AN ALT ASSIGNMENT WAS OVERSHOT WHEN A FLC READ BACK THE WRONG ALT CLRNC DURING DEP IN WX AND ICING CONDITIONS. BOTH CREW MEMBERS RPT THAT THE DIVISION OF ATTN BTWN TOO MANY TASKS AND DISTR WITH THE USE OF ACFT ANTI-ICE PROCS WERE THE MAIN CAUSAL FACTORS. READBACK HEARBACK PROBS CONTRIBUTED.

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WE WERE DEPARTING GRR RWY 8R. DEP WX WAS LIGHT RAIN; 700 FT OVCST AND A TEMP OF 32 DEGS F. RADAR ON TKOF SHOWED SOME AREAS OF MORE INTENSE PRECIP. ICING WAS ANTICIPATED AND ENCOUNTERED DURING THE CLBOUT. SHORTLY AFTER TKOF WHILE PERFORMING THE AFTER TKOF CHKLIST AND STARTING AIRFOIL ANTI-ICE; WE WERE ISSUED A R TURN TO 170 DEGS DIRECT TO FWA WHEN ABLE AND APPARENTLY WE WERE CLRED TO CLB TO 10000 FT. THE ALT CLRNC WAS MISUNDERSTOOD AND 17000 FT WAS SET IN THE ALT WINDOW; AND STATED IN THE READBACK. THE ERROR WAS NOT CAUGHT UNTIL THE FO FELT THAT 17000 FT WAS UNUSUAL AND CALLED TO CONFIRM THE ALT AS WE WERE CLBING THROUGH 12000 FT. WE WERE INFORMED THAT 10000 FT WAS THE CLRNC ALT. I DISCONNECTED THE AUTOPLT AND INITIATED AN IMMEDIATE DSCNT. ALT AT THIS POINT WAS 13000 FT. BEFORE WE COULD DSND BACK TO 10000 FT; THE DEP CTLR GOT FL230 FOR US AND WE WERE HANDED OFF TO ZAU. THERE WAS NO TFC CONFLICT; THE CLB WAS RESUMED AND NO FURTHER PROBS WERE ENCOUNTERED. I FEEL THAT THE ERROR WAS CAUSED BY THE CREW HAVING TO DIVIDE ATTN BTWN THE ACFT (CHKLIST; ANTI- ICE; AND PERFORMANCE); THE WX (ICING; TEMP INVERSION; CONTINUOUS LIGHT TURB); AND ATC (DEP AND ENRTE CLRNCS). IN FACT THE ANTI-ICE WAS INSTITUTED TWICE DURING THE FIRST 10000 FT OF CLBOUT. THE FIRST TIME WAS IN CONJUNCTION WITH THE AFTER TKOF CHKLIST AT 3000 FT THEN IT WAS ALL SECURED AT APPROX 6000 FT DUE TO INCREASING RAM AIR TEMP; AND THEN RE-INSTITUTED AT 10000 FT DUE TO DECREASING RAM AIR TEMP AT THAT POINT. I THINK THE 170 DEG HDG THAT WAS ISSUED WAS MISTAKENLY ACCEPTED AND READ BACK AS 17000 FT. THIS OCCURRED DURING THE AFTER TKOF CHKLIST. SUPPLEMENTAL INFO FROM ACN 332990: COCKPIT VERY BUSY ON CLBOUT. FO BUSY WITH MONITORING ACFT PERFORMANCE; ACFT ICING FOR WING AND ENG; COMPLETING AFTER TKOF CHKLIST; OBSERVING RADAR; AND HANDLING COMS. NO RESPONSE FROM CTLR REGARDING INCORRECT ALT READBACK. NO CORRECTION FROM DEP CTL ONLY REAFFIRMED THAT I HAD UNDERSTOOD THE CLRNC.