Entropy generation in turbulent mixed convection heat transfer to highly variable property pipe flow of supercritical fluids

Energy Conversion and Management 87 (2014) 552–558

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Entropy generation in turbulent mixed convection heat transfer to highly variable property pipe flow of supercritical fluids Mahdi Mohseni a, Majid Bazargan b,⇑ a b

Department of Mechanical Engineering, Qom University of Technology, Qom, Iran Department of Mechanical Engineering, K.N. Toosi University of Technology, 15 Pardis St, Mollasadra Ave, P.O. Box 19395-1999, Tehran 1999 143 344, Iran

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 17 June 2010 Accepted 2 July 2014

In this study, a two dimensional CFD code has been developed to investigate entropy generation in turbulent mixed convection heat transfer flow of supercritical fluids. Since the fluid properties vary significantly under supercritical conditions, the changes of entropy generation are large. The contribution of each of the mechanisms of entropy production (heat transfer and energy dissipation) is compared in different regions of the flow. The results show that the mechanisms of entropy generation act differently in the near wall region within the viscous sub-layer and in the region away from the wall. The effects of the wall heat flux on the entropy generation are also investigated. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Turbulent mixed convection High variable properties Entropy generation Supercritical fluid Numerical solution

1. Introduction The energy crisis in the world has made the optimization of engineering systems an important topic of research in recent decades. One of the popular approaches used to optimize energy systems is the second law of thermodynamics also known as entropy generation analysis [1,2]. In heated turbulent flow in a channel, the total entropy generation is mainly due to the fluid friction (viscous dissipation), the temperature gradient, the turbulent dissipation as well as the turbulent diffusion [3,4]. All these effects occur mainly in the near wall region in which the changes of the field variables such as temperature and velocity are at their maximum. There are a good number of studies related to entropy generation in laminar and turbulent flows [1–28]. Only a few of these studies address the effect of the variations of fluid properties on the rate of entropy generation [5–10]. Their approach is usually to let one property (viscosity or density) vary mildly with temperature while other properties are assumed to be constant. Then, the effect of the variation of the selected property on entropy generation has been examined. In two different studies, Sahin [6,7] showed the significance of the effect of variations of fluid viscosity on entropy generation in turbulent flows. Ben-Mansour and Sahin [8] analyzed the entropy generation in developing laminar flow of engine oil under conditions where the fluid viscosity was varying ⇑ Corresponding author. Tel.: +98 21 84063239; fax: +98 21 88677274. E-mail addresses: (M. Bazargan).

[email protected]

(M.

Mohseni),

http://dx.doi.org/10.1016/j.enconman.2014.07.013 0196-8904/Ó 2014 Elsevier Ltd. All rights reserved.

[email protected]

with temperature. Ben-Mansour et al. [9] have investigated entropy generation for the case of mixed convection heat transfer to a laminar flow considering linear variations of the fluid density with temperature. Flow in a curved square microchannel has been studied by Guo et al. [10] to examine how temperature-dependent viscosity affects the entropy generation. In the current study, entropy generation in highly variable property turbulent flow of mixed convection heat transfer in a heated vertical tube has been investigated numerically. Supercritical carbon dioxide is used to represent a fluid with extremely high property variations. At supercritical pressures, thermodynamic and transport properties of fluids change dramatically near the critical temperature. Nonlinear and nonmonotonic variations of the fluid properties near the critical region makes the supercritical fluid flow a special case of variable property flow which is probably one of the most difficult to analyze [29,30]. Fig. 1 shows the typical variations of the properties of carbon dioxide at supercritical pressure of 8.12 MPa. For calculation of the thermophysical properties of supercritical CO2, the NIST database [31] has been used. Note that the critical temperature and pressure of carbon dioxide are T = 304.13 K and P = 7.37 MPa, respectively. Because of the large property variations, the heat transfer rates are significantly enhanced near the critical region [32–34]. In the current study, the local as well as the total entropy generation are calculated for various wall heat fluxes. In the computational domain, the region with higher entropy generation has been specified. In addition, the contribution of each of the mechanisms

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Nomenclature Be BeT Cp C e1 ; C e2 Cl f1, f2 fl g Gk h H k p Pk Prt q Q Q_ r Sgen SgenF SgenH

Bejan number (SgenH/Sgen) turbulent Bejan number (SgenT/Sgen) specific heat capacity (J kg1 K1) constants in the e equation constant in the eddy viscosity model functions in the e equation damping function in the lt relation gravity acceleration (m s2) turbulent production due to Buoyancy (kg/m1 s3) heat transfer coefficient (W m2 K1) enthalpy (J kg1) turbulent kinetic energy (m2 s2) pressure (N m2) Turbulent production due to shear stresses (kg/m1 s3) turbulent Prandtl number heat flux (W m2) heat transfer rate (W) rate of heat generation per unit volume (W m3) transversal direction of flow (m) entropy generation per unit volume (W m3 K1) entropy generation due to energy dissipation (W m3 K1) entropy generation due to heat transfer (W m3 K1)

of entropy production is determined. The study helps recognize the main sources of system irreversibility and the location where they are produced. Such insight is needed in order to eliminate or minimize disturbances leading to further entropy generation and thus to improve the system efficiency. Furthermore, some differences of entropy generation in flows at normal and supercritical conditions have been discussed. These have been made possible by developing a two-dimensional model to simultaneously solve the governing equations which are highly coupled in the supercritical environment. The partial differential equations of entropy generation are also calculated.

T u, v x y+

temperature (K) velocity components in the x and r directions (m s1) axial direction of flow (m) non-dimensional distance from the wall

Greek symbols e dissipation rate of k (m2 s3) k thermal conductivity (W m1 K1) l molecular viscosity (kg m1 s1) lt turbulent viscosity (kg m1 s1) q density (kg m3) rk, re turbulent Prandtl number in the k and e equations s shear stress (N m2) Subscript b bulk t turbulent w wall

complete forms. The only exception is that the flow is considered to be steady and the temporal terms have been eliminated. The conservation of mass states that

@ðqui Þ ¼0 @xi

ð1Þ

For turbulent flows, by using the Reynolds time averaging method, the momentum equation yields to:

@ @P @ sij @ ðqui uj Þ ¼ qg i  þ þ ðqu0i u0j Þ @xj @xi @xj @xj

ð2Þ

where 2. Governing equations

sij ¼ l

The basic governing equations including the conservations of mass, momentum and energy, together with transport equations modeling the turbulence, are employed to model the flow. The entropy generation equations which are the main part of this study are also applied. Due to large variations of supercritical fluid properties, the governing equations are written and solved in their most

0.06

0.04

25

1000

800

20

15

600

10

400

5 200

0.02

1.4E-04 1.2E-04 1.0E-04 8.0E-05 6.0E-05 4.0E-05 2.0E-05

0 260

280

300

320

340

360

ð3Þ

and

qu0i u0j ¼ lt

  @ui @uj 2 @uk 2  dij qk þ  dij 3 @xj @xi 3 @xk

ð4Þ

where the symbols with primes denote the fluctuation components of the corresponding quantities. The next step is to specify the turbulent viscosity. A popular method is to employ the k–e model in which we have:

lt ¼ qC l fl Viscosity (kg/m.s)

0.08

Density Cp Conductivity Viscosity

Density (kg/m3)

0.12

0.1

1.6E-04

30

Cp (kJ/kg.K)

ThermalConductivity (W/m.K)

0.14

  @ui @uj 2 @uk þ  dij @xj @xi 3 @xk

k

2

ð5Þ

e

The transfer equations for turbulent kinetic energy (k) and the rate of its dissipation (e) are found by the following equations:

@ @ ðqui kÞ ¼ @xi @xi @ @ ðqui eÞ ¼ @xi @xi

 





lþ

lt @k þ Pk þ Gk  qe rk @xi

lþ

lt @ e e e2 þ C e1 f1 ðP k þ Gk Þ  C e2 f2 q re @xi k k



ð6Þ



ð7Þ

where

T (K) Fig. 1. Typical variation of the properties of carbon dioxide at the supercritical pressure of 8.12 MPa.

Pk ¼ qu0i u0j

@ui @xj

ð8Þ

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and

ð9Þ

The energy equation in terms of enthalpy will be in the form of

ð10Þ

Newton’s cooling law, qw = h(Tw  Tb) may be used to calculate the convective heat transfer coefficients. In a fluid flow with heat transfer the entropy generation is the sum of the entropy generation caused by energy dissipation, SgenF, and by the heat transfer, SgenH. Thus, the local entropy generation can be written as follows [35].

ð11Þ

Each of the two mechanisms of entropy generation in a turbulent flow consists of a laminar part and a turbulent part and can be written as follows.

SgenF

  l @ui @uj 2 @uk @ui qe ¼ þ  dij þ T @xj @xi 3 @xk @xj T

SgenH SgenH ¼ Sgen SgenF þ SgenH

8

4

Cross section 1

0 200

240

Cross section 2

280

320

Cross section 3

360

400

440

Hb (kJ/kg) Fig. 2. Comparison of the numerical results with the experiments of Song et al. [38] for Case I.

ð13Þ

ð14Þ

The Be number ranges between 0 and 1. In fact, Be = 0 and Be = 1 are two limiting cases. It is clear from Eq. (14) that when Be = 0.5, the irreversibilities created by each of the two mechanisms of heat transfer and energy dissipation are equal. For Be > 0.5, the irreversibility due to heat transfer is larger than those caused by energy dissipation. Once Be < 0.5, the opposite occurs. Similar to the above definition, we introduce the turbulent Bejan number, BeT, which is a measure of the ratio of turbulent entropy generation to laminar entropy generation. Thus, the BeT is defined as:

SgenT BeT ¼ Sgen

12

Table 1 Flow conditions examined in the present numerical model (P = 8.12 MPa, D = 9 mm).

where at ¼ qlPrt t is the turbulent diffusion of heat. The first term on the right hand side of Eqs. (12) and (13) is associated with the laminar part of the entropy generation while the second term relates to the turbulent part which is modeled by Kock and Herwig [3,4]. The relative importance of the two major mechanisms of entropy generation may be found by dividing the SgenH by Sgen which is known as the Bejan number, Be.

Be ¼

16

ð12Þ

and

    at  k @T @T SgenH ¼ 1 þ a T 2 @xi @xi

present expriment

ð15Þ

More details about the structure of the present code as well as comparison tests showing that the results of the numerical code are reasonable can be found in two recent publications by the authors [36,37]. 3. Results and discussions Before presenting and making discussions on the results of entropy generation under different flow conditions, it is useful to show the variations of heat transfer coefficients for supercritical conditions. Such variations are demonstrated in Fig. 2. The experimental data of Song et al. [38] are also shown for the sake of comparison. A summary of the flow conditions used in this study are stated in Table 1. Fig. 2 shows the results for Case I. The dashed lines are for later use and should be disregarded in this stage. As mentioned before, under supercritical conditions, the heat transfer

q (kW/m2) G (kg/m2 s)

Case I

Case II

Case III

Case IV

Case V

50 1200

60 1200

70 1200

80 1200

30 400

coefficient increases significantly due to large property variations in comparison with constant property flow at normal pressures. Note that it is common practice to use the fluid bulk enthalpy instead of bulk temperature for the horizontal axis in presenting the heat transfer data of supercritical fluids. For the constant heat flux condition, equal differences of the bulk enthalpy values correspond to equal distances along the tube. That is not the case for bulk temperature differences, because the enthalpy does not vary linearly with temperature in the supercritical environment.

3.1. Bulk entropy generation The variations of the bulk entropy generation along the tube for the flow conditions of Case I have been plotted in Fig. 3. The heat

4

16

3.5

14

3

12

2.5

10

2

8

1.5

6

1

4

0.5

2

0 220

Heat Transfer Coefficient (kW/m2.K)

Sgen ¼ SgenF þ SgenH

Bulk Entropy Generation (kW/m3.K)

  @ @ @T lt @H @p þ ui ðqui HÞ ¼ k þ þ Q_ þ u @xi @xi @xi Prt @xi @xi

Heat Transfer Coefficients (kW/m2.K)

20

  k @T Gk ¼ 0:3bg i qu0i u0j @xj e

0 240

260 280

300

320

340

360

380 400

420

440

Hb (kJ/kg) Fig. 3. Variations of bulk entropy generation and the heat transfer coefficient for Case I.

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transfer coefficients are also illustrated for the sake of comparison. Evidently, the bulk entropy generation reaches a minimum value when the heat transfer coefficient has a maximum value. In order to achieve higher efficiency from both points of view, i.e. the first law of thermodynamics (greater heat transfer rate) and the second law of thermodynamics (lower entropy generation), it is strongly recommended to have the thermal system operating as close as possible to pseudo-critical conditions. 3.2. Effect of the wall heat flux In contrast with normal pressure flows, under supercritical conditions, heat transfer rates decrease with increase of the wall heat flux. This can be seen in Fig. 4a for flow conditions of Cases I, II, III and IV. To visualize the effect of the wall heat flux on the entropy generation, the corresponding results are demonstrated in Fig. 4b. As shown, the bulk entropy generation increases as the wall heat flux increases. Thus, the increase of the wall heat flux in a supercritical environment has an adverse effect on the heat transfer rate as well as on entropy generation. To compare constant and variable property flows in this regard, consider the following: the wall heat flux does not affect the heat

16

Case I

14

Case II

12

Case III 10

Case IV 8

6

4

2

220

260

300

340

380

420

Hb (kJ/kg)

(b)

3.3. Heat transfer deterioration It was described in the previous section that for high variable property flow of supercritical fluids the temperature difference is affected significantly with increase of the wall heat flux. Further increase of the heat flux, while other flow conditions are kept constant, may lead to a very large temperature difference between the wall and bulk accompanied with a deterioration in heat transfer. The flow conditions of Case V are related to this situation. The apparent sign of heat transfer deterioration is the large increase in the wall temperature meaning that the heat is not removed well from the wall by the fluid flow. The buoyancy and the thermal acceleration effects are considered to be the main causes of the heat transfer degradation [32,33,39,40]. The buoyancy and the thermal acceleration are promoted due to large variations of the fluid density near the critical region at flows with high heat fluxes. These two effects may cause the velocity profile to be flattened. The shear stress distribution varies accordingly and the turbulence production is reduced. This may lead to relaminarization of the flow and hence deterioration in heat transfers. The predicted wall temperatures for flow conditions of Case V are shown in Fig. 5. The experimental data of Song et al. [38] are also demonstrated for the sake of comparison. The corresponding bulk entropy generation along the tube is also shown in Fig. 5. As expected, in the region of deteriorated heat transfer a large increase in entropy generation is observed. For the deteriorated regime of heat transfer, the present model is capable of predicting the jump in wall temperature. It should be mentioned that the deterioration of heat transfer under supercritical

8 370

10 Tw: Present Tw: experimental data

6

Wall Temperature (K)

Bulk Entropy Generation (kW/m3.K)

7

5

4

3

Case IV

2

Case III Case II

350

8

330

6

310

4

290

2

1 Case I 0

270 220

260

300

340

380

420

Hb (kJ/kg) Fig. 4. Variations of (a) the heat transfer coefficient and (b) bulk entropy generation with the wall heat flux.

Bulk Entropy Generation (kW/m3.K)

Heat Transfer Coefficients (kW/m2.K)

(a)

transfer coefficient for flows at normal pressures as can be found from conventional Nusselt number correlations. From Newton’s cooling law, the constant heat transfer coefficients imply that with a rise of the wall heat flux, the temperature difference between the wall and bulk increases. Such augmentation of the temperature difference results in an increase of entropy generation. It is generally true that in order to lower entropy generation in a thermal system, the wall heat flux needs to be decreased. Under supercritical conditions, lowering the heat flux causes the heat transfer coefficients to increase. It means that two desirable outcomes, better heat transfer coefficients and lower entropy generation, are achieved together. This makes supercritical fluids as excellent medium for heat transfer applications under certain circumstances.

0 200

240

280

320

360

400

Hb (kJ/kg) Fig. 5. Variations of the wall temperature and the bulk entropy generation along the tube for deteriorated heat transfer conditions (Case V).

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conditions is a complicated phenomenon and available numerical models, including the present model, may only predict the flow behavior qualitatively [32,41,42].

It is useful to investigate the details of entropy generation in various locations of the flow and explore the mechanisms involved. Fig. 6 shows the variations of entropy generation in three different flow cross sections, corresponding to dashed lines shown in Fig. 2 for flow conditions of Case I. The three flow cross sections 1, 2 and 3 refer to the locations before, within, and after the region of heat transfer enhancement, respectively. As shown, regardless of the location along the flow, the maximum entropy generation is concentrated near the wall where the variations of the velocity and temperature profiles are the largest. Moving away from the wall towards the pipe centerline, entropy generation diminishes due to the decrease of velocity and temperature profile gradients. It can be seen in Fig. 6 that outside the laminar sub-layer, i.e. for y+  >5, more entropy is generated in the low heat transfer zones (cross sections 1 and 3). Less entropy generation is observed within the region of enhanced heat transfer (cross section 2) because of the reduction in the wall and bulk temperature difference. Once the fluid flows from cross section 1 towards cross section 2, two opposing effects occur. On one hand the heat transfer is improved along the flow and hence the entropy generation tends to decrease. On the other hand, energy dissipation increases and helps the entropy generation to increase. The latter is due to the fact that the Reynolds number and the flow turbulence, and hence the energy dissipation, increase as the fluid properties changes from liquid-like to vapor-like along the tube. In general, the effect of heat transfer enhancement is dominant over the opposing effect of energy dissipation. However, as shown in Fig. 6, the local entropy generation very close to the wall in the laminar sub-layer at cross section 1 is smaller than that of cross section 2. This may be explained as follows: It has been shown by Bazargan and Mohseni [43] that the viscous sub-layer adjacent to the wall does not play an essential role in heat transfer enhancement. The contribution of heat transfer in entropy generation is almost unchanged across the laminar sublayer at different cross sections along the flow. Thus, the energy dissipation is the main factor which makes the differences in entropy generation near the wall at any cross section. To better appreciate the above argument, the corresponding variations of energy dissipation across the flow at cross sections 1, 2 and 3 have

Cross section 1 Cross section 3 3000 2500 2000 `

1500 1000 500 0 0

10

20

30

40

50

60

y+ Fig. 7. Variations of the energy dissipation of the flow at three different cross sections for Case I flow conditions.

been illustrated in Fig. 7. It should be noted that the energy dissipation is the sum of both the viscous and turbulent dissipation. Note also that the turbulent dissipation, e, unlike other turbulent quantities, is not zero at the wall. As shown in Fig. 7, the energy dissipation is increased at the wall neighbourhood. The variation of energy dissipation along the tube, as shown in Fig. 7, can also help explain the difference between the entropy generation at cross sections 1 and 3 appearing in Fig. 6. Note that the heat transfer behavior at cross sections 1 and 3, as can be seen in Fig. 2, is almost identical. Thus, the larger energy dissipation leads to greater entropy generation at cross section 3, compared to cross section 1. Fig. 7 also explains the variations of the Be number along the flow. In order to further clarify the contributions of the two main mechanisms (heat transfer and energy dissipation) leading to entropy generation, the variations of the Be number for various cross sections are plotted in Fig. 8. It can be seen that the Be number is much greater than 0.5 everywhere in the flow and thus the contribution of heat transfer to irreversibilities in the entire cross section is dominant over the irreversibilities due to energy dissipation. It is interesting also to plot the variations of bulk Be number along the flow which is demonstrated in Fig. 9. As shown, within

1.01

800 Cross section 1

1

Cross section 2

700

Cross section 3

0.99

600

Bejan Number

Entropy Generation (kW/m3.K)

Cross section 2

3500

Energy Dissipation (kW/m3 )

3.4. Mechanisms of entropy generation

4000

500 400 `

300 200

0.98 0.97 0.96 `

0.95 0.94

100

0.93

0

0.92

Cross section 1 Cross section 2 Cross section 3

0

10

20

30

40

50

60

y+ Fig. 6. Radial variations of entropy generation at various flow cross sections for flow conditions of Case I.

0

10

20

30

40

50

60

y+ Fig. 8. Radial variations of the Bejan number for various flow cross sections for Case I.

M. Mohseni, M. Bazargan / Energy Conversion and Management 87 (2014) 552–558

developed yet and the difference between the wall and bulk temperatures is small. In order to determine the laminar and turbulent contributions to entropy generation, the variations of the turbulent Bejan number, defined by Eq. (15), are also demonstrated for different flow cross sections in Fig. 11. The results show that in the region away from the wall the entropy generation is almost completely due to the turbulent part. As we move away from the flow centerline towards the pipe wall, the turbulent contribution diminishes, first gradually in the buffer zone and then sharply in the laminar sublayer until it finally vanishes to zero at the wall. Note that the horizontal axis in Fig. 11 shows only a very short distance away from the wall (up to y+ = 60). The BeT numbers will keep their values close to unity beyond the y+ shown up to the flow center.

1.02

Bulk Bejan Number

1

0.98

0.96

0.94

0.92

0.9 220

260

300

340

380

4. Conclusions

420

Hb (kJ/kg) Fig. 9. Variations of the bulk Bejan number for Case I.

Fig. 10. Contour of the Bejan number in the entrance region of the pipe.

1.2

1

Turbulent Bejan Number

557

0.8

0.6 `

0.4

Cross section 1

0.2

Cross section 2 Cross section 3

0 0

10

20

30

40

50

60

y+ Fig. 11. Radial variations of the turbulent Bejan number for various flow cross sections for Case I.

the pseudo-critical region (the high heat transfer zone), the bulk Be numbers have their minimum values. As expected, the Be number in the liquid-like region (the region of lower dissipation rates) is greater than the Be number in the vapor-like region (the region of higher dissipation rates). The Be numbers present a different behavior in the flow entrance region where the entropy generation due to heat transfer in the central zone of the pipe is very less than the entropy generation due to energy dissipation. This becomes clear from the distribution of the constant Bejan number contours shown in Fig. 10. In the entrance region, the thermal boundary layer has not been fully

In this study the entropy generation in highly variable property flow of turbulent mixed convection heat transfer of supercritical carbon dioxide in a vertical pipe has been investigated. To accomplish this, a 2D CFD code has been developed to solve the governing equations which consist of mass, momentum, turbulence and entropy generation equations. The effect of wall heat flux on the local and total entropy generation has been examined. The most important results found are as follows. The bulk entropy generation reaches a minimum value when the heat transfer coefficient has a maximum value. In other words, the large enhancement of the heat transfer rate under supercritical conditions is accompanied with reduction in entropy production. This is while at normal pressures, the enhancement of heat transfer is usually made possible by employing techniques, such as mounting fins, which are accompanied with increase of entropy generation. This makes supercritical fluids a desirable medium for heat transfer applications. 1. It is generally true that in order to transfer the same amount of heat to a flowing fluid, less entropy will be generated once the heat flux at the wall is smaller. This effect is, however, much more pronounced under supercritical conditions where the rate of heat transfer, unlike normal pressure flows, enhances as the heat flux is reduced. 2. In a heated flow, it is observed that the bulk Be number is greater in the liquid-like region than the vapor-like region. While the entropy generation due to heat transfer in these two regions are almost the same, the flow turbulence, and hence the energy dissipation, increases continually from the liquid-like to vapor-like regions along the flow. 3. With increasing wall heat flux, temperature differences in the flow increase. In normal pressure flow, heat transfer coefficients remain constant with increase of the wall heat flux. As the result, the Be number increases. In the supercritical environment, the heat transfer rates decrease with the increase of the wall heat flux and cause further temperature difference in the flow. Thus, the increase of the Be number with the wall heat flux occurs to a larger extent in supercritical fluid flows compared to constant property flows. 4. To explain the contribution of the mechanisms of entropy generation (heat transfer and energy dissipation) in supercritical fluid flow, the flow cross section has been divided in two zones. One is the region very close to the wall (y+ < 10) and the other contains the rest of the flow cross section. 5. In the near wall region, the contribution of heat transfer in entropy generation is almost unchanged as there is no noticeable turbulent activities and the laminar sub-layer demonstrates the same heat transfer behavior at different flow cross sections. In

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this region, the energy dissipation is the factor which can make the difference with respect to entropy generation at various locations along the flow. 6. In the region away from the wall, the rates of heat transfer vary considerably along the flow. The local variation of entropy generation due to heat transfer is correspondingly significant. Such variations of entropy generation are much larger than the variations of entropy generation due to energy dissipation all across the flow (the bulk Be number is close to unity everywhere along the flow). Thus, the lower or higher entropy generation regions can be distinguished from the local heat transfer rates. A higher heat transfer region indicates a lower entropy generation zone and vice versa.

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