A MODEL FOR SOUND PROPAGATION IN THE PRESENCE OF
MICROPOROUS SOLIDS
M. NORI and S. BRANDINI
Department of Chemical Engineering, University College London
Torrington Place, London WC1E 7JE, UK
1. Introduction
The diffusivity of sorbates in microporous solids is a fundamental physical
property that needs to be measured for the development and the design of
adsorption and catalytic processes. Microscopic techniques (NMR, QENS, etc.)
or macroscopic techniques (gravimetric techniques, Zero Length Column,
Frequency Response (FR), etc.) are used for the determination of these data.
The microscopic techniques allow the measurement of fast kinetics, while the
macroscopic methods can be used for kinetics with longer characteristic times.
When the measurement has been possible with both types of techniques, often
there have been discrepancies [1]. There is the need to develop a macroscopic
technique that allows kinetic measurements of fast diffusing strongly adsorbed
systems.
Among the macroscopic techniques, the FR has the fundamental feature
of being able to discriminate between different rate-limiting mechanisms. The
current upper limit of the FR technique is approximately 10-30 Hz (Bourdin et
al. [2] ) as a result of mechanical limitations of volume modulators. This upper
limit of the FR coincides with the lower limit of the acoustic range, 20-20,000
Hz (Everest [3]). Therefore to extend measurements to faster systems a
theoretical investigation of sound propagation in presence of a microporous
solid is presented.
2. Model
The key issues that the mathematical model should address are:
a) The extent of the influence of an adsorbent material on sound
propagation for the development of a new technique based on acoustic
measurements,
b) The validation of the basic assumptions of the FR technique in the
region above 10 Hz.
1
The model is based on the theory of sound propagation in tubes (Rayleigh [4],
Tiejdeman [5] ) and the linearized adsorption-diffusion problem (Sun et al.[6]).
To simplify the problem, the geometry considered is that of two parallel semiinfinite slabs. The Navier-Stokes equations in the vertical and axial directions
are used to describe the motion. The fluid continuity equation and the ideal gas
equation of state combined with the energy balance equation complete the set of
equations needed to describe the gas phase [7].
Fickian diffusion in the solid slab combined with the differential mass balance
and the energy balance in the solid phase are used to describe the kinetics of the
adsorbent phase. These are linearised [6] since the pressure waves associated
with sound propagation are of very small amplitude.
For the solution, one has to solve the wave problem in the fluid phase and the
diffusion problem in the solid phase, and make them meet the boundary
conditions at the interface gas-microporous solid. The full set of differential
equations is summarised as follows.
GAS-PHASE
Momentum balances horizontal and vertical directions:
2
2

 u
p
u
u 
  u  u 1   u v 

  v
u
  2  2 

    0
z
x 
x
3

x

x

z

x

z

 t






(1
2
2

 v
v
v 
p
  v  v 1   u  v 

  v
u  
  2  2 


  0

t

z

x

z
3

z

x

z

x

z

 





(2)
Continuity equation
 u  v 



u
v
     0
t
x
z
 x z 
(3)
Equation of state
p  R o T
(4)
Energy balance
 T
  2 T  2 T  p
T
T 
p
p
C p 
u
 v    2  2  
u
v
 
x
z 
x
z
z  t
 t
 x
(5)
  u  2   v  2    v  u  2 2   u  v  2
  

 v  2       


3  x z 
 x   z    x z 


(5’)
2
MICROPOROUS SOLIDS
Continuity equation
 2 q 2 q 
q
 D 2  2 
t
x 
 z
(6)
Energy balance
 hm Cp hm
  2 Thm Thm
Thm
  hm 

 z 2
t
x 2





(homogeneous
medium
assumption)
(7)
2.1. ACOUSTIC APPROXIMATION
For a planar wave propagating confined between two layers of a microporous
solid, with gas in the channel able to exchange gas molecules with the solid
layers through an adsorption-desorption process, the following simplifying
assumptions are introduced:
(a) homogeneous medium, which means that the wave length and the
distance between the slab and the microporous layer must be large in
comparison with the mean free path; for air of normal atmospheric
temperature and pressure, this condition breaks down for f>108 Hz and
2h<10-5 cm;
(b) no bulk flow, i.e. the average fluid velocity is zero;
(c) small amplitude, sinusoidal perturbations (no circulation and no
turbulence);
(d) slab long enough, so that end effects are negligible.
Upon assuming
u  a o ux, ze it
v  a o vx, z e it


   1  x, z e 
p  p s 1  px, z e it 


T  T 1  Tx, ze 
 s a o2
1  px, z e it

it
it
s
s
q  q s 1  qx, z, t 
Thm  Thms 1  Thm x, z, t 
with all the variable functions (u, v, p, ρ, q, Thm and T) being small sinusoidal
perturbations and the isentropic sound velocity given by
a 0  RTs  
Ps
s
Introducing the dimensionless co-ordinates

x
,
ao

z
h
  t ,
3
equations 1-7 can be rewritten in dimensionless form and the solution is
governed by the following dimensionless groups:
s 
, the shear wave number,

sh
C p

k

, the square root of the Prandtl number,
h
, the reduced frequency, being proportional to the ratio of half distance
ao
slab-microporous layer to wave length,

Cp
Cv
, the ratio of specific heats.
h 2
, diffusive reduced frequency
D
h 2  hm Cp hm
k 
, conductive reduced frequency

kd 
For a given gas σ and γ often can be considered as constants, therefore the four
main parameters are the shear wave number, the reduced frequency, the
diffusive reduced frequency and the conductive reduced frequency.
2.2. LOW REDUCED FREQUENCY APROXIMATION (LRFA) [7]
When the distance between the slabs is small in comparison with the wave
length and the vertical velocity component, v, is small with respect to the
horizontal velocity, u
h
 1
ao
and
v
 1
u
the set of equations can be simplified retaining only the lowest-order terms in k.
1 p 1  2 u

  s 2  2
1 p
0
 
iu  
 u v 
ik    k
 
   
p  T
iT 
1  T
 1
i
p
2

 s 
(8)
(9)
(10)
(11)
2
2 2
q
1  2q

 k d  2
(12)
(13)
4
Thm
1  2 Thm


k   2
(14)
We expect the first condition to be met since the vertical velocity must be zero
at the centre and the displacement per cycle is at most on the order of the half
distance between the slabs.
To obtain the solution, equations 8-14 have to satisfy the following boundary
conditions and assumptions:
a) at the solid surfaces the vertical mass flux must be equal to the diffusive flux
and the horizontal velocity must be zero: i.e.,
  1,1
at
u  0 and v  
a 0 DK q
he it 
where K 
qs
ps
b) at the solid surfaces the following condition on the heat flux must be verified:
i.e.,
HDq s q  hm Thms Thm
T T it

 s
e
h

h

h 
  1,1
at
c) at the solid surface the sorbate concentration is in equilibrium with the fluid
phase and a linearised equilibrium isotherm is used:

q  q s 1  qx,1, t   q s  K p p  ps   K T T  Ts
  1,1 q 
at

ps
T
K p  peit  s K T  Te it
qs
qs
d) at the solid surface there is thermal equilibrium between homogeneous solid
and fluid: i.e.,
at
  1,1 Thm  Teit
e) at the base of the layer diffusive and conductive flux must be zero:

at
h  hs h  hs
,
h
h
Thm
q
 0 and
0


The solution is given by:
p  Ae   Be 
(15)
where  represents the Propagation Constant.
i

n
1


 tanh i s 
1 


i 5 / 2 s 

5/ 2
 D  K  a o  ps
T
 Kp  KT s
n


h

ik
q
q

s
 s
i

n
1
  
 tanh i1 / 2 s
1 

i1 / 2 s

(16)




5/ 2

  1 
  1  
i argdQ i    tanh i s
 1  


      dQi   e
5/ 2
   
i s


 
1
In the LRFA the “equivalent polytropic coefficient” n is a linear combination of
classical contributions and adsorption-desorption effects. The real part of Γ
5
corresponds to the attenuation coefficient (AC), the imaginary part to the phase
shift (PS).
The general solution can be simplified for the following limiting cases for
which analytical solutions are obtained (Tables 1 and 2):
1) Uniform Solid Temperature – infinite thermal conductivity
2) Isothermal Adsorption – infinite solid thermal capacity
3) Ideal Fluid – no sound dissipation in fluid phase – Maximum effect of
adsorption
TABLE 1. Propagation Constants (and n, “polytropic equivalent”)
Model
General
Г Propagation Constant
i

1

1
i
n  tanh i 5 / 2 s 
n  tanh i 1 / 2 s 
1  i 5 / 2 s 
1  i 1 / 2 s 








 D  K  ao  ps
  tanh i 5 / 2s    1 
T    1 
 K p  KT s  
  dQi   ei arg dQi   1   5 / 2
n





qs 
 
i s
  
 h  ik  qs
 
Uniform solid
temperature
i

1
n  tanh i 5 / 2 s
1 
i5/ 2s


 
i



1
n  tanh i 1 / 2 s
1 
i1 / 2 s


1
 


 D  K  ao  ps
 tanh i 5 / 2s    1 
T 
  1 
i arg dQ i  
 K p  KT s ut 
n
 1  ut


   dQi   e


h

ik
q
q

i 5 / 2s
  


s 
 s
 
Isothermal
adsorption
i

1
n  tanh i 5 / 2 s
1 
i5/ 2s


 

i


   1  tanh i s
1 
n  1 
 
i 5 / 2s

Ideal fluid Isothermal
Classical
absorption
5/ 2

1
n  tanh i 1 / 2 s
1 
i1 / 2 s

 



1
 



1 DKa 0
 Qs e i arg Qs  
ik  h

1

DKa o2 Qi   e i arg Q i  

  i 1  i


  h2


i

n
1


tanh i 5 / 2 s
1 

i 5 / 2s

 


i

n
1


tanh i 1 / 2 s
1 

i1 / 2 s

 



  1 
tanh i 5 / 2 s
1 
n  1 
 
i 5 / 2 s

 
1


In figures 1 and 2 the AC and the PS are reported using the isothermal
adsorption model, the classical sound absorption model and the ideal fluid
solutions for a system similar to Benzene-NaX (T = 353K, P = 10 Torr, channel
void fraction= 0.999, hs1m ).
6
TABLE 2. Non-Isothermal Factor
Ξ Non-Isothermal Factor
Model
General
 H  D  q s  p s

T  1 
 T  1
 K p  KT s
 dQi  e i arg dQ(i )   hm hms
di  e i arg di  

q
h
q s  
h

 s


 H  D  q 

  T
T
T
s

 K T s dQi  e i arg dQi    hm hms di  e i arg d i   s i1 / 2s tanh( i 5 / 2s) 


h
qs
h
h




Uniform
temperature (ut)
Solid
Isothermal Adsorption
Ideal Fluid-Isothermal
Classical Absorption


H  D  q s  p s
T  1

 dQi  e i arg dQ(i ) 
K p  KT s


h
q s  
 qs


 H  D  q 


T

T
s
 K T s dQi  e i arg dQi    s i 1 / 2s tanh( i 5 / 2s ) 

h
qs
h





 1

 1

 1

Figure 1. Attenuation constant as a function of frequency.
Figure 2. Phase shift as a function of frequency.
7
3. Conclusions
There is a clear analogy between the attenuation and phase shift of the pressure
wave due to adsorption and the characteristic functions of the FR technique [7].
This investigation has shown that it is theoretically possible to distinguish
adsorption kinetics using acoustic techniques and that it is possible to obtain
closed analytical approximations that can be used to interpret the experimental
results. The development of an experimental apparatus to verify these results is
the aim of future research. In this case, particular attention needs to be given to
the definition of the appropriate finite medium boundary conditions.
Acknowledgements.
Financial support from the Leverhulme Trust (Philip Leverhulme Prize) and
EPSRC is gratefully acknowledged.
8
4. References.
1. Kärger J. and Ruthven D.M. (1992) Diffusion in Zeolites, Wiley, New York.
2. Bourdin V., Gray P.G., Grenier Ph.and Terrier M.F. (1998) Rev. Sci.
Instrum, 69, 5, 2130-21363. Alton Everest F. (1994) The Master
Handbook of Acoustics, Chapter III, , TAB Books, New York
4. Rayleigh J.W.S. (1945) The Theory of Sound, Volume Two,Chapter XIX, ,
Dover Publications, New York
5. Tijdeman H. (1975), J. Sound Vib 39, 1-336. Sun L.M., Meunier F. and
Karger J. (1993) Chem. Engng Scie..48, No.4, 715-722
7. Raspet R., Hickey C.J. and Sabatier J.M. (1999) J. Acoust. Soc. Am. 105, 6573
8. Yasuda Y. (1976) J. Phys. Chem. 80, 17, 1867-1869
9