5. The maximum number of small balls with a diameter of 2 that can be placed in a cylindrical container with an inner diameter of $\frac{4 \sqrt{3}}{3}+2$ and a height of 20 is ( ).
(A) 30
(B) 33
(C) 36
(D) 39
5. C.

The bottom of the cylinder can accommodate three small balls that are pairwise tangent. The next layer also holds three small balls that are pairwise tangent, with each ball touching two balls from the layer below. This method is repeated layer by layer, with a total of \( k \) layers. The total height is given by
\[
H=2+\frac{2 \sqrt{6}}{3}(k-1) \text{. }
\]

When \( k=12 \), \( H=2+\frac{22 \sqrt{6}}{3} \approx 19.9620 \).
Therefore, the maximum number of layers that can be placed is 12, totaling 36 balls.