**3. Thermophysical Characteristics of Al2O3 and** *γ***-Al2O3 Nanofluids**

The heat capacitance (*ρCp*)*n f* and the effective dynamic density *ρn f* of the nanofluids have been provided by the following expressions

$$\rho\_{nf} = (1 - \varrho)\rho\_f + \varrho\rho\_{s\star}$$

$$(\rho\mathbb{C}\_p)\_{nf} = (1 - \varrho)(\rho\mathbb{C}\_p)\_f + \varrho(\rho\mathbb{C}\_p)\_{s\star} \tag{9}$$

where the solid volume fraction of nanofluids is denoted by *ϕ* whereas the nanofluid's dynamic viscosity is characterized by

$$\frac{\mu\_{nf}}{\mu\_f} = (1 - q)^{-2.5} (for \ Al\_2O\_3 - water)$$

$$\frac{\mu\_{nf}}{\mu\_f} = 123 \,\mu^2 + 7.3 \,\upmu + 1 (for \ \gamma - Al\_2O\_3 - water)$$

$$\frac{\mu\_{nf}}{\mu\_f} = 306 \,\upmu^2 - 0.19 \,\upmu + 1 (for \ \gamma - Al\_2O\_3 - C\_2H\_6O\_2). \tag{10}$$

An expression for the nanofluid's effective thermal conductivity is

$$\frac{k\_{nf}}{k\_f} = \frac{k\_s + 2k\_f - 2\varrho(k\_f - k\_s)}{k\_s + 2k\_f + \varrho(k\_f - k\_s)} \text{ (for } Al\_2O\_3 - water)$$

$$\frac{k\_{nf}}{k\_f} = 4.97\varrho^2 + 2.72\varrho + 1 \text{ (for } \gamma - Al\_2O\_3 - water)$$

$$\frac{k\_{nf}}{k\_f} = 28.905\varrho^2 + 2.87\varrho + 1 \text{ (for } \gamma - Al\_2O\_3 - C\_2H\_6O\_2)$$
