INTRODUCTION
The optimal design criteria for thermal systems can be achieved by minimizing
entropy generation in the systems. This problem has been the topic of great
interest in fields such as heat exchangers, energystorage systems, power plants,
refrigeration and space applications. There may exist an optimal thermodynamic
design of these systems which minimizes the amount of entropy generation. Increased
effort is being directed at producing more efficient heat exchangers to affect
savings of energy, material and labor. Improvements in heattransfer augmentation
depend on performance and manufacturing cost (Kakac et al.,
1981). Consequently, there is increased need for utilization of a variety
of duct geometries for heattransfer applications with forced convection and
internal flow. Ko and Ting (2006a) analyzed entropy generation
induced by forced convection in a curved rectangular duct with external heating
by numerical methods. The problem is assumed as steady, threedimensional and
laminar. They analyzed the local entropy generation distributions as well as
the overall entropy generation in the whole flow fields, including the entrance
region and fully developed region. Ko (2006) analyzed
the optimal mass flow rate for steady, laminar, fully developed, forced convection
in a helical coiled tube with fixed size and constant wall heat flux by the
thermodynamic second law based on the minimal entropy generation principle.
Two working fluids, namely air and water, were considered. Ko
and Ting (2006b) analyzed the optimal Reynolds for the steady, laminar,
fully developed forced convection in a helical coiled tube with constant wall
heat flux based on minimal entropy generation principle. Two working fluids,
water and air, were considered. It is found that the entropy generation distributions
are relatively insensitive to coil pitch. Also they proposed a correlation equation
for the optimal Reynolds as a function of curvature ratio and two dimensionless
duty parameters, after a leastsquareerror analysis. Ko
(2006) analyzed the optimal curvature ratio for steady, laminar, fully developed
forced convection in a helical coiled tube with constant wall heat flux. Also
he proposed a correlation equation for the optimal curvature ratio as a function
of Reynolds and two dimensionless duty parameters through a leastsquareerror
analysis. Different criteria for selecting and optimizing the heatexchanger
passage geometries were outlined by Bergles (1997). Because
of size and volume constraints in applications to aerospace, nuclear, biomedical
engineering and electronics, it may be required to use noncircular flowpassage
geometries, particularly in compact heat exchangers and solar collectors. Analytical
solutions of heat transfer and pressure drop for laminar flows in many duct
geometries are available in the literature (Shah, 1975).
In order to enhance the heat transfer, longitudinal fins and twisted tapes are
being used in circular and noncircular ducts. Heat transfer enhancement is achieved
at the expense of pumping power due to the increased friction factor. Efficient
utilization of exergy has become one of the primary objectives in designing
any thermodynamic system. Minimization of entropy generation in thermodynamic
systems leads to efficient use of exergy (Bejan, 1996).
The irreversibilities associated with fluid flows through ducts are usually
related to heat transfer and viscous friction. Different mechanisms and design
features contributing to irreversibilities often compete with one another (Bejan,
2006). Therefore, there may exist an optimal thermodynamic design which
minimizes entropy generation. Irreversibilities in various duct geometries with
constant wall heat flux and laminar flow were studied by Sahin
(1998). Correlations of optimal body sizes with external forcedconvection
heat transfer were discussed by Fowler and Bejan (1994).
Bejan (1980, 1982) presented the
procedure for entropygeneration minimization at the systemcomponent level.
Nag and Kumar (1989) studied second law optimization
for convective heat transfer through a duct with constant heat flux and plotted
the variation of entropy generation vs. the temperature difference between the
bulk flow and surface using a duty parameter. For this case (Nag
and Kumar, 1989), the product of Stanton number and temperature difference
between the bulk and surface is constant owing to the constant heat flux imposed
on the surface. Laminar heat convection in ducts of irregular cross sections
was studied by Uzun and Unsal (1997). They analyzed forced
convective heat transfer during hydrodynamic fully developed laminar fluid flow
inside ducts of irregular cross section utilizing numerical methods.
Here, we discuss various duct geometries. The entropy generation and
pumping power required are compared in order to find the optimum duct
geometry which minimizes exergetic losses for a range of laminar flows
and specified wall heat flux. Rhombic crosssection risers are being used
by some manufactures of solar flatplate collectors. It is customary in
design to treat those cross sections as similar to circular tubes only
using an equivalent hydraulic diameter. Due to difference in geometry
between the two crosssections i.e., circular and rhombic it is reasonable
to doubt their exactly similar thermal and hydraulic behaviors. The present
study is intended to shed some light on this matter.
Entropy generation and required pumping power analysis: In order
to calculate the entropy generation, we consider an axially uniform duct
of arbitrary cross section with a specified heat flux imposed on its surface.
An incompressible viscous fluid with mass flow rate
and inlet temperature T_{O} enters the duct of length L. Heat
transfer to the bulk of the fluid occurs through the heat transfer coefficient
h, which is taken as constant along the surface of the duct for fully
developed laminar flow and constant thermophysical properties.
The entropy generation within a control volume of thickness dx along the duct
is (Bejan, 2006):
For an incompressible fluid
and
where, p is the perimeter of the duct. Integrating Eq. 3,
the bulk temperature variation of the fluid and the total heat transfer
along the duct are:
and
respectively. The pressure drop in Eq. 2 is (Kreith
and Bohn, 1993):
where, f is the friction factor. A dimensionless total entropy generation based
on the flow stream heat capacity rate
is defined as (Sahin, 1998):
where, ΔT is the increase of the fluid temperature, T_{(x=L)}–T_{o}
(Sahin, 1996). Integrating Eq. 1 along
the duct length L and by using Eq. 26, the
dimensionless total entropy generation becomes:
where,
the Stanton number is
and the Eckert number is:
Equation 8 may be written as:
Therefore
Table 1: 
Hydraulic diameters of the ducts used 

Where:
and also .
ψ_{T} and ψ_{P} represent the contribution
of entropy generation from heat transfer irreversibility and fluid friction
irreversibility, respectively.
For a given duct geometry and a stream with constant properties, the dimensionless
numbers II _{1} and II _{2} are constant. Therefore, ψ is a function
of Re and τ only. Values of Nu and f Re for fully developed laminar flow are
given by Shah (1975) for a variety of duct geometries.
The hydraulic diameter may be calculated from D_{H} = 4A_{C}/P
for all geometries (Table 1).
A modified dimensionless entropy generation may be based on the total
rate of heat transfer, viz.,
Where:
The power required to overcome fluid friction in the duct is related
to the total heat transfer rate by:
where, ΔP is the total pressure drop in the duct. From Eq.
5 and 6, the pumping power to heat transfer rate
ratio for fully developed laminar flow becomes:
DISCUSSION
Water has been used as working fluid. The thermophysical parameters used
are shown in Table 2.
In order to understand the detailed contribution of entropy generation
from frictional irreversibility and heat transfer irreversibility, investigations
of the effects of Re on ψ_{P} and ψ_{T} have
been carried out for duct with circular crosssection in Fig.
1.

Fig. 1: 
The influence of Re on ψ_{P} and ψ_{T}
of a baseline case (circular) with A_{C} = 4×10^{6}
m^{2} and q" = 500 W m^{2} 
The competition between the entropy generation from frictional
irreversibility and heat transfer irreversibility can be seen very different
in various Re numbers.
As can be seen, the value of ψ_{P} is nearly zero for low
Re, which indicates the major entropy generation comes from heat transfer
irreversibility. As Reynolds increased, it can be seen that ψ_{P}
increases but ψ_{T} decreases. However, ψ_{T}
is still much larger than ψ_{P}, which implies the entropy
generation is dominated by the heat transfer irreversibility for low Reynolds
numbers. When Re further increases to 1050, the curves of ψ_{P}
and ψ_{T} intersect. When Re is larger than the intersection
Re, ψ_{P} is larger than ψ_{T} and vice verse.
It indicates the entropy generation due to frictional irreversibility
becomes the dominant source.
Since heat transfer irreversibility dominates in lower Re region, the
change of ψ_{T} is more appreciable in lower Re cases. For
the similar reason, the changes of ψ_{P} are more appreciable
in larger Re cases since frictional irreversibility dominates in larger
Re region.
Such influences of Re on entropy generation could be understood from
the effects of Re on fluid friction and heat transfer performance.

Fig. 2: 
Dimensionless entropy generation ψ versus Re for
various duct geometries; A_{C} = 4x10^{6} m^{2}
and q" = 500 W m^{2} 
As
Re increases, the fluid friction in flow fields will increase and thus
results in the increase of entropy generation due to friction irreversibility.
On the contrary, the increase of Re will enhance the heat transfer performance,
making the temperature gradient in flow fields become gentle and in turn
results in the decrease of heat transfer irreversibility and entropy generation.
Figure 2 shows the variation of ψ with Re for
the five selected duct geometries. The cross section A and T_{O}
were constant.
As can be shown, in the lower Reynolds region, with increase of Re, the
entropy generation will decrease and in the larger Reynolds region, with
increase of Re, the entropy generation will increase.
ψ includes terms from heat transfer and viscous friction. Since
the entropy generation in the lower Reynolds cases is dominated by heat
transfer irreversibility, the change of ψ is mainly determined by
ψ_{T}. As discussed previously, the increase of Re would
enhance heat transfer performance and in turn reduce the heat transfer
irreversibility, therefore, ψ in these cases can be seen to decrease
monotonically with increase of Re.
As Reynolds further increases, in the larger Re, the entropy generation
takes a totally different trend, while with the increase of Reynolds,
the entropy generation increases. Such results come from that, for larger
Reynolds, the frictional irreversibility becomes dominant as Re increases
and the more serious entropy generation due to frictional irreversibility
would be raised when Re increases. Although the heat transfer irreversibility
would decrease simultaneously, its contribution is relatively minor in
these cases and therefore the resultant entropy generation is determined
by frictional irreversibility.

Fig. 3: 
Dimensionless entropy generation ψ versus Re for
various duct geometries; A_{C} = 6.34×10^{5} m^{2}
and q" = 500 W m^{2} 
As can be shown, the rhombic geometry with II, = 90, gives the lowest
dimensionless entropy generation and circle geometry has a largest dimensionless
entropy generation.
As Re increases, the effect of geometry disappears and the dimensionless
entropy generation becomes a function of (τ), i.e.,
Also, as A_{c} → ∞ the effect of geometry disappears and the dimensionless
entropy generation becomes the same function mentioned above. To clarify this,
Fig. 3 shows the variation of ψ with Re for the five selected
duct geometries for A_{C} = 6.34e–5.
Comparison of duct geometries using Φ may be appropriate when the
total heat transfer rate is important as shown in Fig. 4.
As Re is increased, the entropy generation will increase. In the Reynolds
greater than 550, the value of Φ will be decreased with the increase
of II, such that for the range of laminar flow, this trend will be
prevailed.
The pumping power required to overcome viscous friction is shown in Fig.
5. The rhombic geometry with Φ = 90° is superior to all other
geometries.
Figure 68 show the effect of cross
section A_{C} on ψ, Φ and P_{r}, respectively.
The rhombic geometry with Φ = 90° has been selected as a representative
case. The minima that exist for ψ are functions of Re as shown in
Fig. 6. Therefore, minimum entropy generation for laminar
flows may not exist for all geometries. Both entropy generation and pumping
power generally increase as the cross section area is decreased.

Fig. 4: 
Modified dimensionless entropy generation ψ versus
RE for various duct geometries; A_{C} = 4x10^{6}
m^{2} and q" = 500 W m^{2} 

Fig. 5: 
Pumping power to heat transfer ratio P_{r} versus
Re for various duct geometries; A_{C} = 4x10^{6}
m^{2} and q" = 500 W m^{2} 

Fig. 6: 
Dimensionless entropy generation ψ versus Re for
rhombic geometry with Φ = 90° of different cross sectional
area; q" = 500 W m^{2} 

Fig. 7: 
Modified dimensionless entropy generation versus Φ
Re for rhombic geometry with of different cross sectional area; q"
= 500 W m^{2} 

Fig. 8: 
Pumping power to heat transfer ratio P _{r} versus
Re for rhombic geometry with Φ = 90° of different cross sectional
area; q" = 500 W m ^{2} 

Fig. 9: 
Dimensionless entropy generation ψ versus Re for
rhombic geometry with Φ = 90° with different wall heat fluxes;
A_{C} = 4×10^{6} m^{2} 

Fig. 10: 
Modified dimensionless entropy generation Φ versus
Re for rhombic geometry with Φ = 90° with different wall
heat fluxes; A_{C} = 4×10^{6} m^{2} 

Fig. 11: 
Pumping power to heat transfer ratio P _{r} versus
Re for rhombic geometry with Φ = 90° with different wall
heat fluxes; A _{C} = 4×10 ^{6} m ^{2} 
Figure 911 show the effects of
wall heat flux on ψ, Φ and P_{r}, respectively. Both
entropy generation and pumping power are influenced by wall heat flux.
As the heat flux is increased, the entropy generation increases but the
pumping power per unit heat transfer rate decreases. In the limit when
τ → 0, the entropy generation becomes a linear function of Re,
viz.,
CONCLUSION
A numerical investigation of entropy generation in ducts of rhombic cross
section has been performed and the results compared to circular tube.
It was found that the rhombic duct performs better than the circular duct
and among ducts of various side angles the one with a 90° angle can
be considered the best. The effect of area cross section and the value
of heat flux were also investigated. Increasing cross sectional area increased
entropy generation. The same was found to be true for increased heat flux.
In some cases there seemed to exist minimum value of entropy generation
at a certain Re.
NOMENCLATURE
A_{C} 
= 
Flow cross sectional area (m^{2}) 
C_{P} 
= 
Specific heat capacity (J kg^{1} K^{1}) 
D_{H} 
= 
Hydraulic diameter (m) 
E_{c} 
= 
Eckert No. 
f 
= 
Friction factor 
h 
= 
Heat transfer coefficient (W m^{2} K^{1}) 
k 
= 
Thermal conductivity (W m^{1} K^{1}) 
L 
= 
Length of the duct (m) 

= 
Mass flow rate (kg sec^{1}) 
Nu 
= 
Nusselt No. (hD_{H}/k) 
p 
= 
Perimeter (m) 
P 
= 
Pressure (N m^{2}) 
P_{r} 
= 
Pumping power to heat transfer ratio 
Pr 
= 
Prandtl No. (μC_{P}/k) 
q" 
= 
Wall heat flux (W m^{2}) 

= 
Total heat transfer (W) 
Re 
= 
Reynolds No. 
S 
= 
Entropy (J kg^{1} K^{1}) 

= 
Entropy generation (W K^{1}) 
St 
= 
Stanton No. 
T 
= 
Temperature (K) 
T_{O} 
= 
Inlet temperature (K) 
T_{W} 
= 
Wall temperature (K) 
U 
= 
Fluid bulk velocity (m sec^{1}) 
x 
= 
Axial distance (m) 
ΔP 
= 
Total pressure drop (N m^{2}) 
ΔT 
= 
Increase of fluid bulk temperature (K) 
μ 
= 
Viscosity (Ns^{1} m^{2}) 
λ 
= 
Dimensionless axial distance (L/D_{H}) 
II _{1} 
= 
Dimensionless group (4Nuλ/Pr) 
II _{2} 
= 
Dimensionless group 
Φ 
= 
Modified dimensionless entropy generation 
ψ 
= 
Dimensionless entropy generation 
ψ_{T} 
= 
Contribution of entropy generation from heat transfer irreversibility 
ψ_{P} 
= 
Contribution of entropy generation from fluid friction irreversibility 
ρ 
= 
Density (kg m^{3}) 
τ 
= 
Dimensionless wall heat flux ((q"/h)/T_{O}) 