Team:Cornell/testing/project/drylab/4

Mathematical Modeling

Theory

So for a given river width W, how do we find the distance L to place our biosensor? One way to do that would be to go to the river in question, measure its height, width, length and volumetric flow rate and numerically solve the advection diffusion equation. However the Alberta Oil Sands are roughly 4,000 km away from Cornell so it was impossible for us to go there and measure the rivers. Instead we decided to use dimensional analysis to estimate what the value of L would be for different rivers. To do this we first defined a time scale τ which is the time scale for advection. τ can be written as τ= L/u where u is the average velocity of the river and L is the length scale for advection. Similarly we defined a time scale for turbulent diffusion which can be written as τ= l^2/D where l is the length scale for turbulent diffusion and D is the turbulent diffusion coefficient. Since advection and diffusion are happening at the same time we can set both time scales equal to each other and solve for L which yields L= 〖u*l〗^2/D The average velocity u can be rewritten as u=Q/(w*h) where Q is the volumetric flow rate of the river, w is the width of the river and h is the depth of the river. Next, the turbulent diffusion coefficient can be rewritten as D=k*h*√(g*h*s) where s is the slope of the river, g is the acceleration due to gravity and k is an experimentally determined constant. This constant differs for turbulent diffusion across a river (horizontal) and turbulent diffusion within a river (vertical). Since we want to place our biosensor at the point where the chemical is completely mixed in the river, we need to take into account both horizontal and vertical turbulent diffusion. Putting both of these definitions into our equation for L gives us L= (Q*l^2)/(k*w*h^2*√(g*h*s)) Finally, we need to define what k and l are. If we want to know how far downstream the chemicals will be mixed horizontally across the river (LY) then l = w and k = 0.6 [1]. If we want to know how far downstream the chemicals will be mixed vertically in the river (LZ) then l = h and k = 0.067 [1]. The larger of these L values will determine where we need to place our biosensor for a giver river. This gives us the following two equations L_Y= (Q*w)/(0.6*h^2*√(g*h*s)) L_Z= Q/(0.067*w*√(g*h*s))