Performance Improvement in a Concentric Circular Heat Exchanger with External Recycle under Uniform Wall Fluxes

Tamkang Journal of Science and Engineering, Vol. 12, No. 3, pp. 231 238 (2009) 231 Performance Improvement in a Concentric Circular Heat Exchanger with External Recycle under Uniform Wall Fluxes Chii-Dong Ho*, Shih-Cheng Yeh, Jia-Jan Guo and Jr-Wei Tu Energy and Opto-Electronic Materials Research Center, Department of Chemical and Materials Engineering, Tamkang University, Tamsui, Taiwan 251, R.O.C. Abstract A considerable improvement of heat transfer efficiency is obtainable by inserting an impermeable tube into a laminar counterflow concentric circular heat exchanger with external recycle under uniform wall fluxes. The mathematical formulation has been developed theoretically and the analytical solutions were achieved by combining a homogeneous solution and an asymptotic solution. The calculating results are represented graphically and are compared with those in a single-pass device and previous work [29]. The influences of subchannel thickness ratio and recycle ratio on the heat-transfer efficiency improvement and power consumption increment were also discussed in this study. Key Words: Heat-Transfer Efficiency Improvement, External Recycle, Double-Pass Operation, Uniform Wall Fluxes 1. Introduction The analysis of heat transfer phenomena in a bounded conduit has widely considered in recent years to engineers. The convective heat transfer problems of laminar internal flows at steady state with negligible axial conduction are as referred to the classical Graetz problem [1,2]. However, for the fluid with low Peclet number, such as liquid metals in compact heat exchangers, the axial heat conduction can not be neglected and this kind of heat transfer problems are known as the extended Graetz problems [3 7]. Many investigations have been dealt with the solution of the multi-stream or multiphase problems, which takes into account the conjugated conduction-convection conditions at the boundaries [8 14]. The conjugated Graetz problems have been simplified drastically to Sturm-Liouville systems and solved analytically by expansions in term of a pair of orthogonal eigenfunctions [15 19]. Based on the above studies, most of *Corresponding author. E-mail: cdho@mail.tku.edu.tw the investigations dealt with the two heat transfer cases of uniform wall temperature (Dirichlet problem) and heat flux (Neumann boundary condition) in the axial direction. The present study is made on the mathematical treatment of double-pass operations with mutual conditions of the contiguous phases under uniform heat fluxes. The calculation procedure presented in this study can be applied to the heat and mass transfer problems of any arbitrary wall temperature and flux distributions. Some authors have reported that the application of the recycle-effect concept to heat and mass exchangers leads to improved device performance [20 22] in producing the desirable effect of the convective transfer coefficient increment. The economical and technical feasibilities of recycling of fluid at the ends have been widely used in many separation processes and reactor designs, such as loop reactors [23,24], air-lift reactors [25,26] and draft-tube bubble columns [27,28]. The purposes of the present study are to extend the previous work [29] on double-pass concentric circular heat exchangers in an alternative arrangement by investi-

232 Chii-Dong Ho et al. gating the device performance improvement and develop the analytical treatment for uniform heat flux distribution in the axial direction based on the superposition technique by using an eigenfunction expansion in power series of the homogeneous nature. This work also includes the influences of the heat-transfer efficiency improvement on such recyclic countercurrent double-pass devices with the subchannel thickness ratio, recycle ratio and Graetz number as the parameters. 2. Theoretical Statement 2.1. Temperature Distributions in Double-Pass Devices with Recycle Consider a circular tube with length L and inside diameter 2R, as shown in Figure 1, The device is divided into two subchannels a and b with thickness 2 R and 2(1 )R, respectively, by inserting an impermeable sheet with negligible thickness and thermal resistance in it. The fluid with volumetric flow rate V and inlet temperature T i firstly feeds in to the subchannel b, as shown in Figure 1, and before entering the subchannel a, the fluid will premix with the fluid with volumetric flow rate MV and outlet temperature T F which exits from the subchannel a. The recycle flow rate is regulated by a conventional pump and the fluid is completely mixed at both the inlet and outlet of the subchannels a and b. The following assumptions are made in the present analysis: constant physical properties of fluid; purely fully-developed laminar flow; negligible entry length, end effects, axial conduction and thermal resistance of the impermeable tube. Introducing the following dimensionless variables (1) the energy equations and velocity distributions may be obtained as (2) (3) (4) (5) Figure 1. Schematic diagram of the double-pass concentric circular heat exchangers with external recycle at both ends.

Performance Improvement in a Concentric Circular Heat Exchanger with External Recycle under Uniform Wall Fluxes 233 in which (11) where is the asymptotic solutions of the inhomogeneous part: are The boundary conditions for solving Eqs. (2) and (3) (12) (6) (7) (8) (9) (13) and is the homogeneous solution of the homogeneous part: (14) The temperature distribution of the double-pass concentric circular heat exchanger with external recycle under uniform heat fluxes can be solved by following the same calculation procedure performed in our previous work [29]. The analytical solution was obtained by separating the inhomogeneous boundary value problems into the inhomogeneous part and homogeneous part. The complete solutions are in the form (10) (15) where a 1, a 2, a 3, b 1, b 2,andb 3 are the undetermined constants, S a,m and S b,m are the expansion coefficients, and F a,m and F b,m are the eigenfunctions. 2.2. Temperature Distributions in a Single-Pass Device The single-pass device of the same working dimen- Figure 2. Schematic diagram of the single-pass concentric circular heat exchangers.

234 Chii-Dong Ho et al. sions without recycle is shown in Figure 2. The velocity distributions and energy equations of the circular tube in dimensionless form may be written as (16) (25) Similarly, for a single-pass operation without recycle (17) The initial and boundary conditions for solving Eq. (16) are 3. Heat-Transfer Efficiency Improvement (18) (19) (20) The average Nusselt number for double-pass operations with recycle may be defined as (26) The improvement of device performance by employing a double-pass operation with recycle is best illustrated by calculating the percentage increase in heattransfer rate, based on that of single-pass operations without inserting an impermeable tube and external recycle under the same working dimensions and operating conditions as (27) (21) in which the average heat-transfer coefficient is defined as or where (22) (23) 4. Case Study A numerical example for heating kerosene by a stainless concentric circular tube with external recycle is studied in this study. The working dimensions of the conduit are L = 1.47 m, R = 0.2 m and = 0.5. The heat conductivity of the stainless conduit is k = 14.9 W/m K and the wall heat flux of the conduit is q = 130 kw/m 2. The inlet temperature of kerosene is T i = 298 K and the thermal conductivity of kerosene is = 8.645 10-8 m 2 /s. The calculating results are shown in Table 1. 5. Results and Discussion thus (24) By following the same calculation methods and procedure performed in concentric circular heat exchangers of single- and double-pass devices as those in the previous work [29], except the recycle type, the calculating results thus obtained and will be discussed.

Performance Improvement in a Concentric Circular Heat Exchanger with External Recycle under Uniform Wall Fluxes 235 Table 1. The theoretical predictions of case study for heating kerosene V 10 6 (m 3 /s) Gz T F (K) T w,i (K) T w,l (K) T w,avg (K) Nu single-pass device 1 10 333 377.98 517.57 447.78 2.33 M = 1 325.27 533.89 429.58 2.65 M = 3 325.95 528.19 427.07 2.70 M = 5 326.15 526.52 426.34 2.72 single-pass device 5 50 305 377.98 405.90 391.94 3.72 M = 1 329.64 363.19 346.42 7.21 M = 3 329.71 363.12 346.42 7.21 M = 5 329.74 363.10 346.42 7.21 The average Nusselt numbers, Nu (or Nu 0 ), as well as the heat transfer coefficients, h (or h 0 ), can be calculated from Eqs. (21) and (23), respectively, for a doublepass device with recycle (or a single-pass device without recycle). Figure 3 gives the graphical representations of the average Nusselt numbers Nuand Nu 0 vs. Gz obtained in the present and previous [29] studies of double-pass devices with recycle under uniform wall fluxes. Both the average Nusselt numbers Nu and Nu 0 increase with Gz because h and h 0 will be enhanced as the fluid velocity V increases in the device. The increase in the present device is rather sensitive at some operating condition, say 10 < Gz < 100. In addition, Nu is much larger than Nu 0, except for very small Gz, say Gz < 10. This is because the amount of flowing fluid MV in the present device is heated in the inner tube by recycle and the extent of further improvement in heat transfer efficiency is rather limited. The values of h, and hence Nu, in the present device are higher, as reasonably inferred from Figure 3, since the annulus of present device is employed for heating the recycle fluid only while that of the previous one is provided for heating the whole fluid. Moreover, the desirable effect of the forced convection increment cannot compensate for the larger amount of flowing fluid, say (M +1)V, through the annulus and the undesirable effect of driving temperature decrement in the previous study [29], as compared to the heat-transfer rate improvement obtained in the present one. The convective heat-transfer coefficient enhancements in both present and previous works [29], and hence the heat-transfer efficiency improvement, are obtained if double-pass operations with external recycle are provided. The suitable selection of the subchannel thickness ratio and Graetz number Gz with consideration of both heat-transfer efficiency improvement I h and power consumption increment I p on economical feasibility has made and presented graphically in Figure 4. The power consumption increment in double-pass device is obtained as follows: (28) The results of I h /I p vs. Graetz number Gz with subchannel thickness ratio as a parameter show that the difference value of I h /I p on both present and previous studies increases with increasing the subchannel thickness ratio, especially for = 0.7. A case study of heating kerosene by employing the present device is studied in this work. The physical properties of kerosene and working dimensions of conduit are given in Sec. 4 and the calculating results are shown in Table 1. Under the uniform wall flux q = 1.3 10 5 W/m 2, the kerosene with inlet temperature T i = 298 K will be heated to T F = 333 K and 305 K as V =1 10-6 m 3 /s and V =5 10-6 m 3 /s, i.e. Gz = 10 and 50, respectively. As indicated in Table 1, the average wall temperatures T w,avg of employing the present device decrease with increasing recycle ratio M and Graetz number Gz, and are lower than those of employing single-pass device, as observing from Table 1. Moreover, as shown in Table 1, the average Nusselt number increases with the increasing recycle ratio M and the Graetz number Gz. 6. Conclusion The influences of subchannel thickness ratio on the heat-transfer efficiency improvement in the double-pass circular tubes under uniform wall fluxes with inserting an impermeable tube of negligible thermal resistance were investigated theoretically. The tempera-

236 Chii-Dong Ho et al. Figure 3. Theoretical average Nusselt number vs. Gz with the subchannel thickness ratio as a parameter. Figure 4. The values of I h /I p vs. Gz with the subchannel thickness ratio as a parameter. ture distribution was solved analytically by separating the inhomogeneous boundary value problems into the inhomogeneous part and homogeneous part. The mathematical analysis provides a relatively straightforward strategy to the solution of multistream heat- or masstransfer devices and the flow pattern with recycle. The present study is actually the extension of our previous work [29] except the recycle type. The comparison with the same parameter values was made to explain how the present device improves heat transfer efficiency on the previous one. The advantage of the present device is evident for considering both device performance and economic sense, as confirmed from Figures 3 and 4. Moreover, the calculating results of the case study of heating kerosene show that the average wall temperatures T w,avg decrease with increasing recycle ratio M and Graetz number Gz, and are lower than those in a single-pass device. Acknowledgement The authors wish to thank the National Science Council of the Republic of China for its financial support. Nomenclature a constants b constants D hydraulic mean diameter, m F m eigenfunction associated with eigenvalue m Gz Graetz number, 4V/ L G m function defined during the use of orthogonal expansion method h average heat transfer coefficient, W/m 2 K I h heat-transfer efficiency improvement, defined by Eq. (27) I P power consumption increment, defined by Eq. (28)

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