## 1 Introduction
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. [PERSON] 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. [PERSON] 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. [PERSON] 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 [PERSON] [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. [PERSON] 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. [PERSON] and [PERSON] [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. [PERSON] and [PERSON] [24] investigated the uncertain response of a convection diffusion problem by employing IM based on the Taylor and Neumann expansion procedure. [PERSON] 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].