Funnel teaching method and apparatus

Abstract

A method and apparatus for teaching physical principles of fluid flow and .[.the.]. mathematical principles .[.of.]..Iadd., such as .Iaddend.integration.Iadd., .Iaddend.by concrete, measurable examples comprised the use of a plurality of teaching funnels each capable of being supported by a mounting mechanism in a stable position over a receptacle. Each funnel has perimetric sidewalls with interior surfaces that conform to equations that will .Iadd.allow various flow characteristics to be computed, and in preferred embodiments will .Iaddend.provide an integrable function when plugged into the equation for computing the time it takes for the funnel to empty of liquid. The funnels can have a base covering the lower end of the funnel with an outlet orifice formed in the base. The orifice is spaced a predetermined distance away from the sidewalls to achieve a desired outflow rate. The different sidewalls can be formed of different shapes as long as each conforms with a formula that .[.yields an.]. .Iadd.allows various flow characteristics to be computed, and in preferred embodiments that yield .Iaddend.integrable .[ƒ]. .Iadd.functions .Iaddend.when plugged into the equation for computing the time it takes for the funnel to empty of liquid. A simple configuration is a funnel with a rectangular horizontal cross-section, three straight sidewalls, and one curvilinear sidewall. The funnels also can be formed with a circular or otherwise formed horizontal cross-section and can be formed in multiple vertical levels. The funnel top can be open or closed and pressurized to vary the pressure on the liquid in the funnel.

The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows: 
     
       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 .Iadd.either analytically or numerically; .Iaddend. filling the funnel with a liquid to a preselected level in the funnel; and   measuring .[.the time it takes for all.]. .Iadd.a characteristic .Iaddend.of the liquid .[.to flow.]. .Iadd.as said liquid flows .Iaddend.out of the funnel outlet orifice, the measured .[.time.]. .Iadd.characteristic .Iaddend.value being comparable with a .[.time.]. .Iadd.theoretical characteristic .Iaddend.value obtainable theoretically .[.by integrating the equation for the theoretical time it takes the funnel to empty.]. .Iadd.by mathematical formulation.Iaddend., a comparison of the measured and theoretical values using mathematical .[.integration.]. .Iadd.calculation .Iaddend.providing a concrete experience that facilitates learning the principles of fluid flow.Iadd., science .Iaddend.and .[.integration.]. .Iadd.mathematical operations .Iaddend.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.]. .Iadd.capable of yielding an analytically or numerically .Iaddend.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.]. .Iadd.capable of yielding an analytically or numerically .Iaddend.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.]. .Iadd.capable of analytically or numerically integrated .Iaddend.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 .Iadd.capable of being analytically or numerically .Iaddend.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 .Iadd.capable of being analytically or numerically .Iaddend.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 .Iadd.capable of being analytically or numerically integrated.Iaddend.. 
     
     
       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. 
     
     
       21. Teaching apparatus for teaching the principles of fluid flow .[.and integration.]. .Iadd.science and mathematics .Iaddend.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.]. .Iadd.capable of being analytically or numerically integrated.Iaddend., 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.]. .Iadd.a flow characteristic of said funnel as said funnel empties .Iaddend.and .[.integrating.]. .Iadd.performing mathematical operations thereon.Iaddend.; 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. 
     
     
       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.]. .Iadd.said flow characteristic .Iaddend.yields an .[.integrable.]. function .Iadd.capable of being analytically or numerically integrated.Iaddend.. 
     
     
       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.]. .Iadd.said flow characteristic .Iaddend.and yield an .[.integrable.]. equation .Iadd.capable of being analytically or numerically integrated.Iaddend.. 
     
     
       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.]. .Iadd.capable of being analytically or numerically integrated .Iaddend.in the formula for calculating .[.the time for emptying the funnel of liquid.]. .Iadd.said flow characteristic.Iaddend.. 
     
     
       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.]. .Iadd.capable of being analytically or numerically integrated .Iaddend.in the formula for calculating .[.the time for emptying the funnel of liquid.]. .Iadd.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 .Iadd.for calculating said flow characteristic.Iaddend., yield an .[.integrable.]. integrand .Iadd.capable of being analytically or numerically integrated.Iaddend.. 
     
     
       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. .Iadd. 
     
     
       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..Iaddend. .Iadd.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. .Iaddend. .Iadd.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. .Iaddend. .Iadd.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. .Iaddend. .Iadd.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. .Iaddend.