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Sistema de Información Científica
Red de Revistas Científicas de América Latina y el Caribe, España y Portugal
REVISTA MEXICANA DE INGENIERIA QUIMICA
Vol. 4
(2005)
201212
Publicado por la Academia Mexicana de Investigación y Docencia en Ingeniería Química, A. C.
201
AN ANALYTICAL STUDY OF THE LOGARITHMIC MEAN TEMPERATURE
DIFFERENCE
ESTUDIO ANALITICO DE LA MEDIA LOGARITMICA DE DIFERENCIA DE
TEMPERATURAS
A. ZavalaRío*, R. Femat and R. SantiestebanCos
Instituto Potosino de Investigación Científica y Tecnológica, Apdo. Postal 266, C. P. 78216, San Luis Potosí, S. L. P.,
México
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 of
the incomplete model derivation generally developed in the textbooks. Heat exchanger dynamic analysis and control
synthesis through lumpedparameter 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 welldefined
expression. Then, several interesting analytical properties of the resulting expression, like continuous differentiability on
its domain, are proved. The usefulness of the results is highlighted throughout the text.
Keywords
: logarithmic mean temperature difference, heat exchangers.
Resumen
La media logarítmica de diferencia de temperaturas (LMTD, por sus siglas en inglés) ha causado inconveniencias en
diversas aplicaciones tales como ciertos programas de simulación de procesos. Su forma indeterminada es una de las
principales causas de tales inconveniencias. Tal indeterminación es una consecuencia del procedimiento incompleto que
generalmente se desarrolla para su obtención en los libros de texto. El análisis dinámico y el diseño de control de
intercambiadores de calor a través de modelos de parámetros agrupados que usan la LMTD como
fuerza conductora
(de
intercambio de calor,
i.e.
diferencia promedio de temperatura entre los fluidos) pueden también ser víctimas de tales
inconsistencias. Este trabajo está dedicado a dar una solución a tales inconveniencias a través de un análisis matemático
formal de la LMTD. Primero, un procedimiento completo para su obtención es desarrollado, dando como resultado una
expresión completa bien definida. Posteriormente, diversas propiedades analíticas de la expresión resultante, tales como
la 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 logarítmica de diferencia de temperatura, intercambiadores de calor.
1. Introduction
The logarithmic mean temperature
difference (LMTD)
21
2
1
ln
TT
T
T
T
Δ−
Δ
Δ=
Δ
Δ
A
(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); steadystate analysis (Kern,
1950; Mathisen, 1994); dynamic modelling
(Reimann, 1986; Steiner, 1989; Steiner, 1987),
simulation (Papastratos,
et al.
; Zeghal
et al.
,
1991), and characterization (ZavalaRío
et al.
,
2003, ZavalaRío and
SantiestebanCos,
2004); (closedloop) stabilitylimit analysis
(Khambanonda
et al.
, 1991; Khambanonda
et
AMIDIQ
*Corresponding author:
Email:
azavala@ipicyt.edu.mx
Phone:
(44) 48342000, Fax: (44) 48342010
A. ZavalaRío et al.
/
Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
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
12
2
a
TT
T
Δ+
Δ
Δ=
(2)
or the geometric one
12
g
TT
T
Δ=ΔΔ
(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 exponential
distribution of the fluid temperatures over the
tube length in stationary conditions (Incropera
and DeWitt, 1990). Consequently, more
appropriate steadystate values are computed
using (1) than (2) or (3). Nevertheless, the
LMTD 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 of
stream
temperatures,
and
hence
zero
temperature differences where (1) reduces to an
indeterminate
form;
moreover,
taking
12
TT
Δ=
Δ
, the derivatives of
Δ
T
A
with respect to
Δ
T
1
and
Δ
T
2
, needed in the Newton iterative
solution of the equations, are undefined. On the
other hand, heat exchanger dynamic analysis
and control design may suffer from such
inconveniences too. In (Alsop and Edgar, 1989;
Malleswararao and Chidambaram, 1992), for
instance, the authors take the following 2nd
order lumpedparameter model
2
(

)
2
(

)
co
cc
i
c
o
cp
c
ho
hh
i
h
o
hp
h
dT
UA
FT T
T
dt
M
C
dT
UA
FT T
T
dt
M
C
⎡⎤
=+
Δ
⎢⎥
⎢⎥
⎣⎦
⎡⎤
=+
Δ
⎢⎥
⎢⎥
⎣⎦
(4)
and define
Δ
T
as in (1) to design control
schemes for countercurrent heat exchangers.
Using
F
c
as control input, they propose an
algorithm to stabilize
T
ho
at a desired value.
Nevertheless, under such a modelling, existence
of solutions going through (or starting at) state
values such that
12
TT
Δ
=Δ
is undefined due to
the undeterminate form of (1). Moreover, the
proposed control schemes make use of the
derivatives of
Δ
T
A
with respect to
Δ
T
1
and
Δ
T
2
which, as above mentioned, are undefined when
12
TT
Δ
=Δ
(and it is not clear in such works how
to handle such an inconsistency). Thus, well
posedness of the LMTD turns out to be
important.
In order to deal with the above
mentioned inconveniences, some authors state
that
Δ
T
A
→
Δ
T
a
as
21

0
TT
Δ
Δ→
(see
e.g.
(Kern, 1959; Gardner and Taborek, 1977)), or
simply suggest to replace
Δ
T
A
by
Δ
T
a
when
12
TT
Δ
=Δ
(see
e.g.
(Faires, 1957)), and others
claim further that
1
2
i
T
T
∂Δ
→
∂Δ
A
,
i
= 1, 2, as
21

0
TT
Δ
Δ→
(see
e.g.
(Paterson, 1984)).
However, a mathematical proof of such
assertions still seems to be lacking. Some others
suggest to replace (1) by an infinite power series
expansion that yields
Δ
T
a
when
12
TT
Δ=
Δ
(see
e.g.
(Steiner, 1989)). Unfortunately, performing
calculations through infinite power series imply
the consideration of an infinite number of
arithmetical operations, while through a
truncated version of the series, accuracy is lost.
Other authors have proposed welldefined
replacement expressions that approximate
Δ
T
A
over an acceptable range of
Δ
T
1
and
Δ
T
2
(Paterson, 1984; Underwood, 1970; Chen,
1987). Paterson (1984), for example, proposes
2
3
ag
P
TT
TT
Δ +Δ
Δ
=≈
Δ
AA
(5)
A. ZavalaRío et al.
/
Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
203
while
1/3
1/3
1/3
1/3
12
2
U
TT
TT
Δ+
Δ
Δ=
≈
Δ
AA
(6)
is proposed by Underwood (1970), and
1/3
2/3
1
Ca
g
TT
T
y
T
Δ=
Δ
Δ
≈
Δ
AA
(7)
and
0.3275
0.3275
0.3275
0.3725
12
2
2
C
TT
TT
Δ+
Δ
Δ=
≈
Δ
AA
(8)
(the latter being a refined modification of
Underwood’s 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, the
results 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) when
12
TT
Δ≠
Δ
and
welldefined when
12
TT
Δ=
Δ
. Consequently,
approximating the LMTD through replacement
expressions may henceforth be avoided. Second,
analytical properties of the resulting
complete
expression
,
such
as
continuity
and
differentiability, are proved for every physically
reasonable combination of values of
Δ
T
1
and
Δ
T
2
, including those where
12
TT
Δ=
Δ
. Heat
exchanger lumpedparameter dynamic models
(like
(4)) using the LMTD, and control schemes
using its derivative with respect to
Δ
T
1
and
Δ
T
2
,
will now be mathematically coherent through
the 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 defined
for its use throughout this work:
F
mass flow rate
Cp
specific heat
M
total mass inside the tube
U
overall heat transfer coefficient
A
heat transfer surface area
T
temperature
Δ
T
temperature difference
Q
rate of heat transfer
t
time
R
set of real numbers
R
2
set of 2tuples (
x
i
)
i
=1,2
with
x
i
∈
R
R
+
set of positive real numbers
2
+
R
set of 2tuples (
x
i
)
i
=1,2
with
x
i
∈
R
Subscripts
:
c
cold
h
hot
i
inlet
o
outlet
Fig.1. Counterflow (left) and parallel flow (right) heat exchangers.
A. ZavalaRío et al.
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Revista Mexicana de Ingeniería Química
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(2005)
201212
204
In particular,
Q
h
,
Q
c
, 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. The
variables
T
h
and
T
c
generically denote the
temperature of the hot and cold fluids,
respectively, at any point along the exchanger,
and
Δ
T
=
T
h

T
c
. Furthermore, let
Δ
T
1
and
Δ
T
2
represent the temperature difference at
each terminal side of the heat exchanger,
i.e.
(see Fig. 1)
⎩
⎨
⎧
−
−
=
Δ
ci
hi
co
hi
T
T
T
T
T
1
and
⎩
⎨
⎧
−
−
=
Δ
co
ho
ci
ho
T
T
T
T
T
2
taken in this work as conventional definitions.
Notice that under such conventional definitions,
Δ
T
1
and
Δ
T
2
shall be considered positive. To
differentiate from
Δ
T
A
in (1), we denote
Δ
T
L
the
wellposed
logarithmic
mean
temperature
difference to be derived in the following section.
Δ
T
a
still stands (as in the previous section) for
the arithmetic model. Consider the sets
B
,
C
,
and
D
with
BC
⊂
, and a function
:
f
CD
→
.
We denote

B
f
the restriction of
f
to
B
,
i.e.
:
:
 ()
()
,
BB
f
BD
x
f
xf
xx
B
→=
∀
∈
6
.
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
C
pc
,
C
ph
, and
U
, are considered
flow and temperature invariant, that is, their
value 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 auxiliary
parameter
α
. Any expression where
α
is not
present is valid for both flow configurations. In
those where
α
appears, such parameter
determines the configuration they are valid for
in the following way
1
1
α
⎧
=
⎨
−
⎩
At any point along the exchanger, the heat
transfer equations are
hh
p
h
h
dQ
F C dT
=
cc
p
c
c
dQ
F C dT
=
hc
dQ
dQ
dQ
α
=
=−
dQ
U TdA
=
Δ
From the constant physical property assumption,
and considering a mean temperature difference,
Δ
T
L
, throughout the exchanger, we have, after
integration over the tube length,
()
h
h
ph
ho
hi
QF
C
TT
=
−
()
cc
p
c
c
i
c
o
QF
C
TT
=
−
hc
QQQ
α
=
=−
L
QU
AT
=
Δ
The next developments follow
hc
dT d
T d
T
Δ
=−
hc
hp
h
cp
c
dQ
dQ
FC
FC
=−
11
cp
c
hp
h
dQ
FC
FC
α
⎛⎞
=−
⎜⎟
⎜⎟
⎝⎠
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)
A. ZavalaRío et al.
/
Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
205
11
cp
c
hp
h
dT UT
d
A
FC
FC
α
⎛⎞
Δ=Δ
−
⎜⎟
⎜⎟
⎝⎠
At this point, we consider two possible
situations:
The general case
. We begin by assuming that
11
0
cp
c
hp
h
T
FC
FC
α
⎛⎞
Δ−
≠
⎜⎟
⎜⎟
⎝⎠
Let us note that this is the situation that is most
often found in physical heat exchangers
1
. Then,
from (20), we have
11
cp
c
hp
h
dT
Ud
A
TF
C
F
C
α
⎛⎞
Δ
=−
⎜⎟
⎜⎟
Δ
⎝⎠
yielding
2
1
11
cp
c
hp
h
T
ln
UA
TF
C
F
C
α
⎛⎞
Δ
=−
⎜⎟
⎜⎟
Δ
⎝⎠
after integration over the tube length. Notice that
since the righthand side of (22) is a nonzero
scalar,
Δ
T
1
and
Δ
T
2
(in the lefthand side) must
be such that
21
(/)
ln
T
T
ΔΔ
be a nonzero scalar
too, which is satisfied if and only if
12
TT
Δ≠
Δ
.
Then, one sees from these developments that
12
(21)
TT
⇔Δ ≠Δ
Now, notice that
11
ci
co
ho
hi
cp
c
hp
h
c
h
TT TT
FC
FC
Q
Q
α
−−
−=
−
1
This may explain why it is in fact the case that is
generally developed in the textbooks (for a specific
flow configuration), implicitly taking (21) as a fact
(which is actually the origin of the inconsistency
problem of (1)).
( )
ci
co
ho
hi
TT
TT
QQ
α
−
−
=−
21
TT
Q
Δ
−Δ
=
which can be substituted into (22) to get
22
1
1
TT
T
ln
UA
TQ
Δ
Δ−
Δ
=
Δ
From this expression, we get
21
2
1
TT
QU
A
T
ln
T
Δ
−Δ
=
Δ
Δ
which, compared to (19) and taking into account
(23), shows that
A
T
T
T
T
T
T
L
Δ
=
Δ
Δ
Δ
−
Δ
=
Δ
1
2
1
2
ln
if
12
TT
Δ≠
Δ
The special case
. Let us now suppose that
11
0
cp
c
hp
h
T
FC
FC
α
⎛⎞
Δ
−=
⎜⎟
⎜⎟
⎝⎠
Although this situation is seldom found in
practice
2
, it is considered here since its
development is needed for the wellposedness of
the LMTD. From (20), we have
0
dT
Δ
=
implying that the temperature difference
hc
TTT
Δ
=−
has a constant value throughout the
exchanger. Hence, after integration of (26) and
(15)
over
the
tube
length,
we
have
21
TTT
Δ
=Δ =Δ
(equivalent to (25)) and
2
This may explain why this case is apparently never
taken into account; it thus constitutes the part of the
derivation that is generally lacking in the textbooks.
(20)
(21)
(23)
(from (18))
(from (9), (10), and (11))
(24)
(25)
(26)
(22)
(from (16) and (17))
A. ZavalaRío et al.
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Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
206
QU
AT
=Δ
. From these expressions, comparing
the latter to (19), we have
L
TT
Δ=
Δ
if
21
TTT
Δ=
Δ=
Δ
The wellposed LMTD.
Finally, from the results
gotten in both, the general and special cases, i.e.
from (24) and (27), we get
21
2
1
L
TT
T
ln
T
T
T
Δ−
Δ
⎧
⎪
Δ
⎪
Δ=
⎨
Δ
⎪
⎪Δ
⎩
4. Analytical properties
In this section we consider the mean
temperature difference a bivariable function,
whether we refer to the logarithmic model in
(28),
Δ
T
L
(
Δ
T
1
,
Δ
T
2
), or the arithmetic one in (2),
Δ
T
a
(
Δ
T
1
,
Δ
T
2
). Furthermore, just the cases where
heat is transferred between fluids are considered
of practical interest in the present section.
Hence, for analysis purposes, we choose to
disregard the noheattransfer trivial case
21
0
TT
Δ=
Δ=
. Under such perspective,
2
+
R
is
considered to constitute the domain of
Δ
T
L
. We
begin by stating a useful equivalent expression.
Lemma 1.
Let
2
21
12
1
21
1
(,
)
1
21
i
i
TT
LT T
iT
T
∞
=
⎛⎞
Δ−
Δ
ΔΔ
+
⎜⎟
+Δ+
Δ
⎝⎠
∑
±
for all
(
∆
T
1
,
∆
T
2
)
such that
∆
T
1
+
∆
T
2
≠
0
.
Then
12
12
12
(,
)
(,
)
(,
)
a
L
TTT
TTT
L
TT
ΔΔΔ
ΔΔΔ≡
ΔΔ
2
2
1
)
,
(
+
∈
Δ
Δ
∀
R
T
T
.
Proof
.
We divide the proof in two parts:
1
)
12
TT
Δ
≠Δ
.
From Formula 4.1.27 in
3
(Abramowitz and Stegun, 1972), we have
21
22
1
0
12
1
1
2
21
i
i
TT
T
ln
Ti
T
T
+
∞
=
⎛⎞
ΔΔ
−
Δ
=
⎜⎟
Δ+
Δ
+
Δ
⎝⎠
∑
2
2
1
)
,
(
+
∈
Δ
Δ
∀
R
T
T
. Then, for all
2
2
1
)
,
(
+
∈
Δ
Δ
R
T
T
such that
12
TT
Δ
≠Δ
, we get
21
21
21
2
21
1
0
21
1
ln
2
21
i
i
TT
TT
T
TT
T
iT
T
+
∞
=
Δ
−Δ
Δ −Δ
=
Δ
⎛⎞
Δ−
Δ
⎜⎟
Δ
+Δ+
Δ
⎝⎠
∑
21
2
21
21
1
21
21
1
21
21
i
i
TT
TT
TT
TT
i
TT
∞
=
Δ
−Δ
=
⎡
⎤
⎛⎞
⎛⎞
Δ−
Δ
Δ−
Δ
+
⎢
⎥
⎜⎟
⎜⎟
Δ+
Δ
+ Δ+
Δ
⎢
⎥
⎝⎠
⎝⎠
⎣
⎦
∑
12
2
21
1
21
2
1
1
21
i
i
TT
TT
iT
T
∞
=
Δ
+Δ
=
⎛⎞
Δ−
Δ
+
⎜⎟
+Δ+
Δ
⎝⎠
∑
12
12
(,
)
(,
)
a
TTT
L
TT
Δ
ΔΔ
=
ΔΔ
2
)
12
TT
Δ
=Δ
.
Notice from (29) and (2) that
(,)
1
LTT
Δ
Δ=
and
(,)
a
TT
T
T
ΔΔΔ=
Δ
,
0
T
∀
Δ≠
. Then, for
12
0
TTT
Δ
=Δ =Δ >
, we
have
(,)
(,)
(,)
a
L
TT
T
TT
T
T
LTT
Δ
ΔΔ
=
Δ=
Δ ΔΔ
ΔΔ
.
Remark 1.
Lemma 1 is of great help in the
analysis of
Δ
T
L
. Indeed, since
(,)
1
LTT
ΔΔ ≥
3
Formula 4.1.27 in (Abramowitz and Stegun, 1972)
states the following wellknown (infinite) series
expansion
of
the
logarithmic
function:
∑
∞
=
+
≠
≥
ℜ
∀
⎟
⎠
⎞
⎜
⎝
⎛
+
−
+
=
0
1
2
.
0
,
0
:
,
1
1
1
2
1
2
ln
i
i
z
z
z
z
z
i
z
if
12
TT
Δ≠
Δ
if
21
TTT
Δ=
Δ=
Δ
(from (2) and (29))
(27)
(28)
(29)
(30)
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Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
207
for all
12
(,
)
TT
ΔΔ
such that
12
0
TT
Δ+
Δ ≠
(see
(29)), one sees readily that
12
12
(,
)
(,
)
a
TTT
L
TT
ΔΔΔ
ΔΔ
is
welldefined at every point on
2
+
R
. Actually, it is
easy
to
see
that
12
12
(, )(,) (, ) (,)1
TT
TT LTT LTT
ΔΔ →ΔΔ⇒ΔΔ →ΔΔ=
,
0
T
∀Δ
≠
. Consequently, one sees from (30) that
1
2
12
12
0(
,
)(
,
)
La
TT
T
T
T
T
T
T
Δ−
Δ→⇒
Δ ΔΔ →
ΔΔΔ
explaining why such an assertion is suggested
by several authors, like in (Kern, 1950; Gardner
and Taborek, 1977; Faires, 1957). Moreover,
from (30), we have
2
2
a
i
L
i
L
L
T
T
T
TL
∂
−Δ
∂Δ
∂Δ
=
∂Δ
i
=1,2, where the arguments have been dropped
for the sake of simplicity, and from (29), one
sees
that
21
1
2
21
i
L
i
ii
T
iS
S
Ti
T
∞
−
=
∂Δ
∂
=
∂Δ
+
∂Δ
∑
,
with
21
21
TT
S
TT
Δ−
Δ
=
Δ+
Δ
and
3
2
21
(1
)2
()
i
i
i
T
S
TT
T
−
−Δ
∂
=
∂Δ
Δ
+ Δ
.
Hence,
12
1
0,
(
0
,
1
)
2
L
ii
T
L
TT
L
TT
⎛⎞
∂Δ
∂
Δ−
Δ→⇒
→
⇒
→
⎜⎟
∂Δ
∂Δ
⎝⎠
,
i
=1,2, as claimed by authors like in [16].
Furthermore,
synthetically
dividing
Δ
T
a
(
Δ
T
1
,
Δ
T
2
) by
L
(
Δ
T
1
,
Δ
T
2
), one gets
24
12
12
12
12
3
1
2
12
12
(, )
()
()
12
(
,
)
...
(, )
6
4
5
(
)
a
a
TTT
TT
TT
TTT
LT T
T T
T T
ΔΔΔ
Δ−Δ
Δ−Δ
=Δ Δ Δ −
−
−
ΔΔ
Δ+
Δ
Δ+
Δ
coinciding with the power series expansion
suggested in (Steiner, 1989) as a replacement
for the LMTD. The proofs of some other
analytical features of
Δ
T
L
stated in the present
section are simplified through the use of (30).
Remark 2.
Expression (32) is helpful to see
the link of the replacement formulas (5)–(8)
with the LMTD. Indeed, notice that for close
enough values of
Δ
T
1
and
Δ
T
2
, the LMTD
may be approximated neglecting the third and
upper terms of the series,
i.e.
2
12
12
12
12
1
2
(, )
()
1
(, )
(, )
6
a
a
TTT
TT
TTT
LT T
T T
ΔΔΔ
Δ−
Δ
≈Δ
Δ
Δ
−
ΔΔ
Δ+
Δ
. For
instance,
Δ
T
1
and
Δ
T
2
resulting in a sufficiently
small
value
of
2
12
()
TT
Δ−Δ
,
say
2
12
()
0
TT
Δ −Δ
≈
(observe
that
2
12
1
2
()
0
TT
T
T
Δ −Δ
≈⇔Δ ≈Δ
),
may
be
considered to apply for such an approximation.
Thus:
2
12
1
2
1
2
()
0
2
TT
T
T
T
T
Δ −Δ
≈⇔ ΔΔ ≈Δ+Δ ⇔
11
2
2
1
2
22
(
)
TT
T
TT
T
Δ
+Δ
Δ
+
Δ
≈
Δ
+
Δ⇔
2
12
2
12
1
2
12
()
()
2
(
)
2(
)
TT
TT
T
T
TT
Δ+Δ
Δ+Δ
≈Δ+
Δ ⇔
Δ+
Δ
22
12
12
12
()
()
1
32
(
)
TT
TT
TT
Δ+
Δ
Δ+
Δ
≈⇔
Δ+
Δ
2
2
12
12
12
()
()
1
36
TT
TT
TT
Δ+Δ
Δ+
Δ
−≈−
⇔ −
≈
Δ+
Δ
12
12
12
2
32
2
TT
TT
TT
Δ+
Δ
Δ+
Δ
⎛⎞
−
−ΔΔ
⇔
−
⎜⎟
⎝⎠
2
12
12
2
()
1
63
ag
TT
TT
TT
Δ +Δ
Δ+
Δ
≈
Δ+
Δ
,
i.e.
P
a
T
T
T
L
T
T
T
A
Δ
≈
Δ
Δ
Δ
Δ
Δ
)
,
(
)
,
(
2
1
2
1
, (see (5)). Alternatively, for
Δ
T
1
and
Δ
T
2
resulting
in
( )
×
Δ
+
Δ
Δ
+
Δ
=
3
/
2
2
2
/
1
2
3
/
1
1
3
/
2
1
1
5
11
5
T
T
T
T
a
( )
2
1/3
1/3
12
0
TT
Δ
−Δ
≈
(observe that
1
0
a
≈
⇒
( )
2
1/3
1/3
12
1
2
0
TT
T
T
Δ
−Δ
≈ ⇔Δ ≈Δ
),
we
have
()
2
12
12
1
12
1
0
26
TT
TT
a
TT
Δ−
Δ
Δ+
Δ
≈⇔
−
Δ+
Δ
3
1/3
1/3
12
2
TT
⎛⎞
Δ+
Δ
≈
⎜⎟
⎝⎠
(
s
e
e
A
p
p
e
n
d
i
x
A
)
,
i.e.
U
a
T
T
T
L
T
T
T
A
Δ
≈
Δ
Δ
Δ
Δ
Δ
)
,
(
)
,
(
2
1
2
1
(see (6)). Furthermore, for
Δ
T
1
and
Δ
T
2
such
that
()
3
/
2
2
1
3
/
1
2
1
3
/
2
2
1
2
2
2
2
2
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
Δ
+
Δ
⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
Δ
Δ
+
⎟
⎠
⎞
⎜
⎝
⎛
Δ
+
Δ
=
T
T
T
T
T
T
a
1/3
12
2(
)
0
TT
⎤
−ΔΔ
≈
⎦
(observe
that
2/3
1/3
12
21
2
02
2
(
)
0
2
TT
aT
T
⎡⎤
Δ+
Δ
⎛⎞
≈
⇒−
Δ
Δ
≈
⇔
⎢⎥
⎜⎟
⎝⎠
⎢⎥
⎣⎦
(31)
(32)
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Revista Mexicana de Ingeniería Química
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208
2/3
2
1/3
12
12
12
()
22
TT
TT
TT
Δ+
Δ
Δ+
Δ
⎛⎞
⎛⎞
≈Δ Δ
⇔
⎜⎟
⎜⎟
⎝⎠
⎝⎠
22
12
1
12
2
12
24
TT
T
TT
T
TT
≈Δ Δ ⇔Δ
+ Δ Δ +Δ
≈ Δ Δ
2
12
1 2
()
0
TT
TT
⇔Δ −Δ
≈ ⇔Δ ≈Δ
),
we
have
()
2
12
1/3
2/3
12
2
12
1
0
26
ag
TT
TT
aT
T
TT
Δ−
Δ
Δ+
Δ
≈⇔
−
≈
Δ Δ
Δ+
Δ
(see Appendix B),
i.e.
1
2
1
2
1
)
,
(
)
,
(
C
a
T
T
T
L
T
T
T
A
Δ
≈
Δ
Δ
Δ
Δ
Δ
(see
(7)). From
these developments we see that as
long as close enough values of
Δ
T
1
and
Δ
T
2
are
considered, relatively good approximations of
the LMTD are gotten through the replacement
expressions (5)–(8) (recall that (8) is just a
refinement of (6)). On the contrary, such
approximations deteriorate as
Δ
T
1
and
Δ
T
2
are
taken far from each other.
Remark 3.
Let us point out that the quotient
function
12
12
12
(,
)
(,
)
(,
)
a
a
TTT
LT T
T
TT
L
ΔΔΔ
ΔΔ
Δ
ΔΔ
=
is
defined on a subset wider than the domain of
Δ
T
L
.
Actually,
{}
R
T
T
R
T
T
L
T
a
→
≠
Δ
+
Δ
∈
Δ
Δ
Δ
0
)
,
(
:
2
1
2
2
1
.
Then
a
T
L
Δ
is an extension of
L
T
Δ
(actually Lemma 1
can be synthesized as
2
+
Δ
≡
Δ
R
a
L
L
T
T
). Therefore,
a
T
L
Δ
may be used to extrapolate
L
T
Δ
to points
on
R
2
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
∑
∑
∞
=
∞
=
∂
−
=
+
=
+
1
0
1
2
1
1
2
1
2
i
j
R
i
j
L
, and since
11
,1
212
i
ii
>∀
≥
−
, then
1
1
21
i
i
∞
=
−
∑
is divergent
according to theorems 3.28 and 3.25 in
4
(Rudin,
4
In (Rudin, 1976), Theorem 3.28 states that
0
1/
p
n
n
∞
=
Σ
converges if p > 1 and diverges if
1
p
≤
,
while point (b) of Theorem 3.25 states that if
0
nn
ad
≥≥
for
0
nN
≥
(for some
0
N
), and if
0
nn
d
∞
=
Σ
diverges, then
0
nn
a
∞
=
Σ
diverges.
1976).
Therefore
12
12
(,
)
(,
)
0
a
TTT
LT T
ΔΔΔ
ΔΔ
→
as
2
2
1
)
,
(
+
∂
→
Δ
Δ
R
T
T
which, from Lemma 1, implies
that
12
(,
)0
L
TTT
Δ
ΔΔ →
as
12
(,
)
TT
ΔΔ
approaches
2
+
∂
R
(from the interior of
2
+
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 (ZavalaRío
et al.
, 2003;
ZavalaRío, 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
T
hi
or lower than
T
ci
(see Fig. 1).
Lemma 2.
The LMTD model in (28) is
continuously differentiable and positive on
2
+
R
.
Proof.
Since
2
2
1
2
1
)
,
(
,
1
)
,
(
+
∈
Δ
Δ
∀
≥
Δ
Δ
R
T
T
T
T
L
(see (29)), one sees from (30) (and (2)) that
L
T
Δ
exists and is continuous on
2
+
R
(also
verifiable from (32), according to Lemma 1).
Moreover, from (31), one observes that
L
i
T
T
∂Δ
∂Δ
,
i
= 1,2, exist and are continuous on
2
+
R
,
proving continuous differentiability. On the
other hand, notice (from (2)) that
2
2
1
2
1
)
,
(
,
0
)
,
(
+
∈
Δ
Δ
∀
>
Δ
Δ
Δ
R
T
T
T
T
T
a
(the average
of two positive numbers is positive).
Consequently,
2
2
1
2
1
2
1
2
1
)
,
(
),
,
(
)
,
(
)
,
(
0
+
∈
Δ
Δ
∀
Δ
Δ
Δ
≤
Δ
Δ
Δ
Δ
Δ
<
R
T
T
T
T
T
T
T
L
T
T
T
a
a
.
Then, from Lemma 1, positivity of
L
T
Δ
follows too.
Lemma 2 is specially interesting when
12
TT
Δ
=Δ
. It proves to be essential when the
LMTD is involved in dynamical analysis and
control synthesis of heat exchangers. In (Zavala
Río and SantiestebanCos, 2004), for example, it
is shown that thanks to such continuous
differentiability
property,
existence
and
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Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
209
uniqueness
of
solutions
of
heat exchanger
lumpedparameter dynamical models as in (4),
using the wellposed LMTD as driving force,
are guaranteed. Furthermore, as mentioned in
Section 1,
the
partial
derivatives
of
the
LMTD
(with
respect
to
its
arguments)
are
used in the control laws synthesized in (Alsop
and
Edgar,
1989;
Malleswararao
and
Chidambaram, 1992). Provided that the well
posed LMTD in (28) is used, their closedloop
analyses are correct (according to the respective
assumptions made in each of those works). On
the contrary, such control strategies could cause
inconvenience if
Δ
T
L
were not continuously
differentiable on
2
+
R
, or
Δ
T
A
in (1) were used
instead. This is true for any kind of (numerical)
algorithm using the partial derivatives of the
LMTD, like the equationoriented flowsheeting
program application mentioned by Paterson
(1984) (see Section 1 above).
Lemma 3.
The
LMTD
model
in
(28)
is
strictly
increasing
in
its
arguments, i e.
0
L
i
T
T
Δ
Δ
∂
>
∂
,
i
=1,2,
2
2
1
)
,
(
+
∈
Δ
Δ
∀
R
T
T
.
Proof.
From (28) (for
Δ
T
2
≠
Δ
T
1
) and (31) (for
Δ
T
2
=
Δ
T
1
; recall Remark 1), we have
221
1
2
2
1
(1
)
1
2
i
i
L
i
TTT
ln
TT
T
T
ln
T
T
⎧
⎡⎤
ΔΔ
−
Δ
−−
⎪
⎢⎥
ΔΔ
⎣⎦
⎪
⎪
Δ
⎪
⎡⎤
Δ
=
⎨
⎢⎥
Δ
Δ
⎪
⎣⎦
⎪
⎪
⎪
⎩
∂
∂
i
=1,2, existing and being continuous on
2
+
R
according to Lemma 2. Notice from (33) that
the proof of the lemma amounts to
demonstrate
positivity
of
22
1
1
(1
)
i
i
TTT
ln
TT
⎡⎤
ΔΔ
−
Δ
−−
⎢⎥
ΔΔ
⎣⎦
,
i
=1,2,
for
all
2
2
1
)
,
(
+
∈
Δ
Δ
∀
R
T
T
such that
12
TT
Δ≠
Δ
. Then,
from Formula 4.1.33 in [23], we have, for all
such
)
,
(
2
1
T
T
Δ
Δ
that
⇔
Δ
Δ
−
Δ
<
Δ
Δ
<
Δ
Δ
−
Δ
1
1
2
1
2
2
1
2
ln
T
T
T
T
T
T
T
T
0
ln
)
1
(
1
2
1
2
>
⎥
⎦
⎤
⎢
⎣
⎡
Δ
Δ
−
Δ
−
Δ
Δ
−
i
i
T
T
T
T
T
,
i
=1,2, proving
the lemma.
Lemma 3 has been helpful in (Zavala
Río and SantiestebanCos, 2004) to prove the
existence of a unique equilibrium solution of
system (4) to which every trajectory converges.
Furthermore, it has been helpful in (ZavalaRío
et al.
, 2003) to characterize heat exchangers as
cooperative systems (under the assumptions
made therein). This means that a temperature
raise at any of the outlets entails a temperature
increase at the other outlet. The dynamics
of
heat
exchangers
may
then
be
analyzed
under
the
framework
of cooperative systems, which
constitutes
a
complementary
way
to
comprehend their behaviour.
Discussion and conclusions
Some insights on the usefulness of the
results presented above are in order. For
instance, a wellknown calculation problem is
that of finding a suitable
Δ
T
2
satisfying an
LMTD relation to a given
Δ
T
1
(Paterson, 1984).
An exact solution to such a problem may be
gotten
considering
the
complete
LMTD
expression in (28). Indeed, departing from (16)
(19), (28), one gets
1
21
cp
c
hp
h
UA
FC
FC
Te
T
α
⎛⎞
⎜−⎟
⎜⎟
⎝⎠
Δ=
Δ
,
which is a simple linear expression. Numerical
algorithms (whose convergence is not even
guaranteed)
or
nonlinear
complex
approximation expressions (Paterson, 1984;
Chen, 1987), are not any longer needed to
perform such a calculation. Furthermore, the
results developed in Sections 3 and 4 have been
very helpful to characterize dynamic properties
of heat exchangers through simple lumped
parameter dynamic models (ZavalaRío and
SantiestebanCos, 2004; ZavalaRío
et al.
,
2003). In (ZavalaRío and SantiestebanCos,
i
f
12
TT
Δ≠
Δ
if
21
TT
Δ=
Δ
(33)
A. ZavalaRío et al.
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Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
210
2004), for instance, dynamic properties of
model (4), using (28), have been analyzed and
proved to be
equivalent
to those of the
distributedparameter model where (4) comes
from. 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,
T
o
*
=
(
T
co
*
,
T
ho
*
)
T
, linearly related to the inlet
temperatures,
T
i
= (
T
ci
,
T
hi
)
T
,
i.e.
T
o
*
=
AT
i
, where
the complete expressions of the elements of
A
∈
R
2
×
2
have been obtained in terms of the
system 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 statespace domain. On the other
hand, based on the results in Section 3, other
interesting dynamic properties of system (4)
have been proved in (ZavalaRío
et al.
, 2003).
Basically, under certain standard assumptions,
model (4) has been characterized as a positive,
compartmental, cooperative system. These
features provide a new framework for the
comprehension of the dynamic behavior of heat
exchangers. Furthermore, the use of the LMTD
has 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 simple
model as (4), like numerical rigidity, for
instance. Nevertheless, such problems may be
avoided if more than one
bicompartmental cell
is
considered
in
the
lumpedparameter
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
heat
exchangers too. For instance, based on (4),
taking
F
c
as input and
T
ho
as output, input
output (partial) linearization control algorithms
have 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 partial
derivatives of
Δ
T
1
and
Δ
T
2
. The authors use (1)
but do not treat its inconsistency either. Again,
numerical results avoiding such inconsistencies
are 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 important
role that the LMTD plays in several theoretical
and practical aspects of heat exchangers. The
results developed here are intended to support
all 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 Underwood’s approximation
( )( )
2
2/3
1/3
1/3
2/3
1/3
1/3
1
1
2
212
51
1
5
0
TT
T
T
T
T
Δ+
Δ
Δ+
Δ
Δ−
Δ
≈
2/3
1/3
1/3
2/3
1/3
1/3
11
2
2
1
2
[5(
5
) 6
]
TT
T
T
T
T
⇔Δ
+
Δ
Δ
+
Δ
+
Δ
Δ
1/3
1/3
2
2/3
1/3
12
1
2
1
2
()
0
[
5
(
)
6
(
TT
T
T
T
T
Δ−
Δ
≈
⇔
Δ
−
Δ
+
Δ
Δ
1/3
2 /3
1/3
1/3
4 /3
1/3
12
1
2
1
1
2
)](
)
0
5
TT
T
T
T
T
T
−Δ
Δ
Δ
− Δ
≈
⇔
Δ
+ Δ
Δ
2/3
2/3
1/3
4/3
4/3
12
1
2
2
1
12
5
0
4
TT
T
T
T
T
−Δ
Δ
+
Δ
Δ+
Δ
≈⇔Δ
+
2/3
2/3
4/3
1/3
2/3
2/3
12
2
1
2
12
44
8
8
TT
T
T
T
TT
ΔΔ
+
Δ
+
Δ
Δ
+
ΔΔ
A. ZavalaRío et al.
/
Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
211
1/3
4/3
1/3
1/3
4/3
12
1
1
2
12
2
89
9
99
TT
T
T
T
TT
T
+Δ
Δ
≈
Δ
+
Δ
Δ
+
Δ
Δ
+
Δ
2 /3
1/3
1/3
2 /3
2
1/3
11
2
2
1
2
1
4(
)
9(
)(
TT
T
T
T
T
T
⇔Δ
+
Δ
Δ
+
Δ
≈
Δ
+
Δ
Δ
2/3
1/3
1/3
2/3
2
1/3
1/3
11
22
12
12
1/3
2
4(
)
9(
)
)
TT
TT
TT
TT
T
Δ+
Δ
Δ+
Δ
≈Δ
+Δ
Δ+
Δ
+Δ
⇔
()
(
)
)
(
9
4
8
3
2
1
2
3
/
2
2
3
/
1
2
3
/
1
1
3
/
2
1
2
3
/
1
2
3
/
1
1
T
T
T
T
T
T
T
T
Δ
+
Δ
Δ
+
Δ
Δ
+
Δ
⋅
Δ
+
Δ
−
⇔
2
1/3
1/3
2
1/3
1/3
12
12 12
12
()
3
()
()
8
1
6
TT
TT
TT
TT
ΔΔ
ΔΔ
Δ
+
Δ
⇔
−
Δ+
Δ
−
≈−
−
1/3
1/3
2/3
1/3
1/3
2/3
12
11
2
2
)
1
(3
6
3
24
TT
TT
T
T
Δ+
Δ
ΔΔ
Δ
Δ
⎡
⎤
≈−
+
−
⎦
⎢
⎣
1/3
1/3
2/3
1/3
1/3
2/3
12
11
2
2
1
(2
)
24
TT
TT
T
T
Δ+
Δ
ΔΔ
Δ
Δ
⎡
=+
+
⎢
⎣
]
×
Δ
+
Δ
=
Δ
−
Δ
Δ
+
Δ
−
2
3
/
1
2
3
/
1
1
3
/
2
2
3
/
1
2
3
/
1
1
3
/
2
1
T
T
T
T
T
T
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
Δ
+
Δ
Δ
−
Δ
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
+
Δ
3
/
2
2
3
/
1
2
3
/
1
1
3
/
2
1
2
3
/
1
2
3
/
1
1
2
T
T
T
T
T
T
1/3
1/3
12
3
12
22
TT
T
T
ΔΔ
Δ
+
Δ
⎛⎞
+
=
−⇔
⎜⎟
⎝⎠
()
1/3
1/3
12
3
2
12
12
12
1
26
2
TT
TT
TT
TT
ΔΔ
Δ+
Δ
ΔΔ
Δ+
Δ
−
⎛⎞
+
−≈
⎜⎟
⎝⎠
.
B. Developments for Chen’s approximation
2/3
2/3
1/3
12
12
12
2(
)
2
22
TT
TT
TT
Δ+
Δ
Δ+
Δ
ΔΔ
⎡⎤
⎡
⎛⎞
⎛⎞
+
⎢⎥
⎢
⎜⎟
⎜⎟
⎝⎠
⎝⎠
⎢⎥
⎢
⎣⎦
⎣
2/3
1/3
12
12
2(
)
0
2
2
TT
TT
Δ+
Δ
ΔΔ
⎡
⎛⎞
⎤
−
≈⇔
−
⎢
⎜⎟
⎦
⎝⎠
⎢
⎣
4/3
1/3
2/3
12
1
2
12
()
9
()
0
4
24
2
TT
T T
TT
ΔΔ
Δ +Δ
ΔΔ
⎤
⎛⎞
−−
≈
⇔
⎜⎟
⎥
⎝⎠
⎦
2/3
1/3
2/3
12
12
12
()
2(
)
2(
)
2
TT
TT
TT
ΔΔ
ΔΔ
ΔΔ
+
−−
4/3
2/3
1/3
12
12
12
()
02
(
)
22
TT
TT
TT
Δ+
Δ
Δ Δ
ΔΔ
⎡
+
⎛⎞
≈⇔
+
⎢
⎜⎟
⎝⎠
⎢
⎣
4/3
2/3
12
12
()
6
2
TT
TT
Δ+
Δ
ΔΔ
⎛⎞
⎤
+
≈=
⎜⎟
⎦
⎝⎠
()
×
Δ
+
Δ
Δ
+
Δ
−
⇔
⎟
⎠
⎞
⎜
⎝
⎛
Δ
+
Δ
Δ
+
Δ
2
1
2
2
1
3
/
1
2
1
2
1
6
1
2
)
(
3
T
T
T
T
T
T
T
T
4/3
2/3
1/3
12
12
12
()
()
22
TT
TT
TT
Δ+
Δ
Δ Δ
ΔΔ
⎡
+
⎛⎞
+
⎢
⎜⎟
⎝⎠
⎢
⎣
21
/
3
2/3
12
12
12
()
22
TT
TT
TT
ΔΔ
ΔΔ
ΔΔ
−+
⎛⎞
⎛⎞
⎤
+≈
−
⎜⎟
⎜⎟
⎦
⎝⎠
⎝⎠
21
/
3
12
12
12
22
TT
TT
TT
ΔΔ
ΔΔ
ΔΔ
⎡⎤
++
⎛⎞
⎛⎞
=−
⎢⎥
⎜⎟
⎜⎟
⎝⎠
⎝⎠
⎢⎥
⎣⎦
()
()
×
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
Δ
+
Δ
−
Δ
Δ
≈
Δ
+
Δ
Δ
−
Δ
−
⇔
3
/
2
2
1
3
/
1
2
1
2
1
2
2
1
2
6
1
T
T
T
T
T
T
T
T
()
−
Δ
Δ
⎟
⎠
⎞
⎜
⎝
⎛
Δ
+
Δ
=
⎟
⎠
⎞
⎜
⎝
⎛
Δ
+
Δ
3
/
1
2
1
3
/
1
2
1
3
/
1
2
1
2
2
T
T
T
T
T
T
()
2
12
12
12
12
1
22
6
TT
TT
TT
TT
ΔΔ
ΔΔ
ΔΔ
ΔΔ
−
++
⇔−
+
()
1/3
1/3
12
12
.
2
TT
TT
ΔΔ
ΔΔ
+
⎛⎞
≈
⎜⎟
⎝⎠
A. ZavalaRío et al.
/
Revista Mexicana de Ingeniería Química
Vol. 4
(2005)
201212
212
References
Abramowitz, M. and Stegun, I.A. (1972).
Handbook of mathematical functions
(9
th
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
, 777–786.
Holman, J.P. (1997).
Heat transfer
(8th ed.).
McGrawHill, USA.
Incropera, F.P. and DeWitt, D.P. (1990).
Fundamentals of heat and mass transfer
(3
rd
ed.). John Wiley & Sons, New York.
Kern, D.Q. (1950).
Process heat transfer
.
McGrawHill, Tokyo.
Khambanonda, T., Palazoglu, A., and Romagnoli,
J.A. (1990). The stability analysis of
nonlinear feedback systems.
AIChE Journal
36
, 66–74.
Khambanonda, T., Palazoglu, A. and Romagnoli,
J.A. (1991). A transformation approach to
nonlinear process control.
AIChE Journal
37
, 1082–1092.
Malleswararao, Y.S.N. and Chidambaram, M.
(1992). Nonlinear controllers for a heat
exchanger.
Journal of Process Control 2
,
17–21.
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.). McGrawHill, Tokyo.
Papastratos, S., Isambert, A. and Depeyre, D.
(1993). Computerized optimum design and
dynamic simulation of heat exchanger
networks.
Computers
and
Chemical
Engineering 17
, S329–S334.
Paterson, W.R. (1984). A replacement for the
logarithmic mean.
Chemical Engineering
Science 39
, 1635–1636.
Reimann, K.A. (1986).
The consideration of
dynamics and control in the design of heat
exchanger networks
. Ph.D. Thesis, Swiss
Federal Institute of Technology Zürich
(ETH), Switzerland.
Rudin, W. (1976).
Principles of mathematical
analysis
. (3rd ed.). McGrawHill, USA.
Steiner, M. (1987). Dynamic models of heat
exchangers. In:
Chemical engineering
fundamentals, XVIII Congress The Use of
Computers in Chemical Engineering
, pp.
809–814. Giardini Naxos, Sicily, Italy.
Steiner, M. (1989). Low order dynamic models of
heat
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
.
ButterworthHeinemann, Stoneham.
ZavalaRío, A. and SantiestebanCos, R. (2004).
Qualitatively
reliable
compartmental
models for doublepipe heat exchangers. In:
Proc. of the
2nd IFAC symposium on
system, structure, and control
, pp. 406–411.
Oaxaca, Mexico.
ZavalaRío, A., Femat, R. and RomeroMéndez,
R. (2003). Countercurrent doublepipe 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. 385–392. 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 Computeroriented
process engineering, EFChE Working
Party
, pp. 165–170. Barcelona, Spain.