Int. J. Exergy, Vol. 3, No. 4, 2006 419

Copyright © 2006 Inderscience Enterprises Ltd.

Entropy generation due to heat and fluid flow in backward facing step flow with various expansion ratios

Eiyad Abu-Nada Department of Mechanical Engineering, Hashemite University, Zarqa, 13115, Jordan E-mail: eiyad@hu.edu.jo

Abstract: The present research investigated second law analysis of laminar flow over a backward facing step with various expansion ratios (ER). As Reynolds number (Re) increased the value of total entropy generation (Ns) increased. For smaller values of Re as the expansion ratio increased the value of Ns decreased. However, for higher values of Re as ER increased the value of Ns decreased but at smaller rates. The total Bejan number increased as the ER increased. At the bottom wall the minimum value of Ns occurred at the step location and the maximum value of Ns occurred inside the recirculation zone.

Keywords: entropy generation; second-law analysis; separated flow; heat transfer.

Reference to this paper should be made as follows: Abu-Nada, E. (2006) ‘Entropy generation due to heat and fluid flow in backward facing step flow with various expansion ratios’, Int. J. Exergy, Vol. 3, No. 4, pp.419–435.

Biographical notes: Eiyad Abu-Nada is an Assistant Professor in the Department of Mechanical Engineering at the Hashemite University in Jordan. Recently he has assumed the position of Head of the Department. He joined the Department of Mechanical Engineering on February 2002. He got his PhD from New Mexico State University in Mechanical Engineering in 2001. His areas of research are entropy generation, viscous fluid flow, computational heat transfer, and flow in micro channels.

1 Introduction

Heat Transfer and fluid flow processes are inherently irreversible, which leads to an increase in entropy generation and thus, destruction of useful energy. The optimal second law design criteria depend on the minimisation of entropy generation encountered in fluid flow and heat transfer processes.

In the last three decades several studies have focused on second law analysis of heat and fluid flow processes. Bejan (1979, 1987) showed that entropy generation in convective fluid flow is due to heat transfer and viscous shear stresses. Numerical studies on the entropy generation in convective heat transfer problems were carried out by different researchers. Drost and White (1991) developed a numerical solution procedure for predicting local entropy generation for a fluid impinging on a heated wall.

420 E. Abu-Nada

Abu-Hijleh et al. (1998, 1999) studied entropy generation due to natural convection around a horizontal cylinder. Extensive literature search indicated very limited work dealing with entropy generation in separated flows. For example, Cheng and Huang (1989) predicted entropy generation within a channel onto whose walls one or two pairs of transverse fins are attached symmetrically. However, they did not consider the effect of separation, reattachment and recirculation on total entropy generation.

Separated flows, accompanied with heat transfer, are frequently encountered in various engineering applications, such as microelectronic circuit boards, heat exchangers, ducts used in industrial applications and gas turbines. These separated flows are intrinsically irreversible because of viscous dissipation, separation, reattachment and recirculation that generate entropy. The flow over a backward facing step (BFS) has the most features of separated flows, such as separation, reattachment, recirculation, and development of shear layers. However, most of the published work on BFS has been extensively investigated, from a fluid mechanics or a heat transfer perspective. For example, Armaly et al. (1983) studied laminar, transition, and turbulent isothermal flow over a BFS, experimentally. Also, numerical studies in the laminar regime for isothermal flow were conducted by Armaly et al. (1983), Gartling (1990), Kim and Moin (1985). Flow over a BFS with heat transfer was conducted by Vradis et al. (1992), Pepper et al. (1992) and Vradis and VanNostrand (1992). Thangam and Knight (1989) studied the effect of step height on the separated flow past a backward facing step. Also, three-dimensional studies over a BFS were conducted by different researchers (Tylli et al., 2002; Barkely et al., 2002; Kaiktsis et al., 1991).

The analysis of entropy generation due to laminar forced convection over a BFS has been conducted by Abu-Nada (2005). Abu-Nada (2005) analysed the flow over BFS and evaluated the destruction of useful energy due to separation, reattachment and recalculating bubbles for an expansion ratio of 1/2. The objective of the present work is to perform a numerical study on the effect of expansion ratio on the entropy generation and on the Bejan number for a flow over a BFS. Also, to gain a better understanding of the effect of the expansion ratio in destroying useful energy in separated flows.

2 Governing equations

The continuity, momentum, energy and entropy generation equations in dimensionless form are given as (White, 1991; Bejan, 1982):

0u vx y

∂ ∂+ =∂ ∂

(1)

2 2

2 21

Reu u p u uu vx y x x y

∂ ∂ ∂ ∂ ∂+ = − + + ∂ ∂ ∂ ∂ ∂ (2)

2 2

2 21

Rev v p v vu vx y y x y

∂ ∂ ∂ ∂ ∂+ = − + + ∂ ∂ ∂ ∂ ∂ (3)

Entropy generation due to heat and fluid flow 421

2 2

2 21

Re Pru v

x y x yθ θ θ θ ∂ ∂ ∂ ∂+ = + ∂ ∂ ∂ ∂

(4)

22 22 2

Ns 2 .u v u vx y x y y xθ θ ϕ

∂ ∂ ∂ ∂ ∂ ∂ = + + + + + ∂ ∂ ∂ ∂ ∂ ∂ (5)

The following dimensionless quantities are defined as:

2, , , , and ,wc

H H m m wh wcm

T Tx y u v px y u v pD D u u T Tu

θρ

∗ ∗ ∗ ∗ ∗−

= = = = = =−

where Twh is the constant temperature of the hot wall and Twc is the constant temperature of the cold wall.

Equations (4) and (5) were derived by assuming that the temperature difference (Twh – Twc) is not relatively high so that the assumption of constant fluid physical properties is valid. Equation (5) consists of two parts. The first part is the entropy generation due to finite temperature difference (Nscond) and the second part is the entropy generation due to viscous effects (Nsvisc). The Bejan number is defined as:

cond

cond visc

NsBe .

Ns Ns=

+ (6)

3 Problem description

The basic flow configuration, under study, is shown in Figure 1. The expansion ratio (ER = S/H) is chosen as: 1/4, 1/3, 1/2, 2/3, and 3/4. The physical domain in the x direction is bounded by 0 < x < (30H). In the y direction it is bounded by –S/DH < y < 1/2. Table 1 gives the limits of the physical domain in the y direction as a function of the expansion ratio. The flow is assumed to be two-dimensional, laminar, steady and incompressible and to have constant fluid properties.

The flow at the inlet, at x = 0, is assumed to be hydro-dynamically fully developed, where a dimensionless parabolic velocity distribution is given as (Bejan, 1994):

2( ) 12( ).u y y y= − (7)

A no-slip velocity boundary condition is applied along the top wall, the bottom wall and the vertical wall of the step. A fully developed outlet velocity boundary condition is assumed:

0.u vx x

∂ ∂= =∂ ∂

(8)

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Figure 1 Problem geometry and boundary conditions

Table 1 S/DH, H/DH, and L as a function of ER

ER S/DH H/DH L (DH)

1/4 1/6 2/3 20

1/3 1/4 3/4 22.5 1/2 1/2 1 30 2/3 1 3/2 45 3/4 3/2 2 60

The temperature at the inlet is assumed to be fully developed and is given as (Bejan, 1994):

1 2 .yθ = − (9)

The temperature outlet boundary condition is assumed to be fully developed and is given as:

0.xθ∂ =

∂ (10)

The temperature is set to a constant value (θ = 1) on the step vertical wall and on the bottom wall downstream of the step. The temperature at the top flat wall is set equal to zero (θ = 0). The length of the computational domain is taken as (L = 30H) to ensure fully developed outlet boundary conditions (Gartling, 1990; Vradis et al., 1992; Pepper et al., 1992).

4 Numerical implementation

Equations (1)–(4), with the corresponding boundary conditions (equations (7)–(10)), are solved using the finite volume approach (Patankar, 1980; Versteeg and Malalasekera, 1995). The SIMPLE algorithm (Patankar, 1980) is used as the computational algorithm

Entropy generation due to heat and fluid flow 423

(Patankar, 1980; Versteeg and Malalasekera, 1995). The diffusion term in the momentum and energy equations is approximated by second-order central difference, which gives a very stable solution. However a hybrid differencing scheme is adopted for the convective terms, which utilises the favourable properties of the upwinding differencing scheme and central differencing scheme (Patankar, 1980; Versteeg and Malalasekera, 1995). A fine grid is used in the regions near the point of reattachment to resolve the steep velocity gradients, while a coarser grid is used downstream of the point (Anderson, 1995). This is done using a grid stretching technique that results in considerable savings in terms of the grid size, and thus, in computational time.

The resulting algebraic finite volume equations are solved with a tri-diagonal matrix algorithm (Thomas algorithm) with the line-by-line relaxation technique. The convergence criteria were defined by the following expression:

5, ,

1 1 1 1

resid 10 ,j M j Mi N i N

i j i jj i j i

φ= == =

−

= = = =

<∑∑ ∑∑ (11)

where resid is the residual; M and N are the number of grid points in the axial and the vertical directions respectively; and φ resembles u, v, or T.

After the temperature and velocity fields are obtained, equation (5) is used to solve for the entropy generation number at each grid point in the flow domain. The rate of entropy generation number at each cross section in the x direction, Ns (x), is calculated as:

1/ 2

/

Ns( ) Ns( , )d .H

y

y S D

x x y y=

=−

= ∫ (12)

Similarly, Be(x), is calculated as:

1/ 2

/

Be( ) Be( , )d .H

y

y S D

x x y y=

=−

= ∫ (13)

Simpson’s rule of integration was employed in the y-direction. The total entropy generation over the entire flow domain is calculated as:

1/ 2

0 /

Ns Ns( , )d d .H

yx L

x y S D

x y y x==

= =−

= ∫ ∫ (14)

Similarly, the total Bejan number for the entire flow domain is calculated as:

1/ 2

0 /

Be Be( , )d d .H

yx L

x y S D

x y y x==

= =−

= ∫ ∫ (15)

Because of the non-uniformity in grid distribution in the x direction, a cubic spline interpolation technique was used in the x direction followed by the application of the 1/3rd Simpson’s rule of integration.

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The Nusselt number can be expressed as:

( )Nu .Hh Dk

= (16)

The heat transfer coefficient is expressed as:

.( )

w

w b

qh

T T=

− (17)

The thermal conductivity is expressed as:

./wq

kT y

= −∂ ∂

(18)

By substituting equations (17) and (18) into equation (16), and using the non dimensional quantities, the Nusselt number on the bottom wall is written as:

/

1Nu .( )

Hw b y S Dy

θθ θ =−

∂= −− ∂

(19)

Similarly, the Nusselt number on the top wall is written as:

1/ 2

1Nu( )w b yy

θθ θ =

∂= −− ∂

(20)

where θb is bulk temperature and it is defined as:

1/ 2

/1/ 2

/

d.

d

H

H

S Db

S D

u y

u y

θθ −

−

=∫∫

A modified Nusselt number is defined as

0

NuNu .Nu x

∗

=

= (21)

4.1 Grid testing

Extensive mesh testing was performed to guarantee a grid independent solution. Eight different meshes were used, using ER = 0.5 and Re = 800, as shown in Table 2. This problem is a well known benchmark problem and is accepted as a benchmark problem by the ‘Benchmark Solutions to Heat Transfer Problems’ organised by the K-12 committee of the ASME for code validation and assessment (Pepper et al., 1992; Vradis et al., 1992). The present code was tested for grid independence by calculating the reattachment length (xr), x2, and x3; see Figure 1. Table 2 reports the results obtained for the grid independence study as shown in Table 2, a grid size of 125 × 250 (125 grid points in y and 250 grid points in x) ensure a grid independent solution.

Entropy generation due to heat and fluid flow 425

Table 2 Grid independence, Re = 800, ER = 0.5

Grid size xr x2 x3

13 × 24 6.50 Not predicted Not predicted 25 × 50 4.00 2.50 5.20 37 × 75 3.35 2.35 5.47 49 × 100 5.77 4.65 9.45 75 × 150 5.90 4.81 9.76 101 × 199 6.00 4.81 10.10 125 × 250 6.03 4.81 10.14 151 × 299 6.03 4.81 10.15

4.2 Code validation

The present numerical solution is validated by comparing the present results for the benchmark problem, for Re = 800 and ER = 1/2, with the experiment of Armaly et al. (1983) and with other numerical published data (Gartling, 1990; Kim and Moin, 1985; Vradis et al., 1992; Pepper et al., 1992). As shown in Table 3, the present results are very close to the other numerical published results. However, all of the numerical published works, including the present work, underestimate the reattachment length. According to Armaly et al. (1983), the flow at Re = 800 has three dimensional features. So, the underestimation of xr, by all numerical published data is due to the two-dimensional assumption embedded in the numerical solution (Armaly et al., 1983; Vradis et al., 1992; Abu-Nada, 2005).

Table 3 Validation of present numerical solution, Re = 800

Authors Type of work xr x2 x3

Armaly et al. (1983) Experimental 7.20 5.30 9.40 Vradis et al. (1992) Numerical 6.13 4.95 8.82 Kim and Moin (1985) Numerical 6.00 No data No data Gartling (1990) Numerical 6.10 4.85 10.48 Pepper et al. (1992) Numerical 5.88 4.75 9.80 Present work Numerical 6.03 4.81 10.15

On the other hand, further comparison between present results and previously published data, for ER = 1/3, 1/2, 2/3, is given in Figure 2. The figure shows excellent agreement between the present results and the experiment of Armaly et al. (1983) for Re ≤ 600. Also, it shows excellent agreement with all numerical published data for the whole range of Reynolds numbers and the whole range of expansion ratios. For more details on the validation of the x component of velocity and temperature profiles, see Abu-Nada (2005).

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Figure 2 Variation of the reattachment length with the Reynolds number for present and previous work, ER = 1/3, 1/2, 2/3

5 Results and discussion

The results are presented for expansion ratio range of 0.25–0.75 where the Reynolds number is always kept in the laminar regime. The Prandtl number is kept constant at 0.71. For the various expansion ratios, Reynolds number is given as follows (to ensure laminar flow):

• for ER = 3/4, Re ≤ 200 (Vradis and VanNostrand, 1992)

• for ER = 2/3, Re ≤ 400 (Vradis and VanNostrand, 1992)

• for ER = 1/2 Re ≤ 800 (Armaly et al., 1983; Kim and Moin, 1985; Thangam and Knight, 1989; Vradis and VanNostrand, 1992; Vradis et al., 1992; Pepper et al., 1992)

• for ER = 1/3, and 1/4, Re ≤ 800.

The variation of total entropy generation number (Ns) with Reynolds number is shown in Figure 3. The figure shows that the value of Ns increases as Reynolds number increases for all expansion ratios. Also, the figure reveals that the rate at which Ns increases for high expansion ratios is much greater than that for low expansion ratios. This can be explained by noting that the value of Ns is influenced by the following two major factors:

• the adverse pressure gradients that are due to the sudden expansion and are related to the expansion ratio

• the inertia effects that are related to the flow Reynolds number.

Entropy generation due to heat and fluid flow 427

Figure 3 Variation of Ns with Reynolds number for various values of ER

For high expansion ratio (ER = 3/4), the flow is mostly affected by the adverse pressure gradient and the velocity gradients are small. So, any change in Reynolds number increases inertia effects and velocity gradients and therefore increases Ns considerably. However, for low expansion ratio (ER = 1/4), the flow is mostly affected by inertia effects and pressure gradients effects are relatively small. Thus, the inertia effects play a dominant factor that contributes to Ns. Thus, any change in Reynolds number will not change the value of Ns significantly because the flow is already influenced by inertia effects. This explains the reason for the small rates at which Ns increases for low expansion ratios. On the other hand, for intermediate values of expansion ratio, both the adverse pressure and inertia effects have comparable effects on the value of Ns. So, the rate at which Ns increases is moderate.

Figure 4 shows the variation of Ns with ER. For low Reynolds number (Re ≤ 300) an increase in expansion ratio increases the adverse pressure gradient that causes the flow to slow down. Hence, the velocity gradients become smaller and the values of Ns are reduced. Thus, the slope at which Ns decreases with expansion ratio is large. On the other hand, for higher values of Reynolds number the magnitude of inertia effects increases, increasing the value of Ns. Thus, for Re ≥ 400 both inertia effects and adverse pressure gradients have equal effect on Ns. Thus, the decrease in Ns due to the adverse pressure gradient is countered by an increase in Ns due to the inertia effects. Thus, Figure 4 shows that the rate at which Ns decreases with ER for Re ≥ 400 is much smaller than that of Re ≤ 300.

Figure 5 shows the effect of Reynolds number on the Bejan number. The Bejan number decreases when Reynolds number increases, due to the increase in Nsvisc. Figure 5 reveals new features for the viscous and the conduction contribution to Ns. The figure shows that for high values of ER (i.e., ER = 3/4), approximately 60% of Ns is contributed by conduction and the rest by viscous effects. However, for small values of ER (i.e., ER = 1/4), approximately 20% is contributed by conduction and the rest by viscous effects. Thus, it is clearly seen that as ER increases the percentage of conduction contribution to Ns is increased.

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Figure 4 Variation of Ns with ER for various values of Reynolds number

Figure 5 Variation of Be with Reynolds number for various values of ER

The effect of expansion ratio on Bejan number is shown in Figure 6. There is an increase in Bejan number with an increase in expansion ratio. The relatively high values of adverse pressure gradient encountered at high expansion ratios tend to slow down the flow and make Nsvisc smaller. Thus, the Bejan number increases.

Figure 6 Variation of Be with ER for various values of Re

Entropy generation due to heat and fluid flow 429

The variation of cross-sectional entropy generation number along the channel length is shown in Figure 7(a) for ER = 1/3. The value of Ns(x) has a maximum at the step location and decreases downstream of the step. The separation at the step produces large velocity gradients that increase the viscous contribution to Ns(x) and therefore the value of Ns(x) increases. Moreover, the laminar boundary layer development at the leading edge of the top flat wall produces large velocity gradients and large shear stresses which increase the Ns(x) along the top wall. The dependence of Ns(x) on Reynolds number is illustrated in the same figure. There is an increase in Ns(x) with Reynolds number due to the increase in inertial effects. Moreover, the figure shows that above x = 10 the value of Ns(x) does not change significantly because the flow is approaching the fully developed condition.

Figure 7 (a) Variation of Ns(x) along the channel, ER = 1/3 and (b) variation of Be(x) along the channel, ER = 1/3

(a) (b)

Figure 7(b) shows the variation of Be(x) along the channel length for ER = 1/3. The figure shows that Be increases downstream of the step and reaches a constant value towards the end of the channel. Also, it is clear that low Reynolds number corresponds to higher values of Bejan number, due to the reduction in inertia and viscous effects. Also, Figures 8 and 9 show a similar trend for ER = 1/2 and 2/3 respectively. Figures 7–9 show that the value of Be fluctuates in the recirculation zone because the pressure gradients increase inside the recirculation zone and their orders of magnitude become comparable to the inertia effect.

Figure 8 (a) Variation of Ns(x) along the channel, ER = 1/2 and (b) variation of Be(x) along the channel, ER = 1/2

(a) (b)

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Figure 9 (a) Variation of Ns(x) along the channel, ER = 2/3 and (b) variation of Be(x) along the channel, ER = 2/3

(a) (b)

Figure 10(a) shows the axial variation of Ns along the bottom wall of the channel for Re = 400 and various expansion ratios. The minimum value of Ns occurs at x = 0, at the bottom left corner, since at this point there is no motion and no heat transfer is taking place. Also, Figure 10(a) shows that the maximum value of Ns occurs inside the recirculation zone and then it drops sharply to a very low value at the point of reattachment (compare Figure 10(a) with Figure 11). This behaviour can be explained by noting that the vortices increase dramatically inside the recirculation zone, which tends to maximise viscous contribution to Ns in this zone. However, at the point of reattachment there are no shear stresses. This eliminates viscous contribution and leaves only the conduction contribution to Ns. This explains the sharp drop in Ns after reaching its maximum value.

Figure 10 (a) Variation of Ns along the bottom wall and (b) variation of Ns along the top wall

(a) (b)

Figure 11 Contour plots for u-velocity Re = 400, ER = 2/3

Figure 10(a) reveals the appearance of a second peak after the point of reattachment. The beginning of the second peak occurs when the secondary recirculation bubble, on the top wall of the channel, appears; see Figure 11. Also, the end of the second peak occurs at

Entropy generation due to heat and fluid flow 431

the end of the secondary recirculation bubble. Thus, it is very clear that this second peak, at the bottom wall, is related to the secondary recirculation bubble. The secondary recirculation bubble does not influence the value of Ns on the top wall of the channel as shown in Figure 10(b). However, its main influence is to narrow down the flow passage between the secondary bubble and the bottom wall of the channel, as shown in Figure 11. This would increase velocity gradients between the secondary recirculation bubble and the bottom wall. These high values of velocity gradients at the wall are responsible for high rates of entropy generation numbers that cause the appearance of the second peak zone. Thus, it can be concluded that the effect of the secondary recirculation zone is crucial in increasing local rates of entropy generation on the bottom wall of the channel and, accordingly, the total rates of entropy generation over the entire flow domain. Thus, any second law analysis improvement on the problem in hand must take into account the secondary recirculation zone.

Figure 10(b) shows the Ns variation along the top wall of the channel, where a maximum value is detected at x = 0. The reason for having a maximum value of Ns at the leading edge of the top wall is due to the development of the viscous boundary layer, flow separation and high heat transfer rates where the highest values of local Nusselt numbers are recorded at the leading edge of the top wall (See Figure 12(a) and 12(b)). Also, Figure 10(b) shows that after the point of reattachment the value of Ns drops to very low values.

Figure 12 Variation of modified Nusselt number along the channel, Re = 400 (a) ER = 1/3 and (b) ER = 2/3

(a) (b)

Figure 13 shows the variation of the Bejan number along the bottom wall of the channel. The Bejan number starts from an undefined value at the x = 0 since no flow or heat transfer is taking places. As shown in Figure 13, the Bejan number takes the value of unity at the point of reattachment because of negligible viscous effects and the value of Ns is due to Nscond only. Also, toward the end of the channel the Bejan number starts decreasing since the flow is approaching the fully developed flow condition. Figure 13(b) shows the variation of Bejan number along the top wall where Be starts with low values due to the development of the boundary layer. For ER = 1/3, the Bejan number starts increasing after x = 0 and reaches a maximum value and drops to very low values near the end of the channel due to the fully developed condition. For ER = 2/3, the Bejan number shows two peaks: one occurs at the beginning of the secondary recirculation bubble and the second peak at the end of the secondary recirculation bubble

432 E. Abu-Nada

(See Figure 11). Between the two peaks the Be drops to low values due to the circulation inside the secondary recirculation bubble.

Figure 13 (a) Variation of Bejan number along the bottom wall and (b) variation of Bejan number along the top wall

(a) (b)

Figure 13 can be used to identify the places where conduction entropy generation and the viscous entropy generation are dominant, along the top and bottom walls. Also, it helps in specifying the places where both conduction and viscous entropy generation are of the same magnitude. It can be concluded that the bad regions that have high values of Ns are the point of separation, primary recirculation zone and secondary recirculation zone. The high production of Ns in these regions needs to be controlled to reduce entropy generation, thus conserving useful energy. Possible control methods are: using suction/blowing at the top and at the bottom walls or imposing magnetic fields at the top and bottom walls.

6 Conclusion

Entropy generation over a backward facing step with various expansion ratios was studied numerically. The results show that as Reynolds number increases, the value of total Ns increases. For lower values of Reynolds number, as the expansion ratio (ER) increases the value of Ns decreases. However, for higher values of Reynolds number, as ER increases the value of Ns decreases but at lower rates as compared to smaller Reynolds number. The total Bejan number decreases as Reynolds number increases. Also, the total Bejan number increases as the ER increases.

The maximum value of Ns(x) occurs at step and it decreases downstream of the step until it reaches a minimum value towards the channel outlet. However, the value of Be(x) increases in the downstream direction. At the bottom wall the minimum value of Ns occurs at the step and the maximum value of Ns occurs inside the recirculation zone. The value of Ns drops sharply to a very low value at the point of reattachment. However, at the top wall the value of Ns is maximum at x = 0 and drops to very low values after the point of reattachment. The effect of the secondary recirculation zone is very crucial in increasing local rates of entropy generation on the bottom wall of the channel.

At the bottom wall the value of Bejan number starts with high values near x = 0 and reaches a value of unity at the point of reattachment and starts decreasing toward the channel outlet.

Entropy generation due to heat and fluid flow 433

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Nomenclature

A Constant in the grid stretching equation a Upstream channel height (m) Be Bejan number (Be = (Nscond/Nscond + Nsvisc))

Br Brinkman number 2(Br ( / ( )))m wh wcu k T Tµ= −

D Location of grid clustering (m) DH Hydraulic diameter at inlet (DH = 2a) (m) ER Expansion ratio (S/H), dimensionless H Downstream channel height (m) h Local convection heat transfer coefficient (W m–2 K–1) k Thermal conductivity (W m–1 K–1) L Length of the channel (m) M Number of points in horizontal direction N Number of points in vertical direction

Ns Entropy generation number 2 2gen( / )HS D kτ′′′

Nu Nusselt number (h(DH)/k) *

Nu Modified Nusselt number (Nu/Nu|x=0)

p Dimensionless pressure *p Pressure (N m–2)

Pr Prandtl number (υ/α) Re Reynolds number (umDH/υ) qw Heat flux (W m–2) S Step height (m)

genS′′′ Volume rate of entropy generation (W K–1 m–3)

T Temperature (K) Tw Wall temperature (K) Tb Bulk temperature (K) um Average velocity of the incoming flow at inlet (m/s) u Dimensionless x component of velocity *u x component of velocity (m/s)

v Dimensionless y component of velocity

Entropy generation due to heat and fluid flow 435

*v y component of velocity (m/s)

x Dimensionless horizontal coordinate *x Horizontal distance (m)

x2 Beginning of the secondary recirculation bubble, dimensionless x3 End of the secondary recirculation bubble, dimensionless xr Reattachment length, dimensionless Y Dimensionless vertical coordinate *y Vertical distance (m)

Greek letters α Thermal diffusivity (m2/s)

β Clustering parameter θ Dimensionless temperature µ Dynamic viscosity (N.s m–2) ρ Density (kg m–3) τ Dimensionless temperature parameter (Twh – Twc))/Twc) υ Kinematic viscosity (m2/s) φ Viscous dissipation number (Br/τ) Subscripts b Bulk value cond Conduction LW Bottom wall m Mean value UW Top wall visc Viscous w Wall wc Cold wall wh Hot wall * Dimensional quantities