An analytical study of heat-transfer efficiency in laminar counterflow concentric circular tubes with external refluxes

Chemical Engineering Science 58 (2003) 1235 – 1250

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An analytical study of heat-transfer e$ciency in laminar counter%ow concentric circular tubes with external re%uxes Chii-Dong Ho∗ , Wen-Yi Yang Department of Chemical Engineering, Tamkang University, Tamsui, Taipei 251, Taiwan, R.O.C. Received 8 October 2001; received in revised form 24 June 2002; accepted 12 November 2002

Abstract The mathematical formulation of a new device of counter%ow heaters or coolers in concentric circular tubes with uniform wall temperature and external re%uxes is developed and the analytical solution has been investigated with the use of an orthogonal expansion technique by expanding eigenfunctions in terms of power series. Comparisons of enhancement in heat-transfer e$ciency are made with single-pass operations of the same size (without an inner tube inserted). Considerable improvement in the heat-transfer e$ciency for large Graetz numbers is obtainable by introducing the recycle-e8ect concept in designing such double-pass operations. Analytical results show that suitable adjustment of impermeable-sheet location could have a substantially improving the heat-transfer e$ciency. The increment of power consumption has been also discussed. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Concentric circular tubes; Conjugated Graetz problem; External re%uxes; Orthogonal expansion techniques

1. Introduction The classical Graetz problem (Shah & London, 1978; Dang & Steinberg, 1980) neglects the e8ect of axial conduction and studies laminar forced convection heat transfer in a viscous %uid %owing in conduit of various cross-sectional geometries. The classical Graetz problem is extended to more complicated systems through conjugated conduction–convection conditions at the boundaries is called conjugated Graetz problems. Perelman (1961) proposed a general theory of equations on conjugated problems of the hyperbolic-elliptic type. Murkerjee and Davis (1972) developed a simple method to analyze the temperature distribution of a stratiBed two-phase laminar %ow. Kim and Cooney (1976) studied the conjugated boundary-value problem involving chemical reaction in hollow-Bber enzyme reactor. Davis and Venkatesh (1979) treated the di$culties associated with solutions of conjugated boundary-value problems in chemical engineering. Papoutsakis and Ramkrishna (1981a), Papoutsakis and Ramkrishna (1981b) have developed and presented a general formalism for problems involving ∗ Corresponding author. Tel.: +886-2621-5656; fax: +886-2620-9887. E-mail address: [email protected] (C.-D. Ho).

any dual combination of %uid and solid phases. Yin and Bau (1996) dealt with the simultaneous energy equations in the %uid and the solid regions by utilizing eigenfunction expansions. Applications of such devices with external or internal re%uxes have led to improved performance in separation processes and reactor designs. The recycle-e8ect concept makes a signiBcant in%uence on heat and mass transfer and hence plays a remarkable role in the design and operation of the equipment, such as loop reactors (Korpijarvi, Oinas, & Reunanen, 1998; Santacesaria, Serio, & Iengo, 1999), air-lift reactors (Garcia-Calvo, Rodriguez, Prados, & Klein, 1998; Atenas, Clark, & Lazarova, 1999) and draft-tube bubble columns (Goto & Gaspillo, 1992; Kikuchi, Takahashi, Takeda, & Sugawara, 1999). The theoretical formulations of Graetz problems and conjugated Graetz problems have been solved by the use of orthogonal expansion techniques (Singh, 1958; Brown, 1960; Nunge & Gill, 1966; Nunge, Porta, & Gill, 1967; Tsai & Yeh, 1985; Yeh, Tsai, & Lin, 1986; Yeh, Tsai, & Chiang, 1987; Ho, Yeh, & Sheu, 1998; Ebadian & Zhang, 1989) resulting in an inBnite number of eigenvalues and only the Brst negative eigenvalue was considered for the rapid convergence in the present paper. It is believed that the availability of such a simplifying mathematical formulation as developed here for concentric circular tubes is the value in the present work and will be an

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(02)00653-X

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C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

important contribution to design and analysis multistream or multiphase problems with internal or external re%uxes coupling mutual conditions at the boundaries. The purposes of the present study are to derive the mathematical formulation and to analytically investigate the improvement of transfer e$ciency of the concentric circular heaters or coolers with external re%uxes. The device performance with re%ux ratio, sheet position and Graetz number is also discussed. The previous works were conBned to a parallel-plate channel (Ho et al., 1998) while the present paper deals with the conjugated problems in a concentric circular tube with external re%uxes. An exact solution of the problem on heat transfer of a laminar countercurrent in annulus with external re%uxes is given in Section 2. The present device has some practical manner of double-pipe heat exchangers in industrial engineering. Countercurrent operations, however, are fundamentally di8erent since the velocity must change sign. The convergence of the type of eigenfunction expansions used in the present study to solve them is brie%y considered. The analytical solution of the equivalent problem in an annulus follows the parallel-plate channel very closely except the additional variable of the radius ratio. Since the inBnite series appearing in the solution of the problem converges rapidly, it is possible to solve the annulus problem in a more straightforward bearing by utilizing the method of eigenfunction expansion technique with eigenfunctions expanding in terms of an extended power series and the velocity distribution is expanded by Taylor series. A more direct method is described herein for determining that the ratio of subchannel thickness and recycle ratio for double-pass operation, which allows the speciBcation set by the designer, of values directly dependent on heat-transfer e$ciency, to be met with the higher value of the ratio of improvement of heat transfer and the increment of power consumption. The same procedure undoubtedly occurs in dealing with many further possible boundary conditions. Therefore, the same method can be applied easily to the solution of other conjugated Graetz problems in heat- and mass-transfer devices with mutual conditions at the boundaries in two or more contiguous streams and phases of multistream or multiphase problems. 2. The governing equations for temperature distributions 2.1. Double-pass concentric tubes with recycle An impermeable sheet with negligible thickness
and thermal resistance is inserted in parallel as the inner tube with diameter of 2 R into a circular tube with length L, inside diameter 2R (

) of the outer tube to divide the open duct into two parts, inner channel and annular channel, as shown in Figs. 1(a) and (b) for %ow pattern A and %ow pattern B, respectively. Before entering the inner tube for a double-pass operation (%ow pattern A) as shown in Fig. 1(a), the %uid with volume %ow rate V and the inlet temper-

ature Ti will mix with the %uid %owing out from the end of the inner tube with the volume %ow rate MV and the outlet temperature TL , which is regulated by using a conventional pump. The inlet %uid may %ow through the annulus with premixing the external re%ux exiting from the annulus (%ow pattern B) as shown in Fig. 1(b). In each %ow pattern, the %uid is assumed to mix completely at the inlet and outlet of the tube. The following assumptions are made in the present analysis: constant physical properties and wall temperatures of the outer tube; purely fully developed laminar %ow on the entire length in the inner and annular channels; negligible end e8ect, axial conduction and thermal resistance of the impermeable sheet. After the following dimensionless variables are introduced: =

r ; R

=

z ; L

a

=

a = 1 −

a

=

T a − Ti ; Tw − T i

b = 1 −

b

=

T b − Ti ; Tw − T i

T a − Tw ; Ti − T w

Gz =

b

=

T b − Tw ; Ti − T w

4V : L

(1)

The velocity distributions and equations of energy in dimensionless form may be obtained as
  @ a ( ; ) va ( )R2 @ a ( ; ) 1 @ ; (2) = L @ @ @
  vb ( )R2 @ b ( ; ) 1 @ @ b ( ; ) ; = L @ @ @

(3)

  2  va ( ) = 2va 1 − ; 

(4)

vb ( ) =

0 5 5 ;

2vb [(1 − 4 )=(1 − 2 ) − (1 − 2 )=ln (1=)]  
 1 − 2 ln ; × 1 − 2 + ln (1=)  5 5 1;

(5)

in which va =(M +1)V=(R)2 and vb =−V=(R2 −(R)2 ) for %ow pattern A while va = −V=(R)2 and vb = (M + 1)V=(R2 − (R)2 ) for %ow pattern B. The boundary conditions for solving Eqs. (2) and (3) are @ a (0; ) = 0; @ b (1; )

= 0;

@ a (; ) @ b (; ) ; = @ @ a (; )

=

b (; )

(6) (7) (8) (9)

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

1237

Fig. 1. Schematic diagrams of concentric circular heat exchangers with external re%uxes at both ends.

and the dimensionless outlet temperature is F = 1 −

F

=

T F − Ti : Tw − T i

(10)

Inspection of Eqs. (2), (3) and (6)–(9) shows that the inlet conditions for both inner and annular channels are not speciBed a priori and reverse %ow occurs. The dimensionless outlet temperatures referred to as the bulk temperature, are obtained by making the overall energy balances on both inner and annular channels. The analytical solutions to both %ow patterns may be obtained by the use of an orthogonal expansion technique with the eigenfunction expanding in terms of an extended power series. Separation of variables in the form a ( ; )

b ( ; )

= =

∞  m=0 ∞ 

Sa; m Fa; m ( )Gm ( );

(11)

Sb; m Fb; m ( )Gm ( )

(12)

m=0

applied to Eqs. (2) and (3) leads to Gm ( ) = e−m (1− ) ;

(13)

Fa; m ( ) +

Fa; m ( ) va ( )R2 m − Fa; m ( ) = 0; L

(14)

Fb; m ( ) +

Fb; m ( ) vb ( )R2 m − Fb; m ( ) = 0 L

(15)

and also the boundary conditions in Eqs. (6)–(9) can be rewritten as Fa; m (0) = 0;

(16)

Fb; m (1) = 0;

(17)

Sa; m Fa; m () = Sb; m Fb; m ();

(18)

Sa; m Fa; m () = Sb; m Fb; m ()

(19)

combination of Eqs. (18) and (19) yields Fa; m () Fb; m () = ; Fa; m () Fb; m ()

(20)

in which the eigenfunctions Fa; m ( ) and Fb; m ( ) were assumed to be polynomials to avoid the loss of generality.

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C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

With the use of Eqs. (16) and (17), we have Fa; m ( ) =

∞ 

dmn n ;

dm1 = 0;

n=0

dm0 = 1 (selected); Fb; m ( ) =

∞  n=0

emn n ;

(21) ∞ 

emn = 0:

(22)

n=0

Substituting Eqs. (21) and (22) into Eqs. (14) and (15), all the coe$cients dmn and emn may be expressed in terms of eigenvalues m by using Eqs. (16) and (17), as referred to in Appendix A. The mathematical treatment is similar to that in the previous work (Ho et al., 1998), except that the orthogonality conditions of concentric problem is replaced by   
va R2 Sa; m Sa; n Fa; m Fa; n d L 0   1
vb R2 Sb; m Sb; n Fb; m Fb; n d = 0 + (23) L  when n = m. The dimensionless inlet and outlet temperatures (a ( ; 0) and F ) for double-pass devices were thus obtained in terms of the Graetz number (Gz), eigenvalues (m ), expansion coe$cients (Sa; m and Sb; m ), location of impermeable sheet () and eigenfunction (Fa; m and Fb; m ). The results are 1 v 2R2 b ( ; 0) d  b F =− V

  1 2L  e−m Sb; m   =− (Fb; m + Fb; m ) d V m  8  e−m Sb; m  =− [Fb; m (1) − Fb; m ()] Gz m

(24)

for %ow pattern A, and  va 2R2 a ( ; 0) d 0 F =− V     e−m Sa; m 2L   =− (Fa; m + Fa; m ) d V m 0 =−

8  e−m Sa; m  Fa; m () Gz m

(25)

for %ow pattern B, and may be examined by Eq. (26) which is readily obtained from the following overall energy balance on outer tube:    1 @ b; m (1; ) 2L − d F = 1 − F = V @ 0 =

∞ 8m  Sb; m Fb; m (1)(1 − e−m ): Gz m=0

(26)

In Eq. (26) the left-hand side refers to the net outlet energy while the right-hand side is the total amount of heat transfer from outer hot tube to the %uid. The mixed inlet temperature is calculated after the coe$cients, Sa; m and Sb; m are obtained as follows:  (M=(M + 1)) 0 va 2R2 L; a ( ; 1) d + V · 1 a; 0 = V (M + 1) =

M 2L  Sa; m 2 (M + 1) V m    × (Fa; m + Fa; m ) d + 0

=

1 M +1



1 M +1

 Sa; m 8M Fa; m () + 1 : (M + 1)Gz m

(27)

Similarly, for %ow pattern B 1 (M=(M + 1))  vb 2R2 L; b ( ; 1) d + V · 1 b; 0 = V (M + 1) =

M 2L  Sb; m (M + 1)2 V m

  ×

1



1 = M +1

(Fb; m + Fb; m ) d



+

1 M +1

 Sb; m 8M (M + 1)Gz m 

×[Fb; m (1)

−

Fb; m ()]

+1 :

(28)

It is seen from Eqs. (A.4) and (A.5) for %ow pattern A (Eqs. (A.6) and (A.7) for %ow pattern B) in appendix that Gz and m always appear together and one may further deBne a modiBed eigenvalue as m∗ = Gzm = constant. In this case m∗ is independent of Gz and one needs not repeat the computation of m for di8erent Gz. There are two unique aspects to be considered beyond the theoretical studies of the single-pass problem in dealing with the double-pass problem which are used broadly in industrial applications. First, an orthogonality relation for the positive and negative sets of eigenfunctions associated with eigenvalues is required. A demonstration of completeness and its justiBcation based on Sturm–Liouville theorem have been developed, as referred to in Appendix B. Second, a calculating procedure with the general expression for the expansion coe$cients has been derived, as referred to in Appendix C, for evaluating the expansion coe$cient and the complete set of eigenfunctions associated with eigenvalues is developed to satisfy inlet conditions. Since the velocity distributions in two subchannels change sign over the interval in question, Eqs. (14) and (15) are the special cases of Sturm–Liouville problem for which there may exist both positive and negative

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

1239

Table 1 Eigenvalues and expansion coe$cients as well as dimensionless outlet temperatures in double-pass devices with recycle for =0:7 and M =5: Gz1 =−12:024 and Gz2 = −15:575

Gz

m

m

Sa; m

Sb; m

F (1 )

F (1 ; 2 )

1

1 2

−12:024 −15:575

7:04 × 10−12 3:64 × 10−27

2:53 × 10−13 0

0.50

0.50

10

1 2

−1:2024 −1:5575

4:15 × 10−7 −1:56 × 10−21

1:49 × 10−8 3:81 × 10−26

0.58840

0.58840

100

1 2

−0:1202 −0:1558

1:87 × 10−6 0.0

6:73 × 10−8 0.0

0.89824

0.89824

1000

1 2

−0:0120 −0:0156

2:29 × 10−6 1:09 × 10−20

8:24 × 10−8 −3:14 × 10−25

0.98819

0.98819

Fig. 2. Single-pass heat exchangers without recycle.

sets of real eigenvalues of having limit points +∞ and −∞, respectively. The eigenvalues indicated in Table 1 are the ones dominate in the system.

0 (1; )

2.2. Single-pass devices without recycle For the single-pass device of the same size without recycle the impermeable sheet in Fig. 1(a) or (b) is removed, as shown in Fig. 2 and thus,  = 1 and 0, respectively. The velocity distribution and equation of energy in dimensionless form may be written as
  v0 ( )R2 @ 0 ( ; ) 1 @ @ 0 ( ; ) = ; (29) L @ @ @ v0 ( ) = 2v0 (1 − 2 ); =

r ; R

0 = 1 −

= a

=

z ; L

0 5 5 1;

0

=

T 0 − Ti ; Ts − T i

(30)

T 0 − Tw ; Ti − Tw GZ =

4V : L

The boundary conditions for solving Eq. (29) are @ 0 (0; ) = 0; @

0 ( 0 ; 0)

= 1:

(33) (34)

The calculation procedure for a single-pass device is rather simpler than that for a double-pass device in the previous section. The dimensionless outlet temperature (0; F = 1 − 0; F ) for single-pass devices was obtained in terms of the Graetz number (Gz), eigenvalues (0; m ), expansion coe$cients (S0; m ) and eigenfunction (F0; m ( 0 )). The dimensionless outlet temperature is calculated by

  1 2L  S0; m   (F0; m + F0; m ) d 0; F = V m 0 =

(31)

= 0;

(32)

8  S0; m · 1 · F0; m (1) Gz m

(35)

and may be examined by Eq. (36), which is readily obtained from the following overall energy balance on the outer

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C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

hot tube:



0; F = 1 − =

0; F

=

1

0

2L V

 −

@

0; m (1; )

@

 d

∞ 80; m  S0; m F0; m (1)(1 − e−m ): Gz

(36)

m=0

3. Improvement of transfer eciency

friction factor which is the function of Reynolds number, Re. For the laminar %ow in tubes 16 : (45) f= Re Combination of Eqs. (44) and (45) yields vQ hfs ˙ 2 : (46) D For single- and double-pass devices, the average velocity of the %uid and the equivalent diameters of conduits are D0 = 2R;

Da = 2R;

Db = 2(R − R)

(47)

The Nusselt number for double-pass devices with recycle may be deBned as Q Q h(2R) hD Nu = = ; (37) k k

and the average velocity are as follows: V (M + 1)V V ; va = ; vb = v0 = 2 2 2 R (R) (R − (R)2 )

in which the average heat-transfer coe$cient is deBned as

for %ow pattern A. Similarly, for %ow pattern B V (M + 1)V : (49) ; vb = va = (R)2 (R2 − (R)2 ) The increment of power consumption, Ip , may be deBned as V (M + 1))(hfs; a + hfs; b ) − V)hfs; 0 P − P0 = ; Ip = P0 V)hfs; 0 (50)

Q q = h(2RL)(T w − Ti ):

(38)

Since Q h2RL(T w − Ti ) = V)Cp (TF − Ti )

(39)

or V)Cp (TF − Ti ) V)Cp hQ = (1 − = 2RL(Tw − Ti ) 2RL

F)

(40)

thus Nu =

V (1 − L

F)

=

1 Gz(1 − 4

F)

=

1 GzF : 4

(41)

Similarly, for a single-%ow operation without recycle included V 1 (1 − 0; F ) = Gz(1 − 0; F ) Nu0 = L 4 =

1 Gz0; F : 4

(42)

The improvement of performance by employing double-pass devices with external re%uxes is best illustrated by calculating the percentage increase in heat-transfer rate, based on the heat transfer of single-pass devices with same device dimension and operating conditions, but without impermeable sheet inserted and without external re%uxes included, as Ih =

Nu − Nu0 F − 0; F 0; F − F = = : 1 − 0; F 0; F Nu0

(43)

4. Increment of power consumption

L vQ2 ; D 2gc

where P = V (M + 1))(hfs; a + hfs; b ). Substitutions of Eqs. (47)–(49) into Eq. (50) results in Eqs. (51) and (52) for %ow pattern A and %ow pattern B in double-pass devices, respectively, (M + 1)2 1 Ip = + −1 (51) 4 (1 − 2 )(1 − )2 and 1 (M + 1)2 Ip = 4 + − 1: (52)  (1 − 2 )(1 − )2 As an illustration, the power consumption of the single-pass device will be illustrated by the working dimensions as follows: L = 1:2 m; R = 0:2 m; V = 1 × 10−4 m3 =s; 1 = 8:94 × 10−4 kg=m s; ) = 997:08 kg=m3 . From those numerical values, the friction loss in conduit of a single-pass device was calculated by the appropriate equations and the result is P0 = V)hfs; 0 = 1:71 × 10−8 W = 2:29 × 10−11 hp:

(53)

Some results for Ip of double-pass devices are presented in Tables 6 and 7 for %ow pattern A and %ow pattern B, respectively. It is seen from Eqs. (51) and (52) that the increment of power consumption does not depend on Graetz number but increases with the re%ux ratio or as  goes away from 0.5, especially for  ¡ 0:5 in %ow pattern A. 5. Numerical examples

The friction loss in conduits may be estimated by hfs = 4f

(48)

(44)

where vQ and D denote the bulk velocity in the conduit and the diameters of the conduit, respectively, while f is the

The improvement in heat-transfer e$ciency by arranging the recycle e8ect will be illustrated by the following two case studies. Consider the heat transfer for a %uid %owing through a parallel conduit with recycle. The working dimensions are: R = 0:2 m;  = 0:5.

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

Case 1: Pure water of 62◦ C is %owing through the counter%ow concentric circular tubes with constant wall temperature of 16◦ C. The numerical values are assigned as Ti = 62◦ C;

Tw = 16◦ C;

water = 1:524 × 10−7 m2 =s:

1241

well with the terms of n = 25 for an extended power series and N = 2 for Taylor series, and thus these two series with terms truncated were employed in the calculation procedure in this study.

Case 2: Oil at 62◦ C is %owing through the counter%ow concentric circular tubes with constant wall temperature of 16◦ C. The numerical values are assigned as

6.1. Dimensionless outlet temperature and transfer e2ciency in double-pass devices of 3ow pattern A

Ti = 62◦ C;

Comparisons of dimensionless outlet temperatures, F and 0; F , as well as average Nusselt numbers, Nu and Nu0 , were made and represented in Figs. 3–6. Fig. 3 shows another more practical form of dimensionless outlet temperature F (or 0; F ) vs. re%ux ratio M with Graetz number Gz a parameter for  = 0:5 while Fig. 4 with the ratio of channel thickness  as a parameter for M = 1 and 5. The application of the recycle-e8ect concept to heat-transfer devices results in two con%icting e8ect: the desirable preheating e8ect of the inlet %uid and the undesirable e8ect of decreasing temperature gradient. It was also found in Figs. 3 and 4 and Table 4 that the dimensionless average outlet temperature increases with decreasing the Graetz number Gz and the re%ux ratio M owing to increasing temperature gradient, but decreases as the ratio of channel thickness  goes away from 0.5, especially for  ¿ 0:5. The reason why  ¡ 0:5 is better than  ¿ 0:5 may be considered as that the enhancement of heat transfer in inner channel due to decreasing the thickness of inner channel to increase the %ow velocity can compensate for increasing the thickness of annular channel with re%ux ratio M to increase the residence time. It is seen from Fig. 4 that the di8erence (F − 0; F ) of outlet temperatures is of minus sign and decreasing with Gz, and then turns to plus sign for  ¡ 0:5 and with some Graetz number, say Gz ¿ 40.

Tw = 16◦ C;

oil = 8:816 × 10−8 m2 =s:

From these values, the improvements in transfer e$ciency in laminar counter%ow concentric circular tubes operated with recycle arrangement under various %ow rates of %uid and re%ux ratios, were calculated by the appropriate equations and the results are presented in Tables 8 and 9. 6. Results and discussion The equation of counter%ow heaters or coolers in concentric circular tubes with uniform outer wall temperature and with external re%uxes has been formulated and solved by the use of the orthogonal expansion technique with the eigenfunction expanding in terms of an extended power series. As an illustration of the dominant eigenvalue, two eigenvalues and their associated expansion coe$cients as well as the dimensionless outlet temperatures were calculated for  = 0:5; M = 1, and Gz = 1; 10; 100 and 1000, as shown in Table 1. It was found in Table 1 that due to the rapid convergence, only the Brst negative eigenvalues necessitates to be considered during the calculation of temperature distributions. The eigenfunctions are expanded in terms of an extended power series as well as ln y term in the velocity distribution is expanded by Taylor series. Comparisons were made to illustrate the di8erence between these two series with terms truncated after n = 25 and 30 for an extended power series and N = 2 and 3 for Taylor series. The accuracy of those comparisons was examined and some results were represented in Tables 2 and 3 for an extended power series and Taylor series, respectively. It is seen from Tables 2 and 3 that two series agree reasonably

6.2. Improvement in transfer e2ciency based on a single-pass device without recycle The Nusselt numbers (Nu and Nu0 ) and hence the improvement of transfer e$ciencies (Ih ) are proportional to F (or 0; F ), as shown in Eqs. (47)–(49), so the higher improvement of performance is really obtained by

Table 2 Convergence of power series in Eqs. (21) and (22) with n = 25 and 30 with  = 0:3 M = 5 (%ow pattern A) Gz

n

m

Sa; m

Sb; m

F

10−4

10−5

1

25 30

−8:42154 −8:44661

−1:49 × −1:45 × 10−4

7:33 × 7:14 × 10−5

0.48952 0.48952

10

25 30

−0:84215 −0:84466

−3:73 × 10−1 −3:72 × 10−1

1:84 × 10−1 1:83 × 10−1

0.62746 0.62703

100

25 30

−0:08422 −0:08447

−1:17 × 100 −1:17 × 100

5:77 × 10−1 5:76 × 10−1

0.92231 0.92210

1000

25 30

−0:00842 −0:00845

−1:36 × 100 −1:36 × 100

6:69 × 10−1 6:68 × 10−1

0.99133 0.99131

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C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

Table 3 Convergence of Taylor series in Eq. (A.3) with N = 2 and 3 (n = 25) with  = 0:7 M = 5 (%ow pattern A)

Gz

n

m

Sa; m

Sb; m 10−12

F

10−13

1

2 3

−12:0241 −11:8426

7:04 × 8:49 × 10−12

2:53 × 8:81 × 10−15

0.50 0.50

10

2 3

−1:20241 −1:18426

4:15 × 10−7 4:27 × 10−7

1:49 × 10−8 4:42 × 10−10

0.58839 0.59031

100

2 3

−0:12024 −0:11843

1:87 × 10−6 1:89 × 10−6

6:73 × 10−8 1:96 × 10−9

0.89824 0.89954

1000

2 3

−0:01202 −0:01184

2:29 × 10−6 2:31 × 10−6

8:25 × 10−8 2:39 × 10−9

0.98819 0.98836

Fig. 3. Dimensionless outlet temperatures vs. M with Gz as a parameter;  = 0:5 (%ow pattern A).

Fig. 4. Dimensionless average outlet temperatures vs. Gz with  as a parameter; M = 1 and 5 (%ow pattern A).

employing a double-pass device, instead of using a single-pass device for large Graetz numbers with  ¡ 0:5, if the volumetric %ow rate in both devices are kept same. Fig. 5 shows the theoretical average Nusselt numbers Nu vs. M , with Graetz number Gz as a parameter for  = 0:5 while Fig. 6 with re%ux ratio and the ratio of the channel thickness  as parameters. On the other hand, as shown in Fig. 6, (Nu − Nu0 ) increases with increasing Gz for

Gz ¿ 40 and  ¡ 0:5. It is concluded that Nusselt number decreases with the ratio of channel thickness  going away 0.5, especially for  ¿ 0:5, but increases with decreasing M as well as with decreasing Graetz number. Some numerical values of the improvement in performance Ih were given in Table 4. The minus signs in Table 4 indicate that no improvement in transfer e$ciency can be achieved as M ¿ 1 and  ¿ 0:5 for Gz ¡ 100, and in this case, the single-pass

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

1243

Fig. 5. Average Nusselt number vs. M with Gz as a parameter;  = 0:5 (%ow pattern A).

Fig. 6. Average Nusselt number vs. Gz with  as a parameter; M = 1 and 5 (%ow pattern A).

device is preferred to be employed rather than using the four-pass one operating at such conditions.

as the improvement of heat transfer decrease as re%ux ratio M goes away some value, say M = 1:5, for  = 0:5 while the improvement of heat transfer in Table 5 increases with Graetz number but decrease with the ratio of the thickness. A comparison of Figs. 5 and 6 with Figs. 9 and 10 indicates an enhancement of heat-transfer rate occurs for %ow pattern B with  ¡ 0:5 and M ¿ 2 as well as with  ¡ 0:3 and M ¡ 1 for any Graetz numbers. Tables 6 and 7 show that the power consumption calculated for both %ow pattern A and %ow pattern B, respectively, by assuming only the friction losses to the walls were signiBcant. It is of interest to note that %ow pattern B yields an enhancement of the average Nusselt number with no increase in power consumption for  ¡ 0:5, and in designing both %ow pattern A and B with a proper selection of re%ux ratio and the ratio of channel thickness should be economically feasible in the device of double-pass concentric coolers and heaters.

6.3. Transfer e2ciency in double-pass devices of 3ow pattern B The calculation methods are similar to those in the previous section of %ow pattern A. The temperature of re%ux %uid increases with the residence time, which is inversely proportional to the inlet volume rate (or the Graetz number) and hence the mixed dimensionless inlet temperature increases with the amount of the re%ux %uid (or re%ux ratio). Accordingly, it is shown in Figs. 7 and 8 that the dimensionless inlet temperature of %uid after mixing increases with re%ux ratio and with decreasing  but decreases with Graetz number. Since the position of the impermeable sheet has much in%uence on the heat-transfer behavior, average Nusselt number has been calculated and presented in Figs. 9 and 10 with Graetz number and the ratio of the thickness  as parameters, respectively, while the percentage of the improvement of heat transfer were shown in Table 5 with Graetz number, re%ux ratio and the ratio of the thickness  as parameters. It is shown that Nusselt number as well

7. Conclusion The analytical solution of heat transfer through double-pass concentric circular tubes with subchannel

1244

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

Table 4 Improvement of the transfer e$ciency with re%ux ratio and sheet position as parameters (%ow pattern A)

Ih (%)

Gz = 1 10 100 1000

M = 0:5

M =1

M =3

M =5









0.3

0.5

0.7

0.3

0.5

0.7

0.3

0.5

0.7

0.3

0.5

0.7

−49:1 −35:1

−50:0 −34:9

−50:4 −66:8 −71:1 −71:3

−49:3 −36:9

−50:0 −34:9

−50:3 −66:1 −69:9 −70:2

−49:1 −36:9 −19:8 −11:9

−49:9 −40:1

−50:0 −46:4 −25:2 −18:6

−48:9 −51:5 −42:9 −40:3

−48:5 −48:2 −34:2 −30:0

−50:0 −46:4 −25:2 −18:7

63.6 120.1

247.9 1072

34.8 68.3

244.9 1019

8.32 27.1

Fig. 7. Dimensionless average inlet temperatures of %uid after mixing. The ratio of channel thickness as a parameter; M = 1 and 5 (%ow pattern B).

Fig. 8. Dimensionless average inlet temperatures of %uid after mixing. Re%ux ratio as a parameter;  = 0:5 (%ow pattern B).

thickness varied, has been investigated and solved by the use of the orthogonal expansion technique with the eigenfunction expanding in terms of an extended power series. The analytical solution of heat transfer through a single-pass device of the same size with an impermeable sheet inserted has also been presented for comparison.

Two case studies were given for the improvement of heat-transfer e$ciency and the results are shown in Tables 8 and 9. From these two tables we see that Ih increases with Gz, or with decreasing M . For large L or small V , the residence time is large enough and the re%ux e8ect is no more important, therefore Ih decreases as we proceed down Tables 8 and 9. The minus signs in Tables 8 and 9 indicate that,

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

1245

Fig. 9. Average Nusselt number vs. M with Gz as a parameter;  = 0:5 (%ow pattern B).

Fig. 10. Average Nusselt number vs. Gz with  as a parameter; M = 1 and 5 (%ow pattern B).

no improvement in mass transfer can be achieved at low Graetz number and large re%ux ratio, and in this case, the device without external re%uxes (single-pass operations) is preferred to be employed rather than using the device with external re%uxes operating at such conditions. The plots of Ih =Ip as function of Gz, M and  have been presented in Figs. 12(a) and (b) for %ow pattern A and %ow pattern B, respectively. It is well known that in selecting the design parameter, , and operating parameters, Gz

and M , The quantity of interest is not the improvement of transfer e$ciency Ih alone, but the ratio of the improvement of transfer e$ciency to the increment of power consumption, which is presented here in the form Ih =Ip . In this case, the suitable selections on consideration of both improvement of heat transfer, Ih , and the increment of power consumption, Ip for %ow pattern A and B are presented in Figs. 11 and 12, respectively. It is seen from Figs. 11 and 12 that the values of Ih =Ip ¿ 0 for  ¡ 0:5 and M ¡ 5 with Gzm ¿ 20 in %ow

Table 5 Improvement of the transfer e$ciency with re%ux ratio and sheet position as parameters (%ow pattern B) Ih (%)

Gz = 1 10 100 1000

M = 0:5

M =1

M =3

M =5









0.3

0.5

0.7

0.3

0.5

0.7

0.3

0.5

0.7

0.3

0.5

0.7

0.0 29.2 241.9 351.9

0.0 −8:13 −39:5 −47:8

−4:73 −49:1 −71:3 −72:5

0.0 29.3 287 471

−0:23 −22:8 −25:3 −24:7

−13:0 −60:2 −71:6 −72:7

0.0 29.5 383 850

0.0 29.40 376.8 423.3

−13:0 −61:9 −72:1 −73:0

0.0 29.51 428.9 1131

0.0 22.26 230.5 392.9

−15:3 −62:5 −72:2 −73:1

1246

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

pattern A while the values of Ih =Ip ¿ 0 for  6 0:3 as well as for  = 0:5 and M ¿ 1 with any Graetz number in %ow pattern B. It is concluded that there exists suitable enhancement of heat-transfer e$ciency by somewhat decreasing the ratio of subchannel thickness and with adding the external re%uxes at both ends should be economically feasible in the design of double-pass heat exchangers. Operation under the new devices has a positive e8ect on enhancement of heat-transfer e$ciency in circular heat exchangers. In addition, the comparison of transfer e$ciencies between doubleand single-pass devices is observed from Figs. 4,6 and 10. At small Graetz number, the residence time of a single-pass device is long enough to change the temperature of %owing streams and should be kept good performance. Under this operation situation, the double-pass e8ect plays no signiBcant role on enhancement of heat-transfer e$ciency in circular heat exchangers. It is apparent that the mathematical formulations developed in this study with concentric circular tubes are only conducted in a heat-transfer sense with constant wall temperature, the complete theory with orthogonal expansion

Table 6 Increment of power consumption with re%ux ratio and barrier position as parameters for %ow pattern A M 0.5 1.0 3.0 5.0

Ip  = 0:3

 = 0:5

 = 0:7

279 495 1976 4445

40 68 260 580

30 37 87 170

Table 7 Increment of power consumption with re%ux ratio and barrier position as parameters for %ow pattern B M

Ip  = 0:3

 = 0:5

 = 0:7

0.5 1.0 3.0 5.0

127 131 158 203

27 36 100 207

52 90 351 787

Table 8 Results of Case 1 (Flow pattern A) L × 102 (m)

V × 106 (m3 =s)

Gz

16.7

1 5 10

50 250 500

3.17 3.55 3.60

95.81 544.92 800.09

−5:34 19.92 24.52

−37:72 −31:49 −30:52

55.7

1 5 10

15 75 150

2.34 3.32 3.48

−19:87 174.19 366.24

−32:97 3.21 14.46

−46:13 −35:44 −32:70

83.5

1 5 10

10 50 100

1.92 3.17 3.40

−34:93 95.81 244.89

−40:13 −5:34 8.32

−48:25 −37:72 −34:20

Nu0

Ih (%) M =3

M =5

Nu0

Ih (%) M =1

M =3

M =5

Table 9 Results of Case 2 (Flow pattern A) L × 102 (m)

V × 106 (m3 =s)

Gz

14.44

1 5 10

100 500 1000

3.40 3.60 3.63

244.89 800.09 1019.2

8.32 24.52 27.25

−34:20 −30:52 −29:99

57.77

1 5 10

25 125 250

2.77 3.45 3.55

12.81 308.65 544.92

−21:60 11.84 19.92

−42:49 −33:35 −31:49

96.28

1 5 10

15 75 150

2.34 3.32 3.48

−19:87 174.19 366.24

−32:97 3.21 14.46

−46:13 −35:44 −32:70

M =1

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

1247

Fig. 11. The values of Ih =Ip vs. Gz with  as a parameter (%ow pattern A).

Fig. 12. The values of Ih =Ip vs. Gz with  as a parameter (%ow pattern B).

techniques may also be applied to other conjugated Graetz problems in heat- or mass-transfer devices with external re%uxes at both ends.

kmn L M

Notation Cp D dmn emn Fm f gc Gz Gm hQ hfs Ih Ip k

heat capacity, J kg−1 K −1 hydraulic radius, m coe$cient in the eigenfunction Fa; m coe$cient in the eigenfunction Fb; m eigenfunction associated with eigenvalue m friction factor conversion factor, kg m s−2 N−1 Graetz number, 4V=L function deBned during the use of orthogonal expansion method average heat-transfer coe$cient, W m−2 K −1 friction loss in conduit, m2 s−2 improvement of heat transfer, deBned by Eq. (43) increment of power consumption, deBned by Eq. (50) thermal conductivity of the %uid, W m−1 K −1

Nu P0 R R1 Re Sm T V v vQ r z

coe$cient in the eigenfunction F0; m conduit length, m re%ux ratio, reverse volume %ow rate divided by input volume %ow rate Nusselt number hydraulic dissipated energy inside diameter of the outer tube, m inside diameter of the inner tube, m Reynolds number expansion coe$cient associated with eigenvalue m temperature of %uid, K input volume %ow rate of conduit, m3 s−1 velocity distribution of %uid, m s−1 average velocity of %uid, m s−1 radial coordinate, m axial coordinate, m

Greek letters 


thermal di8usivity of %uid, m2 s−1 thickness of the impermeable sheet, m longitudinal coordinate, z=L transversal coordinate, r=R

1248

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

  m 1 )

dimensionless temperature, (T − Ti )=(Tw − Ti ) ratio of channel thickness eigenvalue viscosity of %uid, kg m−1 s−1 density of the %uid, kg m−3 dimensionless temperature, (T − Tw )=(Ti − Tw )

Superscripts and subscripts a b F i L 0 w

dm3 = 0; .. . n = 4; 5 : : : ∞; dmn =

  (M + 1)Gz dmn−4  d ; − m mn−2 22 [n(n − 1) + n] 2

(A.4)

em0 = 1;

in inner channel in annular channel at the outlet of a double-pass device at the inlet at the outlet, = 1 in a single-pass device without recycle at the wall surface

em1 = 0;



em2 = −

1 Gzm 8 S(1 − 2 )

em3 = −

Gz 1 m T; 9 S(1 − 2 )

1−

3T 2

 ;

.. . Acknowledgements

n = 4; 5 : : : ∞;

The author wishes to thank the National Science Council of the Republic of China for the Bnancial support.

emn = −

Gzm 2S(1 − − 1) + n − 1]   3T emn−2 + 2Temn−3 × 1− 2    T emn−4 ; − 1+ 2

Appendix A. Coecients in the eigenfunctions Eqs. (14) and (15) can be rewritten as Fa; m ( ) (M + 1)Gzm Fa; m ( ) + − 22
 2  Fa; m ( ) = 0; × 1− 

(A.1)

+··· +

dm2 = − (A.2)

(y − 1)3 (y − 1)2 + 2 3

(y − 1)N : N

dm0 = 1; dm1 = 0;

in which the term ln y in velocity distributions can be expressed in terms of Taylor series as follows: ln y = (y − 1) −

.. . n = 4; 5 : : : ∞; dmn =

Gz m 2 2 [n(n − 1) + n]

em0 = 1;

Combining Eqs. (A.1)–(A.3), (16), (17), (21), and (22) with two-term Taylor series yields

em1 = 0; 1 (M + 1)Gzm 8 S(1 − 2 )

em3 =

1 (M + 1)Gz m T; 9 S(1 − 2 )

(M + 1)Gz m ; 82

.. . n = 4; 5 : : : ∞;





em2 =

dm0 = 1;

dm2 =

Gz m ; 82

dm3 = 0;

(A.3)

dm1 = 0;

(A.5)

for %ow pattern A. Similarly, for %ow pattern B

Fb; m ( ) Gzm Fb; m ( ) + + 2S(1 − 2 ) ×(1 − 2 + T ln )Fb; m ( ) = 0;

2 )[n(n

1−

 dmn−4 ; − d mn−2 2

3T 2

 ;

(A.6)

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

emn =

(M + 1)Gzm 2S(1 − 2 )[n(n − 1) + n − 1]   3T emn−2 + 2Temn−3 × 1− 2  −

T 1+ 2

Then Eq. (B.6) can be rewritten as Sa; m Sa; n [ (Fa; n Fa; m − Fa; m Fa; n )]0 + (n − m )   
va R2 Sa; m Sa; n Fa; m Fa; n d = 0: × L 0





emn−4 ;

in which
 1 − 4 1 − 2 S= ; − 2 1 −  2 ln(1=) 1− T= : ln(1=)

Similarly, one can obtain the same equations for subchannel b as follows:

(A.8)

Sb; m Sb; n [ (Fb; n Fb; m − Fb; m Fb; n )]1 + (n − m )   1
vb R2 Sb; m Sb; n Fb; m Fb; n d = 0: × L 

(A.9)

From Eq. (14) for subchannel a Fa; m ( ) Sa; m Fa; m ( ) + Sa; m 
va R2 Sa; m Fa; m ( ) = 0; − m (B.1) L  Fa; n ( ) Sa; n Fa; n ( ) + Sa; n 
va R2 Sa; n Fa; n ( ) = 0: − n (B.2) L Eq. (B.1) and (B.2) can be rewritten in Sturm–Liouville equations as follows: 
d va R2 (Sa; m Fa; m ( )) − m Sa; m Fa; m ( ) = 0: (B.3) d
L2  d va R Sa; n Fa; n ( ) = 0: (B.4) (Sa; n Fa; n ( )) − n d L Multiplying Eq. (B.3) by Sa; n Fa; n and Eq. (B.4) by Sa; m Fa; m , and subtract the results to get

0

+ (n − m )

 0






 va R2 Sa; m Sa; n Fa; m Fa; n d = 0: L

d ( Fa; n ) + [Fa; n Fa; m − Fa; m Fa; n ]: d

(B.9)

Adding the results of Eqs. (B.8) and (B.9) yield Sa; m Sa; n [ (Fa; n Fa; m − Fa; m Fa; n )]0 + Sb; m Sb; n ×[ (Fb; n Fb; m − Fb; m Fb; n )]1 + (n − m )   
va R2 Sa; m Sa; n Fa; m Fa; n d + (n − m ) × L 0   1
vb R2 Sb; m Sb; n Fb; m Fb; n d = 0; × (B.10) L  in which Sa; m Sa; n [ (Fa; n Fa; m − Fa; m Fa; n )]0 + Sb; m Sb; n [ (Fb; n Fb; m − Fb; m Fb; n )]1 =Sa; m Sa; n [(Fa; n ()Fa; m () − Fa; m ()Fa; n ())] + Sb; m Sb; n [1 × (Fb; n (1)Fb; m (1) − Fb; m (1)Fb; n (1))] − Sa; m Sa; n [0 × (Fa; n (0)Fa; m (0) − Fa; m (0)Fa; n (0))] − Sb; m Sb; n [(Fb; n ()Fb; m () − Fb; m ()Fb; n ())]

(B.11)

using Eqs. (16)–(19) in Eq. (B.11) and hence with n = m in Eq. (B.10) give the orthogonality conditions as shown in Eq. (23).

(B.5) Appendix C. The general expression for the expansion coecients The temperature (B.6)

Integrating by parts in the Brst term of Eq. (B.6) d d [ (Fa; n Fa; m − Fa; m Fa; n )] = Fa; n ( Fa; m ) d d − Fa; m

(B.8)

(A.7)

Appendix B. A demonstration of completeness and its justi+cation based on Sturm–Liouville theorem

Sa; m Sa; n [Fa; n ( Fa; m ) − Fa; m ( Fa; n ) ] + (n − m ) 
va R2 Sa; m Sa; n Fa; m Fa; n = 0: × L Integrating over the region of interest gives   Sa; m Sa; n [Fa; n ( Fa; m ) − Fa; m ( Fa; n ) ] d

1249

L

=

∞ 

Sa; m Fa; m

L

at = 1 can be calculated as follows: (C.1)

m=0

=

∞ 

Sb; m Fb; m :

(C.2)

m=0

(B.7)

Multiplying Eq. (C.1) by (va R2 =L) Sa; m Fa; m d and integrate from 0 to , Eq. (C.2) by (vb R2 =L) Sb; m Fb; m d and

1250

C.-D. Ho, W.-Y. Yang / Chemical Engineering Science 58 (2003) 1235 – 1250

integrate from  to 1, and then add the results to get 
  va R2 Sa; m Fa; m d L L 0 
 1 vb R2 Sb; m Fb; m d + L L    
va R2 2 Sa; m Fa;2 m d = L 0   1
vb R2 2 Sb; m Fb;2 m d ; + (C.3) L  in which   
va R2 1 Fa; m ( ) d = Fa; m ( )|0 L  m 0 1 = Fa; m (); m   1
vb R2 1 Fb; m ( ) d = Fb; m ( )|1 L  m  1 = (Fb; m (1) − Fa; m ()); m   
@Fa; m () va R2 Fa;2 m d = Fa; m () L @m 0 @Fa; m () − Fa; m () ; @m   1
vb R2 Fb;2 m d L 
 @Fa; m () @Fa; m ()  = − Fa; m () : − Fa; m () @m @m

(C.4)

(C.5)

(C.6)

(C.7)

Substitution of Eqs. (C.4)–(C.7) into Eq. (C.3) yields
 Sa; m Sb; m    Fa; m () + (F (1) − Fa; m ()) L m m b; m
 @Fa; m () @Fa; m () = Sa;2 m Fa; m () − Fa; m () @m @m
 @Fa; m () @Fa; m () 2  − Sb; m Fa; m () : − Fa; m () @m @m (C.8) References Atenas, M., Clark, M., & Lazarova, V. (1999). Holdup and liquid circulation velocity in a rectangular air-lift bioreactor. Industrial and Engineering Chemistry Research, 38, 944.

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