IntroductionThe Purpose of this lab is to study thephenomena of tank draining and figuring out a mathematical model to accuratelydescribe what happened in the experiment. This lab experiment is useful tounderstand on a small-scale, how tank draining works on theoretical andtechnical aspect. This is important topic since it covers different areas ofChemical Engineering.
Onearea that tank draining is important is the instance that you are goingimplement a nozzle or pipe into a storage tank. Before you can implement it,you would need to know how fast will the water draining from the tank to withan exit velocity. Another application that values this project is if there is atank rupture, it is important to understand all the factors that was used in thislab in order to perform investigations as well as conduct Process Hazard Analysisand improve on Process Safety Management. Experimental Equipment and ProceduresFigure 1: Stainless Tank Atank that has a cylindrical top with a conical bottom shape (Figure 1) was usedfor this experiment. The bottom of the tank is connected to the to a schedule40, 1.25″ tube. Which is connected to a tee reducer which is then have a tubeconnected to six different tube sizes and Diameters. Torun the experiment, water will be filled up to the initial 30″ water heightabove the tube.
After the water has been filled to the set height, one of thesix tubes will be connected to the bottom of the pipe. To record data the, teethat was connected to the pipe was opened and data was recorded at every inchtill a set water height was reached. Toget a estimate theoretical model for the data for the tank draining, Severalassumptions were made here were the following assumptions: 1. The fluid is incompressible2.
The fluid is also inviscid3. The friction in the cylinder section and the conicalsection, as well as the pipe and reducer are negligible. 4. It is at constant density.With these starting assumptions, themodel was begun by using the mass balance which is:. Since there is no mass flow in the mass balance simplifies to .
The Bernoulli Equation used for thismodel was the head form of the Bernoulli Equation is the head form theBernoulli which is (EQ.2). Several terms can be canceled out withthe following reasons: · Both pressure terms can be canceled out since both areheld at atmospheric pressure· The work term can also be canceled out since there isno work being done on the system· Since the water has an initial height an assumption wasmade that one of the height is zero. Thesimplified equation will end up being .
The assumption can also be made thatthe velocity at the free surface (v2) is smaller than the velocity at the tube(v1) so that term can go away so all is left to solve for is the velocity whichcomes out to be . Since this a constant density drainingand has continuity, a valid equation to use is and set as since the water height is decreasing andcan solve for with the equation: . From this equation two separate modelswere made for the cylindrical section and the conical section. is the area of the tube and is the area of the cylinder section/conicalsection. One term that needed to be defined is .
To get this term, a velocity termneeds to be defined. The velocity term was determined by taking the derivativeof the best fit equation from the experimental plot of water height withrespect to time and plugging either the time used with the experimental data(the time to drain the cylindrical section/conical section can also be used).Once a velocity is found, that velocity can be used to find the Reynolds numberfrom the equation (EQ.7) with D being the diameter of the tube and µ being the viscosity of thewater. The calculated Reynolds number used to find the friction factor.
From the calculated for the Reynoldnumber in both the cylindrical section and conical section was calculated to belaminar flow with each pipe which then the equation: 16/NRE to find the friction factor to find the head friction of thetube which was calculated from the equation . For the first 8 initial fluid heightswhich is in the cylindrical section of the tank, the area of the tubes is setto be and the areas of both the cylinder section andthe conical section are both . For this lab, height was used to solvefor the time taken to drain the water. For the cylindrical section, EQ.9 andEQ.
10 was plugged into EQ.6 to get . This equation is aseparable differential equation and was solved for t to get However, got the friction from the tubeother than Tube No.1 (1/8″ ID, 24″ length) had very miniscule friction so itcould be safely assumed from the data that the term for the rest of the terms werenegligible. For the conical section it was assumed that the conical section wasshaped like a true cone. This is useful for the fact that you can find thetangential angel of the cone in order to find the height of the fluid at anygiven radius.
The tangential angle was solved from the equation . The angle was found to be 31.34.
Thiswas plugged in to find the radius of the cone in . The squared tangential angle turnedout to be .36. EQ.12 was plugged into EQ.6 to get . This is also a separable differentialequation and was solved for t. Since the friction can not be assumed negligiblein the first tube, the equation to solve for t was evaluated to be t .
For the other tubes the equationbecomes Foreach tube, the cylindrical derived equation will be used for the first 8 inchesof fluid height until there is no water left in the cylinder section. Aftertotal drainage in the cylinder section, the equation for the conical section oftank will be used for the rest of the water heights. Experimental Results and Discussion (Tube No.1)Thisis one of the scatter plot of the Tube (No.1) that is graphed as water height withrespect to time in height. The experimental data was compared to thetheoretical model that was generated from the mathematical model.
The followinggraphs were formatted the exact way as the graph above. (Tube No.2 D datawith repsect ot timeFromthe plotted data given above, one observation can be made in that the tank tooka substantial amount of time to drain rather than the experimental data points.Another conclusion that can be made based off these graphs is that the lengthof the tube seems to play a role in the accuracy between the theoretical andthe experimental. The most accurate that the theoretical model was Tube No.3 Relationship between theory and experimental resultsAs observed earlier, the theoreticalmodel data has some significant variation for each tube. However, it would besafe to assume that the model closely models the draining of the water in thecylindrical section than the conical section.
This model is on the basis thatthis is an unsteady state model for the fact that the cross-section area of thetank is decreasing as the fluid height decreases. With the cylindrical sectionof the tank it free surface and had constant cross-sectional area throughoutits section. On the other hand, the equation becomes unsteady when the waterheight reaches the conical part of the section since the radius decreases asheight decreases. One Equation/Law used for tank draining situations is Torricelli’sLaw. It theorizes that the velocity of the fluid is the same as the velocity atrest (Nevers, 2005).
It can be definedfrom the Bernoulli as (Nevers, 2005). This equation doesnot work with this model because this tank has two different shaped sections aswell as there is friction. The model uses a modified version of the Torricelli’sEquation that factors in friction. The Graphs also indicates that the velocity of the water exitingthe pipe increases as the height of the fluid decreases.
This phenomenon can bedescribed for the Coriolis effect. The Coriolis effect describes the motion ofthe fluid in the x,y,z gradient as well as the radial direction in cylindricalcoordinates (McCabe, Smith, & Harriot, 2005). As the fluid flowsdown the tube the velocity in cylindrical coordinate directions which causes a Corioliseffect to occur in the conical section as the channel for the streamline of thewater decreases. This causes interaction in the boundaries of the conical sectionand momentum to occur in the tank results in an increase in turbulence (Moore, 1967).
Conclusion and RecommendationsInConclusion, a general mathematical model for tank draining of water was created.The results from the theoretical data varied with the experimental data given.Some recommendations would be to analyze the conical section more in depth aswell as studying the effect of having an initial flow rate and investigating theconical section since the theoretical time was relatively long compared to the timein the experimental data ID1