## 4 Application of Nanofluid in PTC
### Numerical Studies
#### 4.2.2 Investigations
70% reduction in entropy generation rate was observed due to the application of SWCNT. Pumping work, caused by pressure drop, did not increase too much for flow rates lower than 36.74 m\({}^{3}\)/h. They, finally, concluded that higher thermal conductivity does not necessarily lead to higher thermal efficiency, since the effect of specific heat capacity also should be considered. [PERSON] et al. [81] numerically modeled the effect of TiO\({}_{2}\), CuO, and Al\({}_{2}\)O\({}_{3}\) nanoparticles dispersed in Therminol VP-1 at 1-5 vol. % on thermal performance of PTC. Slight thermal efficiency enhancement was predicted by their model for 3 vol. %. CuO/ Therminol VP-1 nanofluid showed better exergy and energy performance comparing with other nanofluids. More specifically, it can enhance the convection heat transfer by a maximum 83%. [PERSON] et al. [82] numerically investigated the effect of TiO\({}_{2}\)/Syltherm 800, Al\({}_{2}\)O\({}_{3}\)/Syltherm 800, and TiO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\)/Syltherm 800 (hybrid nanofluid) at 3 vol. % for each TiO\({}_{2}\) and Al\({}_{2}\)O\({}_{3}\) as well as 1.5 vol. % for TiO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) on PTC thermal performance at different temperature (300-650 K) and 150 L/min flow rate. It was found that the efficiency of PTC can be enhanced up to 0.74% when using hybrid nanofluid and 0.34 for both mono nanofluids. Enhancements of 142.1%, 35.2%, and 34.9% in heat transfer coefficient was observed for hybrid nanofluid, Al\({}_{2}\)O\({}_{3}\)/Syltherm 800, and TiO\({}_{2}\)/Syltherm, respectively. Finally, the model suggested higher exergy efficiency for nanofluids in comparison with base fluid at higher temperatures. [PERSON] et al. [38] studied numerically the effect of different nanoparticles, Cu, CuO, SiO\({}_{2}\), Al\({}_{2}\)O\({}_{3}\), Fe\({}_{2}\)O\({}_{3}\), and TiO\({}_{2}\) dispersed into Syltherm 800 with different volume concentrations ranging 1-6 vol. % on PTC thermal efficiency. They found that the Cu/syltherm 800 nanofluid had the best thermal performance with the maximum thermal efficiency enhancement of 2.2% at 6 vol. %, 600 K inlet temperature, and 50 L/min flow rate. At 4% volume fraction Cu had the highest efficiency increase of 0.54% followed by 0.46% for CuO, 41% for Fe\({}_{2}\)O\({}_{3}\), 0.35% for TiO\({}_{2}\), 0.35% for Al\({}_{2}\)O\({}_{3}\), and 0.19% for SiO\({}_{2}\). The same trend is observed for enhancement in heat transfer coefficient with  
Cu is having the highest enhancement with 24.42% and SiO\({}_{2}\) with 7.28% the lowest enhancement at 4% volumetric concentration. [PERSON] et al. [83] modeled the effect of MWCNT/thermal oil nanofluid at 3, 6 vol. % of MWCNT on thermal performance of the PTC. They found 15% enhancement in heat transfer coefficient. [PERSON] et al. [84] modeled the effect of Fe\({}_{3}\)O\({}_{4}\)/water nanofluid at 0.6% vol. of Fe\({}_{3}\)O\({}_{4}\) on heat transfer behavior of the PTC. It was shown that the nanofluid could enhance the Nusselt number up to 56% at 6 vol. % of Fe\({}_{3}\)O\({}_{4}\). Also, the maximum thermal efficiency enhancement (combination of nanofluid application with twisted tape) was 1.6%. [PERSON] et al. [30] conducted numerical modeling in which they examined the effect of 0.05 vol. % Al\({}_{2}\)O\({}_{3}\) in Al\({}_{2}\)O\({}_{3}\)/Therminol VP-1 nanofluid on PTC thermal behavior. They suggested 5-10% enhancement in thermal efficiency due to the application of nanofluid. A CFD simulation was conducted by [PERSON] et al. [85] using 8% vol. of Cu, Fe\({}_{3}\)O\({}_{4}\), Al\({}_{2}\)O\({}_{3}\), and TiO\({}_{2}\) dispersed in water and investigated the effect of nanofluid in heat transfer of PTC. It was shown that Cu nanoparticles show better thermal enhancement comparing with Fe\({}_{3}\)O\({}_{4}\), Al\({}_{2}\)O\({}_{3}\), and TiO\({}_{2}\). Finally, [PERSON] et al. [86] simulated the effect of hybrid nanofluid on thermal performance of PTC. Three different hybrid nanofluids were selected: Ag/MgO (1.5 and 2 vol. %), Al\({}_{2}\)O\({}_{3}\)/Cu (0.1 and 1 vol.