A B727 FLT CREW EXPERIENCED AN ENG FIRE WARNING ON CLIMBOUT. THEY DECLARED AN EMERGENCY AND DIVERTED TO A SUITABLE AIRPORT.

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TKOF WT: 166069 LBS. FUEL ON BOARD: 34000 LBS. PAYLOAD: 42430 LBS. ACFT DEPARTED WITH FLAPS 25 DEGS. DURING ENRTE CLB PASSING FL210; #2 ENG FIRE HANDLE ILLUMINATED WITH SIMULTANEOUS FIRE BELL ALARM. I SILENCED THE FIRE BELL AND OBSERVED NO CHANGES WITH ENG #2'S ENG INSTS. THE FE INDICATED NO ABNORMAL INDICATIONS ON HIS PANEL; INCLUDING NO INDICATION OF AN ENG STRUT OR LOWER AFT OVERHEAT. I LEVELED OFF THE ACFT AT FL215 WHEN A SECOND FIRE BELL ALARM SOUNDED. I INSTRUCTED FO (IOE CAPT) TO DECLARE AN EMER TO CTR AND REQUEST AN IMMEDIATE DSCNT. CTR GRANTED OUR REQUEST FOR AN IMMEDIATE DSCNT AND ACKNOWLEDGED OUR EMER DECLARATION. WE REQUESTED THE PROX OF NEAREST ARPTS FROM CTR AND AT THE SAME TIME I SCROLLED THROUGH MY GPS UNIT FOR NEAREST ARPTS. CTR INFORMED US OF 2 ARPTS IN THE AREA AND I ELECTED TO GO TO ZZZ; AS IT WAS APPROX 80 NM AWAY AND I WAS FAMILIAR WITH THE ARPT RWYS AND CONFIDENT THAT CFR SVCS WERE AVAILABLE. SINCE WE NEEDED TO LOSE 21000 FT; I BELIEVED THAT A NORMAL DSCNT WOULD PUT US IN POS FOR A NORMAL LNDG AT ZZZ AND NOT OVERSTRESS THE ACFT (IN THE EVENT OF STRUCTURAL DAMAGE); AND GIVE THE CREW TIME TO PERFORM THE PROPER PROC(S) AND SECURE THE #2 ENG. I CALLED FOR THE ENG FIRE CHKLIST AND INSTRUCTED THE FE TO WORK WITH THE FO ON THE CHKLIST(S) WHILE I FLEW THE ACFT AND HANDLED THE RADIO. I INFORMED CTR OF MY INTENTION TO GO TO ZZZ. CTR CLRED US DIRECT TO ZZZ AND I TURNED THE ACFT DIRECT TOWARDS IT. DURING OUR DSCNT; THE ENG FIRE BELL SOUNDED OFF NUMEROUS TIMES. THE #2 FIRE HANDLE CONTINUED TO REMAIN ILLUMINATED. NOT KNOWING THE ACTUAL CONDITION OF THE EVENT; THE CREW FOLLOWED PROC AND DISCHARGED THE FIRST FIRE BOTTLE. THE #2 ENG FIRE HANDLE CONTINUED TO REMAIN ILLUMINATED AFTER 30 SECONDS; SO THE CREW DISCHARGED THE SECOND FIRE BOTTLE. THE #2 FIRE HANDLE REMAINED ILLUMINATED AND THE FIRE ALARM REPEATEDLY KEPT SOUNDING. PER PROC; WE ELECTED TO PERFORM A PRECAUTIONARY SHUTDOWN OF ENG #2. AFTER ENG SHUTDOWN; THE FIRE ALARM CONTINUED TO SOUND INTERMITTENTLY. WHEN WITHIN VISUAL RANGE OF ZZZ; WE ASKED THE TWR CTLR IF HE COULD SEE ANY SMOKE OR FIRE FROM THE ACFT. THE TWR CTLR INFORMED US THAT HE COULD NOT SEE ANY SMOKE OR FIRE FROM THE ACFT. THE TWR CTLR ADVISED US THAT HE HAD BEEN INFORMED BY CTR OF OUR SITUATION AND THAT CFR IS PREPARED AND IN PLACE. HE HAD INFORMED CTR THAT WE WOULD ACCEPT RWY XX (THE LONGEST RWY AND THE MOST DIRECT ENTRY FOR AN IMMEDIATE LNDG). BEING IN CTL OF THE ACFT AT ALL TIMES AND WITH AN APPROX LNDG WT OF 155000 LBS (NOT TO MENTION A POSSIBLE FIRE AT THE ENG); I ELECTED NOT TO DUMP FUEL. I PERFORMED A VISUAL APCH TO RWY XX; WITH A 1-MIN INOP PROC. THE FIRE ALARM BELL CONTINUED TO SOUND EVEN TO THE GND AT LNDG. 2 CFR VEHICLES MET THE ACFT AS WE TURNED ONTO THE TXWY AND; AT OUR REQUEST; PERFORMED A VISUAL INSPECTION AROUND THE ACFT. CFR RPTED NO SMOKE/FIRE DAMAGE. SO I ELECTED TO TAXI THE ACFT ACROSS THE FIELD TO THE FBO. WE WERE ESCORTED BY THE 2 CFR VEHICLES BEHIND US AND AN ARPT VEHICLE IN FRONT OF US. WE TAXIED TO PARKING WITH NO INCIDENT.