Analytical solution for combined heat and mass transfer in laminar falling film absorption with uniform film velocity – Isothermal and adiabatic wall

Accepted Manuscript Analytical solution for combined heat and mass transfer in laminar falling film absorption with uniform film velocity - Isothermal and adiabatic wall T. Meyer PII:

S0140-7007(14)00207-2

DOI:

10.1016/j.ijrefrig.2014.08.005

Reference:

JIJR 2854

To appear in:

International Journal of Refrigeration

Received Date: 13 June 2014 Revised Date:

31 July 2014

Accepted Date: 8 August 2014

Please cite this article as: Meyer, T., Analytical solution for combined heat and mass transfer in laminar falling film absorption with uniform film velocity - Isothermal and adiabatic wall, International Journal of Refrigeration (2014), doi: 10.1016/j.ijrefrig.2014.08.005. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights (for review)

ACCEPTED MANUSCRIPT

RI PT SC M AN U TE D



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

Introduction of a more versatile analytical solving procedure using the Laplace transform Analytical solution to the combined heat and mass transfer for the isothermal and adiabatic wall Evaluation of the influence of the thermal wall boundary condition on the absorbed mass flux Discussion of the influence of the Lewis number on the absorbed mass flux

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 

*Manuscript Click here to view linked References

ACCEPTED MANUSCRIPT

Analytical solution for combined heat and mass transfer in laminar falling film absorption with uniform film velocity - isothermal and adiabatic wall T.Meyer∗

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Technische Universitaet Berlin, Institiut fuer Energietechnik, Sekreteriat KT 2, Marchstrasse 18, 10587 Berlin, Germany

Abstract

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In the present study the Laplace transform is applied to the partial differential equations obtained from the differential energy and absorbate balances for the combined heat and mass transfer problem in laminar falling

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films with uniform film velocity.

By means of the inverse Laplace transform an analytical solution is provided for the isothermal as well as for the adiabatic wall boundary condition. Temperature and mass fraction profiles across the film as well as the evolution of the absorbed mass flux as a function of the flow length are presented for the adiabatic wall condition as well as for the isothermal wall with different wall temperatures. Furthermore, the influence of the Lewis number on the absorbed mass flux is discussed.

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In addition, the present method allows to apply other wall boundary conditions than the isothermal or the adiabatic wall boundary, which will be addressed in a subsequent study. Keywords: absorption, analytical solution, laminar falling film, heat transfer, mass transfer, Laplace

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Nomenclature

Dimensionless numbers   Le Lewis number Le = a · D−1 ˜ St

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modified Stefan number   ˜ = cs · ∆T · ∆h−1 · ∆c−1 St abs

Greek letters α, β ∗

eigenvalues

corresponding author:

mail:[email protected] phone: +493031422933 Preprint submitted to International Journal of Refrigeration

∆

difference

δ

film thickness

η

dimensionless film thickness

γ

dimensionless mass fraction

λ

thermal conductivity

µ

dimensionless mass flux

ρ

density

Θ

dimensionsless temperature

ξ

normalized flow coordinate

[m]

h i W · m−1 K−1 h i kg · m−3

Latin letters July 31, 2014

ACCEPTED MANUSCRIPT thermal diffusivity

A, B

constants

h i m2 · s−1 h i kg · kg−1 h i kJ · kg−1 K−1 h i m2 · s−1 h i kJ · kg−1

c

mass fraction (absorbate)

c

specific heat capacity

D

mass diffusivity

h

specific enthalpy

i

imaginary unit

k

index

m ˙

mass flux

p

pressure

T

temperature

u

streamwise film velocity

v

transverse film velocity

v

specific volume

x

streamwise direction

y

transverse direction to film flow

z

complex Laplace variable

s

solution

W

wall

1. Introduction By means of the Fourier method Grigor’eva and

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a

Nakoryakov (1977),(Nakoryakov et al., 1997; Nakoryakov and Grigor’eva, 2010) presented an analytical solution for the combined heat and mass transfer in laminar falling film absorption with constant film ve-

SC

h i kg · m−2 s−1

locity. Nevertheless, their solution did not match the

[Pa]

inlet conditions for small dimensionless flow lengths

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[K] h i m · s−1 h i m · s−1 h i m3 · kg−1

ξ and wall temperatures, that are different from the film inlet temperature (Nakoryakov and Grigor’eva, 2010). This mathematical problem originates in the domain restrictions of the tangent function which has been used to determine the eigenvalues by Nakorya-

[m]

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Sub-/Superscripts/Symbols

kov and Grigor’eva (2010). By arranging these tan-

[m]

gent functions to non-restricted sine and cosine functions (Meyer, 2014), more eigenvalues are found and the improved Fourier method matches the inlet con-

inlet values

∞

asymptotic value for ξ → ∞

lengths ξ.

( )

mean value

However, the orthogonality relation, which has been

abs

absorption

derived and used by Grigor’eva and Nakoryakov (1977),

e

energy

necessitates either the dimensionless wall tempera-

eq

equilibrium (at inlet condition)

i

interface

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0

dition for all boundary conditions even for small flow

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ture or the dimensionless wall temperature gradient to be zero. Thus, it is impossible to use any other wall boundary condition for the temperature than the

i, j, k index m

mass

mA

mass fraction absorbate

isothermal or adiabatic wall, as the orthogonality relation is indispensable in order to apply the Fourier method. 2

ACCEPTED MANUSCRIPT For that reason in the present study the partial differential equations for energy and mass fraction are solved without any restrictions for the boundary conditions by means of the Laplace transform. However,

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the objective of this study is to introduce and validate the Laplace method by applying the isothermal and

y

adiabatic wall only. Other boundary conditions will be presented in a subsequent study. The inverse Laplace transform presented by Baehr (1955) is applied

x,u u

Ti T0

TW

ci

c

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and mass transfer problem in the real domain.

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T0 ,c 0

in order to obtain the solutions to the combined heat

2. Film Model

0

T

c0

x1

In order to illustrate the modelling assumptions, the differential absorbate and energy balance are ap-

x2

plied to the film flow, leading to the usual partial

y

0 x,u u

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differential equations for energy and mass fraction, which are the starting point for most of the analytical solutions.

Figure 1 depicts the model of the film flowing down

c

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an isothermal, vertical wall for which, beside the adi-

TW

T0 ci c0

abatic wall, the combined heat and mass transfer is considered.

Ti

T

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In Figure 2 an arbitrary infinitesimal volume ele-

Figure 1: Scheme of the isothermal wall model depicting the

ment within the falling film is depicted and the streams

temperature and mass fraction profile qualitatively for different film flow coordinates x1
marked by the arrows and labelled with j are fluxes of any arbitrary conservation quantity, e.g. mass or energy. Balancing this quantity j for steady state conditions leads to the following equation:   0 = j(x) − j(x + dx) · dydz   + j(y) − j(y + dy) · dxdz.

(1) 3

ACCEPTED MANUSCRIPT The sign of the respective stream is determined by

respective quantity within the control volume is not

the direction of the arrow compared to the volume

possible due to steady state. A change in flux in x

element as well as its direction compared to the di-

direction has to be balanced by an opposite change

rection of the coordinate. By means of a Taylor ex-

in flux in y direction. So far only the steady state condition has been as-

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z

sumed for the film flow. Starting with the overall

y

j(x)

x

mass balance, some more assumptions are applied.

z+dz

2.1. Differential mass balance

x,z

The overall mass fluxes are convective and can

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j(y)

j(y+dy)

x+dx y+dy

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be modelled as follows:

y

jm,x = u(x) · ρ(x),

(5)

jm,y = v(y) · ρ(y),

(6)

which are introduced to (4) leading to:

j(x+dx)

0=−

Figure 2: Infinitesimal volume element within the falling film

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for general differential balancing of any conservation quantity j

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∂ j j(x + dx) = j(x) + dx, ∂x x ∂ j j(y + dy) = j(y) + dy. ∂y y

(7)

sumed to be constant as well as the film’s streamwise velocity u = u. For that reason the transverse film ve-

possible to approximate the streams leaving the element as follows:

∂ (u · ρ) ∂ (v · ρ) − . ∂x ∂y

All thermophysical properties of the solution are as-

pansion neglecting the terms of an order n > 1, it is

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locity v must not change in y direction in order not to violate the conservation of mass.

(2)

It is assumed that the transverse film velocity at the inlet is v = 0, so that for the whole film flow there is

(3)

no convective flow in the transverse direction y, but

Introducing (2) and (3) to (1) only the derivatives of

a constant mean film velocity u in the streamwise di-

the conservation quantity remain:

rection x only.

0=−

∂j ∂j − . ∂x ∂y

The assumption of a uniform mean film velocity u is (4)

rather simplifying, since the steady state momentum

This is trivial for the steady state condition consid-

transfer of the film flow results in the usual parabolic

ered here. Every conservation quantity entering the

velocity profile (Nusselt, 1923). As Grossman (1983)

control volume has to leave it, since a change in the

applied the parabolic velocity profile to the energy 4

ACCEPTED MANUSCRIPT diffusion despite these modelling inconsistencies. The

differential equations (ODEs) by means of the Fou-

unidirectional diffusion model would violate the sim-

rier method. Applying a recursion formula Gross-

plifying assumptions for the hydrodynamics, due to

man (1983) solved these non-linear ODEs and he ob-

the convective mass flow in transverse direction y,

tained the usual infinite power series as eigenfunc-

changing with the local mass fraction gradients. In

tions. Nevertheless, the radius of convergence of

addition, the differential component balance turns to

these power series seems to be very limited as Gross-

a non-linear differential equation by applying uni-

man (1983) stated that his analytical solution con-

directional diffusion. Hence, the counter diffusion

verges for moderate and large film flow lengths only.

model is applied in order to handle the problem ana-

For that reason he used numerical methods in order

lytically.

to solve the problem for small film flow lengths.

Moreover, the diffusive transport in the streamwise

Therefore, within the present study a uniform mean

direction x is neglected compared to the convective

film velocity u is applied in order to obtain an analyt-

flow as well as to the diffusive flow in transverse di-

ical solution which covers the whole film flow range.

rection.

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and mass balances he obtained non-linear ordinary

Introducing these absorbate fluxes to (4) the differen-

2.2. Differential absorbate balance

tial absorbate balance is obtained:

The flux of both, the absorbate and absorbent,

u·

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can be convective as well as diffusive. Taking into ac-

∂c ∂2 c = D · 2. ∂x ∂y

(10)

count the previous assumptions, the absorbate mass

2.3. Differential energy balance

fluxes for the different coordinates can be modelled

jmA,x jmA,y

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as follows:

" # ∂c(x) = ρ · u · c(x) − D , ∂x ∂c(y) = −ρ · D . ∂y

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Energy can be transported convectively with the film flow as well as diffusively through conduction.

(8)

Taking into account all the previous assumptions, the energy fluxes for the different coordinates are the fol-

(9)

lowing:

Here c represents the mass fraction of absorbate. An je,x = u · ρ · h(x) − λ ·

equimolar counter diffusion with constant solution density is assumed. In case of a non volatile ab-

je,y = −λ ·

sorbent, this counter diffusion assumption is prob-

∂T . ∂y

∂T , ∂x

(11) (12)

lematic, as a convective flow from the interface to the

Similar to the absorbate balance, the diffusive energy

bulk film flow is induced. Consequently, the model

flux in streamwise direction is neglected compared

of unidirectional diffusion should be applied. How-

to the convective energy flux and to the transversal

ever, there are some reasons in order to apply counter

conductive heat flux. The enthalpy of the solution is 5

ACCEPTED MANUSCRIPT modelled as an ideal liquid:

the starting point for the analytical solving procedure. The application of the Laplace transform to

dh = cs · dT + vs · d p.

(13) (19) and (20) will be introduced, after the inlet and boundary conditions are determined.

Here vs is the specific volume and cs is the specific heat capacity of the solution. Assuming constant pres-

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2.5. Inlet and boundary conditions

sure during the absorption process, the change in en-

The dimensionless inlet temperature and mass frac-

thalpy is a function of the liquid’s temperature only.

tion equal zero, since the inlet conditions T 0 and c0

Introducing these simplified fluxes to (4), the partial

are used for normalization:

differential energy balance is obtained: 2

2.4. Normalization

(14)

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∂T λ ∂T ∂T u· · 2 =a· 2. = ∂x ρcs ∂y ∂y

(21) (22)

The interface boundary conditions are the only link

In laminar falling film absorption the following normalized variables have been established:

between the temperature and mass fraction profile within the present model. In a previous analytical model Meyer and Ziegler (2014) have set first type

(15)

boundary conditions at the interface. On the con-

(16)

trary the interface temperature and the interface mass

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y η= , δ x λ/δ x a = · ξ= · , δ u · δ δ u · ρ · cs T − T0 Θ= , T eq − T 0 c − c0 . γ= ceq − c0

(17)

fraction in the present study are not constant but are linked by a linear approximation for the isobaric sat-

(18)

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uration temperature of the solution as proposed by

Note that in the present study the dimensionless film

Grigor’eva and Nakoryakov (1977):

thickness η varies from 0 at the film surface to 1 at the wall.

T0 − T0 = 0, T eq − T 0 c0 − c0 γ0 = = 0. ceq − c0

Θ0 =

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2

T i = A − B · ci .

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The interface gradients of temperature and mass frac-

Introducing these dimensionless variables to the par-

tion are linked in terms of an interface energy balance

tial differential equations (10) and (14) the following

accounting for the latent heat. The heat of absorption

dimensionless equations are obtained: ∂γ 1 ∂2 γ = , ∂ξ Le ∂η2 ∂Θ ∂2 Θ = 2. ∂ξ ∂η

(23)

depends on the interface mass fraction gradient and (19)

has to be transported into the bulk film flow by means of heat conduction, which in turn depends on the in-

(20)

These partial differential equations for the dimensionless mass fraction (19) and temperature (20) are 6

terface temperature gradient: ∂c ∂T ρ s · D s · · ∆habs = λ s · . ∂y i ∂y i

(24)

ACCEPTED MANUSCRIPT Inserting dimensionless variables into the interface

3. Laplace transform

boundary conditions (23) and (24), the following di-

The Laplace transform F(z) of a function f (x) is

mensionless relations are obtained:

defined as follows: (25)

F(z) =

(26)

(32)

with z as the complex Laplace variable and x as the corresponding variable in the real domain. The application of the Laplace transform to the dimensionless

(27)

energy (20) and absorbate balance (19) with regard

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a , D c · (T eq − T 0 ) cs · B ˜ = s St = . ∆habs · (ceq − c0 ) ∆habs

f (x) · e −z·x dx,

0

with the dimensionless numbers: Le =

Z∞

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Θi + γi = 1, ∂γ ∂Θ ˜ · = Le · St , ∂η i ∂η i

(28)

to the flow length ξ leads to the following ordinary differential equations (ODEs) in the Laplace domain:

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Here the Lewis number, Le, relates the diffusive heat

transport to the diffusive mass transport. The mod-

∂2 Θ(z, η) − z · Θ, ∂η2 1 ∂2 γ(z, η) − z · γ. −γ0 = Le ∂η2

pacity of the film to the specific heat of absorption set

free at the interface, similarly to Ziegler (2010), who

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due to the definition of the dimensionless temperature and mass fraction and (33) and (34) turn into

Within the present model either the isothermal or the

homogeneous ODEs. Thus, the roots of the charac-

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adiabatic temperature wall boundary condition are applied.

T W = const. → ΘW = const. or ∂Θ ∂T =0→ =0 ∂y W ∂η W

(34)

The dimensionless inlet values Θ0 and γ0 are zero

presented a relative of the Stefan number for vapour compression cycles.

(33)

−Θ0 =

˜ relates the specific heat caified Stefan number, St,

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teristic polynomials and their corresponding formal solutions are:

(29)

√ zΘ = ± z, √ zγ = ± zLe,

(30)

Θ(z, η) = C1 · e γ(z, η) = C3 · e

The absorbate mass fraction gradient at the wall is zero, as the wall is impermeable for the absorbate. ∂c ∂γ (31) =0→ =0 ∂y W ∂η W

√ √

zη

(35) (36) + C2 · e −

zLeη

√

zη

+ C4 · e −

,

√

zLeη

(37) .

(38)

By means of any physically consistent boundary conditions, the integration constants in (37) and (38) can be determined. These boundary conditions have to be transformed to the Laplace domain also. At first those boundary conditions are transformed, which 7

ACCEPTED MANUSCRIPT are identical for both, the isothermal and the adia-

adiabatic wall boundary condition are obtained.

batic wall: (39)

#−1 √ √ √ ˜ + LeSt sinh( z) cosh( zLe)

(40)

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∂γ (z) = 0, ∂η η=1     1 Θ z, η = 0 + γ z, η = 0 = , z 1 ∂γ ∂Θ (z). (z) = ˜ ∂η η=0 ∂η η=0 LeSt

√ √ √ ˜ · sinh( z) · cosh( zLe(η − 1)) γ(z, η) = LeSt "  √ √ · z · cosh( z) sinh( zLe)

(41)

√ √ Θ(z, η) = sinh( zLe) · cosh( z(η − 1)) "  √ √ · z · cosh( z) sinh( zLe)

The impermeable wall boundary (39) and the inter-

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face energy balance (41) in the Laplace domain are formally identical to the boundary conditions in the real domain (31) and (26) respectively. However,

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the dimensionless phase equilibrium (25) changes to (40) in the Laplace domain.

(43)

(44)

#−1 √ √ √ ˜ sinh( z) cosh( zLe) + LeSt 3.2. Isothermal wall The isothermal wall boundary condition in the

Laplace domain results as follows:

3.1. Adiabatic wall

ary condition results as follows: ∂Θ (z) = 0. ∂η η=1

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In the Laplace domain the adiabatic wall bound-

  ΘW Θ z, η = 1 = . z

(45)

In the same way as for the adiabatic wall (AppendixA),

(42)

the boundary conditions are applied in order to obtain the mass fraction (46) and the temperature pro-

In AppendixA the general solving procedure by ap-

file (47) in the Laplace domain for the isothermal

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plication of the boundary conditions in the Laplace domain is exemplified for the adiabatic wall case. As a result the mass fraction profile (43) and the tem-

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perature profile (44) in the Laplace domain for the

8

ACCEPTED MANUSCRIPT Baehr (1955) gave a remarkable introduction of the

wall boundary condition.

application of the inverse Laplace transform to heat

γ(z, η) = h√   √ i √ ˜ · cosh( z) − ΘW cosh zLe(η − 1) LeSt "  √ √ · z · sinh( z) sinh( zLe)

plex integral (48) can be solved by means of Cauchy’s integral as well as the residue theorem:

#−1 √ √ ˜ LeSt · cosh( z) cosh( zLe)

f (x) =

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+

√

transfer problems. Following his instruction, the com-

∞ X

Res [ez·x F(z)]z=zk .

(49)

k=0

Here, zk are the poles of the complex function F(z). In order to identify these poles, it is useful to arrange

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(46)

F(z) as a rational function: Θ(z, η) = "  √ √ ΘW · sinh( zLe) sinh( zη)

f (z) . g(z)

(50)

Consequently, the roots of the denominator function g(z) are the poles of F(z). The corresponding residues

(47)

for simple roots of g(z) can be determined as follows (Baehr, 1955):

f (zk )ezk ·x Res [ez·x F(z)]z=zk =   . dg(z)/dz z=zk

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√ √ √  ˜ + LeSt · cosh( zLe) cosh( zη) √ # √ − sinh( zLe) sinh z(η − 1) "  √ √ · z · sinh( z) sinh( zLe)

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F(z) =

#−1 √ √ √ ˜ + LeSt · cosh( z) cosh( zLe)

(51)

In case of a multiple root for the denominator function g(z), e.g. for z0 = 0, (51) is inapplicable and a

perature (47) profile have the same denominator func-

Laurent series (52) has to be expanded (Baehr, 1955).

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Note that the mass fraction (46) as well as the tem-

tion. This denominator function is essential for the e zx · F(z) = ... +

application of the inverse Laplace transform, which

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is introduced now.

D−1 + D0 + D1 · z + ... z

(52)

From this Laurent series D−1 is obtained as the residue of a multiple root in z0 = 0.

4. Inverse Laplace Transform

Consequently, the overall solution in the real domain

The general inverse Laplace transform of a func-

results by introducing all residues to (49), which is

tion F(z) in the Laplace domain is given by a com-

shown now for the adiabatic wall as well as for the

plex integral:

isothermal wall. At first a general procedure apply-

f (x) =

1 2πi

ω+i∞ Z

ez·x F (z) dz.

ing a numerical root finding is introduced in order to obtain the solution in the real domain. Subsequently

(48)

ω−i∞

a special, fully analytical case is provided. 9

ACCEPTED MANUSCRIPT 4.1. Adiabatic wall

tion is needed, which is as follows: √  √ ˜ +1 √  ∂g(z) LeSt = −i sin i z(1 + Le) ∂z 2 √  √ ˜ −1 √  LeSt −i sin i z(1 − Le) 2 √ √ √ ˜  √ √  z( LeSt + 1)(1 + Le) + cos i z(1 + Le) 4 √ √ √ ˜  √ √  z( LeSt − 1)(1 − Le) + cos i z(1 − Le) . 4 (57)

4.1.1. Numerical root finding Regarding (50), the numerator functions of the mass fraction profile (43) and the temperature profile

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(44) for the adiabatic wall boundary in the Laplace domain are as follows: fγ (z) =

√ √ √ ˜ · sinh( z) · cosh( zLe(η − 1)), LeSt (53)

The periodic roots are iterated numerically using (56) (54)

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√ √ fΘ (z) = sinh( zLe) · cosh( z(η − 1)).

and introduced to (51) in order to obtain the periodic residues.

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The denominator functions of the mass fraction (43) and temperature profile (44) for the adiabatic

The root z0 = 0 is a multiple root of the denominator

wall are identical.  √ √ g(z) = z · cosh( z) sinh( zLe)  √ √ √ ˜ sinh( z) cosh( zLe) + LeSt

function g(z). Hence, a Laurent series expansion is conducted in order to identify D−1 as the residue for z0 = 0 for the adiabatic wall case:

(55)

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In order to determine the roots of (55), it is simplified and rearranged using product to sum identities as well as the identities between the hyperbolic func-

˜ 1 St e zξ · γ(z, η) = ˜ z St  + 1 √ 2   √2 2 z 1 + 3! + ... · 1 + zLe2!(η−1) + ... (1 + zξ + ...) · , √2   √ √ z 1 + 3! (1 + Le)3 (1 − Le)3 + ...

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(58) tions and the trigonometric functions (cf. AppendixB). 1 1  √LeSt √ e zξ · Θ(z, η) = ˜ +1 √  ˜ +1z St g(z) = z · sinh z(1 + Le)  √ 2 √2    2 z (η−1)2 zLe √ 1 + 3! + ... · 1 + + ... (1 + zξ + ...) 2! √ ˜ −1 √  LeSt · . √   + sinh z(1 − Le) √ √ z2 3 (1 − 3 + ... 2 1 + (1 + Le) Le) (56) 3!  √LeSt  √ ˜ +1 √  (59) = −i · z · sin i z(1 + Le) 2 √  √ ˜ −1 √  LeSt Regarding (58) and (59), the asymptotic residues for + sin i z(1 − Le) 2 the mass fraction and the temperature profiles for the Regarding (51), the derivative of the denominator funccase of an adiabatic wall are identified as follows:

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10

˜ St , ˜ +1 St 1 = . ˜ +1 St

Res[e zξ · γ(z, η)]z=0 =

(60)

Res[e zξ · Θ(z, η)]z=0

(61)

ACCEPTED MANUSCRIPT In AppendixC these asymptotic values for the adia-

in the real domain is obtained:

batic wall are derived from an integral energy bal-

γ(ξ, η) =

ance for the film and the results are identical with the

˜ St ˜ +1 St 

kπ √ (1+ Le)

2 X sin + π k=1

The overall solution for the adiabatic wall boundary in the real domain is obtained by inserting all the residues into (49).

k·

· cos



(−1)k

 √ kπ Le(η−1) √ (1+ Le)

·e

 −

kπ √ 1+ Le

2

·ξ

.

(68)

Similarly, the dimensionless temperature profile for the adiabatic wall in the real domain results as fol-

√ ˜ = 1 and Le · St

lows:

the adiabatic wall For the special case of

√ ˜ = 1, the denominaLeSt

Θ(ξ, η) =

 √ √  g(z) = −iz sin i z(1 + Le) √  √ √ √  z ∂g(z) = (1 + Le) cos i z(1 + Le) ∂z 2 Hence, the periodic roots of (62) are given by:

1 ˜ +1 St 

∞ 2 X sin

 √ kπ √Le (1+ Le)

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tor term of (56) simplifies to only one sine function.

SC

4.1.2. Fully analytical solution for

+

(62)

π

k=1

k·

· cos

(−1)k



kπ(η−1) √ (1+ Le)



·e

 −

kπ √ 1+ Le

2

·ξ

.

(69)

(63)

4.2. Isothermal wall boundary 4.2.1. Numerical root finding

!2 kπ with k = 1, 2, 3, ... , (64) =− √ 1 + Le ikπ ikπ cos (−kπ) = (−1)k . (65) = 2 2

TE D

zk ∂g(z) ∂z zk



RI PT

∞

values presented by Grossman (1983).

The corresponding numerator functions of the mass fraction profile (46) and the temperature profile (47) for the isothermal wall boundary in the Laplace domain are as follows:

EP

The poles zk are inserted into the numerator functions

fγ (z) =   √  √ √ ˜ · cosh( z) − ΘW cosh zLe(η − 1) , LeSt

of the mass fraction profile (53) and the temperature profile (54) respectively: =1

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 √  !  kπ Le(η − 1)  kπ  , cos  √ √ (1 + Le) (1 + Le)

fk,γ

z√}| { ˜ sin = i LeSt

fk,Θ

√   !  kπ Le  kπ(η − 1) = i sin  . √  cos √ (1 + Le) (1 + Le)

(70) 

√ √ fΘ (z) = ΘW · sinh( zLe) sinh( zη) √ √ √  ˜ + LeSt · cosh( zLe) cosh( zη) √  √ − sinh( zLe) sinh z(η − 1) .

(66) (67)

Adding the asymptotic to the periodic residues with

(71)

The denominator functions of the mass fraction (46)

respect to (49), the following analytical solution for

and the temperature (47) profile in the Laplace do-

the mass fraction profile for the adiabatic wall case 11

ACCEPTED MANUSCRIPT

# √ √ √ ˜ · cosh( z) cosh( zLe) + LeSt

Hence, the residues of z0 = 0 for temperature and mass fraction are as follows for the isothermal wall case:

(72)

fΘ (z0 = 0) = ΘW , ∂g(z) ∂z

By means of product to sum identities the following

z0 =0

fγ (z0 = 0) = 1 − ΘW . ∂g(z)

simplified relation for g(z) is obtained in order to ap-

∂z

(77)

z0 =0

The overall solution in the real domain is obtained by

SC

introducing all the residues (51) to (49). 4.2.2. Fully analytical solution for (73)

√ ˜ = 1 and Le · St

the isothermal wall

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ply a numerical root finding routine: "√ √ ˜ +1 √  LeSt g(z) = z · cosh z(1 + Le) 2 √ # √ ˜ −1 √  LeSt + cosh z(1 − Le) 2 "√  √ ˜ +1 √  LeSt cos i z(1 + Le) =z· 2 √ #  √ ˜ −1 √  LeSt + cos i z(1 − Le) . 2

(76)

RI PT

main for the isothermal wall are identical. " √ √ g(z) = z · sinh( z) sinh( zLe)

For the special case of

√ ˜ = 1, the denomLe · St

inator function (73) and its derivative in the Laplace domain simplifies to:

The derivative of the denominator function g(z) re-

TE D

sults as follows:

EP

"√  √ ˜ +1 √  LeSt ∂g(z) = cos i z(1 + Le) ∂z 2 √ #  √ ˜ −1 √  LeSt + cos i z(1 − Le) 2 √ " √  √ ˜ + 1)(1 + Le) √  √ ( LeSt −i z· sin i z(1 + Le) 4 √ √ #  √ ˜ − 1)(1 − Le) √  ( LeSt + sin i z(1 − Le) . 4

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The roots of (78) are as follows: z0 = 0, zk = −

(74)

(80) (2k + 1) π2 √ (1 + Le)

!2

;

k = 0, 1, 2, ....

(81)

The asymptotic value for z0 = 0 remains the same as

For the isothermal wall, z0 = 0 is a simple root. Consequently, (51) can be used to determine the residue

shown before in (76) and (77). Inserting the periodic roots into the derivative of the denominator yields:  ∂g(z) π π  = (2k + 1) · sin −(2k + 1) ) . (82) ∂z z=zk 4 | 2} {z

of z0 = 0. For z0 = 0 the derivative of the denominator turns into: √ ∂g(z) ˜ = LeSt. ∂z z0 =0

 √ √  (78) g(z) = z · cos i z(1 + Le) ,  √ √  ∂g(z) = cos i z(1 + Le) ∂z √  √ √  √ (1 + Le) −i z· · sin i z(1 + Le) . 2 (79)

(−1)k+1

(75) 12

ACCEPTED MANUSCRIPT According to (70) and (71) the numerator functions √ ˜ = 1 result as follows: for Le · St

5. Results and Discussion The present analytical solution obtained by means

  √ √ fγ (z) = cos(i z) − ΘW cos(i zLe(η − 1)), (83)

of the Laplace transform has been validated by comparing it to a numerical solution using the Finite Dif-

fΘ (z) = ΘW

For the same simplifying assumptions the results obtained with the numerical method as well as with the present analytical method exhibited no deviations for dimensionless flow lengths of 1 · 10−2 < ξ < 1 · 103

SC

(84)

RI PT

ference method presented by Mittermaier et al. (2014b).

√ √ cosh( z( Le+η))

z }| {  √ √ √ √  · sinh( zLe) sinh( zη) + cosh( zLe) cosh( zη) √  √ − sinh( zLe) sinh z(η − 1) √ √ = ΘW cos(i z( Le + η))  √  √ + sin(i zLe) sin i z(η − 1) .

and the case of an isothermal wall boundary (Mitter-

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maier et al., 2014a; Meyer et al., 2014).

The fully analytical mass fraction profile for the isother- This validated analytical Laplace solution is now used mal wall in the real domain is obtained by inserting

in order to analyze the combined heat and mass trans-

all poles into (49) using (51):

fer for two different Lewis numbers and different wall boundary conditions for the temperature profile. At

γ(ξ, η) = 1 − ΘW ! ! " ∞ (2k + 1) π2 4 X (−1)k+1 + cos − ΘW √ π k=0 2k + 1 (1 + Le) !2 !# (2k+1) π √ 2 (2k + 1) π2 √ − ·ξ · cos · Le · (η − 1) · e (1+ Le) . √ (1 + Le)

TE D

first the results for the fully analytical solution with

EP

(85)

The fully analytical temperature profile for the isother-

as follows:

·e

−

(2k+1) π √ 2 (1+ Le)

·ξ

˜ = 0.1 will be discussed. Subsequently, number of St the Lewis number is increased to a value of Le = 200 and the results are compared to the case of Le = 100. For all infinite series 104 terms are applied, as using

results obtained for the flow lengths of 10−2 < ξ < 103 considered in this study. The solid lines repre-

Θ(ξ, η) = ΘW " ! ∞ (2k + 1) π2 √ 4 X (−1)k+1 ΘW cos + √ ( Le + η) π k=0 2k + 1 (1 + Le) ! !# (2k + 1) π2 √ (2k + 1) π2 + sin Le sin √ √ (η − 1) (1 + Le) (1 + Le) !2

a Lewis number of Le = 100 and a modified Stefan

more terms showed no remarkable deviations to the

mal wall boundary condition in the real domain forms

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sent the solution for the adiabatic wall and the dashed lines correspond to the results obtained for a constant wall temperature of ΘW = −1. Please note, that the wall is at η = 1 and the free surface at η = 0.

. (86) 13

ACCEPTED MANUSCRIPT ˜ = 0.1 5.1. Results for Le = 100 and St

1 ξ=2

5.1.1. Temperature profile 0.5

The evolution of the temperature profile across

ξ=0.2

0

Θ

the film for different flow lengths ξ = 0.02, 0.2 and 2

ξ=0.02 ξ=0.2

−0.5

For the small flow length of ξ = 0.02 there is no

ξ=2

Le=100 St=0.1

−1 0

difference between the isothermal and the adiabatic

RI PT

is shown in Figure 3.

0.2

profile from near the surface half way into the film,

0.4

η

0.6

0.8

1

Figure 3: Evolving temperature profiles across the film for a

since the interface boundary layer is not yet affected

SC

wall temperature of ΘW = −1 (dashed lines) and the adiabatic

by the wall boundary condition.

wall (solid lines)

Corresponding to the short term solution presented

since no heat can be transferred through the adiabatic

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by Nakoryakov et al. (1997), the interface tempera-

wall.

ture and mass fraction at η = 0 are Θi = γi = 0.5

The different evolution of the temperature profiles for

for the dimensionless numbers considered. These

the adiabatic and the isothermal wall increases when

short term interface values for the unaffected interface boundary layer are determined as follows:

γi,st

different asymptotic values for both profiles result.

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Θi,st

1 = √ , ˜ +1 LeSt √ ˜ LeSt . = √ ˜ +1 LeSt

increasing the flow length to ξ = 2. Further on two

(87)

Obviously the temperature for the isothermal wall

(88)

will reach the boundary condition Θ∞ = ΘW = −1 as a final value for ξ → ∞. In AppendixC the asymptotic value for the adiabatic wall boundary condition

ryakov et al. (1997) and adapted to the present defi-

is derived, which is Θ∞ ≈ 0.9 for a modified Stefan

EP

Equation (87) and (88) have been taken from Nako-

˜ = 0.1. number of St

nition of dimensionless numbers.

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At a flow length of ξ = 0.2 the interface boundary 5.1.2. Mass fraction profile

layer for the isothermal wall is obviously affected by

At the beginning of the absorption process the

the wall boundary inducing the interface temperature to decrease.

wall boundary condition for the temperature profile has not yet affected the interface boundary (cf. Fig-

On the contrary, the interface temperature for the adi-

ure 3). For that reason the mass fraction profile, which

abatic wall case remains unaffected for ξ = 0.2, as

develops from the interface only, is identical for the

only one thermal boundary layer propagates from the

isothermal and the adiabatic wall for a flow length of

interface to the wall. The heat of absorption induces

ξ = 0.02 in Figure 4.

the wall temperature at η = 1 to increase for ξ = 0.2, 14

ACCEPTED MANUSCRIPT At a flow length of ξ = 0.2, the interface mass frac-

the adiabatic wall boundary all heat of absorption remains within the film.

1.5 Le=100 St=0.1

1

5.1.3. Dimensionless absorbed mass flux The dimensionless mass fraction gradient at the

ξ=2

0.5

interface µi is defined similarly to Grossman (1983) and can be interpreted as the dimensionless absorbed

0 ξ=0.2

ξ=0.02

0.2

mass flux: 0.4

η

0.6

0.8

1

m ˙ abs · δ ∂γ µi =
. = − ∂η i ρ · D · ceq − c0

(89)

SC

−0.5 0

RI PT

γ

Figure 4: Evolving mass fraction profiles across the film for a

Figure 5 depicts the evolution of the dimensionless

wall (solid lines)

mass flux µi as a function of the flow length ξ for

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wall temperature of ΘW = −1 (dashed lines) and the adiabatic

different temperature wall boundary conditions in a

tion for the isothermal wall is slightly higher than the

double logarithmic diagram. As discussed in the pre-

one for the adiabatic wall case. This corresponds to

vious section, the interface mass fraction gradient is

the slight decrease in interface temperature at ξ = 0.2

not affected by the wall boundary condition for the

in Figure 3 for the isothermal wall case with ΘW =

temperature profile at small flow lengths ξ < 0.1,

TE D

−1.

Increasing the flow length to ξ = 2 the deviations be-

tween the mass fraction profile for the isothermal and

the adiabatic wall become evident. The slope of the

EP

interface mass fraction gradient for the isothermal

which can be seen in Figure 5. For ΘW = −1 depicted by the dashed line in Figure 5, the mass fraction gradient at the interface remains almost constant for the flow length between ξ = 0.1 and ξ = 2, as already described in Figure 4.

wall case remains almost constant between ξ = 0.2

The dotted line represents the interface mass frac-

and ξ = 2 due to the rising interface absorbate mass

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tion gradient for a wall temperature which equals the

fraction, which is induced by the decreasing interface

film inlet temperature ΘW = 0. The dimensionless

temperature due to the cooled wall (cf. Figure 3).

mass flux µi for ΘW = 0 is lower than for ΘW = −1,

On the contrary, the interface mass fraction gradient

since the range for lowering the interface tempera-

for the adiabatic wall drastically decreases between

ture (cf. Fig. 3) is smaller for ΘW = 0 as compared

ξ = 0.2 and ξ = 2 due to a decrease of the interface

to the lower wall temperature of ΘW = −1.

absorbate mass fraction. According to Figure 3 this

The enhanced decrease of the interface mass frac-

decrease of the absorbate mass fraction is caused by

tion gradient for the adiabatic wall compared to the

the rise in interface temperature at ξ = 2, since for

isothermal wall is depicted by the solid line in Figure 15

ACCEPTED MANUSCRIPT 2

1

10

Le=100 St=0.1

ΘW=−1

1

i

St=0.1

Le=100

0.5 Θ

10

µ

0

10

0

Le=200

Le=100

ξ=2

Le=200 ξ=0.02

adiabatic wall

−0.5

Le=100

10

ξ=2

Le=200

−1 0

−2

10 −2 10

RI PT

ΘW=0

−1

−1

10

0

10

1

ξ

2

10

10

0.2

3

10

0.4

η

0.6

0.8

Figure 6: Evolving temperature profiles across the film for a wall temperature of ΘW = −1 (dashed lines) and the adiabatic

temperature of ΘW = −1 (dashed line), ΘW = 0 (dotted line)

wall (solid lines)

SC

Figure 5: Dimensionless mass flux at the interface µi for a wall

and the adiabatic wall (solid line)

5 also.

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5.2.1. Temperature profile Figure 6 depicts the temperature profiles for two

˜ = 0.1 5.2. Results for Le = 200 and St

different flow lengths ξ = 0.02 and ξ = 2 for a Lewis

number of Le = 100 as well as Le = 200. At the

The increase of the Lewis number can be induced

beginning of the absorption process at ξ = 0.02 the

by an increase of the thermal diffusivity a or a de-

only difference between the different Lewis numbers

TE D

crease of the mass diffusivity D. However, a change

in thermal diffusivity would affect the dimensionless flow length ξ also and therefore would shift the ac-

EP

tual flow length x for a constant dimensionless flow length ξ. For that reason, in the present study only

from Θi,st = 0.5 for Le = 100 to Θi,st h 0.414 for Le = 200. The drop in interface temperature for the larger Lewis number is comprehensible, since less

the lower mass diffusivity. Consequently the inter-

the Lewis number is considered in order to evaluate

face temperature decreases for Le = 200.

physically comparable cases, e.g. for a given tube diameter.

is the change in the short term interface temperature

heat of absorption is set free at the interface due to

the case of halving the mass diffusivity D to double

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In Fig.6 at ξ = 2, the temperature profiles corresponding to Le = 200 for the adiabatic as well as for

Due to the different Lewis number the short term in-

the isothermal wall are shifted to lower temperatures

terface values change as follows (cf. (87) and (88)):

as compared to a Lewis number of Le = 100. Again, Θi,st h 0.414,

(90)

this is due to a reduced mass absorption involving

γi,st h 0.586.

(91)

less heat of absorption set free at the interface.

16

1

ACCEPTED MANUSCRIPT 5.2.2. Mass fraction profile

5.2.3. Interface mass fraction gradient The larger mass fraction gradients due to the el-

different Lewis numbers Le = 100 and Le = 200 at

evated Lewis number can be seen in Figure 8 for

two flow length ξ = 0.02 and ξ = 2 for the adia-

the adiabatic as well as the isothermal wall. For the

batic as well as the isothermal wall. At a flow length

Lewis number of Le = 200 the dimensionless mass

RI PT

Figure 7 depicts the mass fraction profiles for two

flux µi is almost doubled as compared to Le = 100.

1.5

According to (89) this involves that halving the mass

St=0.1

1 2

10

ξ=2

0.5

Le=200

Le=200 Le=100

1

10

0

SC

γ

Le=100

0.2

µi

−0.5 0

ξ=0.02

0.4

η

0.6

0.8

0

10

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Le=100 Le=200

1

St=0.1

−1

10

Figure 7: Evolving mass fraction profiles across the film for a

wall temperature of ΘW = −1 (dashed lines) and the adiabatic

W

−2

10 −2 10

wall (solid lines)

Θ =−1 adiabatic wall

of ξ = 0.02 the extent of the mass fraction inter-

−1

10

0

10

1

ξ

10

2

10

TE D

face boundary layer is smaller for Le = 200 than for

temperature of ΘW = −1 (dashed line) and the adiabatic wall

Le = 100, as the mass diffusivity is lower.

(solid line)

The difference in penetration depth increases with increasing flow length. At ξ = 2, the mass fraction

diffusivity by no means would lead to a decrease in

EP

absorbed mass flux by the same factor, since the de-

profiles corresponding Le = 200 are obviously less

crease in mass diffusivity is almost balanced by an

penetrated as compared to the profiles for Le = 100.

increase in the mass fraction gradients at the inter-

The elevated interface mass fractions for Le = 200

face.

result from the lower interface temperatures as dis-

In Fig.8 the constant course of the mass flux for Le =

cussed in Figure 6. Due to the smaller extent of the

100 and ΘW = −1 is at about µi h 10. In comparison

mass fraction boundary layer and the elevated inter-

the mass flux for Le = 200 and ΘW = −1 is at about

face absorbate mass fraction the slope of the inter-

µi h 18. In fact, halving the mass diffusivity would

face mass fraction gradients for both the isothermal

lead to a 10% decrease in mass flux for Le = 200 as

as well as the adiabatic wall boundary have increased

compared to Le = 100 at this particular flow range

for the larger Lewis number of Le = 200 as com-

0.1 < ξ < 1.

pared to Le = 100. 17

3

10

Figure 8: Dimensionless mass flux at the interface µi for a wall

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ACCEPTED MANUSCRIPT 6. Conclusion

AppendixA. Solving procedure in the Laplace domain: Adiabatic wall boundary

The present study introduces the Laplace trans-

The adiabatic wall boundary condition (42) is ap-

ergy and mass fraction for the isothermal as well as

plied to the general solution for temperature (37) in

for the adiabatic wall boundary condition. In princi-

(A.1). Thus, the coefficient C2 can be eliminated.

ple, the Laplace transform allows to apply any bound-

In AppendixB some frequently used mathematical

ary condition in contrast to the Fourier method.

identities are summarized.

The temperature and mass fraction profiles are com-

√ √ √ √z ze − C2 · ze − z √ √ ⇒ Θ(z, η) = 2 · C1 · e z cosh( z(η − 1)) 0 = C1 ·

(A.1)

SC

pared for different flow lengths and two different Lewis

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form to solve the partial differential equations for en-

numbers. The two limiting cases for the heat transfer,

The coefficient C4 in the formal solution to the mass

the isothermal and the adiabatic wall, are discussed

M AN U

fraction profile in the Laplace domain (38) is elim-

in terms of their effect on the dimensionless absorbed

inated by applying the impermeable wall boundary

mass flux.

condition (39).

As expected, the isothermal wall leads to an increase

√ √ √ √ zLe · e zLe − C4 · zLe · e − zLe √ √ ⇒ γ(z, η) = 2 · C3 · e zLe · cosh( zLe(η − 1))

of the absorbed mass flux compared to the adiabatic

0 = C3 ·

wall as soon as the wall boundary starts to affect the

TE D

interface boundary.

(A.2)

Halving the mass diffusivity and hence doubling the Lewis number from Le = 100 to Le = 200 firstly

EP

affects the initial leap of the interface values of temperature and mass fraction. Secondly the propaga-

Inserting (A.1) and (A.2) to the dimensionless phase equilibrium (40), the constant C1 can be expressed as follows: √ √ 1 = 2 · C1 · e z · cosh( z) z √ √ + 2 · C3 · e zLe cosh( zLe),

tion speed of the mass fraction profile is decreased for the lower mass diffusivity.

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The increase of the interface absorbate mass fraction

⇒ 2 · C1 · e

and the decrease of the extent of the interface mass

√

z

=

1 √ z cosh( z)

√ cosh( zLe) − 2 · C3 · e zLe · √ . cosh( z √

fraction boundary layer for Le = 200 leads to larger mass fraction gradients at the interface compared to

(A.3)

Le = 100. In fact, this almost balances the reduced Applying the derivatives of (A.2) and (A.1) to the

mass diffusivity. The absorbed mass flux for half the

dimensionless interface energy balance, (41) result

mass diffusivity is only ≈ 10% lower as compared to the reference case for Le = 100. 18

ACCEPTED MANUSCRIPT to:

sinh(ax) · sinh(bx)

√ √ z sinh( z) = √ √ √ 2 · C3 · e zLe z sinh( zLe), √ ˜ LeSt ⇒ 2 · C1 · e

√

=

(B.5)

   
1 = · cosh (a + b)x + cosh (a − b)x 2

(B.6)

cosh(ax) · cosh(bx)

√

z

   
1 · cosh (a + b)x − cosh (a − b)x 2

=

2 · C3 · e √ ˜ LeSt

zLe

√ sinh zLe √ . sinh z

cosh(ax) · sinh(bx)

(A.4)

= Note that the symmetry of the hyperbolic functions has been taken into account.

   
1 · sinh (a + b)x − sinh (a − b)x 2

(B.7)

AppendixC. Asymptotic values for the adiabatic

SC

Regarding (A.3) and (A.4), the unknown coefficients

wall

C3 and C1 can be determined as follows: √ √ ˜ · sinh( z) = LeSt

2 · C3 · e zLe "  √ √ · z · cosh( z) sinh( zLe)

absorption is absorbed by the film, the film temperature increases while at the same time the absorbent

(A.5)

#−1 √ √ √ ˜ sinh( z) cosh( zLe) , + LeSt

mass fraction at the interface decreases leading to less absorption until a final homogeneous equilibrium state is reached. The integral energy balance

TE D

√ √ 2 · C1 · e z = sinh( zLe) "  √ √ · z · cosh( z) sinh( zLe)

Since for the adiabatic wall condition all heat of

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√

RI PT

2 · C1 · e

√ z

(A.6)

EP

#−1 √ √ √ ˜ sinh( z) cosh( zLe) + LeSt .

from the beginning to this final state results as follows: M s,0 · cs · (T ∞ − T 0 ) = M s,0 · (c∞ − c0 ) · ∆habs . (C.1)

AppendixB. Mathematical identities

Note that the model inherently neglects the mass change

Some frequently used mathematical identities are presented here.

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1 · e x − e −x 2
1 cosh(x) = · e x + e −x 2 sinh(x) =

due to absorption. Introducing dimensionless variables and rearranging this energy balance leads to the following relation of the final values for the dimen-

(B.1)

sionless film temperature and mass fraction: (B.2)

sinh(x) = −i · sin (i · x)

(B.3)

cosh(x) = cos (i · x)

(B.4)

cs · (T eq − T 0 ) · Θ∞ = (ceq − c0 ) · ∆habs · γ∞ ,

(C.2)

˜ · Θ∞ = γ ∞ . St As the asymptotic values Θ∞ and γ∞ constitute uniform profiles, the phase equilibrium at the interface 19

ACCEPTED MANUSCRIPT Baehr,

H.D.,

1955.

Wärmeleitungsprobleme

(C.3)

Transformation.

Die

Lösung

mit

Hilfe

nichtstationärer der

Laplace-

Forschung auf dem Gebiete des Inge-

nieurwesens 21, 33–40.

(C.4)

Grigor’eva, N., Nakoryakov, V., 1977. Exact solution of com-

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bined heat-and mass-transfer problem during film absorption. Journal of Engineering Physics and Thermophysics 33, 1349–1353.

Grossman, G., 1983. Simultaneous heat and mass transfer in Film absorption under laminar flow. Int. J. Heat Mass Trans-

SC

fer 26, 357–371.

Meyer, T., 2014. Improvement of the exact analytical solutions for combined heat and mass transfer problems obtained with

M AN U

the fourier method. Int. J. Refrig. 43, 133–142.

Meyer, T., Mittermaier, M., Ziegler, F., 2014. Comparison of Various Methods for Evaluation of Combined Heat and Mass Transfer in Laminar Falling Films with Constant Film Velocity Part II - Analytical methods. International Sorption Heat Pump Conference 2014 .

Meyer, T., Ziegler, F., 2014. Analytical solution for combined

EP

TE D

1 Θi + γi = 1 is extended to the whole film: 2 3 ˜ · Θ∞ = 1 − Θ∞ → Θ∞ = 1 , St 4 ˜ +1 St 5 ˜ 6 ˜ → γ∞ = St . γ∞ = (1 − γ∞ ) · St 7 ˜ +1 St 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

heat and mass transfer in laminar falling film absorption using first type boundary conditions at the interface. Int. J. Heat Mass Transfer 73, 141–151. Mittermaier, M., Meyer, T., Ziegler, F., 2014a. Comparison of Various Methods for Evaluation of Combined Heat and Mass Transfer in Laminar Falling Films with Constant Film Velocity Part I - Numerical method. International Sorption

AC C

Heat Pump Conference 2014 . Mittermaier, M., Schulze, P., Ziegler, F., 2014b. A numerical model for combined heat and mass transfer in a laminar liquid falling film with simplified hydrodynamics. Int. J. Heat Mass Transfer 70, 990–1002. Nakoryakov, V., Grigor’eva, N., 2010. Nonisothermal absorption in thermotransformers. Journal of Engineering Thermophysics 19, 196–271. Nakoryakov, V., Grigor’eva, N., Potaturkina, L., 1997. Analysis of exact solutions to heat-and mass-transfer problems for absorption with films or streams. Theor. Found. Chem. Eng.

20

AC C

EP

TE D

SC

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31, 119–126. 1 2 Nusselt, W., 1923. Der Wärmeaustausch am Berieselungsküh3 4 ler. Zeitschrift des Vereines deutscher Ingenieure 67, 206– 5 210. 6 7 Ziegler, F., 2010. The multiple meanings of the Stefan-number 8 (and relatives) in refrigeration. Int. J. Refrig. 33, 1343–1349. 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 21 62 63 64 65

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