Introduction

Pulsating heat pipes (PHPs) have been designed for efficiently transporting heat over large distances through the use of capillary action and oscillatory fluid movement. They have gained considerable attention due to their simplicity, cost-effectiveness, and cooling capabilities. These pipes offer several advantages, including a compact size, simple design, low cost, and efficient heat transfer, making them suitable for a wide range of applications such as refrigeration, aerospace, waste heat recovery, and low-grade energy utilization [1,2,3,4]. They also exhibit a high heat transfer to mass ratio and have a strong resistance to drying out, making them particularly well-suited for high-power applications [5, 6]. To ensure optimal performance in practical applications, it is crucial to understand the impact of various operating conditions, such as inclination, filling ratios, and working fluids [7,8,9,10], and to optimize start-up characteristics, including cross-sectional shape, physical properties, and external surface motion [11, 12]. The angle of inclination, or the angle at which the heat pipe is tilted with respect to the horizontal plane, can have a significant effect on the oscillation, fluid flow, heat transfer performance, and temperature uniformity of PHPs [13,14,15,16,17,18].

Literature survey

At the forefront of current development is sustainable energy, with a crucial focus on designing environmentally friendly devices that consume low energy. Key to achieving this goal is the miniaturization of heating devices. The advancement of nanotechnology and production of integrated circuits on a large scale has spurred research into modeling heat transfer in micro-devices. As electronics continue to shrink in size, thermal management of micro components becomes increasingly crucial [19]. To address this, researchers have turned to a three-zone system in the design of PHPs. This system consists of an evaporator zone, a condenser zone, and an adiabatic zone with a smaller fluid volume. In a PHP, heat is transferred through the evaporator section as latent heat energy and then condensed in the cooling section. The primary function of PHPs is to regulate heat and temperature transfer. The concept was first introduced by Akachi [20], and it involves alternating evaporation and condensation sections. While PHPs have shown promise in various energy systems such as solar energy [21], waste heat recovery [22], electronics cooling [23], and more, their use is limited due to their operational mechanisms. Despite this, they have been successfully utilized in a range of applications such as hybrid vehicles [24], fuel cells [25], heat spreaders [26, 27], Stirling coolers [28], energy harvesting [29], battery thermal management [30], space applications [31], geothermal and ocean thermal energy conversion, and aircraft engine cooling [32]. Qu et al. [33] conducted an analysis of the primary factors that influence PHP performance. These include the condition of the wall surface, bubble growth, evaporation, superheat, and vapor bubbles, all of which affect the oscillation of the PHP. Additionally, Pai et al. [34] observed that the heat transfer coefficient increases with the oscillation of PHPs.

The key factors for operating PHP include heating power and gravitational potential energy [35], with a focus on physical and geometrical parameters to ensure efficiency and stability [36,37,38]. Materials such as stainless steel, copper, and aluminium are commonly used for PHP fabrication. At higher power levels, convection is the dominant heat transfer mechanism rather than conduction [39], and a smaller inner diameter can decrease surface tension and enhance the effects of gravity [40]. The shape of the channel cross-section also plays a critical role in determining the flow pattern, with circular cross-sections being more advantageous [41, 42]. The use of a non-return valve can further improve heat transfer in PHP [43]. A PHP with a small number of revolutions will have minimal performance [44], while a larger number of revolutions can yield significant results [45]. Shorter evaporator lengths have been found to enhance the performance of PHP [46]. In terms of input heat load, close-loop PHP is generally preferred [47], and the optimal fill ratio is dependent on the properties of the liquid and operating temperatures [48]. Higher heating powers can rapidly decrease thermal resistance (TR) [49], and the overall heat transfer coefficient (OHTC) is improved by increasing the oscillation frequency [8]. According to Kim and Kim [50], the choice of an appropriate working fluid is crucial in improving the thermal efficiency of a PHP. Thermal conductivity is a key factor to consider when selecting a working fluid for a PHP, as it directly affects its thermal performance [51]. In addition, as highlighted in [52,53,54,55,56,57,58,59,60,61], there are several other important factors that must be taken into account, including thermal conductivity [52, 56], filling ratios [52,53,54, 56, 61], zeotropic mixtures [53, 54], nanofluids [56], and the presence of surfactants [57, 58].

Wang et al. [62] utilized an ANN model to predict the thermal resistance (TR) of a PHP using input parameters like number of turns, heat input, charge ratio, and inner diameter. Haque et al. [63, 64] developed an empirical correlation for estimating the overall heat transfer coefficient (OHTC) of a PHP. Babu et al. [65] employed the Taguchi method, utilizing an L25 orthogonal array, and determined that input heat significantly influences PHP performance. Haque et al. [66] conducted experiments varying filling ratios (40–80%) and inclination angles (0°–180°) to evaluate the performance of a PHP. They measured the heat transfer rate (HTR), TR, and OHTC. Their findings revealed that HTR was lower for all filling ratios at inclinations of 90° and 180°. HTR was highest for a 40% filling ratio at a 30° inclination angle. TR was higher for all filling ratios at inclinations of 90° or greater. TR was lowest at a filling ratio of 60% and a 30° inclination angle. For all filling ratios, OHTC was low at a 90° inclination. At this orientation, bubbles lacked sufficient buoyancy to ascend and transfer heat to the condenser section. At 180° inclination, OHTC values were also very low for all filling ratios due to the limited bubble production in the evaporator section. Therefore, a filling ratio of 40 or 60%, combined with a 30° inclination, resulted in higher OHTC values for the PHP. Different sets of optimal input parameters identified for achieving high HTC, low TR, and high OHTC. A reliable multi-objective optimizations technique should be adopted for a set of optimal input parameters to achieve high HTC, low TR, and high OHTC.

Research gap

Investigations into PHP at 90° and 180° inclination angles yielded unsatisfactory results. Various optimal input parameter combinations have been reported to enhance HTR, reduce TR, and increase OHTC. The conventional Taguchi method is suitable for optimizing a single objective. A robust multi-objective optimization technique is required to address PHP optimization problems and determine optimal input variables that maximize HTR, minimize TR, and maximize OHTC.

Objective of the present study

This article proposes a modified Taguchi approach integrated with a robust multi-objective optimization technique to determine optimal input parameters for maximizing heat transfer rate (HTR), minimizing thermal resistance (TR), and maximizing overall heat transfer coefficient (OHTC) in a pulsating heat pipe (PHP). The modified Taguchi approach utilizes an orthogonal array (OA) based on the number of input parameters and levels to minimize experimental runs and gather data for a comprehensive factorial design of experiments. Empirical models relating PHP performance indicators (HTR, TR, and OHTC) to input parameters (filling ratio and inclination angle) are developed and validated using experimental data. Experimental results fall within the predicted range of HTR, TR, and OHTC. The methodology presented is straightforward and can be implemented on Excel Sheets without requiring specialized software such as Minitab.

Materials and methods

Haque et al. [66] designed a closed loop pulsating heat pipe (CLPHP) system as depicted in Fig. 1. The CLPHP system utilized an aluminium tube with a thickness of 0.5 mm, an inner diameter of 3 mm, and a length of 8862 mm, forming 14 parallel channels with 13 bends. The CLPHP system consisted of an evaporator section measuring 210 mm in length, an adiabatic section measuring 120 mm in length, and a condenser section measuring 324 mm in length. The evaporator section of the heat pipe was situated within an aluminium block (350 × 210 × 35 mm3) with grooves matching the dimensions of the heat pipe to prevent gaps and facilitate seamless heat transfer to the CLPHP. A heating-coil was constructed by winding 0.25 mm diameter nicrome wire at regular intervals of 1.5 mm on ceramic beads, which was then placed in a slot within the aluminium block to serve as the heat source.

Fig. 1
figure 1

Schematic of the CLPHP system and experimental setup [66]

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Heat is provided directly to the working fluid using a variac connected to the AC power supply. Fourteen K-type thermocouples were secured with insulation tape at various locations of the experimental setup to record temperature measurements during testing. A pressure gauge with a max range of 2 MPa is incorporated into the setup to monitor pressure fluctuations at different points. A gate valve is installed on the heat pipe to facilitate the removal of trapped air and the introduction of the working fluid. The apparatus is mounted on a wooden stand that allows for rotation to different orientations. Layers of glass wool (390 × 350 × 75 mm3) are used for insulation of the evaporator and adiabatic sections, while the condenser section remains exposed to the environment for cooling by natural air flow.

Experiments conducted varying filling ratios (40–80%) and inclination angles (0°–180°) to evaluate the performance of a PHP measuring HTR, TR, and OHTC. Set of optimal parameters from filling ratio, \(A\,\, \in \,\,\left\{ {40\% ,\,\,\,60\% ,\,\,\,80\% } \right\}\) and inclination angle,\(B\, \in \,\left\{ {30^{ \circ } ,\,\,45^{ \circ } ,\,\,60^{ \circ } } \right\}\) for high HTC, low TR, and high OHTC are found to be different. There is a need for a reliable multi-objective optimizations technique to find a set of optimal input parameters to achieve high HTC, low TR, and high OHTC.

Modified taguchi method

Fill ratio (A) and tilt angle (B) are the two parameters considered in the experiments to identify the optimal set of parameters A and B to obtain high HTR (heat transfer rate), low TR (thermal resistance), and high OHTC (overall heat transfer coefficient). Taguchi’s L9 OA (orthogonal array) is selected for 2 input parameters and 3 levels to each parameter. As per the Taguchi’s L9, assigning 3 levels to each input variable, one can introduce 4 input variables without increasing the test runs [67]:

$$ N_{{{\text{Exp}}}} = 1 + \,n_{{\text{p}}} \times \left( {n_{l} - 1} \right) \Rightarrow n_{{\text{p}}} = \frac{{N_{{{\text{Exp}}}} - 1}}{{\left( {n_{{\text{l}}} - 1} \right)}} = \frac{9 - 1}{{\left( {3 - 1} \right)}} = 4 $$
(1)

Here, \(N_{{{\text{Exp}}}}\) = 9, is the number of test runs; \(n_{{\text{p}}}\) is the number of input variables and ni = 3, is the number of levels.

For the 2 input variables (A and B), 2 dummy (fictitious) parameters (C and D) were introduced to act as errors (if any) and 3 levels were assigned for each input variable to perform Taguchi’s L9 OA tests for better PHP performance by identifying optimal input variables [10]. Table 1 shows the PHP performance in terms of HTR, TR and OHTC for the specified sets of levels of input variables (A and B). An analysis of variance (ANOVA) was carried out using the measured data of of HTR, TR and OHTC in Table 1, and summarizes the results of ANOVA in Table 2. The fill ratio (A), tilt angle (B), dummy1 (C) and dummy 2 (D) have a contribution of 49.56, 43.1, 0.68 and 6.66% to the HTR. In the case of TR, the variables (A, B, C, D) have 54.45, 25.51, 10.1 and 9.95%. In the case of OHTC, the variables (A, B, C, D) have 60.21, 35.48, 2.22 and 2.09% contribution. ANOVA results in Table 2 indicate the major % contribution of fill ratio (A) on HTR, TR and OHTC. It should be noted that the sum of the % contributions of A, B, C, and D on the performance indicators (HTR, TR and OHTC) is 100%. The fictitious parameters (C and D) have contribution of 7.34% to HTR, while 20.05% to TR and 4.31% to OHTC.

Table 1 Performance of PHP (viz., heat transfer rate (HTR), thermal resistance (TR) and overall heat transfer coefficient (OHTC)) for the inputs (fill ratio (A) and tilt angle (B)) as per L9 orthogonal array
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Table 2 ANOVA for the performance of PHP (viz., HTR, TR, and OHTC)
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Majority in engineering optimization problems depend on experimentation to observe the phenomenon prior to mathematical modeling. Some unknown parameters may influence the process resulting scatter in the performance characteristics in repeated tests. Though the Taguchi’s L9 OA is applicable to 4 parameters with 3 levels, the present problem specifies only 2 parameters (A and B) with 3 levels. Considering the unknown 3rd and 4th (C and D) parameters, ANOVA is performed and arrived the %Contribution of C and D in Table 2. The parameters C and D can be treated as the unknown influencing parameters (or errors in measurements, if any) of the performance characteristics (HTR, TR and OHTC).

From the ANOVA Table 2, the optimal set of input variables for achieving max HTR, min TR, and max OHTC are A1B1, A2B1, and A2B1, respectively. Here, subscripts to A and B denote the level of that variable. The optimal set of input variables (A1B1) for HTR is found to be different to those (A2B1) of TR and OHTC. It is preferable to have a set of optimal input variables which yield the requirement of max HTR, min TR, and max OHTC. This requirement demands for a reliable multi-objective optimization scheme.

Multi-objective optimization

In fact, the Taguchi method is useful for optimization of single objective problems [67], whereas a few researchers [68,69,70] have used it for optimizing the multiple responses. A simple and reliable multi-objective optimization method [71, 72] basing the Taguchi concept is followed here. Introducing the positive weighing factors \(\omega_{1}\), \(\omega_{2}\) and ω3 (which satisfy \(\omega_{1}\) + \(\omega_{2}\) + ω3 = 1), a single objective function (\(\zeta\)) is defined in the form

$$ \zeta = \omega_{1} \left( {1 - \frac{{{\text{HTR}}}}{{{\text{HTR}}_{\max } }}} \right) + \omega_{2} \left( {\frac{{{\text{TR}}}}{{{\text{TR}}_{\max } }}} \right) + \omega_{3} \left( {1 - \frac{{{\text{OHTC}}}}{{{\text{OHTC}}_{\max } }}} \right) $$
(2)

Here, HTRmax, TRmax, and OHTCmax are the max values within the range of input variables. These are obtained from Table 1: HTRmax = 34.451 W; TRmax = 0.261 °C W−1; and OHTCmax = 835.796 W m−2 K−1, which can be confirmed from ANOVA Table 2 by finding the levels of A and B corresponding to the max mean values. In the present study, A1B1 is the set of parameters corresponding to HTRmax; A3B3 is the set of parameters corresponding to TRmax; and A2B1 is the set of parameters corresponding to OHTCmax. Minimization of \(\zeta\) in Eq. (2) yields max HTR, min TR, and max OHTC.

Using the max values of HTR, TR, and OHTC (i.e. HTRmax = 34.451 W; TRmax = 0.261 °C W−1; and OHTCmax = 835.796 W m−2 K−1), the test data of Table 1 are converted into dimensionless form and the multi-objective function (\(\zeta\)) in Table 3 is evaluated using Eq. (2) for optimization. ANOVA conducted in Table 4. A2B1 (Fill ratio, A2 = 60; and tilt angle, B1 = 30°) provides the optimal solution to achieve the min \(\zeta\)(which corresponds to max HTR, min TR, and max OHTC). This way, the optimal set of parameters to the present problem is identified. Table 4 shows the ANOVA for the multi-objective optimization function.

Table 3 Multi-objective function \(\zeta\) for the performance indicators
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Table 4 ANOVA for the multi-objective optimization function
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Empirical relationships

The modified Taguchi method enables the determination of the range of estimates in performance indicators (such as HTR, TR and OHTC). By considering the specified input variables, namely, fill ratio (A) and tilt angle (B), this approach facilitates process design by providing insights into the expected scatter in repeated experiments. By considering \(\Psi\) as a measure of performance (HTR, or TR, or OHTC) and employing the additive law [67], the Eq. (2) allows for the prediction of \(\hat{\Psi }\) based on the levels of the input variables.

$$ \hat{\Psi } = \overline{\Psi }_{{\text{g}}} + \sum\limits_{i = 1}^{{n_{p} }} {\left( {\overline{\Psi }_{{{\text{il}}}} - \vec{\Psi }_{{\text{g}}} } \right) = \sum\limits_{i = 1}^{{n_{p} }} {\overline{\Psi }_{{{\text{il}}}} - \left( {n_{{\text{p}}} - 1} \right)\hat{\Psi }_{{\text{g}}} } } $$
(3)

\(\overline{\Psi }_{{{\text{il}}}}\) can be obtained from ANOVA Table 3. Subscript ‘\({\text{i}}\)’ with \(i = 1,\,2,\,3,\,4\) stands for the input variables (A, B, C, and D), and ‘\(l\)’ with \(l = 1,\,2,\,3\) stands for the 1-mean, 2-mean, 3-mean of the performance indicator (HTR, or TR, or OHTC) to the levels of (A, B, C, and D). \(\vec{\Psi }_{{\text{g}}}\) is the grand mean of \(\Psi\) for all 9 test runs. Considering HTR as the performance indicator, \(\overline{\Psi }_{{\text{g}}}\) = 33.64 W and \(\overline{\Psi }_{22}\) = 33.53 W in ANOVA Table 2, which corresponds to the 2-mean of HTR for the level-2 of the input variable, B. Here, the estimates of HTR, TR and OHTC in Table 5 are compared with experimental results of 9 test runs in Table 1 as per the Taguchi's L9 OA. \(n_{{\text{p}}}\) = 2 and \(n_{{\text{p}}}\) = 4 in Eq. (3) provide the estimates of the performance characteristics by excluding and including the contribution of the fictitious parameters (C and D). It is clearly visible about the discrepancy in the estimates in Table 5 for \(n_{{\text{p}}}\) = 2, whereas in case of \(n_{{\text{p}}}\) = 4, the estimates are identical to the test data. The corrections to the lower bound estimates for \(n_{{\text{p}}}\) = 2 are determined by calculating the sum of deviation of min mean values related to the dummy parameters C and D from the grand mean of the performance indicator (HTR, or TR, or OHTC). Similarly, the corrections to the upper bound estimates for \(n_{{\text{p}}}\) = 2 are determined by calculating the sum of deviation of max mean values related to the dummy parameters C and D from the grand mean of the performance indicator (HTR, or TR, or OHTC). Test data [66] in Table 5 fall within the estimated range.

Table 5 Estimates of HTR, TR and OHTC
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The corrections (\(\Delta {\text{THR}}_{\min }\) and \(\Delta {\text{THR}}_{\max }\)) to the lower and upper bound estimates of HTR for \(n_{p}\) = 2 are obtained using the mean values to the dummy parameters C and D in ANOVA Table 2 and the grand mean value as follows.

$$ \begin{aligned} \Delta {\text{THR}}_{\min } = & \min \left\{ {33.686,33.620,33.610} \right\} - 33.639 + \min \left\{ {33.758,33.501,33.657} \right\} - 33.639 \\ = & 33.610 - 33.639 + 33.501 - 33.639 = - 0.167\;{\text{W}} \\ \end{aligned} $$
$$ \begin{aligned} \Delta {\text{THR}}_{\max } = & \max \left\{ {33.686,33.620,33.610} \right\} - 33.639 + \max \left\{ {33.758,33.501,33.657} \right\} - 33.639 \\ = & 33.686 - 33.639 + 33.758 - 33.639 = 0.166\,{\text{W}} \\ \end{aligned} $$

The corrections (\(\Delta {\text{TR}}_{\min }\) and \(\Delta {\text{TR}}_{\max }\)) to the lower and upper bound estimates of TR for np = 2 are obtained using the mean values to the dummy parameters C and D in ANOVA Table 2 and the grand mean value as follows.

$$ \begin{aligned} \Delta {\text{TR}}_{\min } = & \min \left\{ {0.1203,0.1433,0.0987} \right\} - 0.1208 + \min \left\{ {0.1427,0.0983,0.1213} \right\} - 0.1208 \\ = & 0.0987 - 0.1208 + 0.0983 - 0.1208 = - 0.0446\,^\circ {\text{C W}}^{{ - 1}} \\ \end{aligned} $$
$$ \begin{aligned} \Delta {\text{TR}}_{\max } = & \max \left\{ {0.1203,0.1433,0.0987} \right\} - 0.1208 + \max \left\{ {0.1427,0.0983,0.1213} \right\} - 0.1208 \\ = & 0.1433 - 0.1208 + 0.1427 - 0.1208 = 0.0444\;^{ \circ } {\text{C}}\;{\text{W}}^{ - 1} \\ \end{aligned} $$

The corrections (\(\Delta {\text{OHTC}}_{\min }\) and \(\Delta {\text{OHTC}}_{\max }\)) to the lower and upper bound estimates of OHTC for \(n_{{\text{p}}}\) = 2 are obtained using the mean values to the dummy parameters C and D in ANOVA Table 2 and the grand mean value as follows.

$$ \begin{aligned} \Delta {\text{OHTC}}_{\min } = & \min \left\{ {610.43,599.24,663.45} \right\} - 624.37 + \min \left\{ {606.07,662.77,604.34} \right\} - 624.37 \\ = & 599.24 - 624.37 + 604.34 - 624.37 = - 45.16\;{\text{W}}\;{\text{m}}^{ - 2} \;{\text{K}}^{{ - {1}}} \\ \end{aligned} $$
$$ \begin{aligned} \Delta {\text{OHTC}}_{\max } = & \max \left\{ {610.43,599.24,663.45} \right\} - 624.37 + \max \left\{ {606.07,662.77,604.34} \right\} - 624.37 \\ = & 663.45 - 624.37 + 662.77 - 624.37 = 77.48\;{\text{W}}\;{\text{m}}^{ - 2} \;{\text{K}}^{ - 1} \\ \end{aligned} $$

The estimated range of the performance indicator (HTR, or TR, or OHTC) can be obtained after application of these corrections to the estimates. Applying corrections (− 0.167 and 0.166 W to the estimates of HTR), (− 0.0446 °C W−1 and 0.0444 °C W−1 to the estimates of TR), and (− 45.16 W m−2 K−1 and 77.48 W m−2 K−1 to the estimates of OHTC), the lower and upper bound values of HTR, TR, and OHTC are obtained.

To illustrate the simplicity of the present analysis, the estimates of \(\Psi\)(= HTR) in Test Run-1 of Table 5 using Eq. (3) are presented below. The levels of (A, B, C, and D) in Test Run-1 are (1, 1, 1, 1) and the mean values of \(\Psi\)(= HTR) from ANOVA Table 2 are: \(\overline{\Psi }_{11} =\) 33.92 W; \(\overline{\Psi }_{21} =\) 34.01 W; \(\overline{\Psi }_{31} =\) 33.69 W; and \(\overline{\Psi }_{41} =\) 33.76 W. The grand mean, \(\overline{\Psi }_{{\text{g}}}\) = 33.64 W.

Using Eq. (3), estimate of HTR with exclusion of C and D (\(n_{{\text{p}}} = 2\)) in Test Run-1:

$$ \mathop{\Psi }\limits^{\leftarrow} = \overline{\Psi }_{11} + \overline{\Psi }_{21} - \overline{\Psi }_{{\text{g}}} = {33}.{92} + {34}.0{1}{-}{33}.{64} = {34}.{29}\;{\text{W}} $$

Using Eq. (3), estimate of HTR with inclusion of C and D (\(n_{{\text{p}}}\) = 4) in Test Run-1:

$$ \hat{\Psi } = \overline{\Psi }_{11} + \overline{\Psi }_{21} + \overline{\Psi }_{31} + \overline{\Psi }_{41} - 3 \times \overline{\Psi }_{{\text{g}}} = 33.92 + 34.01 + 33.69 + 33.76 - 3 \times 33.64 = 34.46\;{\text{W}} $$

The corrections to the estimates of HTR for \(n_{{\text{p}}}\) = 2 are − 0.167 W and 0.166 W. Applying these corrections to the estimate of HTR, the lower and upper bound estimates obtained for Test Run-1 are: 34.123 (= 34.29–0.167) and 34.456 (= 34.29 + 0.166).

Transforming the three levels of the input variables (A and B) to − 1, 0 and 1, it is possible to represent mean values (see ANOVA Table 2) of the performance indicator (HTR or TR or OHTC) in a quadratic polynomial form and use them in the additive law (3) to obtain empirical relationship for HTR, TR and OHTC in terms of input variables (A, B). The transformed process parameters are: \(\xi_{1} = \frac{A}{20} - 3\); and \(\xi_{2} = \frac{B}{15} - 3\). Using the additive law (3) and the mean values of HTR, TR and OHTC (in ANOVA Table 2), the following empirical relationships are developed.

$$ {\text{HTR}} = 33.65 - 0.3367\xi_{1} - 0.1797\xi_{1}^{2} - 0.3153\xi_{2} + 0.1623\xi_{2}^{2} $$
(4)
$$ {\text{TR}} = 0.0892 + 0.0445\xi_{1} + 0.0462\xi_{1}^{2} + 0.0355\xi_{2} + 0.0012\xi_{2}^{2} $$
(5)
$$ {\text{OHTC}} = 712.87 - 151.12\xi_{1} - 164.98\xi_{1}^{2} - 135.85\xi_{2} + 32.24\xi_{2}^{2} $$
(6)

To illustrate the effectiveness of the developed empirical relationships (4)–(6), the two levels of Test Run-1 having a set of parameters (A = 40, B = 30) are considered. The transformed parameters for this set of parameters are (\(\xi_{1} = - 1,\,\xi_{2} = - 1\)). Using them in Eqs. (4)–(6) result the same values in Test Run-1 ofTable 5 for \(n_{{\text{p}}} = 2\):

$$ {\text{HTR}} = 33.65 - 0.3367 \times ( - 1) - 0.1797 \times ( - 1)^{2} - 0.3153 \times ( - 1) + 0.1623 \times ( - 1)^{2} = {34}.{285}\;{\text{W}} $$
$$ {\text{TR}} = 0.0892 + 0.044 \times ( - 1) + 0.0462 \times ( - 1)^{2} + 0.0355 \times ( - 1) + 0.0012 \times ( - 1)^{2} = 0.057\;^\circ {\text{C}}\;{\text{W}}^{{ - {1}}} $$
$$ {\text{OHTC}} = 712.87 - 151.12 \times ( - 1) - 164.98 \times ( - 1)^{2} - 135.85 \times ( - 1) + 32.24 \times ( - 1)^{2} = {867}.{1}\;{\text{W}}\;{\text{m}}^{{ - {2}}} \;{\text{K}}^{{ - }{1}} $$

Applying corrections to the developed empirical relationships, the lower and upper bound values of HTR, Tr and OHTC can be obtained. The above illustration confirms the validity of the developed empirical relationships (4)–(6). Table 6 shows the PHP working parameters for specific optimal conditions.

Table 6 PHP working parameters for specific optimal conditions
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Results and discussion

The angle of inclination, or the angle at which the heat pipe is tilted with respect to the horizontal plane, can have a significant effect on the performance of pulsed heat pipe (PHP). Investigations into PHP at 90° and 180° inclination angles yielded unsatisfactory results [66]. Different sets of optimal input variables, namely, fill ratio (A) and inclination angle (B) were obtained for a PHP while maximizing the heat transfer rate (HTR), minimizing the thermal resistance (TR), and maximizing the overall heat transfer coefficient (OHTC). Modified Taguchi method combining with a simple multi-objective optimization is briefly highlighted in the preceding section and recommended a set of optimal input variables (Fill ratio, A2 = 60%; and tilt angle, B1 = 30°). Table 6 gives PHP working parameters for specific optimal conditions. Empirical relationships (4)–(6) are developed for HTR, TR and OHTC in terms of input variables (A and B) and validated in preceding section with test data.

For specified input variables (A and B), HTR, TR, and OHTC are evaluated using empirical relations (4)–(6). Appropriate corrections are applied to obtain the lower and upper limits of HTR, TR, and OHTC. From relations (4)–(6), the performance indicators (HTR, TR and OHTC) are obtained for 9 combinations of two input variables with three levels: \((((A_{{\text{i}}} ,B_{{\text{j}}} ),\;j = 1\;{\text{to}}\;3),\;\;i = 1\;{\text{to}}\;3)\). The heat transfer rate (HTR), thermal resistance (TR), and overall heat transfer rate (OHTC) for the specified 9 sets of input variables are estimated from empirical relationships (4)–(6), and compared with test results in Figs. 2–4. The test data is found to be within the lower and upper bound estimates.

Fig. 2
figure 2

Heat transfer rate (HTR) estimates for all combinations of PHP working parameters

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Fig. 3
figure 3

Thermal resistance (TR) estimates for all combinations of PHP working parameters

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Fig. 4
figure 4

Overall heat transfer coefficient (OHTC) estimates for all combinations of PHP working parameters

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Figures 2–4 shows PHP performance indicators for a sequence of experiments. Estimated data falls between the lower and upper limits.

Figure 2 shows the variation of heat transfer rate at different fill ratio and inclination angle. The heat transfer rate (HTR) for the specified sets of input variables in 9 test runs are estimated and compared with test results. It is found that for all fill ratios the heat transfer rate, Qphp at inclinations 30°, 45°and 60° are very close to each other though it is slightly lower for fill ratio 0.8. It is also observed that for all fill ratios heat transfer rate, Qphp at inclinations 30° is much lower and it is lowest for fill ratio 0.6 [62].

From Fig. 2, it follows that for the 40% fill ratio and 30° tilt angle, the max heat transfer rate from the test is 34.45 W, which is in between the estimated range of 32.12–34.45 W. Test results will fall under the upper and lower bound value of the results obtained from empirical relationship (4). The test data is found to be within the lower and upper bound estimates.

Figure 3 shows the variation of thermal resistance of pulsating heat pipe with all the parameters considered. The thermal resistance (THR) for the specified sets of input variables in 9 test runs are estimated and compared with test results. Here, the lowest value in each graph specifies the optimum level of that particular parameter. From the Fig. 3, it is observed that among all the parameters considered the thermal resistance is lower at filling ratio of 60% and 30º inclination angles. The test data is found to be within the lower and upper bound estimates. Observed results will fall under the upper and lower bound value of the results obtained from empirical relationship (5). From the Fig. 3 it is found that for the 60% fill ratio and 30° tilt angle, the min thermal resistance from the test is 0.078 °C W−1, which is in between the estimated range of 0.0207–0.1097 °C W−1.

The overall heat transfer coefficient (OHTC) for the specified sets of input variables in 9 test runs are estimated and compared with test results in Fig. 4. From Fig. 4, it is evident that overall heat transfer coefficient, U for a fixed heat input is low for fill ratio 0.8 This is happening because at fill ratio 0.4 and 0.6, CLPHP gets more space to produce bubbles for heat transfer to the condenser section. So for the better performance of PHP, recommended set of optimal input variables is as follows: fill ratio = 60% and tilt angle = 300 [66]. Presented results fall under the upper and lower bound of the results results obtained from empirical relationship (6). From the Fig. 4 it follows that for 60% fill ratio and 30° tilt angle, the max overall heat transfer coefficient from the test is 835.79 W m−2K−1, which is in between the estimated range of 835.8–958.42 W m−2K−1. From the multi-objective optimization, max value of HTR (i.e. 34.45 W), min value of THR (i.e. 0.078 °C W−1) and max value of OHTC (i.e. 835.79 W m−2K−1) were possible to achieve for the specified parameters (i.e. 60% fill ratio and 30° tilt angle). These achieved characteristics were within the estimated range. The test data is found to be within the lower and upper bound estimates.

Concluding remarks

The modified Taguchi approach combining with a simple multi-objective optimization is adopted in the present study to obtain optimal working parameters of pulsed heat pipe (PHP), viz., fill ratio (A), and tilt angle (B). The heat transfer rate (HTR), thermal resistance (TR) and overall heat transfer coefficient (OHTC) are the performance indicators of PHP. In a PHP performance evaluation, investigations were made on the experimental data of HTR, TR, and OHTC by varying filling ratios (40–80%) and inclination angles (0°–180°). Unsatisfactory results were noted at 90° and 180° inclination angles of PHP. Taguchi’s L9 OA is considered for 3 levels of PHP working parameters: \(A \in \left\{ {40\% ,60\% ,80\% } \right\}\) and \(B \in \left\{ {30^\circ ,45^\circ ,60^\circ } \right\}\). Set of optimal parameters (A and B), for high HTC, low TR, and high OHTC are found to be different. Modified Taguchi method combining with a simple multi-objective optimization is used for identifying a set of optimal PHP working parameters. Outcomes of the present study are as follows.

  • Developed empirical relationships for HTR, TR and OHTC in terms of PHP working parameters (A and B) using the additive law and the mean values of the output responses from the ANOVA table. Corrections to the relationships are arrived by following the modified Taguchi approach. Test data is found to be within the limits of the estimates.

  • ANOVA results in Table 2 indicate the major % contribution of fill ratio (A) on HTR, TR and OHTC.

  • Recommended a set of optimal input variables (Fill ratio, A2 = 60%; and tilt angle, B1 = 30°) to achieve high HTR, low TR, and high OHTC.

  • For the identified set of optimal PHP working parameters, the range of HTR estimates from empirical relationship (4) is from 33.96 to 34.29 W. The test data of 34.124 W is within the range of estimates.

  • For the identified set of optimal PHP working parameters, the range of TR estimates from empirical relationship (5) is from 0.0207 to 0.1097 °C W−1. The test result of 0.078 °C W−1 is within the range of estimates.

  • For the identified set of optimal PHP working parameters, the range of OHTC estimates from empirical relationship (6) is from 835.8 to 958.42 W m−2 K−1. The test result of 835.79 W m−2 K−1 is within the range of estimates.

Future work is directed to study the influence of different working fluids in lower/ sub-ambient temperature zones, and geometrical parameters for distinct cooling applications.