Energy and Power Engineering, 2011, 3, 144-149
doi:10.4236/epe.2011.32018 Published Online May 2011 (http://www.SciRP.org/journal/epe)
A Cell Model to Describe and Optimize Heat and Mass
Transfer in Contact Heat Exchangers
Vadim Mizonov1, Nickolay Yelin2, Piotr Yakimychev2
1Ivanovo State Power Engineering University, Ivanovo, Russia
2Ivanovo State University of Architecture and Civil Engineering, Ivanovo, Russia
Received February 28, 2011; revised March 29, 2011; accepted April 2, 2011
Abstract
A cell model to describe and optimize heat and mass transfer in contact heat exchangers for utilization of
exhaust gases heat is proposed. The model is based on the theory of Markov chains and allows calculating
heat and mass transfer at local moving force of the processes in each cell. The total process is presented as
two parallel chains of cells (one for water flow and one for gas flow). The corresponding cells of the chains
can exchange heat and mass, and water and gas can travel along their chains according to their transition ma-
trices. The results of numerical experiments showed that the most part of heat transfer occurs due to moisture
condensation from gas and the most intense heat transfer goes near the inlet of gas. Experimental validation
of the model showed a good correlation between calculated and experimental data for an industrial contact
heat exchanger if appropriate empirical equations were used to calculate heat and mass transfer coefficient. It
was also shown that there exists the optimum height of heat exchanger that gave the maximum gain in heat
energy utilization.
Keywords: Direct Contact Heat Exchanger, Heat and Mass Transfer, Condensation, State Vector, Transition
Matrix, Optimization
1. Introduction
Contact heat exchangers are widely used in different
industries for different technological purposes connected
with heat and mass transfer. Therefore, different aspects
of their experimental investigation, mathematical mod-
eling and optimization can be found in literature. It can
be seen even from titles of the papers on the problem in
question recently published in various journals [1-7]. The
objective of the present study is to model and optimize
the process in a contact heat exchanger from the view-
point of using them to utilize heat of exhaust gases with
high moisture content. It can be a process of contact
heating water by combustion products of natural gas, by
exhaust air-vapor flow in processes of fabrics treatment
in textile industry, and so on. All papers on the problem
can be roughly subdivided into three groups. The first
one deals with fundamentals of the local heat and mass
transfer though a contact surface. The results obtained
here are very important but can hardly be directly used
for the engineering calculation of the process. The sec-
ond group deals with experimental investigation and
mathematical description of total heat and mass balances
in an apparatus. Within this approach, heat and mass flux
is calculated for average difference of transfer potential:
average temperature difference, and average vapor par-
tial pressure difference. The heat and mass transfer coef-
ficients are also calculated for some average parameters
of the interacting media. In this case, these coefficients
are sooner calibrating parameters of such models than
the real characteristics of heat and mass transfer. It is
obvious that the approach is very easy and convenient for
engineering calculation but it can hardly be used when
the process parameters go out of the range that was used
to obtain experimental correlations. This approach is
fully represented in the book [8] where a lot of data on
the exchangers design, experimental investigations and
their generalization on the basis of total balances is pre-
sented. At last, the third group of researches deals with
process simulation on the basis of local heat, mass and
momentum balances, i.e., with differential equations of
the process. However, these equations are not always
easy to solve-particularly if stochastic components of
media motion are taken into account when the equations

V. MIZONOV ET AL.
145
(or a part of them) become partial differential equations.
Some of examples of such approach application can be
found in papers [9-12].
According to our viewpoint, an effective tool to model
such processes is the theory of Markov chains. The basic
principles of its application in process engineering are
described in our paper [13]. The paper [14] shows how
this theory can be applied to model heat and mass trans-
fer between stochastically moving gas and particulate
flows. This approach is based on separation of operating
volume of an apparatus into a chain of perfectly mixed
cells of small but finite volume. The travel of a medium
over the cells is controlled by the matrix of transition
probabilities. If there are several chains of cell that can
exchange heat or/and mass, each chain is considered as a
heat or mass source for another one. Thus, this approach
is based on the local heat and mass balance traveling
along and between the chains. On the other hand, it does
not require advanced mathematical tools except rudi-
ments of matrix algebra.
2. Structure of the Model and Governing
Equations
The scheme of the process is shown in Figure 1. The
operating volume is filled with a packing (for instance,
with Rashig rings). The cold water is supplied at the top
of the apparatus and flows down as a thin film over the
packing surface that is rather high due to the shape of its
elements. The hot gas is fed at the bottom of the appara-
tus and flows up to the top interacting with the water
flow. Let us separate in the volume two parallel vertical
channels: one for the water flow and another one for the
gas flow. The channels can be presented as two one-di-
mensional chains of cells of the length Δx. The number
of cells in each chain is m = H/Δx, and the serial number
of the cell i is the integer argument of its spatial position.
The thermophysical state of the flows can be presented
by column vectors. For example, the vectors of heat,
temperature and mass for the water chain have the form
Qw = {Qwi}, tw = {twi}, mw = {mwi}, etc, where i = 1, 2,,
m and the size of each vector is mx1.
Let at certain moment of time τk the heat state of the
media is characterized by the set of distributions ,
, , end so on. After the small time interval Δτ,
during which heat and mass can transit only to
neighboring cells, the k-th distribution will transit into
the (k + 1)-th one. At such presentation, the process time
is also expressed by the integer argument k (the transition
number) when the real moments of time are calculated as
τk= (k – 1)Δτ.
k
w
Q
k
w
tk
w
m
During the time Δτ the following amounts of mass and
heat are transferred between the parallel cells of the
Figure 1. Design model of the process and its cell presenta-
tion.
chains:


kkk
wv
S

 mpp

 . (1)


*
kkk
gw
S

Qtt

 . (2)
where


ppt
kkk kkk
www
и vv are the vectors of
partial vapor pressure above the water surface at water
temperature and in the gas that are calculated from em-
pirical expressions, dk is the moisture content in the gas,
β is the vector of mass transfer coefficients, S = SsFΔx is
the surface of exchange in the cell (Ss is the specific sur-
face of the packing, F is the cross section area of the ap-
paratus), α is the vector of heat emission coefficients, the
operator .* mean s element by element multiplication of
vectors.

ppd
The further kinetics of the process can be described by
the following set of recurrent matrix equalities:
kk k
g
gv gg
mmm (3)


1kkk kk
g
vg gvgvf
mPmm m
(4)
1kkk k
g
ggggggf
mPm m (5)


1kkk k
wwgv
mPmm m k
wf
(6)


1kkkk k
g
gg gf
QPQQ Q (7)
Water Gas

V. MIZONOV ET AL.
146
k
wf

1kkkkkk
www
.QPQQrmQ (8) The stochastic parameter in the matrix (9) depends on
the dispersion coefficient Dw and can be calculated as
follows:
where the indices gv and gg are related to the vapor and
gas phase in the gas-vapor mixture, the index w – to the
water flow, r is the vector of latent heat of evapora-
tion/condensation in the cells. The Equations (4)-(8) de-
scribe the following heat or mass balances. During each
transition a cell gets or gives out a certain amount of heat
or mass calculated from Equations (1), (2). Then due to
motion of heat carriers this heat or mass transits from the
cell to the neighboring cell, and the inlet cell of each
chain gets a portion of fresh heat carrier with its input
parameters.
2
ww
sD
x


 (12)
The transition matrix for the gas-vapor flow can be
constructed on the basis of the same rule.
The vectors with the index f in Equations (4) - (8) are
the feed vectors. All their elements are zeros except for
the cells, the media are fed to which






00 0
00
0
1,
1,
1
k
gvf g
k
ggf g
k
wf w
mmG dd
mmGd
mG






(13)
The matrices Pw and Pg are the matrices of transition
probabilities, or the transition matrices. They describe
the longitudinal transitions of the media along the chains.
Such matrix is the basic operator of a Markov chain
model. It consists of transition probabilities and can be
constructed using the following rule. The i-th column of
the matrix consists of probabilities related to the i-th cell.
The probability to transit into the j-th cell is placed in the
j-th row of this column.
where d0 is the initial moisture content in the gas, Gw0 are
Gg0 are the flow rates of water and gas-vapor mixture.
In order to turn back from the heat distributions over
the cells of the chains to the temperature distributions
and calculate the moisture content distribution the fol-
lowing relationships are to be used:
In many practical cases it is convenient to subdivide
the transitions into completely random (symmetrical)
transitions and deterministic transition. For instance, if
the forward transition probability from a cell is equal to
0.5 and the backward transition probability is equal to
0.2, the value of the forward transition probability can be
presented as 0.2 + 0.3 where 0.2 is the symmetrical part
of the transition probabilities and 0.3 is the non-sym-
metrical (deterministic) part. The matrix for the water
flow has the following form (9):
k
w
P


11 1kk k
ww wwffc
V

 
tQ mc.. f (14)


11 11kk kk
wg ggggv
 
tQ mcmc.. .
v

(15)
11kkk
mv gg
1


dmm. (16)
where Vf, ρf, and cf is the volume of packing in cell, its
density and specific heat respectively, and the operator
.means element by element division of vectors.
In order to start the recurrent computational procedure
given by Equations (1)-(15), it is necessary to know the
initial distributions of all the parameters over the cells.
They can be taken homogeneous and equal to the pa-
rameters of inlet flows. The model given by Equations
(1)-(15) fully describes transient process and steady-state
distributions of the parameters in a contact heat ex-
changer. The heat loss into outside medium can be easily
taken into account by adding to the right hand part of
Equation (7) the item with a heat transfer coefficient
though the apparatus wall and corresponding temperature
difference. The model is very easy for programming,
particularly in programming environment oriented to
manipulation with matrices like MATLAB.
where vw is the part of the mass of water that transits
from a cell to the lower cell during Δτ due to the deter-
ministic (average) component of water motion, sw is the
same for stochastic component of water motion that oc-
curs to the upper and lower cell. The value of can
be found from the continuity equation as follows:
k
wi
v
kk
ww
vG m  .k
w
(10)
where is the local water flow rate trough the i-th
cell that varies from cell to cell due to mass transfer
k
wi
G
0
1
k
i
ki
wi w
m
G=G



 (11)
1
12
23
34
10
12 0
=01
01
k
ww w
kk
www ww
kkk
wwwww w
kk
www w
vs s
vsvss
vsvss
0vsvs
0
2

























P

(9)

V. MIZONOV ET AL.
147
3. Some Results and Discussion
This section describes some results of numerical experi-
ments carried out with the described above model. Cal-
culations were done for the contact heat exchanger of the
height 1m and cross-section area 1.53 m2 with packing of
Rashig rings of the size 25 × 25 × 3 mm. The following
parameters of heat carries were used: Gw0 = 10 m3/h,
tw0 = 18˚C, Gg0 = 1 kg/s, tg0 = 100˚C.
Figure 2 shows the steady state temperature distribu-
tions of heat carriers and evaporation/condensation mass
flow rate distribution for several moisture contents in the
inlet gas. If the gas is dry (d0 = 0), intense evaporation of
water occurs near its inlet, the gas is saturated with water,
and rather soon the process comes to equilibrium when
evaporation practically stops. The increase of water
temperature is very small that is the evidence that the
convection heat flux from gas to water brings small con-
tribution into heating. At d0 = 0.1 there is no water
evaporation at all, and condensation of water vapor from
gas occurs. The equilibrium is reached near the middle of
the apparatus, and water is heated up to 40˚C that shows
that the main contribution in heating is brought by the
latent heat of condensation. These processes are more
marked at d0 = 0.2 when water is heated up to 55˚C.
Thus, under other conditions being equal, the heat capac-
ity of the process grows with increase of initial moisture
content in gas. Figure 3 shows the distribution of the
heat transferred during one transition in steady-state re-
gime by convection and by vapor condensation from gas
at d0 = 0.2. It can be seen that that the heat with vapor
condensation is much bigger than the heat with convec-
tion.
For experimental validation of the model the extensive
experimental data on testing industrial contact heat ex-
changes for furnace gases presented in [8] were used.
The following pure empirical correlations for calculation
the heat and mass transfer coefficients gave the best fit to
experimental data:
1.3 0.33
0.016RePrfor Re200
gg
Nu   (17)
0.670.330.17
0.035Re PrforRe 200
gwg
Nu g (18)
where Reg = vgdr/νg, gw = Gw0/Gg0, dr is the equivalent
diameter of the packing element that is equal to its vol-
ume multiplied by 6 and divided by its surface. The effi-
cient gw appears in Equation (18) because beginning with
the certain value of Reg the coefficients of heat and mass
transfer increases due to intense formation of waves on
the water film surface. On the basis of the triple analogy
the same equations were used for mass transfer but with
the diffusion Nusselt and Prandtl number (NuD = βdr/Dg,
PrD = ν/Dvg where Dvg is the diffusion coefficient of wa-
ter vapor in gas).
Figure 2. Distribution of some process parameters at vari-
ous initial moisture content in gas.
Figure 3. Distribution of parts of heat that is transferred
from gas to water by convec tion (dark bars) a nd w ith vapor
condensation (light bars).

148 V. MIZONOV ET AL.
Figure 4 shows measured and calculated dependence
of the gas outlet temperature (tgout = tg1) on the water
outlet temperature (twout = twm) that was varied by chang-
ing the rate of water flow. The other process parameters
were: H = 1m, tw0=12˚C, tg0 = 230˚C, Rashig rings 35 ×
35 × 4 mm, d0 = 0.11. It can be seen that the calculated
curve is in good correlation with the experimental data.
Let us turn back to Figure 2, which shows that very
often only a part of the apparatus that is close to the inlet
of gas is in active heat exchange work. The part that is
close to the inlet of water practically does not work but
causes the gas pressure drop and worthless fan power for
gas transportation. Under other conditions being equal,
the fan power depends on specific resistance of packing,
the apparatus height H, and just slightly depends on ini-
tial moisture content in gas. Even at identical specific
surface of packing, its resistance can vary strongly de-
pending on its shape. As concerns the influence of the
apparatus height, the necessary fan power can be taken
directly proportional to the height. In order to estimate
the total efficiency of such apparatus use in a technology,
the following decision function can be used as the crite-
rion: P(H) = NQ(H) – NF(H), where NQ(H) is the heat
power of the heat exchanger, NF(H) is the fan power for
gas transportation. At the gas flow rate given, the fan
power can be written as NF(H) = BH where B is the pro-
portional coefficient that depends on specific gas resis-
tance of the packing. The heat power of an exchanger as
function of H can be easily calculated using the model
described above.
Figure 5 shows the function P(H) calculated for the
same conditions as in Figure 2 for d0 = 0.1. At B = 0 we
get the function NQ(H), which shows how the heat power
of the apparatus varies with its height. It is easy to see
that the growth of heat power practically stops beginning
with H = 0.8 m, and the heat power reaches its nominal
value. On the other hand the necessary fan power line-
arly grows with H, and the difference between NQ and NF
has the optimum at certain value of H that depends on B.
Thus, we get a tool to optimize a contact heat exchange
at certain conditions of its technological use.
4. Conclusions
The proposed cell model of heat and mass transfer in
contact heat exchangers based on the theory of Markov
chains allows more precise calculating of distributed
over the apparatus height heat and mass fluxes between
humid hot gas and water to be heated. It is shown that the
most part of heat utilization goes due to moisture con-
densation and latent heat of condensation caused by this.
This process is mostly concentrated near the gas inlet,
and the zones of the apparatus that are distant from the
Figure 4. Comparison of calculated (line) and measured
(circles) temperatures of heat carriers at outlet.
Figure 5. To optimization of the apparatus height.
inlet give very small contribution into the whole process.
If we estimate the total efficiency of such heat exchange
use in a technology, an appropriate objective function
can be the difference between the apparatus heat power
and the fan power that is necessary for gas transportation.

V. MIZONOV ET AL.
149
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It is shown that this objective function has maximum at a
certain height of apparatus that can be calculated using
the model. The appropriately chosen empirical equations
for calculating the heat and mass transfer coefficients
allow obtaining good coincidence of calculated and ex-
perimental data for industrial heat exchanger. The model
is open for taking into account more detailed factors of
the process without changing the general algorithm of
modeling and calculation.
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