estudio analitico de la media mica de diferencia de temperatura
TRANSCRIPT
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
1/12
REVISTA MEXICANA DE INGENIERIA QUIMICA Vol. 4 (2005) 201-212
Publicado por la Academia Mexicana de Investigacin y Docencia en Ingeniera Qumica, A. C. 201
AN ANALYTICAL STUDY OF THE LOGARITHMIC MEAN TEMPERATURE
DIFFERENCE
ESTUDIO ANALITICO DE LA MEDIA LOGARITMICA DE DIFERENCIA DETEMPERATURAS
A. Zavala-Ro*, R. Femat and R. Santiesteban-Cos
Instituto Potosino de Investigacin Cientfica y Tecnolgica, Apdo. Postal 2-66, C. P. 78216, San Luis Potos, S. L. P.,
Mxico
Received May 19 2005; Accepted August 29 2005
Abstract
The logarithmic mean temperature difference (LMTD) has caused inconveniences in several applications like equation-
oriented flow sheeting programs. Such inconveniences have arisen from its indeterminate form. This is a consequence ofthe incomplete model derivation generally developed in the textbooks. Heat exchanger dynamic analysis and control
synthesis through lumped-parameter models using the LMTD as driving force (fluid mean temperature difference) may
suffer from such inconsistencies too. This paper is devoted to give a solution to such inconveniences by providing a
formal mathematical treatment of the LMTD. First, a complete derivation is restated resulting in a complete well-defined
expression. Then, several interesting analytical properties of the resulting expression, like continuous differentiability onits domain, are proved. The usefulness of the results is highlighted throughout the text.
Keywords: logarithmic mean temperature difference, heat exchangers.
Resumen
La media logartmica de diferencia de temperaturas (LMTD, por sus siglas en ingls) ha causado inconveniencias en
diversas aplicaciones tales como ciertos programas de simulacin de procesos. Su forma indeterminada es una de lasprincipales causas de tales inconveniencias. Tal indeterminacin es una consecuencia del procedimiento incompleto que
generalmente se desarrolla para su obtencin en los libros de texto. El anlisis dinmico y el diseo de control de
intercambiadores de calor a travs de modelos de parmetros agrupados que usan la LMTD comofuerza conductora (de
intercambio de calor, i.e. diferencia promedio de temperatura entre los fluidos) pueden tambin ser vctimas de talesinconsistencias. Este trabajo est dedicado a dar una solucin a tales inconveniencias a travs de un anlisis matemtico
formal de la LMTD. Primero, un procedimiento completo para su obtencin es desarrollado, dando como resultado una
expresin completa bien definida. Posteriormente, diversas propiedades analticas de la expresin resultante, tales comola continuidad y la diferenciabilidad en todo su dominio, son probadas. La utilidad de los resultados es comentada a lo
largo del texto.
Palabras clave: media logartmica de diferencia de temperatura, intercambiadores de calor.
1. Introduction
The logarithmic mean temperature
difference (LMTD)
2 1
2
1
ln
T TT
T
T
=
(1)
plays an important role in theoretical and
practical aspects of heat exchangers. It is
involved in their design (Reimann, 1986;Incropera and DeWitt, 1990); performance
calculation (Incropera and DeWitt, 1990;Holman, 1997); steady-state analysis (Kern,
1950; Mathisen, 1994); dynamic modelling
(Reimann, 1986; Steiner, 1989; Steiner, 1987),simulation (Papastratos, et al.; Zeghal et al.,
1991), and characterization (Zavala-Ro et al.,
2003, Zavala-Ro and Santiesteban-Cos,2004); (closed-loop) stability-limit analysis
(Khambanonda et al., 1991; Khambanonda et
AMIDIQ
Corresponding author:E-mail: [email protected]
Phone: (44) 48342000, Fax: (44) 48342010
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
2/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
202
al., 1990); and control synthesis (Alsop and
Edgar, 1989; Malleswararao and Chidambaram,
1992). For the sake of simplicity, the use of less
involved expressions, like the arithmetic model
1 2
2a
T TT
+ = (2)
or the geometric one
1 2gT T T = (3)
is sometimes preferred to approximate the mean
temperature difference along the exchanger.
However, among these expressions, it is just the
LMTD that takes into account the exponentialdistribution of the fluid temperatures over the
tube length in stationary conditions (Incroperaand DeWitt, 1990). Consequently, more
appropriate steady-state values are computed
using (1) than (2) or (3). Nevertheless, theLMTD may cause inconvenience if it is taken
simply as shown in (1). Paterson (1984), for
instance, points out that the iterative equation-solving procedures performed by equation-
oriented flow sheeting programs, commonly
take starting values that involve the equality ofstream temperatures, and hence zero
temperature differences where (1) reduces to an
indeterminate form; moreover, taking
1 2T T = , the derivatives ofT with respect to
T1 and T2, needed in the Newton iterative
solution of the equations, are undefined. On theother hand, heat exchanger dynamic analysis
and control design may suffer from such
inconveniences too. In (Alsop and Edgar, 1989;
Malleswararao and Chidambaram, 1992), forinstance, the authors take the following 2nd-
order lumped-parameter model
2( - )
2( - )
coc ci co
c pc
hoh hi ho
h ph
dT UAF T T T
dt M C
dT UAF T T T
dt M C
= +
= +
(4)
and define T as in (1) to design control
schemes for countercurrent heat exchangers.
Using Fc as control input, they propose an
algorithm to stabilize Tho at a desired value.Nevertheless, under such a modelling, existence
of solutions going through (or starting at) state
values such that 1 2T T = is undefined due to
the undeterminate form of (1). Moreover, the
proposed control schemes make use of the
derivatives ofT with respect to T1 and T2
which, as above mentioned, are undefined when
1 2T T = (and it is not clear in such works how
to handle such an inconsistency). Thus, well
posedness of the LMTD turns out to beimportant.
In order to deal with the above
mentioned inconveniences, some authors state
that T Ta as 2 1- 0T T (see e.g.
(Kern, 1959; Gardner and Taborek, 1977)), or
simply suggest to replace T
by Ta when
1 2T T = (see e.g. (Faires, 1957)), and others
claim further that1
2i
T
T
, i = 1, 2, as
2 1- 0T T (see e.g. (Paterson, 1984)).
However, a mathematical proof of such
assertions still seems to be lacking. Some otherssuggest to replace (1) by an infinite power series
expansion that yieldsTa when 1 2T T = (see
e.g. (Steiner, 1989)). Unfortunately, performing
calculations through infinite power series imply
the consideration of an infinite number of
arithmetical operations, while through atruncated version of the series, accuracy is lost.
Other authors have proposed well-defined
replacement expressions that approximate T
over an acceptable range of T1 and T2(Paterson, 1984; Underwood, 1970; Chen,1987). Paterson (1984), for example, proposes
2
3
a g
P
T TT T
+ =
(5)
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
3/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
203
while
1/3 1/ 31/3 1/3
1 22
U
T TT T
+ = (6)
is proposed by Underwood (1970), and
1/3 2/ 3
1C a gT T T y T = (7)
and0.3275 0.3275
0.3275 0.37251 22
2C
T TT T
+ =
(8)
(the latter being a refined modification of
Underwoods approximation) by Chen (1987)(a comparison study of these four
approximations is presented in (Chen, 1987)).
This work is devoted to provide a formal
mathematical treatment of the LMTD, theresults of which are intended to give a solution
of the above mentioned inconveniences. First,
we show that the indeterminate form of (1)is a consequence of the incomplete derivation of
the LMTD generally presented in the textbooks
(see for instance (Incropera and DeWitt, 1990;
Holman, 1997; Kern, 1950; Faires, 1957;McAdams, 1954; Walas, 1991)). By restating a
complete derivation, an expression is gotten
being equivalent to (1) when1 2
T T and
well-defined when 1 2T T = . Consequently,
approximating the LMTD through replacementexpressions may henceforth be avoided. Second,
analytical properties of the resulting complete
expression, such as continuity anddifferentiability, are proved for every physically
reasonable combination of values of T1 and
T2, including those where 1 2T T = . Heat
exchanger lumped-parameter dynamic models
(like (4)) using the LMTD, and control schemes
using its derivative with respect to T1 and T2,
will now be mathematically coherent throughthe use of such a complete expression.
The work is organized as follows:
Section 2 states the notation adopted in the present study. In Section 3, the complete
derivation of the LMTD is developed.Analytical properties of the resulting model are
stated in Section 4. Finally, conclusions are
given in Section 5.
2. Nomenclature and notation
The following nomenclature is definedfor its use throughout this work:
F mass flow rateCp specific heatM total mass inside the tube
U overall heat transfer coefficientA heat transfer surface areaT temperature
T temperature differenceQ rate of heat transfer
t timeR set of real numbers
R2 set of 2-tuples (xi)i=1,2 withxiRR+ set of positive real numbers
2+R set of 2-tuples (xi)i=1,2 withxiR
Subscripts:
c cold
h hoti inlet
o outlet
Fig.1. Counterflow (left) and parallel flow (right) heat exchangers.
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
4/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
204
In particular, Qh, Qc, and Q represent
the rates of heat convected through the hot and
cold fluid tubes and that transferred from the
hot to the cold fluid respectively. Thevariables Th and Tc generically denote the
temperature of the hot and cold fluids,
respectively, at any point along the exchanger,and T = Th - Tc. Furthermore, let T1 and
T2 represent the temperature difference ateach terminal side of the heat exchanger, i.e.
(see Fig. 1)
=
cihi
cohi
TT
TTT1
and
=
coho
ciho
TT
TTT2
taken in this work as conventional definitions.
Notice that under such conventional definitions,T1 and T2 shall be considered positive. To
differentiate from T
in (1), we denote TL the
well-posed logarithmic mean temperature
difference to be derived in the following section.Ta still stands (as in the previous section) forthe arithmetic model. Consider the sets B, C,
and D with B C , and a function :f C D .
We denote |Bf the restriction off to B, i.e.
| : : | ( ) ( ),B Bf B D x f x f x x B = .
The boundary of a subset, say B, is represented
as B .
3. Complete derivation of the LMTD
We recall that the present derivation
assumes that Cpc, Cph, and U, are considered
flow and temperature invariant, that is, theirvalue is considered to be constant throughout
the exchanger; see for instance (Incropera and
DeWitt, 1990; Holman, 1997; Kern, 1950;Faires, 1957; McAdams, 1954; Wallas, 1991);
for a longer (plus exhaustive) list of the
assumptions involved, see for example (Kern,
1950; Incropera and DeWitt, 1990).
Furthermore, the following developments take
into account both flow configurations of heat
exchangers simultaneously through an auxiliaryparameter . Any expression where is not
present is valid for both flow configurations. In
those where appears, such parameterdetermines the configuration they are valid for
in the following way
1
1
=
At any point along the exchanger, the heattransfer equations are
h h ph hdQ F C dT =
c c pc cdQ F C dT =
h cdQ dQ dQ= =
dQ U TdA=
From the constant physical property assumption,
and considering a mean temperature difference,
TL, throughout the exchanger, we have, afterintegration over the tube length,
( )h h ph ho hiQ F C T T =
( )c c pc ci coQ F C T T =
h cQ Q Q= =
LQ UA T =
The next developments follow
h cd T dT dT =
h c
h ph c pc
dQ dQ
F C F C =
1 1
c pc h ph
dQF C F C
=
and, from (15), we get
if counterflow
if parallel flow
(9)
if counterflow
if parallel flow(10)
if counterflow
if parallel flow(11)
(16)
(17)
(18)
(19)
(from (12) and (13))
(from (14))
(12)
(13)
(14)
(15)
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
5/12
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
6/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
206
Q UA T = . From these expressions, comparing
the latter to (19), we have
LT T = if 2 1T T T = =
The well-posed LMTD. Finally, from the resultsgotten in both, the general and special cases, i.e.
from (24) and (27), we get
2 1
2
1L
T T
TlnT
T
T
=
4. Analytical properties
In this section we consider the mean
temperature difference a bivariable function,whether we refer to the logarithmic model in
(28), TL(T1,T2), or the arithmetic one in (2),
Ta(T1,T2). Furthermore, just the cases whereheat is transferred between fluids are considered
of practical interest in the present section.
Hence, for analysis purposes, we choose to
disregard the no-heat-transfer trivial case
2 1 0T T = = . Under such perspective,2+R is
considered to constitute the domain ofTL. We
begin by stating a useful equivalent expression.
Lemma 1.Let2
2 11 2
1 2 1
1( , ) 1
2 1
i
i
T TL T T
i T T
=
+
+ +
for all(T1, T2)such thatT1+T2 0.Then
1 21 2
1 2
( , )( , )
( , )
aL
T T TT T T
L T T
221 ),( + RTT .
Proof. We divide the proof in two parts:
1) 1 2T T . From Formula 4.1.27 in3
(Abramowitz and Stegun, 1972), we have
2 1
2 2 1
01 2 1
12
2 1
i
i
T T Tln
T i T T
+
=
=
+ +
221 ),( + RTT . Then, for all
221 ),( + RTT
such that 1 2T T , we get
2 1 2 1
2 12
2 1
1
0 2 1
1ln 2
2 1
i
i
T T T T
T T TT
i T T
+
=
=
+ +
2 1
2
2 1 2 1
12 1 2 1
12 1
2 1
i
i
T T
T T T T
T T i T T
=
=
+
+ + +
1 2
2
2 1
1 2 1
2
11
2 1
i
i
T T
T T
i T T
=
+
=
+
+ +
1 2
1 2
( , )
( , )
aT T T
L T T
=
2) 1 2T T = . Notice from (29) and (2) that
( , ) 1L T T = and ( , )aT T T T = ,
0T . Then, for 1 2 0T T T = = > , we
have( , )
( , )
( , )
aL
T T TT T T T
L T T
= =
.
Remark 1. Lemma 1 is of great help in the
analysis of TL. Indeed, since ( , ) 1L T T
3 Formula 4.1.27 in (Abramowitz and Stegun, 1972)
states the following well-known (infinite) series
expansion of the logarithmic function:
=
+
+
+=
0
12
.0,0:,1
1
12
12ln
i
i
zzzz
z
iz
if1 2T T
if 2 1T T T = =
(from (2) and (29))
(27)
(28)
29
(30)
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
7/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
207
for all 1 2( , )T T such that 1 2 0T T + (see
(29)), one sees readily that 1 2
1 2
( , )
( , )
aT T T
L T T
is
well-defined at every point on 2+R . Actually, it is
easy to see that
1 2 1 2( , ) ( , ) ( , ) ( , ) 1T T T T L T T L T T = ,
0T . Consequently, one sees from (30) that
1 2 1 2 1 20 ( , ) ( , )L aT T T T T T T T
explaining why such an assertion is suggested
by several authors, like in (Kern, 1950; Gardnerand Taborek, 1977; Faires, 1957). Moreover,
from (30), we have
2
2a
iL
i
L LT
TT
T L
=
i=1,2, where the arguments have been dropped
for the sake of simplicity, and from (29), one
sees that 2 1
1
2
2 1
iL
ii i
T i SS
T i T
=
=
+ , with
2 1
2 1
T TS
T T
=
+
and 32
2 1
( 1) 2
( )
i
i
i
TS
T T T
=
+
.
Hence,
1 2
10 , (0,1)
2
L
i i
TLT T L
T T
,
i=1,2, as claimed by authors like in [16].
Furthermore, synthetically dividingTa(T1,T2) byL(T1,T2), one gets
2 4
1 2 1 2 1 21 2 3
1 2 1 2 1 2
( , ) ( ) ( )1 2( , ) ...
( , ) 6 45( )
aa
T T T T T T T T T T
L T T T T T T
=
+ +
coinciding with the power series expansion
suggested in (Steiner, 1989) as a replacement
for the LMTD. The proofs of some other
analytical features ofTL stated in the presentsection are simplified through the use of (30).
Remark 2. Expression (32) is helpful to seethe link of the replacement formulas (5)(8)
with the LMTD. Indeed, notice that for closeenough values of T1 and T2, the LMTD
may be approximated neglecting the third and
upper terms of the series, i.e.2
1 2 1 2
1 2
1 2 1 2
( , ) ( )1
( , )( , ) 6
a
a
T T T T T
T T TL T T T T
+ . For
instance, T1 and T2 resulting in a sufficiently
small value of 21 2( )T T , say
2
1 2( ) 0T T (observe that
2
1 2 1 2( ) 0T T T T ), may be
considered to apply for such an approximation.
Thus:2
1 2 1 2 1 2( ) 0 2T T T T T T +
1 1 2 2 1 22 2( )T T T T T T + + + 2
1 22
1 2 1 2
1 2
( )( ) 2( )
2( )
T TT T T T
T T
+ + +
+ 2 2
1 2 1 2
1 2
( ) ( )1
3 2 ( )
T T T T
T T
+ +
+ 2 2
1 2 1 2
1 2
( ) ( )1
3 6
T T T T
T T
+ +
+
1 2 1 21 2
2
3 2 2
T T T T T T
+ +
2
1 2
1 2
2( )1
6 3
a gT TT T
T T
+ +
+ , i.e.
Pa T
TTLTTT
),(
),(
21
21 , (see (5)). Alternatively, for
T1 and T2 resulting in
( )++= 3/222/123/113/211 5115 TTTTa ( )
21/3 1/3
1 2 0T T (observe that 1 0a
( )2
1/3 1/3
1 2 1 20T T T T ), we have
( )2
1 21 21
1 2
10
2 6
T TT Ta
T T
+
+
31/ 3 1/ 3
1 2
2
T T +
(see Appendix A), i.e.
Ua T
TTL
TTT
),(
),(
21
21 (see (6)). Furthermore, for
T1 and T2 such that
( )3/2
213/121
3/2
212
22
22
+
+
+=
TTTT
TTa
1/ 3
1 22( ) 0T T (observe that2 / 3
1/ 31 22 1 20 2 2( ) 0
2
T Ta T T
+
(31)
32
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
8/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
208
2 / 3 2
1 / 31 2 1 21 2
( )2 2
T T T T T T
+ +
2 2
1 2 1 1 2 2 1 22 4T T T T T T T T + + 2
1 2 1 2( ) 0T T T T ), we have
( )2
1 2 1/3 2/31 22
1 2
10
2 6a g
T TT Ta T T
T T
+
+
(see Appendix B), i.e. 121
21
),(
),(C
a TTTL
TTT
(see
(7)). From these developments we see that as
long as close enough values ofT1 and T2 areconsidered, relatively good approximations of
the LMTD are gotten through the replacement
expressions (5)(8) (recall that (8) is just arefinement of (6)). On the contrary, such
approximations deteriorate as T1 and T2 aretaken far from each other.
Remark 3. Let us point out that the quotient
function 1 21 2
1 2
( , )
( , )( , ) aa
T T T
L T T
TT T
L
=
is
defined on a subset wider than the domain ofTL. Actually,
{ } RTTRTTL
Ta +
0),(: 212
21. Then
aT
L is an extension of LT (actually Lemma 1
can be synthesized as2+
R
aL
L
TT ). Therefore,
aT
L
may be used to extrapolate LT to points
on R2
where the latter is not defined. This may
not make physical sense but could be helpful for
analysis purposes. For example, one sees from
(29) that
=
=
=
+=
+
10 12
1
12
12
ij
R
ij
L, and since
1 1, 1
2 1 2i
i i>
, then
1
1
2 1i i
= is divergent
according to theorems 3.28 and 3.25 in4
(Rudin,
4 In (Rudin, 1976), Theorem 3.28 states that
01/p
n n
= converges if p > 1 and diverges if 1p ,while point (b) of Theorem 3.25 states that if
0n na d for 0n N (for some 0N ), and if
0n nd
= diverges, then 0n na
= diverges.
1976). Therefore 1 21 2
( , )
( , )0a
T T T
L T T
as
221 ),( + RTT which, from Lemma 1, implies
that 1 2( , ) 0LT T T as 1 2( , )T T
approaches 2+R (from the interior of2+R ). Then,
zero can be considered the value that the LMTD
(as a bivariable function) would take at any
point on 2+R . This has been useful in the
analyses developed in (Zavala-Ro et al., 2003;Zavala-Ro, 2004) to prove that the solutions of
system (4) remain within a physically coherent
(according to the assumptions made therein)domain where the outlet temperatures cannot
become either higher than Thi or lower than Tci(see Fig. 1).
Lemma 2. The LMTD model in (28) is
continuously differentiable and positive on 2+R .
Proof. Since 22121 ),(,1),( + RTTTTL
(see (29)), one sees from (30) (and (2)) that
LT exists and is continuous on2+R (also
verifiable from (32), according to Lemma 1).
Moreover, from (31), one observes that L
i
TT
,
i = 1,2, exist and are continuous on 2+R ,
proving continuous differentiability. On theother hand, notice (from (2)) that
22121 ),(,0),( +> RTTTTTa (the average
of two positive numbers is positive).
Consequently,
22121
21
21 ),(),,(),(
),(0 +
< RTTTTT
TTL
TTTa
a .
Then, from Lemma 1, positivity of LT follows too.
Lemma 2 is specially interesting when
1 2T T = . It proves to be essential when the
LMTD is involved in dynamical analysis and
control synthesis of heat exchangers. In (Zavala-
Ro and Santiesteban-Cos, 2004), for example, itis shown that thanks to such continuous
differentiability property, existence and
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
9/12
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
10/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
210
2004), for instance, dynamic properties of
model (4), using (28), have been analyzed and
proved to be equivalent to those of the
distributed-parameter model where (4) comesfrom. The complete expression of (28) and its
analytical features have played a key role for
such results in three main directions: 1)existence and uniqueness of solutions of (4)
have been demonstrated for any physically possible initial state conditions (such is not a
property of (4) if (1) is used); 2) (4) was proved
to have a unique equilibrium point, To*
=(Tco
*,Tho
*)T, linearly related to the inlet
temperatures, Ti = (Tci,Thi)T, i.e.To
*=ATi, where
the complete expressions of the elements ofAR22 have been obtained in terms of thesystem properties, whatever value these ones
take; 3) such a unique equilibrium point was proved to be exponentially stable, and to keep
its asymptotical stability character globally on
the system state-space domain. On the otherhand, based on the results in Section 3, other
interesting dynamic properties of system (4)
have been proved in (Zavala-Ro et al., 2003).Basically, under certain standard assumptions,
model (4) has been characterized as a positive,compartmental, cooperative system. Thesefeatures provide a new framework for the
comprehension of the dynamic behavior of heat
exchangers. Furthermore, the use of the LMTDhas found an important application in dynamic
simulation too (Papastratos et al.; Zeghal et al.,
1991; Reimann, 1986). However, none of the previous works on the subject treat the
inconsistency of (1). Results avoiding such an
inconsistency are generally shown. The
complete expression in (28) may now be used
avoiding such a problem. Other numerical
problems arising from the choice of the system
parameter values may still take place if
simulations are performed through a simplemodel as (4), like numerical rigidity, for
instance. Nevertheless, such problems may be
avoided if more than one bi-compartmental cellis considered in the lumped-parameter
modelling, as suggested in (Papastratos, et al.;Zeghal et al., 1991; Reimann, 1986). Finally,
the results proposed in this work find potential
applications in control design of heatexchangers too. For instance, based on (4),
taking Fc as input and Tho as output, input-
output (partial) linearization control algorithmshave been proposed in (Alsop and Edgar, 1989;Malleswararao and Chidambaram, 1992). In a
natural way, such design methods lead the
proposed control laws to involve the partialderivatives ofT1 and T2. The authors use (1)
but do not treat its inconsistency either. Again,
numerical results avoiding such inconsistenciesare presented. The results developed here give
sense to their results if (28), and (33), are
considered in their analysis and their proposed
algorithms.Thus, an analytical study of the LMTD
turns out to be essential in view of the importantrole that the LMTD plays in several theoretical
and practical aspects of heat exchangers. The
results developed here are intended to supportall those works where the LMTD is involved
and the consideration of its analytical properties
is important, like dynamical modelling,simulation, characterization, and control of heat
exchangers.
A. Developments for Underwoods approximation
( ) ( )2
2/3 1/3 1/3 2/3 1/3 1/3
1 1 2 2 1 25 11 5 0T T T T T T + + 2/3 1/3 1/3 2/3 1/3 1/3
1 1 2 2 1 2[5( 5 ) 6 ]T T T T T T + + + 1 / 3 1 / 3 2 2 / 3 1 / 3
1 2 1 2 1 2( ) 0 [ 5 ( ) 6 (T T T T T T + 1 / 3 2 / 3 1 / 3 1 / 3 4 / 3 1 / 3
1 2 1 2 1 1 2) ] ( ) 0 5T T T T T T T + 2 / 3 2 / 3 1 / 3 4 / 3 4 / 3
1 2 1 2 2 11 2 5 0 4T T T T T T + + +2 / 3 2 / 3 4 / 3 1 / 3 2 / 3 2 / 3
1 2 2 1 2 1 24 4 8 8T T T T T T T + + +
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
11/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
211
1 / 3 4 / 3 1 / 3 1 / 3 4 / 3
1 2 1 1 2 1 2 28 9 9 9 9T T T T T T T T + + + + 2 / 3 1 / 3 1 / 3 2 / 3 2 1 / 3
1 1 2 2 1 2 14 ( ) 9 ( ) (T T T T T T T + + +
2 / 3 1 / 3 1 / 3 2 / 3 21 / 3 1 / 31 1 2 2
1 2
1 2
1 / 3
2
4 ( )
9 ( )) T T T T T T
T TT + + +
+ +
( ) ( ))(9
4
8
3
21
23/22
3/12
3/11
3/2123/1
23/1
1TT
TTTTTT
+
+++
2
1 / 3 1 / 3 2 1 / 3 1 / 3 1 2
1 2 1 2
1 2
( )3( ) ( )
8
1
6
T TT T T T
T T
+
+
1/3 1/ 3
2 /3 1/3 1/3 2 /31 2
1 1 2 2)
1( 3 6 3
2 4
T TT T T T
+
+
1 / 3 1 / 3
2 / 3 1/ 3 1/ 3 2 / 31 2
1 1 2 2
1( 2 )
2 4
T TT T T T
+
= + +
] +
=+ 2
3/12
3/113/2
2
3/1
2
3/1
1
3/2
1
TT
TTTT
( )
+
+ 3/22
3/12
3/11
3/21
23/1
23/1
1
2TTTT
TT1 / 3 1 / 3
1 2
3
1 2
2 2
T T T T + +=
( ) 1 / 3 1 / 31 2
32
1 21 2
1 2
1
2 6 2
T TT T T T
T T
+
+
+
.
B. Developments for Chens approximation
2/3 2/3
1/31 2 1 21 22 ( ) 2
2 2
T T T T T T
+ +
+
2 / 3
1/ 3 1 21 22( ) 0 2
2
T TT T
+
4/31/32/31 2 1 2
1 2
( ) 9( ) 0 4
2 4 2
T T T T T T
+
2 / 31/ 3 2 / 31 2
1 2 1 2
( )2( ) 2( )
2
T TT T T T
+
4/ 3 2/ 31/31 2 1 2
1 2
( )0 2 ( )
2 2
T T T T T T
+
+ +
4 / 3
2 / 3 1 21 2( ) 6
2
T TT T
+
+ =
( )
+
+
++
21
221
3/1
2121
6
1
2)(3
TT
TTTTTT
4 / 3 2 / 31/ 31 2 1 2
1 2
( )( )
2 2
T T T T T T
+
+ +
2 1/ 3
2/ 3 1 2 1 21 2( ) 2 2
T T T T
T T
+
+
2 1 / 3
1 2 1 21 2
2 2
T T T T
T T
+ +
=
( )( )
+
+
3/2
213/121
21
221
26
1 TTTT
TT
TT ( )
+=
+ 3/121
3/1
21
3/1
21
22TT
TTTT
( )2
1 21 2 1 2
1 2
1
2 2 6
T TT T T T
T T
+ +
+( )
1/ 31/ 31 2
1 2 .2
T TT T
+
-
8/14/2019 Estudio Analitico de La Media mica de Diferencia de Temperatura
12/12
A. Zavala-Ro et al. / Revista Mexicana de Ingeniera Qumica Vol. 4 (2005) 201-212
212
References
Abramowitz, M. and Stegun, I.A. (1972).
Handbook of mathematical functions (9th
printing). Dover Publications, New York.
Alsop, A.W. and Edgar, T.F. (1989). Nonlinear
heat exchanger control through the use of
partially linearized control variables.
Chemical Engineering Communications 75,
115.
Chen, J.J.J. (1987). Comments on improvements
on a replacement for the logarithmic mean.
Chemical Engineering Science 42, 2488
2489.
Faires, V.M. (1957). Thermodynamics (3rd ed.).
The MacMillan Company, New York.
Gardner, K. and Taborek, J. (1977). Mean
temperature difference: a reappraisal,
AIChE Journal 23, 777786.
Holman, J.P. (1997). Heat transfer(8th ed.).
McGraw-Hill, USA.
Incropera, F.P. and DeWitt, D.P. (1990).
Fundamentals of heat and mass transfer
(3rd ed.). John Wiley & Sons, New York.
Kern, D.Q. (1950). Process heat transfer.
McGraw-Hill, Tokyo.
Khambanonda, T., Palazoglu, A., and Romagnoli,
J.A. (1990). The stability analysis of
nonlinear feedback systems.AIChE Journal36, 6674.
Khambanonda, T., Palazoglu, A. and Romagnoli,
J.A. (1991). A transformation approach tononlinear process control. AIChE Journal
37, 10821092.
Malleswararao, Y.S.N. and Chidambaram, M.
(1992). Nonlinear controllers for a heat
exchanger. Journal of Process Control 2,1721.
Mathisen, K.W. (1994). Integrated design and
control of heateExchanger networks. Ph.D.
Thesis, University of Trondheim, NIT,Norway.
McAdams, W.H. (1954). Heat transmission (3rd
ed.). McGraw-Hill, Tokyo.
Papastratos, S., Isambert, A. and Depeyre, D.
(1993). Computerized optimum design and
dynamic simulation of heat exchanger
networks. Computers and Chemical
Engineering 17, S329S334.
Paterson, W.R. (1984). A replacement for the
logarithmic mean. Chemical Engineering
Science 39, 16351636.
Reimann, K.A. (1986). The consideration ofdynamics and control in the design of heat
exchanger networks. Ph.D. Thesis, Swiss
Federal Institute of Technology Zrich
(ETH), Switzerland.
Rudin, W. (1976). Principles of mathematical
analysis. (3rd ed.). McGraw-Hill, USA.
Steiner, M. (1987). Dynamic models of heat
exchangers. In: Chemical engineering
fundamentals, XVIII Congress The Use of
Computers in Chemical Engineering, pp.
809814. Giardini Naxos, Sicily, Italy.
Steiner, M. (1989). Low order dynamic models ofheat exchangers. In: International
symposium on district Heat Simulations.
Reykjavik, Iceland.
Underwood, A.J.V. (1970). Simple formula to
calculate mean temperature difference.
Chemical Engineering, June 15, 192.
Walas, S.M. (1991). Modelling with differential
equations in chemical engineering.
Butterworth-Heinemann, Stoneham.
Zavala-Ro, A. and Santiesteban-Cos, R. (2004).
Qualitatively reliable compartmental
models for double-pipe heat exchangers. In:Proc. of the 2nd IFAC symposium on
system, structure, and control, pp. 406411.
Oaxaca, Mexico.
Zavala-Ro, A., Femat, R. and Romero-Mndez,
R. (2003). Countercurrent double-pipe heat
exchangers are a special type of positive
systems. In:Positive systems proceedings
of the first multidisciplinary international
symposium on positive systems: Theory and
applications (L. Benvenuti, A. de Santis,
and L. Farina, eds.), pp. 385392. LNCIS
294, Springer, Rome, Italy.Zeghal, S., Isambert, A., Laouilleau, P., Boudehen,
A. and Depeyre, D. (1991). Dynamic
simulation: a tool for process analysis. In:
Proceedings of the Computer-oriented
process engineering, EFChE WorkingParty, pp. 165170. Barcelona, Spain.