] A. Bejan, The thermodynamic design of heat and mass transfer processes and devices, Int. J. Heat. Fluid Flow 8 (4) (1987) 258–276. [7] J.Y. San, W.M. Worek, Z. Lavan, Entropy generation in combined heat and mass transfer, Int. J. Heat Mass Transfer 30 (7) (1987) 1359–1369. [8] C.G. Carrington, Z.F. Sun, Second law analysis of combined heat and mass transfer in internal and external flows, Int. J. Heat. Fluid Flow 13 (1) (1992) 65– 70.

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[9] G. Prakash Narayan, J.H. Lienhard V, S.M. Zubair, Entropy generation minimization of combined heat and mass transfer devices, Int. J. Therm. Sci. 49 (10) (2010) 2057–2066. [10] I. Chermiti, N. Hidouri, A.B. Brahim, Entropy generation in gas absorption into a falling liquid film, Mech. Res. Commun. 38 (8) (2011) 586–593. [11] N. Hidouri, I. Chermiti, A.B. Brahim, Second law analysis of a gas–liquid absorption film, J. Thermodyn. (2013) 10. ID 909162. [12] S.K. Choudhury, D. Hisajima, T. Ohuchi, A. Nishiguchi, T. Fukushima, S. Sakaguchi, Absorption of vapors into liquid films flowing over cooled horizontal tubes, ASHRAE Trans. 99 (1993) 81–89. [13] L. Harikrishnan, Shaligram Tiwari, M.P. Maiya, Numerical study of heat and mass transfer characteristics on a falling film horizontal tubular absorber for R-134a-DMAC, Int. J. Therm. Sci. 50 (2) (2011) 149–159. [14] W.G. Vincenti, C.H. Kruger Jr., Introduction to Physical Gas Dynamics, Wiley, New York, 1965. [15] O.F. Properties, Lithium bromide-water solutions at high temperatures and condensations - Part I. thermal conductivity, ASHRAE Trans. 96 (1990). [16] G. Grazzini, F. Gori, Entropy parameters for heat exchanger design, Int. J. Heat Mass Transfer 31 (12) (1988) 2547–2554. [17] A. Bejan, The concept of irreversibility in heat exchanger design: counterflow heat exchangers for gas-to-gas applications, J. Heat Transfer 99 (1977) 374– 380. [18] D.P. Sekulic´, C.V. Herman, One approach to irreversibility minimization in compact crossflow heat exchanger design, Int. Commun. Heat Mass Transfer 13 (1) (1986) 23–32. [19] H. Pahlavanzadeh, P. Nooriasl, Entropy generation in liquid desiccant dehumidification system, Energy Procedia 14 (2012) 1855–1860. [20] D. La, Y. Li, Y. Dai, T. Ge, R. Wang, Effect of irreversible processes on the thermodynamic performance of open-cycle desiccant cooling cycles, Energy Convers. Manage. 67 (2013) 44–56. [21] A. Myat, K. Thu, N. Kim Choon, The experimental investigation on the performance of a low temperature waste heat-driven multi-bed desiccant dehumidifier (MBDD) and minimization of entropy generation, Appl. Therm. Eng. 39 (2012) 70–77. [22] N. Giannetti, A. Rocchetti, K. Saito, S. Yamaguchi, Entropy parameters for desiccant wheel design, Appl. Therm. Eng. 75 (2015) 826–838.