Introduction

Today, with the advancement of technology, one of the most important engineering challenges is to increase heat transfer and improve the performance of heating systems. By growing and development of modern technologies, special attention has been paid to heat transfer, reducing heat transfer time, minimizing the size of heat exchangers and finally maximizing the efficiency of heating equipment [1]. As mentioned, due to energy constraints and the fact that the world is moving toward energy storage and using net-zero energy systems, it is necessary to optimize heating systems and increase heat transfer, one of which is heat exchangers [2,3,4]. One of the techniques to increase the heat transfer rate is to increase the level of heat exchange by using tubes with protrusions and dimples [5]. Dimple tubes are in fact smooth tubes that have indentation on their walls that increase the level of heat transfer and thus increase heat transfer [6]. Dimples act as vortices to create turbulent flow in the flow to increase heat transfer. Recent researches show that dimpled surfaces can increase the rate of convective heat transfer and decrease the relative pressure [5], [7]. Due to the suitability of submerged surfaces in improving heat transfer, many experimental and numerical studies have been performed in recent decades [8].

Wang et al. [9] investigated the performance of protruding and dimpled tubes using air-operated fluid and proposed relationships to predict heat transfer and pressure drop. Kumar et al. [10] performed experiments on dimpled tubes. They found that most tube hydraulics have a indentation of 14–56% more than flat tubes. Li et al. [11] experimentally examined dimpled tubes in in-tube heat exchangers. They found that the increase in heat transfer compared to a flat tube was 200%. Kukulka et al. [12] performed experiments on the condensation and evaporation of dimpled tubes, which they called Vipertex tubes. They reported that Vipertex tubes evaporate well because indentations increase surface turbulence and boundary layer distribution. Shafaee et al. [13] investigated the thermal performance of dimpled tubes for both deep and shallow indentations using R-600 operating fluid. Experiments have shown that tubes with dents (dimples) increase the heat transfer coefficient by 29–100% compared to simple tubes. Aroonrat and Wongwises [14], [15] experimentally studied biphasic density on hydraulic heat performance in dimpled tubes. They found that dimpled tubes resulted in the highest Nusselt number being about 84% higher than flat tubes. Chen et al. [16] performed experiments on tubes with two three-dimensional diagonal grooves and proposed relationships to predict the heat transfer coefficient. Cheng et al. [17] measured the Nusselt numerical distribution and the pressure drop coefficient for four hexagonal channels with dimpled walls and obtained relationships for each of the tubes. Xie et al. [18] developed a complex geometry using the plastic forming method for tube reinforced with indentations and ridges and discussed its heat-hydraulic performance. They found that the protrusions cause the flow of the fluid to rot and block the flow in the ridges and indentations, thus increasing the rate of heat transfer.

Another way to increase heat transfer is to use turbulator which are one of the tools that increase heat transfer by disturbing the flow of fluid. Turbulators are usually spiral blades that are inserted into boiler tubes, heat exchangers, boiler chimneys, aircraft fins, and fiberglass boilers to rotate the flow [19]. Turbulators are fixed or vibrant mechanical components that increase the heat transfer coefficient by creating turbulence in the flow and destroying the boundary layer, preventing sediment on the inner surface of the tube and prolonging the flow path in the tube, resulting in more fluid flow time [20]. Contact is achieved by the heat exchange surface and the heat transfer rate increases. Some studies focused on the properties of the operating fluid and improving it, and some others focused on active and inactive methods for heat exchanger optimization. Changes in geometric parameters and other effective factors to improve heat transfer have been extensively investigated by scholars. When a fluid passes through a tube, the part of the fluid that is close to the tube wall is cooled due to friction with the wall and the low passage rate, and the rest of the fluid have less heat transfer, as a result, heat transfer around the tube will be less. In addition, the fluid flow in the tube and heat exchangers tends toward the wall of the tube, which acts both as an insulating layer and causes sediment in the walls. The turbulator breaks this layer by creating turbulence in the flow and converts the flow regime from laminar to turbulent regime, thus increasing heat transfer and less sedimentCitationID="CR18">18. Eiamsa-ard and Promvonge [22] performed an experimental study of the effect of a helical strip inside a circular tube on the Nusselt number at Reynolds 2300 to 8800. They showed that the highest heat transfer rate occurs by placing the full length of the helical strip with the central rod in a flat tube. Jasinski [23, 24 ]performed two numerical studies on the thermo-hydraulic properties of a circular tube equipped with a ball-shaped turbulator. These researches consisted of two phases (the second phase focuses on the heat transfer process), the first of which presents the numerical simulation results in a circular tube with an internal ball flowing. The obtained results show a good correlation between computer testing and modeling. Several studies that focused on using turbulators to improve the heat transfer rate in heat exchangers are illustrated in Table 1.

After comprehensively considering aforementioned studies, we can see that there is rare studies focused on using turbulators in dimpled tubes based on the first and second law of thermodynamics. Adding dimples on the inner bodies of the tubes can be favorable to increase the heat transfer coefficient with a consequent rise in the friction factor and interruption of the boundary layer, increase the turbulence and effective heat transfer area and generation of secondary flows [34], [35]. Generally, by reviewing similar studies published in the mentioned researches reviewed above, it can be said that there is no complete and comprehensive study that investigates heat transfer in dimple tubes equipped with turbulators. The effect of pitch (S) and the number of dimples (N) in three heat fluxes applied to the dimple tube has been investigated. The important point in this research is the numerical investigation of the Nusselt number and the first and second laws of thermodynamics to obtain the effect of the Reynolds number and the mentioned parameters on the heat transfer rate and entropy generation. The novelty of this research is the integration of both turbulence factors in the tube, i.e., dimpling on the tube and also placing the turbulator in the centerline of tube. Dimpling the tube leads to disrupting the boundary layer, creating turbulence, as well as increasing the effective surface area, friction coefficient, and ultimately improving heat transfer. On the other hand, the presence of the turbulator leads to the prolonging the flow path in the tube in addition to the effects mentioned for the dimples. Many studies in the field of only dimpling the tube and only placing the turbulator in the tubes can be found in the literature. However, the integration of these two passive methods to intensify the heat transfer in the tubes under the heat flux has not been done numerically so far. Investigating this issue from the point of view of the first and second laws of thermodynamics is a significant issue. On the other hand, this idea can be considered as a net-zero issue because it is passive and does not use any external energy to improve heat transfer. Briefly, it can be said that the objectives of this research are as follows:

  • This research focused on optimizing the geometric parameters in a dimpled tube equipped with a conical turbulator based on the first and second laws of thermodynamics.

  • The investigated dimpled tube is equipped the cones, which are installed on a narrow rod in the center of the tube.

  • The first main aim is to investigate the effect of geometric parameters such as pitch (S) and number of dimples (N), Reynolds number (Re = in the range of 5000 to 20,000) and heat flux (3000, 5000, 10,000 (W m−2)) on entropy generation, Nusselt number and the pressure drop in the dimpled tube with turbulator.

  • The second aim is to investigate the effect of turbulator geometric parameters on entropy generation and to find the relationship between them and also to evaluate heat transfer and pressure drop to achieve more optimal conditions.

Table 1 Summarized list of some studies that used turbulators in heat exchangers
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Numerical modeling

Physical model and geometry

Cone turbulence means that the cones are mounted on a narrow bar in the center of the dimpled tube. For this purpose, a tube with a length of 100 mm and diameter of 20 mm with a turbulator with a diameter of 1 mm in the center is investigated, and its schematic is shown in Fig. 1.

Fig. 1
figure 1

a Schematic of the dimpled tube and a constant heat flux applied to its sidewall, b facing view of turbulator and tube, c side view of the turbulator

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The governing equations

The numerical simulation involves solving basic conservation equations. The governing equations in this research are as follows[33,34,]–[38]:

Continuity equation:

$$\frac{\partial }{\partial {x}_\text{i}}\left(\rho {u}_\text{i}\right)=0$$
(1)

Momentum equation:

$$\frac{\partial }{{\partial x_\text{j} }}\left( {u_\text{j} \rho u_\text{i} } \right) = - \frac{\partial p}{{\partial x_\text{i} }} + \frac{\partial }{{\partial x_\text{j} }}\left[ { - \rho \overline{{u_\text{i}^{\prime } u_\text{j}^{\prime } }} } \right] + \frac{\partial }{{\partial x_\text{i} }}\left\{ {\mu \left[ {\frac{{\partial u_\text{i} }}{{\partial x_\text{j} }} + \frac{{\partial u_\text{j} }}{{\partial x_\text{i} }}} \right]} \right\}$$
(2)

Energy equation:

$$\frac{\partial }{\partial {x}_\text{j}}\left(\rho {u}_\text{i}T\right)=\frac{\partial }{\partial {x}_\text{i}}\left\{\left[\frac{\mu }{Pr}+\frac{{\mu }_\text{t}}{{Pr}_\text{t}}\right]\frac{\partial T}{\partial {x}_\text{i}}\right\}$$
(3)

Reynolds stress:

$$- \rho \overline{{u_\text{i}^{\prime } u_\text{j}^{\prime } }} = \mu_\text{t} \left[ {\frac{{\partial u_\text{i} }}{{\partial x_\text{j} }} + \frac{{\partial u_\text{j} }}{{\partial x_\text{i} }}} \right] - \frac{2}{3}\rho k\delta_\text{ij} - \frac{2}{3}\mu_\text{t} \frac{{\partial u_\text{k} }}{{\partial x_\text{k} }}\delta_\text{ij}$$
(4)

Confused viscosity:

$${\mu }_\text{t}=\rho {C}_{\mu }\frac{{K}^{2}}{\varepsilon }$$
(5)

Equations of k and ε for turbulent flow model:

$$\frac{\partial }{\partial {x}_\text{i}}\left(\rho {u}_\text{i}k\right)=\frac{\partial }{\partial {x}_\text{j}}\left\{\left[\frac{{\mu }_\text{t}}{{\sigma }_\text{k}}+\mu \right]\frac{\partial k}{\partial {x}_\text{i}}\right\}+{G}_\text{k}-\rho \varepsilon$$
(6)
$$\begin{gathered} \frac{\partial }{{\partial x_\text{i} }}\left( {\rho u_\text{i} \varepsilon } \right) = \frac{\partial }{{\partial x_\text{j} }}\left\{ {\left[ {\frac{{\mu_\text{t} }}{{\sigma_{\varepsilon } }} + \mu } \right]\frac{\partial \varepsilon }{{\partial x_\text{i} }}} \right\} + \frac{\varepsilon }{k}\left[ {C_{1\varepsilon } G_\text{k} - \rho C_{2\varepsilon } \varepsilon } \right] \hfill \\ G_\text{k} = - \rho \overline{{u_\text{i}^{\prime } u_\text{j}^{\prime } }} \frac{{\partial u_\text{j} }}{{\partial x_\text{i} }} \hfill \\ \end{gathered}$$
(7)

The constant coefficients selected in the k-ε equation (RNG model) are as follows:

$${C}_{1\varepsilon }=1.42, {C}_{2\varepsilon }=1.68, {C}_{\mu }=0.0845, {Pr}_\text{t}=0.85, {\sigma }_\text{k}=1, {\sigma }_{\varepsilon }=1.3$$

Boundary conditions

The boundary conditions include uniform inlet velocity at the inlet of the dimpled tube, non-slip condition in the walls, and atmospheric pressure at the outlet of the dimpled tube. Temperature boundary conditions also include the application of a uniform heat flux on the tube wall and an inlet temperature of 300 K. The mathematical form of boundary conditions is as follows:

Inputs:

$$u={u}_{0}; T={T}_{0}$$
(8)

Boundary between solid and liquid:

$$u=0; -k\frac{\partial T}{\partial y}={q}^{\prime\prime}$$
(9)

Outputs:

$$p={p}_{0}$$
(10)

Heat transfer performance and entropy generation

Using Eq. (11), the Reynolds number (Re) can be calculated, which is an indicator of the type of flow (laminar or turbulent):

$$Re=\frac{\rho uD}{\mu }$$
(11)

where μ is the dynamic viscosity, ρ is the density, u is the flow velocity, and D is the diameter of the tube [39], [40].

Equation (12) is used to calculate the Nusselt number (Nu). Here, h is the convective heat transfer coefficient and k is the conduction heat transfer coefficient [36,37,]–[41]:

$$Nu=\frac{hD}{k}$$
(12)

Using Eq. (13), the friction coefficient (f) can be calculated as follows [36,37,]–[41]:

$$f=\frac{2D}{L}\frac{\Delta p}{\rho {u}^{2}}$$
(13)

where Δp is the pressure drop, u velocity and L is the length of the tube.

Overall entropy [42]:

$${S}_\text{t}={S}_\text{f}+{S}_\text{th}$$
(14)

Thermal entropy [43]:

$${S}_\text{th}=\frac{k}{{T}^{2}}[{\left(\frac{\partial T}{\partial x}\right)}^{2}+{\left(\frac{\partial T}{\partial y}\right)}^{2}+{\left(\frac{\partial T}{\partial z}\right)}^{2}$$
(15)

Frictional entropy [43]:

$${S}_{\text{f}}=\frac{\mu }{T}\left\{2\left[{\left(\frac{{\partial u}_{\text{x}}}{\partial x}\right)}^{2}+{\left(\frac{{\partial u}_{\text{y}}}{\partial y}\right)}^{2}+{\left(\frac{\partial {u}_{\text{z}}}{\partial z}\right)}^{2}\right]+{\left(\frac{\partial {u}_\text{x}}{\partial y}+\frac{\partial {u}_{\text{y}}}{\partial x}\right)}^{2}+{\left(\frac{\partial {u}_{\text{x}}}{\partial z}+\frac{\partial {u}_{\text{z}}}{\partial x}\right)}^{2}+{\left(\frac{\partial {u}_{\text{y}}}{\partial z}+\frac{\partial {u}_{\text{z}}}{\partial y}\right)}^{2}\right\}$$
(16)

Grid independence and model validation

In order to find the correct grid number to solve the equations numerically, it is necessary to examine the dependence of the results on the number of grids. For this purpose, the number of computational domain grids gradually is increased, and the results in each case are recorded. In the end, when the results do not change, the number of grids is acceptable, and there is no need to make the grid smaller and increase the computational cost. In Table 2, grid independence is evaluated in two Re = 10,000 and 15,000. According to the diagram shown in Fig. 2, there is no change in the Nusselt number by increasing the number of cells. In addition, according to the diagram in Fig. 3, the friction coefficient does not change or these changes are so small that they are ignored, so that to save time and reduce the computational cost, approximately 804,963 cells are enough for the computing grid, and reliable answers can be obtained. Therefore, this number of grids is considered in this study. The following table is for pitch (S) = 14 mm and the number of dimples (N) = 2.

Table 2 Investigation of Nusselt number changes and friction factor
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Fig. 2
figure 2

Changes in Nu versus the number of cells for two different Reynolds numbers

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Fig. 3
figure 3

Changes in the friction factor versus the number of cells for two different Reynolds numbers

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After modeling the system geometry and checking the independence of the grid, the results should be compared and evaluated with the results that have already been obtained numerically or experimentally. Therefore, the results were compared with the results presented by Xie et al. [18], which is a study on the characteristics of flow and heat transfer in dimple tubes with tear dimples. The Reynolds number changed between 5000 and 30,000.

According to Fig. 4, the numerical results obtained for the Nusselt number are well-consistent with the results presented by Xie et al. [18], in which case the deviation rate of these results is 5.54%.

Fig. 4
figure 4

Comparison of Nusselt number versus Reynolds number in Xie et al. [18] with numerical results obtained in this study

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Results and discussion

Effect of pitch

Effect of pitch on pressure drop, velocity distribution and Nusselt number

Figure 5 shows the pressure drop changes versus pitch (S) in different Reynolds numbers for the flux of 10,000 W m−2 at the number of dimples of 3 (N = 3). As can be seen, the pressure drop decreases with increasing pitch (S). In fact, with increasing pitch, the number of cones decreases and the pressure drop decreases.

Fig. 5
figure 5

Pressure drop changes versus pitch (S) in different Reynolds numbers for flux (W m−2) of 10,000 and N = 3

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Figure 6 shows the pressure contours for pitches, S = 14, 22 and 30 mm at Re = 20,000. According to this figure, in the pitch of S = 14 mm, the maximum pressure drop is observed, which occurs due to the friction between the fluid with the wall and the cones. As the pitch decreases, the number of conical turbulator increases due to the turbulence of the flow and fluid friction. Therefore, increasing the pitch reduces the friction coefficient and pressure drop.

Fig. 6
figure 6

Pressure contour for different pitches in Re = 20,000: a S = 14 mm, b S = 22 mm, c S = 30 mm

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Figure 7 shows the velocity contours for different pitches at N = 3 in Re = 10,000. Figure 8 shows the changes of the Nusselt number versus pitch in different Reynolds numbers at the heat flux (W m−2) of 10,000. It is clear that as the pitch increases, the Nusselt number decreases. In fact, as the pitch increases, the number of cones decreases and the flow turbulence decreases. In addition, reducing the temperature gradient near the wall reduces the heat transfer coefficient and reduces the heat transfer. As the pitch decreases, in fact, the number of cones increases, and consequently, the turbulence of the flow and the turbulence of the thermal boundary layer increase. The rate of heat exchange also increases, which increases the displacement heat transfer coefficient, which is directly related to the Nusselt number. Therefore, decreasing the pitch increases the Nusselt number and increases the heat transfer.

Fig. 7
figure 7

Velocity contours for different pitches in Re = 10,000: a S = 14 mm, b S = 22 mm, c S = 30 mm

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Fig. 8
figure 8

Variation in Nusselt number versus pitch (S) in different Reynolds numbers for q = 10,000 (W m−2) and N = 4

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Effect of pitch on entropy generation

As mentioned in the previous sections, variations in the friction entropy are a function of changes in velocity gradient. Here, the process of changing the friction entropy based on the pitch is investigated. Figure 9 shows the frictional and thermal entropy changes in Reynolds numbers at different pitches for N = 2 and q″ = 3000 W m−2. The pitch is actually the distance between the two cones inserted on the center bar, and as the number of pitches increases, the number of cones decreases. Therefore, it can be concluded that increasing the pitch or decreasing the number of cones reduces the velocity gradient. As the number of cones decreases, the number of obstacles in front of the flow decreases and the velocity gradient, which represents velocity changes, decreases, causing the frictional entropy to decrease. Therefore, as shown in Fig. 9a, at a constant Reynolds number, as the pitch increases, the friction entropy decreases so that the minimum friction entropy is related to the pitch of S = 30 mm.

Fig. 9
figure 9

Changes in a Frictional entropy generation, and b thermal entropy generation changes versus Re in different pitches for q″ = 3000 W m−2 and N = 2

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Figure 9b shows a graph of thermal entropy changes under the mentioned conditions in Fig. 9a. At a constant Reynolds number, the higher the pitch results in the higher the thermal entropy. According to Eq. (15), thermal entropy is directly related to temperature gradient. The greater pitch leads to the higher the temperature gradient in the wall and the higher the thermal entropy. Figure 9b shows that the maximum thermal entropy is related to the pitch of S = 30 mm and the Re = 5000.

Investigating the effect of Reynolds number

Figure 10 shows the pressure drop changes versus Reynolds number at different N for S = 14 mm at q″ = 5000 W m−2. The pressure drop increases with increasing Reynolds number. In fact, the higher the Reynolds number causes increasing the flow friction with the tube wall and turbulator. Figure 11 shows the velocity contours in different Reynolds numbers for S = 22 mm and N = 4.

Fig. 10
figure 10

Variation of the pressure drop versus Reynolds number for different N for S = 14 mm and q = 5000 W m−2

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Fig. 11
figure 11

Velocity contour in different Reynolds numbers for S = 22 mm and N = 4 for a Re = 5000, b Re = 10,000, c Re = 15,000, d Re = 20,000

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Figure 12 shows the variation of the Nusselt number versus Reynolds number in the different number of dimples for q″ = 5000 W m−2. As can be seen, with increasing Reynolds number, Nusselt number rises because this boosts the turbulence of the flow and thus the heat transfer rate. The reason can be explained by the fact that since the inlet temperature and heat flux have constant values, so with reducing the outlet temperature, the temperature difference decreases and based on the relationship of heat transfer coefficient of convection (h = q ''/∆T), which can be said to increase h.

Fig. 12
figure 12

Variations of the Nusselt number versus Reynolds number in different N for q″ = 5000 W m−2 and S = 30 mm

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The total entropy is equal to the sum of the frictional and thermal entropies. First, the effect of Reynolds number on the generation of friction entropy is investigated. According to Eq. (16), changes in velocity gradient determine the amount and trend of friction entropy generation. As the Reynolds number increases, the gradient increases, and the frictional entropy also increases. Figure 13a shows the changes in friction entropy versus Reynolds number at S = 30 mm and different N. It is clear that as the Reynolds number increases, the frictional entropy also increases.

Fig. 13
figure 13

Changes of a frictional entropy, b thermal entropy versus Reynolds number in different N for q″ = 10,000 W m−2 and S = 30 mm

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The second part of total entropy generating is thermal entropy, which is a function of temperature gradient. According to Eq. (15), increasing the Reynolds number reduces the wall temperature and temperature gradient; and improves the thermal performance. Figure 13b shows that the slope of the graphs decreases with increasing Reynolds number. In fact, the higher the Reynolds number, it has the less effect on entropy generation, indicating that greater Reynolds numbers outcomes less entropy in the same pitch compared to smaller Reynolds numbers.

Figure 14 shows the total entropy changes versus Reynolds number at N = 3 and different pitches. It is clear that as the Reynolds number increases, the total entropy also decreases. As mentioned, the total entropy generation consists of two terms: thermal entropy and frictional entropy. It is obvious that with increasing Reynolds number, the total entropy generation decreases, which is due to the total entropy formula mentioned in Eq. (14).

Fig. 14
figure 14

Changes in total entropy generation versus Reynolds number in different S for q″ = 3000 W m−2 and N = 3

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The trend of changing the total entropy goes back to the share of each of the heat and friction entropies in the generation of total entropy. Figure 15 is a circular chart, which shows the contribution of each entropy to the total entropy generation, and shows that the role of thermal entropy generation is greater. As explained so far, the effect of thermal entropy is greater. Therefore, the total entropy change is a function of the thermal entropy change, so as the Reynolds number increases, the total entropy decreases.

Fig. 15
figure 15

Circular chart to investigate the contribution of heat and friction entropy on the total entropy generation

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Investigation of the number of dimples (N) in the cross section

Figure 16 shows the pressure drop changes in the number of different dimples. As can be seen, with increasing number of dimples in the cross section, the pressure drop has also increased. Due to the friction of the fluid with the walls, a pressure drop occurs. In fact, increasing the number of dimples increases the friction and consequently increases the pressure drop.

Fig. 16
figure 16

Variation of pressure drop in pitches at different N at Re = 1000 and q″ = 3000 W m−2

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As shown in Fig. 17, the Nusselt number increases with the number of dimples on the cross section of the tube. In fact, the greater the number of these dimples, the more obstacles are placed against the flow and the longer the flow path of the fluid. The greater the number of dimples, the greater the fluid flow path, resulting in more opportunity for heat transfer and increasing heat transfer. Also, increasing the number of dimples reduces the cross-sectional area, which means a reduction in hydraulic diameter.

Fig. 17
figure 17

Nusselt number changes versus number of pitches for different N at Re = 5000 and q″ = 5000 W m−2

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From Eq. (11) (u = (Re μ)/Dρ), in a constant Reynolds, the reduction in the hydraulic diameter increases the velocity, which is also shown in Fig. 18. Increasing the velocity reduces the thickness of the thermal and hydraulic boundary layer and also increases the convection heat transfer coefficient which is directly related to the Nusselt number, thus increasing the Nusselt number and increasing the heat transfer.

Fig. 18
figure 18

Velocity contour in Re = 15,000, S = 14 mm and in different N on the tube; a N = 4, b N = 3, c N = 2

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Figure 19 shows the total entropy changes versus Reynolds numbers at q″ = 3000 W m−2 and S = 30 mm. In a constant Reynolds number, the total entropy decreases with increasing number of dimples, so that the highest total entropy is related to N = 2. In fact, thermal entropy determines the overall entropy trend, and as the thermal entropy trend was examined, with increasing number of dimples, the temperature gradient decreased and, consequently, reduced thermal entropy.

Fig. 19
figure 19

Total entropy generation changes versus Reynolds number at q″ = 3000 W m−2 and S = 30 mm and different N

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Investigation of the effect of applied heat flux

One way to increase entropy production is to increase heat. As the heat flux increases, the temperature of the walls actually increases and the temperature gradient increases. Figure 20 shows the entropy changes generation versus Reynolds number at different heat fluxes at S = 14 mm and N = 2. It can be seen that with increasing Reynolds number, the total entropy decreases and in a constant Reynolds number, the higher the heat flux causes the higher the entropy. In the previous sections, it was explained that thermal entropy has a greater effect on friction entropy than total entropy generation.

Fig. 20
figure 20

Variation in entropy: a total entropy, b frictional entropy, and c thermal entropy based on Reynolds number in different heat fluxes, S = 14 mm and N = 2

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Conclusions

This research focused on comprehensive numerical study of heat transfer in a dimple tube equipped with turbulators with regard to consider the first and second laws of thermodynamic. The important point in this research is the numerical investigation of the heat transfer rate and entropy generation by examining the effect of the heat flux, Reynolds number, the pitch (S) and number of dimples (N). The results of this study include the following results:

  • As the Reynolds number increases, the frictional entropy increases, the thermal entropy decreases, and since the share of thermal entropy in the total entropy generation is greater, the total entropy decreases.

  • Increasing the pitch (S) or decreasing the number of inserted cones (N) reduces the Nusselt number and heat transfer.

  • Increasing the pitch (S) reduces the pressure drop and reduces the friction coefficient.

  • As the number of dimples or the number of indentations on the tube increases, the Nusselt number increases and causes a decrease in the pressure drop and an increase in the friction coefficient.

  • As the number of dimples increases, the frictional entropy increases and the thermal entropy decreases, thereby increasing the overall entropy.

  • In low Reynolds numbers, the share of thermal entropy generation is much higher than the friction entropy generation. In fact, as the Reynolds number increases, the share of friction entropy generation in total entropy generation increases.

  • The maximum Nusselt number and friction coefficient are at Re = 20,000 and the S = 14 mm and the N = 4, and the heat flux has not significant effect on heat transfer and pressure drop.

  • The maximum thermal entropy and total entropy are at Re = 5000 and S = 14 mm and the N = 2, and the maximum frictional entropy is at Re = 5000, and S = 14 mm and N = 4 at q″ = 10,000 W m−2.

The results of this modeling can be used to obtain the optimal condition to reach the maximum heat transfer rate. For future work, we aim to optimize in order to find the optimal value of each variable.