91Ó°ÊÓ

In this step, need to understand that the problem is basically about calculating the flow rate of the wort being siphoned from one tank to another and then graphing the relationship between the flow rate and the height difference. Torricelli's law, which states that the speed v of efflux of a fluid under the force of gravity from a hole in a large, open tank is \(v = \sqrt{2gh}\), where g is acceleration due to gravity and h is the height of fluid above the hole, is applied here as the principle. Since the hose has a circular cross section, the flow rate Q can be calculated by multiplying the speed of efflux \(v\) and the cross-sectional area \(A\) of the hole (which is the ID tubing in this case). So, \(Q=vA\).

The cross-sectional area of the hose is calculated using the formula for area of a circle, which is \(\pi r^2\), where r is the radius. As the diameter is given, the radius is obtained by dividing the diameter by 2. Hence, \( r = \frac{5}{2} = 2.5 mm = 0.0025 m \). Simplifying further, the expression for \(Q = \pi r^2 \sqrt{2gh}\) is obtained.

For the graph plotting, the unit of flow rate should be in liters per minute while the unit of \(h\) should be in millimeters. The standard units in Torricelli's theorem are m/s for velocity, m for height. So, multiply the formula by a conversion factor to match the desired unit. The conversion factor between cubic meters and liters is 1000 and between seconds and minutes is 60. Hence, the formula becomes \(Q =1000 *60 * \pi r^2 \sqrt{2gh}\). With this formula, a graph can be plotted between \(Q\) and \(h\).

Now, the above formula should be used to solve for \(h\) when given \(Q= 2L/min\). The gravitational constant \(g = 9.81 m/s^2 \). By substituting the known values in the formula, the value of \(h\) can be derived.