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.