B737-800 CREW DID NOT CLB AS FAST AS WANTED IN ZHU CLASS A AIRSPACE.

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THIS EVENT AROSE BECAUSE THE SECTOR CTLR WAS NOT SATISFIED WITH OUR RATE OF CLB AND WE WANTED TO HELP. AT ABOUT FL310 THE RATE OF CLB WAS STEADY AT 1000-1200 FPM BECAUSE WE HAD JUST DEPARTED IAH AND THE ACFT WAS HVY. WE WERE CLRED TO FL370 BY THE PREVIOUS CTLR. CHKING OUR COMPUTED PERFORMANCE OUR MAX ALT CAPABILITY WAS FL372. THERE WAS VERY LITTLE LEEWAY IN OUR CLB CAPABILITY. WE WERE ASKED WHAT OUR RATE OF CLB WAS AND WE STATED 1000-1200 FPM. THE CTLR STATED 'NOT GOOD ENOUGH; NEED TO DO BETTER.' WE WERE NOT ADVISED OF ANY TFC PROBS AT THAT TIME. WITH THE AUTOPLT ENGAGED; THE FO SELECTED VERT SPD AND CAUSED OUR RATE OF CLB TO INCREASE TO ABOUT 1800-2000 FPM BY TRADING OFF SOME AIRSPD FOR A FASTER RATE OF CLB. WE WERE TRADING OPTIMUM CLB AIRSPD FOR ALT AND APCHING OUR MINIMUM SLOW SPD FOR CLB. PASSING FL360 THE FMC DROPPED OUT OF 'VERT SPD AND REVERTED TO A SPD MODE.' THE FO DISCONNECTED THE AUTOPLT AND PITCHED THE NOSE OVER TO REGAIN SOME OF THE AIRSPD LOST IN THE EXPEDITED CLB. WE LOST 500 FT ALT TO ABOUT FL355 AND QUICKLY LEVELED OFF AT FL360 AND INCREASED AIRSPD MORE SUITABLE FOR THE REST OF THE CLB TO FL370 AT A MUCH SLOWER RATE OF CLB. THE CTLR ASKED US OUR ALT; THEN TOLD US THAT WE SHOULD HAVE TOLD HIM WE WERE DSNDING. WE REPLIED THAT WE WERE LEVELING OFF AT FL360; SO THAT WE COULD ACCELERATE. AT FL360 WE WERE THEN ADVISED TO TURN R 90 DEGS AS THE CTLR ALSO ADVISED ANOTHER ACFT TO TURN 90 DEGS THE OPPOSITE DIRECTION. CHKING THE TCASII; I OBSERVED ANOTHER ACFT IN EXCESS OF 20 MI AT FL350. IN MY OPINION; THERE WAS NOT A CONFLICT WITH THE OTHER ACFT. I BELIEVE WE WERE TRYING TO BE GOOD PLTS AND HELP ATC IN AN EXPEDITIOUS CLB TO ALT. IN HINDSIGHT; WITH A HVY ACFT; I WILL NOT ATTEMPT TO EXPEDITE A CLB. I FEEL THAT THE CTLR SHOULD HAVE STATED THAT HE HAD A CONFLICT AND RESTR US TO A LOWER ALT SINCE WE WERE NOT CLBING AS FAST AS HE WOULD HAVE LIKED. WE WERE NOT AWARE OF THE OTHER ACFT UNTIL THE CTLR TURNED US OFF THE AIRWAY AND I CHKED THE TCASII. THAT TARGET WAS WELL OUTSIDE THE ALERTING PERIMETERS OF THE TCASII. THE REST OF THE FLT WAS UNEVENTFUL. SUPPLEMENTAL INFO FROM ACN 517452: ACFT WAS CLBING OUT OF HOUSTON ENRTE TO CARRACAS; VENEZUELA. PASSING FL310; ATC QUERIED US ABOUT RATE OF CLB. WE STATED WE WERE DOING APPROX 1000 FPM. CTLR STATED WE HAVE TO DO BETTER THAN THAT. WE THEN PROCEEDED TO INCREASE RATE OF CLB TO 1500-1800 FPM. PASSING THROUGH FL355 THE AUTOPLT LEVELED OFF ACFT MOMENTARILY TO REGAIN AIRSPD. AT THAT POINT ATC ASKED US OUR RATE OF CLB. HE THEN TOLD US WE HAD OPPOSITE DIRECTION TFC AND GAVE US A 90 DEG TURN OFF COURSE. THE CONFLICTING TFC WAS ALSO GIVEN A 90 DEG TURN. OBSERVING THE TCASII THE CONFLICTING ACFT APPEARED TO BE 20 MI AWAY AT THE CLOSEST POINT. IN RETROSPECT; WE SHOULD HAVE TOLD ATC THAT THE ACFT WAS VERY HVY AND THAT 1000 FPM WAS THE BEST WE COULD DO FOR RATE OF CLB. THIS WOULD HAVE ALLOWED ATC TO PLAN AHEAD A LITTLE MORE AND WOULD HAVE PRECLUDED ANY SENSE OF URGENCY ON HIS PART. A TFC CALL TO US WOULD ALSO HAVE GIVEN US SOME ORIENTATION TO THE CONFLICT TFC. THE CONFLICT ACFT WAS NON RADAR WHICH ALSO CONTRIBUTED TO THE PROBS.