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Optimizing the parameters of heat transmission in a small heat exchanger with spiral tapes cut as triangles and Aluminum oxide nanofluid using central composite design method

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Abstract

The present study aims at optimizing the heat transmission parameters such as Nusselt number and friction factor in a small double pipe heat exchanger equipped with rotating spiral tapes cut as triangles and filled with aluminum oxide nanofluid. The effects of Reynolds number, twist ratio (y/w), rotating twisted tape and concentration (w%) on the Nusselt number and friction factor are also investigated. The central composite design and the response surface methodology are used for evaluating the responses necessary for optimization. According to the optimal curves, the most optimized value obtained for Nusselt number and friction factor was 146.6675 and 0.06020, respectively. Finally, an appropriate correlation is also provided to achieve the optimal model of the minimum cost. Optimization results showed that the cost has decreased in the best case.

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Abbreviations

A:

Heat transfer area (m2)

Cp:

Specific heat (kJ kg−1 °C−1)

D:

Tube diameter (m)

d:

Nanoparticle diameter (m)

h:

Convective heat transfer coefficient (W m-2oC−1)

k:

Thermal conductivity (W m-1oC−1)

L:

Tube length (m)

mo :

Mass flow rate (kg s−1)

Nu:

Nusselt number (Dimensionless)

Pe:

Peclet number (Dimensionless)

Pr:

Prandtl number (Dimensionless)

Re:

Reynolds number (Dimensionless)

T:

Temperature (°C)

U:

Overall heat transfer coefficient (W m-2oC−1)

w:

Width

R2 reg :

Registered correlation coefficient

R2 Adj :

Adjusted correlation coefficient

R2 Pred :

Predicted correlation coefficient

F Value:

F distribution-a mathematical function

P value:

P distribution -a mathematical function

w:

Weight fraction (%)

V:

Velocity (m2 s−1)

F:

Friction factor

Nu:

Nusselt number

Δ:

Thickness

∆P nf :

Pressure drop (Pa)

α :

Thermal diffusivity (m2/s)

ρ :

Density (kg m−3)

ϑ :

Kinematic viscosity (m2/s)

φ V :

Nanoparticle volume concentration (%)

ANOVA:

Analysis of variance

CCD:

Central composite design

CFD:

Computational fluid dynamics

SEM:

Scanning electron microscope

TEM:

Transmission electronic microscope

PRESS:

Prediction error of sum of squares

3D:

Three dimensional

2D:

two dimensional

ANN:

Artificial neural network

GPR:

Geometrical progression ratio

RGPR:

Reducer geometrical progression ratio

RSM:

Response surface methodology

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Author information

Authors and Affiliations

  1. Department of Chemistry, Arak Branch, Islamic Azad University, Arak, Iran

    Nahid Ghasemi

  2. Department of Chemistry, Payame Noor University (PNU), P.O.Box, 19395-3697, Tehran, Iran

    Reza Aghayari & Heydar Maddah

Authors
  1. Nahid Ghasemi
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  2. Reza Aghayari
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  3. Heydar Maddah
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Corresponding author

Correspondence to Nahid Ghasemi.

Appendix: Uncertainty analysis

Appendix: Uncertainty analysis

A systematic error analysis was used to estimate the errors in the experimental analysis. The uncertainties calculated with the maximum possible error for the parameters and various instruments are given in Tables 6 and 7, respectively.

  1. (a)

    Reynolds number

Table 6 Uncertainties of parameters
Full size table
Table 7 Uncertainties of experimental instruments
Full size table
$$ \mathit{\operatorname{Re}}=\frac{4{m}^{\bullet }}{\pi D\mu},\kern2em \frac{U_{Re}}{\operatorname{Re}}=\left(\sqrt{{\left(\frac{U_{\dot{m}}}{\dot{m}}\right)}^2}+\sqrt{{\left(\frac{U_{\mu }}{\mu}\right)}^2}\right)\sqrt{{\left(\frac{U_{\dot{m}}}{\dot{m}}\right)}^2+{\left(\frac{U_{\mu }}{\mu}\right)}^2}=\sqrt{\left({0.0001}^2\right)+\left({0.01}^2\right)}=0.01\% $$
  1. (b)

    Heat transfer coefficient

$$ \mathrm{h}=\frac{q}{T_w-{T}_b},\frac{U_h}{\mathrm{h}}=\sqrt{{\left(\frac{U_q}{q}\right)}^2+{\left(\frac{U_{T_w}-{T}_b}{T_w-{T}_b}\right)}^2}=\sqrt{(0.2130)^2+{(0.0687)}^2}=0.2238\% $$
  1. (c)

    Nusselt number

$$ Nu=\frac{hD}{K}\kern1em ,\kern0.75em \frac{U_{Nu}}{Nu}=\sqrt{{\left(\frac{U_h}{h}\right)}^2+{\left(\frac{U_K}{K}\right)}^2}=\sqrt{(0.1876)^2+{(0.08)}^2}=0.2039\% $$

(d) Friction factor

$$ f=\frac{\Delta \mathrm{p}}{\left(\frac{\mathrm{L}}{\mathrm{D}}\right)\left(\frac{\uprho {\mathrm{V}}^2}{2}\right)},\frac{{\mathrm{U}}_{\mathrm{f}}}{\mathrm{f}}=\sqrt{{\left(\frac{{\mathrm{U}}_{\Delta \mathrm{P}}}{\Delta \mathrm{P}}\right)}^2+{\left(\frac{{\mathrm{U}}_{\uprho}}{\uprho}\right)}^2+{\left(\frac{2{\mathrm{U}}_{\mathrm{V}}}{\mathrm{V}}\right)}^2}=\sqrt{(0.002114)^2+{(0.081)}^2+{\left(2\times 0.0001\right)}^2}=0.08102\% $$

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Ghasemi, N., Aghayari, R. & Maddah, H. Optimizing the parameters of heat transmission in a small heat exchanger with spiral tapes cut as triangles and Aluminum oxide nanofluid using central composite design method. Heat Mass Transfer 54, 2113–2130 (2018). https://doi.org/10.1007/s00231-018-2292-8

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