Optimal heating strategy for minimization of peak temperature and entropy generation for forced convective flow through a circular pipe

International Journal of Heat and Mass Transfer 150 (2020) 119318

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt

Optimal heating strategy for minimization of peak temperature and entropy generation for forced convective flow through a circular pipe Sukumar Pati a,∗, Rajib Roy a, Nabajit Deka a, Manash Protim Boruah a, Mriganka Nath a, Ritwick Bhargav a, Pitambar R. Randive a, Partha P. Mukherjee b,∗ a b

Department of Mechanical Engineering, National Institute of Technology Silchar, Silchar 788010, India School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, United States

a r t i c l e

i n f o

Article history: Received 2 September 2019 Revised 20 December 2019 Accepted 4 January 2020

Keywords: Laminar flow Forced convection Non-uniform heating Entropy generation

a b s t r a c t Fundamental insight on the minimization of entropy generation and maximum temperature is important for efficient performance and proper design of thermal systems. One of the key factors that dictate the entropy generation and temperature within a thermal system is the applied heat source and therefore appropriate heating strategy that minimizes the peak temperature and entropy generation needs to be determined. In this work, non-uniform distribution of heat flux applied to the wall of a circular tube undergoing laminar forced convective flow of a high Prandtl number fluid has been investigated numerically to optimize the heating strategy which results in minimum entropy generation and peak temperature as compared to the uniform heating. Various heat flux configurations are evaluated for the same amount of total heat rate and comparative assessment between the different cases has been made in terms of peak temperature and entropy generation. In terms of achieving both minimum peak temperature and entropy generation simultaneously, the periodic heat flux condition (sinusoidal with number of waves greater than or equal to 100) can be recommended as the optimum heating strategy for typical thermal system. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Optimization of thermal system has received a high degree of scientific attention in the last few decades owing to the necessity of minimizing peak temperature and entropy generation for efficient performance of the system [1]. The hunt for achieving an optimized thermal system has motivated the researchers to investigate different concomitant factors that are responsible in dictating the temperature attained and the irreversibilities incurred within the system. In the process, researchers have determined that several key factors such as geometrical configuration of the system [2], thermo-physical properties of the materials [3] and other passive inclusions such as baffles [4,5] mediate the optimality of temperature and entropy generation of the system to a significant extent. In the literature, several works have been reported focusing on the control and minimization of temperature, such that the peak temperature (scientifically termed as “hot spots”) remains

∗

Corresponding authors. E-mail addresses: [email protected] (S. Pati), [email protected] (P.P. Mukherjee). https://doi.org/10.1016/j.ijheatmasstransfer.2020.119318 0017-9310/© 2020 Elsevier Ltd. All rights reserved.

lower than a pre-defined value. In order to achieve the same, several techniques of enhancing the heat transfer were devised utilizing different flow configurations. For instance, Bejan and Sciubba [6] obtained minimum peak temperatures by maintaining optimal distance between an array of heat source during forced convection. Another technique was presented by Bejan et al. [7], modifying their previous work [6] by inserting blades in the entrance region and optimizing their space to maximise the heat transfer density. Hajmohammadi et al. [8] utilized the advantage of conjugate heat transfer for optimal conduction of heat by incorporating a conducting substrate over the heaters. Reddy and Balaji [9] used the micro genetic algorithm (MGA) optimization technique to minimize the peak temperature with an optimal combination or arrangement of heating tool in an enclosure with ventilation. Artificial neural network (ANN) was also applied in an article by Sudhakar et al. [10] where they configured five discrete sources and determined an energy efficient configuration for mixed convective flow in a vertical duct. Numerical analysis was also performed in an open square cavity by Muftuoglu and Bilgen [11] to control the peak temperature achieved by placing discrete heaters at optimum positions. Furthermore, da Silva et al. [12] and Hajmohammadi et al. [13] analytically determined the optimal heat-

2

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

Nomenclature Be Cp D h k L Nu p Pr q r R Re Sgen  S˙ gen T Tw Tm vr vz vz,in z Z

Bejan number specific heat capacity of the fluid (J/kg K) diameter of the tube (m) heat transfer coefficient (W/m2 K) thermal conductivity of the fluid (W/mK) length of the tube (m) local Nusselt number pressure (N/m2 ) Prandtl number mean heat flux (W/m2 ) radial distance (m) radius of the tube (m) Reynolds number rate of total entropy generation (W/K) rate of volumetric entropy generation (W/m3 K) temperature (K) wall temperature (K) bulk mean temperature (K) radial velocity (m/s) axial velocity (m/s) inlet axial velocity (m/s) longitudinal distance (m) dimensionless longitudinal distance

Greek symbols μ viscosity of the fluid (Pa s) φ energy dissipation (1/s2 ) ρ density of the fluid (kg/m3 ) ing arrangement for forced convection heat transfer with minimum temperature. Recently, Hajmohammadi et al. [14] proposed a technique of enlarging the thermal entrance length by placing insulators between the heat applied regions along a pipe. This modification could indirectly reduce the peak temperature within the system. Another key parameter that dictates the optimality of a thermal system is entropy generation, which signifies the irreversibility of the system and the same has been rigorously examined by several researchers [15–18]. This approach has been initially applied to pratical devices such as single-phase liquid cooling devices, power plants and refrigeration systems, tube-fin condensers with fans and heat exchangers. Recently, seven different cases of heat flux distributions were investigated by Esfahani and Shahabi [19] for a high Prandtl number fluid in the laminar developing flow regime. Their study indicated minimum entropy generation for an ascending distribution of heat flux at the wall of the pipe. Poddar et al. [20] assessed the performance of infinite triangular and square sub-channel on the basis of minimum entropy generation because of forced convective turbulent flow through it. Pati et al. [21] combined both the laws (first and second) of thermodynamics to exploit the manner in which interfacial slip flow in microscale geometries dictate the associated heat transfer and irreversibility characteristics. Mahian et al. [22] reviewed the studies on entropy generation analysis in nanofluid flow for proper use in thermal systems and focused on the importance of entropy generation analysis. Zimparov [23] investigated the influence of variation in fluid temperature along the flow direction of heat exchanger with constant wall temperature. He proposed performance evaluation criteria for surfaces prone to heat transfer with the help of entropy generation theorem. Several articles [24–31] have also investigated different thermal boundary conditions like constant wall temperature [24], constant heat flux [25, 28, 30, 31], non-uniform heating system [26, 29] and

different geometric configurations [27–30] to manipulate the fluid flow and heat transfer so that the entropy generation or the temperature remains minimum. Sahin [24] compared the second-law irreversibilities in different duct geometries subjected to a constant wall temperature and determined the optimum geometry from the perspective of minimum irreversible losses for a range of laminar flows. In another work, Sahin [25] carried out second law analysis in a duct subjected to constant wall temperature. He found that temperature dependent viscosity accurately dictates the associated entropy generation. Moallemi et al. [26] proposed a novel concept of non-uniform temperature discrete heating system and their analysis revealed that this system showed a high feasibility for being a dominant discrete heating scheme in the future. Sahin et al. [27] numerically analysed the effect of fouling and the incurred irreversibility on the performance cost for pipe flow. Some other notable research effort in the context includes the work by Borah et al. [28] who studied the non-uniform heat flux mediated entropy generation in the thermally developing regime for flow through parallel plates. Arabkoohsar [29] also proposed a non-uniform temperature discrete heating system with decentralized heat pumps and standalone heat storage units and their analysis showed that the rate of heat loss is minimal in such system. Recently, Boruah et al. [30] explored asymmetric distribution of the heat flux on two opposite walls of a microchannel and determined the optimum asymmetricity for minimum entropy generation with maximum heat transfer. From the above discussion, it is imperative to note that most of the work in this context has been carried out to optimize either entropy generation or temperature. However, for safe and efficient operation of a thermal system, both these parameters (entropy generation and temperature) need to be optimized. However, to the best of authors’ knowledge, studies optimizing both the temperature and entropy generation are limited and needs investigation. Moreover, studies on the comparative assessment of uniform and non-uniform heating configuration remains largely overlooked and requires an in-depth investigation due to its relevance in several applications discussed above. Accordingly, the effect of uniform and non-uniform (linearly decreasing, linearly increasing and sinusoidal) heat flux conditions at the wall for a developing laminar flow of fluid through a pipe on the temperature and entropy generation has been assessed in detail. Five different cases of heat flux distributions having the same rate of heat transfer were examined to determine the optimum heat flux distribution that could restrict both the temperature and entropy generation to minimum.

2. Problem statement and theoretical background The present work considers laminar, steady and incompressible flow of a Newtonian fluid in an axisymmetric circular tube of length (L) 1m and diameter (D=2R) 0.025m as shown by the schematic in Fig. 1. A high Prandtl number fluid (Pr=13400) has been used as the working medium having the thermo-physical properties as listed in Table 1. The wall of the tube is subjected to different heat flux boundary conditions to study their effect on the entropy generation and maximum temperature (Tmax ) and thereby determine the optimum heat flux condition for achieving minimum peak temperature and entropy generation as well. To assess the same, the circular duct is subjected to five different heat flux distribution viz. uniform, linearly increasing, linearly decreasing, sinusoidal and periodic heat flux as shown in Fig. 1. It is important to note that, for a comparative assessment of all the five different cases, the value of total heat transfer rate and Reynolds number were fixed.

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

3

Fig. 1. Schematic of different cases of heat flux distribution considered in present investigation.

Table 1 Thermo-physical properties of the fluid. Thermo-physical property

Value

Reference temperature, Tref Inlet fluid temperature, Tin Total heat transfer rate, Q Density, ρ Viscosity, μ Specific heat, Cp Thermal conductivity, k Inlet axial fluid velocity, vz, in

288.16 K 273.15 K 392.7 W 889 kg/m3 1.06 N s/m2 1845 J/kg K 0.146 W/m K 0.02 m/s

2.1. Governing equations Under the assumptions mentioned in the previous section, the relevant governing equations [32] that have been numerically solved are given below. Continuity equation:

∂ vr vr ∂ vz + + =0 ∂r r ∂z

(1)

4

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318 Table 2 Comparison of Nusselt number at the heated wall at different longitudinal locations for different grid sizes for sinusoidal heat flux condition. Number of Elements

Dimensionless distance (z/R)

Nusselt Number (Nu)

Error (%)

500 × 30

0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9

18.754 12.991 10.996 9.864 9.117 18.802 13.022 11.021 9.886 9.138 18.839 13.046 11.041 9.903 9.153

0.45 0.42 0.41 0.39 0.39 0.19 0.18 0.18 0.17 0.16 – – – – –

500 × 50

500 × 70

r-momentum equation:

      ∂v ∂v ∂p ∂ 2 vr ∂ 1 ∂ ( r vr ) ρ vr r + vz r = − + μ + ∂r ∂z ∂r ∂r r ∂r ∂ z2

(2)

Bejan number: The Bejan number defines the fraction of entropy generated due to thermal energy transport as compared to the total irreversibility incurred and it can be mathematically expressed as [32]

z-momentum equation:

      ∂v ∂v ∂p ∂ 2 vz ∂ 1 ∂ ( r vz ) ρ vr z + vz z = − + μ + ∂r ∂z ∂z ∂r r ∂r ∂ z2

(3)

Be =

k T

k T2

 2 ∂T ∂r

 2 ∂T ∂r

2 

+ ∂∂Tz

2 

+ ∂∂Tz

(7)

+ μφ

Energy equation:

      ∂T ∂T 1 ∂ ∂ (T ) ∂ 2T ρ C p vr + vz =k r + + μ ∂r ∂z r ∂r ∂r ∂ z2 where



=2

∂ vr ∂r

2 +

 v 2 r

r



∂ vz + ∂z

2 



∂ vr ∂ vz + + ∂z ∂r

3. Numerical solution methodology and grid independence test

(4)

2

2.2. Boundary conditions The boundary conditions employed to solve Eqs. (1)–(4) are given as follows:

Inlet :

vz = vz,in , vr = 0, T = Tin

(5a)

∂ vz ∂ vr ∂  T − Tw  = 0, = 0, =0 Outlet : ∂z ∂z ∂ z Tm − Tw Axial symmetry :

(5b)

∂ vz ∂ vr ∂  T − Tw  = 0, = 0, vr = 0, =0 ∂r ∂r ∂ r Tm − Tw (5c)

Tube wall :

vz = 0, vr = 0,

q = heat flux depending upon the case

(5d)

As mentioned earlier, the distribution of heat flux for different cases were maintained in such a way that the total heat transfer rate remains the same for all the cases.

The governing equations are discretized using a finite volume method on a structured grid. The pressure and velocity components are distributed at the cell center and cell faces respectively using a staggered grid arrangement. The SIMPLE algorithm is used for the pressure–velocity coupling. The convective fluxes are discretized in the interior points using the deferred QUICK scheme while the convective fluxes at the boundary nodes are discretized using the central difference scheme. Moreover, the solutions are solved iteratively until the summation of residuals for all the variables is lower than 10−6 . In order to determine the sensitivity of the computational grid size on the calculated solutions, extensive grid independence studies have been performed for the case of non-uniform heating with 500 × 30, 500 × 50 and 500 × 70 grid size. Each of these mesh configurations were refined near the wall of the tube to accurately capture the temperature gradient. The results of the grid independence study are shown in Table 2 in terms of the variation of local Nusselt number (Nu) for different grid sizes. It becomes apparent from Table 2 that the local Nusselt number at any location along the heated wall is almost the same for 500 × 50 and 500 × 70 grid sizes. Hence, it can be inferred that 500 × 50 grid size could produce accurate results in minimum computational time and therefore, 500 × 50 grid size has been considered for all the computations involved in the present study. 4. Validation of the numerical model

2.3. Thermo-hydraulic formulation The thermo-hydraulic characteristics have been delineated in terms of total entropy generation and Bejan number and these can be mathematically defined as: Total entropy generation [32]: 

Sgen =

k T2



∂T ∂r

2

 +

∂T ∂z

2 

+

μ T

φ

(6)

In order to verify the accuracy of the numerical methodology employed in the present work, the solver has been validated with the experimental work by Tam et al. [31]. As evident from Fig. 2(a), the results for local Nusselt number obtained along the heated wall of the pipe for a high Prandtl number fluid by the present numeri1/3

cal solver closely satisfies the correlation N ux = 1.24(ReD Pr ) obx tained by Tam et al. [31]. It is also to be noted that the discrepancies between the numerical results and the experimental values

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

5

tal entropy generation and Bejan number at different radial and longitudinal locations of the tube for all the cases and a comparative assessment has been made between the maximum temperature and entropy generation for different cases to determine the efficient heating strategy. 5.1. Effect of wall heat flux condition on radial temperature profiles

Fig. 2. Comparison of the (a) axial variation of local Nusselt number obtained by the present study with that of Tam et al. [31] and (b) total entropy generation obtained by the present study with that of Esfahani and Shahabi [19].

has been found to be less than 10%, thus confirming the accuracy of the solver. A seperate validation has been made for total entropy generation by considering a computational model similar to Esfahani and Sahabi [19]. The results for total entropy generation in case of discrete heating of a channel obtained by the present numerical procedure are presented in Fig. 2(b). As can be seen from Fig. 2(b), the close agreement between the results for total entropy generation obtained by the present procedure with Esfahani and Sahabi [19] justifies the accuracy of our numerical procedure for further investigation. 5. Results and discussion A novel approach of applying non-uniform wall heat flux at the wall of a circular duct for achieving laminar forced convective heat transfer with minimum peak temperature and entropy generation is discussed in this study. Various types of wall heat flux boundary conditions (i.e., uniform, linearly increasing, linearly decreasing and sinusoidal heat flux) have been investigated numerically for an equal amount of heat transfer rate for all the cases for laminar flow through a circular duct. Subsequent sections discusses the results obtained in terms of the variation of temperature, to-

Radial distribution of fluid temperature at five different axial distance of the tube is shown in Fig. 3 for all the cases considered in this study. It can be seen from Fig. 3(c) that the maximum temperature occurs at z=0.5 m for the case of linearly decreasing heat flux distribution which is consistent with the results reported in the literature [19]. This once again reiterates the validity of the results presented herein. It is evident from the distribution of heat flux presented in Fig. 1 that the temperature gradient at the wall also decreases with reduction in heat flux along the flow direction in the case of linearly decreasing heat flux distribution. Also, it can be observed that the temperature at the longitudinally middle location is greater than that near the outlet of the duct (z=0.9m) for the case of linearly decreasing heat flux distribution because the heat flux in this case reduces to zero at the latter location. This also supports the fact that there is less transfer of energy from the wall to the core region of the duct at its near end location since the flowing liquid already gets heated up prior to reaching the near end location resulting in lesser temperature gradient. It can also be noted that, unlike other cases, the temperature at the entrance in the case of linearly decreasing heat flux distribution is more than the near end section. This is because of the higher amount of heat flux at the entrance region of the duct for the case of linearly decreasing heat flux distribution as compared to other cases. Furthermore, the variation of temperature for increasing heat flux boundary condition along the flow direction upto the mid-section of the tube can be observed for the case of linearly increasing heat flux distribution and sinusoidal heat flux distribution with λ=1 as presented in Fig. 3(b) and 3(d) respectively. As evident from the distribution of heat flux, the fluid is relatively hotter at the end of the tube for the case of linearly increasing heat flux distribution, however, the maximum temperature is obtained at the longitudinal location of 0.7m for the case of sinusoidal heat flux distribution with λ=1 because of the increasing heat flux in the first half of the applied sinusoidal heat flux distribution. Interestingly, it can be seen from Fig. 3(d) that the temperature at the end of the tube remains relatively lower in the case of sinusoidal heat flux distribution with λ=1 because the latter half of heating follows a decreasing heat flux input. It can also be adjudged from the distribution of linearly increasing heat flux that the fluid temperature gradient is lower at the entrance locations and higher at the downstream axial locations. Moreover, it is interesting to observe that the maximum temperature at the wall (r/R=1) of the channel is higher for increasing heat flux boundary condition and sinusoidal heat flux condition as compared to cases of uniform, linearly decreasing and periodic (sinusoidal with λ=100) heat flux condition. On the contrary, cases of uniform, linearly decreasing and periodic (sinusoidal with λ=100) heat flux condition are marked by high initial temperature due to sufficiently strong heat flux imposed on the wall. Moreover, it can be seen from Fig. 3 that the radial penetration of heat for the linearly decreasing and periodic heat flux condition is higher than that observed for uniform, linearly increasing and sinusoidal heat flux condition. The higher amount of radial penetration of heat for linearly decreasing and periodic (sinusoidal with λ=100) heat flux condition can be related to the greater amount of heat flux applied at the entrance region of the channel for these cases. Noteworthy to mention that the periodic heat flux condition shows the least temperature, which is very much desirable as far as the design of thermal system is concerned.

6

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

Fig. 3. Radial variation of temperature at five longitudinal locations along the tube for different heat flux configurations.

5.2. Effect of wall heat flux condition on radial entropy generation profile The radial variation of total entropy generation at different axial locations is shown in Fig. 4 for all the cases investigated in this

study. As shown in Fig. 4(c), the total entropy generation remains almost constant in the core region of the duct throughout the length for all the cases. This is primarily due to the viscous irreversibilities dominating at the core of the duct due to greater flow velocity in this region as compared to the near wall region. Since a

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

7

Fig. 4. Radial variation of entropy generation at five longitudinal locations along the tube for different heat flux configurations.

fixed value of Reynolds number is considered in this study, therefore, the viscous irreversibilities in the core region remain unaltered. However, in the near wall region, the irreversibilities are dictated by the temperature gradients that are closely dependent on the distribution of the heat flux applied at the wall of the duct. The dominance of thermal irreversibilities in the near wall region of

the duct is clearly revealed from the difference in the variation of total entropy generation in the near wall region for different cases and at different longitudinal locations. In cases of uniform, linearly decreasing and periodic (sinusoidal with λ=1) heat flux condition, the total entropy generation is higher in the entrance region due to higher heat flux and decreases on proceeding towards the down-

8

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

stream of the tube. However, an opposite trend is observed for the case of linearly increasing heat flux distribution, which is marked by continuous increment in entropy generation from the inlet to the outlet of the duct. Such a variation in entropy generation for different cases of applied heat flux can be attributed to two factors viz. increase in wall heat flux and the heating of colder fluid with higher temperature gradient. For the case of linearly increasing heat flux distribution, the heat flux and hence the temperature gradient increases along the flow direction, thus resulting in higher entropy generation in the downstream location as observed from Fig. 4(b). However, reverse is the case for linearly decreasing heat flux condition as evident from Fig. 4(c). In case of sinusoidal heat flux, the entropy generation increases due to the increase in the radial gradient of temperature up to the mid-longitudinal location of the duct. Moreover, the colder fluid gets heated with higher temperature gradient up to the mid-longitudinal location, resulting in higher entropy generation. However, beyond the mid-longitudinal location, the radial gradient of temperature decreases along the flow direction due to the decrease in the applied heat flux, thereby resulting in a net decrease in the irreversibilities along the downstream direction of the channel as can be seen from Fig. 4(d). Referring to the case of uniform heating, it can be adjudged that the temperature gradient remains constant throughout the length of the duct and therefore, only a minimal variation in the total entropy generation at different longitudinal locations can be observed from Fig. 4(a). This can be interpreted to be due to the decrease in viscous irreversibilities along the downstream direction. However, in case of periodic heating condition, both the axial and radial gradient of temperature are equally significant and their influence from both the upstream and downstream direction of the duct is prominent at the mid-longitudinal location. Therefore, the entropy generation for periodic heating (sinusoidal with λ = 100) is higher at z=0.5m as compared to other longitudinal locations. Furthermore, from a comparative analysis of the values of Sgen from Fig. 4, it can be deduced that the maximum irreversibilities are incurred in case of linearly decreasing heat flux condition, while for the case of uniform heat flux, the irreversibilities incurred is the minimum among all the cases investigated in this study. The linearly decreasing heat flux distribution has very high gradients of temperature and velocity in the entrance region of the duct that causes maximum entropy generation. 5.3. Effect of wall heat flux condition on the radial variation of Bejan number Fig. 5 shows the radial variation of Bejan number at five different longitudinal locations of the channel for different cases of applied heat flux. The values of Be close to one indicates the dominance of thermal irreversibilities, while Be close to zero indicates higher viscous irreversibilities. At the foremost, it can be pointed out from Fig. 5 that the value of Be at all the longitudinal locations is zero at the radially central location of the tube for all the cases of heat flux studied here. The reason behind this is that the heat does not diffuse up to the centre of the tube and the temperature gradient in the region near the radial centre line is negligible. As a result, the viscous irreversibilities dominate over the thermal irreversibilities in the region near the radial centre line of the tube due to greater flow velocity in this region. A better insight into the results presented in Fig. 5 infers the fact that the value of Be at the entrance location remains zero for a greater radial distance from the centre line of the tube as compared to that at the downstream location of the tube because of the larger velocity gradients at the entrance region and the increase in the depth of penetration of heat as the flow proceeds ahead. This observation holds true for all the cases of heat flux, although the radial distance from the centre line of the duct upto which Be remains zero varies with the

applied heat flux condition. It can be noticed from Fig. 5(d) that, at z=0.1 m, Be remains zero for a non-dimensional radial distance (r/R) of 0.7 for the case of sinusoidal heat flux distribution with λ=1, which is the maximum among all the cases. This physically entails from the fact that the heat flux and hence the penetration of heat is minimum near the inlet region for the case of sinusoidal heat flux distribution with λ=1 and therefore, the viscous irreversibilities dominate for a greater radial distance in this case. Moreover, due to decreasing heat flux along the flow direction for the case of linearly decreasing heat flux distribution, although the penetration of heat along the radial direction decreases, there is an increase in the penetration of heat along the axial direction and therefore, Be remains zero for r/R < 0.1, which is the minimum among all the cases. On approaching towards the wall of the tube from its centre line, there is steady increase in Be for all the cases. This shows that there is a steep rise in the contribution of thermal entropy generation in the vicinity of the wall, indicating the dominance of entropy generation due to heat transfer over fluid friction in this region. It is found that the Bejan number is close to one and remains constant near the wall for all cases owing to steep temperature gradient. Moreover, because of the strong influence of heat transfer and higher temperature gradient in the near wall region of the tube, Be tends to one and remains constant for a certain radial distance from the wall towards the centre line of the tube. However, it can be seen that the value of Be at the entrance location remains one for a smaller radial distance from the wall towards the centre line of the tube as compared to that at the downstream location of the tube because of the larger velocity gradients at the entrance region of the duct and the increase in the depth of penetration of heat as the flow proceeds towards the downstream direction. This observation holds true for all the cases of heat flux, although the radial distance from the wall towards the centre line of the tube up to which Be remains close to one varies with the change in the applied heat flux condition. It can be noticed from Fig. 5(b) that Be remains close to one for the minimum radial distance from the wall towards the centre line of the tube at all the longitudinal locations for the case of linearly increasing heat flux distribution. This is in support to the distribution of heat flux which reveals that the irreversibilities due to fluid friction is maximum at the entrance location and those due to thermal transport is maximum, while reverse is the scenario near the outlet. However, Be remains close to one for the maximum radial distance from the wall towards the centre line of the tube at all the longitudinal locations for periodic heating (sinusoidal with λ = 100). This is because of the fluctuation in the longitudinal distribution of heat flux that increases both the axial and radial gradients of temperature, thereby resulting in higher thermal irreversibilities. 5.4. Effect of wall heat flux condition on longitudinal entropy generation profiles Variation of total entropy generation along the longitudinal direction of the duct at three different radial locations is shown in Fig. 6 for all the cases of heat flux distributions studied here. Owing to negligible velocity gradient and zero temperature gradient, the total irreversibility at the centre line is around zero for all the cases. However, irreversibility near the wall is mainly due to the transfer of heat as compared to that due to fluid friction. It can be seen that Sgen has a very high value in the entrance region of the tube and decreases along the longitudinal direction for uniform and linearly decreasing heat flux condition at r/R=1. Moreover, the decrement in Sgen along the longitudinal direction is more significant for the case of linearly decreasing heat flux distribution in comparison to uniform heat flux condition. This is because of the adverse temperature gradient at the entrance location due to the higher amount of applied heat flux that increases the net

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

9

Fig. 5. Radial variation of Bejan number at five longitudinal locations along the tube for different heat flux configurations.

entropy generation at this location. However, due to the decreasing distribution of heat flux towards the downstream location, the radial gradient of temperature decreases, thereby resulting in a significant decrement in Sgen along the longitudinal direction of the tube. Similarly, in the case of sinusoidal heat flux distribution with λ=1, due to the increase of temperature gradient up to the mid-

longitudinal location of the tube, Sgen also increases up to the midlongitudinal location and decreases thereafter due to the reduction in the applied heat flux. On the contrary, as evident from the distribution of linearly increasing heat flux, both the axial and radial gradients of temperature increases along the downstream direction, thus causing a net increase in Sgen along the longitudinal

10

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

Fig. 6. Axial variation of entropy generation at three different radial locations for different heat flux configurations.

direction as can be seen from Fig. 6(b). Interestingly, the dominance of thermal entropy generation in the near wall region is more prominently revealed by the nature of the variation of Sgen along the longitudinal direction of the tube for periodic heating (sinusoidal with λ = 100). Following the sinusoidally fluctuating

variation of applied heat flux, Sgen also varies with the same trend for periodic heating (sinusoidal with λ = 100). The longitudinal variation of Sgen is also shown at r/R=0.8 for all the cases and it is important to note at priori that the longitudinal variation of Sgen at r/R=0.8 is dictated by the competing influ-

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

11

Fig. 7. Variation of (a) maximum temperature and (b) total entropy generation along the tube for different heat flux configuration.

ence of thermal and viscous irreversibilities at this radial location. For the case of linearly increasing heat flux distribution, the axial and radial gradients of temperature increases along the downstream direction due to the nature of the applied heat flux. As a consequence, the thermal irreversibilities dominate over the frictional irreversibilities at r/R=0.8 and therefore, Sgen continuously increases along the direction of the flow. In case of uniform wall

heat flux condition, although there is no variation in the applied heat flux along the longitudinal direction, the fluid that is initially heated at the upstream location transfers the thermal energy to the colder fluid in the downstream location, resulting in an increase in Sgen along the length of the tube at r/R=0.8. One of the most notable effect in terms of entropy generation can be observed with decreasing wall heat flux condition. For example, at the ra-

12

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

Fig. 8. Variation of (a) maximum temperature and (b) total entropy generation along the tube for different periodic heat flux configurations and their comparison with the uniform heat flux configuration.

dial location (r/R) of 0.8, it can be seen that in the cases of linearly decreasing and periodic (sinusoidal with λ=1) heat flux condition, the entropy generation along the tube increases initially, followed by continuous decrease thereafter along the longitudinal direction. This can be attributed to the reduction in the fluid temperature near the wall along the downstream direction of the tube for the case of linearly decreasing heat flux distribution. However, due to the fluctuation in the applied heat flux along the tube for periodic heating (sinusoidal with λ = 100), although the temperature gradient is higher in the inlet region, there occurs a uniformity in the longitudinal distribution of temperature and the temperature

gradient. Therefore, we observe an increment in Sgen in the entrance region for the case of periodic heating, followed by a decrease along the longitudinal direction thereafter at r/R=0.8. 5.5. Comparative assessment of effect of wall heating condition on maximum temperature and entropy generation Although the discussions in the previous sections gives an overview of the nature of the variation of temperature and entropy generation for different cases of applied heat flux, it is essential to make a comparative assessment of both to devise the efficient

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

heating strategy. In this section, observations are made to judge whether the minimum peak temperature could be obtained with minimum entropy generation, which would be an ideal scenario for any thermal system. In order to determine the same, the variation of peak temperature at different axial locations of the tube is depicted in Fig. 7(a) and compared with the corresponding variation of the total entropy generation at different axial locations of the tube as presented in Fig. 7(b) for different wall heating conditions. It can be observed from Fig. 7(a) that the temperature is maximum for linearly increasing wall heat flux boundary condition. Furthermore, in case of linearly decreasing heat flux and sinusoidal wall heating boundary condition (λ = 1), an increase in the temperature can be seen initially upto a certain length of the tube, followed by a decrease in temperature in the downstream portion of the tube. This must be attributed to the decrease in the heat flux in latter portion of the tube. Interestingly, for a particular case of periodic heating (sinusoidal with λ = 100), the temperature is fairly less as compared to that observed in case of uniform heating. Also, Fig. 7(b) indicates that the entropy generation for periodic heating (sinusoidal with λ = 100) is less than the one in case of uniform heating. Nevertheless, the entropy generation in case of linearly increasing heat flux condition is significantly less than the other cases, however it is associated with the maximum temperature among all the cases and therefore cannot be recommended as an optimum heating strategy. A comparative study of the variation of temperature and entropy generation for five different values of λ (=40, 80, 100, 150 and 200) has also been performed and presented in Fig. 8 and the same is also compared with the results for uniform heat flux configuration. Compared to the uniform heat flux condition, it is clearly evident from Fig. 8 that the maximum temperature and total entropy generation is higher for periodic heat flux condition with λ=40 and λ=80, while the same is lower for λ=100, 150 and 200. It is worth mentioning that extensive investigations were also carried out with values of λ close to 100 and it is found that periodic heat flux condition with λ<100 results in higher entropy generation and temperature than the uniform heat flux condition, while the same is lower for λ≥100. Thus, the periodic heating (sinusoidal with λ = 100) condition is a unique case where minimization of entropy generation is achieved simultaneously along with minimum peak temperature. In nutshell, the comparative assessment of temperature and entropy generation for different cases confirms that the sinusoidal heating condition with λ = 100 can be thought of as one of the potential boundary condition for any thermal system where it is expected to keep both entropy generation and peak temperature minimum. Further, it can be adjudged that the any value of λ greater than 100 results in lower entropy generation and temperature and therefore it can be recommended for safe operation of any thermal system.

6. Conclusions A novel approach of applying non-uniform wall heat flux at the walls of a circular duct for achieving laminar forced convective heat transfer with minimum peak temperature and entropy generation is discussed in this study. Various types of wall heat flux boundary conditions (i.e., uniform, linearly increasing, linearly decreasing and sinusoidal heat flux) have been investigated numerically for an equal amount of heat transfer rate for all the cases for laminar flow through a circular duct. Specific findings of the study undertaken are as follows: •

The radial penetration of heat for the linearly decreasing and periodic heat flux condition is higher than the radial penetration of heat for uniform, linearly increasing and sinusoidal heat flux condition.

•

•

•

•

•

13

Minimum peak temperature could be achieved by applying periodic heat flux distribution (sinusoidal with number of waves greater than or equal to 100). Heat flux distribution with linearly decreasing arrangement always generates more entropy than the cases with uniform, linearly increasing, sinusoidal and periodic heat flux distribution. The magnitude of radial distance up to which the viscous and thermal irreversibilities dominate within the tube varies with the nature of heat flux distribution. Precisely, the viscous irreversibilities dominate for the maximum radial distance in case of sinusoidal heat flux distribution and minimum for the case of linearly decreasing heat flux distribution. In the context of thermal irreversibilities, it is dominant for the maximum radial distance from the wall towards the centre line of the tube in case of periodic heat flux distribution and minimum for the case of linearly increasing heat flux distribution. The entropy generation in case of linearly increasing heat flux condition is significantly less than the other cases; however it is associated with the maximum temperature among all the cases and therefore cannot be recommended as an optimum heating strategy. The peak temperature for a particular case of periodic heat flux (sinusoidal with λ = 100) has been found to lesser as compared to uniform heating. Moreover, the entropy generation for this case is less than that of uniform heating case. Therefore, in terms of achieving both minimum temperature and entropy generation simultaneously, the periodic heat flux distribution (sinusoidal with λ = 100 or λ > 100) can be recommended as the optimum heating strategy for any thermal system.

Credit authorship contribution statement Sukumar Pati: Conceptualization, Formal analysis, Writing - review & editing. Rajib Roy: Visualization, Formal analysis. Nabajit Deka: Visualization, Formal analysis. Manash Protim Boruah: Visualization, Formal analysis. Mriganka Nath: Visualization, Formal analysis. Ritwick Bhargav: Visualization, Formal analysis. Pitambar R. Randive: Conceptualization, Formal analysis, Writing - review & editing. Partha P. Mukherjee: Conceptualization, Formal analysis, Writing - review & editing. Declaration of Competing Interest There is no conflict of interest to the best of the authors’ knowledge. CRediT authorship contribution statement Sukumar Pati: Conceptualization, Formal analysis, Writing - review & editing. Rajib Roy: Visualization, Formal analysis. Nabajit Deka: Visualization, Formal analysis. Manash Protim Boruah: Visualization, Formal analysis. Mriganka Nath: Visualization, Formal analysis. Ritwick Bhargav: Visualization, Formal analysis. Pitambar R. Randive: Conceptualization, Formal analysis, Writing - review & editing. Partha P. Mukherjee: Conceptualization, Formal analysis, Writing - review & editing. References [1] A. Bejan, A study of entropy generation in fundamental convective heat transfer, J. Heat Transf. 101 (2010) 718–725. [2] J. Guo, L. Cheng, M. Xu, Optimization design of shell-and-tube heat exchanger by entropy generation minimization and genetic algorithm, Appl. Therm. Eng. 29 (2009) 2954–2960, doi:10.1016/j.applthermaleng.2009.03.011. [3] V.L. Le, A. Kheiri, M. Feidt, S. Pelloux-prayer, Thermodynamic and economic optimizations of a waste heat to power plant driven by a subcritical ORC (Organic Rankine Cycle) using pure or zeotropic working fluid, Energy 78 (2014) 622–638.

14

S. Pati, R. Roy and N. Deka et al. / International Journal of Heat and Mass Transfer 150 (2020) 119318

[4] M.P. Boruah, S. Pati, P.R. Randive, Implication of fluid rheology on the hydrothermal and entropy generation characteristics for mixed convective flow in a backward facing step channel with baffle, Int. J. Heat Mass Transf. 137 (2019) 138–160. [5] M.P. Boruah, P.R. Randive, S. Pati, Hydrothermal performance and entropy generation analysis for mixed convective flows over a backward facing step channel with baffle, Int. J. Heat Mass Transf. 125 (2018) 525–542. [6] A. Bejan, E. Sciubba, The optimal spacing for parallel plates cooled by forced convection, Int. J. Heat Mass Transf. 35 (1992) 3259–3264. [7] A. Bejan, N. Carolina, Y. Fautrelle, Constructal multi-scale structure for maximal heat transfer density, Acta Mech. 49 (2003) 39–49, doi:10.1007/ s00707- 003- 1008- 3. [8] M.R. Hajmohammadi, A. Campo, S. Antonio, Improvement of forced convection cooling due to the attachment of heat sources to a conducting thick plate, Int. J. Heat Mass Transf. 135 (2018) 1–4, doi:10.1115/1.4024897. [9] R.R. Madadi, C. Balaji, Optimization of the location of multiple discrete heat sources in a ventilated cavity using artificial neural networks and micro genetic algorithm, 51 (2008) 2299–2312. doi:10.1016/j.ijheatmasstransfer.2007.08.033. [10] T.V.V Sudhakar, C. Balaji, S.P. Venkateshan, Optimal configuration of discrete heat sources in a vertical duct under conjugate mixed convection using artificial neural networks, Int. J. Therm. Sci. 48 (2009) 881–890, doi:10.1016/j. ijthermalsci.2008.06.013. [11] A. Müftüo, Natural convection in an open square cavity with discrete heaters at their optimized positions, Int. J. Therm. Sci. 47 (2008) 369–377, doi:10.1016/ j.ijthermalsci.2007.03.015. [12] A.K. Silva, S. Lorente, A. Bejan, Optimal distribution of discrete heat sources on a plate with laminar forced convection, Int. J. Heat Mass Transf. 47 (2004) 2139–2148, doi:10.1016/j.ijheatmasstransfer.20 03.12.0 09. [13] M.R. Hajmohammadi, E. Shirani, M.R. Salimpour, A. Campo, Constructal placement of unequal heat sources on a plate cooled by laminar forced convection, Int. J. Therm. Sci. (2012) 60, doi:10.1016/j.ijthermalsci.2012.04.025. [14] M.R. Hajmohammadi, S.S. Nourazar, A. Campo, S. Poozesh, Optimal discrete distribution of heat flux elements for in-tube laminar forced convection, Int. J. Heat Fluid Flow. 40 (2013) 89–96. [15] A. Bejan, Entropy Generation Minimization: the Method of Thermodynamic Optimization of Finite Size Systems and Finite Time Processes, CRC Press, Boca Raton, New York, Florida, 1996. [16] I. Dagtekin, H.F. Oztop, A.Z. Sahin, An analysis of entropy generation through a circular duct with different shaped longitudinal fins for laminar flow, Int. J. Heat Mass Transf. 48 (1) (2005) 171–181. [17] D. Nouri, M. Pasandideh-Fard, M.J. Oboodi, O. Mahian, A.Z. Sahin, Entropy generation analysis of nanofluid flow over a spherical heat source inside a channel with sudden expansion and contraction, Int. J. Heat Mass Transf. 116 (2018) 1036–1043.

[18] C. Wang, M. Liu, Y. Zhao, Y. Qiao, J. Yan, Entropy generation analysis on a heat exchanger with different design and operation factors during transient processes, Energy (2018). [19] J.A. Esfahani, P.B. Shahabi, Effect of non-uniform heating on entropy generation for the laminar developing pipe flow of a high Prandtl number fluid, Energy Conv. Manag. 51 (2010) 2087–2097, doi:10.1016/j.enconman.2010.02.022. [20] A. Poddar, R. Chatterjee, K. Ghosh, A. Mukhopadhyay, S. Sen, Performance assessment of longitudinal flow through rod bundle arrangements using entropy generation minimization approach, Energy Conv. Manag. 99 (2015) 359–373, doi:10.1016/j.enconman.2015.04.075. [21] S. Pati, S.K. Som, S. Chakraborty, Thermodynamic performance of microscale swirling flows with interfacial slip, Int. J. Heat Mass Transf. 57 (2013) 397–401. [22] O. Mahian, A. Kianifar, C. Kleinstreuer, A.N. Moh’d, I. Pop, A.Z. Sahin, S. Wongwises, A review of entropy generation in nanofluid flow, Int. J. Heat Mass Transf. 65 (2013) 514–532. [23] V. Zimparov, Extended performance evaluation criteria for enhanced heat transfer surfaces : heat transfer through ducts with constant wall temperature, Int. J. Heat Mass Transf. 43 (20 0 0) 3137–3155. [24] A.Z. Sahin, Irreversibilities in various duct geometries with constant wall heat flux and laminar flow, Energy 23 (6) (1998) 465–473. [25] A.Z. Sahin, Entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to constant heat flux, ASME J. Heat Transf. 2 (2002) 314–321. [26] A. Moallemi, A. Arabkoohsar, F.J. Pujatti, R.M. Valle, K.A. Ismail, Non-uniform temperature district heating system with decentralized heat storage units, a reliable solution for heat supply, Energy 167 (2019) 80–91. [27] A.Z. Sahin, S.M. Zubair, A.Z. Al-Garni, R. Kahraman, Effect of fouling on operational cost in pipe flow due to entropy generation, Energy Convers. Manag. 41 (20 0 0) 1485–1496. [28] A. Borah, N. Deka, S. Pati, Effect of non-uniform heating on entropy generation for thermally developing flow between parallel plates, in: Proceedings of the International Conference Computer Methods Thermal Problem THERMACOMP2018, Bengaluru, India, 2018, pp. 47–50. [29] A. Arabkoohsar, Non-uniform temperature district heating system with decentralized heat pumps and standalone storage tanks, Energy (2019). [30] M.P. Boruah, P.R. Randive, S. Pati, Effect of non-uniform asymmetric heating on the thermal and entropy generation characteristics for flow of Al2 O3 -water nanofluid in a micro-channel, Int. J. Numer. Meth. Heat Fluid Flow 29 (3) (2019) 981–999. [31] L.M. Tam, A.J. Ghajar, H.K. Tam, Contribution analysis of dimensionless variables for laminar and turbulent flow convection heat transfer in horizontal tube using artificial neural network, Heat Transf. Eng. 29 (2008) 793–804. [32] A.Z. Sahin, R. Ben-Mansour, Entropy generation in laminar fluid flow through a circular pipe, Entropy 5 (5) (2003) 404–416.