4.2.1: Cooling and Mixing (Exercises)

Q4.2.1

1. A thermometer is moved from a room where the temperature is $70^\circ$F to a freezer where the temperature is $12^\circ F$. After $30$ seconds the thermometer reads $40^\circ$F. What does it read after $2$ minutes?

2. A fluid initially at $100^\circ$C is placed outside on a day when the temperature is $-10^\circ$C, and the temperature of the fluid drops $20^\circ$C in one minute. Find the temperature $T(t)$ of the fluid for $t > 0$.

3. At 12:00 pm a thermometer reading $10^\circ$F is placed in a room where the temperature is $70^\circ$F. It reads $56^\circ$ when it is placed outside, where the temperature is $5^\circ$F, at 12:03. What does it read at 12:05 pm?

4. A thermometer initially reading $212^\circ$F is placed in a room where the temperature is $70^\circ$F. After 2 minutes the thermometer reads $125^\circ$F.

1. What does the thermometer read after $4$ minutes?
2. When will the thermometer read $72^\circ$F?
3. When will the thermometer read $69^\circ$F?

5. An object with initial temperature $150^\circ$C is placed outside, where the temperature is $35^\circ$C. Its temperatures at 12:15 and 12:20 are $120^\circ$C and $90^\circ$C, respectively.

1. At what time was the object placed outside?
2. When will its temperature be $40^\circ$C?

6. An object is placed in a room where the temperature is $20^\circ$C. The temperature of the object drops by $5^\circ$C in $4$ minutes and by $7^\circ$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 $152^\circ$F. After another minute its temperature is $112^\circ$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 ft$^3$) produces $q_1$ ft$^3$/min of a noxious gas as a byproduct. The gas is dangerous at concentrations greater than $\overline c$, but harmless at concentrations $\le \overline c$. Intake fans at one end of the laboratory pull in fresh air at the rate of $q_2$ ft$^3$/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 $c_0$ is at a safe level, find the smallest value of $q_2$ 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 $t_0=0$, 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 $T_1$ initially contain $50$ gallons of pure water. Starting at $t_0=0$, water that contains $1$ pound of salt per gallon is poured into $T_1$ at the rate of $2$ gal/min. The mixture is drained from $T_1$ at the same rate into a second tank $T_2$, which initially contains $50$ gallons of pure water. Also starting at $t_0=0$, a mixture from another source that contains $2$ pounds of salt per gallon is poured into $T_2$ at the rate of $2$ gal/min. The mixture is drained from $T_2$ at the rate of $4$ gal/min.

1. Find a differential equation for the quantity $Q(t)$ of salt in tank $T_2$ at time $t > 0$.
2. Solve the equation derived in (a) to determine $Q(t)$.
3. Find $\lim_{t\to\infty}Q(t)$.

16. Suppose an object with initial temperature $T_0$ is placed in a sealed container, which is in turn placed in a medium with temperature $T_m$. Let the initial temperature of the container be $S_0$. 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.)

1. Assuming that the container and the medium have distinct temperature decay constants $k$ and $k_m$ respectively, use Newton’s law of cooling to find the temperatures $S(t)$ and $T(t)$ of the container and object at time $t$.
2. 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 $t$.
3. Find $\lim._{t\to\infty}S(t)$ and $\lim_{t\to\infty}T(t)$.

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 $T_m=T_m(t)$ be the temperatures of the object and the medium, respectively, and let $T_0$ and $T_{m0}$ be their initial values. Again, we assume that $T$ and $T_m$ are related by Newton’s law of cooling,

$$T'=-k(T-T_m). \tag{A}$$

We also assume that the change in heat of the object as its temperature changes from $T_0$ to $T$ is $a(T-T_0)$ and that the change in heat of the medium as its temperature changes from $T_{m0}$ to $T_m$ is $a_m(T_m-T_{m0})$, where $a$ and $a_m$ 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

$$a(T-T_0)+a_m(T_m-T_{m0})=0. \tag{B}$$

1. Equation (A) involves two unknown functions $T$ and $T_m$. Use (A) and (B) to derive a differential equation involving only $T$.
2. Find $T(t)$ and $T_m(t)$ for $t>0$.
3. Find $\lim_{t\to\infty}T(t)$ and $\lim_{t\to\infty}T_m(t)$.

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 $V^2$. Suppose $V(0)=V_0$ and the constants of proportionality are $a$ and $b$, respectively. Find $V(t)$ for $t>0$ and find $\lim_{t\to\infty}V(t)$.

19. Identical tanks $T_1$ and $T_2$ initially contain $W$ gallons each of pure water. Starting at $t_0=0$, a salt solution with constant concentration $c$ is pumped into $T_1$ at $r$ gal/min and drained from $T_1$ into $T_2$ at the same rate. The resulting mixture in $T_2$ is also drained at the same rate. Find the concentrations $c_1(t)$ and $c_2(t)$ in tanks $T_1$ and $T_2$ for $t>0$.

20. An infinite sequence of identical tanks $T_1$, $T_2$, …, $T_n$, …, initially contain $W$ gallons each of pure water. They are hooked together so that fluid drains from $T_n$ into $T_{n+1}\,(n=1,2,\cdots)$. 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 $T_1$.

1. Find the concentration $c_n(t)$ in tank $T_n$ for $t>0$.
2. Find $\lim_{t\to\infty}c_n(t)$ for each $n$.

21. Tanks $T_1$ and $T_2$ have capacities $W_1$ and $W_2$ liters, respectively. Initially they are both full of dye solutions with concentrations $c_{1}$ and $c_2$ grams per liter. Starting at $t_0=0$, the solution from $T_1$ is pumped into $T_2$ at a rate of $r$ liters per minute, and the solution from $T_2$ is pumped into $T_1$ at the same rate.

1. Find the concentrations $c_1(t)$ and $c_2(t)$ of the dye in $T_1$ and $T_2$ for $t>0$.
2. Find $\lim_{t\to\infty}c_1(t)$ and $\lim_{t\to\infty}c_2(t)$.

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 $t\to\infty$. In this case the differential equation for $Q$ is of the form

$$Q'+{a(t)\over150}Q=2 \nonumber$$

where $\lim_{t\to\infty}a(t)=1$.

1. Assuming that $Q(0)=Q_0$, can you guess the value of $\lim_{t\to\infty}Q(t)$?.
2. Use numerical methods to confirm your guess in the these cases:

$$\text{(i) } a(t)=t/(1+t) \quad \text{(ii) } a(t)=1-e^{-t^2} \quad \text{(iii) } a(t)=1-\sin(e^{-t}). \nonumber$$

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 $t\to\infty$. In this case the differential equation for $Q$ is of the form

$$Q'+{a(t)\over t+100}Q=1 \nonumber$$

where $\lim_{t\to\infty}a(t)=1$.

1. Let $K(t)$ be the concentration of salt at time $t$. Assuming that $Q(0)=Q_0$, can you guess the value of $\lim_{t\to\infty}K(t)$?
2. Use numerical methods to confirm your guess in the these cases:

$$\text{(i) } a(t)=t/(1+t)\quad \text{(ii) } a(t)=1-e^{-t^2} \quad \text{(iii) } a(t)=1+\sin(e^{-t}). \nonumber$$