In the traditional numerical investigations on the forced convection problem, the parameters that affect the thermal performance and flow behavior are treated as deterministic values. However, for complicated convection heat transfer problem in the real world, uncertainties exist naturally. To obtain a deeper understanding of the forced convection problem of nanofluids in microchannels, uncertainties are suggested to be considered.
In 2000, Xuan et al. [
1] pointed out that nanofluids had tremendous potential in heat transfer enhancement. Then they investigated the transport and thermal performances of nanofluids and proposed two fitting methods for heat transfer correlations. Maiga et al. [
2] investigated the laminar convection heat transfer performance of nanofluids in two different types of passages. Comparisons were carried out between Al
2O
3-water and Al
2O
3-Ethylene Glycol. As concluded in this study, nanofluids could improve thermal performance and this effect was strengthened with the increase in particle concentration. The latter had better thermal performance than the former. He et al. [
3] experimentally studied the thermal behavior of TiO
2 nanofluids in vertical tubes, covering laminar, turbulence, and transition states. It was pointed out that nanoparticles increased thermal conductivity and the beneficial effect was more significant at a larger particle concentration and a smaller particle size. An experimental system of a circular channel with nanofluids flow was constructed by Heris et al. [
4]. They studied the heat transfer performance under isothermal wall temperature conditions. The experimental results were obviously higher than the prediction results by single phase heat transfer correlation. Nada et al. [
5] performed numerical investigations on the natural convection heat transfer of various water-based nanofluids in horizontal annular tubes, and TiO
2, Ag, Cu, and Al
2O
3 nanoparticles were considered. They suggested that the heat transfer was significantly influenced by the Rayleigh number and coefficient of thermal conduction of nanoparticles. Jung et al. [
6] experimentally studied the heat transfer and flow behavior of Al
2O
3-water nanofluids in rectangular microchannels. Wherein, the diameter of nanoparticles was 170 nm. They found that, under laminar condition, the adoption of 1.8% nanofluids improved the heat transfer coefficient by 32% compared with water, meanwhile, no obvious resistance loss was produced. Hwang et al. [
7] measured the thermal conductivity and pressure drop of Al
2O
3-water nanofluids in uniformly heated tubes. They found that the friction data were in accord with prediction results of the Darcy equation, while the heat transfer enhancement was not in good agreement with the Shah equation. Duangthongsuk et al. [
8,
9] reported an experimental study on turbulence heat transfer of TiO
2-water nanofluids in a double-tube counterflow heat exchanger. In their study, the solid volume fraction is 0.2–2%. The results showed that the heat transfer could be improved by 6–11% using nanofluids, and a larger friction was also produced. Then they presented correlations of the Nusselt number and the friction factor based on experimental results. Akbari et al. [
10] established microchannels with ribs and carried out a numerical investigation on laminar (Re = 10–100) heat transfer behavior of Al
2O
3-water nanofluids of which the maximum volume fraction was 0.04%. Effects of the ribs height and position, nanoparticle concentration, and Re on thermal-hydraulic performance were discussed. It was shown that the increase of rib height and nanoparticle concentration were beneficial to the heat transfer performance. Sekrani et al. [
11] studied the flow and heat transfer of nanofluids in horizontal passages using direct numerical simulation. They found that, compared with the single-phase model, the two-phase model provided higher prediction accuracy. Moreover, effects of the nanoparticle type and diameter were also explored and empirical correlations of the Nusselt number and coefficient of friction were proposed. Hussain et al. [
12] proposed a mathematical model to analyze the thermal performance and flow behavior of nanofluids on porous media plate. It was suggested that the increase of porous media permeability, nanofluids viscosity, and velocity slip parameter could reduce the thermal boundary layer and improve heat transfer performance. Fetecau et al. [
13] obtained the heat transfer and flow behavior of fractional nanofluids in isothermal vertical tubes under heat radiation. Temperature, velocity, Nusselt number, and friction factor were provided, and a comparison was conducted between fractional ordinary nanofluids. Zhao et al. [
14] predicted the viscosity of different types of nanofluids. Experimental results of Ethylene Glycol/water-based CuO, TiO
2, SiO
2, and SiC were used to train the neural network, and high prediction accuracy could be reached.
Valuable results concerning heat transfer and the flow of nanofluids have been obtained through deterministic analysis. However, the uncertainty is inevitable due to manufacturing errors, measurement errors, and variable working conditions, which will lead to uncertain thermal performance and flow behavior. For the purpose of taking uncertainties into account, the probabilistic numerical method [
15], fuzzy theory [
16], and interval method [
17] have been developed. Monte Carlo simulation (MCS) [
18] and polynomial chaos method [
19,
20], which belong to the probabilistic method are a valuable solution procedure that can handle the uncertainties. However, the uncertainties in the probabilistic method are treated as stochastic variables whose probability distribution function should be known in advance, which is tough work in reality. The fuzzy theory introduced by Zadeh [
21] is an efficient method when uncertainty is considered. The uncertainties in the fuzzy theory are subject to the expert opinions, which should be determined beforehand. In the probabilistic method and the fuzzy theory, the information which is not easy to get about uncertainties is required. In some circumstances, assumptions of the probability distribution function are made when applying the above methods, which will lead to unexpected error and is unreliable. IM is such a method that can overcome the disadvantages of determining a large amount of information in advance in the above methods.
IM is widely employed to analyze the uncertainty problem. Only the bounds of uncertainties are required to determine the bounds of the interested properties. The advantage of IM is it is free of the probability distribution function of uncertainties. Pereira et al. [
22] predicted the uncertain temperature of the transient heat conduction in a basin. To avoid overestimations, an element-by-element technique was proposed. Villacci and Vaccaro [
23] employed interval mathematics to analyze the transient temperature tolerance in a power cable. The bounds of temperature were predicted, and the results were proved to be conservative in comparison with those obtained by MCS. Xue and Yang [
24] investigated the uncertain response of a convection diffusion problem by employing IM based on the Taylor and Neumann expansion procedure. Wang et al. [
17] proposed two variations of IM to estimate the bounds of temperature in heat convection-diffusion problems, and different types of uncertainties were taken into account. Then, in their latter work [
25], they regarded the fuzzy parameters as interval variables and analyzed the fuzzy uncertainty propagation in a heat conduction problem.
IM has been widely used in uncertain thermodynamics, uncertain structure response [
26,
27,
28], and uncertain rotor dynamics [
29,
30,
31]. However, few papers about uncertain forced convection of nanofluids in microchannels have been published, which motivates this study. Based on the Chebyshev polynomial approximation, IM is proposed to analyze the uncertain thermal performance and flow behavior with uncertainties in material properties and geometric parameter. A comparison is made between IM and SM, which proves the effectiveness and accuracy of IM. Finally, the bounds of the temperature, velocity, and Nusselt number under different interval uncertainties are predicted.