Dimensional analysis, consider the flow system described in the first paragraph of S14.3, for which dimensional analysis has already given the dimensionless velocity profile (Eq. 6.2-7) and temperature profile (Eq. 14.3-9). 

(a) Use Eqs. 6.2-7 and 14.3-9 and the definition of cup mixing temperature to get the time-averaged expression. 

(b) Use the result just obtained and the definitions of the heat transfer coefficients to derive Eqs. 14.3-12, 13, and 14