A long steel rod $($k$=16$ W$/($m$.$K$)$; a$=1\mathrm{.}1$x$10^-^5$m$^2/$s$),$ with a diameter of $15$ cm$,$ is initially at a temperature of $225\backslash$deg C$.$ It is then exposed to air at a temperature of $25\backslash$deg C and h$=50$ W$/($m$^2$K$).$

Determine the temperature of the rod shaft and its surface $2$ hours after the start of the cooling process.

Problem $2$

A flow of water inside a copper tube measuring $16$ mm and $1$ mm thick has an average speed of $0\mathrm{.}15$ m$/$s$.$

A heat flux of $1000$ W$/$m$^2$ is applied to the outer surface of the tube, with the aim of heating incoming water to $15\backslash$deg C$.$

a$)$ Knowing that the tube is $2$ m long, what is the leaving temperature of the water?

b$)$ What is the temperature of the inner wall of the tube in the inlet and outlet sections?

Problem $3$

A flat horizontal plate $2$ m high and $2\mathrm{.}5$ m wide receives a flux of solar radiation, absorbing $750$ W$/$m$^2,$ at the same time as it is exposed to air at $25\backslash$deg C$.$ On the other side, the plate is thermally insulated.

a$)$ Determine the heat transmission coefficient by convection from the plate to the air. At what temperature is the surface of the plate?

b$)$ What is the heat flow that the plate loses to the air?

Problem $4$

In a simple equicurrent heat exchanger, the hot fluid has a heat capacity of $2000$ W$/$K$,$ entering the exchanger at $80\backslash$deg C$,$ while the cold fluid also has a heat capacity of $2000$ W$/$K and enters the exchanger at $10\backslash$deg C$.$

The heat transmission coefficient of the exchanger is $500$ W$/($m$^2$K$)$ and an area of $5$ m$^2.$

a$)$ Determine the exit temperatures of the two fluids.

b$)$ Determine the calorific power exchanged in the exchanger.

Problem $5$

With the same data as the previous problem, now consider that the heat exchanger would have a simple countercurrent arrangement. What would be the exchange area necessary for the efficiency to be equal to that obtained in the previous problem, assuming that the global heat transmission coefficient is also the same as in the previous problem?