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Situations arise in engineering when we want to control the flow rate of a solution. This solution could be pumped into a holding tank, distillation column, heat exchanger, reactor, etc. This means that the flow rate becomes a function of time. When we turn on a pump, the flow rate is (for the most part) constant. The problem is that we may want the flow rate to be very fast at first, and then very slow towards the end of the process. This is especially true in certain chemical reactors where the concentration of reactants is always important. Control systems that could handle this type of situation are very expensive. Pumps (relatively speaking) are cheap. So, having a bank of pumps will usually solve the problem. The pumps are simply turned on or off to regulate the flow. Mathematically, this amounts to taking the volumetric flow rate function and finding a piecewise-linear approximation. Suppose that, because of the application we are running, we need a flow rate given by the function ,where t is in hours and v(t) is in 100 cubic feet per hour (e.g., v(0)=1 which means that at time t=0, we need a flow rate of 1*100=100 cubic feet per hour). Find a linear approximation and a piecewise-linear approximation using first two lines and then three lines. Think about what using more lines would do to the approximation. You will need to find suitable values of t to be able to find your approximation. As a further exercise, try to find values of t for your approximation that keeps the error (i.e., the difference between the approximation and the real function v(t) ) less than 0.1.Hint: to help with this, ask yourself this question: where would more lines be useful - where the function changes a lot or where the function changes a little?]
The density of an ideal gas decreases by 10% when compressed isothermally from 10 atm to 11 atm. Determine the percent decrease in density of the gas if it compressed isothermailly from 1000 atm to 1001 atm.
If the maximum total acceleration of the train must not exceed 1.5 m/s^2, determine the shortest distance in which the train can reach a speed of 72 km/h, and determine the corresponding constant rate of acceleration at.
What uniform annual series of cash flows over a 12-year period is equivalent to an investment of $5,000 at t= 0, followed by receipts of $600 per year for 11 years and a final receipt of $1,600 at t= 12 of the investor's time value of money is 6% ..
A fish hanging from the bottom of the spring oscillates vertically at a frequency of 2.00 Hz. Ignoring the mass of the spring, what is the mass of the fish?
Calculate carbonate, noncarbonate, and total hardness Na+ = 27.8 mg/L Cl- = 35.6 mg/L Ca+2 = 48mg/L CO3-2 = 30 mg/L Mg+2 = 12mg/L HCO3- = 61 mg/L K+ = 12 mg/L SO4-2 = 48 mg/L
Account for chemical reactions, groundwater flows, streams, precipitation, evapotranspiration. Indicate any assumptions that can be made to simplify your equation.
Three shafts and four gears are used to form a geartrain which will transmit 7.5 kW from the motor at A to a machine tool at F. (Bear- ings for the shafts are omitted from the sketch.) Knowing that the frequency of the motor is 30 Hz
An effluent holding pond to be used as part of a spray irrigations system is being built. The holding pond is located 2.5 miles awary for the 10 mgd WWTP which will supply the effluent
Sketch a streamline, pathline, and streakline, For a given flow the velocity from time t=0 to t=5s was u=2 m/s, v=0. Then from t=5s to t=10s the velocity was u=+3 m/s, v=-4 m/s. A dye streak was started at t=0
Some distance downstream, the velocity profile can be (very) roughly approximated by a region inside the wake with u = U/2 and an exterior region with u = 3U/2.
A steam locomotives had scoops installed that took water from a tank located between the tracks and lifted water into the tender. To lift the water 12 ft with a scoop, neglecting all losses, what speed in mph is required
A certain gas has a molecular mass of 40. It is under a pressure of 2.5atm and a temperatature of 25 Degree Celsius. Determine its density and pressure in CGS absolute units.
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