Suppose that we are interested in the amount of salt (in kg) at a certain moment in time, denoted byA(t), in a well-mixed, infinitely large tank with 1,250-L of water initially. Further, suppose we have set-up the experiment such that a salt solution containing 12 kg/L flows into the tank at a rate of 12 L/sec. Ifwe allow the solution to flow out at 7 L/sec, then what is the volume of water in the tank at time t?V(t) =asin (a)150+ 8 t∞ain litres (DO NOT use commas in your answer)Construct a model for the change of salt in the tank at time t in seconds.[To ensure you are marked correct, please use A to represent the amount of salt in the tank at time tnot A(t)].dAdt

The inflow rate of solution into the tank is 12 L/sec.The outflow rate of solution from the tank is…

Answered: Suppose that we are interested in the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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10.14 Ninety kilograms of sodium nitrate is dissolved in 110 kg of water. When the dissolution is complete (at time t = 0), pure water is fed to the tank at a constant rate m(kg/min), and solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed. (a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value. ANSWER + (b) Write a balance on sodium nitrate, letting x(t, m) equal the mass fraction of NaNO3 in the tank and outlet stream. Convert the balance into an equation for dx/dt and provide an initial condition. (c) On a single graph of x versus t, sketch the shapes of the plots you would expect to obtain for m = 50 kg/min, 100 kg/min, and 200 kg/min. (Don't do any calculations.) Explain your reasoning, using the equation of Part b in your explanation. (d) Separate variables and integrate the balance to obtain an expression for x(t, m). Check your solution. Then…
Consider the cascade of two tanks of brine shown in the figure below, with V1=99 gallons and V2=204 gallons being the volumes of brine in the two tanks. Each tank also initially contains 51 lb of dissolved salt. The three flow rates indicated in the figure are each 4.6 gal/min, with pure water flowing into tank 1. (a) Find the amount x(t) of salt in tank 1 at time t. x(t)=? (b) Suppose that y(t), the amount of salt in tank 2 at time t, is given by the differential equation dy/dt=(4.6x/99)−(4.6y/204). Solve for y(t) using the function x(t) found in part (a) y(t)=?
6. A 50 gallons tank initially contains 10 gal of fresh water. At t = 0, a brine solution containing 1 lb of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the tank at the rate of 2 gal/min. Find a. the amount of time required for overflow to occur b. the amount of salt in the tank at the moment of overflow
2. A tank is initially filled with 100 L of salt solution with a concentration of 0.12 kg/L. At time t = 0, a solution containing 0.2 kg/L of salt is poured into the tank at a rate of 3 L/min. Simultaneously, a drain is opened in the bottom of the tank allowing the solution to leave the tank at a rate of 4 L/min. The tank is well-stirred. Let x(t) represent the amount of salt (in kg) in the tank after t minutes. Find the differential equation and initial condition for which x(t) is a solution. DO NOT SOLVE THE DIFFERENTIAL EQUATION. (t)= S t x' (t) = , x(0) =
The decomposition of nitrogen dioxide (NO2) into nitrogen monoxide (NO) and oxygen is a second-order reaction. This means that the concentration C of NO, at time t is given by 1/C = kt + 1/Co, where Co is the initial concentration and k is the rate constant. Assume the initial concentration is known to be 0.03 mol/L exactly. Assume that time can be measured with negligible uncertainty. After 40 s, the concentration C is measured to be 0.0023 + 2.0 x 10-4 mol/L. Estimate the rate constant k, and find the uncertainty in the estimate. a. After 50 s, the concentration C is measured to be 0.0018 + 2.0 x 10-4 mol/L. Estimate the rate constant k, and find the uncertainty in the estimate. Denote the estimates of the rate constant k in parts (a) and (b) by k, and k, respectively. The average (, + k,)/2 is used as an estimate of k. Find the uncertainty b. C. in this estimate. d. Find the value of c so that the weighted average ck + (1- c)k2 has the smallest uncertainty.

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