Non-adiabatic energy balance

In order for T to operate in a linear fashion. Equation 4.15 must be satisfied. Substituting Equation 4.15 into the expression provides a relation for the heating or cooling duty for linear mixing to be obeyed. 

Two beakers of mass mj = 1 kg and m2 = 1.5 kg are mixed together. The temperature of beaker 1 is Tj = 25 C, and the temperature of beaker 2 is T2 =45°C. What is the heating duty required in order for the final mixture temperature, T, to obey a linear mixing law  

Calculating the energy requirements is straightforward since all relevant information is given. The mixing fraction is determined from mj and m2 as follows  

Substituting these values into Equation 4.17 and solving for q then gives 

A negative value indicates a net release of energy. Therefore, 140.3 kJ of cooling is required per kg of mixture in order for temperature to obey a linear mixing law. 

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