$4\mathrm{.}34.$ The indicator$-$dilution method is a technique used to determine flow rates of fluids in channels for which devices like rotameters and orifice meters cannot be used $($e$.$g$.,$ rivers, blood vessels, and largediameter pipelines$).$ A stream of an easily measured substance $($the tracer$)$ is injected into the channel at a known rate, and the tracer concentration is measured at a point far enough downstream of the injection point for the tracer to be completely mixed with the flowing fluid. The larger the flow rate of the fluid, the lower the tracer concentration at the measurement point.

A gas stream that contains $1\mathrm{.}50$mole$\%C{O}_2$ flows through a pipeline. Twenty $(20\mathrm{.}0)$ kilograms of $C{O}_2$ per minute is injected into the line. A sample of the gas is drawn from a point in the line $150$ meters downstream of the injection point and found to contain $2\mathrm{.}3$ mole $\%C{O}_2.$

$($a$)$ Estimate the gas flow rate $(kmo\frac{l}{m}in)$ upstream of the injection point.

$($b$)$ Eighteen seconds elapse from the instant the additional $C{O}_2$ is first injected to the time the $C{O}_2$ concentration at the measurement point begins to rise. Assuming that the tracer travels at the average velocity of the gas in the pipeline $($i$.$e$.,$ neglecting diffusion of $C{O}_2),$ estimate the average velocity $(\frac{m}{s}).$ If the molar gas density is $0\mathrm{.}123$kmo$\frac{l}{m^3},$ what is the pipe diameter?