Analysis of Ice Cream Melting Due to Ambient Heat and Licking

Problem Setup

We want to determine how quickly an ice cream stick melts under the combined effects of ambient air temperature and repeated licking.

Variables and Equations

Assumptions:

• \( T_{\text{air}} \): Ambient temperature
• \( T_{\text{ice}} \): Ice cream temperature (near freezing point)
• \( Q_{\text{lick}} \): Heat transferred per lick
• \( f \): Frequency of licking (licks per second)
• \( h_{\text{air}} \): Convective heat transfer coefficient
• \( A \): Surface area of ice cream exposed to air
• \( m \): Mass of ice cream remaining at time \( t \)
• \( L \): Latent heat of fusion of ice cream

Heat Transfer Due to Ambient Air

The rate of heat transfer from the ambient air is given by:

\[ \frac{dQ_{\text{air}}}{dt} = h_{\text{air}} A (T_{\text{air}} - T_{\text{ice}}) \]

Heat Transfer Per Lick

Each lick adds a fixed amount of heat \( Q_{\text{lick}} \), so the heat input from licking is:

\[ Q_{\text{lick}} = \text{constant heat transfer per lick} \]

Melting Rate

The melting rate is determined by the total heat input, divided by the latent heat of fusion:

\[ \frac{dm}{dt} = -\frac{1}{L} \left( \frac{dQ_{\text{air}}}{dt} + f \cdot Q_{\text{lick}} \right) \]

Dependence of Melting Time on Licking Frequency

As \( f \) increases, the additional heat input from licking also increases, accelerating the melting rate.