Second Law Analysis of Laminar Flow in a Circular Pipe Immersed in an Isothermal Fluid

Entropy generation and pumping power to heat transfer ratio (PPR) of a laminar flow, for a circular tube immersed in an isothermal fluid, are studied analytically in this paper. Two different fluids, namely, water and ethylene glycol, are chosen to study the influence of fluid properties on entropy generation and PPR.The expressions for dimensionless entropy generation, Bejan number and PPR are derived in a detailed way and their variations with Reynolds number, external Biot number, and the dimensionless temperature difference are illustrated. The results of the analysis are compared with those for a laminar flow in a circular tube with uniform wall temperature boundary condition. Finally, a criterion is established to determine which type of thermal boundary conditions is more suitable for a particular fluid, with respect to its influence on entropy generation.


Introduction
Heat transfer is a fundamental source of thermodynamic irreversibility in all real engineering devices.When heat is transferred across a finite temperature difference, some capacity to do work is lost.In convection, apart from heat transfer, fluid friction is the other source of loss of available work.Both heat transfer and fluid friction generate entropy.This entropy generation must be minimized to reduce the loss of available work.Entropy generation minimization is no longer an avant-garde philosophy but a mainstream one in the design of thermal systems.
In the past, many researchers have studied the problem of entropy generation minimization in fluid flow with heat transfer.Bejan [1,2] outlined the method for evaluating the entropy generation in fluid flow with heat transfer.He found out the entropy generated in fluid flow with heat transfer over a flat plate, in a duct, for cylinders in cross flow and in various other geometrical configurations.S ¸ahin [3] studied the entropy generated in a circular duct with uniform wall temperature for two fluids, namely, water and glycerol.The effect of temperature on viscosity was taken into account in that study.He found out that the total energy loss (due to pumping process and entropy generation) can be minimized with respect to the duct length for viscous fluids with temperature dependent viscosity.S ¸ahin [4,5] also considered the effect of duct geometries on the entropy generation, both for uniform wall temperature and for uniform heat flux boundary conditions.The various duct geometries like circular, square, rectangular, equilateral triangle, and sinusoidal were considered.In general, circular geometry was found to be preferable, especially for high Reynolds number.A similar study for hexagonal cross-section was carried out by Jarungthammachote [6] and for a helical coil by Ko and Ting [7].Dagtekin et al. [8] analyzed the entropy generation in a circular tube with internal fins of different shapes.The study of entropy generation in a pipe was extended to non-Newtonian fluids by Mahmud and Fraser [9,10] and by Hung [11].
As far as the authors are aware, the studies carried out so far have been confined to tubes/ducts with uniform wall temperature or with uniform wall heat flux.The present study is concerned with entropy generation in a circular duct immersed in an isothermal external fluid, which is a more general thermal boundary condition.According to Sparrow and Patankar [12], both the uniform heat flux and uniform wall temperature boundary conditions are the limiting cases of this boundary condition.To study the effect of fluid properties, two different fluids, water and ethylene glycol, have been considered in this paper.The effects of dimensionless temperature difference and external Biot number on the dimensionless entropy generation, Bejan number, and PPR have been investigated.A comparison has been made with uniform wall temperature boundary condition.

Analysis
Consider a circular tube immersed in an isothermal fluid (at temperature   ) as shown in Figure 1.Another fluid flows inside the tube.The flow inside the tube is assumed to be laminar and enters the tube at a uniform temperature   .The energy balance for a control volume (CV) of infinitesimal length  (neglecting axial conduction, changes in kinetic and potential energies, and viscous dissipation) is given by where q  , the rate of heat transfer per unit length from the external fluid to the internal fluid, is given by ( ĥ in (2) is the overall heat transfer coefficient between the external fluid and the internal fluid).Solving the two equations above for (), we get Using nondimensional parameters, we can write this equation as where St is the Stanton number given by St = ĥ   (5) and  is the non-dimensional duct length, given by The temperature at the outlet of the duct,  out , is given by where Writing (7) in terms of mass flow rate, we get The rate of total heat transferred to the duct is given by The entropy generated in this infinitesimal CV is given by Now, Substituting the values of q  and  wall from ( 12) into (11), we get the following: − ĥ (  − )  ĥ (  − ) /ℎ +  + ṁ.
For an incompressible fluid, where the pressure drop per unit length is related to the friction factor as Thus we get Substituting ( 16) to (13) where St is the Stanton number (defined in (5)), Ec is the Eckert number given as In the literature, dimensionless entropy generation has also been used to quantify the entropy generated.It is given by If we take the limit  → 0, that is, ( Nu → Nu), (18) reduces to Using the identity lim  → 0 (ln(1 + )/) = 1, we get in the simplified form the following: which is the same as the expression for entropy generation for fully developed laminar flow in a circular pipe with uniform wall temperature as given in [3].This corroborates the conclusion of Sparrow and Patankar [12] that, as the external Nusselt number tends to duct side Nusselt number, the isothermal external fluid boundary condition tends to the uniform wall temperature boundary condition.Equation ( 22) can also be written as function of Re instead of St (St = Nu/Re Pr): where At Be = 1, all the entropies are generated due to heat transfer and at Be = 0, all the entropies generated are due to fluid friction.
In the current analysis, if we take lim   →   , it gives us the dimensionless entropy generated due to fluid friction only (as   →   , there is no heat transfer, and so entropy generated will be due to fluid friction only): where FF is dimensionless entropy generated due to fluid friction only.So, 2.2.Pumping Power to Heat Transfer Ratio.Another parameter which is a good indicator of the performance of a heat exchanger pipe is the ratio of pumping power to heat transfer (PPR), which can be expressed as Here the pressure drop Δ along the length of the duct can be obtained by integrating the infinitesimal pressure drop  given by Using (32) to calculate pressure drop and (10) to calculate heat transfer rate and inserting the same into (31), we get

Fluid Properties.
To study the effect of fluid properties, two different fluids, water and ethylene glycol, are considered.
Water.The thermophysical properties of water are calculated using the following equations, given in [6]: +2.78178981 × 10 −9 3( + 27) 4 ) ×  0.018 . (37) In the above equations,  is the temperature of water in degree  and  is the universal gas constant.
Ethylene Glycol.The density of ethylene glycol is taken as constant, since it varies roughly by only 1% in the temperature range considered.For determining the viscosity of ethylene glycol, the following empirical correlation given by Sherman [14] is used: In (38),  ref is the reference temperature whose value is taken as 273 K.  and  are fluid dependent parameters, whose numerical values for ethylene glycol are found out using regression.To estimate thermal conductivity and specific heat, the fourth degree polynomial equations are used.The coefficients of these polynomial equations are also determined using regression.The data for regression is taken from [15].The regression is carried out using the commercially available software Microsoft Excel.
The fluid properties are evaluated at  avg = ( out +   )/2 using an iterative procedure.A value of  out is guessed.Using this guessed value, the fluid properties are found out at  avg .These fluid properties are then used to find out the new  out using (7).This process is repeated till the difference in the successive values of  out is less than 1 K.The  out for the last iteration is taken as the final value.The  avg , which is calculated using the final value of  out , is used to estimate the fluid properties.

Results and Discussion
The Nu and Nu are functions of external Bi.Their values are given in tabular form in [12] and are presented in Table 1.The value of friction factor for laminar flow is taken from [16] and is represented as: The numerical values of the constant parameters are given in Table 2.For these parameters, the effect of Bi and  on the dimensionless entropy generation , the Bejan number Be, and the PPR has been studied.The results are detailed below.(40)

Effect on 𝜓.
Therefore, at higher Re the temperature gradients inside the fluid decrease and there is less entropy generated due to heat transfer.Even though as Re increases, the entropy generated due to pressure increases, but as we will see later, the contribution of entropy generated due to pressure to total entropy generated is very less (Be is close to 1), so  decreases when Re increases.
It is also seen from Figure 2(a) that as Re increases, the curves for different Bi tend to merge.Similarly, the curves for different  also tend to merge at higher Re (see Figure 2(b)).This is because, as Re increases, the contribution of heat transfer to entropy generation decreases and total entropy generated is roughly due to pressure only.Since entropy generation due to pressure is independent of Bi or , the curves for different Bi (and for different ) tend to merge at higher Re.We see from Figure 2(a) that  increases with increase in Bi.This is because as Bi increases, Nu increases (see Table 1), which means that heat transfer rate to the fluid increases.This leads to increase in temperature gradients inside the fluid.This increase in temperature gradient inside the fluid leads to increase in entropy generation.
Figure 2(b) shows that, as the dimensionless temperature difference  increases,  increases.This is because an increase in  causes the heat transfer to the fluid to increase, which leads to an increase in temperature gradients and hence in .
Figure 3 shows the variation of  with Re for different values of (a) Bi and (b)  for ethylene glycol.Similar trends, as observed for water, can be observed here.

Effect on Be. Figure 4 shows the variation of Be with
Re for different values of (a) Bi and (b)  for water.As can be seen from the figure, the value of Bejan number is very high (close to 1).This means that heat transfer irreversibility contributes almost 100% to the total entropy generation.This is corroborated by Ben-Mansour and Sahin [17] for circular pipes and by Jarungthammachote [6] for hexagonal pipes.We can also see from the figures that Be decreases with increase in Re.This is because, as pointed out in the previous section, the temperature gradients inside the fluid decrease with increase in Re.This leads to a decrease in heat transfer irreversibility.Moreover, the irreversibility due to pressure difference increases with increase in Re.Due to these two reasons, Be decreases with increase in Re.
We can also see from Figure 4(a) that, as Bi increases, Be increases for a fixed Re.From Figure 4(b), we see that, as  increases, Be increases.The result of these two figures can be explained on the basis of heat transfer irreversibility.As Bi or  increases, the heat transfer to the fluid increases, due to which temperature gradients inside the fluid increase.This leads to increase in heat transfer irreversibility, which further leads to increase in Be.
Figure 5 shows the variation of Be with Re for different values of (a) Bi and (b)  for ethylene glycol.Trends, similar to the case of water, can also be observed for ethylene glycol.to 0. This is because, as Re → 0, there is no flow and so no pumping power is required.As Re increases, PPR increases, because, to maintain a flow at higher Re, more pumping power is required.Figure 6(a) shows that, as Bi increases, the PPR decreases.This is because, as Bi increases, Nu increases (see Table 1), which means more heat transfer to the fluid for the same Re.This leads to decrease in PPR.For the same reason, PPR decreases with increase in , as shown in Figure 6(b).

Effect on PPR.
Figure 7 shows the variation of PPR with Re for different values of (a) Bi and (b)  for ethylene glycol.Trends, similar to the case of water, can be observed.ethylene glycol for the same mass flow rate, same Biot number, and same dimensionless temperature difference is shown in Table 3.
As can be seen from Table 3, the total entropy generated and entropy generated due to heat transfer are higher for water while the entropy generated due to pressure is higher for ethylene glycol.
To investigate the reason for this trend, (9) must be paid attention to.For the same mass flow rate, the temperature at the outlet depends on /  .Even though   of water is higher than that of ethylene glycol, the  for water is still higher, with the result that /  is higher for water.So  out for water is higher than  out for ethylene glycol.Due to this, temperature gradients inside the water are higher which leads to higher entropy generation for water than for ethylene glycol.
Entropy generation due to pressure is higher for ethylene glycol than that of water.This is because the viscosity of ethylene glycol is much higher than that of water.

Comparison with Uniform Wall
Temperature. Figure 8 shows the comparison of (a)  and (b) PPR between the circular duct with uniform wall temperature and circular duct immersed in an isothermal fluid.For comparison purpose, the temperature of isothermal external fluid is kept same as uniform wall temperature.As can be seen from the figure,  is higher for uniform wall temperature condition than for isothermal external fluid condition, while PPR is lower for uniform wall temperature condition than for isothermal external fluid condition; other parameters are kept the same.While comparing  and PPR for different types of boundary conditions, the question arises: which type of boundary conditions is more suitable?To answer this, it should be noted that the total exergy loss in the pipe is the sum of  and PPR.As can be seen from Table 4, for less viscous fluids like water,  is much higher than PPR, a trend which becomes more pronounced for smaller Re.But more viscous fluids like ethylene glycol, on the other hand, have higher PPR than .From this, it can be concluded that, for fluids whose viscosity is low, isothermal external fluid boundary condition should be preferred, while, for more viscous fluids, like ethylene glycol, uniform wall temperature boundary condition is to be preferred.

Figure 1 :
Figure 1: An infinitesimal control volume in a circular duct immersed in an isothermal fluid.

Figure 2
shows the variation of  with Re for different values of (a) Bi and (b)  for water.As can be seen from the figure,  decreases with Re.To explain this, we consider (4), where we find that if we let Re tend to infinity, then the () tends to   (since St = Nu/(Re ⋅Pr)); that is, lim Re → ∞  () =   .

Figure 2 :
Figure 2: Variation of dimensionless entropy generation  of water with Re for different values of (a) Bi and (b) .

Figure 3 :
Figure 3: Variation of dimensionless entropy generation  of ethylene glycol with Re for different values of (a) Bi (b) .

Figure 4 :
Figure 4: Variation of Bejan number of water with Re for different values of (a) Bi and (b) .

Figure 5 :
Figure 5: Variation of Bejan number of ethylene glycol with Re for different values of (a) Bi and (b) .

3. 4 .Figure 6 :
Figure 6: Variation of PPR of water with Re for different values of (a) Bi and (b) .

Figure 7 :
Figure 7: Variation of PPR of ethylene glycol with Re for different values of (a) Bi and (b) .

Figure 8 :
Figure 8: Comparison of (a)  and (b) PPR for different boundary conditions.

Table 2 :
Values of different constant parameters used in the analysis.

Table 3 :
Comparison of entropy generation due to temperature difference and entropy generation due to pressure for water and ethylene glycol.

Table 4 :
Comparison of dimensionless entropy generation  with PPR for water and ethylene glycol.