  The Sunyaev-Zel'dovich effect is a powerful new tool for finding and studying
clusters at high redshift, particularly in combination with their X-ray
properties. In this paper we quantify the expected scaling relations between
these properties using numerical simulations with various models for heating
and cooling of the cluster gas. For a Non-radiative model, we find scaling
relations in good agreement with self-similar predictions: $Y\propto T_X^{5/2}$
and $Y \propto L_X^{5/4}$. Our main results focus on predictions from Cooling
and Preheating simulations, shown by Muanwong et al. (2002) to provide a good
match to the X-ray scaling relations at z=0. For these runs we find slopes of
approximately $Y \proptoT_X^3$ and $Y \propto L_X$, steeper and flatter than
the self-similar scalings respectively. We also study the redshift evolution of
the scaling relations and find the slopes show no evidence of evolution out to
redshifts well beyond one, while the normalizations of relations between the SZ
signal and X-ray properties do show evolution relative to that expected from
self-similarity, particularly at z<1.
