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Article

Analytical Entropy Analysis of Recuperative Heat Exchangers

1
University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, I. Lucica 5, 10000 Zagreb, Croatia
2
University of Osijek, Mechanical Engineering Faculty, Trg I. B. Mazuranic 18, 35000 Slavonski Brod, Croatia
Entropy 2003, 5(5), 482-495; https://doi.org/10.3390/e5050482
Submission received: 22 July 2003 / Accepted: 28 November 2003 / Published: 31 December 2003
(This article belongs to the Special Issue Entropy Generation in Thermal Systems and Processes)

Abstract

:
The analytical solutions for the temperature variation of two streams in parallel flow, counter flow and cross-flow heat exchangers and related entropy generation due to heat exchange between the streams are presented. The analysis of limiting cases for the relative entropy generation is performed, and corresponding analytical expressions are given. The obtained results may be included in a more general procedure concerning optimal heat exchanger design.

1. Introduction

Many years ago Bošnjaković [1] has pointed out the necessity of introducing entropy generation into the analysis of heat process apparatus. Recuperative heat exchangers are typical examples of such apparatus. There are two sources of entropy generation in heat exchangers. One is due to heat exchange between two streams of different temperature, and the second one due to viscosity of moving fluids. The entropy generation rate due to viscosity is usually much smaller than the generation due to heat exchange, so it can be neglected. In [2,3] one can find a detailed analysis of entropy generation calculated from integral balance along streamlines and normalized by the heat capacity rate of one of the streams. Some more recent papers [4,5] deal with the entropy analysis of heat exchangers in a similar way. In this paper the three types of heat exchangers are analyzed and entropy generation rate due to heat exchange is calculated exactly, following the integration of entropy generation rate over the exchanger surface. The relative entropy generation rate is introduced and shown as a function of the same parameters for all types of heat exchangers. Analytical solutions are presented and analyzed.

2. Analytical Relationships for Heat Exchangers

The analyzed heat exchangers may be considered as a dividing wall between two fluids. Any of them is characterized by the wall surface area A0, overall heat exchange coefficient k, heat capacity rates of the weaker and stronger stream C1 and C2 and inlet temperatures of two streams T 1 and T 2 . Instead of these six dimensional variables it is convenient to introduce the following three non-dimensional parameters:
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For the each type of the considered heat exchangers, the analytical solution for the temperature distribution in both streams exists, so it is possible to find the heat transfer rate between two streams
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and the entropy generation rate due to heat transfer. According to [6,7], this part of entropy generation rate is defined as
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Usual dimensionless forms of the heat transfer rate and the entropy generation rate are
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where ε is heat exchanger effectiveness and NS the entropy generation number.

2.1 Solution for the Parallel Flow Heat Exchanger

The temperature distribution in the streams of a parallel flow heat exchanger (PAHE) is defined as [7]
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for the following boundary conditions: for A = 0, T1 = T 1 and for A = A0, T2 = T 2 . By using Eqs. (3), (6) and (7), the expression for the entropy generation number follows
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2.2 Counterflow heat exchanger

The temperature distribution in the streams of a counterflow heat exchanger (COHE) is defined as [7]
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for the following boundary conditions: for A = 0 , T2 = T 2 and for A = A0, T1 = T 1 . By using Eqs. (3), (9) and (10), the expression for the entropy generation number becomes
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2.3 Cross-flow heat exchanger

In a cross flow heat exchanger (CRHE) the streams are at an angle, so the temperatures of both streams depend on two coordinates, the non-dimensional x and y. The analytical solution is then defined by the infinite series [8]
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Using the above formulas it is possible to calculate the temperatures T1 and T2 up to a prescribed accuracy. The same applies for the numerical integration of the total heat flux between streams and the entropy generation rate [9].

2.4 Limiting cases of the entropy number

When one of the streams evaporates or condenses (π3 = 0) the entropy number is the same for all three types of heat exchangers. It follows from Eq. (8) or (11)
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Eq. (11), for a counterflow heat exchanger is the indeterminate form 0/0 for π3 = 1. The limit can be found by using L’Hospital’s rule
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When both streams evaporate or condense (C1 → ∞ and C2 → ∞; π2 = 0 and π3 = 1), the maximal entropy number is the same for all types of heat exchangers. It is defined by
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2.5 Relative entropy generation

If the maximum entropy number is known, the relative entropy generation can be defined as
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It is well known that the heat exchanger effectiveness depends on parameters π2 and π3. It is possible to express the parameter π2 as a function of ε and π2. For the parallel flow heat exchanger the function is given in [7].
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and for counterflow heat exchanger it is
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For π3 = 0 this relationship is the same for the all three types of heat exchangers
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The relative entropy generation depends on the parameters π2 , π3 and πT , but it may be rearranged into a form
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In the next section this relationship is presented graphically for the all three types of heat exchangers.

3. Graphical Presentation of the Entropy Generation Rate

3.1 General case

Figure 1a, Figure 1b, Figure 2a and Figure 2b show the relative entropy generation versus heat exchanger effectiveness for the parallel flow heat exchanger. Fig. 1a and Fig. 1b correspond to π3 = 0.5 , while Fig. 2a and Fig. 2b are for π3 = 1. In Fig. 1a and Fig. 2a the parameter πT varies from 0.2 to 1, and in Fig. 1b and Fig. 2b from 1 to 10.
Figure 1a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=0.5 and πT =0.2 to 1, for a parallel flow heat exchanger
Figure 1a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=0.5 and πT =0.2 to 1, for a parallel flow heat exchanger
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Figure 1b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=0.5 and πT=1 to 10, for a parallel flow heat exchanger
Figure 1b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=0.5 and πT=1 to 10, for a parallel flow heat exchanger
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Eq.(18) clearly indicates that for π3 = 0.5 and π2 → ∞ the effectiveness of a parallel flow heat exchanger tends to 2/3, as it shown in Figure 1a and Figure 1b. From these diagrams it may be seen that εS increases with πT increasing from 0.2 to 1 and decreases again with πT increasing 1 to 10.
For π3 = 1 the limiting value of the parallel flow heat exchanger effectiveness is 0.5, as it is shown in Fig. 2a and Fig. 2b.
Figure 2a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=1.0 and πT =0.2 to 1, for a parallel flow heat exchanger
Figure 2a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=1.0 and πT =0.2 to 1, for a parallel flow heat exchanger
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Figure 2b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=1.0 and πT =1 to 10, for a parallel flow heat exchanger
Figure 2b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=1.0 and πT =1 to 10, for a parallel flow heat exchanger
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Figure 3a, Figure 3b, Figure 4a and Figure 4b show the relative entropy generation as a function of the heat exchanger effectiveness for the counterflow heat exchanger. Fig. 3a and Fig. 3b correspond to π3 = 0.5 , while Fig. 4a and Fig. 4b are for π3 = 1. In Fig. 3a and Fig. 4a the parameter πT varies from 0.2 to 1, and in Fig. 3b and Fig. 4b from 1 to 10.
Figure 3a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=0.5 and πT =0.2 to 1, for a counterflow heat exchanger
Figure 3a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3=0.5 and πT =0.2 to 1, for a counterflow heat exchanger
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Figure 3b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =0.5 and πT =1 to 10, for a counterflow heat exchanger
Figure 3b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =0.5 and πT =1 to 10, for a counterflow heat exchanger
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From Eq.(19) it follows for a counterflow heat exchanger that for π2 → ∞, ε → 1.0 , as shown in Fig. 3a, Fig. 3b, Fig. 4a and Fig. 4b.
Figure 5a, Figure 5b, Figure 6a and Figure 6b show the relative entropy generation as a function of the heat exchanger effectiveness for the cross-flow heat exchanger. Fig. 5a and Fig. 5b correspond to π3 = 0.5 , while Fig. 6a and Fig. 6b are for π3 = 1. In Fig. 5a and Fig. 6a the parameter πT varies from 0.2 to 1 and in Fig. 5b and Fig. 6b from 1 to 10. The plotted results were obtained by using a numerical integration [9] giving accurate results at least in three significant digits.
Figure 4a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =0.2 to 1, for a counterflow heat exchanger
Figure 4a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =0.2 to 1, for a counterflow heat exchanger
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Figure 4b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =1 to 10, for a counterflow heat exchanger
Figure 4b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =1 to 10, for a counterflow heat exchanger
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Figure 5a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =0.5 and πT =0.2 to 1, for a cross-flow heat exchanger
Figure 5a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =0.5 and πT =0.2 to 1, for a cross-flow heat exchanger
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Figure 5b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =0.5 and πT =1 to 10, for a cross-flow heat exchanger
Figure 5b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =0.5 and πT =1 to 10, for a cross-flow heat exchanger
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Figure 6a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =0.2 to 1, for a cross flow heat exchanger
Figure 6a. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =0.2 to 1, for a cross flow heat exchanger
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Figure 6b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =1 to 10, for a cross-flow heat exchanger
Figure 6b. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for π3 =1 and πT =1 to 10, for a cross-flow heat exchanger
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For this type of heat exchanger, the effectiveness also tends to one when π2 → ∞, regardless of the parameter πT. Table 1 shows the values of the relative entropy generation εS and parameter π2 for ε = 0.4 , π3 = 0.5 and 1.0 and πT = 0.2 and 10.
Table 1. Calculated ε and π2 for ε =0.4 and selected π3 and πT
Table 1. Calculated ε and π2 for ε =0.4 and selected π3 and πT
Heat exchanger ε = 0.4
π3 = 0.5π3 = 1.0
πT = 0.2πT = 10πT = 0.2πT = 10
PAHEεS20.400 0.220 0.215 0.165
π20.6100.6100.8050.805
CRHEεS20.415 0.230 0.245 0.185
π20.5900.5900.7100.710
COHEεS20.420 0.235 0.265 0.195
π20.5750.5750.6670.667
It is obvious from Table 1 that, for the given effectiveness, the counterflow heat exchanger generates most of the entropy rate, but it has the smallest surface area (parameter π2).

3.2 Cases of an evaporator or a condenser

In the case of an evaporator or a condenser (C2 → ∞ and π3 = 0) the entropy generation for all three types of heat exchangers becomes identical, as shown in Eq. (14), so the relative entropy generation can be shown as a function of ε and πT, as in Fig. 7.
Figure 7. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for πT =0.2 to 10 and π3=0 (meaning of symbols is the same as in the previous figures)
Figure 7. Relative entropy generation εS as a function of the heat exchanger effectiveness ε for πT =0.2 to 10 and π3=0 (meaning of symbols is the same as in the previous figures)
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For ε → 1 , the heat exchanger surface area obviously tends to infinity, and consequently NS → 0. Since the NS,max takes a finite value depending on πT only, the relative entropy generation tends to zero for A0 → ∞. Further it can be noticed in Fig. 7 that the parametric curves corresponding to πT > 1 are concave, while those for πT < 1 are convex. For the given heat exchanger effectiveness the relative entropy generation grows if drops theπT. For e.g. ε =0.4 , πT = 10 and 0.2, εS becomes 0.275 and 0.735, respectively. For εS = 0.5 and πT = 10 and 0.2, the achieved heat exchanger effectiveness is 0.18 and 0.715, respectively.
Fig. 8 shows the relative entropy generation as a function of heat exchanger effectiveness for πT = 0.5 and for three values of π3(0; 0.5 and 1) and for all three types of heat exchangers. The solution for π3 = 0 is the same for all the three types.
Figure 8. Relative entropy generation εS of the PAHE (triangles) COHE (squares) and CRHE (circles) versus the heat exchanger effectiveness ε for the π3 =0 (dashed line); 0.5 (outlined symbols) and 1 (filled symbols) and for πT =0.5
Figure 8. Relative entropy generation εS of the PAHE (triangles) COHE (squares) and CRHE (circles) versus the heat exchanger effectiveness ε for the π3 =0 (dashed line); 0.5 (outlined symbols) and 1 (filled symbols) and for πT =0.5
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It may be seen that in the range 0 < ε < 0.2 and for a given πT, the type of heat exchanger has no influence on the relative entropy generation. For a given εS evaporator or condenser is the most effective, followed by the counterflow, cross-flow and parallel flow heat exchanger respectively. As concluded earlier, for a given ε the maximal relative entropy generation occurs in an evaporator or condenser, followed by the counterflow heat exchanger, while it becomes minimal in the parallel flow heat exchanger.

4. Conclusion

The relative entropy generation is defined as a ratio of actual entropy generation rate and its maximal value. Analytical expressions of relative entropy generation are given for parallel flow and counterflow heat exchangers and analytical-numerical results for cross-flow heat exchangers. An analysis of obtained expressions is carried out for the case of evaporators or condensers. The entropy generation is related to the heat exchanger effectiveness, so the heat exchangers can be evaluated quantitatively. Obtained results can be used in a more general procedure for heat exchanger optimization.

References

  1. Bošnjaković, F. Kampf den Nichtumkehrbarkeiten; Arch. Wärmewirtsch. Dampfkesselwes, 1938. [Google Scholar]
  2. Bejan, A. Advanced Engineering Thermodynamics; John Wiley & Sons: New York, 1988. [Google Scholar]
  3. Bejan, A. Entropy Generation Minimization; CRC Press: Boca Ratom, 1996. [Google Scholar]
  4. Hesselgreaves, J. E. Rationalization of second law analysis of heat exchangers. International Journal of Heat and Mass Transfer 2000, 43, 4189–4204. [Google Scholar] [CrossRef]
  5. Guo, Z. Y.; Shou, S.; Chen, L. Theoretical analysis and experimental confirmation of the uniformity principle of temperature difference field in heat exchanger. International Journal of Heat and Mass Transfer 2002, 45, 2119–2127. [Google Scholar] [CrossRef]
  6. Bošnjaković, F.; Knoche, K. F. Technische Thermodynamik; Teil I, VEB: Leipzig, 1988. [Google Scholar]
  7. Bošnjaković, F.; Knoche, K. F. Technische Thermodynamik; Teil II, Steinkopf: Darmstadt, 1997. [Google Scholar]
  8. Nusselt, W. Forschung auf dem Gebiete der Ingenieur Wesens. Technische Mechanik und Thermodynamik 1930, 1, 417–425. [Google Scholar]
  9. Galović, A.; Živić, M.; Halasz, B. The analysis of the entropy production and heat transfer efficiency of a crossflow heat exchanger, 13th International Conference on Thermal Engineering and Thermogrammetry (THERMO). 18-20 June 2003, Budapest.

NOMENCLATURE

A0
heat exchanger wall surface area, m2
C
heat capacity rate, W/K
k
overall heat transfer coefficient, W/(m2·K)
NS
entropy generation number
S · gen
entropy generation rate, W/K
T
temperature, K

Greek Letters

Φ
heat transfer rate, W
ε
heat exchanger effectiveness
εS
relative entropy generation
π2
number of transfer units
πT
ratio of inlet temperatures
π3
ratio of heat capacity rates

Subscripts

1
weaker stream
2
stronger stream
max
maximum

Superscripts

inlet

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MDPI and ACS Style

Galovic, A.; Virag, Z.; Zivic, M. Analytical Entropy Analysis of Recuperative Heat Exchangers. Entropy 2003, 5, 482-495. https://doi.org/10.3390/e5050482

AMA Style

Galovic A, Virag Z, Zivic M. Analytical Entropy Analysis of Recuperative Heat Exchangers. Entropy. 2003; 5(5):482-495. https://doi.org/10.3390/e5050482

Chicago/Turabian Style

Galovic, Antun, Zdravko Virag, and Marija Zivic. 2003. "Analytical Entropy Analysis of Recuperative Heat Exchangers" Entropy 5, no. 5: 482-495. https://doi.org/10.3390/e5050482

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