Problem $3$ CL: $30/$Others:optional After successfully modeling the heat loss in a pipe transporting

saturated steam $($in class$),$ you found out that the expected heat loss in the winter is very high. This will

imply the need of producing more steam in the plant with the corresponding increase in fuel

consumption $($in the boilers$).$ Your next task is to propose a solution for this issue by considering

insulating the pipe. Your manager is extremely interested in hearing about it since there are many other

pipes with similar issues and the economic loss that is expected is SIGNIFICANT.

i$)$ Develop a mathematical formulation for this problem. In other words:

a$.$ Find a suitable objective function $($i$.$e$.$ what do you want to minimize or maximize$).$

b$.$ Generate a model that will capture the relationships between key variables.

ii$)$ For mechanical reasons the thickness of the insulation should not be greater than $T{h}_{\text{m}\text{a}\text{x}}.$

How can you add this information inside your formulation?

iii$)$ Does the heat loss depend on the steam flow? Why? If it does not, under what conditions

may the steam flow variable become relevant? $($Mention at least one$)$

iv$)$ Any increase in the fuel consumption implies an increase in the CO$2$ emissions. How does

the objective function change if you want to penalize these emissions?

Considerations:

Please use symbols to represent any parameter that you need in the model $($e$.$g$.$ insulator price

$(p_{IN}),$ conductivity $(k_{IN}),$ etc.$).$

Assume the cost of the insulator per unit of time to be the total cost of the insulator divided by

its lifespan $(t).$

The cost of producing the saturated steam $($given in US$$/$kW$)$ is $S_c$

Any equation in the formulation that does not affect the solution space should be removed

The CO$2$ penalization is given in US$$/$ton of CO$2$ emitted.

Hints:

Heat loss from radiation is not significant if the external temperature of the insulating material is

close to ambient temperature.

The energy carried by the extra steam generated is equal to the heat lost in the pipe.