14. Let $x, y, z$ be positive real numbers. Find
$$
\left(x+\frac{1}{y}+\sqrt{2}\right)\left(y+\frac{1}{z}+\sqrt{2}\right)\left(z+\frac{1}{x}+\sqrt{2}\right)
$$

the minimum value.
14. From the problem, we know

the original algebraic expression
$$
\begin{array}{l}
\geqslant\left(2 \sqrt{\frac{x}{y}}+\sqrt{2}\right)\left(2 \sqrt{\frac{y}{z}}+\sqrt{2}\right)\left(2 \sqrt{\frac{z}{x}}+\sqrt{2}\right) \\
=8+4 \sqrt{2}\left(\sqrt{\frac{x}{z}}+\sqrt{\frac{z}{y}}+\sqrt{\frac{y}{x}}\right)+ \\
4\left(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{z}}+\sqrt{\frac{z}{x}}\right)+2 \sqrt{2} \\
\geqslant 8+4 \sqrt{2} \times 3 \sqrt[3]{\sqrt{\frac{x}{z}} \sqrt{\frac{z}{y}} \sqrt{\frac{y}{x}}}+ \\
4 \times 3 \sqrt[3]{\sqrt{\frac{x}{y}} \sqrt{\frac{y}{z}} \sqrt{\frac{z}{x}}}+2 \sqrt{2} \\
=8+4 \sqrt{2} \times 3+4 \times 3+2 \sqrt{2} \\
=20+14 \sqrt{2} \text {. } \\
\end{array}
$$

When $x=y=z=1$, the minimum value is achieved.
Therefore, the required minimum value is $20+14 \sqrt{2}$.