Problem $5$

In a soap manufacturing process, a cold die initially at $-20C$ compresses soap material

initially at $+20C.$ After a one second contact time, the die is lifted and a conveyor brings a

new piece $($at $20C)$ under the die. This process is repeated continuously. The period between

consecutive operations is $4$ seconds $($Dwell or Contact time: $1$ sec $,$ Period for $1$cycle:$4sec).$

The die is cooled $($by flowing coolant internally$)$ so that before each contact the die

temperature can be assumed to be uniform at $-20C.$ The die surface $($which comes in contact

with soap$)$ is $8cm8cm.$

Model the problem appropriately and calculate cooling requirements $($the extracted energy to

maintain the die temperature at $-20C$ for $1$ sec $)$ for the die $($neglecting all other thermal

interactions except contact with the soap$).$

Assume that the respective penetration depths during contact time are smaller than the height

of the corresponding pieces $($die & soap$).$