Consider a single story $($one floor$),30$ ft x $30$ ft house with $8$ ft high walls.$$ The house contains $6$ double pane windows.$$ The roof and floor are well insulated.$$ The walls are standard construction with $(1)/(2)$ sheetrock, R$19$ insultation in a $2$ x $6$ wall, and $(1)/(2)$ plywood.$$ On a cold, windy winter night the outside air temperature is $0$ oF and the inside air temperature is $65$ oF$.$

When the inside air temperature drops to $60$ oF$,$ the heating system turns on$.$ The house is heated with baseboard hot water heaters.$$ The supply temperature of the hot water is $150$ oF$.$ The water flows through a $$ diameter copper pipe with Aluminum fins and the rate of heat loss to the surroundings is $400$ BTU$/$hr per linear foot of baseboard heater at a volume flow rate of $4$ GPM$.$

$1.$ Compute the rate of heat loss from the air inside to the outside.$$ Consider only conduction and convection.$$ Make assumptions as necessary.

$2.$ Compute the ratio of heat loss through the walls to that through the windows.

$3.$ Determine the linear feet of baseboard heaters required to raise the inside air temperature to $70$ oF in $30$ minutes

$4.$ Determine the temperature on the outside of the copper pipe.

$5.$ Determine the temperature of the water exiting the baseboard heater.

$6.$ Conceptual Questions

$-$Discuss ways in which the rate of heat loss to the outside air can be reduced.

$-$Discuss the mechanisms of heat transfer occurring with the baseboard heating system

$-$Discuss ways in which the heat transfer rate per linear foot could be increased if necessary.

Note: If additional information is needed, please make a reasonable engineering assumption and document all assumptions you make