  We investigate the effect of disorder on the heat transport properties of the
$S=\tfrac{1}{2}$ Heisenberg chain compound Sr$_2$CuO$_3$ upon chemically
substituting Sr by increasing concentrations of Ca. As Ca occupies sites
outside but near the Cu-O-Cu spin chains, bond disorder, i.e. a spatial
variation of the exchange interaction $J$, is expected to be realized in these
chains. We observe that the magnetic heat conductivity
($\kappa_{\mathrm{mag}}$) due to spinons propagating in the chains is gradually
but strongly suppressed with increasing amount of Ca, where the doping
dependence can be understood in terms of increased scattering of spinons due to
Ca-induced disorder. This is also reflected in the spinon mean free path which
can be separated in a doping independent but temperature dependent scattering
length due to spinon-phonon scattering, and a temperature independent but
doping dependent spinon-defect scattering length. The latter spans from very
large ($>$ 1300 lattice spacings) to very short ($\sim$ 12 lattice spacings)
and scales with the average distance between two neighboring Ca atoms. Thus,
the Ca-induced disorder acts as an effective defect within the spin chain, and
the doping scheme allows to cover the whole doping regime between the clean and
the dirty limits. Interestingly, at maximum impurity level we observe, in
Ca-doped Sr$_2$CuO$_3$, an almost linear increase of $\kappa_{\mathrm{mag}}$ at
temperatures above 100 K which reflects the intrinsic low temperature behavior
of heat transport in a Heisenberg spin chain. These findings are quite
different from that observed for the Ca-doped double spin chain compound,
SrCuO$_2$, where the effect of Ca seems to saturate already at intermediate
doping levels.
