Steam is produced in the copper tubes $($k $=223$ Btu$/$h $$ ft $\backslash$deg F$)$ of a heat exchanger at a temperature of $280\backslash$deg F by another fluid condensing on the outside surfaces of the tubes at $350\backslash$deg F$.$ The inner and outer diameters of the tube are $1$ in$.$ and $1\mathrm{.}3$ in$.,$ respectively. When the heat exchanger was new, the rate of heat transfer per foot length of the tube was $2$x${10}^4$ Btu$/$h$.$

$($a$)$ Determine the rate of heat transfer per foot length of the tube when a $0\mathrm{.}01-$in$.-$thick layer of limestone $($k $=1\mathrm{.}7$ Btu$/$h$$ft$\backslash$deg F$)$ has formed on the inner surface of the tube after extended use.