Worked Problem

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Section 1.9 Worked Problem

Activity 1.6.

An object with an initial temperature of $T_0$ is placed in an environment with a constant temperature $T_s$ at $t=0\text{.}$ The time-dependent temperature of the object is given by:

\begin{equation*} T = T_s + (T_0 - T_s)e^{-kt} \end{equation*}

where $k$ is a positive constant known as the thermal diffusivity and depends on the material’s thermal conductivity ($K$), specific heat ($c$), and density ($r$).

Suppose a soda can with thermal diffusivity of $k = 1.25×10^{-4}$ that was left in the sun has an initial temperature of $T = 120ºF$ and was then placed in a refrigerator at $T=38ºF\text{.}$

To the nearest degree, what is the temperature of the can after three hours?

Answer.

$T_{3hr} = 59ºF$

Solution.

>> Ts = 38;        % temperature of fridge
>> T0 = 120;       % initial temperature
>> k = 0.000125;   % thermal diffusivity
>> t = 3*60*60;    % 3 hours in second
>> 
>> T_3hr = round(Ts + (T0-Ts)*exp(-k*t))

T_3hr =
    59

Subsection 1.9.1 Improving Our Solution

Activity 1.7.

We can do much better than just finding the temperature at hour three of the experiment. We can plot its temperature across an array of times! How can we do this?

Solution.

>> t = 0:60:60*60*10;        % array of times
>> T = round(Ts + (T0-Ts)*exp(-k*t));
>> plot(t/3600, T)          % plot the curve

Introduction to MATLAB: As seen in ENGS 20

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Section 1.9 Worked Problem

Activity 1.6.

Subsection 1.9.1 Improving Our Solution

Activity 1.7.