INSTRUCTOR WITH STUDENT DEPARTS WRONG RWY AT CLE.

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I WAS ON A TRIP TO CLE WITH AN INST STUDENT. THIS STUDENT WAS CLOSE TO BEING FINISHED WITH THE TRAINING AND HAD ABOUT 500 HRS TOTAL TIME. WE LANDED IN THE MORNING AND HE LEFT THE FBO AND RETURNED PRIOR TO XA00. I WAITED THE DURATION. IT WAS A GOOD IFR DAY WITH LIGHT DRIZZLE AND LOW CEILINGS WITH A GOOD WIND. WE TAXIED FROM FBO VIA PAPA WITH A R TURN ONTO LIMA GOING TO RWY 23L FOR TKOF. WE DID OUR RUNUP AT THE LIMA/WHISKEY INTXN AND THEN PROCEEDED FORWARD TO JULIET. THIS WAS AN UNFAMILIAR ARPT AND HAD ASKED THE GND CTLR FOR PROGRESSIVE INSTRUCTIONS TO THE RWY. WE CONTACTED TWR AT JULIET AND THEY TOLD US TO TAXI INTO POS AND HOLD ON RWY 23L. WE PULLED OUT THINKING WE WERE ON RWY 23L; WHEN IN ACTUALITY WE WERE ON RWY 28. WE WERE AT THE INTXN OF THE 2 RWYS. I LOOKED AHEAD AND SAW A SIGN THAT SAID RWY 23L/5R CONFIRMING THAT WE WERE ON THE CORRECT RWY. WE WERE CLRED FOR TKOF AND PROCEEDED DOWN RWY 28; BUT WE THOUGHT WE WERE ON RWY 23L; BASED ON THE SIGN. AT ABOUT 500 FT AGL; I LOOKED BACK THROUGH THE GLASS WINDOW IN BACK AND KNEW THAT WE HAD DEPARTED ON THE WRONG RWY BASED ON VISUALLY SEEING THE RWYS FROM ABOVE. I CALLED THE TWR TO CONFIRM THIS AND THEY ACKNOWLEDGED THAT WE HAD; BUT SAID IT WAS NO FACTOR. THE TWR WAS QUITE CALM ABOUT THE WHOLE THING AND I'M GLAD BECAUSE DEPARTING FROM THE WRONG RWY IN A BUSY TCA IS BAD NEWS; ESPECIALLY IF A VIOLATION COULD MEAN THE END OF MY CAREER. I WAS PRETTY DISTURBED ABOUT THE WHOLE INCIDENT BECAUSE THERE COULD HAVE BEEN LNDG TFC ON RWY 23R; WHICH COULD HAVE RESULTED IN A MAJOR ACCIDENT. THE INTXN OF RWY 28/23L WAS VERY CONFUSING TO ME. I HAVE FLOWN INTO HOPKINS ONE OTHER TIME AND LINED UP FOR TKOF ON RWY 28; THINKING IT WAS RWY 23L. THE TWR CAUGHT OUR MISTAKE THAT TIME AND TOLD US TO TAXI FORWARD TO RWY 23L. I HAVE A COLLEGE DEGREE FROM (COLLEGE) AND CONSIDER MYSELF A SAFE INSTRUCTOR. I HAVE NEVER HAD ANY PROBS WITH VIOLATIONS OR RWY INCURSIONS. THE ONLY PLACE THAT I HAVE HAD A PROB WAS AT HOPKINS UNDER THIS CIRCUMSTANCE. I BELIEVE THAT THE FACILITY SHOULD CHANGE PLACEMENT OF THE RWY SIGN FOR RWY 23L/5R; BECAUSE IF YOU MISTAKE RWY 28 FOR RWY 23L/5R; THE SIGN IS CLRLY VISIBLE. THIS LEADS YOU TO BELIEVE THAT YOU ARE ON RWY 23L/5R WHEN YOU ARE ON RWY 28. THE ONLY OTHER WAY TO CATCH THE MISTAKE WOULD HAVE BEEN TO VISUALLY CHK MY DIRECTIONAL GYROSCOPE WITH THE RWY HDG.