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Refrigeration
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Chapter 5
HEAT EXCHANGERS1
5.1 Introduction
5.2 Overall Heat Transfer Coefficient
5.2.1 Heat Transfer Coefficients
5.2.2 Fin Efficiency
5.2.3 Fouling Factor
5.3 Energy Balance
5.4 Mean Temperature Difference
5.5 Effectiveness Method
5.6 Simulation of Heat Exchangers
5.7 Performance Prediction from Empirical Data
5.1 INTRODUCTION
Heat exchangers are a vital part of the refrigeration system. The condenser and the
evaporator are the links between the working fluid (refrigerant) and the refrigerated and ambient
media. Less efficient heat exchangers will result in larger temperature differences and,
consequently, in a lower evaporating temperature and a greater condensing temperature. It has
been seen in the preceding chapters that, under greater condensing to evaporating temperature
differentials, the compressor will present lower capacity and greater energy consumption. Also,
the system coefficient of performance will decrease.
Condensers and evaporators can take different forms in refrigeration systems. The
condenser can be water or air cooled. In the latter, air can be forced through by a fan or heat transfer
occurs by natural convection. Evaporators can be of the air source type or the medium to be cooled
is water. In this case they are called chillers. Figure 5.1 depicts a few examples. Other heat
exchangers can be found in refrigeration systems of all sizes: sub-coolers, liquid line/suction line
heat exchangers, condenser/evaporators, to name but a few.
The objective of this chapter is to present the basic theory of heat exchangers, providing the basic
tools for the simulation of such components.
5.2 OVERALL HEAT TRANSFER COEFFICIENT
Consider two fluids exchanging heat through a flat wall, Figure 2. Heat is transferred by
convection, in both streams, and by conduction, through the wall. The temperature difference that is
likely to occur is depicted in Figure 5.2. The heat flow rate between the hot fluid, 1, and the wall is
given by:
1 w11Q = A( - )h T T� (5.1)
1 José A.R. Parise, PUC-Rio
RefrigerationChapter05v05.doc, 25/10/2005
Heat Exchangers 2
Figure 5.1 - Examples of condensers and evaporators.
And, for the cold fluid, 2:
w2 22Q = A( - )h T T� (5.2)
Figure 5.2 - Heat transfer between two fluids though a flat wall.
Conduction though the wall provides:
Heat Exchangers 3
w1 w2
kAQ = ( - )T T
δ� (5.3)
The combination of equations (5.1) to (5.3) provides the rate of heat exchange between
fluids 1 and 2.
1 2-T TQ =
R� (5.4)
where ΣR is the summation of the thermal resistances R. An overall heat transfer coefficient, U,
can be defined, so that:
1 2Q =UA( - )T T� (5.5)
and, from (5.4),
1UA=
RΣ (5.6)
The product UA is called the heat exchanger overall thermal conductance. For the case of
the flat plate separating the fluids, one has:
1 2
1 1 1= + +
UA A k A Ah h
δ (5.7)
where h1 and h2 are the film coefficients on the fluid sides. Table 5.1 shows typical values of the film
coefficient for various modes of heat transfer. Equation (5.7) shows how the overall conductance of
the heat exchanger is affected by the film coefficients on both sides and by the thickness and thermal
conductivity of the separating wall.
The effect of fouling can be introduced in equation (5.7), leading to:
1 s1 s2 2
1 1 1 1 1= + + + +
UA A A k A A Ah h h h
δ (5.8)
where 1sh and
2sh are the fouling (or scale) coefficients. Table 2 shows typical values for the fouling
coefficient. Usually, for heat exchangers with metallic separating walls, the conduction resistance is
relatively too small, and can be neglected. It should be noted, also, that some thermal resistances
may be lower than the others by factors of ten. This may be found in condensers or evaporators with
natural convection in the air side and two-phase condensing or boiling on the refrigerant side.
Heat Exchangers 4
External condensation over a tube
bundle (condensing temperature of
30oC, 6 rows in vertical direction,
tubes with 25 mm diameter) – R22
1142
Same as above –
R717 (Amonia)
5096
Table 5.1- Typical values for the film coefficient (Holman, 1981; Stoecker and Jaiz Jabardo,
1998).
In most heat exchangers heat is transferred through cylindrical surfaces (tubes), where the
outer area is larger than the inner one. For heat exchangers with bare tubes, the overall
conductance is:
w
21 s1 s2 21 1 2
1 1 1 1 1= + + + +R
UA A A Ah h h h A (5.9)
The wall thermal resistance for a circular tube is:
ln o
iw
w
D( )
D=R2 Lkπ
(5.10)
Heat Exchangers 5
Finally, the concept of different areas can be extended to the case of heat exchangers with
one surface (or even both) with fins. Account must be given to the fact that the whole fin may not be
at the same temperature (1wT or
2wT ). For that, the total surface effectiveness, 0η , is introduced in the
overall conductance equation:
w
s s0 0 0 01 1 2 2
1 1 1 1 1= + + + +R
UA ( h A ( A ( A ( h A) ) ) )h hη η η η (5.11)
If A is the total finned area,
f pA= +A A (5.12)
then the effective total area is:
t f0 fA= +A Aη η (5.13)
pA is the primary area (usually the tube at the fin base), which is at temperature 1p
T or 2p
T .
Therefore, the total surface effectiveness is calculated by:
f
0 f
A= 1- (1- )
Aη η (5.14)
The overall heat transfer coefficient can be defined in terms of either the hot or the cold
fluid surfaces.
1 21 2UA= =U UA A (5.15)
Thus, equation (5.5) can be written as:
1 1 2 2 1 21 2Q = ( - )= ( - )U UA T T A T T� (5.16)
Estimates of typical values for the overall heat transfer coefficient are difficult to obtain as
two flow streams, with corresponding film coefficients and fouling factors, are involved. If a
preliminary calculation is required, it is advisable to estimate the film coefficients and fouling
factors separately and then proceed with the calculation of U .
5.2.1 Heat Transfer Coefficient
Approximate ranges of convection heat transfer coefficients are given in table 5.1. It can be
seen that the film coefficient can vary considerably, as it depends on the geometry, fluid properties
Heat Exchangers 6
and flow conditions. Therefore, resorting to guessed values should be done with extreme caution and
the use of proper correlations, even approximate, should be encouraged instead. The reader is
referred to handbook publications (Rohsenow et al., 1998; ASHRAE, 2001) for an extensive
collection of heat transfer coefficient correlations for different geometries and flow conditions.
5.2.2 Fin Efficiency
Expressions for the efficiency of extended surfaces of different types can be found in the
literature (Schneider, 1974; Holman, 1981, Rohsenow et al., 1985).
5.2.3 Fouling Factor
Estimates for the fouling factor are presented in table 5.2. ASHRAE (1993) summarizes
correlations for the film coefficient for different flow conditions.
Table 5.2 - Typical values for the fouling coefficient (Holman, 1981).
5.3 ENERGY BALANCE
Consider a heat exchanger, as in Figure 5.3. Applying control volumes to each of the fluid
streams, the corresponding energy balances are:
1
11 1 1e1i
dU= - -h Qm m h
dt�� � (5.17)
22 2i 2 2e2
dU= + - hQm h m
dt�� � (5.18)
The left hand side of equations (5.17) and (5.18) are the rate of variation of the internal
energy of fluids 1 and 2 inside the heat exchanger. Equations (5.17) and (5.18) can be further
simplified if the following assumptions are made:
i) steady-state operation;
ii) adiabatic heat transfer, i.e., with no heat losses;
iii) no phase change in the fluid streams;
iv) constant specific heat for both fluid streams.
The energy balance equations thus become:
Heat Exchangers 7
1 1 1i 1epQ = m c ( - )T T� � (5.19)
2 2 2e 2ipQ = m c ( - )T T� � (5.20)
Defining the thermal capacity rate as
pC = mc� (5.21)
then,
1i 1e1Q = ( - )C T T� (5.22)
2e 2i2Q = ( - )C T T� (5.23)
Equations (5.22) and (5.23) simply state that heat has been removed, at a certain rate, from
the hot fluid, 1, and has been transferred to the cold stream, 2. They are employed with the heat
transfer equation, (5.16), which allows for the estimate of the heat transfer rate as a function of the
heat exchanger geometry and operating conditions.
Figure 5.3 - Heat balance in a heat exchanger.
5.4 OVERALL TEMPERATURE DIFFERENCE
Figure 5.4 shows the temperature distribution, of the cold and hot fluid, along the area of
different heat exchangers. Clearly, the temperature difference between the fluids varies and equation
(5) must be substituted by:
mQ =UA T∆� (5.24)
where ∆Tm is a suitable overall temperature difference.
Heat Exchangers 8
Figure 5.4 - Temperature distribution in typical heat exchangers. (a) Double pipe heat exchangers
(Welty et al., 1969); (b) 1-2 (one shell pass and two tube passes) heat exchanger (Shah, 1988).
The objective of this section is to develop an expression for the overall temperature
difference. Consider the simple case of a double pipe parallel flow heat exchanger, Figure 5.5a. For
an element of the heat exchanger, energy balances and the heat transfer equation, (5.5), can be
applied.
1 1dQ = -C dT� (5.25)
2 2dQ= C dT� (5.26)
1 2dQ =U( - )dAT T� (5.27)
From equations (5.25) and (5.26),
1
1
dQd = -T
C
�
(5.28)
2
2
dQd =T
C
�
(5.29)
Heat Exchangers 9
Figure 5.5 - Temperature difference in a double pipe heat exchanger. (a) Parallel Flow; (b) Counter
Flow.
Subtracting (5.29) from (5.28),
1 2 1 2
1 2
1 1d - d = d( - )= -dQ( + )T T T T
C C
� (5.30)
Taking equation (5.27) into (5.30),
1 2 1 2
1 2
1 1d( - )= -[U( - )dA] +T T T T
C C
(5.31)
or,
1 2
1 2 1 2
d( - ) 1 1T T= -U + dA
( - ) C CT T
(5.32)
Integrating from inlet to outlet,
ln1e 2e
1i 2i 1 2
- 1 1T T= -UA( + )
- C CT T (5.33)
From equations (5.22) and (5.23),
1i 1e
1
1 -T T=
QC � (5.34)
2e 2i
2
1 -T T=
QC � (5.35)
Taking equations (5.34) and (5.35) into (5.33),
( )ln
1e 2e 1i 2i
1e 2e
1i 2i
( - )- ( - )T T T TQ =UA
( - )T T( )
-T T
� (5.36)
Heat Exchangers 10
Comparison of equation (5.36) with equation (5.24) provides the logarithmic mean overall
temperature difference, LMTD, which applies for the double tube parallel flow heat exchanger:
( )
ln
ln
1e 2e 1i 2im
1e 2e
1i 2i
( - ) - ( - )T T T T= =T T
( - )T T( )
-T T
∆ ∆ (5.37)
It can be shown that the LMTD also applies for the double tube counter flow heat
exchanger (Figure 5.5b). Referring to Figure 5.5, the LMTD can be written in a form that applies for
both counter and parallel flow configurations (Kreith, 1973).
ln
ln
a bm
a
b
-T T= =T T
T
T
∆ ∆∆ ∆
∆ ∆
(5.38)
Other configurations lead to overall temperature differences that are different from the
LMTD. For those, a correction factor, F, is employed:
lnmQ =UA = UAFT T∆ ∆� (5.39)
Figure 5.6 depicts two examples of charts for the correction factor. However, for simulation
purposes, it is more convenient to have F in terms of an equation. This has been provided by Roetzel
and Nicole (1975), as follows:
( )sin tank -1i=1 k=1
m n m mikF = (1- 2i)a r R Σ Σ (5.40)
where,
1i 1em
2e 2i
-T T=R
-T T (5.41)
and
lnm
1i 2i
T=r
-T T
∆ (5.42)
Values for coefficient aik, for different heat exchanger configurations, are provided by
Roetzel and Nicole (1975).
TABLE FOR aik VALUES.
Heat Exchangers 11
Figure 5.6 - Correction factor for heat exchanger: (a) with one shell pass and multiple of 2 tube
passes; (b) with two shell passes and multiple of 4 tube passes (Welty et al., 1969).
For moderate temperature differences (Eastop and McConkey, 1978), an approximation to
the LMTD can be used. It employs the arithmetic mean temperature difference.
( ) ( )1i 1e 2i 2ea bm a
+ ++ T T T TT T= = -T T
2 2 2
∆ ∆∆ ≈ ∆ (5.43)
5.5 THE EFFECTIVENESS METHOD
The system of equations (5.22), (5.23) and (5.39) describe the heat exchanger behavior. For
simulation purposes, when the heat transfer rate and exit temperatures are the unknown values, the
non-linearity of equation (5.39) requires a numerical method to have the system solved. Although
this may not be a major problem, computing time and handling of equations may increase
dramatically, once more complex refrigeration systems are studied. An alternative for the corrected
logarithmic mean temperature difference equation is provided by the effectiveness-NTU method.
The idea is to "remove" all unknown variables from the non-linearity of the heat transfer rate
equation.
Consider the effectiveness of a heat exchanger, defined as the ratio between the actual heat
transfer rate and the maximum heat transfer rate that can be obtained for that given inlet
temperatures and thermal capacity rates.
max
Q=
Qε
�
� (5.44)
Figure 5.7 shows two possibilities for the maximum heat transfer rate to occur. Taking the
heat transfer area to infinity, either the hot fluid exit temperature becomes equal to the inlet
temperature of the cold fluid or the cold fluid exit temperature becomes equal to the hot fluid inlet
temperature. Referring to equations (5.22) and (5.23), and due to energy conservation (the heat
released by 1 is equal to the heat received by 2), the maximum temperature difference must occur
with the fluid with the smaller capacity rate. Therefore,
Heat Exchangers 12
maxminmax=Q C T∆� (5.45)
min min 1 2= ( , )C C C (5.46)
max 1i 2i-T T T∆ = (5.47)
Figure 5.7 - Maximum heat transfer rate.
From the definition of the effectiveness,
min 1i 2iQ = ( - )C T Tε� (5.48)
Equation (5.48) substitutes the heat transfer equation (5.39), which employs the corrected
logarithmic mean temperature. From the definition of the effectiveness, equation (5.44), and solving
for the first possible case, where C2=Cmin, one has:
max
2e 2i 2e 2i 22
1i 2i 1i 2i2
( - ) ( - )C T T T T T= = =
( - ) ( - )C T T T T Tε
∆
∆ (5.49)
From the energy balance equations, (5.22) and (5.23),
1i 1e 2e 2i1 2( - )= ( - )C CT T T T (5.50)
As an example, from equation (5.33), it is possible to develop an expression for the
effectiveness of a double tube parallel flow heat exchanger.
exp1e 2e
1i 2i 1 2
- 1 1T T= -UA( + )
- C CT T
(5.51)
Taking equations (5.50) and (5.51) into (5.49),
Heat Exchangers 13
exp 2
2 1
2
1
UA C1- - (1+ )
C C=
C1+
C
ε
(5.52)
Equation (5.52) gives the effectiveness of a double tube parallel flow heat exchanger, for
the case of the cold fluid having the minimum capacity rate. Analogously, for the minimum capacity
rate in the hot stream,
exp 1
1 2
1
2
UA C1- - (1+ )
C C=
C1+
C
ε
(5.53)
Equations (5.52) and (5.53) can be combined into a single one.
( )exp *
*
1- -NTU (1+ )C=
1+Cε
(5.54)
where
min
max
* C=C
C (5.55)
and NTU is the number of transfer units, defined as:
min
UANTU =
C (5.56)
Table 5.3 summarizes the expression for the effectiveness of heat exchanger in various
configurations. Note that the effectiveness does not depend on any of the fluids' temperatures (inlet
or exit). It is only a function of NTU and C*.
*= f(NTU, )Cε (5.57)
In all cases, when C* is equal to zero, the effectiveness reduces to:
exp= 1- (-NTU)ε (5.58)
Equation (5.58) applies for heat exchangers when one of the fluids is undergoing a
phase change.
Heat Exchangers 14
Table 5.3 - Effectiveness of heat exchangers (Holman, 1981).
5.6 SIMULATION OF HEAT EXCHANGERS
Trocadores de calor constituem uma parte vital do ciclo de refrigeração. O condensador e o
evaporador são os elementos de contato entre o fluido refrigerante e o meio externo ao ciclo
(fonte fria ou o meio ambiente). Trocadores de calor menos eficientes, impondo maiores
diferenças de temperatura entre fluido externo e refrigerante, resultarão em um maior
distanciamento entre as temperaturas (e pressões) de condensação e evaporação, com o
conseqüente aumento no consumo do compressor e queda no COP do ciclo. Para uma dada
geometria e condições de operação, a simulação isolada de um trocador de calor permitirá uma
previsão de seu desempenho térmico. Quando parte de um sistema de refrigeração, uma
modelagem adequada do condensador e evaporador levará à determinação das temperaturas de
condensação e evaporação, fatores determinantes no desempenho do ciclo, porém desconhecidos
a priori em uma simulação. Os métodos de análise de condensadores e evaporadores dividem-se
em três categorias básicas (Parise, 2004; Braun, 2004), a saber: (i) método de parâmetros
concentrados, (ii) método de multizona ou de fronteira móvel e (iii) método de análise local ou de
volumes finitos. Apresentam-se, a seguir, descrições sumárias destes métodos.
The basic simulation of a heat exchanger can be summarized in four equations, as follows:
1) Energy balance in the hot fluid: equation (5.22);
2) Energy balance in the cold fluid: equation (5.23);
3) Heat transfer rate equation: equation (5.39), overall temperature difference method, or equation
(5.48) , effectiveness-NTU method;
4) Overall temperature difference equation, (5.38), or the effectiveness equation, (5.57).
Heat Exchangers 15
For the reasons already explained, the performance prediction (exit temperatures and rate
of heat transfer as unknowns) of a heat exchanger is better served by the effectiveness-NTU method.
On the other hand, if the area of the heat exchanger, for a prescribed performance, is required, then
the overall temperature difference method may be also appropriate.
Consider the modeling of the steady-state performance of a heat exchanger, with no phase-
change in the fluid streams. Input parameters are: inlet temperatures of both fluids, T1i and T2i,
thermal capacity rates, C1and C2 and the thermal conductance of the heat exchanger, UA. Solution
for the system of equations (5.22), (5.23), (5.48) and (5.57) is straightforward:
Equations (5.57) and (5.48) provide, in that order, the effectiveness of the heat exchanger
and the heat transfer rate. Then, from equations (5.22) and (5.23):
1e 1i
1
Q= -T T
C
�
(5.59)
and
2 21 e i
2
Q= +T T
C
�
(5.60)
When one of the fluids undergoes change of phase, a different set of equations result. At
this stage, only "pure" condensers and evaporators are considered. The simulation of heat
exchangers with combined latent and sensible heat, typical of real condensers (desuperheating,
condensation and subcooling) and evaporators (evaporation and superheating), will be dealt with
further on the text. For the condenser of Figure 3a, the energy balance in the refrigerant is:
1 lv 1i 1eQ = ( - )m h x x�� (5.61)
which provides the exit vapor quality of the refrigerant:
1e 1i
lv
Q= -x x
mh
�
� (5.62)
Similarly, for the evaporator:
2 lv 2e 2iQ = ( - )m h x x�� (5.63)
and
2e 2i
lv
Q= +x x
mh
�
� (5.64)
For the condenser and evaporator, the heat transfer equation are, respectively:
Heat Exchangers 16
( )2 2 1 2p i iQ = m c T Tε −� � (5.65)
( )1 1 1 2p i iQ = m c T Tε −� � (5.66)
The effectiveness equation is, irrespective of the heat exchanger arrangement:
1 exp( )NTUε = − − (5.67)
4.1 Modelos de parâmetros concentrados
Neste modelo, a teoria básica de trocadores de calor é empregada. Utiliza-se, como
característica do trocador de calor, sua condutância global, ( )UA [kW/oC].
Condensadores. Para o condensador a ar, esquematizado na Fig.12, têm-se as equações de
conservação de energia, aplicadas aos volumes de controle do refrigerante, Eq.(5.68), e do ar,
Eq.(5.69). O método da efetividade, Eq.(5.70), é preferido em relação ao da diferença média de
temperaturas, pois resulta, para a simulação de desempenho térmico, conhecidas a geometria e as
condições de entrada do refrigerante e do ar, em um sistema de equações algébricas lineares
facilmente resolvível (situação adequada para a solução de um sistema completo de refrigeração).
( )2 3cd rQ m h h= −� � (5.68)
( )cd a pa ae aiQ m c T T= −� � (5.69)
( )cd cd a pa cd aiQ m c T Tε= −� � (5.70)
onde cdQ� é a taxa de transferência de calor no condensador [kW], rm� e am� são as vazões
mássicas do refrigerante e do ar [kg/s], respectivamente, 2h , a entalpia específica [kJ/kg] do
refrigerante superaquecido que entra no condensador e 3h , a do líquido, saturado ou sub-
resfriado, de saída, aeT e aiT , as temperaturas de entrada e saída do ar [oC] e cdT , a temperatura de
condensação. A efetividade do condensador, cdε , é calculada supondo-se todo o trocador de
calor, no lado do refrigerante, tomado por mudança de fase, isto é, desprezando-se a região de
dessuperaquecimento.
( )1 exp cd
cd
a pa
UA
m cε
= − −
� (5.71)
Heat Exchangers 17
Figura 12 – Esquema do condensador a ar (método dos parâmetros concentrados).
Evaporadores. O balanço de energia no refrigerante, Fig. 13, é dado por:
( )1 4ev rQ m h h= −� � (5.72)
onde evQ� é a taxa de transferência de calor no evaporador [kW], 1h e 4h , as entalpias específicas
de saída –vapor saturado ou superaquecido - e de entrada – mistura líquido/vapor,
respectivamente [kJ/kg]. Pode-se supor a superfície externa da serpentina como estando seca ou
úmida, com a condensação da umidade d ar. No primeiro caso, as equações de conservação de
energia para o lado do ar e da taxa de troca de calor são dadas por:
( )ev a pa ai aeQ m c T T= −� � (5.73)
( )ev ev a pa ai evQ m c T Tε= −� � (5.74)
onde evT é a temperatura de evaporação [
oC]. Da mesma forma que no condensador, a efetividade
do evaporador de serpentina seca é calculada desprezando-se o efeito da região de
superaquecimento do refrigerante.
( )1 exp ev
ev
a pa
UA
m cε
= − −
� (5.75)
No segundo caso, com a serpentina úmida, a equação de conservação de energia aplicada
ao fluxo de ar é escrita em função das entalpias específicas do ar úmido, à entrada e à saída do
evaporador, aih e aeh , respectivamente:
( )ev a ae aiQ m h h= −� � (5.76)
rm�
am�
CD
2h
aeT
3h
aiT
Heat Exchangers 18
Figura 13 – Esquema do evaporador a ar (método dos parâmetros concentrados).
A equação da taxa de transferência de calor é escrita supondo-se que a serpentina esteja
molhada em toda sua extensão (Braun, 2004):
*
,( )ev ev a ai s evQ m h hε= −� � (5.77)
onde ,s evh é a entalpia específica do ar úmido saturado a uma temperatura igual à temperatura de
evaporação do refrigerante e *
evε é a efetividade de troca de massa e de calor do evaporador, dada
em função da condutância média de transferência de massa e de calor da seção bifásica do
evaporador, ( )*
evUA [kg/s]:
( )*
* 1 exp evev
a
UA
mε
= − �
(5.78)
5.7 PERFORMANCE PREDICTION FROM EMPIRICAL DATA
This section deals with the situation when the simulation of an existing heat exchanger is
needed and experimental data from only one existing operating condition are available. Consider a
single-phase heat exchanger from which all fluid temperatures were measured. The conditions for
which these temperatures were taken are defined as standard conditions and are denoted by the
superscript *. If the configuration is known, then the heat transfer equation provides the heat
exchanger conductance for the standard condition.
** *
m1= (UA T)Q ∆� (5.79)
where,
ln
**m = (F T )T ∆∆ (5.80)
rm�
am�
EV 4h
h
aeh
1h
aih
Heat Exchangers 19
In this analysis, for simplicity, the refrigerant stream is denoted by subscript 1, not
necessarily meaning that the refrigerant is the hot fluid. The other fluid, air, water or brine, is
represented by subscript 2. Also, it is assumed that, in heat exchangers where superheated vapor or
subcooled liquid zones are present, the effect of the single-phase heat transfer and pressure drop is
neglected. From equation (5.9), the overall conductance of the heat exchanger is:
1
1 f1 k f2 2
1(UA =)
+ + + +R R R R R (5.81)
The thermal resistance on the refrigerant side depends on the refrigerant, its transport and
thermodynamic properties and the mass flow rate. Assuming that Rf1, Rf, Rf2 and R2 are fixed
values, i.e., they do not vary whatever refrigerant or operating conditions (refrigerant-side) are
applied, equation (5.81) can be written as:
*
1 *1
1(UA =)
+ RR Σ (5.82)
All thermal resistances that are supposed not to vary under any refrigerant condition are
grouped under the parameter ΣR, which can be calculated from equation (5.82).
*1*
1
1R = - R
(UA)Σ (5.83)
From equation (5.81), for any refrigerant condition (which even includes a different
refrigerant), one has:
1
1
1(UA =)
+ RR Σ (5.84)
Substituting (5.84) in (5.83),
1*
1 1 *
1
1(UA =)
1( - )+R R
(UA)
(5.85)
Equation (5.85) provides the required correction of the overall heat conductance for
refrigerant conditions other than the standard ones. A basic assumption was that ΣR is the same for
both standard and desired conditions. Typically, this implies that air or water mass flow rate is fixed.
Regarding 1sR and 2sR , it is reasonable to assume that fouling factors on both sides and the tube
thermal conductivity will not vary significantly with any changes in refrigerant or operating
conditions.
The overall heat conductance at standard conditions, ( )*
1UA , remains an input value for the
model. On the other hand, the refrigerant-side thermal resistance at standard conditions, *
1R , has to
be either calculated or estimated.
Heat Exchangers 20
*1 *
1
1=R
h (5.86)
REFERENCES
ASHRAE Fundamentals, 1993.
Eastop T.D. and McConkey A., Applied Thermodynamics for Engineering Technologists, third
edition, Longman, 1978.
Holman, J.P., Heat Transfer, McGraw-Hill, 1981.
Kreith, F., Principles of Heat Transfer, 3rd edition, International Educational Publishers, 1973.
Rosehnow W.M., Hartnett, J.P., Ganic, E.N., Handbook of Heat Transfer Applications, second
edition, McGraw-Hill Company, 1985.
Roetzel, W. and Nicole, F.J.L., Mean Temperature Difference for Heat Exchanger Design -A
General Approximate Explicit Equation, Transactions of ASME, Journal of Heat Transfer, pp 5-8,
February 1975.
Schneider, P.J., Conduction Heat Transfer, Addison-Wesley Pub. Company, 1974.
Shah, R.K., Single-Phase Process Heat Exchangers, Short Course, Rio de Janeiro, 1988.
Welty, J.R., Wilson,R.E. and Wicks, C.E., Fundamentals of Momentum, Heat and Mass Transfer,
Wiley International, 1969.
Braun, J.E., Air-cooled condenser and direct-expansion evaporator modeling, USNC/IIR Short
Course on “Simulation Tools for Vapor Compression Systems and Component Analysis”,
International Refrigeration Conference at Purdue, Purdue University, West Lafayette, EUA, July
10-11, 2004.
Stoecker, W.F., Saiz Jabardo, J.M., Refrigeração Industrial, Editora Edgard Blücher Ltda., São
Paulo, 1998.
ASHRAE, Fundamentals Handbook, (SI Edition), 2001.
Rohsenow, W.M., Hartnett, J.P., Cho, Y.I., Handbook of Heat Transfer, Third Edition, New
York, 1998.
Heat Exchangers 21
NOMENCLATURE
A area [m2]
ika coefficient in LMTD correction factor equation
C thermal capacity rate [kW/K] C� ??
C* thermal capacity rate ratio [-]
pc specific heat at constant pressure [kJ/kg K]
D tube diameter [m]
F LMTD correction factor
h heat transfer coefficient [kW/m2 K] DOUBLE SIGNIFICANCE
h specific enthalpy [kJ/kg]
hlv latent heat of vaporization [kJ/kg]
k thermal conductivity [kW/m K]
L tube length [m]
NTU number of transfer units [-]
Q� heat transfer rate [kW]
R thermal resistance [K/kW]
Rm temperature ratio in LMTD correction factor equation
rm temperature ratio in LMTD correction factor equation
T temperature [K]
U overall heat transfer coefficient [kW/m2 K]
U internal energy [kJ/kg]
x vapor quality [-]
Greek
aT∆ arithmetic mean overall temperature difference [K]
lnT∆ logarithmic mean overall temperature difference [K]
mT∆ overall temperature difference [K]
δ wall thickness [m]
ε heat exchanger effectiveness [-]
0η total surface effectiveness of an extended surface [-]
fη fin efficiency [-]
Subscripts a side a of heat exchanger
b side b of heat exchanger
e exit
f fin
i inner DOUBLE SIGNIFICANCE
i inlet
max maximum
min minimum
Heat Exchangers 22
o outer
p primary surface (or plain tube)
s fouling (or scale)
t total
w wall
1 hot fluid
2 cold fluid
PROBLEMS
1) Tem-se água, a 230 kg/h e 35oC, para resfriar óleo (cpo= 2.1 kJ/kg
oC) de uma temperatura de
90oC para 60
oC. O trocador de calor disponível é do tipo tubo duplo contra-corrente, com
diâmetros interno e externo e comprimento iguais a 9 cm, 13 cm e 5 m, respectivamente. A vazão
de óleo é de 4 L/min. Determine:
a) o coeficiente global de troca de calor, kW/m2 o
C;
b) a taxa de calor trocado, kW;
c) as temperaturas de saída do óleo e da água, oC.
Compare os valores obtidos em (b) e (c) utilizando uma diferença média aritmética, logarítmica
ou o método da efetividade.
2) Considere dois trocadores de calor de mesma efetividade, dispostos em contra-corrente.
a) Determine a efetividade do trocador de calor equivalente;
b) A partir do item acima determine a efetividade de um trocador de calor líquido-gás, do tipo
serpentina aletada, com duas fileiras de n tubos cada, de circuito simples em contra-corrente
global. Sugestão: considere cada tubo como sendo um trocador de calor independente. A
serpentina será, então, um arranjo de vários trocadores de calor, isto é, tubos.
3) Estabeleça as equações básicas (balanço de energia e equação de troca) para um evaporador-
condensador de um ciclo de compressão de vapor em cascata. Despreze as regiões de
dessuperaquecimento e sub-resfriamento.
4) Desenvolva o modelo matemático de um trocador de calor sucção-linha de líquido. Este
trocador é, geralmente do tipo tubo-duplo contra-corrente e promove o superaquecimento do
vapor na sucção do compressor a partir do subresfriamento do refrigerante saindo do
condensador. Este trocador de calor permite um superaquecimento do vapor de sucção com um
aumento do efeito refrigerante.
5) Considere um sistema indireto de transferência de calor ("run around coil") conforme a Figura
5.8P. Suponha conhecidos os seguintes parâmetros:
a) vazão mássica de ar nos dois dutos (fluxo frio e fluxo quente);
b) vazão mássica do fluido de acoplamento;
c) geometria dos dois trocadores de calor.
Desenvolva uma expressão para a efetividade equivalente do conjunto ("run around coil").
Heat Exchangers 23
Figura 5.8P – Sistema indireto de transferência de calor (“run-around-coil”).
6) Desenvolver planilha de cálculo para trocadores de calor de uma instalação operando de
acordo com as seguintes condições.
R-134a; Tev = 0
oC; Tcd = 40
oC; Tent,fluido = 20
oC; ∆Tsuperaq = 10
oC; ∆Tsub-resf = 15
oC;
1kg/minnominalm =�
Definir capacidade frigorífica
Trocador de calor a escolher:
- condensador a água tipo tubo e carcaça
- condensador a ar forçado
- evaporador a ar forçado
- evaporador a água
- condensador tubo e arame com convecção natural
Proposta para o desenvolvimento:
Dados de entrada:
- Condições de entrada do refrigerante;
- Temperatura e vazão do fluído (ar ou água);
- Geometria do Trocador de calor.
Planilha
Refrigerante
Fluido Geometria Desempenho
↓ Tentativas para o trocador de calor
7) Calcular a economia (%) de energia ao se instalar um recuperador de calor em um sistema
utilizando chuveiro elétrico, conforme Figura 5.9P. Uma bomba de calor elétrica (em substituição
ao chuveiro) seria mais eficiente?
Heat Exchangers 24
Figura 5.9P – Sistema de recuperação de calorr de rejeito da água de banho.
8) Projetar ou simular um dos trocadores de calor listados abaixo. Buscar as condições de
operação típicas na literatura técnica disponível.
Condensador a água (tubo-e-carcaça) para uma unidade condensadora.
Condensador a água (tubo-e-serpentina) para uma unidade condensadora.
Condensador a ar de uma unidade condensadora.
Condensador a ar por convecção natural (tubo-e-arame) de um refrigerador doméstico.
Resfriador de líquido (evaporador) tipo tubo-e-carcaça.
Evaporador de refrigeradores domésticos do tipo “roll-bond” (congelador).
Evaporador automotivo.