$1[35$ points$].$ Consider a microchannel heat sink of base dimensions, length $\backslash ($ W $\backslash )$ and width $\backslash ($ W $\backslash ),$ to handle a heat removal rate of $\backslash ($ q$^\{\backslash$prime $\backslash$prime$\}\backslash$mathrm$\{$W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$ Due to high pressure drops encountered in microchannel flows, a relatively large pump is needed to drive fluid through microchannels. A pump is usually characterized in terms of the flow rate and pressure head it can provide.

Assume that the heat sink material, made of copper, has high thermal conductivity $(\backslash ($ k$\_\{$c p$\}\backslash )),$ and the temperature of the fluid entering the microchannels is $\backslash ($ T $\backslash ).$ Assume that all necessary fluid properties are known. The aspect ratio of the microchannels $($height of each microchannel to its width$)$ is equal to one $($i$.$e$.,$ square channels, $\backslash ($ w $/$ h$=1\backslash )).$ The width of the walls separating the microchannels is $\backslash (\backslash$delta $\backslash ).$ The number of microchannels is $\backslash ($ N $\backslash ).$ You can neglect entrance and exit pressure losses in the microchannels. Treat the internal flow as being fully developed and laminar. State all your assumptions clearly.

Consider conditions for which equivalent heat$-$generating systems $($see figure below$)$ are attached to the top and bottom of the cold plate.

a$)$ Sketch the temperature profile along the sections A$-$A and B$-$B$,$ labeling key features.

b$)$ Draw the thermal resistance network between the fluid and the location where copper experiences the highest temperature.

c$)$ Develop expression to determine how the volumetric flow rate $\backslash (\backslash$boldsymbol$\{$Q$\}\backslash )($total through all microchannels$)$ and pressure head $\backslash (\backslash$Delta $\backslash$boldsymbol$\{$p$\}\backslash )$ that the pump needs to handle the required heat removal rate. Consider a heat sink with perfectly insulating cover plate $($i$.$e$.,$ thermal conductivity of cover plate is negligibly small$).$ Thickness of the cover plate is $\backslash (($H$-$h$)/2\backslash ).$

d$)$ Sketch the temperature profile along the sections A$-$A and B$-$B$,$ labeling key features.

e$)$ Draw the thermal resistance network between the fluid and the location where copper experiences the highest temperature.

f$)$ Develop expression to determine how the volumetric flow rate $\backslash (\backslash$boldsymbol$\{$Q$\}\backslash )($total through all microchannels$)$ and pressure head $\backslash (\backslash$Delta $\backslash$boldsymbol$\{$p$\}\backslash )$ that the pump needs to handle the required heat removal rate.