4.2E: Cooling and Mixing (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Q4.2.1
1. A thermometer is moved from a room where the temperature is 70∘F to a freezer where the temperature is 12∘F. After 30 seconds the thermometer reads 40∘F. What does it read after 2 minutes?
2. A fluid initially at 100∘C is placed outside on a day when the temperature is −10∘C, and the temperature of the fluid drops 20∘C in one minute. Find the temperature T(t) of the fluid for t>0.
3. At 12:00 pm a thermometer reading 10∘F is placed in a room where the temperature is 70∘F. It reads 56∘ when it is placed outside, where the temperature is 5∘F, at 12:03. What does it read at 12:05 pm?
4. A thermometer initially reading 212∘F is placed in a room where the temperature is 70∘F. After 2 minutes the thermometer reads 125∘F.
- What does the thermometer read after 4 minutes?
- When will the thermometer read 72∘F?
- When will the thermometer read 69∘F?
5. An object with initial temperature 150∘C is placed outside, where the temperature is 35∘C. Its temperatures at 12:15 and 12:20 are 120∘C and 90∘C, respectively.
- At what time was the object placed outside?
- When will its temperature be 40∘C?
6. An object is placed in a room where the temperature is 20∘C. The temperature of the object drops by 5∘C in 4 minutes and by 7∘C in 8 minutes. What was the temperature of the object when it was initially placed in the room?
7. A cup of boiling water is placed outside at 1:00 pm. One minute later the temperature of the water is 152∘F. After another minute its temperature is 112∘F. What is the outside temperature?
8. A tank initially contains 40 gallons of pure water. A solution with 1 gram of salt per gallon of water is added to the tank at 3 gal/min, and the resulting solution drains out at the same rate. Find the quantity Q(t) of salt in the tank at time t>0.
9. A tank initially contains a solution of 10 pounds of salt in 60 gallons of water. Water with 1/2 pound of salt per gallon is added to the tank at 6 gal/min, and the resulting solution leaves at the same rate. Find the quantity Q(t) of salt in the tank at time t>0.
10. A tank initially contains 100 liters of a salt solution with a concentration of .1 g/liter. A solution with a salt concentration of .3 g/liter is added to the tank at 5 liters/min, and the resulting mixture is drained out at the same rate. Find the concentration K(t) of salt in the tank as a function of t.
11. A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 1/4 pound of salt per gallon is added to the tank at 4 gal/min, and the resulting mixture is drained out at 2 gal/min. Find the quantity of salt in the tank as it is about to overflow.
12. Suppose water is added to a tank at 10 gal/min, but leaks out at the rate of 1/5 gal/min for each gallon in the tank. What is the smallest capacity the tank can have if the process is to continue indefinitely?
13. A chemical reaction in a laboratory with volume V (in ft3) produces q1 ft3/min of a noxious gas as a byproduct. The gas is dangerous at concentrations greater than ¯c, but harmless at concentrations ≤¯c. Intake fans at one end of the laboratory pull in fresh air at the rate of q2 ft3/min and exhaust fans at the other end exhaust the mixture of gas and air from the laboratory at the same rate. Assuming that the gas is always uniformly distributed in the room and its initial concentration c0 is at a safe level, find the smallest value of q2 required to maintain safe conditions in the laboratory for all time.
14. A 1200-gallon tank initially contains 40 pounds of salt dissolved in 600 gallons of water. Starting at t0=0, water that contains 1/2 pound of salt per gallon is added to the tank at the rate of 6 gal/min and the resulting mixture is drained from the tank at 4 gal/min. Find the quantity Q(t) of salt in the tank at any time t>0 prior to overflow.
15. Tank T1 initially contain 50 gallons of pure water. Starting at t0=0, water that contains 1 pound of salt per gallon is poured into T1 at the rate of 2 gal/min. The mixture is drained from T1 at the same rate into a second tank T2, which initially contains 50 gallons of pure water. Also starting at t0=0, a mixture from another source that contains 2 pounds of salt per gallon is poured into T2 at the rate of 2 gal/min. The mixture is drained from T2 at the rate of 4 gal/min.
- Find a differential equation for the quantity Q(t) of salt in tank T2 at time t>0.
- Solve the equation derived in (a) to determine Q(t).
- Find limt→∞Q(t).
16. Suppose an object with initial temperature T0 is placed in a sealed container, which is in turn placed in a medium with temperature Tm. Let the initial temperature of the container be S0. Assume that the temperature of the object does not affect the temperature of the container, which in turn does not affect the temperature of the medium. (These assumptions are reasonable, for example, if the object is a cup of coffee, the container is a house, and the medium is the atmosphere.)
- Assuming that the container and the medium have distinct temperature decay constants k and km respectively, use Newton’s law of cooling to find the temperatures S(t) and T(t) of the container and object at time t.
- Assuming that the container and the medium have the same temperature decay constant k, use Newton’s law of cooling to find the temperatures S(t) and T(t) of the container and object at time t.
- Find lim.t→∞S(t) and limt→∞T(t).
17. In our previous examples and exercises concerning Newton’s law of cooling we assumed that the temperature of the medium remains constant. This model is adequate if the heat lost or gained by the object is insignificant compared to the heat required to cause an appreciable change in the temperature of the medium. If this isn’t so, we must use a model that accounts for the heat exchanged between the object and the medium. Let T=T(t) and Tm=Tm(t) be the temperatures of the object and the medium, respectively, and let T0 and Tm0 be their initial values. Again, we assume that T and Tm are related by Newton’s law of cooling,
T′=−k(T−Tm).
We also assume that the change in heat of the object as its temperature changes from T0 to T is a(T−T0) and that the change in heat of the medium as its temperature changes from Tm0 to Tm is am(Tm−Tm0), where a and am are positive constants depending upon the masses and thermal properties of the object and medium, respectively. If we assume that the total heat of the system consisting of the object and the medium remains constant (that is, energy is conserved), then
a(T−T0)+am(Tm−Tm0)=0.
- Equation (A) involves two unknown functions T and Tm. Use (A) and (B) to derive a differential equation involving only T.
- Find T(t) and Tm(t) for t>0.
- Find limt→∞T(t) and limt→∞Tm(t).
18. Control mechanisms allow fluid to flow into a tank at a rate proportional to the volume V of fluid in the tank, and to flow out at a rate proportional to V2. Suppose V(0)=V0 and the constants of proportionality are a and b, respectively. Find V(t) for t>0 and find limt→∞V(t).
19. Identical tanks T1 and T2 initially contain W gallons each of pure water. Starting at t0=0, a salt solution with constant concentration c is pumped into T1 at r gal/min and drained from T1 into T2 at the same rate. The resulting mixture in T2 is also drained at the same rate. Find the concentrations c1(t) and c2(t) in tanks T1 and T2 for t>0.
20. An infinite sequence of identical tanks T1, T2, …, Tn, …, initially contain W gallons each of pure water. They are hooked together so that fluid drains from Tn into Tn+1(n=1,2,⋯). A salt solution is circulated through the tanks so that it enters and leaves each tank at the constant rate of r gal/min. The solution has a concentration of c pounds of salt per gallon when it enters T1.
- Find the concentration cn(t) in tank Tn for t>0.
- Find limt→∞cn(t) for each n.
21. Tanks T1 and T2 have capacities W1 and W2 liters, respectively. Initially they are both full of dye solutions with concentrations c1 and c2 grams per liter. Starting at t0=0, the solution from T1 is pumped into T2 at a rate of r liters per minute, and the solution from T2 is pumped into T1 at the same rate.
- Find the concentrations c1(t) and c2(t) of the dye in T1 and T2 for t>0.
- Find limt→∞c1(t) and limt→∞c2(t).
22. Consider the mixing problem of Example 4.2.3, but without the assumption that the mixture is stirred instantly so that the salt is always uniformly distributed throughout the mixture. Assume instead that the distribution approaches uniformity as t→∞. In this case the differential equation for Q is of the form
Q′+a(t)150Q=2
where limt→∞a(t)=1.- Assuming that Q(0)=Q0, can you guess the value of limt→∞Q(t)?.
- Use numerical methods to confirm your guess in the these cases:
(i) a(t)=t/(1+t)(ii) a(t)=1−e−t2(iii) a(t)=1−sin(e−t).
23. Consider the mixing problem of Example 4.2.4 in a tank with infinite capacity, but without the assumption that the mixture is stirred instantly so that the salt is always uniformly distributed throughout the mixture. Assume instead that the distribution approaches uniformity as t→∞. In this case the differential equation for Q is of the form
Q′+a(t)t+100Q=1
where limt→∞a(t)=1.- Let K(t) be the concentration of salt at time t. Assuming that Q(0)=Q0, can you guess the value of limt→∞K(t)?
- Use numerical methods to confirm your guess in the these cases:
(i) a(t)=t/(1+t)(ii) a(t)=1−e−t2(iii) a(t)=1+sin(e−t).