# Design and Optimization of a Spiral-Tube Instantaneous Water Heater Using Response Surface Methodology

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Heat Transfer in STHEs

#### 1.2. RSM for the Optimization of HEs

_{2}O

_{3}-H

_{2}O nanofluid. The authors claimed that the pitch ratio has the most significant effect on the Nusselt number increment and the friction factor reduction and is able to increase efficiency up to five times. Xie et al. [23] worked on the optimal design of fin-tube HEs equipped with vortex generators using RSM and Artificial Neural Network (ANN). In their research, the explanatory variables included arc angle, attack angle, and length of the vortex generators, and target variables included the friction factor and Nusselt number. The results showed that both optimization approaches achieved reliable data. Zhou et al. [24] investigated the optimization of a perforated-finned heat sink. The design variables included the fins’ size and cross-sectional shape and the Reynolds number. Additionally, the Nusselt number and drag forces were considered as target parameters. The authors suggested the optimal design variables based on computational fluid dynamics and RSM methods. Additionally, a multi-objective RSM was used for a new design of a heat sink [25]. In this study [25], the effects of heights, angles, and circumferences of the fins were examined, and the optimal values were introduced. The results stated that heat-dissipation efficiency would be increased by the synchromesh of the display field [25]. Chananipoor et al. [26] optimized a double-pipe HE, which used nano-encapsulated phase change material (NPCM) slurry as the working flow. The authors used RSM to improve the convection heat transfer caused by NPCM slurry. The input parameters were NPCM concentration, inlet temperature, and Reynolds number (Re). The maximum improvement of thermal efficiency was considered the target parameter. The results showed that the inlet temperature and the mass fraction of NPCM have the most influence on the heat transfer behavior. Liu et al. [27] worked on the optimization of a fuel-air tube-in-tube helical coil HE using a combination of RSM and a multi-objective genetic algorithm. The results demonstrated that the hydraulic diameter of the HE’s annulus side and the inner diameter of the HE’s inner tube were the most influential parameters in the thermal and hydraulic characteristics of this kind of HE. Additionally, an optimization study on the same HE was conducted by Liu et al. [28] using RSM. Dagdevir [29] used Taguchi-based Grey relational analysis and RSM to obtain the lowest pressure drop and the highest convective coefficient in a dimpled HE tube. The design parameters of dimple diameter, dimple pitch length, and dimple height were considered. The results indicated that dimple pitch length has the most effect on the target parameters. Furthermore, the RSM optimization of spiral ground HEs was focused on in many studies [30,31].

## 2. Materials and Methods

_{c}) in an ideal case was set to 40 L/min. A thermometer was used to calculate the temperature difference between the water and the HE’s surface. In the experimental work, tests were performed at different levels of the three major factors, i.e., fin number, exhaust, and flow rate. The observed values of water temperature difference, WFR, and hot water output time were recorded (Table 1).

#### 2.1. RSM

#### 2.2. DoE Using CCD

#### 2.3. DoE and Presentation of the Mathematical Model

_{0}is a constant coefficient, n is the number of variables, B

_{i}is a linear coefficient, B

_{ij}denotes an interaction coefficient, B

_{ii}refers to quadratic coefficients, and X

_{i}and X

_{j}are coded values of the independent variables (e.g., factors) [40]. Each variable must be coded within its range to normalize it into a [–1, 1] interval. This is conducted to ensure that the regression analysis works properly, because independent variables may come in different units or different ranges. The normalization can be achieved through Equation (4) [39].

_{i}is the natural value of the ith variable, and X

_{max}and X

_{min}are the maximum and minimum levels of the ith variable, respectively.

^{2}), while its statistical significance was examined by an F-test (i.e., F-value). The significance of different regression coefficients was also calculated using the so-called t-test. It should be considered that R

^{2}alone is not able to clarify the model accuracy adequately, as it expresses the variations around the mean response [41]. Accordingly, practitioners use another index called adjusted R

^{2}(R

^{2}

_{adj}). For calculating this index, in contrast to R

^{2}, the average sum of squares is utilized as a substitute for the sum of squares. Equations (5) and (6) express the relations for calculating these parameters.

^{2}measures the quality of fitting the experimental data to the model, while R

^{2}

_{adj}is an adjusted representation of R

^{2}, into which the degree of freedom is further incorporated. The higher the number of data points and the closer the values of R

^{2}and R

^{2}

_{adj}are to 1, the more acceptable the fitting results will be.

## 3. Results and Discussion

#### 3.1. Final Equations for Obtaining the Effect of Factors on Responses

#### 3.2. Results of ANOVA for the Efficiency Response

^{2}), exhaust outlet diameter (B

^{2}), and WFR (C

^{2}) were close to zero, indicating their strong impacts on the response variable. The p-value for the mutual effect of fin number and exhaust outlet diameter (AB) was calculated at 0.022, i.e., below the significance level of 0.05, indicating the acceptable significance of this factor. The obtained p-values for the exhaust outlet diameter (B) and the mutual effect of the exhaust outlet diameter and WFR (BC) were higher than 0.05, indicating the insignificant impacts of these factors, especially for the exhaust outlet diameter.

^{2}

_{adj}of 0.9835, indicating proper agreement of the experimental data with the predicted results by the model developed for optimizing the HE.

#### 3.3. Summary of Statistical Results for the HTR Response

^{2}), exhaust outlet diameter (B

^{2}), and WFR (C

^{2}) were close to zero, while p-value for the mutual effect of fin number and exhaust outlet diameter (AB) was calculated at 0.0427, i.e., below the significance level of 0.05. These results indicate the significance of the mentioned variables for optimizing the HE. The exhaust outlet diameter (B) and the mutual effect of the exhaust outlet diameter and WFR (BC) exhibited p-values higher than 0.05, indicating their insignificant effects.

^{2}

_{adj}of 0.9992, indicating proper agreement of the experimental data with the predicted results by the developed model.

#### 3.4. Graphical Demonstration of Response Surface and Contour Lines

#### 3.5. Response Surface and Contour Lines for Efficiency Response

#### 3.6. Response Surface and Contour Lines for HTR Response

#### 3.7. Optimization

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

η | Efficiency (%) |

m | Water mass flow rate (L/min) |

m_{c} | Gas consumption (kcal/min) |

R | Fuel heat value (kcal/m^{3}) |

C | Specific heat capacity of water (J/kg.°C) |

Q | Heat transfer rate (W) |

h | Heat transfer coefficient (W/m^{2}k) |

$\underset{A}{\xaf}$ | Heat transfer area (m^{2}) |

ΔT | Temperature difference (k) |

## Abbreviations

CCD | Central Composite Design |

DoE | Design of Experiments |

HE | Heat Exchanger |

HTR | Heat Transfer Rate |

NPCM | Nano-Encapsulated Phase Change Material |

RSM | Response Surface Methodology |

STHE | Spiral-Tube Heat Exchanger |

WFR | Water Flow Rate |

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**Figure 2.**A view of the internal structure of the aluminum threaded core. The figure on the right is a magnification of the “A” region.

**Figure 3.**(

**a**) Response surface representing efficiency as a function of fin number and WFR, and (

**b**) contour lines representing efficiency as a function of fin number and WFR.

**Figure 4.**(

**a**) Response surface representing HTR as a function of fin number and WFR, (

**b**) contour lines representing HTR as a function of fin number and WFR, (

**c**) response surface representing HTR as a function of fin number and exhaust outlet diameter, and (

**d**) contour lines representing HTR as a function of fin number and exhaust outlet diameter.

Test No. | Tube’s Internal Area (m^{2}) | Fin Area (m^{2}) | Total Area (m^{2}) | Fin Numbers | Exhaust Outlet Diameter (cm) | Flow Rate(L/min) | Efficiency (%) | HTR (W) |
---|---|---|---|---|---|---|---|---|

1 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 2 | 8 | 40.8616780 | 7101.352304 |

2 | 0.0753 | 0.02868 | 0.10398 | 6 | 4 | 14 | 16.8253968 | 7838.943021 |

3 | 0.0753 | 0.02868 | 0.10398 | 6 | 2 | 14 | 25.2380952 | 7867.329561 |

4 | 0.0753 | 0.00478 | 0.08008 | 1 | 4 | 14 | 23.5555556 | 6054.636588 |

5 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 4 | 8 | 43.2653061 | 7113.914399 |

6 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 3 | 8 | 64.8979592 | 7226.973254 |

7 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 3 | 14 | 21.031746 | 6950.607164 |

8 | 0.0753 | 0.02868 | 0.10398 | 6 | 4 | 2 | 37.2562358 | 8662.152681 |

9 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 3 | 2 | 36.0544218 | 7641.522389 |

10 | 0.0753 | 0.00478 | 0.08008 | 1 | 3 | 8 | 72.1088435 | 6321.351036 |

11 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 3 | 8 | 65.3786848 | 7229.485673 |

12 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 3 | 8 | 65.3786848 | 7229.485673 |

13 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 3 | 8 | 65.3786848 | 7229.485673 |

14 | 0.0753 | 0.00478 | 0.08008 | 1 | 4 | 2 | 24.0362812 | 6430.660236 |

15 | 0.0753 | 0.00478 | 0.08008 | 1 | 2 | 14 | 12.6190476 | 6026.216196 |

16 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 3 | 8 | 65.3786848 | 7229.485673 |

17 | 0.0753 | 0.02868 | 0.10398 | 6 | 2 | 2 | 40.861678 | 8747.312301 |

18 | 0.0753 | 0.00478 | 0.08008 | 1 | 2 | 2 | 25.2380952 | 6452.522076 |

19 | 0.0753 | 0.01673 | 0.09203 | 3.5 | 2 | 8 | 67.3015873 | 7239.535349 |

20 | 0.0753 | 0.02868 | 0.10398 | 6 | 2 | 8 | 86.5306122 | 8293.127661 |

Factor | Unit | Min. | Max. |
---|---|---|---|

Fins | Count per unit area | 1 | 6 |

Water flow rate | L/min | 2 | 14 |

Exhaust | cm | 2 | 4 |

Std | Run | Space Type | Factor 1 | Factor 2 | Factor 3 | Response 1 | Response 2 | ||
---|---|---|---|---|---|---|---|---|---|

A: Fin (Number) | B: Exhaust Diameter (cm) | C: Water Flow (L/min) | Predicted Value | Efficiency (%) | Predicted Value | Q (W) | |||

11 | 1 | Axial | 3.5 | 2 | 8 | 43.58 | 40.8617 | 7118.83 | 7101.35 |

8 | 2 | Factorial | 6 | 4 | 14 | 19.16 | 16.8254 | 7829.27 | 7838.94 |

6 | 3 | Factorial | 6 | 2 | 14 | 22.74 | 25.2381 | 7846.17 | 7867.33 |

7 | 4 | Factorial | 1 | 4 | 14 | 20.50 | 23.5556 | 6078.18 | 6080.87 |

12 | 5 | Axial | 3.5 | 4 | 8 | 43.61 | 43.2653 | 7105.19 | 7113.91 |

20 | 6 | Center | 3.5 | 3 | 8 | 64.60 | 64.898 | 7227.82 | 7226.97 |

14 | 7 | Axial | 3.5 | 3 | 14 | 23.66 | 21.0317 | 6975.89 | 6938.05 |

4 | 8 | Factorial | 6 | 4 | 2 | 35.90 | 37.2562 | 8664.27 | 8662.15 |

13 | 9 | Axial | 3.5 | 3 | 2 | 36.49 | 36.0544 | 7612.44 | 7641.52 |

9 | 10 | Axial | 1 | 3 | 8 | 75.93 | 72.1088 | 6301.90 | 6321.35 |

19 | 11 | Center | 3.5 | 3 | 8 | 64.60 | 65.3787 | 7227.82 | 7229.49 |

18 | 12 | Center | 3.5 | 3 | 8 | 64.60 | 65.3787 | 7227.82 | 7229.49 |

16 | 13 | Center | 3.5 | 3 | 8 | 64.60 | 65.3787 | 7227.82 | 7229.49 |

3 | 14 | Factorial | 1 | 4 | 2 | 25.77 | 24.0363 | 6449.63 | 6430.66 |

5 | 15 | Factorial | 1 | 2 | 14 | 13.21 | 12.619 | 6021.91 | 6026.22 |

17 | 16 | Center | 3.5 | 3 | 8 | 64.60 | 65.3787 | 7227.82 | 7229.49 |

2 | 17 | Factorial | 6 | 2 | 2 | 43.15 | 40.8617 | 8747.82 | 8747.31 |

1 | 18 | Factorial | 1 | 2 | 2 | 22.14 | 25.2381 | 6460.01 | 6452.52 |

15 | 19 | Center | 3.5 | 3 | 8 | 64.60 | 67.3016 | 7227.82 | 7239.54 |

10 | 20 | Axial | 6 | 3 | 8 | 85.77 | 86.5306 | 8321.34 | 8293.13 |

Source | Sum of Squares | df | Mean Square | F-Value | p-Value | |
---|---|---|---|---|---|---|

Model | 0.9208 | 9 | 0.1023 | 126.85 | <0.0001 | significant |

A-Fin | 0.0242 | 1 | 0.0242 | 29.96 | 0.0003 | |

B-Exhaust diameter | 1.44 × 10^{−7} | 1 | 1.44 × 10^{−7} | 0.0002 | 0.9896 | |

C-Water flow | 0.0412 | 1 | 0.0412 | 51.07 | <0.0001 | |

AB | 0.0059 | 1 | 0.0059 | 7.33 | 0.0220 | |

AC | 0.0066 | 1 | 0.0066 | 8.17 | 0.0170 | |

BC | 0.0007 | 1 | 0.0007 | 0.8329 | 0.3829 | |

A^{2} | 0.0726 | 1 | 0.0726 | 90.05 | <0.0001 | |

B^{2} | 0.1213 | 1 | 0.1213 | 150.43 | <0.0001 | |

C^{2} | 0.3278 | 1 | 0.3278 | 406.41 | <0.0001 | |

Residual | 0.0081 | 10 | 0.0008 | |||

Lack of Fit | 0.0077 | 5 | 0.0015 | 21.52 | 0.0022 | significant |

Pure error | 0.0004 | 5 | 0.0001 | |||

Cor total | 0.9289 | 19 |

Summary of Statistical Results for the Efficiency Response | ||||
---|---|---|---|---|

Std. Dev. | 0.0284 | R^{2} | 0.9913 | |

Mean | 0.4496 | Adjusted R^{2} | 0.9835 | |

C.V. % | 6.32 | Predicted R^{2} | 0.8830 | |

Adeq. Precision | 36.1303 |

Source | Sum of Squares | df | Mean Square | F-Value | p-Value | |
---|---|---|---|---|---|---|

Model | 1.14 × 10^{7} | 9 | 1.26 × 10^{6} | 2542.29 | <0.0001 | significant |

A-Fin | 1.02 × 10^{7} | 1 | 1.02 × 10^{7} | 20,508.95 | <0.0001 | |

B-Exhaust diameter | 465.01 | 1 | 465.01 | 0.9354 | 0.3563 | |

C-Water flow | 1.01 × 10^{6} | 1 | 1.01 × 10^{6} | 2037.73 | <0.0001 | |

AB | 2676.88 | 1 | 2676.88 | 5.38 | 0.0427 | |

AC | 1.08 × 10^{5} | 1 | 1.07 × 10^{5} | 216.12 | <0.0001 | |

BC | 2220.76 | 1 | 2220.76 | 4.47 | 0.0607 | |

A^{2} | 19,311.08 | 1 | 19,311.08 | 38.85 | <0.0001 | |

B^{2} | 36,881.20 | 1 | 36,881.20 | 74.19 | <0.0001 | |

C^{2} | 12,103.84 | 1 | 12,103.84 | 24.35 | 0.0006 | |

Residual | 4971.21 | 10 | 497.12 | |||

Lack of fit | 4873.37 | 5 | 974.67 | 49.81 | 0.0003 | significant |

Pure error | 97.84 | 5 | 19.57 | |||

Cor total | 1.14 × 10^{7} | 19 |

Summary of Statistical Results for the HTR Response | |||
---|---|---|---|

Std. Dev. | 22.30 | R^{2} | 0.9996 |

Mean | 7244.99 | Adjusted R^{2} | 0.9992 |

C.V. % | 0.3077 | Predicted R^{2} | 0.9967 |

Adeq Precision | 172.9001 |

Number | Fin | Exhaust Diameter (cm) | Water Flow (L/min) | Efficiency (%) | Q (W) | Desirability |
---|---|---|---|---|---|---|

1 | 6.000 | 2.920 | 5.952 | 85.0 | 8479.001 | 0.940 |

2 | 6.000 | 2.910 | 5.959 | 85.0 | 8478.532 | 0.940 |

3 | 6.000 | 2.929 | 5.934 | 85.0 | 8480.356 | 0.940 |

4 | 6.000 | 2.931 | 5.991 | 85.1 | 8475.718 | 0.940 |

5 | 6.000 | 2.906 | 5.923 | 84.9 | 8481.438 | 0.940 |

6 | 6.000 | 2.940 | 5.984 | 85.1 | 8476.132 | 0.940 |

7 | 6.000 | 2.929 | 5.851 | 84.8 | 8486.981 | 0.940 |

8 | 6.000 | 2.895 | 5.950 | 85.0 | 8479.350 | 0.940 |

9 | 6.000 | 2.905 | 6.065 | 85.3 | 8470.079 | 0.940 |

10 | 6.000 | 2.945 | 6.041 | 85.2 | 8471.528 | 0.940 |

11 | 6.000 | 2.937 | 6.082 | 85.3 | 8468.378 | 0.940 |

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## Share and Cite

**MDPI and ACS Style**

Rezaei, P.; Moheghi, H.R.; Amiri Delouei, A.
Design and Optimization of a Spiral-Tube Instantaneous Water Heater Using Response Surface Methodology. *Water* **2023**, *15*, 1458.
https://doi.org/10.3390/w15081458

**AMA Style**

Rezaei P, Moheghi HR, Amiri Delouei A.
Design and Optimization of a Spiral-Tube Instantaneous Water Heater Using Response Surface Methodology. *Water*. 2023; 15(8):1458.
https://doi.org/10.3390/w15081458

**Chicago/Turabian Style**

Rezaei, Pedram, Hamid Reza Moheghi, and Amin Amiri Delouei.
2023. "Design and Optimization of a Spiral-Tube Instantaneous Water Heater Using Response Surface Methodology" *Water* 15, no. 8: 1458.
https://doi.org/10.3390/w15081458