BE55 pilot reports multiple attempts to start both engines using normal start procedures. The left engine eventually starts but the right engine does not and catches fire. A fire extinguisher is used to put the fire out.

Narrative

I gave the left engine prime to 12 gallons per hour and then tried to crank using a normal start. It didn't start; primed to 12 gallons per hour then used a normal start didn't start. Went to the right engine gave it prime to 12 gallons per hour. [I] used normal start procedures. It didn't start. [I] re-primed to 12 gallons per hour. [I] used normal start procedures. It didn't start. Went to the left engine gave it prime to 12 gallons per hour. Used normal procedures then pushed the throttle to full open and mixture to a lean setting around where the M is at. Left engine started. [I] leaned mixture to peak; checked engine instruments; ok on left engine. Tried to start the right engine; prime to 12 gallons per hour; started with normal procedures and then moved the throttle to full open and leaned the mixture. Engine was in the process of starting but didn't. Tired to restart the right engine; gave it prime to 16 gallons per hour and tried to use a hot start procedure; engine didn't start. Noticed there was smoke coming from the right engine; went mixture idle cutoff and turned off all magnetos; alternators and battery. Notified passenger there was a fire and we needed to leave the aircraft. Went and looked at the smoke and to see if it was just smoke or if there were other signs; noticed discoloration on right engine. [I] went to the office to notify them of the fire on the right engine. They called the fire department. I used a fire extinguisher to put out the fire on the right engine. Fire was out when fire department arrived. When trying to start the right engine; I pumped the throttle twice. When cranking the engine; I feel that this action put fuel in the carburetor that ended up running to the air box. When the engine tried to start and it didn't and without the starter still being engaged this caused the fire to start. The fire was discovered by the smoke rising out of the top of the cowling. I feel that the throttle shouldn't have been pumped thus helping to break the chain of events.