  Magnetic insulators have proven to be usable as quantum simulators for
itinerant interacting quantum systems. In particular the compound
(C$_{5}$H$_{12}$N)$_{2}$CuBr$_{4}$ (short (Hpip)$_{2}$CuBr$_{4}$) was shown to
be a remarkable realization of a Tomonaga-Luttinger liquid (TLL) and allowed to
quantitatively test the TLL theory. Substitution weakly disorders this class of
compounds and allows thus to use them to tackle questions pertaining to the
effect of disorder in TLL as well, such as the formation of the Bose glass. As
a first step in this direction we present in this paper a study of the
properties of the related (Hpip)$_{2}$CuCl$_{4}$ compound. We determine the
exchange couplings and compute the temperature and magnetic field dependence of
the specific heat, using a finite temperature Density Matrix Renormalization
group (DMRG) procedure. Comparison with the measured specific heat at zero
magnetic field confirms the exchange parameters and Hamiltonian for the
(Hpip)$_{2}$CuCl$_{4}$ compound, giving the basis needed to start studying the
disorder effects.
