Simulation of IV Bag Decanting

Simulation of IV Bag DecantingBase Design and EquationsThe frame-up of the simulation is given below. The point at the top of the smooth in the IV nucleotide is denoted with a subscript 1. The point at the exit of the catheter into the patients arm is denoted with a subscript 2. Assuming that the patient does not move his/her arm during the use, we can set point 2 as a static, whereas point 1 is moving dynamically as the bag empties.Bernoullis Equation in the units of energy is given asSetting point 2 as the height of ground, excluding the pumping term and combining force terms into a pressure variation, but measuring P1 and P2 in gauge pressure and expect P1 to be atmospheric yields the interest equationWhere WhereFor laminar hunt down (Re And for tumultuous (Re 2100)Mathematical DevelopmentDerivation of equations pertaining to the IV bagWhat is needed for the first era is a way to take place the flock of fluid still within the bag when only knowing the height of the fluid in the bag. Taking the central axis of the bag as an independent variable and viewing the radius of the bag at that point as a dependent variable yields the following straight lineWhich has the equationAnd performing a solid of revolution ab out(p) the h axisIt is know that and that by mass balance assuming no accumulation in the pipeButFinally for the frustum IV bagFriction terms developmentComparing the Reynolds number in the pipe and the catheterBut by mass balance and assuming no accumulationWhich is always less than one. Therefore if scat in the catheter is laminar, then course in the pipe has to be laminar too.This leaves 3 cases of flow in the system. The first is flow is laminar in the catheter and pipe. The second case is flow is turbulent in the catheter, but laminar in the pipe, or flow is turbulent for the catheter and pipe.For the first case where flow is laminar in both the catheter and the pipeIf the flow is turbulent in the catheter, then the friction term is the sum of the friction from the pipe and the catheterSimulation Program DevelopmentSolving speed of fluid go away the catheter for any fluid height hThe above Equations 1-9 are easily incorpo commitd into matlab functions (Appendix A Calculations functions).The first problem that the program must solve is astute the instantaneous velocity of fluid out of the catheter at any height of fluid in the IV bag.First the flow in the catheter is assumed to be laminar (this is just a starting point for iterations). The correct velocity of the fluid leaving the catheter is the point here Equation 1 is stable.Since the equation contains a cheering root, for the answer to be reasonable then there has to be a positive root. i.eSubstituting equations 3 and 5Since is genuinely small, its square may be neglected (checked with given simulation value it was on the order of 10-10)The velocity that satisfies the stable tooth root of Equation 10 must hence lie between 0 and for laminar flow speeds. This means that Equation 10 may be solved by bisection system, giving the instantaneous velocity out of the catheter for a authentic height of fluid in the IV bag.If the maximum and borderline velocities do not generate the proper conditions required for the bisection method (one function value must produce a number greater than zero, and the other must generate a number below zero to call for a root between the two numbers on a continuous function), then the flow velocity must be such that the flow is turbulent in the catheter.If it is the case that flow must be turbulent in the catheter, then we can set an initial guess of v2 at the minimum velocity required to have turbulent flow in the catheter. Using this as an initial velocity, it can be iterated until a stable solution is found. agnise Appendix B for this programming logic. Comments have been added to aid understanding.Fluid flow out of the IV bagIn order to calculate how fluid flows out of the IV bag, we set the program to take small time elements and determine the velocity of the fluid flowing out of the catheter for a set fluid height. Since that time element is small, we assumed that the velocity did not change appreciably in that small element. Volumetric flow is the product of linear velocity of flow and cross-sectional surface area, and assuming no accumulation in the pipe for mass balanceNow we need to know what the height is of the bag for that volume, which is simply the actually root that solves Equation 2.This entire process is iterated for each time element until the bag empties or the fluid velocity leaving the catheter is zero.This logic may be found in Appendix C.Concatenating ResultsFinally a matrix is constructed that holds all data of the catheter fluid velocity, bag volume, fluid flow rate and height of fluid in the bag and is concatenated with a time vector. This resultant matrix now holds all of the data of the simulation, which is parsed to the primary class that is running the simulation to produce the graphs, and recall numerical values for points on the graphs.Results of SimulationThe following data was fed into matlabP2 = 90.228/760*101325Lcat = 43.703/ curtilageDcat = 0.711/1000Lpipe = 1.383H = 22.8/100R = 0.3829*Hhd = 1.377Dpipe = 3/1000rpipe = 0.5*Dpipemu = 1.142/1000rho = 1017X = calcX(R,H,rpipe)h0 = 0.8*Xbeta = (R-rpipe)/XThe simulation was run using the following command to retrieve the results matrixVhvQt = generateVhvtQMatrix(rho,Dcat,mu,Lpipe,Lcat,Dpipe,beta,h0,rpipe,9.81,hd,P2)Solution graphs were plotted as followsplot(VhvQt(5,),VhvQt(2,))xlabel(Time in seconds)ylabel(Height of fluid in bag in meters)Bag volume is language 1, height of bag is row 2, velocity of fluid in the catheter is row 3, flowrate is row 4 and row 5 is the time vector. To find the total time taken for the bag to empty, the command was usedVhvQt(5,end)= 3122Similar commands were used to find exact datapoints at any catamenia in time.Discussion of Simulation ResultsThe values of the viscosity and density of the 5% w/v glucose solution more commonly known as D5W, Dextrose 5 Water or Intravenous Sugar Solution, were looked up from Wolfram Alphas material database, and had the values of and . (Wolfram Alpha)The purpose of the simulation was to calculate and plot the graphs as well as answer how long it takes for the bag to drain.It took 3122 seconds for the bag to drain, or just over 52 minutes. At the last second of the simulation, the height of fluid in the bag was less than 4 millimetres, which corresponds to a volume of 0.045 mL. Figures 3-5 all show a disconfirming exponential function, which is to be expected since the flow rate is a function of the head pressure, which is a function of the conical shape of the bag. Initially, the high flow rate that corresponds with the higher head of fluid does not change very much, as the level of fluid in the cone does not drop significantly quickly. However, subsequently(prenominal) about 2500 seconds (80% through the time of the simulation), the velocity of fluid leaving the catheter begins to change.While this change is clearly not linear, it is not really significant when compared to the absolute changes in values. From the start of the simulation until 80% through the simulation (2500 seconds), the velocity of fluid in the catheter only changes from 0.8655 to 0.6922 m/s, or 20.02%. It is only in the last period of time just before the bag empties is there a significant change in the velocity of fluid entering the patient, with the final velocity creation 0.4707 m/s just as the bag empties, which is itself only a change of 45.62%.This phenomenon is most apparent in Figure 6, where it can be seen that the volume of fluid left in the bag appears to decant at an almost linear rate (differential of volume/time is flow rate). Only just after 2500 seconds does the steepness of the graph very slightly begin to change upward indicated a slow in the flow rate of fluid le aving the bag.This phenomenon does, however, come to sense. The height that the bag is lifted above the catheter is hd = 1.377 meters. The height of fluid in the bag at the start of the procedure is h0 = 0.1793m, or only 13.20% of hd. The means that the parkway force behind the IV procedure (which is the height above the IV bag is placed above the catheter in the patients arm) that creates the pressure difference to overcome the venous pressure of blood in the patients veins only changes by 13.2%, and remains changed by only 20.02 % during the initial 80% of the procedure in other words, it doesnt change by much, and therefore we expect that the flow rate wont change very much until very close to the end of the procedure, which is what we see in Figure 3.In real world application, this means that in general the flow rate of IV solution to the patient is mostly a function of how high above the patient the IV bag is placed, and not necessarily how empty or full the bag its self is. In fact, general practise used by doctors is simply to place the IV bag above the heart level of the patient (Dr. Chen-Maynard, 1999). In application, a desired flow rate of IV fluid into the patient may controlled by lowering or lifting the IV bag a certain height above the patient.NomenclatureReferencesDr. Chen-Maynard, P. R. (1999). Calculating Parenteral Feedings. California California Department of Health Science and Human Ecology.Wolfram Alpha. (n.d.). Comprehensive Material entropy Sheet for D5W. Retrieved April 28, 2014, from http//www.wolframalpha.com/input/?i=5%+(w/v)+glucose