A Cell Model to Describe and Optimize Heat and Mass Transfer in Contact Heat Exchangers

doi:10.4236/epe.2011.32018

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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

E-mail: [email protected]

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.

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



 mpp





. (1)





kkk



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

gv gg

mmm (3)





1kkk kk

vg gvgvf

mPmm m

(4)

1kkk k

ggggggf

mPm m (5)





1kkk k

wwgv

mPmm m

(6)





1kkkk k

gg gf

QPQQ Q (7)

Water Gas

V. MIZONOV ET AL.

146



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.





 (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

gvf g

ggf g

wf w

mmG dd

mmGd







(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):





11 1kk k

ww wwffc



 

tQ mc.. f

(14)





11 11kk kk

wg ggggv

 

tQ mcmc.. .



(15)

11kkk

mv gg



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:

vG m



(10)

where is the local water flow rate trough the i-th

cell that varies from cell to cell due to mass transfer

wi w

G=G







 (11)

12 0

=01

ww w

www ww

kkk

wwwww w

www w

vs s

vsvss

0vsvs















































(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

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

[7] M. C. De Andrés, E. Hoo, and F. Zangrando, “Perform-

ance of Direct-Contact Heat and Mass Exchangers with

Steam-Gas Mixtures at Subatmospheric Pressures,” In-

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No. 5, 1996, pp. 965-973.

doi:10.1016/0017-9310(95)00173-5

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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doi:10.1016/0017-9310(95)00357-6

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