@Deusovi was a few hours earlier, but perhaps this answer gives a simpler argument for the lower bound and a more practical description of an optimal strategy.
Let N (=52) be the number of cards and k (=6) the smallest integer such that $N \le 2^k$.
First, let us show that at least k operations are required:
There are already a couple of answers, but I think this question is a bit better than just a straightforward calculation problem, so here's one more.
You can reword the question into 5 separate questions like this:
Here is a card of which you know nothing, except that it was the smallest in a random sample of 5 cards. Do you wish to exchange it for a card ...
I wasn't fast enough, but here's some code confirming Jerry's answer:
from itertools import combinations
wins = *6
total = *6
for A in combinations(range(10),5):
rem = set(range(10))-set(A)
for N in range(6):
for draw in combinations(rem,N):
total[N] += 1
if sum(A[N:]+draw) > 22: wins[N] += 1
for i in range(6):...