Problem $5$

In a soap manufacturing process, a cold die initially at $\backslash (-20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ compresses soap material initially at $\backslash (+20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ After a one second contact time, the die is lifted and a conveyor brings a new piece $($at $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ))$ under the die. This process is repeated continuously. The period between consecutive operations is $4$ seconds $($Dwell or Contact time: $1$ sec $,$ Period for $\backslash (1\backslash$mathrm$\{$cycle$\}$: $4\backslash$mathrm$\{$sec$\}\backslash )).$ The die is cooled $($by flowing coolant internally$)$ so that before each contact the die temperature can be assumed to be uniform at $\backslash (-20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The die surface $($which comes in contact with soap$)$ is $\backslash (8\backslash$mathrm$\{$~cm$\}\backslash$times $8\backslash$mathrm$\{$~cm$\}\backslash ).$

Model the problem appropriately and calculate cooling requirements $($the extracted energy to maintain the die temperature at $\backslash (-20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ 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 $\backslash$& soap$).$

Heat and mass transfer