Optimization of Dimensions of Smooth and Twisted-Tape-Inserted Tubes for Heat Transfer with NaCl/KCl/MgCl₂ Molten Salts by Principle of Entropy Generation Minimization

In the effort toward providing the world with clean, safe, and reliable energy, renewable energy must have a larger presence in the world’s energy portfolio. Energy Information Administration (EIA) predicted in an outlook that solar energy will generate 20% of the total electricity in the U.S. by 2050 [1]. However, to further expand the use of solar energy and minimize the use of fossil fuels, efficiency and energy storage capacity in solar energy technology must increase in addition to lowering the cost. The analysis in this paper is an effort on improving the solar-to-electricity efficiency of concentrated solar power systems (CSP).

There are two major types of systems of concentrating solar thermal energy for electrical power: one is based on a solar trough, and the other is based on a solar tower. Some analyses have been performed to optimize the optical system of CSP systems such as optimizing the parabolic trough, determining the optimal heliostat field and solar tower, and determining the heliostat sunlight blockage for different layouts [2–4]. Beyond the optical system, the optimization of the flow and heat transfer in the heat collection system is also important. As seen in the study by Mills and Coimbra [5] changing the flow cross section results in a change in the heat transfer Nusselt number. Furthermore, Jankowski [6] optimized cross-sectional flow as a method of minimizing entropy generation, which can minimize the loss of useful work in a power generation system [7]. Hooman et al. [8] examined the heat transfer and entropy generation of a rectangular duct with different boundary conditions for different height-to-width ratios.

One can also increase the efficiency of CSP systems by choosing the optimal heat transfer fluid (HTF). The best HTF can be chosen by performing an analysis comparing the minimum entropy generation of several HTFs in an existing CSP. Xu et al. [9] performed a comparison of the entropy generation of solar salt and several chloride salts. This analysis is not limited to molten salts but all types of HTF as studied by Zhang and Li [10]. Most recently, Flesch et al. [11] performed an analysis in which Solar Salt, Sodium Nitrate, and Lead-Bismuth were compared to one another to determine which HTF has the minimum entropy generation, thus allowing CSP plant designers to choose the most optimal HTF.

In the present study, the minimum entropy generation will be related to the optimal Reynolds number for both a smooth tube and one with a twisted-tape insert. Twisted-tape-inserts have been shown to increase heat transfer [12]. Recently, with the most advanced additive manufacturing technology, internal fins or twisted tape can be easily manufactured with tubes as one integrated unit through 3D printing. There have been various experimental tests to observe the effect of twisted-tape insert on heat transfer, including using different configurations and augmentations such as conical rings, delta winglets, V-cuts, multiple tape-inserts, tapered tapes, and rectangular cuts [13–18]. Furthermore, several authors have calculated entropy generation for smooth and enhanced tubes, including helical fins and twisted tapes, using computational fluid dynamics methods [19–23]. In the present paper, a new method of calculating the minimum entropy generation rate for both a smooth and rough flow for HTF is introduced. The minimum entropy generation rate for smooth tube and twisted-tape-inserted flow will be determined using the Gnielinski and Petukhov correlations as well as the correlations developed by Manglik and Bergles, respectively. The Gnielinski and Petukhov correlations for the Nusselt number and Moody friction factor are more accurate and valid in a wide range of Reynolds numbers than Dittus–Boelter and Blasius correlations, with 3000 ≤ Re_D ≤ 5 × 10⁶ and 0.5 < Pr < 2000 [24]. The Manglik and Bergles correlation [25] is valid for Re ≥ 10⁴ and showed a very good agreement with experimental data within $±10%$⁠.

The present analysis will first express the entropy generation rate as a function of the Reynolds number. The optimum Reynolds number for minimum entropy generation rate is then found by drawing the curves of entropy generation rate versus Reynolds number. Other optimum parameters that will be determined as a consequence of the obtained optimum Reynolds number, include Nusselt number, convective heat transfer coefficient, friction factor, and tube diameter. The analysis will be repeated for different heat fluxes to simulate the operation of a solar power plant. The HTFs in this study are NaCl–KCl–MgCl₂ and KCl–MgCl₂ because of their stability at high temperatures and reasonably good thermal and transport properties, although the corrosion issue to high-temperature alloys needs reliable solutions in the future [26]. Therefore, the analysis is aimed at determining more accurate minimum entropy generation rates for a smooth tube and a tube with a twisted-tape insert at their optimum Reynolds numbers. The study will also observe the change of these optimum values for varied thermal powers and heat fluxes for the eutectic molten salts NaCl–KCl–MgCl₂ and KCl–MgCl₂. Furthermore, the minimum entropy generation rate of the two eutectic molten salts will be compared to determine the more suitable HTF.

The optimization analysis has the objective of finding the optimal Reynolds number that makes the heat transfer system entropy generation rate the minimum. The given conditions or restrictions are the mass flowrate of the fluid, the inlet and outlet fluid temperatures, and thus the total heat transfer rate, heat transfer surface area of the tube or the heat flux based on the inner wall of the bare tube. Therefore, at different Reynolds numbers $(Re=4m˙/πDμ)$⁠, the corresponding inner diameter and length of the tube are determined based on the relevant heat transfer correlations. The Reynolds number that makes the minimum entropy production of the system is optimal to be considered. In extreme situations where the optimal Reynolds number requires a small tube diameter and very high flow velocity, extra care, and discussion will be applied from the perspective of erosion and corrosion of metals due to high fluid speed.

The entropy generation rate is analyzed for a control volume, which is simply a section of the heat transfer tube with heat flux added from the wall to the fluid flowing inside, as shown in Fig. 1 for a smooth tube, and Fig. 2 for a twisted-tape-inserted tube. The cross-sectional view of the twisted-tape-inserted tube is shown in Fig. 3.

The following assumptions are made to set up the basic conditions for the analysis of the entropy generation in the control volume:

1. The flow in the tube is steady-state and fully developed turbulent flow so that the internal heat transfer coefficients in turbulent flow are roughly close between uniformly and non-uniformly heated tubes.

2. A uniform heat flux along the entire length and circumference of the tube will be considered as seen in Figs. 1 and 2. In the actual operation, the tubes in both the solar tower and parabolic trough systems typically have heating on half side of the tube. For turbulent flow heat transfer, Okafor et al. [27] showed that the average internal heat transfer coefficient in tubes was roughly the same for a uniformly heated tube and non-uniformly heated tube.

3. The tubes in a solar tower and parabolic trough CSP system have bends along the flow path as the flow heads toward the heat exchanger. The solar tower has a bend at the top of the tower, and the parabolic trough has turns to go from one line of troughs to the next. In this study, only the straight tube is considered for analysis. The pressure loss in the bends and turns is not considered or interested in the analysis.

4. There are several examples of the conditions of temperatures and heat flux for heat receiving tubes of concentrated solar power tower receiver tubes [11,28,29]. The boundary conditions, including the inlet and outlet temperature, heat flux, and thermal power, set by Flesch et al. [11] were determined referencing the Solar Two Project by Sandia National Laboratory [28]. In this study, the inlet temperature T_in and outlet temperature T_out were fixed per the request for fluid temperature in CSP system. Based on the average of T_in and T_out, the thermophysical properties of both salts were determined through the equations provided by Xu and Wang [26,30]. Based on the melting temperature of both salts, it is reasonable for the operating temperature to be between 550 °C and $700∘C$⁠. Thus, we let $Tin=550∘C$ and $Tout=700∘C$⁠. Furthermore, this analysis will be performed for an array of heat fluxes and thermal powers to see the difference in the entropy generation rate. These values will be determined by first dividing the thermal power by the heat flux (set by Flesch et al. [11]) to determine the surface area of the tube which is determined to be $Q˙/q˙″=660kW/(500kW/m2)=1.32m2$ [11]. In the analysis, the surface area will be taken as the inner surface area of the tube, which is defined as SA = πDL, where D is the maximum inner tube diameter and L is the tube length. Then, for simplicity, the array of $Q˙$ will range from 100 kW to 1000 kW and thus $q˙″$ can be determined by $q˙″=Q˙/(1.32m2)$⁠. The thermal power and heat flux values to be studied are listed in Table 1. These heat fluxes cover the typical heat flux of solar receivers in the current industry.

5. As seen in Fig. 2, the pitch of the twisted tape has a 180 deg twist. The dimensionless pitch, y, which will be seen later in the friction and heat transfer correlations, is a constant defined as y = H/D, where H is the pitch of the twisted tape. The choice of y consequently determines the number of twists within the control volume. Furthermore, the tape thickness to maximum inner diameter ratio is defined as C = δ/D. Nair and Al-Fahed [31,32] both tested the heat transfer effects of twisted tape using C = 0.03 and 0.036, respectively. Thus, for this analysis, it is reasonable to choose C to be 0.03 with five different dimensionless pitches (y = 0.25, 0.5, 1, 3, 4).

Since all the geometric and thermophysical parameters have been defined, the friction and heat transfer correlations for smooth and twisted-tape-inserted tubes can now be selected for the computation of the entropy generation rate in the tube under the heat transfer rates given in Table 1.

Essentially, the optimization of the heat transfer system is to find the best Re, which renders the minimum entropy generation rate. From the optimal Re, the optimal diameter of the tube, the length of the tube, and thus the flow velocity can be found for the defined heat transfer rate and heat transfer flux as given in Table 1. The following steps are exercised to find optimal Re:

1. Define T_in, T_out, $Q˙$⁠, $q˙″$⁠, and then determine $m˙$ as known parameters or constants.

2. Calculate T_m and T_bulk, and thus find all properties, μ, c_p, k, Pr, and ρ at the temperature of T_bulk.

3. Re_opt is found from the derivative of Eq. (5) by setting the derivative equal to zero and then solving for Re as the Re_opt.

4. Determine minimum $S˙gen,min$⁠, and the optimal tube diameter from $m˙$ and Re_opt by plugging in Re_opt back into the corresponding equations.

1. Define T_in, T_out, $Q˙,q˙″$⁠, y, and C and then determine $m˙$ as known parameters or constants.

2. Calculate T_m and T_bulk, provide an initial guess for T_w (1000 °C) and thus find all properties, μ, c_p, k, Pr, and ρ at the temperature of T_bulk as well as μ_w at the temperature of T_w.

3. An iterative method is used by creating an iteration loop until the wall temperature converges, then Re_opt is calculated. Here, Re_opt is found from the derivative of Eq. (5) and setting it equal to zero and then solving for Re as the Re_opt.

4. T_w and μ_w are redefined at the bottom of the iteration loop by $Tw=q˙″/hopt+Tbulk$ and by the corresponding thermophysical equation, respectively, until T_w converges. The calculated wall temperature is the temperature of the inner wall of the tape-inserted tube.

5. Once T_w converges, $S˙gen,min$ and the optimal hydraulic diameter as well as maximum inner diameter of the tube are determined from $m˙$ and Re_opt by plugging in Re_opt back into the corresponding equations.

The thermophysical properties of molten salts NaCl–KCl–MgCl₂ and KCl–MgCl₂ are given in Tables 2 and 3 from the work reported by Wang and Xu, respectively [26,30]. These properties are calculated using the bulk temperature of the fluid in the control volume, T_bulk = T_in + T_out/2, except μ_w, which is calculated using the wall temperature T_w. The equations are valid for the operating temperature between 400 °C and 800 °C.

For the given temperatures of $Tin=550∘C$ and $Tout=700∘C$⁠, as well as $Q˙$ and $q˙″$ in Table 1, the heat transfer system entropy generation at different Reynolds numbers is obtained for the ternary molten salt NaCl–KCl–MgCl₂ (wt% 45.98–38.91–15.11) and binary molten salt KCl–MgCl₂ (wt%, 62.5–37.5) with both the smooth tube and twisted-tape-inserted tubes. Results for smooth tube entropy production using our model and code have been compared with the results that we used in the method reported in Ref. [11], where different heat transfer and friction factor correlations were used. For the molten salt NaCl–KCl–MgCl2, our model found the optimum Reynolds number being Re_opt = 112,004 at 600 kW, and our results using Flesch’s analytical solution in Ref. [11] show Re_opt = 114,574 at 600 kW. This difference is because we used more accurate heat transfer and friction factor correlations. Nonetheless, the results are not too far away from each other, which cross-checked the accuracy of our model.

From the obtained curves of the system entropy generation rate against Reynolds number, the minimum entropy generation rate can be found at an optimal Reynolds number. Based on the optimized results of the Reynolds number and the given mass flowrate for each case, the optimal tube diameter, pitch, tape thickness, friction factor, and Nusselt number as well as the minimum entropy generation rate are decided for the molten salt fluid.

The variation of the entropy generation rate $S˙gen$ is plotted as function versus Reynolds number for smooth and twisted-tape-inserted tubes. The plots in Fig. 4 are for the ternary molten salt NaCl–KCl–MgCl₂ (wt%, 45.98–38.91–15.11), and plots in Fig. 5 are for the binary molten salt KCl–MgCl₂ (wt%, 62.5–37.5).

It is important to examine and compare the results in Figs. 4 and 5 from the perspectives of the entropy generation rate against the Reynolds number, the difference of minimum entropy generation rate between smooth tube and twisted-tape-inserted tubes, the entropy generation rate from various types of cases of twisted-tape-inserted tubes, and the difference from using two different types of molten chloride salts for the studied tubes.

For each case of a fixed heating rate or heat flux, there is an optimal Reynolds number where the entropy generation rate is the minimum. Based on the critical Reynolds number, we can decide a tube diameter and length as well as the velocity of the fluid in the tube. At below the critical Reynolds number, the increase of heat transfer coefficient from the increase of Reynolds number contributes to the decrease of entropy generation rate. However, beyond the critical Reynolds number, the increase in Reynolds number makes the entropy generation rate increase. This is because the pressure-loss-introduced entropy generation rate increases dramatically and overcomes the decrease of entropy generation caused by the increase in heat transfer coefficient. The entropy generation rate caused by heat transfer and pressure loss is both plotted in Fig. 6. One can see that when the curve of decreased entropy generation (due to increased heat transfer coefficient) intercepts with the curve of increased entropy generation (due to increased pressure loss), the total entropy generation reaches the minimum. A similar phenomenon has been reported in the literature [19].

Observation of the curves in Figs. 4 and 5 at different heating rates $Q˙$ also found that the optimal Reynolds number becomes larger when the heating rate $Q˙$⁠, or heat flux $q˙″$ is higher. This implies that for higher heat flux, the higher heat transfer coefficient at a larger Reynolds number is important to contribute more to the decrease of the entropy generation than to the increase of entropy production from friction loss, and therefore, the optimal Reynolds number is higher. It is also important to observe that at the low heating rate cases (100–200 kW), the heating rate change from one case to another case is more significant, for example, from 100 kW to 200 kW is a 100% increase; however, from 900 kW to 1000 kW, the increase of heating rate is 1.1 times. This is the reason that from 100 kW to 200 kW there is a significant shift of the optimal Reynolds number, while at higher heating rates, the shift of optimal Reynolds number is less significant.

Figures 4 and 5 also show that for larger heating rate $Q˙$⁠, or heat flux $q˙″$⁠, the minimum entropy generation rate is also higher, which is easy to understand that the entropy generation rate in the heat transfer process is generally proportional to the amount of heat to be transferred.

Shown in Fig. 7 is the minimum entropy generation for the various dimensions (1/y) of tape insertion for the heat transfer enhancement. First, the minimum entropy generation is higher for larger heat transfer rates, as the entropy generation is proportional to the magnitude of heat transfer rate. Second, it is seen that the minimum entropy generation rate shows a decrease with the decrease of y, which is the pitch of the twist of tape versus the inner diameter of the tube. This is understandable that with a smaller y there is more twist of the tape and therefore the insertion can introduce more turbulence and enhanced heat transfer. The larger the pitch of the twist, the tape insertion is less effective and approaches that of a smooth tube.

The properties of fluid influence the heat transfer coefficients and also the friction loss; therefore, different heat transfer fluids will have different levels of entropy production in the process of transferring the same amount of heat. Detailed theoretical analysis of fluid selection for less entropy production has also been reported in the literature [10]. Comparing the properties of NaCl–KCl–MgCl2 with KCl–MgCl2, the former is favorable in general regarding better heat transfer and less friction loss.

It is important to note that the Reynolds number in operation should definitely not exceed the optimal Reynolds number as that is related to excessive pumping power for heat transfer improvement, which is not worthy. Other factors such as fluid speed may also impact the decision of choosing a Reynolds number. The representative data of the optimal Reynolds number and the corresponding dimensions of tube diameter and fluid velocity for the ternary chloride molten salt NaCl–KCl–MgCl₂ are listed in Table 4.

It is seen From Table 4 that the optimal tube diameters for smooth tubes with various heating rates (100–1000 kW) are at the level of 11.6–19.0 mm, and the fluid velocity is at the level from 4 m/s to 14.8 m/s; while the minimum entropy generation rate is at the level from 1.84 W/K to 63.33 W/K. For the tube with twisted-tape-inserts with y = 0.25, the diameter of the tube is at the level from 22.7 mm to 36.5 mm, the flow velocity is at the level from 1.0 m/s to 4.0 m/s, and the minimum entropy generation rate is at the level from 1.44 W/K to 53.07 W/K. Similarly, to examine the results for the tube with twisted-tape-inserts with y = 0.5, the flow velocity ranges from 1.5 m/s to 5.7 m/s, and the entropy generation rate ranges 1.7–62.63 W/K. When the twisted-tape-inserts have y from 1 to 4, the minimum entropy generation rate can be slightly larger than that of the smooth tube.

Figure 8 also shows that for the same amount of heat transfer rate, the optimal Reynolds number of the smooth tube is much higher than that of the twisted-tape-inserted tubes, while the minimum entropy generation rate of the smooth tube at the optimal Reynolds number is about the same as that of the tube with twisted-tape insertion. On the other hand, at the respective optimal Reynolds number, the fluid velocity for the smooth tube is much higher than the counterpart of the twisted-tape-inserted tube and may not be acceptable considering the erosion and corrosion of the molten salt to the material of the tubes. This also means that if the flow velocity is selected to be below 4 m/s for the smooth tube, then the entropy generation from the smooth tube can be much higher than that of the twisted-tape-inserted tubes, since the latter has an enhanced heat transfer effect. Therefore, the benefit of using twisted-tape-inserted tube is that the flow velocity is much lower while the entropy generation rate is also low compared to that from a smooth tube. It is concluded that through the heat transfer enhancement approach, one can obtain low entropy generation at a relatively low Reynolds number like the case of y = 0.25. It is also important to know that if heat transfer enhancement effect is not significant, the heat transfer tube may behave approximately like a smooth tube. This is observed that when the dimensionless pitch y is much bigger, the heat transfer effect is much less, and at the same time, the twisted-tape insertion causes higher pressure loss, which does not help the reduction of the entropy generation significantly.

In summary, the optimal tube diameters for the smooth tube and the twisted-tape-inserted tube are given in Fig. 9 for selection at different heat transfer rates and heating flux listed in Table 1. Obviously, at higher heating flux, larger tube diameter is better to have minimized entropy generation. The twisted-tape-inserted tube with a small twist pitch y can have a small entropy generation rate and low speed of fluid flow, which is a very good option. In the meantime, it needs a larger tube diameter. For a smooth tube, in order to have the minimum entropy generation rate at the same level as that of twisted-tape-inserted tube, a small tube diameter and a very large Reynolds number, and fluid speed are needed, which may not be the suitable option.

The studies in this work are aimed at searching for the optimal Reynolds number and tube diameter for the operation of molten chloride salt heat transfer in solar receivers at certain heat flux. The optimization is based on the principle of minimization of entropy generation. Optimal Reynolds number and the corresponding optimal diameter of the heat transfer tube at different heat transfer rates and heat flux can be found where the entropy generation rate of the heat transfer system is the minimum. The analysis has been carried out for smooth tube and twisted-tape-inserted tube.

For smooth tubes, relatively more accurate heat transfer correlations, Gnielinski correlation for Nusselt number versus Reynolds number, and Moody friction factor as a function of Reynolds number given by Petukhov, were used to have improved accuracy over existing work from the literature [11].

For twisted-tape-inserted tube, correlations for Nusselt number and friction factor in the literature by Manglik and Bergles [25] were used. It has been found that the optimal tube diameters for smooth tubes with various heating rates and heat flux (100–1000 kW and correspondingly 75.75–757.57 kW/m²) is at the level of 11.6–19 mm, and the fluid velocity is at the level from 4 m/s to 14.8 m/s, while the minimum entropy generation rate is at the level from 1.84 W/K to 63.33 W/K. For the tube with twisted-tape-inserts with y = 0.25, the diameter of the tube is at the level from 22.7 mm to 36.58 mm, the flow velocity is at the level from 1.0 m/s to 4.0 m/s, and the minimum entropy generation rate is at the level from 1.44 W/K to 53.07 W/K. Considering that the molten salt flow velocity shall not be over 4 m/s, the smooth tube cannot actually operate at the optimal Reynolds number. Therefore, the entropy production rate for the smooth tube will be higher than that of the twisted-tape-inserted tube if the flow velocity is limited to below 4.0 m/s. The latter has enhanced heat transfer over smooth tubes and the flow velocity is also acceptable from the perspective of erosion and corrosion of molten salt to tubes at high temperatures. Nevertheless, a smaller entropy generation for a twisted-tape-inserted tube compared to that of a smooth tube is not a general conclusion but depends on the design of the internal fin, the size of the twist pitch as well as the condition of the internal flow.

Comparing the two types of molten chloride salts, NaCl–KCl–MgCl₂ and KCl–MgCl₂, the ternary salt shows a slightly better performance with slightly lower entropy generation. The optimal tube diameter is almost identical for both salts either in a smooth tube or in a twisted-tape-inserted tube.

The general form of entropy generation rate equation and solution approach to determine the minimum entropy generation rate for heat transfer enhanced tube is very useful for the optimization of solar received using molten salts.

The authors are grateful for the support from the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office (SETO) Award No. DE-EE0009380.

There are no conflicts of interest.

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

f =

friction factor

h =

convective heat transfer Coefficient, W/m^2K

k =

thermal conductivity of the fluid, W/m K

u =

velocity of fluid, m/s

y =

H/D = dimensionless pitch

C = δ/D =

tape thickness to inner diameter ratio

D =

$4m˙/Reπμ=$ maximum inner diameter of tube determined from the Reynolds number, m

H =

pitch of tape (180 deg twist)

L =

length of tube, m

T =

temperature of working fluid, K

$m˙$ =

mass flow rate, kg/s

$q˙$ =

heat flux, W/m²

$Q˙$ =

heat rate, W

c_p =

specific heat capacity at constant pressure of the fluid, J/kg K

A_c =

cross-sectional area, m²

D_h =

hydraulic diameter, m

T_bulk =

bulk temperature of fluid, $∘C$

T_in =

inlet temperature of fluid, $∘C$ (converted to K for Eqs. (4) and (5))

T_m = (T_out − T_in)/ln(T_out/T_in) =

log mean temperature, K

T_out =

outlet temperature of fluid, $∘C$ (converted to K for Eqs. (4) and (5))

T_w =

wall temperature, $∘C$

$S˙gen$ =

entropy generation rate, W/K

Nu =

Nusselt number, hD/k

Pr =

Prandtl number, μc_p/k

Re =

Reynolds number, ρuD/μ

δ =

thickness of twisted tape, m

μ =

dynamic viscosity of fluid, Pa · s

ρ =

density of fluid, kg/m³