$1.(25)$ Water needs to be heated at the Easy Spray Car Wash. The water flows through a tube $($assume drawn tubing$)1$ cm in inner diameter at a volume flow rate of $4$ liters per minute. The tube $(1\mathrm{.}4$ cm outer diameter$)$ is heated on the outside by a natural gas heater that provides a constant heat flux of $\backslash (50\backslash$mathrm$\{$~kW$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash )$ on the outside wall.

a$.$ How long $($ L $)$ does the tube need to be to heat the water from $15$ C to $45$ C $?($Note that in situations involving heating or cooling of fluids in tubes the tubes are often coiled or otherwise bent to minimize volume occupied by the heater, which changes the heat transfer some, but we will neglect that effect here.

Assume a straight tube.$)$

b$.$ How long is the entry length compared to the overall tube length in this case?

c$.$ Plot the tube wall temperature from $0$ to L $($Neglect the entrance region based on what you found in part b$.)$

d$.$ What is the pressure drop over the length of this tube?

e$.$ Theoretically, how much power is needed by the pump? $($this will set a minimum pump size$)$

f$.$ Comment on the overall energy balance of the Easy Spray, comparing how much power is used to heat the water compared to how much is used to pump it$.$