This is the first part of a study of the detailed X-ray properties of the cores of nearby clusters. We have used the flux-limited sample of 55 clusters of Edge et al. (1990) and archival and proprietary data from the {\it ROSAT} observatory. In this paper an X-ray spatial analysis based on the surface-brightness-deprojection technique is applied to the clusters in the sample with the aim of studying their cooling flow properties. We determine the fraction of cooling flows in this sample to be 70-90 percent and estimate the contribution of the flow region to the cluster X-ray luminosity. We show that the luminosity within a strong cooling flow can account for up to 70 percent of a cluster X-ray bolometric luminosity. Our analysis indicates that about 40 percent of the clusters in the sample have flows depositing more than 100 \Msunpyr throughout the cooling region and that these possibly have been undisturbed for many Gyr, confirming that cooling flows are the natural state of cluster cores. New cooling flows in the sample are presented and previously ambiguous ones are clarified. We have constructed a catalogue of some intracluster medium properties for the clusters in this sample. The profiles of the mass deposited from cooling flows are analyzed and evidence presented for the existence of breaks in some of the profiles. Comparison is made to recent optical and radio data. We cross-correlate our sample with the Green Bank, NVSS and FIRST surveys and to the volume-limited sample of brightest cluster galaxies presented by Lauer and Postman (1994). Although weak trends exist, no strong correlation between optical magnitude or radio power of the brightest cluster galaxy and the strength of the flow is found.
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