Lesson 99 — Newton's Law of Cooling

$T_0 = 100$ °C, $T_{\text{amb}} = 22$ °C, $k = 0{,}04$ min$^{-1}$. Write $T(t)$ and calculate $T(15)$.

$T_0 = 100$ °C, $T_{\text{amb}} = 20$ °C, $T(10) = 55$ °C. Determine $k$.

$T_0 = 5$ °C (cold object), $T_{\text{amb}} = 25$ °C, $k = 0{,}06$ min$^{-1}$. Calculate $T(20)$.

$T_0 = 90$ °C, $T_{\text{amb}} = 20$ °C, $k = 0{,}05$ min$^{-1}$. Calculate the half-life of the temperature difference and the temperature at that instant.

$T_0 = 80$ °C, $T_{\text{amb}} = -10$ °C, $k = 0{,}03$ min$^{-1}$. Calculate $\tau$ and $T(\tau)$.

$T_0 = 100$ °C, $T_{\text{amb}} = 20$ °C, $k = 0{,}05$ min$^{-1}$. How long until $T = 40$ °C?

$k = 0{,}04$ min$^{-1}$. How long until the temperature difference drops to less than 1% of the initial value?

Body found at 10:00 PM: $T = 33$ °C. $T_{\text{amb}} = 18$ °C, $T_{\text{normal}} = 37$ °C, $k = 0{,}06$ h$^{-1}$. Estimate the time of death.

Container with liquid: $h = 20$ W/(m²K), $A = 0{,}04$ m², $m = 0{,}3$ kg, $c_p = 4000$ J/(kgK). Calculate $k$ and the time constant $\tau$.

Derive the formula for $k$ from two temperature measurements $T_1$ (at $t_1$) and $T_2$ (at $t_2$) with known $T_{\text{amb}}$.

$T_0 = 100$ °C, $T_{\text{amb}} = 25$ °C, $k = 0{,}05$ min$^{-1}$. Use Euler with $h = 5$ min to estimate $T(15)$ and compare with the exact value.

The temperature difference between an object and the environment drops from 80 °C to 40 °C in 10 min. How much additional time until it drops from 40 to 20 °C?

Show that if $T_0 = T_{\text{amb}}$, the solution is constant. Interpret physically.

Milk: $T_0 = 72$ °C, $T_{\text{amb}}^{\text{cold}} = 0$ °C, $k = 0{,}15$ s$^{-1}$. How long to cool to 4 °C?

Forensic case. Body found at 11:00 PM with $T = 30$ °C. $T_{\text{amb}} = 20$ °C, $k = 0{,}07$ h$^{-1}$. Estimate the time of death. Discuss the method's uncertainties.

Object with constant internal heat source: $T' = -k(T - T_a) + H$, where $H = Q/(mc_p)$. With $T_a = 22$ °C, $k = 0{,}05$ min$^{-1}$, $H = 5$ °C/min. What is the equilibrium temperature?

Object warming: measurements $T(0) = 20$, $T(30) = 40$, $T(60) = 55$ °C. Estimate $T_{\text{amb}}$ and $k$ assuming one of the three equations might be noisy.

Processor with dissipation $H = 2$ °C/min, $T_a = 25$ °C. To keep $T \leq 40$ °C, what is the minimum $k$ required in the cooling system?

How does the cooling rate $|T'(t)|$ vary over time for an object with $T_0 > T_{\text{amb}}$?

How does $k$ depend on the physical properties of the system? What happens to the time constant $\tau$ when $k$ increases?

In what situations does Newton's law of cooling cease to be valid?

Two measurements: $T(0) = 30$ °C, $T(1) = 28$ °C, $T_{\text{amb}} = 20$ °C. Determine $k$ and estimate $T(5)$.

$T(0) = 30$ °C, $T(1) = 28$ °C, $T_{\text{amb}} = 20$ °C. Determine $k$ and calculate $T(5)$.

Server: $P = 2000$ W, $hA = 200$ W/K, $T_a = 24$ °C. What is the equilibrium temperature? What is needed to keep it below 27 °C?

$T_a(t) = 20 + 8\cos(\pi t/12)$ °C (daily variation with 24 h period). Write the formal solution of $T' = -k(T - T_a(t))$ and discuss how the oscillation amplitude of $T$ compares to that of $T_a$.

Show that the IVP $T' = -k(T - T_a)$, $T(0) = T_0$ has a unique solution for all $t \geq 0$.

Verify by direct substitution that $T(t) = T_{\text{amb}} + (T_0 - T_{\text{amb}})e^{-kt}$ satisfies the ODE and the initial condition.

Mutual cooling. Two objects exchange heat with each other: $T_1' = -k_1(T_1-T_2)$, $T_2' = -k_2(T_2-T_1)$. $T_1(0) = 100$ °C, $T_2(0) = 20$ °C. Find the equilibrium temperature and the rate of approach.

Steel part: $T_0 = 850$ °C, $T_{\text{amb}} = 30$ °C, $k = 0{,}02$ min$^{-1}$. How long to cool to 200 °C?

Compare Newton's law of cooling with radioactive decay. What are the mathematical similarities? What is the difference in equilibrium?