Exercise 1.1.21

Question

You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.

Khagan Gulusoy · Accepted Answer

Since the temperature of cold water is higher than the temperature of ice, the temperature of cold water starts decreasing at $t_0$. This happens as the result of heat transfer from ice to cold water, which is at first stronger than the heat transfer from the outside. 
However, as ice cools water, the surface area of ice cubes reduces. As a result of this, the heat transfer from the environment to the water will eventually be higher than the heat transfer from ice cubes to the water (time point $t_1$) and the temperature will stop decreasing and start rising.
The temperature of the water will keep rising until it is equal to the temperature of the environment (time point $t_2$).

\begin{figure}[H] \centering
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
            width=6cm,height=6cm,
             scale only axis,
             axis lines=left,
             xmin=0,xmax=9,
             ymin=0,ymax=9,
             ytick={0, 3, 7}, yticklabels={$0$, $T_{0}$, $T_{	ext{outside}}$ },
             xtick={0, 3.13729, 6.28319},
             xticklabels={$t_0$, $t_1$, $t_2$},
             axis line style = thick,
             x label style={anchor=west},
             y label style={anchor=south},
             axis on top,
             samples=50
            ]
\addplot [domain=0:3.142, samples=50, color=blue, ultra thick] {-sin(deg(x+1.5*pi))+2};
\addplot [domain=3.142:6.284, samples=50, color=blue, ultra thick] {3*sin(deg(x+0.5*pi))+4};
\addplot [domain=6.284:9, samples=2, color=blue, ultra thick] {7};

\addplot +[dashed,gray] coordinates {(3.13729,0) (3.13729,1)};
         \addplot +[dashed,gray] coordinates {(6.28319,0) (6.28319,7)};
         \addplot +[dashed,gray] coordinates {(0,7) (6.28319,7)};  
\end{axis}
\end{tikzpicture}
\end{figure}

In this graph, $T_0$ is the initial temperature of the water, $T_{	ext{outside}}$ is the environment temperature such that $T_{	ext{outside}} > T_0$.
\begin{remark} 
ormalfont
The temperature of water also depends on several other factors which we have kept constant for simplicity such as the initial temperature of the water in the glass $T_0$, environment temperature $T_{	ext{outside}}$, temperature of ice and quantity of ice and water. $f$ will cross zero if these constants are low enough. 
\end{remark}

Exercise 1.1.21

Answers

Comments

Exercise 1.1.21

Answers

Comments

Add answer