A concentric tube $($parallel flow$)$ heat exchanger is built and operated as shown in the illustration be$-$

low. $$e purpose of the heat exchanger is to use heat from a hot$-$water waste stream to heat a stream

of cold water from the municipal supply. $$e cold tube$-$side flow $(0\mathrm{.}35$ kg$/$s$)$ passes through the entire

heat exchanger, while the hot shell$-$side flow $(0\mathrm{.}92$ kg$/$s$)$ enters at the middle and splits evenly

$(0\mathrm{.}46$ kg$/$s in each direction$).$e two halves of the heat exchanger are identical, except for the flow

direction on the shell side.

$$e cold inlet temperature is $8$C$,$ and the hot inlet temperature is $92$C$.$e hot outlet temperature

from the side with countercurrent flow is $73$C$.$ Use the heat capacity at the average of the hot inlet

and cold inlet temperatures throughout your calculations, for both the hot$-$ and cold$-$sides.

$($a$)$ Determine the cold$-$side temperature at the mid$-$point in the tube, by considering only the coun$-$

tercurrent side of the heat exchanger. Determine the log$-$mean temperature di$$erence on that side

and estimate the overall heat transfer coe$$cient for that side of the heat exchanger.

$($b$)$ Explain why you expect that the overall heat transfer coe$$cient for the other half of the heat ex$-$

changer will almost identical to the value above. $$en explain separately why the overall heat

transfer coe$$cient for the other half of the heat exchanger will be very slightly di$$erent$.$

$($c$)$ Assuming that the overall heat transfer coe$$cients are the same for both sides of the heat ex$-$

changer, estimate the outlet temperatures from the other half of the heat exchanger using the

NTU method $($on the second half of the heat exchanger$).$

$($d$)$ An outlet cold temperature of $40$C is desired, which can be achieved by altering the right$-$le$$

split of flow on the hot side. Consider the worst case for heat exchange with the same total flow

rates and inlet temperatures. Assuming that the overall heat transfer coe$$cient for each half does

not change: what is the minimum cold$-$side outlet temperature that can be achieved with such an

approach? Hint: using the NTU method again is the easiest approach.