Math SL INTERNAL ASSESMENT
Funnel and Various Inflow Rates For many years, we have been using a wide conical shape that has a narrow stem called a Funnel for various purposes. Antonio Berlese invented the funnel. The funnel originally was used to extract insects from the soil and microorganisms from litters of a leaf. 1 In this day and age, a funnel can be used in various ways such as it can be used as a Filter funnel, a thistle funnels, a dropping funnel, a filter funnel or a Büchner funnel. My fascination with funnels was first sparked in our chemistry class when we were told to pour a chemical from one container to another. While I was doing the experiment, it made me think about the inflow rate at which the liquid flows into the funnel and the outflow rate at which liquid flows out of a funnel and the factors that affect it. Therefore leading to this exploration about the water level in a funnel with variable inflow rates. To investigate the water level in a funnel with variable inflow rate differential equation is formulated for the water level as a function of time. I used Maxima 2 to solve the differential equation for various inflow rates and plot the corresponding water level. Cogitate a funnel in the shape of a frustum of a circular cone with a lower radius a, upper radius b and height H, as shown in Figure 1. Water flows in at a volume rate of
1
Q(t) and
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.
leaves at a volume rate of q. The dependence of Q (t) on time t is prescribed. The outflow rate q depends on the height h of water in the funnel. It is found that the flow rates Q(t) and q are not equal in general. Consequently, the level of water, h(t), in the funnel varies with
time. Therefore, the goal of this exploration is to determine h as a function of time. Q(t)
b
Q(t)
A(t)
H h(t)
Figure 1 Water flow
a q(h)
If this topic is mathematically scrutinized then to find the water level in the funnel, we need to determine the funnel’s outflow rate. According to Torricelli's Law 3states, that speed at
which fluid is ejected from a hole is given by
where h is the height of the fluid above
the hole and g is the acceleration of gravity. Because if Torricelli's law ignores the fluid's viscosity. It gives reasonable estimates for outflow velocity for low viscosity fluids such as water or gasoline but it is not applicable at all to high viscosity fluids, such as honey or molasses. Think of identical barrels of water and molasses with holes punched in their bottoms. We don’t expect equal rates of flow, right.
In this funnel, the area of the exit hole is
(1)
therefore:
3
"Torricelli's Law." Boundless. N.p., n.d. Web. 01 Apr. 2014.
its-applications/bernoulli-s-equation/torricelli-s-law/>.
Let V (t) is the volume of the water in the funnel at time t. The rate of change of V(t) equals the rate of inflow minus the rate of outflow:
Its already known that q(h) is related to h. The next objective is to relate dV/dt to h. From
where A(h) is the funnel's horizontal cross and the previous equations changes to: sectional area at level h. Therefore calculus we are aware that
(2)
It remains Obtain an expression for A(h). Toward this end, let r(h) be radius of the funnel's horizontal cross-sectional at height h above the bottom. In Figure 2, we see a vertical cross- section of the funnel. The “ edge" of the funnel is signified by the slanted line that connects the points with coordinates (a,0) and (b,H). The equation of that line is:
y (b,H) Figure 2: To determine the relationship between r and h. The points (b,H) and (a,0) lie on the top and bottom rims. For an arbitrary point
(r,h)
with coordinates (r,h) as shown, we wish to determine the relationship between r and h
(a,0)
x
To find the radius at level r, we plug in y=h and solve for x which equals the desired r(h). It can be seen that:
(3)
It can be noted that this equation implies that r(0)=a and r(H)=b , as expected. For example:
If Q(t) =0 if plotted on h versus t which is corresponding to the constant input rates. 10.0 7.5
h
5.0
2.5 0.0 20 t
0
40
IF Q(t) =6 is plotted on h versus t which is corresponding to the constant input
rates. The graph would be something like this:
10.0 7.5 h
5.0
2.5 0.0 0
100 t
200
If Q(t) =10 is plotted on h versus t which is corresponding to the constant input rates the graph would look something like this:
10.0
7.5
h
5.0
2.5
0.0 200 t
0
In an overview of equation (1), (3) and
400
, equation (2) reduces to a first order
differential equation in the unknown h(t). Solving this equation in MAXIMA we see what we are dealing with: (4)
This needs to solved with the initial condition where
What happens when there is no incoming flow, that is Q (t) is identically zero, the differential equation (2) is separable and can be solved by hand. For nonzero inflow, (2) is not solvable in terms of elementary functions. We use Maxima’s numerical solver to compute and plot solutions. For example, we use the following values: a = 1, b = 7, H = 10, g = 1
and it is assumed that the funnel is full to the brim at t = 0. In Figure 3 the graphs of h(t) for three different choices of Q(t). The leftmost graph corresponds to no inflow. It is also observed that the water level drops to zero in a finite time. The two other graphs correspond to constant inflow rates of Q = 6 and Q = 10. And in each case, the water level stabilizes to a fixed level after sufficiently long wait.
Figure 3: h(t) for three different choices of Q(t) This makes us question that: What determines the stabilization level? That is, what is the horizontal asymptote of h (t) for a constant inflow rate of
?
The answer can be found by simple algebra. To see this, it is noted that h(t) approaches its asymptote, its slope which is
,goes to zero. Then by looking at the different equation in it
(4) it is clear that the left hand side goes to zero, therefore the right hand size will have to go to zero too. Setting the right hand size to zero and solving for h we get the asymptotic value,
: ()
For example, for example, for the previously used parameter values a = 1, g = 1 and Q0 = 10, we get h1 = 5:066 which agrees with the asymptotic limit of the rightmost graph in Figure 3. To find the critical flow rate that keeps the funnel filled to the brim, take a flow rate, which equals that critical rate in the time interval graphs of
Q
and drops to thereafter. The
and the corresponding are shown in Figure 6.
10.0
10.0
7.5
7.5 h
5.0
5.0
2.5
2.5
0.0
0.0 200 t
0
200 t
0
400
400
Figure 4: On the left is the in ow rate versus time. On the right is the corresponding water level, h(t), versus time.
20
10.0
15
7.5 h
10 Q
5.0
5.0
2.5
0.0
0.0 0
200 t
400
0
500 t
1000
Figure 5: On the left is the inflow rate versus time. On the right is the corresponding water level, h(t), versus time.
Therefore, the critical flow rate as the constant rate of inflow that keeps the funnel full to the brim without overflowing.
