Tank Draining - Tank Draining Modeling Using Mathematical...

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Tank Draining Modeling Using Mathematical Approximations Eric Burbach, Alex McKinney, Mason Merritt, Jasa Zunaibi University of Nebraska-Lincoln
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Table of Contents Table of Contents 2 Abstract 3 Introduction 3 Experimental Apparatus 5 Safety 6 Operating Procedures 6 Personal and Chemical Safety 7 Process Safety 7 Results and Discussions 7 Data Fitting 7 Model Validity 12 Conclusion and Recommendations 13 References 14 Appendix 15 2
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Abstract Water draining from a tank can be modeled with the famous Bernoulli’s equation, and through experimental data, the friction coefficient of the valves and fittings can be quantified. In order to find the unknown resistance coefficient, the Torricelli’s Theorem must be derived from Bernoulli’s Equation. Torricelli’s Theorem can then be applied to an unsteady state material balance and further solved to yield height as a function of time. A linear regression using experimental data can be used to find the unknown resistance and the results can be compared with the derived model. A system of three tanks of differing heights and diameters were used to gather experimental data.This system allowed for solving of an unknown friction coefficient from a manually opened valve and fittings by collecting the height of the fluid with respect to time using a process control system called DeltaV and fitting this in the model. The experimental data followed the created linear regression model to an acceptable level when end effects at the beginning and end of the trial were ignored. The model showed that how open the valves were directly correlated to the value of K in a manner that was intuitive. More specifically, as the valve was opened to larger percentages the values of c 1 increased as well. Since the K value is inversely proportional to c 1 , the percentage open is as well. For Tank 1, as the percent open changed from 32 to 64 the c 1 value went from 0.2561 to 0.3208 which corroborates what one would intuitively think. The model fits the data and allows for an approximate value of K to be found from c 1 . Introduction Chemical engineering calculations are different depending on the system that is being observed. Certain systems such as batch and semibatch processes can be studied through mass and energy balances. These processes are never in a steady state and complicate the balances with the necessary usage of differentials. A tank draining from a certain level due to gravity is a specific example of a semibatch process that requires the use of calculus to accurately model. A famous energy balance known as the Bernoulli’s Equation is typically used in systems with an incompressible flowing liquid. It was created based on the principle of the conservation of energy. This principle states that energy can neither be created nor destroyed. Energy only exists in various forms and can be transformed from one kind to another. This fact is better quantified by stating that the total energy in an isolated system is constant. More specifically, the
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