SERIES OF ERRORS; BOTH BY THE DEP CTLR AND THE SF34 PLTS; CAUSED A CONFLICT TO OCCUR.

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I WAS PIC OF FLT ACR X ON DATE. WE WERE POS AND HOLD ON RWY 21R AT DTW ARPT. TWR ISSUED A TKOF CLRNC 'FLT ACR Y CLR FOR TKOF RWY 21R TURN L TO A HEADING OF 120 DEGS.' WE REALIZED IT WAS THE WRONG FLT NUMBER BUT WE WERE THE ONLY ACFT POS AND HOLD ON RWY 21R. THEREFORE; THE FO READ BACK THE TKOF CLRNC WITHOUT INCLUDING THE HEADING ASSIGNMENT. THE TWR DID NOT QUESTION OUR READBACK. I THOUGHT I HEARD A TURN TO A 190 DEG HDG; THE FO DID NOT RECALL ANY HEADING ASSIGNMENT. THE FO WAS FLYING AND AFTER A FREQ CHANGE TO DEP I NOTICED WE WERE STILL ON RWY HEADING. I ASKED DEP IF THE HEADING ASSIGNMENT WAS 190 DEGS. THE CTLR REPLIED NEGATIVE AND THEN ASSIGNED A L TURN TO 090 DEGS. AFTER WE WERE ESTABLISHED ON THE 090 DEGS WE RECEIVED AN RA. DEP THEN ALERTED US TO THE TFC. THE ASSIGNED TURN TO 090 DEGS DIRECTLY RESULTED IN A LOSS OF SEPARATION AND AN RA FOR BOTH ACFT INVOLVED. WHILE THE ASSIGNED TURN BY TRACON WAS DIRECTLY RESPONSIBLE FOR THIS INCIDENT; THERE WERE SEVERAL CONTRIBUTING FACTORS AND OPPORTUNITIES TO CHANGE THE OUTCOME. 1) OUR FAILURE TO CLARIFY THE CLRNC CONTAINING THE WRONG FLT NUMBER. HAD WE CLARIFIED WE WOULD HAVE HAD ANOTHER CHANCE TO HEAR THE PROPER HEADING ASSIGNMENT. 2) A HEADING REMINDER FROM THE TWR WITH THE FREQ CHANGE WOULD HAVE ALERTED US TO THE ERROR. MANY TWR CTLRS GRATEFULLY USE THIS TECHNIQUE. 3) A FLT CREW REMINDER IN THE FORM OF A NOTE ON THE YOKE CLIP; SETTING THE ASSIGNED HEADING AS A COURSE IN A STANDBY VOR; OR SETTING THE HEADING BUG TO THE ASSIGNED HEADING. 4) IN MY CONVERSATION WITH A TRACON SUPVR AFTER THE INCIDENT I FOUND WHEN I INQUIRED WHETHER OUR ASSIGNED HEADING WAS 190 DEGS THE CTLR ASSUMED WE WERE ON A 190 DEG HDG. OUR ACTUAL HEADING WAS STILL A RWY HEADING OF 210 DEGS. SHE THEN ISSUED THE TURN TO 090 DEGS; WHICH WAS A 100 DEG CHANGE IN DIRECTION FROM THE HEADING SHE THOUGHT WE WERE ON. SHE WOULD HAVE ISSUED A TURN TO 110 DEGS IF SHE HAD KNOWN OUR ACTUAL HEADING WAS 210 DEGS. 5) RECENT CHANGE IN PROC CHANGED OUR CLBOUT PROFILE. THE NEW PROC IS TO CLB AT V2 + 10 KTS (134 KTS IN THIS CASE) UNTIL 400 FT AGL AND THEN VCL + 15 KTS (144 KTS) UNTIL THE MSA OF 2800 FT MSL. NORMAL CLBOUT PREVIOUSLY WOULD HAVE BEEN 160-180 KTS RESULTING IN A LOWER ALT AND FURTHER DISTANCE FROM THE ARPT AND CONFLICTING TFC. TFACON MAY NOT BE FULLY AWARE OF THIS CHANGE; WHICH HAS JUST BEEN IMPLEMENTED WITHIN THE LAST MONTH.