B737 BLEEDS OFF TKOF FROM MMMX. CREW IS DISTR BY WX DEVS; FORGETS TO RECONFIGURE; AND GETS CABIN PRESSURE WARNING AT FL240.

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CAPT TKOF; IMPROVED CLB; BLEEDS OFF; RWY 23R; MEM; SW WIND AT ABOUT 10 KTS; NUMEROUS VICINITY TSTMS AND THUNDERSHOWERS. AFTER TKOF; THE ENG BLEED SWITCHES WERE NOT REPOSITIONED FOR NORMAL OPS DURING THE AT/O FLOW AND THE MISTAKE WAS NOT TRAPPED; AS IT SHOULD HAVE BEEN BY THE CHKLIST; OR BY THE CAPT WITH FOLLOW-UP OBSERVATION. AT APPROX FL240; THE CABIN ALT WARNING HORN SOUNDED AND I IMMEDIATELY LEVELED THE ACFT AT AROUND FL245 (WE HAD BEEN CLRED TO FL380) HAVING ASCERTAINED THE PROB FAIRLY QUICKLY. THE FO EXPEDITIOUSLY REPOSITIONED THE BLEEDS BUT THE CABIN INITIALLY DID NOT RESPOND WITH AN INDICATED DSCNT. AT THAT POINT; WE ELECTED TO SELECT MANUAL AND DRIVE THE OUTFLOW VALVE TOWARDS CLOSE; WHICH CAUSED AN IMMEDIATE DOWNWARD DEFLECTION OF THE NEEDLE AND A PRETTY SIGNIFICANT PRESSURE BUMP. I THINK THAT ALL OF THIS TOOK PLACE IN A TIMEFRAME OF LESS THAN 45 SECONDS. THE CABIN ASCENT RATE HAD BEEN ABOUT 200-300 FPM WHEN WE FIRST ATTACKED THE PROB AND WITH MANUAL MOVEMENT OF THE OUTFLOW VALVE; THE WARNING HORN ALMOST IMMEDIATELY CEASED. FROM MY VANTAGE POINT; THE CABIN ALT NEVER EXCEEDED 10000 FT ON THE GAUGE. I SPOKE WITH THE FLT ATTENDANT AND EXPLAINED WHAT HAD OCCURRED. THE DEP WX WAS A DEFINITE FACTOR IN THE DEVELOPMENT OF THIS INCIDENT. IMMEDIATELY FOLLOWING LIFTOFF; IT WAS APPARENT THAT THERE WAS A LINE OF TSTM CELLS EXTENDING RIGHT ACROSS THE MIDLINE OF OUR DEP COURSE WHICH WAS A R TURN AT 4 DME TO INTERCEPT A COURSE TO MATEO VOR. I MADE THE DECISION TO DEVIATE N (R) BUT STILL HAD TO WEAVE BTWN CELLS IN IMC USING RADAR ONLY FOR AVOIDANCE. WE ENCOUNTERED MODERATE RAIN AND OCCASIONAL MODERATE TURB FOR A FAIRLY EXTENDED TIME PERIOD AND WERE IN CONSTANT COMPETITION WITH OTHER ACFT FOR COM WITH DEP CTL FOR CLRNCS TO DEVIATE. WE DID NOT BREAK OUT OF THE CLOUDS UNTIL PASSING FL200 WHERE IT WAS STILL NECESSARY TO MANEUVER IN ORDER TO VISUALLY AVOID LARGE CUMULONIMBUS CLOUD COLUMNS. ALL OF THIS ATTN TO CELL AVOIDANCE HELPED TO CREATE A DISTR THAT CULMINATED IN A RUSHED AT/O FLOW AND CHKLIST WHERE VITAL ELEMENTS WERE MISSED. LOOKING BACK; I SHOULD HAVE MORE SPECIFICALLY ADDRESSED THE BLEED CONFIGN DURING MY TKOF BRIEF AND ALSO SHOULD HAVE INQUIRED SPECIFICALLY ABOUT IT IN THE INITIAL PERIOD OF TIME AFTER CALLING FOR THE AT/O CHKLIST. ON FURTHER REFLECTION; DUE TO THE VERY NATURE OF BLEEDS OFF TKOFS; IT WOULD SEEM TO ME THAT A CLB CHKLIST ITEM REGARDING 'APU AS REQUIRED' WOULD HELP AS A SECONDARY BACKUP TO THE AT/O CHK TO REFOCUS CREWS ON ABNORMAL CONFIGN AND POSSIBLY HELP FURTHER TO TRAP THIS TYPE OF ERROR.