Applied Mathematics and Mechanics Accuracy ... - Springer Link

Appl. Math. Mech. -Engl. Ed., 34(8), 971–984 (2013) DOI 10.1007/s10483-013-1721-8 c Shanghai University and Springer-Verlag Berlin Heidelberg 2013

Applied Mathematics and Mechanics (English Edition)

Accuracy analysis of predicted velocity profiles of laminar duct flow with entropy generation method∗ J. A. ESFAHANI,

M. MODIRKHAZENI, S. MOHAMMADI

(Center of Excellence on Modeling and Control Systems (CEMCS) & Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran)

Abstract The objective of this work is to estimate the accuracy of a predicted velocity profile which can be gained from experimental results, in comparison with the exact ones by the methodology of entropy generation. The analysis is concerned with the entropy generation rate in hydrodynamic, steady, laminar, and incompressible flow for Newtonian fluids in the insulated channels of arbitrary cross section. The entropy generation can be calculated from two local and overall techniques. Adaptation of the results of these techniques depends on the used velocity profile. Results express that in experimental works, whatever the values of local and overall entropy generation rates are close to each other, the results are more accuracy. In order to extent the subject, different geometries have been investigated. Also, the influence of geometry on the entropy generation rate is studied, and the distribution of volumetric local entropy generation rate for the selected geometries is drawn. Key words internal flow

entropy generation, velocity profile, duct, arbitrary cross section, laminar

Chinese Library Classification O357.3 2010 Mathematics Subject Classification

41A46, 54C70, 83C15

Nomenclature a, vertical length of cross section; b, horizontal length of cross section; Dh , hydraulic diameter; F1 , F2 , functions of (a, b); G1 , G2 , functions of (a, b); H1 , H2 , functions of (a, b); K1 , K2 , functions of (a, b); L, axial length of duct; m, ˙ mass flow rate; p, pressure;

r, S˙ 0 , ∗ , S˙ gen

radial coordinate; characteristic entropy generation rate; non-dimensional entropy generation rate; S˙ gen , entropy generation rate per volume; S˙ gen , entropy generation rate; T, static temperature; U, average fluid velocity; u, v, w, velocity components; x, y, z, cross sectional coordinates.

∗ Received Nov. 29, 2012 / Revised Feb. 13, 2013 Corresponding author J. A. ESFAHANI, Professor, Ph. D., E-mail: [email protected]

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J. A. ESFAHANI, M. MODIRKHAZENI, and S. MOHAMMADI

Greek symbols ρ, μ,

1

fluid density; dynamic fluid viscosity;

Δp,

pressure drop.

Introduction

In recent years, the optimization of industrial processes for maximum utilization of the available energy has become a very active line of scientific research. Conservation of useful energy depends on the design of efficient thermodynamic heat transfer processes, i.e., minimization of entropy generation due to heat transfer and flow friction. As we know, the first law of thermodynamics is about of the conservation of energy principle while the second law of thermodynamics states that all real processes are irreversible. Entropy generation in real processes is the criterion of irreversibility which wastes the available energy of systems and causes efficiency reduction. Studying the distribution of the entropy generation rate within the fluid volume is important if the preservation of energy quality in a fluid flow process is needed. Decreasing the considered entropy generation rate needs a better understanding of how and where the entropy is generated. Considerable research studies were carried out to examine the entropy generation rate in the flow systems. Bejan[1] presented the basis for the entropy generation minimization method. In his book, Bejan[2] also conducted the second law analysis of thermodynamics via the minimization of the entropy generation for the single-phase convection heat transfer. Furthermore, Bejan[3] devised concrete methods for minimizing entropy generation in the engineering equipment for heat transfer. Many researchers have continued Bejan’s works and used the entropy generation method in analyzing their problems. Saouli and Aiboud-Saouli[4] performed entropy analysis of laminar liquid falling film along an inclined heated plate and found that fluid friction irreversibility dominates over heat transfer irreversibility. Also, Teymourtash et al.[5] used second law analysis and studied the effects of rate of expansion and injection of water droplets on the entropy generation of nucleating steam flow in a Laval nozzle. Moreover, the entropy generation in a vertical concentric isothermal channel with a temperature-dependent viscosity was presented by Tasnim and Mahmud[6] . Furthermore, Mahmud and Fraser[7] analyzed the irreversibility of concentrically rotating annuli. They presented the distributions of volumetric average entropy generation rate for both isothermal and iso-flux conditions. Dung and Yang[8] , Li and Yang[9] , Yang et al.[10] , Li and Yang[11] , and Chen et al.[12] considered the entropy generation of film condensation on the outer surface of a circular and elliptical tube. They investigated the parameters which influence the entropy generation number. Furthermore, the entropy generation analysis in film condensation on an elliptical tube with interfacial shear stress at high bond numbers are studied by Esfahani and Koohi-Fayegh[13]. They tried to discuss the optimum condition related to bond numbers by introducing two sets of dimensionless numbers by using the entropy generation minimization method. An analysis of the mechanism of entropy generation on an elliptical tube encountered in a forced convective film condensation heat transfer was studied by Esfahani and Modirkhazeni[14–15] . A complete thermodynamic analysis, including first and second law was done by them. They concluded that decreasing Brinkman number descended the total entropy generation. The attempt of Narusawa[16] was to study the second law analysis of mixed convection in the rectangular ducts. The trial of Ibanez et al.[17] to minimize the entropy generation by the asymmetric convective-cooling can also be referenced as a part of the entropy generation works. It would be mentioned that recently Chen published some papers which are concerned with entropy generation. The systematically investigation of entropy generation of turbulent double-diffusive natural convection in a rectangle cavity[18] , numerical solving the effect of Richardson number on entropy generation

Accuracy analysis of predicted velocity profiles of laminar duct flow

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over backward facing step[19] , and the analysis of entropy generation in counter-flow premixed hydrogen-air combustion[20] were some of his studies. Considering the non-adiabatic ducts, Sahin[21] presented the second law analysis to minimize losses for different shaped ducts such as triangular, sinusoidal, etc, in laminar flow with the constant wall temperature boundary conditions. After comparing different ducts to find an optimum shape that has minimum losses, he reported that the circular duct is the best choice among the various geometries. Also, he did another study on investigating the constant heat flux effects on the ducts of mentioned cross section without taking into account the viscosity variation in the analysis[22]. Furthermore, a study on entropy generation through hexagonal ¨ cross-sectional duct for the constant wall temperature in laminar flow was applied by Oztop ¨ et al.[23] . Also, Oztop et al.[24] worked on the second law analysis of fully developed laminar flow for rectangular ducts with semicircular corners. They considered two different boundaries for ducts as the constant wall temperature and the constant heat flux. The effects of some parameters like Reynolds and aspect ratio of cross section were analytically analyzed by them. Minimizing heat transfer and fluid flow entropy generation in the internal flows by adjusting the shape of the cross-section was done by Jankowski[25]. He presented a design correlation that allows a determination of the optimal shape of a duct by the given flow rate, the heat transfer rate, the available cross section, and the fluid properties. Recently, effect of non-uniform heating on the entropy generation for the laminar developing pipe flow of a high Prandtl number fluid was investigated by Esfahani and Shahabi[26] . Their attempt was to discover the optimum case in which minimum entropy is generated. They concluded that the generated entropy for the cases with decreasing heat flux distribution is always more than the case of uniform heat flux. To add more, the entropy generation analysis of a flat plate boundary layer with various solution methods was carried out by Esfahani and Malek-Jafarian[27]. Their results showed that the exact solution (similarity solution) is the one that minimizes the rate of total entropy generation in the boundary layer. Also, they stated that if the exact solution for a specified problem is not available, the evaluation of the approximate solutions and recognizing the best one among them can be done with the help of the entropy generation method. Esfahani and Roohi[28] studied different approximate solutions of heat conduction equation, integral, and variation Ritz and Kantorovich methods in a plate with internal heat generation. The entropy generation analysis was carried out by them for each approximate solution as well as the exact solution. Their study resulted that the entropy generation analysis is a good check for the validity of approximate solutions. Also, they found that a new parameter which is defined as “average normalized entropy generation” behaves exactly like “average error” between exact and approximate solution. Furthermore, Esfahani and Koohi-Fayegh[29] worked on the entropy generation analysis in error estimation of an approximate solution for a semi infinite conduction problem. They tried to introduce an average normalized entropy generation in order to prove that the error of the integral solution is in the same order as the values calculated for the normalized entropy generation. They concluded that when no exact solution is available for a similar problem, one can verify the error of the available approximate solutions by applying an entropy generation analysis to the problem. In the present study, it has been tried to prove that the methodology of the entropy generation is a good way for estimating the accuracy of a predicted velocity profile which may be obtained from an experimental work or may be guessed as an approximate profile. These profiles are related to the laminar internal flow in different ducts which can be easily extended to different class of fluid flow. To affirm this idea, the entropy generation rate has been calculated with the help of two local and overall techniques for six different geometries (circular, elliptical, quadrangular, rectangular, triangular, and annular). As the flow friction irreversibility is considered, only a velocity profile is needed for calculating the entropy generation rate. The required velocity profile can be one of two types, exact or approximate. At the first step, the local entropy generation rate has been defined by using exact and approximate velocity

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profiles of different geometries. The overall entropy generation rate which is proportional to pressure drop has been calculated for the exact and approximate velocity profiles at the next step. Finally, the results of computed local and overall entropy generation rates have been compared with each other for exact and approximate velocity profiles separately. The results show that if the exact profile is used, the concluded local and overall entropy generation rates are equal to each other, while they are not the same for the approximate velocity profile. In the following, the effect of geometry on entropy generation rate has been investigated. The relationship between non-dimensional entropy generation rate and number of corners expresses that increasing the number of corners will decrease the value of entropy generation rate. Also, to study the effect of the cross section of a duct on irreversibility, contours of distribution of volumetric local entropy generation rates have been drawn.

2

Problem statements

Consider a viscous steady fully developed laminar flow in an adiabatic channel with an arbitrary constant cross-sectional area A, and a constant length L bounded by a closed curve Γ as shown in Fig. 1. The flow is assumed to be incompressible, and the fluid properties such as ρ and k are constants. Also, it is considered that there is no mass diffusion, chemical reaction, and electromagnetic effects. Body forces such as gravity, centrifugal, and Coriolis do not exist. The rarefaction and surface effects are assumed to be negligible, and the fluid is considered to be continuum[30] .

Fig. 1

Arbitrary cross-section channel subjected to constant pressure gradient

According to the proposed physical model, the governing equation for a steady state and laminar fluid can be written as follows: −∇p + μ∇2 V = ρV · ∇V ,

Pressure

Friction

(1)

Inertia

which is a balance of three inertia, pressure, and friction forces. For the fully developed flow that the length of the tube is much larger than the hydrodynamic entrance length in a duct of arbitrary but constant cross section, Eq. (1) will be simplified to Poisson’s equation[31] ∇p = μ∇2 V .

(2)

For a one-dimensional problem, Poisson’s equation can be simplified to ∇2 u =

1 dp = constant, μ dx

(3)

where x and u are the flow direction and the related fluid velocity, respectively. This equation is always used to determine the exact velocity profiles.


3

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Exact solutions

As it was mentioned before, the velocity profile for an arbitrary cross-sectional duct can be obtained by integrating Poisson’s equation. This velocity profile is needed to compute the fluid flow entropy generation rate. The process for obtaining the velocity profile for a circular duct is presented here by details[32–33] . The other profiles for elliptical, quadrangular, rectangular, triangular, and annular ducts are just listed in Table 1, which are rederived by the authors themselves in the same way as follows. Table 1

Analytical velocity profiles for selected geometries

Cross section

Analytical velocity profile “ “ r ”2 ” 2U 1 − a

Circular

“ “ y ”2 “ z ”2 ” 2U 1 − − a b

Elliptical

Rectangular

∞ “ P 1 “ Δp ”“ y 2 − a2 (−1)n a2 + 16 ` π(2n+1)b ´ 2 μ L 2 n=0 (2n + 1) π 2 cosh 2a

· cos

“ π(2n + 1)y ” 2a

cosh

“ π(2n + 1)z ””” 2a

∞ “ P 7.092U “ 2 (−1)n (2a)2 2) + 2(y − a ` (2n+1)π ´ a2 n=0 (2n + 1)2 π 2 cosh 2

Quadrangular

· cos

Triangular

“ π(2n + 1)y ” 2a 0.07U

Annular

2U

cosh

“ π(2n + 1)z ””” 2a

“ 27y 3 − 81yz 2 − 54a(y 2 + z 2 ) + 32a3 ” a3

“ (r 2 − a2 ) ln ` a ´ + (a2 − b2 ) ln ` a ´ ” b ` ´ r (a2 + b2 ) ln ab + b2 − a2

The physical models for various cross sections have been shown in Fig. 2. Consider a circular cross-section duct shown in Fig. 2(a) with the corresponding boundary condition as r = a → u(r) = 0, r=0→

d u(r) = 0. dr

(4) (5)

By double integrating Eq. (3) and substituting the boundary conditions in the updated equation, an analytical velocity profile will be gained as follows: r 2 u(r) = 2U 1 − , a

(6)

where U is an average velocity which can be introduced as U = 0.125

a2 Δp . μ L

(7)

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Fig. 2


Cross sections of (a) circular tube, (b) elliptical tube, (c) rectangular tube, (d) quadrangular duct, (e) triangular duct, and (f) annular duct

This equation can be rearranged due to the pressure drop (Δp), i.e., Δp =

8μU L , a2

(8)

which is necessary for overall method of computing the entropy generation rate that will be explained later. The method which has been presented for obtaining the circular velocity profile can be applied to gain the velocity profile of the other shapes. The analytical velocity profiles are presented in Table 1. Also, mean velocities and pressure drops for all considered geometries are shown in Table 2. It should be mentioned that the complete formulas of the analytical mean velocity and the pressure drop for a rectangular duct are extremely long. Due to this matter, these formulas have been neglected to be listed in Table 2, and only the closed forms of them (F (a, b)) have been reported. Responses for F1 and F2 that are reported in Table 2 are just as a sample for the specific case of a = 2 and b = 3. Table 2 Cross section Circular

Analytical mean velocities and pressure drops

Analytical mean velocity

.“ 1 “ Δp ”” μ

0.125a2

Analytical pressure drop/(μU L) 8 a2

a2

4(a2 + b2 ) a2 b2

Rectangular

F1 (a, b) · F1 (2, 3) = 0.783

F2 (a, b) · F2 (2, 3) = 1.277

Quadrangular

0.141a2

7.092 a2

Triangular

0.067a2

Annular

“ b2 − a2 ” ` ´ 0.125 a2 + b2 + ln ab

15 a2 ` ´ 8 ln ab à´ (a2 + b2 ) ln b + b2 − a2

Elliptical

0.25

a2 b2 + b2

L

*Complete formulae of F1 and F2 are very long so they have just been reported for special case of a and b


4

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Approximate solutions

In different case studies, where the exact profile is not available or using it causes trouble in calculations, one may use a predicted profile to simplify the problem. This predicted profile may be the result of an experimental test whose accuracy is unknown or this profile can be guessed according to the behavior of the problem. In this paper, due to the treatment of the exact profile, the approximate velocity profile for the selected geometries has been selected. For example, with respect to the boundary conditions for a circular duct (see Fig. 2(a)), it is assumed that the velocity field can be introduced by the following profile: π r u(r) = A cos . (9) 2a Since this velocity profile should satisfy Poisson’s equation, the constant coefficient A can be gained as A = 1.161U,

(10)

where U is an average velocity for this approximate profile which is defined as U = 0.147

a2 Δp . μ L

(11)

Similar to the exact solution method, Eq. (11) can be rearranged with respect to the following pressure drop: Δp =

6.790μU L . a2

(12)

The optional velocity profiles and their corresponding mean velocities and pressure drops are listed in Table 3 and Table 4. Therefore, it is left to the reader to apply the above steps for other cross-sections. Table 3 Cross section Circular

Approximate velocity profiles Approximate velocity profile “π r ” 1.161U cos 2a

Elliptical

6U (105πb2 + 60πy 2 − 80by 2 + 16a2 b)(a2 b2 − a2 z 2 − b2 y 2 ) a2 b2 (8a2 b + 30πa2 + 315πb2 )

Rectangular

“ y ”2 ”“ “ z ”2 ” 9 “ U 1− 1− 4 a b

Quadrangular

“ y ”2 ”“ “ z ”2 ” 9 “ U 1− 1− 4 a a

Triangular

U (2a + 3y)(−27z 2 + 9y 2 − 24ya + 16a2 ) 4 a2 (−3y + 4a)

Annular

` ´ ` ´ 2.465U (a2 − b2 )3 cos π2 ar cos π2 rb ` ` ` ´ ` ´´ ` ` ´ ` ´´´ a2 b2 π(b2 − a2 ) cos π2 ab + cos π2 ab + 4ab sin π2 ab − sin π2 ab

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J. A. ESFAHANI, M. MODIRKHAZENI, and S. MOHAMMADI Table 4

Cross section

Approximate mean velocities and pressure drops

Approximate mean velocity

.“ 1 “ Δp ”” μ

Approximate pressure drop/(μU L)

L

Circular

0.147a2

6.790 a2

Elliptical

0.083a2 b(8a2 b + 30πa2 + 315πb2 ) 45πa2 b + 16a4 + 105πb3 + 96a2 b2

12(45πa2 b + 16a4 + 105πb3 + 96a2 b2 ) a2 b(8a2 b + 30πa2 + 315πb2 )

Rectangular

0.333

a2 b2 a2 + b2

3

(a2 + b2 ) a2 b2

Quadrangular

0.167a2

6 a2

Triangular

0.048a2

Annular

“ “ “π a” 0.129a2 b2 π(b2 − a2 ) cos 2 b “ “π a” “ π b ””” “ π b ”” + 4ab sin − sin + cos 2a 2 b 2a “ π a ””−1 “ “π b ” − cos ·(a2 − b2 )−2 cos 2a 2 b

21 a2 “

5

“π b” 2a “ π a ””“ “ − cos a2 b2 π(b2 − a2 ) 2 b “ π b ”” “ “π a” + cos · cos 2 b 2a “ π b ””””−1 “ “π a” − sin +4ab sin 2 b 2a 7.752(a2 − b2 )2 cos

Computation of local entropy generation rate

The main purpose of this study is to prove that the entropy generation is a good method for estimating the accuracy of a velocity profile which can be used in any work. Therefore, at the beginning of the analysis, the entropy generation should be computed. One way to gain the entropy generation rate is the local method which is based on integrating the local entropy generation rate. The volumetric entropy generation rate at each point of the flow field for a circular duct is[1] k ∂T 2 ∂T 2 μ (13) + Φ, = 2 + S˙ gen T ∂x ∂r T where Φ=2

∂v 2 ∂r

+

v 2 r

+

∂u 2 ∂x

+

∂v ∂x

+

∂u 2 . ∂r

(14)

The first term on the right-hand side of Eq. (13) is the thermal entropy generation due to the finite temperature differences in thermal contacts, and the second one is the frictional entropy generation due to viscous effects of the fluid. Because an adiabatic duct has been chosen for this paper, the thermal entropy generation rate will be omitted. Therefore, Eq. (13) for the proposed physical model can be simplified to μ ∂u 2 S˙ gen = . (15) T ∂r 5.1 Local entropy generation rate for exact solution By substituting Eq. (6) into Eq. (15), the entropy generation per volume for a circular duct can be obtained as μ 16U 2 r2 S˙ gen = . T a4

(16)


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The entropy generation rate can be gained by the triple integration of Eq. (16). Therefore, the entropy generation rate from the local method for a circular duct can be defined as 25.133μU 2L S˙ gen,loc = . (17) S˙ gen rdrdθdx = T In order to avoid repeating of the method, the local entropy generation rate just for a circular duct has been explained here and it has been presented for all considered geometries in Table 5. Table 5

Local and overall entropy generation rates for exact solutions

Cross section

.“ μU 2 L ” Local entropy generation rate T

Circular

25.133

25.133

Elliptical

4(a2 + b2 ) ab

4(a2 + b2 ) ab

Rectangular

G1 (a, b) · G1 (2, 3) = 30.650

G2 (a, b) · G2 (2, 3) = 30.650

Quadrangular

28.454

28.454

Triangular

34.641 ` 25.133 ln ab )(a2 − b2 ) ` ´ (b2 + a2 ) ln ab + b2 − a2

34.641 ` ´ 25.133 ln ab (a2 − b2 ) ` ´ (b2 + a2 ) ln ab + b2 − a2

Annular

Overall entropy generation rate

.“ μU 2 L ” T

*Complete formulae of G1 and G2 are very long so they have just been reported for special case of a and b

5.2 Local entropy generation rate for approximate solution The same steps that are followed for the exact solution for obtaining the local entropy generation rate for the circular duct can also be applied here to compute the local entropy generation rate for the approximate profiles. Therefore, the local entropy generation per volume can be obtained by substituting Eq. (9) into Eq. (13) for the approximate velocity profile as follows: 11.526μU 2 2 π r S˙ gen sin . (18) = a2 T 2a and the local entropy generation rate after the triple integration of Eq. (18) will be gained as 25.444μU 2L ˙ Sgen,loc = . (19) S˙ gen rdrdθdx = T The local entropy generation rates for all arbitrary cross sectional ducts are depicted in Table 6. Table 6

Local and overall entropy generation rates for approximate solutions

Cross section

.“ μU 2 L ” Local entropy generation rate T

Circular

25.444

Elliptical

H1 (a, b) · H1 (2, 3) = 28.768

.“ μU 2 L ”

b2 )

H2 (a, b) · H2 (2, 3) = 27.245

ab

Quadrangular

28.8

24

Triangular

45.726

48.498

Annular

K1 (a, b) · K1 (2, 3) = 188.243

K2 (a, b) · K2 (2, 3) = 182

+

T

21.332

12(a2 + b2 ) ab

Rectangular

14.4(a2

Overall entropy generation rate

*Complete formulae of H1 , H2 , K1 , and K2 are very long so they have just been reported for special case of a and b

980

6


Computation of overall entropy generation rate

The entropy generation can also be computed from the overall method which does not need any integration from the governing equation. This method is based on using pressure drop and mass flow rate for obtaining the irreversibility. The corresponding equation for obtaining the entropy generation rate can be defined as[2] m ˙ Δp . S˙ gen,ove = ρ T

(20)

6.1 Overall entropy generation rate for exact solution By substituting the pressure drop from Eq. (8) into Eq. (20), the overall entropy generation rate can be gained as 25.133μU 2L S˙ gen,ove = . T

(21)

The overall entropy generation rates for all selected geometries are written in Table 5. 6.2 Overall entropy generation rate for approximate solution There are many auxiliary instruments for measuring the pressure drop during an experiment. The measured pressure drop can be used for calculating the amount of irreversibility. The overall entropy generation rate for a measured pressure drop or for Δp which is gained from the guessed approximate profile, can be described with the help of Eq. (20) for the circular duct as Δp 21.332μU 2L S˙ gen,ove = ρU (πa2 ) = . ρT T

(22)

The amounts of overall entropy generation rates for all of the approximated solutions of considered ducts have been reported in Table 6.

7

Results and discussion

The main attempt in this study was to find a new method to measure the accuracy of experimental works. Due to this idea, the differences between the amount of local and overall entropy generation rates of exact and approximate velocity profiles have been discussed. As it was mentioned before, the entropy generation rates resulted from two local and overall techniques have been presented in Table 5 for the exact velocity profile and Table 6 for the approximate one. Comparing the amounts of entropy generation rates demonstrates that for exact velocity profile of any arbitrary cross sections, the local and overall entropy generation rates are equal to each other while if the approximate velocity profile is used, the resulted local and overall entropy generations are not the same. These results state that if the local and overall entropy generations produced by a profile are equal to each other, the used profile is an exact one. Also, it can be concluded that the difference between the local and overall irreversibility is a good scale for estimating the accuracy of a velocity profile because for an exact profile, the local and overall amounts are the same. Thus, whatever the local and overall entropy generation rates close to each other, the experiment has more accuracy. Therefore, if an exact profile is not available, an approximate one can be substituted and its exactitude can be estimated with the help of the entropy generation method. As the mentioned results are correct for all the optional geometries, this result can be extended to each arbitrary cross sectional duct in any analytical or experimental works. Because of the long length of the formulas for the entropy generation rate of elliptical and annular ducts, the complete function has not been mentioned in Tables 5 and 6, and they have just been reported for a specified case.


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It can be resulted from Table 6 that for most of the selected cross sections, the overall entropy generation rate is less than the local one. Furthermore, by comparing with the values of Tables 5 and 6, it can be seen that for all the optional ducts, the amount of approximate local entropy generation rate, in contrast with the amount of approximate overall entropy generation rate, is closer to the amount of exact entropy generation rate. Therefore, it may be concluded that the overall method for computing irreversibility, causes more error in solutions and has less exactness than local method due to its larger difference to the exact values. It should be noted that this result is valid for all arbitrary ducts. In order to consider the effect of cross section on irreversibility, a comparison between the amounts of non-dimensional entropy generation rates of a circular, quadrangular, and triangular duct has been done and it is shown in Fig. 3. This figure shows the variation of non-dimensional entropy generation rate via the number of the edges of cross section of a duct. It would be mentioned that the non-dimensional entropy generation rate can be defined as Eq. (23) in order to compare with the entropy generation of cross sections[22] , i.e., S˙ gen ∗ = , S˙ gen S˙ o

(23)

where S˙ o is the characteristic entropy generation rate, and due to the formulas of the computed entropy generation rate, it is defined as D4 Δp 2 S˙ o = h L . μT L

(24)

Figure 3 expresses that the circular cross section produces the least amount of nondimensional entropy generation while the triangular cross section has the largest value. These results are predictable because the circular duct causes small resistance for the flow path, and consequently, it generates less entropy generation. It can be resulted from this figure that whatever the number of corners increases or in other words how much the cross section profile of a duct approaches to a circle, the irreversibility decreases. Therefore, from this point of the view, the circular cross section is an optimum selection in the design of a duct. It would be noted that the entropy generation rate is not a comprehensive parameter to choose the best case because in real process there are some other parameters such as economics and executive dictates which influence the design calculations.

Fig. 3

Variation of entropy generation rate versus number of sides of duct

The distribution of local entropy generation rate per unit volume resulted from the exact velocity profile for different cross sections (for specified a and b) are shown in Fig. 4. This figure leads us to the better understanding of how geometry influences the irreversibility. Figure 4 illustrates that in triangle, quadrangle, and rectangle, the corners have less amount of entropy generation in compared with the sides of the duct. This treatment can be explicated according to the shear stress. The shear stress increases by approaching to the sides of duct. The

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physical relationship between the shear stress and the friction factor, and on the other hand, the relationship between the friction factor and the entropy generation rate states that the shear stress affects the entropy generation rate. Therefore, the entropy generation rate also increases by approaching to the sides of the duct.

Fig. 4

8

Distribution of volumetric local entropy generation rates in duct of (a) triangular (b) quadrangular (c) rectangular (d) circular (e) elliptical cross sections

Conclusions

In the present study, the entropy generation method has been used to estimate the accuracy of an approximate velocity profile which can be gained from experimental works. These profiles related to the class of laminar internal flow in different ducts which can be easily extended to different class of fluid flow. Two local and overall techniques have been used to compute the rate of flow friction entropy generation in the duct of arbitrary cross section. The results show that if the exact velocity profile is used, the concluded local and overall entropy generations will be equal to each other, while they are not the same as the approximate velocity profile. This result means that in experimental works, whatever the amount of local and overall entropy generation rates are close to each other, the obtained velocity profile has more accuracy. Consequently, it


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can be said that the entropy generation method is a helpful tool to experimental works. Also, the effect of geometry on entropy generation has been investigated. Results express that as the number of corners increases or in other words how much the cross section profile of a duct approaches to a circle, the irreversibility decreases. Acknowledgements This research is partly supported by a grant from the Center of Excellence on Modeling and Control Systems (CEMCS) of Ferdowsi University of Mashhad, Iran.

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