Shodhan Uday Mayekar*
| Milind Shashikant Yadav
© 2025 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
This work reports an experimental study on the thermo-hydraulic behavior of a tubular heat exchanger fitted with variable-diameter helical coils. Three coil arrangements, namely convergent, convergent–divergent and divergent, were examined under constant heat flux to determine their influence on heat transfer and flow resistance. The investigation considered the effects of conical length ratio (CLR = 2, 3, 4), diameter ratio (DR = 0.12, 0.16, 0.20), and Reynolds number (Re = 4000 – 10000). Performance was evaluated in terms of the Nusselt number (Nu), friction factor (f), and thermal performance factor (TPF). All coil configurations enhanced heat transfer compared to a plain tube, with average Nu improvements of 37.13%–74.17% for convergent coils, 47.18%–88.18% for convergent-divergent coils, and 58.11%–100.79% for divergent coils. The corresponding increase in f ranged from 1.98–2.97, 2.55–3.80, and 3.22–4.55 times, respectively. Among the three, divergent coils delivered the greatest improvement. A larger DR and smaller CLR promoted Nu enhancement but resulted in greater flow resistance. As TPF values exceeded unity in all cases, variable-diameter coils demonstrated significant potential for heat transfer augmentation. Using Response Surface Methodology (RSM), the optimal parameters were identified as CLR = 2, DR = 0.2, and Re = 10000. The novelty of this work lies in combining variable-diameter coil geometries with RSM-based optimization to generate new insights and practical guidelines for performance improvement of the heat exchanger.
heat transfer enhancement, variable diameter helical coils, Nusselt number, friction factor, thermal performance factor, response surface methodology
In recent years, the rising population, expanding industrial activities, and accelerating urbanization have collectively led to higher energy demand [1]. Clean energy utilization and efficiency improvements are becoming a major focus. At the same time, enhancing the performance of current heat transfer systems remains an active area of research. Heat exchangers are ubiquitous as they find use in chemical and food industries, petrochemical processing, pharmaceutical drug development, refrigeration and waste heat recovery devices [2, 3]. The energy-efficient heat exchangers help to scale down surface-area requirement, minimize material cost, and reduce weight. Further improving heat exchanger performance also addresses the concern of global energy sustainability.
Making the heat exchanger efficient by increasing the heat transfer coefficient is termed heat transfer augmentation. Different techniques of heat transfer enhancement typically increase the heat duty of a heat exchanger and allow for closer approach temperatures. Various heat augmentation techniques are grouped into two main types-passive and active. Passive techniques do not require any power for their functioning. They depend on modifying the geometry of the tube carrying the fluid or changing the flow physics. These techniques include the use of inserts [4-9], rough surfaces [10-13], extended surfaces [14-16], etc. Conversely, active methods require external power to operate. The power is imparted to either a heated surface or to fluids. These methods include the use of electric or magnetic fields, fluid or surface vibration, mechanical aids, etc. [17, 18]. By and large, active methods are more intricate and expensive. Hence, they are typically not utilized extensively. However, they come up with an opportunity to regulate the heat transfer enhancement.
Inserts, often called turbulators, offer a cost-effective means of enhancing heat transfer compared to redesigning or replacing the entire heat exchanger. They can be retrofitted into existing systems with minimal modifications. Helical wire coils are recognized as an effective passive method for intensifying the heat transfer and are often applied to improve the thermal performance of heat exchangers [19]. Significant advancements have been made in the past two decades in studying the contribution of wire coil inserts towards heat transfer enhancement of heat exchangers. Garcia et al. [20] experimentally demonstrated the thermo-hydraulic behavior of a round pipe containing helical wire coil inserts in three different regimes of the flow, namely laminar, transition, and turbulent. The effect of pitch and diameter of coil on heat transfer characteristics was described in detail. Effects of coil-wire inserts with different coil pitches on the heat transfer and pressure drop characteristics in a horizontal double pipe using air as the working fluid were investigated by Naphon [21]. The findings revealed that an increased heat transfer rate was offered by a shorter pitch length. A comparative study of the heat transfer intensification between the tube heat exchanger with wire coil having square and circular shapes in the turbulent air flow was carried out by Promvonge [22]. It was concluded that wire coils with a square cross-section are more capable of inducing turbulence in the flow, resulting in a greater enhancement of heat transfer compared to those with a circular cross-section. The friction factor, heat transfer rate and thermal enhancement efficiency were increased by the shorter pitch length.
Gunes et al. [23] figured out the effect of a wire coil having an equilateral triangular cross-section placed at a distance from the tube wall on the convective heat transfer rate and friction loss due to pressure drop in the turbulent regime of air flow. The study revealed that reducing the pitch length of the equilateral triangle resulted in higher pressure drop and enhanced heat transfer. Also, it was concluded that increasing the side length resulted in a proportional rise in both. Further, Gunes et al. [24] experimentally demonstrated the impact of distances from the tube at which equilateral triangular coiled wires were placed. The findings showed that a lower separation distance and coil pitch length yielded a superior overall enhancement ratio than the others. The influence of variation of the diameter of wires and lengths of pitch placed in a horizontal pipe was demonstrated by Akhavan-Behabadi et al. [25]. Improved thermal performance was noted in tubes using wire coils of smaller diameter. Additionally, it was found that low values of pitch lengths improve the heat transfer performance.
Chandrasekar et al. [26] determined the heat transfer rate and friction loss of a tubular heat exchanger using a combination of Al2O3-water nanofluid and wire coil inserts. The concentration of nanoparticles in the fluid was limited to 0.1% by volume. As the thermal conductivity of the working fluid was improved by the addition of Al2O3 nanoparticles, the combination of nanofluid with wire coils increased the Nusselt number by 15.91% and 21.53% at pitch ratios of 2 and 3, respectively. Additionally, the pressure drop associated with nanofluid was found to be almost equivalent to that of pure water. Eiamsa-ard et al. [27] reported the thermal characteristics heat exchanger having a square cross section with a tandem wire coil insert. Different insert lengths and free spacing configurations of wire coils were examined, and their performance was compared against that of a continuous, full-length wire coil. It was found that the heat transfer improvement in a square duct using a continuous coil was greater than that achieved with tandem wire insertions. Saeedinia et al. [28] recommended a nanofluid containing CuO with a higher particle concentration along with conventional wire coil inserts to enhance the heat transfer rate in the laminar region. The heat transfer and fluid flow characteristics of a pipe inserted with wire coil inserts in the laminar and transitional flow conditions were reported by Martinez et al. [29]. Compared to a plain tube, the heat transfer rate showed an enhancement of up to 4.5 times, while the friction factor increased by approximately 3.5 times.
The experiments were conducted by Chang et al. [30] to identify the influence of square wire coils with grooved or ribbed structures having different pitch configurations on the heat transfer augmentation. According to their findings, the modified square wire coils with grooves or ribs outperformed the smooth square wire coil in terms of performance factor across all pitch ratio variations. Syam Sundar et al. [31] analyzed the performance of a double pipe heat exchanger with U–bend by calculating values of the effectiveness and number of transfer units (NTU) in two scenarios- in the presence and absence of wire coil inserts with Fe3O4-water nanofluid. The results indicated that the NTU and effectiveness are directly proportional to the concentration of Fe3O4 particles. Both performance parameters were significantly reduced by the wire coils having a longer pitch. Du et al. [32] evaluated the thermo-hydraulic flow behavior in a corrugated tube containing modified wire coil inserts arranged at regular intervals. Results showed that the combination performed better than the standalone corrugated tube. Also, it was described that the increased friction caused by the introduction of coils resulted in a lower overall thermal performance.
Abdul Hamid et al. [33] investigated the influence of both pitch ratio and nanoparticle concentration on the heat transfer and flow characteristics of wire coil-inserted tubes using TiO₂–SiO₂ nanofluid. A pitch ratio of 1.5 for the wire coil and a 2.5% nanofluid volume concentration were found to be optimized performance affecting parameters. The effect of different wire coil orientations and spacing ratios on heat transfer in a transverse corrugated tube was experimentally studied by Hong et al. [34]. Findings of the study showed that a decrease in the spacing ratio significantly enhanced the convective heat transfer performance, along with a corresponding rise in the friction factor due to increased flow resistance. The effect of the circular, square and triangular cross-section of wire coil was evaluated by Yu et al. [35]. Due to the generation of strong turbulence compared to others, the square cross-section emerged as an efficient one. The average enhancement in heat transfer was found to be 26.25% for wire coils with a circular cross-section, 57.46% for those with a square cross-section, and 45.92% for coils with an equilateral triangular cross-section. Chompookham et al. [36] reported results obtained by introducing a serrated wire coil. A rise in Nusselt number was observed with a decrease in pitch length and an increase in coil diameter. But the friction factor showed a declining trend with larger coil diameter and extended pitch length. Within the tested range, the growth of 1.75–2.46 times and 3.31–8.16 times in Nu and f was observed.
The effect of combinations of helical wire coil-twisted tape [37, 38], rectangular wire coil-twisted tape [39] and wire coil-perforated conical ring [40] on the thermohydraulic performance of the heat exchanger was also reported in the literature. Although the combination resulted in enhanced thermal performance compared to the individual methods, it was accompanied by a comparatively higher pressure drop.
In-depth literature analysis indicates that incorporating helical wire coils improves convective heat transfer in heat exchangers, with considerable attention given to factors such as flow velocity, working fluid properties, pitch ratio, and coil geometry. The present study investigates the convective heat transfer and flow resistance behavior in a circular tube equipped with three variable diameter coils, namely convergent (C), divergent (D) and alternate convergent-divergent (CD) is investigated. The geometric parameters associated with these variable diameter coils are conical length ratio (CLR) and diameter ratio (DR) and CLR is defined as the ratio of conical length (L) to maximum diameter (Dm) of the coil, whereas DR is expressed as the ratio of wire diameter (dw) to maximum coil diameter (Dm). The geometry of a conical element of variable diameter coil is represented in Figure 1. To maintain dimensional accuracy, 3D printing technology was used to manufacture the coils.
Figure 1. Geometry of a conical element of variable diameter helical coil
Previous investigations of conical passive enhancement techniques have primarily focused on the influence of pitch ratio and spacing length on heat transfer and pressure drop characteristics. However, the effect of the conical length ratio, which directly governs the slant height and apex angle of the conical element, on the thermo-hydraulic performance of tubular heat exchangers has not been reported in the literature, to the best of the authors’ knowledge. So, in this study, efforts are taken to investigate the effect of CLR along with DR and Reynolds number (Re) on Nusselt number (Nu), friction factor (f) and thermal performance factor (TPF). Also, multi-criterion optimization is carried out for each coil to determine the appropriate level of performance affecting parameters. The geometry of variable diameter helical coils is shown in Figure 2. The overall length of every coil used is 1000 mm, and the maximum diameter is 25 mm. Values of other geometric parameters associated with all three configurations of variable diameter helical coils are listed in Table 1.
Figure 2. Actual geometries of variable diameter helical coils used in experimentation
Table 1. Values of geometric parameters of variable diameter helical coils
|
|
Case-I |
Case-II |
Case-III |
|
Conical length (L) in mm |
50 |
75 |
100 |
|
Conical length ratio (CLR) |
2 |
3 |
4 |
|
Wire diameter (d) in mm |
3 |
4 |
5 |
|
Diameter ratio (DR) |
0.12 |
0.16 |
0.2 |
A schematic layout of the experimentation facility is shown in Figure 3. A copper tube measuring 1000 mm in length, with an internal diameter of 26 mm and a wall thickness of 1 mm, was utilized as the test section. To ensure uniform heat input, an electric coil was spirally wound around the length of the test section. The electrical input was adjusted using a variac transformer. To minimize convective losses to the environment, the outer surface was insulated. Seven thermocouples were tapped on the outer wall of the tube to measure the temperature variation along the length of the tube. An orifice meter, designed as per ASME standards [41], was used to measure the volumetric flow rate of the water. A flow control valve was used to adjust the deflection of mercury across the orifice meter, which ultimately helped to set the desired value of Re. During experimentation, the Re was varied from 4000 to 10000. To accurately capture minor pressure fluctuations across the test length, a U-tube manometer filled with carbon tetrachloride was employed. For each test run, water from the reservoir was pumped through an orifice meter and subsequently into the test section. After attaining the steady state, surface temperatures of the tube, inlet and outlet temperatures of water and pressure drop across the test section were recorded. Properties of water required for calculations were considered at the mean bulk temperature.
Figure 3. Schematic diagram of the experimental setup
Two key performance indicators, Nusselt number (Nu) and friction factor (f) are used to evaluate the efficacy of any heat transfer enhancement method. While the Nusselt number evaluates the rate of convective heat transfer, the friction factor indicates the associated pressure drop.
The rate of convective heat transfer is expressed by Eq. (1):
$\text{Heat }\!\!~\!\!\text{ convected}={{\text{Q}}_{\text{conv}.}}=\text{h}{{\text{A}}_{\text{s}}}\left( {{\text{T}}_{\text{s}}}-{{\text{T}}_{\text{b}}} \right)$ (1)
where,
h = Heat transfer coefficient in w/m2K
As = Surface area of the tube = πdl
d = Diameter of the tube in m
l = Length of the tube in m
Ts = Mean surface temperature and it is given by Eq. (2):
${{\text{T}}_{\text{s}}}=\frac{{{\text{T}}_{1}}+{{\text{T}}_{2}}+{{\text{T}}_{3}}+{{\text{T}}_{4}}+{{\text{T}}_{5}}+{{\text{T}}_{6}}+{{\text{T}}_{7}}}{7}$ (2)
Tb = Mean bulk temperature, which is given by Eq. (3):
${{\text{T}}_{\text{b}}}=\frac{{{\text{T}}_{\text{in}}}+{{\text{T}}_{\text{out}}}}{2}$ (3)
The heat transferred by the heating coil to the water is given by Eq. (4):
$\text{Heat }\!\!~\!\!\text{ absorbed }\!\!~\!\!\text{ by }\!\!~\!\!\text{ water}={{\text{Q}}_{\text{a}}}=\text{m}{{\text{C}}_{\text{p}}}\left( {{\text{T}}_{\text{out}}}-{{\text{T}}_{\text{in}}} \right)$ (4)
where,
m = Mass flow rate of water in Kg/sec
Cp = Specific heat of water at constant pressure
Tout = Temperature of water at outlet
Tin = Temperature of water at inlet
At steady state,
${{Q}_{a}}$ = ${{Q}_{conv.}}$
So, the heat transfer coefficient is given by Eq. (5):
$\text{h}=\frac{\text{m}{{\text{C}}_{\text{p}}}\left( {{\text{T}}_{\text{out}}}-{{\text{T}}_{\text{in}}} \right)}{{{\text{A}}_{\text{s}}}\left( {{\text{T}}_{\text{s}}}-{{\text{T}}_{\text{b}}} \right)}$ (5)
Experimental Nusselt number (Nu) is given by Eq. (6):
$\text{Nu}=\frac{\text{hd}}{\text{k}}$ (6)
where, k = Thermal conductivity of water.
Experimental friction factor is calculated by Eq. (7):
$\text{f}=\text{ }\!\!~\!\!\text{ }\frac{\text{p}}{\frac{\text{l}}{\text{d}}\times \frac{\text{ }\!\!\rho\!\!\text{ }{{\text{v}}^{2}}}{2}}$ (7)
where,
∆p = Pressure drop in the test section in N/m2
ρ = Density of water in Kg/m3
v = Velocity of water in m/sec
The thermal performance factor is calculated by Eq. (8):
$T\text{PF}=\text{ }\!\!~\!\!\text{ }\frac{\frac{\text{Nu}}{\text{N}{{\text{u}}_{\text{o}}}}}{{{(\frac{\text{f}}{{{\text{f}}_{\text{o}}}})}^{\frac{1}{3}}}}$ (8)
where, Nu and f are Nusselt number and friction factor for the tube equipped with coil inserts, while Nuo and fo represent the corresponding values for the plain tube.
The methodology for predicting the uncertainty of experimental results given by Kline and McClintock [42] is used. The method is effective in analyzing the errors arising in experimental work due to deviations in primary measurements. It determines the overall uncertainty of a derived quantity by combining the uncertainties of the individual primary measurements. The formulation is based on a first-order Taylor series expansion, which assumes small deviations in the measured variables. The distribution of error is assumed to be normal and the uncertainty in each variable (Y) is described as
$Y=X\pm x$
The above equation states that the best value of variable, Y is believed to be X and its true value lies within the interval (X + x, X - x).
The maximum uncertainties for dimensionless parameters Re, Nu, f and TPF are about ± 2.01%, ± 3.3%, ± 2.94 and ± 3.44% respectively. Eqs. (9)-(12) are used to determine the uncertainties.
$\frac{Re}{Re}={{\left[ {{\left( \frac{m}{m} \right)}^{2}}+{{\left( \frac{d}{d} \right)}^{2}} \right]}^{{}^{1}/{}_{2}}}$ (9)
$\frac{\Delta N u}{N u}=\left[\left(\frac{\Delta h}{h}\right)^2+\left(\frac{\Delta d}{d}\right)^2\right]^{1 / 2}$ (10)
$\frac{\Delta f}{f}=\left[\left(\frac{\Delta d}{d}\right)^2+\left(\frac{\Delta v}{v}\right)^2+\left(\frac{\Delta l}{l}\right)^2+\left(\frac{\Delta(\Delta p)}{\Delta p}\right)^2\right]^{1 / 2}$ (11)
$\frac{\triangle T P F}{T P F}=\left[\left(\frac{\Delta N u}{N u}\right)^2+\left(\frac{1}{3} \times \frac{\Delta f}{f}\right)^2\right]^{1 / 2}$ (12)
4.1 Validation of experimental setup
The accuracy of the experimental setup was validated by comparing experimental values of Nusselt number and friction factor for a plain tube with those calculated using the Dittus-Boelter and Blasius correlations [43-45]. Figures 4 and 5 illustrate the results obtained. It can be concluded that the experimental values agree reasonably with the values obtained from correlations. The mean absolute percentage deviations are 3.42% and 3.01% for Nu and f, respectively.
Figure 4. Validation of Nu for plain tube
Figure 5. Validation of f for plain tube
Dittus-Boelter correlation is given by Eq. (13):
$Nu=0.023\times R{{e}^{0.8}}\times P{{r}^{0.4}}$ (13)
Blassius correlation is given by Eq. (14):
$f=\frac{0.3164}{R{{e}^{0.25}}}$ (14)
where,
Re = Reynolds number and Pr = Prandtl number.
4.2 The influence of variable diameter helical coils on heat transfer rate and pressure drop
After carrying out validation of the experimental setup, the experiments were performed by introducing three configurations of variable diameter wire coils with three different CLR (2,3 and 4), three different DR (0.12, 0.16 and 0.2). The Re varied from 4000 to 10000 and water was used as a working fluid.
It has been revealed that the Nu is directly proportional to Re in each case. All configurations of variable diameter wire coils contribute to the rise in heat transfer rate compared to the plain tube. Figures 6, 7, and 8 illustrate the change in Nu with Re for all configurations of variable diameter helical coils at different DR. Within the tested range, the average increment in Nu compared to plain tube for convergent coils is 37.13%-74.17%, for convergent-divergent coils it is 47.18%-88.18% and for divergent coils it is 58.11%-100.79%. The increase in turbulent intensity is the main reason behind the improvement. The increased turbulent intensity results in better mixing in the flow field and it interrupts the formation of the boundary layer in which viscous effects are dominant. Also, the increase in the chaotic nature of the flow increases the residence time of fluid in the flow domain, facilitating extended heat exchange between the tube wall and the fluid.
Figure 6. Variation of Nu with Re at DR = 0.12
Figure 7. Variation of Nu with Re at DR = 0.16
Figure 8. Variation of Nu with Re at DR = 0.2
It is seen from Figures 6, 7, and 8 that at the same CLR, DR and Re, the values of Nu are highest for the divergent variable diameter helical coils. The convergent-divergent coils show the relatively lower values of Nu and the convergent coils configuration shows the least values of Nu. The divergent variable diameter helical coils are efficient in moving the fluid from the core region of the flow to the peripheral region close to the wall. It destroys the boundary layer in which fluid experiences deceleration that ultimately increases turbulence intensity, thereby contributing a remarkable upswing in the heat transfer rate. For example, at the same DR = 0.12 and CLR = 4 average increment in Nu, in contrast to the plain tube, for the divergent coil is 58.11%, for the convergent-divergent coil it is 47.18% and for the convergent coil it is 37.13%.
Figure 9. Variation of f with Re at DR = 0.12
Figure 10. Variation of f with Re at DR = 0.16
Figure 11. Variation of f with Re at DR = 0.2
All three embedding types of variable diameter coils increase the pressure drop when introduced in the flow field. This results in an increase in the values of the friction factors. Figures 9, 10, and 11 depict the variation of f with Re for all configurations of variable diameter helical coils and different DR. ivergent variable diameter coils show the highest values of f at the same CLR, DR and Re. The values of f for convergent-divergent coils are in between those of divergent and convergent coils. And convergent coils show the least values of f. Also, as expected, the value of f decreases with an increase in Re. For example, at DR = 0.12 and CLR = 4 as compared to plain tube for divergent coil, f increases by 3.22 times, in case of convergent-divergent coil, it increases by 2.55 times and in case of convergent coil, it increases by 1.98 times.
4.3 Effect of conical length ratio
It is observed that for all three profiles of variable diameter coils, the lowest CLR results in high values of Nu. It is important to note that secondary flows are encouraged, and flow disturbance rises as the number of conical elements in the flow region increases because of lower CLR values. In the present study, three values of CLR = 2, 3, and 4 are considered. Compared with the empty tube at DR = 0.12, the average increment in Nu for convergent coils at CLR = 4 is 37.13%, at CLR = 3 it is 44.94% and at CLR = 2 it is 52.89%. In case of convergent-divergent coils, the improvement in Nu at CLR = 4 is 47.18%, at CLR = 3 it is 57.14% and at CLR = 2 it is 66.95%. As earlier stated, divergent coils are most effective. For them, the average percentage augmentation in Nu over plain tube at CLR = 4 is 58.09%, at CLR = 3 it is 66.03% and at CLR = 2 it is 74.55%. At DR = 0.16 and for the same three CLRs, the average rise in Nu for convergent coils is observed as 43.81%, 53.44% and 61.26% respectively. For convergent-divergent coils, the values are 55.42%, 68.05% and 74.93% respectively. And for divergent coils, the values are 65.06%, 74.57% and 84.36% respectively. A similar trend is observed at DR = 0.2 also. At the same three CLRs, the average growth in Nu for convergent coils is seen as 53.51%, 64.84% and 74.19% respectively. The values of increments for convergent-divergent coils are 68.89%, 79.52% and 88.22% and for divergent coils, the values are 77.7%, 89.91% and 100.79% respectively.
Increased resistance to flow due to lower values of CLR results in higher values of f. Friction factor values are maximum at the lowest CLR for all three combinations at constant DR. Increased residence time of fluid in the test section due to more conical elements causes an upsurge in the pressure drop. In comparison with the plain tube at DR = 0.12, for convergent coils, f is increased by 1.98 times, 2.22 times and 2.47 times for CLR = 4, 3 and 2, respectively. Convergent-divergent coils show a hike in f by 2.55 times, 2.92 times and 3.33 times, whereas divergent coils show an increase in f by 3.22 times, 3.47 times and 3.75 times at the same CLRs. At DR = 0.16, the penalties of friction factor contrast to plain tube for convergent coils are 2.21 times, 2.45 times and 2.72 times, respectively. Results about convergent-divergent coils show growth of 2.91 times, 3.33 times and 3.55 times in f compared to the plain tube. In the case of divergent coils, the values of rise in f are 3.47 times, 3.77 times and 4.19 times at the same CLRs. Similar patterns are also observed at DR = 0.2. For convergent coils, the increase in f is 2.47 times, 2.72 times and 2.97 times, respectively. In case of convergent-divergent coils, the values of rise in f are 3.33 times, 3.54 times and 3.8 times, respectively. The divergent coils exhibit the enhancement in f by 3.75 times, 4.2 times and 4.55 times within the tested range of CLRs.
4.4 Effect of diameter ratio (DR)
All three types of variable diameter coils show an increase in Nu with an increase in DR at the same CLR. The increase in DR is due to an increase in wire diameter, which results in increased obstruction to the flow. And it causes a rise in turbulent intensity, which enhances the heat transfer rate. At CLR = 4 for convergent coils average rise in Nu at DR = 0.12 relative to plain tube is 37.15%, at DR = 0.16 it is 43.81% and at DR = 0.2 it is 53.48%. Similarly, for convergent-divergent coils the observed improvements in Nu are 47.18%, 55.41% and 68.86% respectively. In case of divergent coils, values of Nu are raised by 58.09%, 65.06% and 77.66% respectively for the same DRs. The resembling pattern is observed at CLR = 3 and 2. At CLR = 3 and for the same DRs, convergent coils show a rise of 44.94%, 53.44% and 64.84% in Nu. Whereas convergent-divergent coils indicate the boost of 57.14%, 68.05% and 79.52% in Nu. Also, divergent coils show augmentation of 66.01%, 74.57% and 89.91% in Nu. At CLR = 2, increments of 52.89%, 61.26% and 74.15% in Nu are noticed for convergent coils. In case of convergent-divergent coils, the values of Nu are improved by 66.95%, 74.93% and 88.18%. For divergent coils, the increments of 74.55%, 84.36% and 100.79% are observed in values of Nu within the tested range of DRs.
Figure 12. Variation of TPF with Re at DR = 0.12
Figure 13. Variation of TPF with Re at DR = 0.16
4.5 Evaluation of thermal performance factor
Passive heat transfer enhancement techniques, while increasing the heat transfer rate, also lead to a higher pressure drop, which consequently results in an increased pumping power requirement. Therefore, the thermal performance factor (TPF) is assessed to examine the usefulness of the employed technique. It is stated that the applied technique is beneficial if the TPF is greater than unity. In this study, TPF values for all three configurations at all CLR and DR are calculated. Variation of TPFs with Re at different DR is demonstrated in Figures 12, 13, and 14. It is observed that the values of TPF for all inserts are greater than unity, which highlights the promising nature of these variable diameter coils as a heat transfer augmentation technique. Also, as Re increases, TPF decreases. Highest values of TPF are observed at minimum values of CLR and maximum values of DR. Divergent variable diameter coils outperform when compared with their counterparts. For convergent coils, the highest and lowest values of TPF are 1.2694 and 1.0693, for convergent-divergent coils, they are 1.2807 and 1.073 and for divergent coils, the values are 1.2973 and 1.0842, respectively.
Figure 14. Variation of TPF with Re at DR = 0.2
The comparison of thermal performance factors (sometimes called enhancement efficiency) of various wire coil inserts is provided in Table 2.
Table 2. Maximum value of TPF for various wire coil inserts
|
Reference |
Conditions |
TPF |
|
[46] |
Wire coil having a triangular section positioned separately at a distance = 1 mm and having pitch ratio = 1 at Re ≈ 3430 |
1.82 |
|
[36] |
Serrated wire coils having the pitch ratio = 0.1969 and diameter ratio = 0.94 at Re = 5114. |
1.41 |
|
[23] |
Wire coil having triangular cross section with ratio of triangle length side to tube diameter = 0.0892 and pitch ratio = 1 at Re ≈ 3860 |
1.37 |
|
This work |
Divergent coil at CLR = 2, DR = 0.2 and ate Re = 4000 |
1.2973 |
|
This work |
Convergent-divergent coil at CLR = 2, DR = 0.2 and ate Re = 4000 |
1.2807 |
|
This work |
Convergent coil at CLR = 2, DR = 0.2 and ate Re = 4000 |
1.2694 |
|
[34] |
Twin wire coils having a space ratio = 18.1 and Reynolds number around 9700 inserted in a traverse corrugated tube |
1.09 |
|
[47] |
Conventional wire coil having pitch ratio = 3 and at Re = 4600 |
1.01 |
|
[48] |
The wire coil has changing pitch ratio ranging from 0.172 to 0.690 mm and Reynolds number close to 6200 |
0.99. |
It is very challenging to conduct experiments for every possible combination of input variables when a process is influenced by numerous variables. Design of experiments is a structured approach used to plan experiments so that appropriate data can be gathered and analyzed by statistical methods [49-51]. Response surface methodology (RSM) Hence, in the present analysis, RSM is applied to formulate predictive models for the Nusselt number (Nu) and friction factor (f), and to determine their optimal values under different fluid flow conditions and geometric parameters associated with variable diameter helical coils.
Figure 15. Flow chart for response surface methodology
RSM generally follows three essential steps. Initially, the significant independent variables and their respective levels are identified. Next, a mathematical model is developed to describe the relationship between these variables and the response, followed by validation of the model. Finally, contour and surface plots are generated to interpret the results and by applying constraints, the optimal operating conditions are obtained. The steps carried out in RSM analysis are described in Figure 15.
In the present study, Re, CLR, and DR are considered as input parameters and Nu and f are considered as response parameters. In the analysis, experimental data are gathered using the Central Composite Design (CCD). The factorial component of CCD includes a full factorial arrangement, covering every possible combination of the selected factors at two levels, high and low [55]. The design includes six axial points and six center points to adequately estimate the curvature of the response surface. The star points are located at the faces of the cube. This kind of configuration is called the face-centered CCD [56]. Commercially available Design Expert software was used for analysis. The arrangement consists of 20 runs, and it is shown in Table 3. Values of Nu and f are taken from experimental results.
Table 3. Matrix of experiments along with results
|
Re |
CLR |
DR |
Convergent Coils |
Convergent-Divergent Coils |
Divergent Coils |
|||
|
Nu |
f |
Nu |
f |
Nu |
f |
|||
|
4000 |
2 |
0.12 |
64 |
0.12 |
74 |
0.19 |
82 |
0.25 |
|
10000 |
2 |
0.12 |
106 |
0.06 |
110 |
0.07 |
114 |
0.08 |
|
4000 |
4 |
0.12 |
54 |
0.08 |
65 |
0.15 |
74 |
0.2 |
|
10000 |
4 |
0.12 |
98 |
0.06 |
100 |
0.06 |
104 |
0.07 |
|
4000 |
2 |
0.2 |
76 |
0.17 |
85 |
0.23 |
94 |
0.29 |
|
10000 |
2 |
0.2 |
118 |
0.07 |
122 |
0.08 |
129 |
0.09 |
|
4000 |
4 |
0.2 |
65 |
0.12 |
75 |
0.19 |
84 |
0.25 |
|
10000 |
4 |
0.2 |
105 |
0.06 |
110 |
0.07 |
115 |
0.08 |
|
4000 |
3 |
0.16 |
65 |
0.12 |
75 |
0.19 |
83 |
0.25 |
|
10000 |
3 |
0.16 |
105 |
0.06 |
110 |
0.07 |
114 |
0.08 |
|
7000 |
2 |
0.16 |
88 |
0.09 |
97 |
0.12 |
102 |
0.13 |
|
7000 |
4 |
0.16 |
79 |
0.07 |
86 |
0.09 |
91 |
0.11 |
|
7000 |
3 |
0.12 |
80 |
0.07 |
87 |
0.09 |
92 |
0.11 |
|
7000 |
3 |
0.2 |
91 |
0.09 |
100 |
0.12 |
105 |
0.13 |
|
7000 |
3 |
0.16 |
85 |
0.08 |
94 |
0.11 |
96 |
0.12 |
|
7000 |
3 |
0.16 |
85 |
0.08 |
94 |
0.11 |
96 |
0.12 |
|
7000 |
3 |
0.16 |
85 |
0.08 |
94 |
0.11 |
96 |
0.12 |
|
7000 |
3 |
0.16 |
85 |
0.08 |
94 |
0.11 |
96 |
0.12 |
|
7000 |
3 |
0.16 |
85 |
0.08 |
94 |
0.11 |
96 |
0.12 |
|
7000 |
3 |
0.16 |
85 |
0.08 |
94 |
0.11 |
96 |
0.12 |
5.1 ANOVA analysis
To analyze the effects of the input variables, a polynomial regression approach is adopted. As a first-order (linear) model lacks the ability to capture the complex nature of the responses effectively, a second-order (quadratic) model is used. The choice of quadratic models is justified by the observation that responses have been impacted by squared and interactive terms in addition to linear terms. Analysis of Variance (ANOVA) was conducted to determine the statistical significance of the parameters affecting the responses. Tables 4-9 show the ANOVA tables for responses Nu and f for three configurations of variable diameter coils.
Table 4. ANOVA table of Nu for convergent coils
|
Source |
SS |
df |
MS |
F-Value |
p-Value |
|
|
Model |
4936 |
9 |
548.45 |
1515.61 |
< 0.0001 |
significant |
|
A-Re |
4360 |
1 |
4359.74 |
12047.88 |
< 0.0001 |
|
|
B-CLR |
276.9 |
1 |
276.89 |
765.16 |
< 0.0001 |
|
|
C-DR |
286.6 |
1 |
286.55 |
791.85 |
< 0.0001 |
|
|
AB |
0.047 |
1 |
0.0465 |
0.1285 |
0.7274 |
|
|
AC |
2.28 |
1 |
2.28 |
6.3 |
0.0309 |
|
|
BC |
3.34 |
1 |
3.34 |
9.23 |
0.0125 |
|
|
A² |
0.818 |
1 |
0.8182 |
2.26 |
0.1636 |
|
|
B² |
0.7 |
1 |
0.7001 |
1.93 |
0.1944 |
|
|
C² |
3.21 |
1 |
3.21 |
8.87 |
0.0138 |
|
|
Residual |
3.62 |
10 |
0.3619 |
|
|
|
|
Lack of Fit |
3.62 |
5 |
0.7237 |
|
|
|
|
Pure Error |
0 |
5 |
0 |
|
|
|
|
Cor Total |
4940 |
19 |
|
|
|
|
Table 5. ANOVA table of f for convergent coils
|
Source |
SS |
df |
MS |
F-Value |
p-Value |
|
|
Model |
0.013 |
9 |
0.0014 |
175.38 |
< 0.0001 |
significant |
|
A-Re |
0.008 |
1 |
0.008 |
986.65 |
< 0.0001 |
|
|
B-CLR |
0.001 |
1 |
0.0012 |
150.12 |
< 0.0001 |
|
|
C-DR |
0.001 |
1 |
0.0012 |
150.12 |
< 0.0001 |
|
|
AB |
6E-04 |
1 |
0.0006 |
75.99 |
< 0.0001 |
|
|
AC |
6E-04 |
1 |
0.0006 |
75.99 |
< 0.0001 |
|
|
BC |
0 |
1 |
0 |
0 |
1 |
|
|
A² |
6E-04 |
1 |
0.0006 |
76.77 |
< 0.0001 |
|
|
B² |
0 |
1 |
0 |
0 |
1 |
|
|
C² |
0 |
1 |
0 |
0 |
1 |
|
|
Residual |
1E-04 |
10 |
8.06E-06 |
|
|
|
|
Lack of Fit |
1E-04 |
5 |
0 |
|
|
|
|
Pure Error |
0 |
5 |
0 |
|
|
|
|
Cor Total |
0.013 |
19 |
|
|
|
|
Table 6. ANOVA table of Nu for convergent-divergent coils
|
Source |
SS |
df |
MS |
F-Value |
p-Value |
|
|
Model |
3798 |
9 |
422 |
972.62 |
< 0.0001 |
significant |
|
A-Re |
3172 |
1 |
3171.96 |
7310.73 |
< 0.0001 |
|
|
B-CLR |
283.6 |
1 |
283.56 |
653.54 |
< 0.0001 |
|
|
C-DR |
330.7 |
1 |
330.74 |
762.29 |
< 0.0001 |
|
|
AB |
1.3 |
1 |
1.3 |
3.01 |
0.1136 |
|
|
AC |
0.113 |
1 |
0.1128 |
0.26 |
0.6212 |
|
|
BC |
2.13 |
1 |
2.13 |
4.91 |
0.051 |
|
|
A² |
1.34 |
1 |
1.34 |
3.09 |
0.1093 |
|
|
B² |
3.72 |
1 |
3.72 |
8.58 |
0.0151 |
|
|
C² |
2.02 |
1 |
2.02 |
4.65 |
0.0564 |
|
|
Residual |
4.34 |
10 |
0.4339 |
|
|
|
|
Lack of Fit |
4.34 |
5 |
0.8678 |
|
|
|
|
Pure Error |
0 |
5 |
0 |
|
|
|
|
Cor Total |
3802 |
19 |
|
|
|
|
Table 7. ANOVA table of f for convergent-divergent coils
|
Source |
SS |
df |
MS |
F-Value |
p-Value |
|
|
Model |
0.039 |
9 |
0.0043 |
1344.75 |
< 0.0001 |
significant |
|
A-Re |
0.032 |
1 |
0.032 |
10008.99 |
< 0.0001 |
|
|
B-CLR |
0.001 |
1 |
0.0014 |
442.44 |
< 0.0001 |
|
|
C-DR |
0.001 |
1 |
0.0014 |
442.44 |
< 0.0001 |
|
|
AB |
6E-04 |
1 |
0.0006 |
175.31 |
< 0.0001 |
|
|
AC |
6E-04 |
1 |
0.0006 |
175.31 |
< 0.0001 |
|
|
BC |
1.13E-06 |
1 |
1.13E-06 |
0.3515 |
0.5664 |
|
|
A² |
0.002 |
1 |
0.0017 |
517.65 |
< 0.0001 |
|
|
B² |
2.51E-06 |
1 |
2.51E-06 |
0.7829 |
0.397 |
|
|
C² |
2.51E-06 |
1 |
2.51E-06 |
0.7829 |
0.397 |
|
|
Residual |
0 |
10 |
3.20E-06 |
|
|
|
|
Lack of Fit |
0 |
5 |
6.40E-06 |
|
|
|
|
Pure Error |
0 |
5 |
0 |
|
|
|
|
Cor Total |
0.039 |
19 |
|
|
|
|
Table 8. ANOVA table of Nu for divergent coils
|
Source |
SS |
df |
MS |
F-Value |
p-Value |
|
|
Model |
3207 |
9 |
356.35 |
1744.96 |
< 0.0001 |
significant |
|
A-Re |
2523 |
1 |
2523.33 |
12356.31 |
< 0.0001 |
|
|
B-CLR |
273.8 |
1 |
273.84 |
1340.96 |
< 0.0001 |
|
|
C-DR |
367.6 |
1 |
367.6 |
1800.07 |
< 0.0001 |
|
|
AB |
3.1 |
1 |
3.1 |
15.18 |
0.003 |
|
|
AC |
2.58 |
1 |
2.58 |
12.62 |
0.0053 |
|
|
BC |
2.95 |
1 |
2.95 |
14.46 |
0.0035 |
|
|
A² |
5.21 |
1 |
5.21 |
25.53 |
0.0005 |
|
|
B² |
0.305 |
1 |
0.3053 |
1.49 |
0.2495 |
|
|
C² |
8.49 |
1 |
8.49 |
41.56 |
< 0.0001 |
|
|
Residual |
2.04 |
10 |
0.2042 |
|
|
|
|
Lack of Fit |
2.04 |
5 |
0.4084 |
|
|
|
|
Pure Error |
0 |
5 |
0 |
|
|
|
|
Cor Total |
3209 |
19 |
|
|
|
Table 9. ANOVA table of f for divergent coils
|
Source |
SS |
df |
MS |
F-Value |
p-Value |
|
|
Model |
0.0859 |
9 |
0.0095 |
2054.15 |
< 0.0001 |
significant |
|
A-Re |
0.0699 |
1 |
0.0699 |
15035.9 |
< 0.0001 |
|
|
B-CLR |
0.0016 |
1 |
0.0016 |
347 |
< 0.0001 |
|
|
C-DR |
0.0016 |
1 |
0.0016 |
347 |
< 0.0001 |
|
|
AB |
0.0005 |
1 |
0.0005 |
96.81 |
< 0.0001 |
|
|
AC |
0.0005 |
1 |
0.0005 |
96.81 |
< 0.0001 |
|
|
BC |
5.00E-07 |
1 |
5.00E-07 |
0.1076 |
0.7497 |
|
|
A² |
0.0062 |
1 |
0.0062 |
1337.42 |
< 0.0001 |
|
|
B² |
3.01E-06 |
1 |
3.01E-06 |
0.6466 |
0.44 |
|
|
C² |
3.01E-06 |
1 |
3.01E-06 |
0.6466 |
0.44 |
|
|
Residual |
0 |
10 |
4.65E-06 |
|
|
|
|
Lack of Fit |
0 |
5 |
9.30E-06 |
|
|
|
|
Pure Error |
0 |
5 |
0 |
|
|
|
|
Cor Total |
0.086 |
19 |
|
|
|
|
The quadratic model is used to develop the relationship between input and response variables. The equations for response variables consist of linear, interaction and squared terms and are given by Eqs. (15)-(20).
For convergent coils,
$\begin{gathered}N u=+33.61496+0.006747 \times R e+0.172356 \times \\ C L R-2.66174 \times D R+0.000025 \times R e \times C L R- \\ 0.004448 R e \times D R-16.15625 \times C L R \times D R+ \\ 6.06061 \exp -08 \times R e^2-0.504545 \times C L R^2+ \\ 675.28409 \times D R^2\end{gathered}$ (15)
$\begin{gathered}f=+0.197050-0.000030 \times R e-0.031417 \times \\ C L R+0.785417 \times D R+2.91667 \exp -06 \times \\ R e \times C L R-0.000073 \times R e \times D R- \\ 1.14174 \exp -15 C L R \times D R+1.66667 E- \\ 09 \times R e^2-2.64742 \exp -17 \times C L R^2- \\ 7.19178 \exp -15 \times D R^2\end{gathered}$ (16)
For convergent-divergent coils,
$\begin{gathered}N u=+36.31393+0.007268 \times R e+4.66117 \times \\ C L R+4.20303 \times D R-0.000135 \times R e \times C L R+ \\ 0.000990 \times R e \times D R-12.90625 \times C L R \times D R- \\ 7.75758 \exp -08 R e^2-1.16318 \times C L R^2+ \\ 535.51136 \times D R^2\end{gathered}$ (17)
$\begin{gathered}f=+0.322080-0.000054 \times R e-0.027214 \times \\ C L R+0.948826 \times D R+2.79167 \exp -06 \times \\ R e \times C L R-0.000070 \times R e \times D R+0.009375 \times \\ C L R \times D R+2.72727 \exp -09 \times R e^2- \\ 0.000955 \times C L R^2-0.596591 \times D R^2\end{gathered}$ (18)
For divergent coils,
$\begin{gathered}N u=+77.20890+0.003019 \times R e+0.648591 \times \\ C L R-187.33030 \times D R-0.000207 \times R e \times \\ C L R+0.004729 \times R e \times D R-15.18750 \times C L R \times \\ D R+1.52980 \exp -07 \times R e^2-0.333182 \times \\ C L R^2+1098.01136 \times D R^2\end{gathered}$ (19)
$\begin{gathered}f=+0.562843-0.000099 \times R e-0.035473 \times \\ C L R+0.564659 \times D R+2.50000 \exp -06 \times \\ R e \times C L R-0.000062 \times R e \times D R-0.006250 \times \\ C L R \times D R+5.28283 \exp -09 \times R e^2+ \\ 0.001045 \times C L R^2+0.653409 \times D R^2\end{gathered}$ (20)
After several refinement cycles, the most suitable models were identified and their performance was assessed by calculating values of coefficients of determination (R²) to ensure satisfactory predictive accuracy. It is calculated by Eq. (21).
${{R}^{2}}=1-\frac{Residual}{CorTotal}$ (21)
The model design fit statistics show values of (R2) as 0.9993 and 0.9937 for Nu and f, respectively, in the case of convergent coils. For convergent-divergent coils, the values are 0.9989 and 0.9992, respectively and for divergent coils, the values are 0.9994 and 0.9995, respectively. R2 value suggests the extent to which the model accurately captures the interconnection between independent and response variables, making it reliable for prediction and optimization [57, 58]. Adjusted R2 and predicted R2 are also determined by Eq. (22) and Eq. (23), respectively.
$Adjusted~{{R}^{2}}=1-\frac{Residual/Residual~df~}{\left( Residual+ModelSS \right)/\left( Residualdf+modeldf \right)}$ (22)
$Predicted~{{R}^{2}}=1-\frac{Predicted~error~sum~of~squares}{Residual+Model~SS}$ (23)
The adjusted R² and predicted R² values are summarized in Table 10.
Table 10. Values of adjusted R2 and predicted R2
|
Model |
Adjusted R2 |
Predicted R2 |
|
Nu for convergent coils |
0.9986 |
0.9904 |
|
f for convergent coils |
0.988 |
0.9505 |
|
Nu for convergent-divergent coils |
0.9978 |
0.9926 |
|
f for convergent-divergent coils |
0.9984 |
0.9948 |
|
Nu for divergent coils |
0.9988 |
0.9893 |
|
f for divergent coils |
0.999 |
0.9963 |
Adjusted R2 is a revised version of R2 that penalizes the inclusion of irrelevant variables. Predicted R2 helps to assess the predictive ability of a regression model for new, unseen data.
The significant terms affecting the responses are decided based on p-value of the ANOVA table. If p-value is greater than 0.1, then those model terms are not significant. For convergent variable diameter coils for determination of values of Nu, all linear terms Re, CLR, DR, interaction terms Re × DR and CLR ×DR and the squared term DR2 are significant model terms. Whereas for the determination of f, all linear terms Re, CLR, DR, interaction terms Re × CLR and Re × DR and one squared term Re2 are significant. In case of the convergent-divergent variable diameter coils for determination of Nu the significant terms are Re, CLR, DR, CLR × DR, CLR2 and DR2. For determination of f the significant terms are Re, CLR, DR, Re × CLR and Re × DR and Re2. For the third configuration, i.e. in case of divergent variable diameter wire coils for the prediction of Nu, the significant terms are Re, CLR, DR, Re × CLR, Re × DR, CLR × DR, Re2 and DR2. The significant terms to predict values of f are Re, CLR, DR, Re × CLR and Re × DR and Re2.
Figure 16. Combined effect of Re and CLR on Nu
Figure 17. Combined effect of Re and DR on Nu
Figure 18. Combined effect of CLR and DR on Nu
Figure 19. Combined effect of Re and CLR on f
Figure 20. Combined effect of Re and DR on f
Figure 21. Combined effect of CLR and DR on f
The 3D surface plots obtained from RSM analysis provide insights into the variation of response variables with respect to the input variables. With 3D plots, it becomes easy to understand the combined effect of factors on the response variable. 3D surface plots for Nu for all three configurations of variable diameter helical coils are shown in Figures 16, 17, and 18. Higher values of Re and DR and lower values of CLR promote turbulence inside the flow field and hence result in higher values of Nu. 3D surface plots for f for all three configurations of variable diameter helical coils are shown in Figures 19, 20, and 21. As far as f is concerned, lower values of Re and CLR and higher values of DR increase resistance to flow and hence higher values of f are observed in these conditions.
5.2 Optimization
The primary objective of any heat transfer enhancement method is to attain higher values of Nu with minimal increases in the f. For optimization, the desirability function approach is employed with the objective of maximizing Nu and minimizing f. Every response is transformed into a corresponding desirability value (D). It evaluates the degree to which the set of input variables meets the objective specified for the chosen response. The value of desirability ranges from 0 to 1. A desirability value of 1 indicates that the selected set of input factors perfectly meets the optimization objectives decided for the responses. Desirability functions for maximizing and minimizing the goals by using RSM [59] are given by Eq. (24) and Eq. (25).
$D_i=\left\{\begin{array}{cc}0 & \text { If } P R<L V \\ \left(\frac{P R-L V}{T V-L V}\right)^r & L V<P R<T V \\ 1 & P R>T V\end{array}\right.$ (24)
$D_i=\left\{\begin{array}{cc}1 & \text { If } P R<T V \\ \left(\frac{P R-U V}{T V-U V}\right)^r & T V<P R<U V \\ 0 & P R>U V\end{array}\right.$ (25)
where, PR represents predicted values, LV denotes the minimum response value observed across all cases, UV indicates the maximum response value observed, TV refers to the desired or target value of the response and r is a weighting factor that reflects the relative importance of the response. The combination of input variables with the highest desirability is chosen as the optimum input variables. The optimal operating conditions are identified at the highest Re, the lowest CLR, and the maximum DR. The values of optimum input variables for Re, CLR, and DR are 10000, 2, and 0.2, respectively. The results of optimization are shown in Figure 22. Optimized values of Nu and f, along with the value of desirability for all three categories of variable diameter helical coils, are given in Table 11.
Figure 22. Optimum levels of input variables
Table 11. Results of optimization
|
|
Optimized Values of Response Variables |
Desirability |
|
|
Nu |
f |
||
|
Divergent variable diameter helical coils |
128.35 |
0.0926 |
0.946 |
|
Convergent-divergent variable diameter helical coils |
122.21 |
0.0798 |
0.953 |
|
Convergent variable diameter helical coils |
117.1 |
0.0723 |
0.94 |
5.3 Confirmation experiment
To ensure the reliability of the proposed model, confirmation tests are carried out for each category of variable diameter helical coil. The confirmation experiments were performed at Re = 7000, CLR = 3, and DR = 0.16. The results obtained are shown in Table 12. Corresponding values of response variables predicted by models are also mentioned in Table 12. The average deviation for Nu is found to be 0.2596%, and for f, it is 1.91%. These values are significantly lower than the typical uncertainty margins encountered in experimental heat transfer research, which highlights the reliability of the present model. The close agreement between prediction and observation demonstrates that the RSM-based equations successfully capture the combined effects of input variables. Furthermore, the minimal discrepancies confirm that no major systematic errors are present in the modeling framework. This high level of consistency not only increases confidence in the model’s predictive capability but also suggests its suitability for practical applications where accurate estimation of Nu and f is essential. This confirms that the response equations developed using RSM are reliable for predicting Nu and f across the range of input parameters studied.
Table 12. Results of confirmation experiments
|
|
Convergent Coils |
Convergent-Divergent Coils |
Divergent Coils |
|||
|
Nu |
f |
Nu |
f |
Nu |
f |
|
|
Value predicted by model |
84.39 |
0.079 |
93.36 |
0.109 |
96.72 |
0.118 |
|
Experimental result |
84.58 |
0.081 |
93.65 |
0.108 |
96.48 |
0.115 |
|
Percentage variation |
0.21 |
1.79 |
0.31 |
0.64 |
0.25 |
3.30. |
An experimental investigation is carried out to analyze the thermo-hydraulic performance of a circular tube equipped with three configurations of variable diameter wire coils, namely convergent, convergent-divergent and divergent. The effect of conical length ratio, diameter ratio and Reynolds number on Nusselt number, friction factor and thermal performance factor is demonstrated. The analysis is followed by multicriteria optimization by using RSM. The important conclusions of the study are as follows:
This study demonstrates that variable-diameter helical coils offer a practical and effective method to enhance heat transfer in tubular heat exchangers, providing design guidance for industrial applications where improved thermal performance and energy efficiency are critical.
While this study demonstrates the effectiveness of variable-diameter helical coils in enhancing thermal performance, future work could explore their integration with other heat transfer enhancement techniques, as well as their application in non-circular ducts, multi-phase flows, and low-Reynolds-number regimes to broaden practical applicability.
|
As |
Surface area of the tube, m2 |
|
CCD |
Central composite design |
|
CLR |
Conical length ratio |
|
CorTotal |
Corrected total sum of squares |
|
Cp |
Specific heat of water at constant pressure, J.Kg-1.K-1 |
|
d |
Diameter of the tube, m |
|
df |
Degrees of freedom |
|
DR |
Diameter ratio |
|
f |
Friction factor |
|
h |
Heat transfer coefficient, W.m-2.K-1) |
|
l |
Length of the tube, m |
|
m |
Mass flow rate of water, Kg.sec-1 |
|
Model SS |
Model sum of squares |
|
Nu |
Nusselt number |
|
PR |
Predicted response |
|
Pr |
Prandtl number |
|
R2 |
Coefficient of determination |
|
Re |
Reynolds number |
|
Tb |
Mean bulk temperature, K |
|
Tin |
Temperature of water at inlet, K |
|
Tout |
Temperature of water at outlet, K |
|
Ts |
Mean surface temperature, K |
|
v |
Velocity of water, m.sec-1 |
|
∆p |
Pressure drop in the test section, N.m-2 |
|
Greek symbols |
|
|
ρ |
Density of water, Kg.m-3 |
|
Subscripts |
|
|
b |
Bulk |
|
s |
Surface |
|
Superscripts |
|
|
INV |
Inverse |
|
T |
Transpose |
Appendix-I
Calculations involved in ANOVA and RSM
To model the experimental results, three matrices are used: the matrix of model terms [A], the matrix containing the regression coefficients [B] and the response matrix [C]. These three matrices are related by Eq. (26).
$\left[ A \right]\times \left[ B \right]=\left[ C \right]$ (26)
To simplify the regression analysis and interpretation, the model terms were converted into coded form, assigning +1 to the highest level and -1 to the lowest level. Intermediate levels are coded using equal intervals within the specified range. In order to obtain model term coefficients, the matrix operations given by Eq. (27) are performed [56].
$\left[ B \right]={{[{{A}^{T}}A]}^{INV}}\times \left[ {{A}^{T}}C \right]$ (27)
where, superscripts INV and T denote inverse and transpose operations of a matrix, respectively.
The corrected total sum of squares (CorTotal) in the ANOVA is obtained by using Eq. (28).
$C\text{orTotal}=\mathop{\sum }_{\text{i}=1}^{\text{n}}\text{y}_{\text{i}}^{2}-{{\text{Z}}^{2}}/\text{N}$ (28)
where, yi is the actual response (AR). The variable Z denotes the addition of all measured values of the response variable. and N is the value of the total count of experiments. Using model Eqs. (17)-(22), predicted responses (PR) are obtained. Then the residual is obtained by using Eq. (29).
$Residual=\mathop{\sum }_{i=1}^{N}{{(AR_{i}^{2}-PR_{i}^{2})}^{2}}$ (29)
The model sum of squares (Model SS) is calculated by deducting the residual from the corrected total sum of squares (CorTotal). It is given by Eq. (30).
$Model~SS=CoTotal-Residual$ (30)
The factor sum of squares is obtained by excluding each model term individually from the model term matrix [A]. Accordingly, the coefficient matrix [B] is adjusted. New model term coefficients and model equations are obtained by repeating all matrix operations described in Eq. (27). For each model term, a new residual sum of squares is computed by excluding the term from the model. The difference between this and the residual from the complete model represents the sum of squares for the removed term. This procedure is repeated for each remaining model term to compute its respective SS values.
Total degrees of freedom (df) in the ANOVA table is calculated by subtracting one from the total number of experiments. Each model term is allocated one degree of freedom. As there is a total of nine terms, the value of df for the model is nine. The residual df is calculated by subtracting the total model df from the total df. Mean square (MS) is obtained by dividing SS by the respective df. The F-value is computed by taking the ratio of the mean square associated with the independent variable to the mean square of the residual error. The confidence level (CL) and p-value for the model are derived from the F-distribution, utilizing the model’s F-statistic along with its associated degrees of freedom for both the model and the residuals.
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