21. teaching apparatus for
teaching the
principles of fluid
flow and integration science and mathematics in terms of fluid
flow comprising one or more funnels having enclosed sidewalls and an
inlet and
outlet at respective
upper and lower
ends, the
mathematical equations for the sidewalls being
functions of the
distance along a
vertical axis of the funnel and being such that when the equations are plugged into the formula for
computing the
time it takes liquid to
flow out of the funnel, the resultant
function is
integration capable of being analytically or numerically integrated, the funnel having a
predetermined and
known mathematical configuration and
known shape and
flow characteristics for a given liquid such that
physical measurement of the
time it takes for the liquid to
flow from the funnel can be substantially duplicated by inserting the equation for the funnel sidewalls into the formula for theoretically calculating
the time it takes for the funnel to empty a flow characteristic of said funnel as said funnel empties and
integrating performing mathematical operations thereon; the apparatus further including support means for supporting the funnel in a
stable position, with the
axis of the funnel maintained in a generally
vertical position.
1. A method for
teaching physical principles of fluid
flow and
mathematical principles of integration by
concrete, measurable examples comprising the steps of: supporting a funnel of
predetermined interior configuration in
position to receive liquid in an
open upper end and
discharge the liquid through an
outlet orifice in a lower
end, the funnel having sidewalls with
interior surfaces that conform with
mathematical formulas as a
function of the
distance along the funnel
axis that, when inserted in the formula for calculating the
time it takes for the funnel to empty, provides a
function capable of being integrated
either analytically or numerically;
filling the funnel with a liquid to a preselected level in the funnel; and
measuring the time it takes for all a characteristic of the liquid to flow as said liquid flows out of the funnel outlet orifice, the measured time characteristic value being comparable with a time theoretical characteristic value obtainable theoretically by integrating the equation for the theoretical time it takes the funnel to empty by mathematical formulation, a comparison of the measured and theoretical values using mathematical integration calculation providing a concrete experience that facilitates learning the principles of fluid flow, science and integration mathematical operations in terms of abstract symbols.
2. A method according to
claim 1 wherein the funnel comprises a closed containment vessel including enclosed perimetric sidewalls and a base with an
outlet orifice covering the lower
end of the sidewalls, the base restricting streamlined
flow through the
orifice and maintaining similar
characteristics of
outlet flow in funnels having different sidewall
configurations.
3. A method according to
claim 1 wherein the funnel has a rectangular horizontal cross-sectional shape and four sidewalls, three sidewalls being
vertical and the fourth sidewall having a curvilinear shape, thereby presenting a simplistic problem for solution by beginning students.
4. A method according to
claim 1 wherein the inner surfaces of the funnel correspond to any of the
mathematical formulas x=f
u (y), where y is a
vertical axis of the funnel, x is the horizontal
distance of the inner surface from the y
axis for any specific value of y, and f
u (y) is one or more of the
functions of y selected from the twenty-eight
functions set forth in the present specification.
5. A method according to
claim 1 wherein the funnel comprises a polygonal cross-section and is formed of a plurality of interconnected sidewalls, each sidewall having an inner surface formed in accordance with the right cylindrical projection of a graph whose
function, when plugged into the formula for
computing the
time it takes liquid to
flow out of the funnel,
yields an capable of yielding an analytically or numerically integrable
function.
6. A method according to
claim 5 wherein the funnel comprises a rectangular cross section with four curvilinear sidewalls.
7. A method according to
claim 1 wherein the funnel has a circular horizontal cross-section, with the
configuration of the sidewalls being formed by rotating about the
axis of the funnel a graph corresponding with a formula that can be plugged into the equation for calculating the
time it takes for the funnel to empty and
yield an capable of yielding an analytically or numerically integrable equation.
8. A method according to
claims 5 or 7 wherein the funnel comprises a horizontal juxtaposition of at least one each of the funnel sections of claims 5 or 7, with the equation for each section separately being
integrable capable of analytically or numerically integrated in the formula for calculating the
time for emptying the funnel of liquid.
9. A method according to
claim 7 wherein the funnel is formed of a horizontal juxtaposition of one or more curvilineal sections, with the equation for each section separately being
capable of being analytically or numerically integrable in the formula for calculating the
time for emptying the funnel of liquid.
10. A method according to
claim 1 wherein the funnel is formed of one or more
vertical levels joined along horizontal seams, with the equations for the sections of each
level being
capable of being analytically or numerically integrable in the formula for calculating the
time for emptying the funnel of liquid.
11. A method according to
claim 1 wherein a funnel includes one or more occlusions in its
interior, with the occlusions being formed of any combination of shapes that correspond to equations that, when plugged into the
time equation, yield an
integrable integrand
capable of being analytically or numerically integrated.
12. A method according to
claim 1 wherein the
orifice of the funnel contains an iris closure capable of variating the effective size of the
orifice.
13. A method according to
claim 1 wherein the fluid medium draining from the funnel is any liquid, any solution of a liquid and a solute, any solution of liquids, or any number of immiscible liquids.
14. A method according to
claim 1 wherein a number of funnels are arranged in series with the efflux of one funnel draining into the next.
15. A method according to
claim 1 wherein the
outlet orifice is positioned at a
predetermined distance from the sidewalls of the funnel, with said
distance being such as to maximize the correlation between the measured and mathematically calculated
times for the funnel to empty.
16. A method according to
claim 1 wherein the
outlet orifice is contoured so as to maximize the correlation between the measured and mathematically calculated
times for the funnel to empty.
17. A method according to
claim 1 wherein the method employs a plurality of funnels of varying shapes and support means for supporting each funnel in a
stable,
vertical position while demonstrating the
time it takes liquid to
flow from the funnel.
18. A method according to
claim 1 wherein the funnel includes a base enclosing the bottom
end of the funnel, an
outlet orifice of
predetermined size being formed in the base, the base restricting streamlined
flow through the
orifice and maintaining similar
characteristics of
outlet flow in funnels having different sidewall
configurations.
19. A method according to
claim 1 wherein the area of the
outlet orifice together with the
configuration of the funnel are such that when the funnel is filled with liquid to its
predetermined depth, the
time for the funnel to empty by the liquid flowing through the
orifice under gravity is about ten (10) to sixty (60) seconds.
20. A method according to
claim 1 and further comprising closing the
upper end of the funnel after filling the funnel and applying pressure other than atmospheric pressure to the
upper surface of the liquid so as to vary the conditions of the fluid
flow.
22. teaching apparatus according to
claim 21 wherein the equations for the
interior surfaces of the sidewalls at each point in the sidewalls about a funnel
axis conform with one or more of the twenty-eight
functions set forth in the present specification.
23. teaching apparatus according to
claim 21 wherein the funnel comprises a polygonal cross-section and is formed of a plurality of interconnected sidewalls, each sidewall having an inner surface formed in accordance with the right cylindrical projection of a graph whose
function, when plugged into the formula for
computing the time it takes liquid to flow out of the funnel said flow characteristic yields an
integrable function capable of being analytically or numerically integrated.
24. teaching apparatus according to
claim 23 wherein the funnel comprises a rectangular cross-section with four curvilinear sidewalls.
25. teaching apparatus according to
claim 23 wherein the funnel comprises a rectangular cross section and has three
vertical side sidewalls and one curvilinear sidewall, thereby presenting a simplistic problem for solution by beginning students.
26. teaching apparatus according to
claim 21 wherein the funnel has a circular horizontal cross-section, with the
configuration of the sidewalls being formed by rotating about an
axis of the funnel a graph corresponding with a formula that can be plugged into the equation for calculating
the time it takes for the funnel to empty said flow characteristic and yield an
integrable equation
capable of being analytically or numerically integrated.
27. teaching apparatus according to
claim 21 wherein the inner surfaces of the funnel correspond to any of the
mathematical formulas x=f
u (y), where y is any
vertical axis of the funnel, x is the horizontal
distance of a surface from the y
axis for any specific value of y, and f
u (y) is one or more of the
functions of y selected from the twenty-eight
functions set forth in the present specification.
28. teaching apparatus according to
claims 23 or 26 wherein the funnel is formed of a horizontal juxtaposition of at least one each of the funnel sections of claims 23 or 26, with the equation for each section separately being integrable in the formula for calculating the
time for emptying the funnel of liquid.
29. teaching apparatus according to
claim 26 wherein the funnel is formed of a horizontal juxtaposition of one or more curvilineal sections, with the equation for each section separately being
integral capable of being analytically or numerically integrated in the formula for calculating
the time for emptying the funnel of liquid said flow characteristic.
30. teaching apparatus according to
claim 21 wherein the funnel is formed of one or more
vertical levels joined along horizontal seams, with the equations for the sections of each
level being
integrable capable of being analytically or numerically integrated in the formula for calculating
the time for emptying the funnel of liquid said flow characteristic.
31. teaching apparatus according to
claim 21 wherein the apparatus includes a plurality of funnels of varying shapes and support means for supporting each funnel in a
stable,
vertical position while demonstrating the
time it takes liquid to
flow from the funnel.
32. teaching apparatus according to
claim 21 wherein the funnel includes a base enclosing the bottom
end of the funnel, an
outlet orifice of
predetermined size being formed in the base, the base restricting streamlined
flow through the
orifice and maintaining similar
characteristics of
outlet flow in funnels having different sidewall
configurations.
33. teaching apparatus according to
claim 32 wherein the
orifice is positioned a
predetermined distance from the sidewalls of the funnel, with the
distance being selected so as to maximize the correlation between the measured and mathematically calculated
times for the funnel to empty.
34. teaching apparatus according to
claim 21 wherein the area of the
outlet orifice together with the
configuration of the funnel are such that when the funnel is filled with liquid to its
predetermined depth, the
time for the funnel to empty by the liquid flowing through the
orifice under gravity is about ten (10) to sixty (60) seconds.
35. teaching apparatus according to
claim 21 wherein a funnel includes any number of occlusions in its
interior, with the occlusions being formed of any combination of shapes that correspond to equations that, when plugged into the
time equation
for calculating said flow characteristic, yield an
integrable integrand
capable of being analytically or numerically integrated.
36. teaching apparatus according to
claim 21 wherein the
orifice of the funnel contains an iris closure capable of variating the effective size of the
orifice.
37. teaching apparatus according to
claim 21 wherein the fluid medium draining from the funnel is any liquid, any solution of a liquid and a solute, any solution of liquids, or any number of immiscible liquids.
38. teaching apparatus according to
claim 21 wherein a number of funnels are arranged in series with the efflux of one funnel draining into the next.
39. teaching apparatus according to
claim 21 wherein the
outlet orifice is positioned at a
predetermined distance from the sidewalls of the funnel, with said
distance being such as to maximize the correlation between the measured and mathematically calculated
times for the funnel to empty.
40. teaching apparatus according to
claim 21 wherein the
outlet orifice is contoured so as to maximize the correlation between the measured and mathematically calculated
times for the funnel to empty.
41. teaching apparatus according to
claim 21 and further comprising means for closing the
open upper end of the funnel after filling the funnel; and means for applying pressure other than atmospheric pressure to the
upper surface of the liquid so as to vary the conditions of the liquid
flow.
42. A method according to
claim 1 wherein said supporting step includes supporting a funnel having sidewalls configured such that said
function provided is capable of being analytically or numerically integrated, said
flow characteristic measuring step includes the measured
flow characteristic value being comparable with a
flow characteristic value obtainable theoretically by manipulating the equation for the theroetical
time it takes the funnel to empty.
43. A method according to claim 4 wherein said supporting step includes supporting a funnel having sidewalls configured such that said function provided is capable of being analytically or numerically integrated, said flow characteristic measuring step includes the measured flow characteristic value being comparable with a flow characteristic value obtainable theoretically by manipulating the equation for the theoretical time it takes the funnel to empty. 44. An apparatus according to claim 21 wherein said funnel is configured such that said resultant function is analytically or numerically integrable and the physical measurement of said flow characteristic can be substantially duplicated by inserting the equation for the funnel sidewalls into the formula for theoretically calculating said flow characteristic and solving said calculation. 45. An apparatus according to claim 22 wherein said funnel is configured such that said resultant function is analytically or numerically integrable and the physical measurement of said flow characteristic can be substantially duplicated by inserting the equation for the funnel sidewalls into the formula for theoretically calculating said flow characteristic and solving said calculation. 46. An apparatus according to claim 27 wherein said funnel is configured such that said resultant function is analytically or numerically integrable and the physical measurement of said flow characteristic can be substantially duplicated by inserting the equation for the funnel sidewalls into the formula for theoretically calculating said flow characteristic and solving said calculation.