16. (12 points) Given the function
$$
f(x)=\frac{2 x^{2}+4 x+a}{x}, x \in[1,+\infty) \text {. }
$$
(1) When $a=2$, find the minimum value of the function $f(x)$;
(2) If for any $x \in[1,+\infty), f(x)>0$ always holds, try to find the range of the real number $a$.

16. (1) When $a=2$, $f(x)=2 x+\frac{2}{x}+4$.

Since $f(x)$ is an increasing function on $[1,+\infty)$, $f(x)$ has a minimum value of $f(1)=8$ on $[1,+\infty)$.
(2) On $[1,+\infty)$, $f(x)=\frac{2 x^{2}+4 x+a}{x}>0$ always holds if and only if $2 x^{2}+4 x+a>0$ always holds. Let
$$
g(x)=2 x^{2}+4 x+a=2(x+1)^{2}+a-2 \text {, }
$$

then $g(x)$ is an increasing function on $[1,+\infty)$.
When $x=1$, it has a minimum value of $6+a$.
For $f(x)>0$ to always hold, we need $6+a>0$. Therefore, $a>-6$.