Consider a horizontal flat plate $($of length $L)$ exposed to a liquid solvent stream in laminar flow parallel to the plate, with a free$-$stream velocity of U $.$ The plate is coated with a solute that dissolves slowly into the flowing solvent. The mass transfer in this system is driven by a concentration gradient resulting from the difference between the solute concentration at the surface $(C_{AS})$ and the solute concentration in the bulk $(C_A)$ over the concentration boundary layer near the surface. Suppose that as the solute diffuses into the liquid, it is also consumed by a first$-$order chemical reaction, with a reaction rate constant $k_r.$

$($a$)$Write the governing mass transfer equation and appropriate boundary conditions needed to completely describe the steady$-$state mass transport of the solute inside the concentration boundary layer.

$($b$)$By integrating both sides of the equation in part $($a$),$ derive the appropriate integral form of the concentration boundary layer equation.

$($c$)$Assume that the solvent has a very small Schmidt number and that the concentration distribution within the boundary layer can be reasonably approximated by a profile of the form:

$C_A=a+b*Cos(c*y+d)$

Where a$,b,c$ and $d$ are coefficients. Obtain an expression for the local mass transfer coefficient $($as a function of distance from the leading edge of the plate$).$

$($d$)$ For the case considered in part $($c$),$ derive an expression for the $($local$)$ mass transfer rate enhancement factor due to reaction.