  We consider integrable quantum spin chains with competing interactions. We
apply the quantum transfer matrix approach to these spin chains. This allowed
us to derive a set of non-linear integral equations for the thermodynamics of
these spin chains. We provide numerical solution of these integral equations
for the entropy as function of magnetic field, temperature and the coupling
constant. This allow us to assess, at low but finite temperature, the picture
describing the ground state diagram for high spin chain and longer range
interchain interactions.
