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1-057-FiguringFluidFlow-ModelingScenario

Author(s): Brian Winkel

SIMIODE - Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations

Keywords: discharge tank Torricelli's Law Conservation of Energy fluid flow height liquid

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Abstract

Resource Image We propose three differential equations models for the height of a column of falling water as the water exits a small bore hole at the bottom of the cylinder and ask students to determine which model is the best of the three.

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Article Context

Resource Type
Differential Equation Type
Technique
Qualitative Analysis
Application Area
Course
Course Level
Lesson Length
Technology
Approach
Skills
Key Scientific Process Skills
Assessment Type
Pedagogical Approaches
Vision and Change Core Competencies - Ability
Principles of How People Learn

Description

Evangelestia Torricelli (1608-1647) was an Italian physicist and mathematician who also served Galileo as a secretary. He investigated the concepts of atmospheric pressure and vacuums and he invented the barometer.

Torricelli was interested in fluids and their flow rates. He studied the following situation. Consider a cylindrical container (e.g., tin can open at top and closed at bottom) containing water. A small hole has been placed in the side of the can and the can is resting at the edge of the surface of a platform (on a table) with the hole in the can over the edge of the platform.

Torricelli was interested in the rate at which the water would flow out of the hole both as a function of height and time. Equivalently he was interested in the change in volume of the water in the cylinder per unit time or the change in the height, h, (for a fixed radius, r, cylinder) per unit time, i.e. he was interested in h'(t) where h(t) is the height of the ater at time t.

Certainly one could reason that with more water in the cylinder, hence more weight on the water above the hole, there would be more force pressing down, thus causing more force on the water going out. This would mean there is more water per unit time exiting the hole when h is big than when h is small.

We use these notions in model building.

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Authors

Author(s): Brian Winkel

SIMIODE - Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations

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