  We use the theory of sutured TQFT to classify contact elements in the sutured
Floer homology, with $\Z$ coefficients, of certain sutured manifolds of the
form $(\Sigma \times S^1, F \times S^1)$ where $\Sigma$ is an annulus or
punctured torus. Using this classification, we give a new proof that the
contact invariant in sutured Floer homology with $\Z$ coefficients of a contact
structure with Giroux torsion vanishes. We also give a new proof of Massot's
theorem that the contact invariant vanishes for a contact structure on $(\Sigma
\times S^1, F \times S^1)$ described by an isolating dividing set.
