By the way, are you sure that your model is correct?

D[T[z, t], z] == 0 /. z -> 0

means there is no flow of heat at the boundary !?

Look at this

fT = TT0 + (Tinfi - TT0) Erfc[x/(2 Sqrt[a t])]

It fulfills the heat-equation

D[fT, t] - a D[fT, x, x] // Simplify

For all t you have Tinfi at x = 0

Limit[fT, x -> 0] /. Sqrt[u_ v_] -> Sqrt[u] Sqrt[v]

Your other condition is matched

Limit[fT, t -> 0, Assumptions -> {x > 0, a > 0}] 

But the derivative is non-zero: there is a flow of heat

D[fT, x] /. x -> 0

A plot looks similar to the result of the numerical Integration

val = {TT0 -> 303, Tinfi -> 373, a -> 10};
p1 = Plot3D[fT /. val, {x, 0, 10}, {t, 0, 50}]