# Team:Cornell/project/drylab/modeling/deployment

## Deployment

### Where should our biosensors be deployed and why does it matter?

One inspiration behind our S.A.F.E.B.E.T. is to fulfill the currently inadequate system to monitor the overall water quality of the Athabasca region. In addition to gaining a better picture of the overall water quality, the system should be cheap enough to be worth implementing. Therefore, choosing strategic locations for our biosensors is very important in order to keep the cost and complexity of our system to a minimum. Imagine for a moment that there is a river and in this river there is a spot where chemicals are seeping into the river from the soil. As these chemicals seep into the river, they are carried downstream via the bulk motion of the stream and spread across the width of the river through turbulent mixing. Taken together, bulk motion and turbulent mixing affect the spread and transport of chemicals as seen in Figure 1. If a biosensor is placed in the wrong part of a river, it may never even make contact with a chemical spilling into the river upstream. On the other hand, a biosensor can be strategically placed so as to maximize the likelihood of making contact with a chemical from an upstream spill. Using a tiny bit of luck and a lot of Fluid Mechanics, we must determine the optimal placement of S.A.F.E.B.E.T. devices to provide a comprehensive picture of the level of toxins in the Athabasca river basin.

### Theory

If S.A.F.E.B.E.T. is too far downstream from a chemical source then the chemical will be too dilute to detect. Therefore, S.A.F.E.B.E.T. needs to be at most some distance $L$ downstream of a chemical source in order to detect the chemical. So for a given river width $W$, how do we find the distance $L$ to place our biosensor? One way to determine $L$ is to go to the river in question, measure its height, width, length and volumetric flow rate and numerically solve for bulk motion. Unfortunately, the Alberta Oil Sands are roughly 4,000 km away from Cornell, so going to Alberta is out of the question. A simpler solution is to use dimensional analysis to estimate what the value of $L$ would be for each river that may be effected by Oil Sands operations. First, the time scale $\tau$ which is the time scale for bulk motion can be written as: $$\tau = \frac{L}{u}$$ where $u$ is the average velocity of the river and $L$ is the length scale for bulk motion. Second, a time scale for turbulent mixing can be written as: $$\tau= \frac{l^2}{D}$$ where $l$ is the length scale for mixing and $D$ is the turbulent mixing coefficient. Since bulk motion and turbulent mixing are happening at the same time, time scales can be set equal to each other allowing $L$ to be solved for: $$L= \frac{ul^2}{D}$$ The average velocity $u$ can be rewritten as: $$u = \frac{Q}{wh}$$ 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 mixing coefficient can be rewritten as: $$D=kh \sqrt{ghs}$$ 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 mixing across a river (horizontal) and turbulent mixing 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 mixing. Putting both of these definitions into our equation for $L$ gives us $$L= \frac{Ql^2}{kwh^2\sqrt{ghs}}$$ 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 ($L_Y$) 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 ($L_Z$) 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 $$L_Y= \frac{Qw}{0.6h^2\sqrt{ghs}}$$ $$L_Z= \frac{Q}{0.067w\sqrt{ghs}}$$

### Calculation of Biosensor Placement

Assuming the above set of equations are accurate, they can be used to determine how far downstream from a chemical source S.A.F.E.B.E.T. need to be placed in order to detect the chemical.Every river has different parameters to contribute to the equations - height, width, slope and the volumetric flow rate. The Alberta Government [2] and the Regional Monitoring Aquatics Program (RAMP) [3] have been conducting hydrologic tests on the Alberta Oil Sands for over a decade. Thanks to their efferts, calculating LY and LZ for several rivers in the Alberta Oil Sands has been made possible. The placement of these biosensors can be seen on the map below. In addition to these biosensors we’ve also marked on the map potential locations for our biosensors for rivers where we couldn’t get data on but are important nonetheless for quantifying the release of toxins into the Athabasca Oil Sands. These sites were chosen based on their proximity to tailing ponds and their similarity to the rivers for which we have data on. The determined locations are marked in green and the potential locations are marked in red.

$\mathrm{Athabasca\ River}$$\mathrm{Tar\ River}$$\mathrm{Steepbank\ River}$$\mathrm{Muskeg\ River}$$\mathrm{Jackpine\ Creek}$
$L_Y$$71.233 \ \mathrm{m} 1.683 \cdot 10^{-4} \ \mathrm{m}$$0.1639 \ \mathrm{m}$$0.0091 \ \mathrm{m}$$12.2495 \ \mathrm{m}$
$L_Z$$198.621 \ \mathrm{m} 0.2893 \ \mathrm{m}$$22.1028 \ \mathrm{m}$$9.5927 \ \mathrm{m}$$1.582 \cdot 10^{-4} \ \mathrm{m}$