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New thermal effectiveness data and formulae for some cross-flow arrangements of practical interest
International Journal of Heat and Mass Transfer 69 (2014) 237–246
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
New thermal effectiveness data and formulae for some cross-flow arrangements of practical interest Luben Cabezas-Gómez a,⇑, José Maria Saiz-Jabardo b, Hélio Aparecido Navarro c, Paulo Eduardo Lopes Barbieri d a Department of Thermal and Fluid Sciences, Universidade Federal de São João del-Rei, UFSJ, Campus Santo Antônio, Praça Frei Orlando, 170, Centro, CEP 36307-352 São João del-Rei, MG, Brazil b Escuela Politécnica Superior, Universidad de la Coruña, Mendizábal s/n Esteiro, 15403 Ferrol, Coruña, Spain c Mechanical Engineering Department, Escola de Engenharia de São Carlos, Universidade de São Paulo, Av. Trabalhador São-carlense, 400, Centro, CEP 13566-590, São Carlos, SP, Brazil d Centro Federal de Educação de Minas Gerais, Departamento Acadêmico de Engenharia Mecânica, Av. Amazonas, 7675, Laboratório de Refrigeração, Nova Gameleira, CEP 30510-000 Belo Horizonte, MG, Brazil
a r t i c l e
i n f o
Article history: Received 8 May 2013 Received in revised form 3 September 2013 Accepted 9 October 2013 Available online 5 November 2013 Keywords: Cross-flow heat exchangers Z-shape flow arrangements Multipass parallel–counter–cross-flow arrangements Thermal effectiveness Number of transfer units
a b s t r a c t This study provides new effectiveness data and closed form relations for some recently proposed crossflow heat exchanger flow arrangements of industrial and research interest. The study also covers the so called Z-shape flow arrangement of widespread use in the refrigeration and automobile industries. The closed form relations as well as the effectiveness data are presented in the standard effectiveness – number of transfer units (e-NTU) format. Slight deviations have been observed of the arithmetic mean effectiveness results, proposed by Pongsoi et al. (2011, 2012) [3,4], with respect to those from both closed form relations and data from the HETE program (Navarro and Cabezas-Gómez (2005) [9], Cabezas-Gómez et al. (2007) [10]) for several flow arrangements, with deviations increasing with NTU and C⁄. Data provided in the present paper, along with the closed form expressions, could be useful in enhancing the external heat transfer coefficient precision based on the procedure of determining the NTU value from the experimentally obtained thermal effectiveness of the heat exchanger. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The experimental study and analysis of the thermal performance of heat transfer surfaces used in heat exchangers is one of the most important objectives of researchers and engineers in the heat transfer field. A significant number of studies have been dedicated to the experimental determination of the heat transfer characteristics of different surfaces. The books by Kays and London [1] and Webb and Kim [2] report some of the results of these investigations. This paper aims at the development of closed form expressions for computing the thermal effectiveness of some cross-flow arrangements that have been recently investigated by Pongsoi et al. [3] and Pongsoi et al. [4–6]. These flow arrangements are
⇑ Corresponding author at: Department of Thermal and Fluid Sciences, Universidade Federal de São João del-Rei, UFSJ, Campus Santo Antônio, Praça Frei Orlando, 170, Centro, CEP 36307-352 São João del-Rei, MG, Brazil. Tel.: +55 31 91940649. E-mail addresses: [email protected], [email protected] (L. Cabezas-Gómez), [email protected] (J.M. Saiz-Jabardo), [email protected] (H.A. Navarro), [email protected] (P.E.L. Barbieri). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.10.022
typical of industrial applications, where sensible heating is a commonplace. One of such applications is waste heat recovering [3]. The so called Z-shape, one of the flow arrangements considered in the present paper, is extensively used in the refrigeration/air conditioning and the automobile industries. Pongsoi et al. [3–6] investigated mixed flow arrangements known as multi-pass parallel and counter cross-flow configurations. Those configurations, except the Z-shape one, present a combination of parallel–cross-flow and counter–cross-flow arrangements in sequence, and could be considered as new configurations. In their investigation, Pongsoi et al. [3–6], due to the unavailability of precise thermal effectiveness correlations for the flow arrangements under consideration, suggested the effectiveness evaluation through the mean arithmetic of the thermal effectiveness of a counter–cross-flow, ec,cf, and a parallel–cross-flow, ep,cf, configurations. This fairly new approach allowed the determination of both the Number of Transfer Units, NTU, from the experimentally measured thermal effectiveness, and, as a consequence, the external convective heat transfer coefficient related to the investigated heat transfer surfaces.
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Nomenclature A C C⁄ NTL Nr NTU P q U UA
exchanger outer total heat transfer area heat capacity rate heat capacity rate ratio, Cmin/Cmax, dimensionless number of tube lines number of tube rows number of transfer units, UA/Cmin heat exchanger effectiveness, related to e heat transfer rate Overall heat transfer coefficient heat exchanger conductance
Greek symbols d relative error e conventional heat exchanger thermal effectiveness, q/ qmax
As previously mentioned, in the present study, closed form (eNTU) relations are developed for the thermal effectiveness of the aforementioned flow arrangements. These theoretical correlations have been developed from the procedure described by Domingos [7], and applied by Shah and Pignotti [8] for similar flow arrangements (see, for example, case 5 of [8]). To the authors’ best knowledge, the proposed closed form expressions have not been previously published in the open literature. The main advantage of closed form expressions is that they allow the determination of the thermal effectiveness of the particular flow configuration preventing the use of either computational or approximate approaches. The proposed closed form relations have been evaluated through the results numerically obtained from the HETE program, Navarro and Cabezas-Gómez [9] and Cabezas-Gómez et al. [10], whose results accuracy has been demonstrated in several previous studies [9–12]. Thermal effectiveness deviations have been found between the results from the procedure suggested by Pongsoi et al. [3–6] and those from either the closed from correlations or HETE program. Though differences were generally low, under certain conditions and flow arrangements, Z-shape flow, for example, they can attain non negligible values. Thus, effectiveness values obtained from either the closed form relations or the HETE program could be very useful for experimental data reduction of new heat transfer surfaces. 2. Mean arithmetic thermal effectiveness computation Fig. 1 displays the two-dimensional scheme of the flow arrangements investigated by Pongsoi et al. [3–6]. These configurations were experimentally studied by Pongsoi et al. [3,4] aiming at the determination of the effect of fin pitches and material and the number of tube rows on the air side thermal performance for crimped spiral fin-and-tube heat exchangers. According to these authors, there are few investigations of this kind of heat exchangers, which find industrial applications as, for example, in waste heat recovering. The studied configurations also are characterized by a mixed internal flow circuitry known as multi-pass parallel and counter cross-flow configurations [3,4]. In fact, these are new configurations, except the first one, Fig. 1(a), and consist of a combination of parallel-cross-flow and counter–cross-flow arrangements in sequence. The arrangement of Fig. 1(a) could be considered as Z-shape flow one. Due to the unavailability of theoretical relations for calculating thermal effectiveness of the configurations of Fig. 1, Pongsoi et al. [3–6] suggested to evaluate them as the mean arithmetic of the
D
absolute error
Subscripts air external unmixed fluid, air c counter cf cross-flow m mean arithmetic value max maximum value min minimum value p parallel th theoretical or closed form TL tube lines
counter–cross-flow, ec,cf, and a parallel–cross-flow, ep,cf, heat exchangers thermal effectiveness, that is,
em ¼
ep;cf þ ec;cf 2
ð1Þ
Note that the average thermal effectiveness is designated as em whereas both ep,cf and ec,cf should be determined for the corresponding number of tube rows of the particular flow arrangement. The tube rows stand for the columns of tubes with respect to the incoming external unmixed fluid. Several theoretical expressions from Cabezas-Gómez et al. [10] and used by Pongsoi et al. [3–6] in their procedure are shown in Table 1. Thus, in order to compute the thermal effectiveness for some flow arrangements, Eq. (1) could be used along with the corresponding equation of Table 1. According to this procedure, the thermal effectiveness of the configuration of Fig. 1(a) is determined from Eq. (1) along with Eqs. (T1.1) and (T1.5). It is worth mentioning at this point that, for more than four rows, Pongsoi et al. [3–6] suggest that ep,cf, and ec,cf must be determined from Eqs. (T1.4) and (T1.8), which correspond to the pure parallel and counter-flow arrangements, respectively. However, Cabezas-Gómez et al. [10] have shown that even for a number of tube rows higher than 10 the use of either the pure parallel or pure counter-flow relations could lead to non-desired effectiveness errors for high values of NTU and C⁄. These errors could be avoided if effectiveness data were obtained either from straight application of the HETE program [10] or from the closed form expressions, to be introduced further on in this paper, with support from the HETE program to determine the effectiveness for five rows parallel and counter–cross flow arrangements. 3. Closed form relations for computing the thermal effectiveness Closed form expressions for the heat exchanger thermal effectiveness are developed in this section for the flow arrangements considered in the present study. To the best knowledge of the authors, these expressions have not been published in the open literature yet. In addition, they could be useful not only in thermal systems analysis but also in thermo-hydraulic evaluations of heat exchangers with similar flow arrangements. Finally these correlations, designated herein as theoretical, will be used to evaluate the precision of the thermal effectiveness obtained from both the mean arithmetic expression, Eq. (1), and the numerical one from the HETE program. Figs. 1 and 2 present the set of flow arrangements considered in the present study, with those of Fig. 2 being for two tube lines per
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External fluid
(a)
(b)
(c)
(d)
(e)
Fig. 1. Schematic representation of the flow arrangements from Pongsoi et al. [3–6]. ( and signs indicates that in-tube fluid flows out or into the paper, respectively).
row, NTL = 2. This approach (Fig. 2 arrangements) allows one to develop the correlations in a coherent form without the use of unnecessary complex geometries. In fact, the correlations for more than two tube lines are an extension of those for just two tubes per row. Note that the configurations of Fig. 2 are for tubes in line since the arrangement of tubes does not affect the thermal effectiveness. The employed procedure for obtaining the closed form solutions is based on the following commonly adopted underlying assumptions: (i) the heat exchanger operates in a steady state, (ii) negligible heat transfer with the surroundings, (iii) no heat sources either in the fluids or in the heat exchanger walls, (iv) the in-tube fluid is considered mixed in each tube cross-section, being the bulk temperature representative, depending only of the position along the fluid path, (v) transport properties of the fluids and heat transfer coefficients are assumed constant for the heat exchanger, (vi) no phase change occur in both fluid streams, (vii) there is no conduction in the fluids and in the heat exchanger walls, in the flow direction, and (viii) the fluids are well distributed in the headers, mainly the external one. The proposed closed forms are based on case 5 of the paper by Shah and Pignoti [8], who used a previously developed procedure by Domingos [7] and Pignoti and Cordero [13]. In their paper, Shah and Pignoti [8] developed thermal effectiveness expressions, in PNTU format, for seven flow arrangements from a heat exchanger with six tube rows with respect to the external fluid and ten tubes per row. Each arrangement would correspond to different tube connections. The flow arrangement of Fig. 2(a), corresponding to the one in Fig. 1(a), known as Z-shape configuration, will be used as a reference for the present development. The flow arrangements of Fig. 3 cover all possible configurations for two rows and two tubes per row. They correspond to the cases (a)–(d) of Fig. 2, which also include flow arrangements for heat exchangers with three to five rows and two tubes per row. A particle of the internal fluid for the Z-shape configuration, Figs. 2(a) and 3(a), passes four times between the inlet and the outlet of the heat exchanger. According to the nomenclature given by Shah and Pignoti [8], an internal fluid particle performs respectively two passes in an over-and-under fashion (serial coupled passes). This is equivalent to say that in each tube line, the in-tube fluid flows in counter flow with respect to the external fluid as it performs two passes. As the in-tube particle passes from the first line (over) to the second (under) one it follows the so called side-by-side pass, i.e., a parallel-coupled pass with respect to the external fluid, Fig. 2(a), Fig. 3(a) and (b). Different streams of the external fluid
flow over this pass [8] which divides the heat exchanger into two smaller heat exchangers each one encompassing a tube line and its corresponding external non-mixed stream. It should be noted that in all the configurations of Fig. 3 there is just one in-tube fluid stream. The fluid in each cross section of this stream is assumed as being well mixed and as such being at a given temperature (the so called average temperature) which varies along the path of the in-tube fluid. On the other hand, the external fluid configuration corresponds to the typical unmixed one, characterized by several streams which in Fig. 3 are represented by discontinuous lines. Each of these streams is characterized by its own temperature, which varies along the heat exchanger, as the stream flows from the inlet to the outlet sections of the heat exchanger. All the streams mix together at the exit of the heat exchanger with the resulting average temperature being considered as the external fluid exit temperature. The configuration of Fig. 3(a) can be represented by the equivalent two-dimensional diagram shown in Fig. 3(b), which can be reduced to the equivalent assembly of two identical exchangers of Fig. 3(c), designated as exchangers C. These exchangers are of the same kind namely counter–cross-flow two-passes with one tube row per pass and one tube per row. According to case 5 from Shah and Pignotti [8] and the matrix formalism of Section 5 from Domingos [7], the following closed-form expression can be written for the overall thermal effectiveness of this Z-shape heat exchanger:
eth
1 ¼ C
(
C ec;cf ðC m ; NTUÞ 1 1 N TL
NTL ) ð2Þ
where
C m ¼
C NTL
ð3Þ
NTL stands for the number of tube lines in each row, 2 in the case of Fig. 3(a), and ec;cf ðC m ; NTUÞ is the thermal effectiveness from Eq. (T1.5) in Table 1, where the capacity rate ratio is given by Eq. (3). Note that the correlations of Table 1 are written for the fluid B, the unmixed one (external in the present case), as being the one with the minimum heat capacity, Cmin. The closed form expressions presented in the text, such as Eq. (2), depend on which of the fluids is the one with Cmin. Equation (2) can also be applied to the configuration of Fig. 1(a), the Z-shape counter–cross-flow configuration, though in this case the number of lines, NTL, would equal to 9. For
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Table 1 Theoretical relationships for ep,cf and ec,cf.? Nr
Side of Cmin
Multi-pass parallel cross-flow, ep,cf 2 B
3
B
4
B
5
A or B
Multi-pass counter cross-flow, ec,cf 2 B
3
Theoretical relationship
1 K ð1 e2KC B Þ; K ¼ 1 eNTUB =2 1 CB 2 " # 2 1 K K K ¼ 1 1 e3KC B K 1 þ KC B 1 eKC B ; CB 2 4 2
eB
" ! # 3 1 K K K2 K K K 1 þ 2KC B 1 e2KC B 1 e4KC B ; 1 þ K 1 1 4 CB 2 2 2 2 2
eB ¼
1 eNTUB ð1þCB Þ 1 þ C B
eB ¼
" 1 # 1 K K 2KC e B þ 1 ; 1 CB 2 2
eB ¼
1 C
(T1.6)
K ¼ 1 eNTU B =3
2 ! !1 3 3 1 4 K K K2 K K K 5; 1 2KC B 1 e2KC B þ 1 e4KC B 1 þ þK 1 1 4 CB 2 2 2 2 2
(T1.7)
K ¼ 1 eNTUB =4
1 eNTUB ð1C B Þ 1 C B eNTUB ð1C B Þ
(T1.8)
eA ¼ eB =C A ; NTU A ¼ NTUB =C A ; C A ¼ C A =C B .
the case of Fig 2(b), the Z-shape parallel–cross-flow arrangement, Eq. (2) also applies though ec;cf ðC m ; NTUÞ must be substituted by ep;cf ðC m ; NTUÞ, and computed as in Eq. (T1.1) of Table 1, with the number of lines per row being equal to 2. The flow arrangements of Fig. 2(c) and (d) present combined configurations for the over and down flows of the in-tube fluid, with the former being counter flow and the latter parallel flow for the case of Fig. 2(c). The opposite occurs in the case of Fig. 2(d). According to the proposed procedure, for both cases, the overall thermal effectiveness of the heat exchanger is given by the following expression:
eth ¼
(T1.5)
K ¼ 1 eNTU B =2
eB ¼
(T1.3)
(T1.4)
2 !1 3 2 1 4 K K K 2 KC 3KC B 5; B 1 K e 1 1 e þ K 1 C B C B 2 4 2
Fluid A mixed (internal), Fluid B unmixed (external), C A ¼ 1=C B ;
(
K ¼ 1 eNTUB =4
B
A or B
(T1.2)
K ¼ 1 eNTU B =3
eB ¼
eB ¼
5
(T1.1)
eB ¼
B
4
Equation
N N C ec;cf C m ; NTU TL;c C ep;cf ðC m ; NTUÞ TL;p 1 1 1 NTL NTL
)
ð4Þ
changer with five rows. However, Eqs. (T1.4) and (T1.8), Table 1, are approximate expressions for five rows and more precise equations are recommended, as will be shown in the next section. As another example of application of the closed form expressions, let us consider the case of Fig. 1(c). This is a three rows heat exchanger with nine tube lines per row. From these lines, 5 are parallel and 4 counter flow. Thus, considering that ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ can be evaluated from Eqs. (T1.2) and (T1.6), respectively, the overall thermal effectiveness of the heat exchanger can be evaluated from the following expression:
eth
1 ¼ C
C ec;cf ðC m ; NTUÞ 1 1 9
4 5 ) C ep;cf ðC m ; NTUÞ 1 9
ð5Þ
where
C m ¼ The exponents, NTL,c and NTL,p, depend on the number of tube lines where the flow is counter-flow and parallel-flow respectively, which is equal to one for the cases of Fig. 2(c) and (d). The total number of tube lines per row is 2 and, as a result, the modified heat capacity, C m , must be determined based on NTL = 2. Equations (T1.5) and (T1.1) can be used for the evaluation of ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ. From Eq. (4), it can be concluded that the thermal effectiveness of the flow arrangements of Fig. 2(c) and (d) is the same since NTL is even and NTL,c and NTL,p are equal. Note that for the flow arrangements (c)–(j) of Fig. 2, NTL,c = 1 and NTL,p = 1. Expressions for ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ from Table 1 must be changed since the number of tube rows is different among the flow arrangements of Fig. 2. A similar procedure could be used for the rest of the cases of Fig. 2, Fig. 2(e)–(j), and Fig. 1, Fig. 1(c)–(e). As an example consider the case of Fig. 2(j) with 5 rows and two lines per row. It can be noted that the in-tube over flow is parallel whereas the under flow is counter-flow. Thus, Eq. (4) can be applied. Equation (4) is still valid for a more complex configuration such as the one of Fig. 1(e). It must be noted that in both cases, Fig. 2(j) and Fig. 1(e), ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ must be evaluated for a heat ex-
(
C 9
ð6Þ
Table 2 sums ups the closed form expressions for the overall thermal effectiveness of the flow arrangements of Figs. 1 and 2. Their development and use for other configurations is straightforward. Note that ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ should be computed from the adequate expressions of Table 1. Table 2 closed form expressions provide effectiveness results that are identical, up to the sixth decimal place after period, to the numerical results computed by the HETE code (see [9–12] for more details). Note that the HETE code performs iterations of the temperature field to a relative error of less than 1010 between two consecutive iterations of the external mean outlet temperature. 4. Thermal effectiveness data comparison The variation of the thermal effectiveness with NTU for the Zshape heat exchanger of Fig. 1(a) is shown in Fig. 4 for two capacity ratios: 0.25 and 1. The arrangement of Fig. 1(a) corresponds to that of Fig. 2(a) with 9 tube lines per row. It must be emphasized that the in-tube fluid flows in a counter-flow manner in all the lines
L. Cabezas-Gómez et al. / International Journal of Heat and Mass Transfer 69 (2014) 237–246
(a)
241
(b) in
in
out
(c)
out
(d) in
in
out
out
(e)
(f)
in
in
out
out
(g)
(h) in
in
out
out
(i) in
out
(j) in out
Fig. 2. Two dimensional schemes of the all studied flow arrangements considering only two tube lines (NTL = 2).
of the arrangement of Fig. 1(a). The results plotted in Fig. 4 correspond to those from the closed form equation, Eq. (2), designated as eth, with NLT = 9, which are overlaid with those from the mean arithmetic thermal effectiveness, em, Eq. (1). The value of em is obtained from Eqs. (T1.1) and (T1.5), from Table 1, whereas the closed form effectiveness value, eth, results from the straight application of Eq. (T1.5). At a first glance, the mean arithmetic thermal effectiveness seems to compare reasonably well with the closed form expression, especially at low NTU and capacity ratio values. In fact, for C⁄ = 0.25 and NTU = 2.5, the absolute error of the mean arithmetic effectiveness from the closed form effectiveness is of the order of 0.7%. However, for the same value of NTU and C⁄ = 1.0 the absolute error increases to 3.1%. Considering the experimental determination of the parameter NTU, it would be interesting to evaluate the error involved when NTU is determined from a given value of the thermal effectiveness, which could have been determined, for example, from experimental results. For that purpose consider the results from Table 3 for a heat exchanger like the one of Fig. 1(a) with C⁄ = 1.0. The first three columns correspond to the assumed NTU values (1.0 and 1.25) and the resulting closed form and mean arithmetic effectiveness, eth and em. The fourth column includes the absolute deviations of the closed form from the mean arithmetic effectiveness. Now, suppose that the closed form effectiveness value (0.4714 or 0.5142) were known (from, for example, an experimental study) and the NTU were to be determined using the mean arithmetic relation, Eq.
(1). The result is shown in the fifth column. Thus, instead of NTU = 1.0 or 2.0 (first column), the results of the fifth column are obtained, with absolute errors with respect to the values of the first column of the order of 4.2% and 9.6% (sixth column). The resulting NTU deviations are clearly higher than those obtained for the effectiveness. In the case of experimentally obtained UA values, NTU deviations could be significant since they could be compounded with measuring errors. It would be interesting at this point to investigate two trends: (i) the asymmetry of parallel–cross-flow and counter–cross-flow exchangers; and (ii) the effect of the number of tube lines per row. Parallel and counter cross flow Z-shape exchangers of two rows will be considered for that purpose, corresponding to Fig. 2(a) and (b), though for different number of tube lines per row. Closed form effectiveness correlations will be applied and compared with the result from the arithmetic mean one, Eq. (1), for the following case: NTU = 2.0; C⁄ = 0.5; and Cmin corresponding to the external fluid. The arithmetic mean effectiveness for the case study, evaluated from Eq. (1) and Eqs. (T1.1) and (T1.5) from Table 1, is equal to 0.69660. This value is independent of the number of tube lines per row, and results from the assumption that the internal fluid alternates between parallel and counter flow with respect to the external fluid. The thermal effectiveness from the closed form expressions for different number of tube lines are shown in Table 4. The relative errors of the arithmetic mean effectiveness with respect to the closed form expressions are also
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(a)
(b)
(c)
C
in
out
C
(d)
(e)
(f)
P
in
out
P
(g)
(h)
(i)
C
in
out
P
(j)
(k)
in
(l)
P
out
C Fig. 3. Two dimensional graphical schemes used to derive the closed form relations.
shown in this table. The following are conclusions that can be drawn from this table. (i) As expected, the counter–cross-flow exchanger effectiveness is higher than the parallel–cross-flow one. (ii) The counter–cross-flow exchanger effectiveness diminishes whereas the parallel–cross-flow one increases with the number of lines up to the limit of 100. (iii) Note that, for the counter–cross-flow arrangement, the relative errors of the arithmetic mean effectiveness, Eq. (1), with respect to the theoretical one (closed form) are relatively high for small number of tube lines per row, decreasing with the number of lines. Similar trend occurs for the parallel–
cross-flow heat exchanger up to 10 lines, though in this case, the relative errors are positive. From this number of lines on, the effectiveness of the parallel–cross-flow exchanger is higher than the arithmetic mean one, the difference increasing with the number of tube lines per row. (iv) Though the maximum number of lines in Table 4 is relatively high (100), the limit effectiveness is not achieved (see also Fig. 5). The infinity number of tube lines per row limit is valid for both Z-shape arrangements and corresponds to the pure mixed/unmixed cross flow heat exchanger (single row) [7], [8]. The pure cross flow heat exchanger effectiveness for mixed/unmixed fluids with unmixed fluid Cmin is given by the following expression:
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L. Cabezas-Gómez et al. / International Journal of Heat and Mass Transfer 69 (2014) 237–246 Table 2 Closed form relationships for eth for the flow arrangements of Figs. 1 and 2. Figure
Side of Cmin
1(a)
B
1(b), (d), (e)
B
1(c)
B
2(a)
B
2(b)
B
2(c)–(j)
B
Theoretical relationships
Fluid A mixed (internal), Fluid B unmixed (external), C A ¼ 1=C B ;
Equation
(
9 ) C ec;cf ðC B =9; NTU B Þ 1 1 B 9
(T2.1)
(
5 4 ) C ec;cf C B =9; NTU B C ep;cf ð9; NTU B Þ 1 1 B 1 B 9 9
(T2.2)
(
4 5 ) C ec;cf ðC B =9; NTU B Þ C ep;cf ð9; NTU B Þ 1 1 B 1 B 9 9
(T2.3)
(
2 ) C ec;cf C B =2; NTU B 1 1 B 2
(T2.4)
(
2 ) C ep;cf ðC B =2; NTU B Þ 1 1 B 2
(T2.5)
eth ¼
1 C B
eth ¼
1 C B
eth ¼
1 C B
eth ¼
1 C B
eth ¼
1 C B
eth ¼
C ep;cf C B =2; NTU B C ec;cf ðC B =2; NTU B Þ 1 1 B 1 1 B 2 C B 2
(T2.6)
eA ¼ eB =C A ; NTU A ¼ NTUB =C A ; C A ¼ C A =C B .
Table 4 Comparison between arithmetic mean (em = 0.69660) and closed form effectiveness values as a function of the number of tube lines for configurations of Fig. 2(a) and (b). (NTU = 2, C⁄ = 0.5 and Cmin = Cair).
Fig. 4. Effectiveness values for the Z-shape configuration from Pongsoi et al. [3], Fig. 1(a) or Fig. 2(a) with NTL = 9 (counter-cross-flow arrangement).
Table 3 NTU absolute errors when determined from the mean arithmetic effectiveness procedure.
a b
NTU
eth
em
De [%]a
NTUnew
DNTU [%]b
1.0 1.25
0.4714 0.5142
0.4639 0.5026
1.042 1.16
1.042 1.346
4.2 9.6
De = (eth em) 100 [%]. DNTU = (NTUnew NTU) 100 [%].
eTL!1 ¼
i 1h NTU 1 eC ð1e Þ C
ð7Þ
The value of eTL?1 for the case of Table 4 is equal to 0.702013 which is very close to the effectiveness from the closed forms for 100 tube lines per row. (v) The behavior suggested in (iv) can be justified as follows for the counter–cross flow configuration. For a single tube line, NTL = 1, the flow arrangement is a true two-pass (in fluid)
a
Number of tube lines
eth (counter– cross-flow)
de [%]a
eth (parallel– cross-flow)
de [%]a
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 50 100
0.72862 0.72007 0.71568 0.71210 0.71120 0.70991 0.70893 0.70817 0.70756 0.70706 0.70664 0.70629 0.70599 0.70572 0.70549 0.70529 0.70511 0.70495 0.70480 0.70313 0.70257
4.394 3.259 2.665 2.176 2.052 1.874 1.739 1.633 1.548 1.479 1.420 1.371 1.329 1.292 1.260 1.232 1.206 1.184 1.163 0.928 0.849
0.67265 0.68272 0.68766 0.69058 0.69252 0.69389 0.69492 0.69572 0.69635 0.69687 0.69730 0.69767 0.69798 0.69825 0.69849 0.69870 0.69888 0.69905 0.69920 0.70089 0.70145
3.561 2.034 1.301 0.872 0.590 0.391 0.242 0.127 0.036 0.038 0.100 0.153 0.197 0.236 0.270 0.300 0.326 0.350 0.371 0.612 0.691
de = ((em eth)/eth) 100 [%].
counter–cross-flow arrangement, whose thermal effectiveness is given by Eq. (T1.5) from Table 1. For two tube lines per row, NTL = 2, the flow arrangement becomes that shown in Fig. 2(a) and consists of two two-passes (in fluid) counter– cross-flow arrangement regarding the external fluid. In this case the in-tube fluid performs the first two passes in counter-flow regarding the air (external fluid) in the first tube line and then performs the second two passes in counterflow regarding the air in the second tube line, below the first one, Fig. 2 (a). Thus, the external fluid, with uniform inlet temperature, is divided into two unmixed streams, which will have different mean temperatures at the outlet of the heat exchanger, i.e., at the exit of each tube line. This process leads to an overall temperature distribution that is different from that for NTL = 1, and, as a result, to a decrement of the thermal effectiveness. As the number of tube lines increases the thermal effectiveness diminishes, though at a reduced
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Fig. 5. Thermal effectiveness for Z configuration from Fig. 2(a) and (b) as a function of NTL for NTU = 2, C⁄ = 0.5 and Cmin = Cair.
rate. The increment of the in-tube fluid passes in the vertical direction as the number of tube lines per row increases leads to a general mixture effect of the in-tube fluid. As a result, the thermal effectiveness of this flow arrangement as the number of tube lines increases tends to the value of the mixed/unmixed cross-flow arrangement heat exchanger with one tube row, given by Eq. (7). Similar arguments could be used for the case of parallel–cross flow arrangements. (vi) The overlaid plots of Fig. 5 summarize the aforementioned trends besides making clear the asymmetry between parallel and counter–cross-flow arrangements. (vii) The results from Table 4 indicate that, for small number of tube lines, the arithmetic mean value could produce nondesired errors for the thermal effectiveness of a Z-shape arrangement, especially for the counter–cross-flow arrangement (Fig. 2(a)). In order to further explore the closed form expressions, in what follows, the thermal effectiveness of flow arrangements of Fig. 1 will be evaluated and the results compared with those from the arithmetic mean expression. The first arrangement is the one of Fig. 1(b), which corresponds to the one of Fig. 2(c) with 9 tube lines per row, and is a combination of counter–cross and parallel–cross flow arrangements. The closed form evaluation requires the use of Eq. (4), with 5 counter–cross-flow and 4 parallel–cross-flow passes, with a total of 9 tube lines per row. Thus, Eq. (T2.2) along with Eqs. (T1.1) and (T1.5) must be applied. The value of em is determined from Eq. (1) along with Eqs. (T1.1) and (T1.5). The effectiveness variation with NTU for two values of the capacity ratio, C⁄ = 0.25 and C⁄ = 1.0, is shown in the plot of Fig. 6. Results from both closed form and arithmetic mean expressions are overlaid in the plot of Fig. 6. It can be noted that differences are more significant for the higher capacity ratio, whereas for C⁄ = 0.25 differences are too small to call. The maximum absolute error of em with respect to eth is of the order of 2.19%, for C⁄ = 1.0 and NTU = 2.5. It is interesting to note that the errors tend to diminish with the increment of the tube lines per row, the limit for both cases being that given by Eq. (7).
Fig. 6. Thermal effectiveness for Fig. 1(b) arrangement for NTU = 2, C⁄ = 0.5 and Cmin = Cair.
Closed form thermal effectiveness, eth, results from Table 5 have been determined for the flow arrangements of Fig. 2(f) and (g), corresponding to the arrangements of Fig. 1(c) and (d) with 9 tube lines per row, for NTU = 2.0 and C⁄ = 0.5, with the minimum capacity ratio being that of the external fluid. The table includes the relative error of the thermal effectiveness from the arithmetic mean expressions with respect to the closed form ones. Equations (T1.2) and T(1.6) were used in the determination of em for the arrangement of Fig. 2(f) (3 rows) whereas Eqs. (T1.3) and (T1.7) were the ones used for the arrangement of Fig. 2(g) (4 rows). The resulting effectiveness values were respectively equal to 0.700995 and 0.702203. According to the results from Table 5, (i) the closed form effectiveness varies in an oscillatory mode with the number of lines per row for both flow arrangements; and (ii) the relative errors of em with respect to eth are rather small and also vary in an oscillatory mode given that em is constant for both arrangements. The oscillatory mode of eth is the result of the alternate parallel and counter cross flows. In fact, considering the arrangement of Fig. 2(f), when one goes from 2 to 3 lines, a parallel pass is added and, as a result, thermal effectiveness must decrease. The opposite occurs from 3 to 4 lines since in this case a counter pass is added. Similar trends are noted for the arrangement of Fig. 2(g). It has been determined that the oscillating mode continues up to the order of 20 lines per row. Beyond this number of tube lines per row the closed form thermal effectiveness is approximately constant and equal to 0.702013 for both flow arrangements. It is interesting to note that this is the same limit value attained by the Z-shape arrangement, though for a much larger number of tube lines, which is the same that would be obtained from Eq. (7) for a mixed/unmixed cross flow heat exchanger with one tube row. The relative errors of em with respect to eth for this condition are 0.145% and +0.027%, respectively, for the arrangements of Fig. 2(f) and (g), though for a high number of tube lines per row. The final arrangement of Fig. 1, not considered so far, is the one of Fig. 1(e) with 5 rows of tubes. This particular arrangement corresponds to the arrangement of Fig. 2(i) but with 9 lines of tubes per row. A closed form expression as the ones proposed herein
L. Cabezas-Gómez et al. / International Journal of Heat and Mass Transfer 69 (2014) 237–246 Table 5 Comparison between arithmetic mean and the closed form effectiveness values as a function of the number of tube lines for configurations of Fig. 2(f) (three rows) and Fig. 2(g) (four rows). (NTU = 2, C⁄ = 0.5 and Cmin = Cair).
a
Number of tube lines
eth (Fig. 2(f))
de [%]*
eth (Fig. 2(g))
de [%]a
2 3 4 5 6 7 8 9 10 20 50 100
0.702152 0.694871 0.702048 0.699443 0.702028 0.700702 0.702021 0.701220 0.702018 0.702014 0.702013 0.702013
0.165 0.881 0.150 0.222 0.147 0.042 0.146 0.032 0.146 0.145 0.145 0.145
0.702507 0.709725 0.702137 0.704798 0.702068 0.703435 0.702044 0.702873 0.702033 0.702018 0.702014 0.702013
0.043 1.060 0.009 0.368 0.019 0.175 0.023 0.095 0.024 0.026 0.027 0.027
de = ((em eth)/eth) 100 [%].
Table 6 Comparison between arithmetic mean and HETE numerical simulated effectiveness values as a function of the number of tube lines for the configuration of Fig. 2(i) (five rows). (NTU = 2, C⁄ = 0.5 and Cmin = Cair).
a
Number of tube lines
e (simulated with HETE) (Fig. 2(i))
de [%]aem by Eqs. (1, T1.4, and T1.8)
de [%]aem by Eq. (1) and HETE values
2 3 4 5 6 7 8 9 10 20 50 100
0.702700 0.709960 0.702186 0.704882 0.702090 0.703478 0.702056 0.702899 0.702041 0.702020 0.702014 0.702013
0.190 0.834 0.264 0.120 0.277 0.080 0.282 0.162 0.284 0.287 0.288 0.288
0.027 0.996 0.100 0.283 0.114 0.084 0.119 0.001 0.121 0.124 0.125 0.125
de = ((em eth)/eth) 100 [%].
could be applied if exact thermal effectiveness correlations were available for the parallel and counter cross flow arrangements for 5 rows. The ones of Table 1, Eqs. (T1.4) and (T1.8), are approxima-
245
tions for an infinite number of rows. Thus, in order to estimate the error in evaluating the effectiveness of the arrangement of Fig. 1(e) (Fig 2(i) when NTL it is varied) through Eq. (1), the program HETE has been used as reference. Column 2 of Table 6 includes the variation thermal effectiveness of arrangement 2(i), determined with HETE, for a different number of tube lines per row, up to 100 lines, for NTU = 2.0 and C⁄ = 0.5, and Cmin being that of the external unmixed fluid. The arithmetic mean effectiveness, Eq. (1), determined from Eqs. (T1.4) and (T1.8) is equal to 0.704038. The relative error of this value with respect to the reference one provided by HETE is shown in the third column of Fig. 6. The fourth column of Table 6 includes the relative error of the arithmetic mean effectiveness with respect to that provided by HETE (second column). However, in this case, the values of parallel and counter cross flow thermal effectiveness were determined with HETE program for a five rows arrangement. The following values were obtained: ep,cf = 0.63483 and ec,cf = 0.770948. It can be noted that, generally, the relative errors of column 4 are lower than those of column 3 of Table 6, especially from the eighth tube line on. In addition, the oscillatory behavior of the thermal effectiveness from HETE is also observed up to line 10, as in the cases of Table 5. The limit thermal effectiveness for a 100 tube lines per row is 0.702013, which is equal to the value that would be obtained from Eq. (7), corresponding to a one row cross flow arrangement. This is also the limit value for the other flow arrangements considered herein, as should be expected. As a final application, charts of thermal effectiveness vs. the Number of Transfer Units, based on the closed form expressions applied to the Z configurations of Fig. 2(a) and (b), are displayed in Fig. 7(a) and (b) for different capacity ratios. Equations (T2.4) and (T1.5) have been used in the development of the chart of Fig. 7(a) whereas the one of Fig. 7(b) is based on Eqs. (T2.5) and (T1.1). 5. Conclusions Closed form expressions for the thermal effectiveness of several cross flow arrangements of practical interest have been proposed and their results evaluated, using as reference the simulation program HETE [9–12], and compared with results from the arithmetic mean procedure proposed by Pongsoi et al. [3–6]. The proposed
Fig. 7. Effectiveness-NTU charts, based on closed form expressions, with Cmin = Cair, for the Z configurations from: (a) Fig. 2(a); and (b) Fig. 2(b).
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closed form correlations fill a gap in the literature for the evaluation of the thermal effectiveness and other performance parameters of flow arrangements like the ones considered herein. They are especially important in experimental investigations that use e-NTU correlations in data reduction, as in the Pongsoi et al. [3– 6] studies. The following general conclusions have been drawn from the present investigation: – The arithmetic mean equation, Pongsoi et al. [3–6], for the thermal effectiveness of cross flow arrangements like the ones considered in this paper, though producing results that are close to those from the closed form correlations, do present deviations that increase with NTU. These deviations could contribute to increase the experimental uncertainty in data reduction. – Flow arrangements such as the ones of Fig. 1 present an asymmetric behavior regarding the thermal effectiveness. The thermal effectiveness of these configurations oscillates with the number of tube lines per row due to the alternation of parallel and counter flows. This trend is significant for small number of tube lines, the range depending on the flow arrangement. – The thermal effectiveness of all the flow arrangements considered in this study tends to that of Eq. (7) for a high number of tube lines. – The flow arrangement that presents the higher thermal effectiveness is the one of Fig. 2(a), corresponding to the counter– cross-flow configuration, whereas the lowest value is attained by the parallel–cross flow configuration, Fig. 2(b). The other arrangements of Fig. 2 present effectiveness values in the range between the aforementioned limits.
Acknowledgments The first author acknowledges the financial support received from CNPq (Conselho Nacional de Desenvolvimento Científico e
Tecnológico) through process 307141/2011-0 and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais) through process PPM-00639/11.
References [1] W.M. Kays, A.L. London, Compact Heat Exchangers, third ed., McGraw Hill, New York, 1998. [2] R.L. Webb, N.H. Kim, Principles of Enhanced Heat Transfer, second ed., Taylor & Francis Group LLC., London, 2005. [3] P. Pongsoi, S. Pikulkajorn, C.C. Wang, S. Wongwises, Effect of fin pitches on the air side performance of crimped spiral fin-and-tube heat exchangers with a multipass parallel and counter cross-flow configuration, Int. J. Heat Mass Transfer 54 (2011) 2234–2240. [4] P. Pongsoi, S. Pikulkajorn, C.C. Wang, S. Wongwises, Effect of number of tube rows on the air side performance of crimped spiral fin-and-tube heat exchangers with a multipass parallel and counter cross-flow configuration, Int. J. Heat Mass Transfer 55 (2012) 1403–1411. [5] P. Pongsoi, S. Pikulkajorn, S. Wongwises, Effect of fin pitches on the optimum heat transfer performance of crimped spiral fin-and-tube heat exchangers, Int. J. Heat Mass Transfer 55 (23–24) (2012) 6555–6566. [6] P. Pongsoi, S. Wongwises, Experimental study on the air-side performance of a multipass parallel and counter cross-flow L-footed spiral fin-and-tube heat exchanger, Heat Transfer Eng. 33 (15) (2012) 1251–1263. [7] J.D. Domingos, Analysis of complex assemblies of heat exchangers, Int. J. Heat Mass Transfer 12 (1969) 537–548. [8] R.K. Shah, A. Pignotti, Thermal analysis of complex cross-flow exchangers in terms of standard configurations, ASME J. Heat Transfer 115 (1993) 353–359. [9] H.A. Navarro, L. Cabezas-Gómez, A new approach for thermal performance calculation of cross-flow heat exchangers, Int. J. Heat Mass Transfer 48 (2005) 3880–3888. [10] L. Cabezas-Gómez, H.A. Navarro, J.M. Saiz-Jabardo, Thermal performance of multipass parallel- and counter–cross-flow heat exchangers, ASME J. Heat Transfer 129 (3) (2007) 282–290. [11] L. Cabezas-Gómez, H.A. Navarro, S.M. Godoy, A. Campo, J.M. Saiz-Jabardo, Thermal characterization of a cross-flow heat exchanger with a new flow arrangement, Int. J. Therm. Sci. 48 (2009) 2165–2170. [12] L. Cabezas-Gómez, H.A. Navarro, J.M. Saiz-Jabardo, S.M. Hanriot, C.B. Maia, Thermal characterization of a cross-flow heat exchanger with a new flow arrangement, Int. J. Therm. Sci. 55 (2012) 122–132. [13] A. Pignotti, G. Cordero, Mean temperature difference in multipass crossflow, ASME J. Heat Transfer 105 (1983) 584–591.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
New thermal effectiveness data and formulae for some cross-flow arrangements of practical interest Luben Cabezas-Gómez a,⇑, José Maria Saiz-Jabardo b, Hélio Aparecido Navarro c, Paulo Eduardo Lopes Barbieri d a Department of Thermal and Fluid Sciences, Universidade Federal de São João del-Rei, UFSJ, Campus Santo Antônio, Praça Frei Orlando, 170, Centro, CEP 36307-352 São João del-Rei, MG, Brazil b Escuela Politécnica Superior, Universidad de la Coruña, Mendizábal s/n Esteiro, 15403 Ferrol, Coruña, Spain c Mechanical Engineering Department, Escola de Engenharia de São Carlos, Universidade de São Paulo, Av. Trabalhador São-carlense, 400, Centro, CEP 13566-590, São Carlos, SP, Brazil d Centro Federal de Educação de Minas Gerais, Departamento Acadêmico de Engenharia Mecânica, Av. Amazonas, 7675, Laboratório de Refrigeração, Nova Gameleira, CEP 30510-000 Belo Horizonte, MG, Brazil
a r t i c l e
i n f o
Article history: Received 8 May 2013 Received in revised form 3 September 2013 Accepted 9 October 2013 Available online 5 November 2013 Keywords: Cross-flow heat exchangers Z-shape flow arrangements Multipass parallel–counter–cross-flow arrangements Thermal effectiveness Number of transfer units
a b s t r a c t This study provides new effectiveness data and closed form relations for some recently proposed crossflow heat exchanger flow arrangements of industrial and research interest. The study also covers the so called Z-shape flow arrangement of widespread use in the refrigeration and automobile industries. The closed form relations as well as the effectiveness data are presented in the standard effectiveness – number of transfer units (e-NTU) format. Slight deviations have been observed of the arithmetic mean effectiveness results, proposed by Pongsoi et al. (2011, 2012) [3,4], with respect to those from both closed form relations and data from the HETE program (Navarro and Cabezas-Gómez (2005) [9], Cabezas-Gómez et al. (2007) [10]) for several flow arrangements, with deviations increasing with NTU and C⁄. Data provided in the present paper, along with the closed form expressions, could be useful in enhancing the external heat transfer coefficient precision based on the procedure of determining the NTU value from the experimentally obtained thermal effectiveness of the heat exchanger. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The experimental study and analysis of the thermal performance of heat transfer surfaces used in heat exchangers is one of the most important objectives of researchers and engineers in the heat transfer field. A significant number of studies have been dedicated to the experimental determination of the heat transfer characteristics of different surfaces. The books by Kays and London [1] and Webb and Kim [2] report some of the results of these investigations. This paper aims at the development of closed form expressions for computing the thermal effectiveness of some cross-flow arrangements that have been recently investigated by Pongsoi et al. [3] and Pongsoi et al. [4–6]. These flow arrangements are
⇑ Corresponding author at: Department of Thermal and Fluid Sciences, Universidade Federal de São João del-Rei, UFSJ, Campus Santo Antônio, Praça Frei Orlando, 170, Centro, CEP 36307-352 São João del-Rei, MG, Brazil. Tel.: +55 31 91940649. E-mail addresses: [email protected], [email protected] (L. Cabezas-Gómez), [email protected] (J.M. Saiz-Jabardo), [email protected] (H.A. Navarro), [email protected] (P.E.L. Barbieri). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.10.022
typical of industrial applications, where sensible heating is a commonplace. One of such applications is waste heat recovering [3]. The so called Z-shape, one of the flow arrangements considered in the present paper, is extensively used in the refrigeration/air conditioning and the automobile industries. Pongsoi et al. [3–6] investigated mixed flow arrangements known as multi-pass parallel and counter cross-flow configurations. Those configurations, except the Z-shape one, present a combination of parallel–cross-flow and counter–cross-flow arrangements in sequence, and could be considered as new configurations. In their investigation, Pongsoi et al. [3–6], due to the unavailability of precise thermal effectiveness correlations for the flow arrangements under consideration, suggested the effectiveness evaluation through the mean arithmetic of the thermal effectiveness of a counter–cross-flow, ec,cf, and a parallel–cross-flow, ep,cf, configurations. This fairly new approach allowed the determination of both the Number of Transfer Units, NTU, from the experimentally measured thermal effectiveness, and, as a consequence, the external convective heat transfer coefficient related to the investigated heat transfer surfaces.
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Nomenclature A C C⁄ NTL Nr NTU P q U UA
exchanger outer total heat transfer area heat capacity rate heat capacity rate ratio, Cmin/Cmax, dimensionless number of tube lines number of tube rows number of transfer units, UA/Cmin heat exchanger effectiveness, related to e heat transfer rate Overall heat transfer coefficient heat exchanger conductance
Greek symbols d relative error e conventional heat exchanger thermal effectiveness, q/ qmax
As previously mentioned, in the present study, closed form (eNTU) relations are developed for the thermal effectiveness of the aforementioned flow arrangements. These theoretical correlations have been developed from the procedure described by Domingos [7], and applied by Shah and Pignotti [8] for similar flow arrangements (see, for example, case 5 of [8]). To the authors’ best knowledge, the proposed closed form expressions have not been previously published in the open literature. The main advantage of closed form expressions is that they allow the determination of the thermal effectiveness of the particular flow configuration preventing the use of either computational or approximate approaches. The proposed closed form relations have been evaluated through the results numerically obtained from the HETE program, Navarro and Cabezas-Gómez [9] and Cabezas-Gómez et al. [10], whose results accuracy has been demonstrated in several previous studies [9–12]. Thermal effectiveness deviations have been found between the results from the procedure suggested by Pongsoi et al. [3–6] and those from either the closed from correlations or HETE program. Though differences were generally low, under certain conditions and flow arrangements, Z-shape flow, for example, they can attain non negligible values. Thus, effectiveness values obtained from either the closed form relations or the HETE program could be very useful for experimental data reduction of new heat transfer surfaces. 2. Mean arithmetic thermal effectiveness computation Fig. 1 displays the two-dimensional scheme of the flow arrangements investigated by Pongsoi et al. [3–6]. These configurations were experimentally studied by Pongsoi et al. [3,4] aiming at the determination of the effect of fin pitches and material and the number of tube rows on the air side thermal performance for crimped spiral fin-and-tube heat exchangers. According to these authors, there are few investigations of this kind of heat exchangers, which find industrial applications as, for example, in waste heat recovering. The studied configurations also are characterized by a mixed internal flow circuitry known as multi-pass parallel and counter cross-flow configurations [3,4]. In fact, these are new configurations, except the first one, Fig. 1(a), and consist of a combination of parallel-cross-flow and counter–cross-flow arrangements in sequence. The arrangement of Fig. 1(a) could be considered as Z-shape flow one. Due to the unavailability of theoretical relations for calculating thermal effectiveness of the configurations of Fig. 1, Pongsoi et al. [3–6] suggested to evaluate them as the mean arithmetic of the
D
absolute error
Subscripts air external unmixed fluid, air c counter cf cross-flow m mean arithmetic value max maximum value min minimum value p parallel th theoretical or closed form TL tube lines
counter–cross-flow, ec,cf, and a parallel–cross-flow, ep,cf, heat exchangers thermal effectiveness, that is,
em ¼
ep;cf þ ec;cf 2
ð1Þ
Note that the average thermal effectiveness is designated as em whereas both ep,cf and ec,cf should be determined for the corresponding number of tube rows of the particular flow arrangement. The tube rows stand for the columns of tubes with respect to the incoming external unmixed fluid. Several theoretical expressions from Cabezas-Gómez et al. [10] and used by Pongsoi et al. [3–6] in their procedure are shown in Table 1. Thus, in order to compute the thermal effectiveness for some flow arrangements, Eq. (1) could be used along with the corresponding equation of Table 1. According to this procedure, the thermal effectiveness of the configuration of Fig. 1(a) is determined from Eq. (1) along with Eqs. (T1.1) and (T1.5). It is worth mentioning at this point that, for more than four rows, Pongsoi et al. [3–6] suggest that ep,cf, and ec,cf must be determined from Eqs. (T1.4) and (T1.8), which correspond to the pure parallel and counter-flow arrangements, respectively. However, Cabezas-Gómez et al. [10] have shown that even for a number of tube rows higher than 10 the use of either the pure parallel or pure counter-flow relations could lead to non-desired effectiveness errors for high values of NTU and C⁄. These errors could be avoided if effectiveness data were obtained either from straight application of the HETE program [10] or from the closed form expressions, to be introduced further on in this paper, with support from the HETE program to determine the effectiveness for five rows parallel and counter–cross flow arrangements. 3. Closed form relations for computing the thermal effectiveness Closed form expressions for the heat exchanger thermal effectiveness are developed in this section for the flow arrangements considered in the present study. To the best knowledge of the authors, these expressions have not been published in the open literature yet. In addition, they could be useful not only in thermal systems analysis but also in thermo-hydraulic evaluations of heat exchangers with similar flow arrangements. Finally these correlations, designated herein as theoretical, will be used to evaluate the precision of the thermal effectiveness obtained from both the mean arithmetic expression, Eq. (1), and the numerical one from the HETE program. Figs. 1 and 2 present the set of flow arrangements considered in the present study, with those of Fig. 2 being for two tube lines per
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External fluid
(a)
(b)
(c)
(d)
(e)
Fig. 1. Schematic representation of the flow arrangements from Pongsoi et al. [3–6]. ( and signs indicates that in-tube fluid flows out or into the paper, respectively).
row, NTL = 2. This approach (Fig. 2 arrangements) allows one to develop the correlations in a coherent form without the use of unnecessary complex geometries. In fact, the correlations for more than two tube lines are an extension of those for just two tubes per row. Note that the configurations of Fig. 2 are for tubes in line since the arrangement of tubes does not affect the thermal effectiveness. The employed procedure for obtaining the closed form solutions is based on the following commonly adopted underlying assumptions: (i) the heat exchanger operates in a steady state, (ii) negligible heat transfer with the surroundings, (iii) no heat sources either in the fluids or in the heat exchanger walls, (iv) the in-tube fluid is considered mixed in each tube cross-section, being the bulk temperature representative, depending only of the position along the fluid path, (v) transport properties of the fluids and heat transfer coefficients are assumed constant for the heat exchanger, (vi) no phase change occur in both fluid streams, (vii) there is no conduction in the fluids and in the heat exchanger walls, in the flow direction, and (viii) the fluids are well distributed in the headers, mainly the external one. The proposed closed forms are based on case 5 of the paper by Shah and Pignoti [8], who used a previously developed procedure by Domingos [7] and Pignoti and Cordero [13]. In their paper, Shah and Pignoti [8] developed thermal effectiveness expressions, in PNTU format, for seven flow arrangements from a heat exchanger with six tube rows with respect to the external fluid and ten tubes per row. Each arrangement would correspond to different tube connections. The flow arrangement of Fig. 2(a), corresponding to the one in Fig. 1(a), known as Z-shape configuration, will be used as a reference for the present development. The flow arrangements of Fig. 3 cover all possible configurations for two rows and two tubes per row. They correspond to the cases (a)–(d) of Fig. 2, which also include flow arrangements for heat exchangers with three to five rows and two tubes per row. A particle of the internal fluid for the Z-shape configuration, Figs. 2(a) and 3(a), passes four times between the inlet and the outlet of the heat exchanger. According to the nomenclature given by Shah and Pignoti [8], an internal fluid particle performs respectively two passes in an over-and-under fashion (serial coupled passes). This is equivalent to say that in each tube line, the in-tube fluid flows in counter flow with respect to the external fluid as it performs two passes. As the in-tube particle passes from the first line (over) to the second (under) one it follows the so called side-by-side pass, i.e., a parallel-coupled pass with respect to the external fluid, Fig. 2(a), Fig. 3(a) and (b). Different streams of the external fluid
flow over this pass [8] which divides the heat exchanger into two smaller heat exchangers each one encompassing a tube line and its corresponding external non-mixed stream. It should be noted that in all the configurations of Fig. 3 there is just one in-tube fluid stream. The fluid in each cross section of this stream is assumed as being well mixed and as such being at a given temperature (the so called average temperature) which varies along the path of the in-tube fluid. On the other hand, the external fluid configuration corresponds to the typical unmixed one, characterized by several streams which in Fig. 3 are represented by discontinuous lines. Each of these streams is characterized by its own temperature, which varies along the heat exchanger, as the stream flows from the inlet to the outlet sections of the heat exchanger. All the streams mix together at the exit of the heat exchanger with the resulting average temperature being considered as the external fluid exit temperature. The configuration of Fig. 3(a) can be represented by the equivalent two-dimensional diagram shown in Fig. 3(b), which can be reduced to the equivalent assembly of two identical exchangers of Fig. 3(c), designated as exchangers C. These exchangers are of the same kind namely counter–cross-flow two-passes with one tube row per pass and one tube per row. According to case 5 from Shah and Pignotti [8] and the matrix formalism of Section 5 from Domingos [7], the following closed-form expression can be written for the overall thermal effectiveness of this Z-shape heat exchanger:
eth
1 ¼ C
(
C ec;cf ðC m ; NTUÞ 1 1 N TL
NTL ) ð2Þ
where
C m ¼
C NTL
ð3Þ
NTL stands for the number of tube lines in each row, 2 in the case of Fig. 3(a), and ec;cf ðC m ; NTUÞ is the thermal effectiveness from Eq. (T1.5) in Table 1, where the capacity rate ratio is given by Eq. (3). Note that the correlations of Table 1 are written for the fluid B, the unmixed one (external in the present case), as being the one with the minimum heat capacity, Cmin. The closed form expressions presented in the text, such as Eq. (2), depend on which of the fluids is the one with Cmin. Equation (2) can also be applied to the configuration of Fig. 1(a), the Z-shape counter–cross-flow configuration, though in this case the number of lines, NTL, would equal to 9. For
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Table 1 Theoretical relationships for ep,cf and ec,cf.? Nr
Side of Cmin
Multi-pass parallel cross-flow, ep,cf 2 B
3
B
4
B
5
A or B
Multi-pass counter cross-flow, ec,cf 2 B
3
Theoretical relationship
1 K ð1 e2KC B Þ; K ¼ 1 eNTUB =2 1 CB 2 " # 2 1 K K K ¼ 1 1 e3KC B K 1 þ KC B 1 eKC B ; CB 2 4 2
eB
" ! # 3 1 K K K2 K K K 1 þ 2KC B 1 e2KC B 1 e4KC B ; 1 þ K 1 1 4 CB 2 2 2 2 2
eB ¼
1 eNTUB ð1þCB Þ 1 þ C B
eB ¼
" 1 # 1 K K 2KC e B þ 1 ; 1 CB 2 2
eB ¼
1 C
(T1.6)
K ¼ 1 eNTU B =3
2 ! !1 3 3 1 4 K K K2 K K K 5; 1 2KC B 1 e2KC B þ 1 e4KC B 1 þ þK 1 1 4 CB 2 2 2 2 2
(T1.7)
K ¼ 1 eNTUB =4
1 eNTUB ð1C B Þ 1 C B eNTUB ð1C B Þ
(T1.8)
eA ¼ eB =C A ; NTU A ¼ NTUB =C A ; C A ¼ C A =C B .
the case of Fig 2(b), the Z-shape parallel–cross-flow arrangement, Eq. (2) also applies though ec;cf ðC m ; NTUÞ must be substituted by ep;cf ðC m ; NTUÞ, and computed as in Eq. (T1.1) of Table 1, with the number of lines per row being equal to 2. The flow arrangements of Fig. 2(c) and (d) present combined configurations for the over and down flows of the in-tube fluid, with the former being counter flow and the latter parallel flow for the case of Fig. 2(c). The opposite occurs in the case of Fig. 2(d). According to the proposed procedure, for both cases, the overall thermal effectiveness of the heat exchanger is given by the following expression:
eth ¼
(T1.5)
K ¼ 1 eNTU B =2
eB ¼
(T1.3)
(T1.4)
2 !1 3 2 1 4 K K K 2 KC 3KC B 5; B 1 K e 1 1 e þ K 1 C B C B 2 4 2
Fluid A mixed (internal), Fluid B unmixed (external), C A ¼ 1=C B ;
(
K ¼ 1 eNTUB =4
B
A or B
(T1.2)
K ¼ 1 eNTU B =3
eB ¼
eB ¼
5
(T1.1)
eB ¼
B
4
Equation
N N C ec;cf C m ; NTU TL;c C ep;cf ðC m ; NTUÞ TL;p 1 1 1 NTL NTL
)
ð4Þ
changer with five rows. However, Eqs. (T1.4) and (T1.8), Table 1, are approximate expressions for five rows and more precise equations are recommended, as will be shown in the next section. As another example of application of the closed form expressions, let us consider the case of Fig. 1(c). This is a three rows heat exchanger with nine tube lines per row. From these lines, 5 are parallel and 4 counter flow. Thus, considering that ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ can be evaluated from Eqs. (T1.2) and (T1.6), respectively, the overall thermal effectiveness of the heat exchanger can be evaluated from the following expression:
eth
1 ¼ C
C ec;cf ðC m ; NTUÞ 1 1 9
4 5 ) C ep;cf ðC m ; NTUÞ 1 9
ð5Þ
where
C m ¼ The exponents, NTL,c and NTL,p, depend on the number of tube lines where the flow is counter-flow and parallel-flow respectively, which is equal to one for the cases of Fig. 2(c) and (d). The total number of tube lines per row is 2 and, as a result, the modified heat capacity, C m , must be determined based on NTL = 2. Equations (T1.5) and (T1.1) can be used for the evaluation of ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ. From Eq. (4), it can be concluded that the thermal effectiveness of the flow arrangements of Fig. 2(c) and (d) is the same since NTL is even and NTL,c and NTL,p are equal. Note that for the flow arrangements (c)–(j) of Fig. 2, NTL,c = 1 and NTL,p = 1. Expressions for ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ from Table 1 must be changed since the number of tube rows is different among the flow arrangements of Fig. 2. A similar procedure could be used for the rest of the cases of Fig. 2, Fig. 2(e)–(j), and Fig. 1, Fig. 1(c)–(e). As an example consider the case of Fig. 2(j) with 5 rows and two lines per row. It can be noted that the in-tube over flow is parallel whereas the under flow is counter-flow. Thus, Eq. (4) can be applied. Equation (4) is still valid for a more complex configuration such as the one of Fig. 1(e). It must be noted that in both cases, Fig. 2(j) and Fig. 1(e), ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ must be evaluated for a heat ex-
(
C 9
ð6Þ
Table 2 sums ups the closed form expressions for the overall thermal effectiveness of the flow arrangements of Figs. 1 and 2. Their development and use for other configurations is straightforward. Note that ec;cf ðC m ; NTUÞ and ep;cf ðC m ; NTUÞ should be computed from the adequate expressions of Table 1. Table 2 closed form expressions provide effectiveness results that are identical, up to the sixth decimal place after period, to the numerical results computed by the HETE code (see [9–12] for more details). Note that the HETE code performs iterations of the temperature field to a relative error of less than 1010 between two consecutive iterations of the external mean outlet temperature. 4. Thermal effectiveness data comparison The variation of the thermal effectiveness with NTU for the Zshape heat exchanger of Fig. 1(a) is shown in Fig. 4 for two capacity ratios: 0.25 and 1. The arrangement of Fig. 1(a) corresponds to that of Fig. 2(a) with 9 tube lines per row. It must be emphasized that the in-tube fluid flows in a counter-flow manner in all the lines
L. Cabezas-Gómez et al. / International Journal of Heat and Mass Transfer 69 (2014) 237–246
(a)
241
(b) in
in
out
(c)
out
(d) in
in
out
out
(e)
(f)
in
in
out
out
(g)
(h) in
in
out
out
(i) in
out
(j) in out
Fig. 2. Two dimensional schemes of the all studied flow arrangements considering only two tube lines (NTL = 2).
of the arrangement of Fig. 1(a). The results plotted in Fig. 4 correspond to those from the closed form equation, Eq. (2), designated as eth, with NLT = 9, which are overlaid with those from the mean arithmetic thermal effectiveness, em, Eq. (1). The value of em is obtained from Eqs. (T1.1) and (T1.5), from Table 1, whereas the closed form effectiveness value, eth, results from the straight application of Eq. (T1.5). At a first glance, the mean arithmetic thermal effectiveness seems to compare reasonably well with the closed form expression, especially at low NTU and capacity ratio values. In fact, for C⁄ = 0.25 and NTU = 2.5, the absolute error of the mean arithmetic effectiveness from the closed form effectiveness is of the order of 0.7%. However, for the same value of NTU and C⁄ = 1.0 the absolute error increases to 3.1%. Considering the experimental determination of the parameter NTU, it would be interesting to evaluate the error involved when NTU is determined from a given value of the thermal effectiveness, which could have been determined, for example, from experimental results. For that purpose consider the results from Table 3 for a heat exchanger like the one of Fig. 1(a) with C⁄ = 1.0. The first three columns correspond to the assumed NTU values (1.0 and 1.25) and the resulting closed form and mean arithmetic effectiveness, eth and em. The fourth column includes the absolute deviations of the closed form from the mean arithmetic effectiveness. Now, suppose that the closed form effectiveness value (0.4714 or 0.5142) were known (from, for example, an experimental study) and the NTU were to be determined using the mean arithmetic relation, Eq.
(1). The result is shown in the fifth column. Thus, instead of NTU = 1.0 or 2.0 (first column), the results of the fifth column are obtained, with absolute errors with respect to the values of the first column of the order of 4.2% and 9.6% (sixth column). The resulting NTU deviations are clearly higher than those obtained for the effectiveness. In the case of experimentally obtained UA values, NTU deviations could be significant since they could be compounded with measuring errors. It would be interesting at this point to investigate two trends: (i) the asymmetry of parallel–cross-flow and counter–cross-flow exchangers; and (ii) the effect of the number of tube lines per row. Parallel and counter cross flow Z-shape exchangers of two rows will be considered for that purpose, corresponding to Fig. 2(a) and (b), though for different number of tube lines per row. Closed form effectiveness correlations will be applied and compared with the result from the arithmetic mean one, Eq. (1), for the following case: NTU = 2.0; C⁄ = 0.5; and Cmin corresponding to the external fluid. The arithmetic mean effectiveness for the case study, evaluated from Eq. (1) and Eqs. (T1.1) and (T1.5) from Table 1, is equal to 0.69660. This value is independent of the number of tube lines per row, and results from the assumption that the internal fluid alternates between parallel and counter flow with respect to the external fluid. The thermal effectiveness from the closed form expressions for different number of tube lines are shown in Table 4. The relative errors of the arithmetic mean effectiveness with respect to the closed form expressions are also
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(a)
(b)
(c)
C
in
out
C
(d)
(e)
(f)
P
in
out
P
(g)
(h)
(i)
C
in
out
P
(j)
(k)
in
(l)
P
out
C Fig. 3. Two dimensional graphical schemes used to derive the closed form relations.
shown in this table. The following are conclusions that can be drawn from this table. (i) As expected, the counter–cross-flow exchanger effectiveness is higher than the parallel–cross-flow one. (ii) The counter–cross-flow exchanger effectiveness diminishes whereas the parallel–cross-flow one increases with the number of lines up to the limit of 100. (iii) Note that, for the counter–cross-flow arrangement, the relative errors of the arithmetic mean effectiveness, Eq. (1), with respect to the theoretical one (closed form) are relatively high for small number of tube lines per row, decreasing with the number of lines. Similar trend occurs for the parallel–
cross-flow heat exchanger up to 10 lines, though in this case, the relative errors are positive. From this number of lines on, the effectiveness of the parallel–cross-flow exchanger is higher than the arithmetic mean one, the difference increasing with the number of tube lines per row. (iv) Though the maximum number of lines in Table 4 is relatively high (100), the limit effectiveness is not achieved (see also Fig. 5). The infinity number of tube lines per row limit is valid for both Z-shape arrangements and corresponds to the pure mixed/unmixed cross flow heat exchanger (single row) [7], [8]. The pure cross flow heat exchanger effectiveness for mixed/unmixed fluids with unmixed fluid Cmin is given by the following expression:
243
L. Cabezas-Gómez et al. / International Journal of Heat and Mass Transfer 69 (2014) 237–246 Table 2 Closed form relationships for eth for the flow arrangements of Figs. 1 and 2. Figure
Side of Cmin
1(a)
B
1(b), (d), (e)
B
1(c)
B
2(a)
B
2(b)
B
2(c)–(j)
B
Theoretical relationships
Fluid A mixed (internal), Fluid B unmixed (external), C A ¼ 1=C B ;
Equation
(
9 ) C ec;cf ðC B =9; NTU B Þ 1 1 B 9
(T2.1)
(
5 4 ) C ec;cf C B =9; NTU B C ep;cf ð9; NTU B Þ 1 1 B 1 B 9 9
(T2.2)
(
4 5 ) C ec;cf ðC B =9; NTU B Þ C ep;cf ð9; NTU B Þ 1 1 B 1 B 9 9
(T2.3)
(
2 ) C ec;cf C B =2; NTU B 1 1 B 2
(T2.4)
(
2 ) C ep;cf ðC B =2; NTU B Þ 1 1 B 2
(T2.5)
eth ¼
1 C B
eth ¼
1 C B
eth ¼
1 C B
eth ¼
1 C B
eth ¼
1 C B
eth ¼
C ep;cf C B =2; NTU B C ec;cf ðC B =2; NTU B Þ 1 1 B 1 1 B 2 C B 2
(T2.6)
eA ¼ eB =C A ; NTU A ¼ NTUB =C A ; C A ¼ C A =C B .
Table 4 Comparison between arithmetic mean (em = 0.69660) and closed form effectiveness values as a function of the number of tube lines for configurations of Fig. 2(a) and (b). (NTU = 2, C⁄ = 0.5 and Cmin = Cair).
Fig. 4. Effectiveness values for the Z-shape configuration from Pongsoi et al. [3], Fig. 1(a) or Fig. 2(a) with NTL = 9 (counter-cross-flow arrangement).
Table 3 NTU absolute errors when determined from the mean arithmetic effectiveness procedure.
a b
NTU
eth
em
De [%]a
NTUnew
DNTU [%]b
1.0 1.25
0.4714 0.5142
0.4639 0.5026
1.042 1.16
1.042 1.346
4.2 9.6
De = (eth em) 100 [%]. DNTU = (NTUnew NTU) 100 [%].
eTL!1 ¼
i 1h NTU 1 eC ð1e Þ C
ð7Þ
The value of eTL?1 for the case of Table 4 is equal to 0.702013 which is very close to the effectiveness from the closed forms for 100 tube lines per row. (v) The behavior suggested in (iv) can be justified as follows for the counter–cross flow configuration. For a single tube line, NTL = 1, the flow arrangement is a true two-pass (in fluid)
a
Number of tube lines
eth (counter– cross-flow)
de [%]a
eth (parallel– cross-flow)
de [%]a
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 50 100
0.72862 0.72007 0.71568 0.71210 0.71120 0.70991 0.70893 0.70817 0.70756 0.70706 0.70664 0.70629 0.70599 0.70572 0.70549 0.70529 0.70511 0.70495 0.70480 0.70313 0.70257
4.394 3.259 2.665 2.176 2.052 1.874 1.739 1.633 1.548 1.479 1.420 1.371 1.329 1.292 1.260 1.232 1.206 1.184 1.163 0.928 0.849
0.67265 0.68272 0.68766 0.69058 0.69252 0.69389 0.69492 0.69572 0.69635 0.69687 0.69730 0.69767 0.69798 0.69825 0.69849 0.69870 0.69888 0.69905 0.69920 0.70089 0.70145
3.561 2.034 1.301 0.872 0.590 0.391 0.242 0.127 0.036 0.038 0.100 0.153 0.197 0.236 0.270 0.300 0.326 0.350 0.371 0.612 0.691
de = ((em eth)/eth) 100 [%].
counter–cross-flow arrangement, whose thermal effectiveness is given by Eq. (T1.5) from Table 1. For two tube lines per row, NTL = 2, the flow arrangement becomes that shown in Fig. 2(a) and consists of two two-passes (in fluid) counter– cross-flow arrangement regarding the external fluid. In this case the in-tube fluid performs the first two passes in counter-flow regarding the air (external fluid) in the first tube line and then performs the second two passes in counterflow regarding the air in the second tube line, below the first one, Fig. 2 (a). Thus, the external fluid, with uniform inlet temperature, is divided into two unmixed streams, which will have different mean temperatures at the outlet of the heat exchanger, i.e., at the exit of each tube line. This process leads to an overall temperature distribution that is different from that for NTL = 1, and, as a result, to a decrement of the thermal effectiveness. As the number of tube lines increases the thermal effectiveness diminishes, though at a reduced
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Fig. 5. Thermal effectiveness for Z configuration from Fig. 2(a) and (b) as a function of NTL for NTU = 2, C⁄ = 0.5 and Cmin = Cair.
rate. The increment of the in-tube fluid passes in the vertical direction as the number of tube lines per row increases leads to a general mixture effect of the in-tube fluid. As a result, the thermal effectiveness of this flow arrangement as the number of tube lines increases tends to the value of the mixed/unmixed cross-flow arrangement heat exchanger with one tube row, given by Eq. (7). Similar arguments could be used for the case of parallel–cross flow arrangements. (vi) The overlaid plots of Fig. 5 summarize the aforementioned trends besides making clear the asymmetry between parallel and counter–cross-flow arrangements. (vii) The results from Table 4 indicate that, for small number of tube lines, the arithmetic mean value could produce nondesired errors for the thermal effectiveness of a Z-shape arrangement, especially for the counter–cross-flow arrangement (Fig. 2(a)). In order to further explore the closed form expressions, in what follows, the thermal effectiveness of flow arrangements of Fig. 1 will be evaluated and the results compared with those from the arithmetic mean expression. The first arrangement is the one of Fig. 1(b), which corresponds to the one of Fig. 2(c) with 9 tube lines per row, and is a combination of counter–cross and parallel–cross flow arrangements. The closed form evaluation requires the use of Eq. (4), with 5 counter–cross-flow and 4 parallel–cross-flow passes, with a total of 9 tube lines per row. Thus, Eq. (T2.2) along with Eqs. (T1.1) and (T1.5) must be applied. The value of em is determined from Eq. (1) along with Eqs. (T1.1) and (T1.5). The effectiveness variation with NTU for two values of the capacity ratio, C⁄ = 0.25 and C⁄ = 1.0, is shown in the plot of Fig. 6. Results from both closed form and arithmetic mean expressions are overlaid in the plot of Fig. 6. It can be noted that differences are more significant for the higher capacity ratio, whereas for C⁄ = 0.25 differences are too small to call. The maximum absolute error of em with respect to eth is of the order of 2.19%, for C⁄ = 1.0 and NTU = 2.5. It is interesting to note that the errors tend to diminish with the increment of the tube lines per row, the limit for both cases being that given by Eq. (7).
Fig. 6. Thermal effectiveness for Fig. 1(b) arrangement for NTU = 2, C⁄ = 0.5 and Cmin = Cair.
Closed form thermal effectiveness, eth, results from Table 5 have been determined for the flow arrangements of Fig. 2(f) and (g), corresponding to the arrangements of Fig. 1(c) and (d) with 9 tube lines per row, for NTU = 2.0 and C⁄ = 0.5, with the minimum capacity ratio being that of the external fluid. The table includes the relative error of the thermal effectiveness from the arithmetic mean expressions with respect to the closed form ones. Equations (T1.2) and T(1.6) were used in the determination of em for the arrangement of Fig. 2(f) (3 rows) whereas Eqs. (T1.3) and (T1.7) were the ones used for the arrangement of Fig. 2(g) (4 rows). The resulting effectiveness values were respectively equal to 0.700995 and 0.702203. According to the results from Table 5, (i) the closed form effectiveness varies in an oscillatory mode with the number of lines per row for both flow arrangements; and (ii) the relative errors of em with respect to eth are rather small and also vary in an oscillatory mode given that em is constant for both arrangements. The oscillatory mode of eth is the result of the alternate parallel and counter cross flows. In fact, considering the arrangement of Fig. 2(f), when one goes from 2 to 3 lines, a parallel pass is added and, as a result, thermal effectiveness must decrease. The opposite occurs from 3 to 4 lines since in this case a counter pass is added. Similar trends are noted for the arrangement of Fig. 2(g). It has been determined that the oscillating mode continues up to the order of 20 lines per row. Beyond this number of tube lines per row the closed form thermal effectiveness is approximately constant and equal to 0.702013 for both flow arrangements. It is interesting to note that this is the same limit value attained by the Z-shape arrangement, though for a much larger number of tube lines, which is the same that would be obtained from Eq. (7) for a mixed/unmixed cross flow heat exchanger with one tube row. The relative errors of em with respect to eth for this condition are 0.145% and +0.027%, respectively, for the arrangements of Fig. 2(f) and (g), though for a high number of tube lines per row. The final arrangement of Fig. 1, not considered so far, is the one of Fig. 1(e) with 5 rows of tubes. This particular arrangement corresponds to the arrangement of Fig. 2(i) but with 9 lines of tubes per row. A closed form expression as the ones proposed herein
L. Cabezas-Gómez et al. / International Journal of Heat and Mass Transfer 69 (2014) 237–246 Table 5 Comparison between arithmetic mean and the closed form effectiveness values as a function of the number of tube lines for configurations of Fig. 2(f) (three rows) and Fig. 2(g) (four rows). (NTU = 2, C⁄ = 0.5 and Cmin = Cair).
a
Number of tube lines
eth (Fig. 2(f))
de [%]*
eth (Fig. 2(g))
de [%]a
2 3 4 5 6 7 8 9 10 20 50 100
0.702152 0.694871 0.702048 0.699443 0.702028 0.700702 0.702021 0.701220 0.702018 0.702014 0.702013 0.702013
0.165 0.881 0.150 0.222 0.147 0.042 0.146 0.032 0.146 0.145 0.145 0.145
0.702507 0.709725 0.702137 0.704798 0.702068 0.703435 0.702044 0.702873 0.702033 0.702018 0.702014 0.702013
0.043 1.060 0.009 0.368 0.019 0.175 0.023 0.095 0.024 0.026 0.027 0.027
de = ((em eth)/eth) 100 [%].
Table 6 Comparison between arithmetic mean and HETE numerical simulated effectiveness values as a function of the number of tube lines for the configuration of Fig. 2(i) (five rows). (NTU = 2, C⁄ = 0.5 and Cmin = Cair).
a
Number of tube lines
e (simulated with HETE) (Fig. 2(i))
de [%]aem by Eqs. (1, T1.4, and T1.8)
de [%]aem by Eq. (1) and HETE values
2 3 4 5 6 7 8 9 10 20 50 100
0.702700 0.709960 0.702186 0.704882 0.702090 0.703478 0.702056 0.702899 0.702041 0.702020 0.702014 0.702013
0.190 0.834 0.264 0.120 0.277 0.080 0.282 0.162 0.284 0.287 0.288 0.288
0.027 0.996 0.100 0.283 0.114 0.084 0.119 0.001 0.121 0.124 0.125 0.125
de = ((em eth)/eth) 100 [%].
could be applied if exact thermal effectiveness correlations were available for the parallel and counter cross flow arrangements for 5 rows. The ones of Table 1, Eqs. (T1.4) and (T1.8), are approxima-
245
tions for an infinite number of rows. Thus, in order to estimate the error in evaluating the effectiveness of the arrangement of Fig. 1(e) (Fig 2(i) when NTL it is varied) through Eq. (1), the program HETE has been used as reference. Column 2 of Table 6 includes the variation thermal effectiveness of arrangement 2(i), determined with HETE, for a different number of tube lines per row, up to 100 lines, for NTU = 2.0 and C⁄ = 0.5, and Cmin being that of the external unmixed fluid. The arithmetic mean effectiveness, Eq. (1), determined from Eqs. (T1.4) and (T1.8) is equal to 0.704038. The relative error of this value with respect to the reference one provided by HETE is shown in the third column of Fig. 6. The fourth column of Table 6 includes the relative error of the arithmetic mean effectiveness with respect to that provided by HETE (second column). However, in this case, the values of parallel and counter cross flow thermal effectiveness were determined with HETE program for a five rows arrangement. The following values were obtained: ep,cf = 0.63483 and ec,cf = 0.770948. It can be noted that, generally, the relative errors of column 4 are lower than those of column 3 of Table 6, especially from the eighth tube line on. In addition, the oscillatory behavior of the thermal effectiveness from HETE is also observed up to line 10, as in the cases of Table 5. The limit thermal effectiveness for a 100 tube lines per row is 0.702013, which is equal to the value that would be obtained from Eq. (7), corresponding to a one row cross flow arrangement. This is also the limit value for the other flow arrangements considered herein, as should be expected. As a final application, charts of thermal effectiveness vs. the Number of Transfer Units, based on the closed form expressions applied to the Z configurations of Fig. 2(a) and (b), are displayed in Fig. 7(a) and (b) for different capacity ratios. Equations (T2.4) and (T1.5) have been used in the development of the chart of Fig. 7(a) whereas the one of Fig. 7(b) is based on Eqs. (T2.5) and (T1.1). 5. Conclusions Closed form expressions for the thermal effectiveness of several cross flow arrangements of practical interest have been proposed and their results evaluated, using as reference the simulation program HETE [9–12], and compared with results from the arithmetic mean procedure proposed by Pongsoi et al. [3–6]. The proposed
Fig. 7. Effectiveness-NTU charts, based on closed form expressions, with Cmin = Cair, for the Z configurations from: (a) Fig. 2(a); and (b) Fig. 2(b).
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closed form correlations fill a gap in the literature for the evaluation of the thermal effectiveness and other performance parameters of flow arrangements like the ones considered herein. They are especially important in experimental investigations that use e-NTU correlations in data reduction, as in the Pongsoi et al. [3– 6] studies. The following general conclusions have been drawn from the present investigation: – The arithmetic mean equation, Pongsoi et al. [3–6], for the thermal effectiveness of cross flow arrangements like the ones considered in this paper, though producing results that are close to those from the closed form correlations, do present deviations that increase with NTU. These deviations could contribute to increase the experimental uncertainty in data reduction. – Flow arrangements such as the ones of Fig. 1 present an asymmetric behavior regarding the thermal effectiveness. The thermal effectiveness of these configurations oscillates with the number of tube lines per row due to the alternation of parallel and counter flows. This trend is significant for small number of tube lines, the range depending on the flow arrangement. – The thermal effectiveness of all the flow arrangements considered in this study tends to that of Eq. (7) for a high number of tube lines. – The flow arrangement that presents the higher thermal effectiveness is the one of Fig. 2(a), corresponding to the counter– cross-flow configuration, whereas the lowest value is attained by the parallel–cross flow configuration, Fig. 2(b). The other arrangements of Fig. 2 present effectiveness values in the range between the aforementioned limits.
Acknowledgments The first author acknowledges the financial support received from CNPq (Conselho Nacional de Desenvolvimento Científico e
Tecnológico) through process 307141/2011-0 and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais) through process PPM-00639/11.
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