Question

Imagine that you have just purchased a coffee in a cup, with a lid. You place your coffee cup on a table and it starts to cool down. We are going to model this cooling process. Choose whatever type of coffee cup you are most familiar with.

2. Do a Ninja Physics estimate of how long it will take for the sides of the cup to warm up.

The specific heat capacity of water (and coffee) is around 4200 J Kg-1 K-1. Cardboard has a thermal conductivity of around k ~ 0.025 W m-2 K-1 , a density of around 700 kg m-3 and a specific heat capacity of around 1400 J kg-1 K-1. Glass has a thermal conductivity of around k ~ 1 W m-2 K-1, a density of around 2500 kg m-3 and a specific heat capacity of around 700 J kg-1 K-1. Styrofoam (foamed plastic) has a thermal conductivity similar to air and a density of around 50 kg m-3, while hard plastics have thermal conductivities of k ~ 0.025 W m-2 K-1 and densities similar to water: both have specific heat capacities of around 1500 J Kg-1 K-1.

Answer 1

Cup is getting warmed up due to Heat conduction.

Heat conduction equation :-

.....................(1)

where dQ is heat conducted in time dT , k is thermal conductivity, A is cross section area through which heat is conducted,

dT is temperature difference between hot coffee surface and outer surface of cup and dx is the thickness of the cup.

we will assume A and dx are same for all cups.   Let us assume C = A/dx, a constant value.

Also we assume , sides of cup to warm up means outer surface should reach a temperature 50^o C and temperature of coffee is at 60^o C so that dT = 10^o C

Hence, using eqn.(1), heat energy dQ absorbed by the cup in a time duration t s is given by,

..........................(2)

Let us assume initially the cup is at ambient temperature, say, 20^oC and now temperature has raised to 50^oC

due to warming up , i.e, temperature rise = 30^oC

Quantity of heat dQ required to raise the temperature by is given by,

dQ = mC_p .......................(3)

where m is mass of cup, C_p is specific heat of material of cup

If we assume all the cups made of different materials are having same volume V, then

eqn.(3) is written as,   dQ = VC_p .........................(4)

where is density of material of cup

If we equate eqn.(2) and (4), required time duration t to warm up is obtained as

............................(5)

we assumed the constant C = A/dx, where A is surface area and dx is thickness of cup.

hence constant C is written as, C = V/(dx)²

Using the above substitution, eqn.(5) is written as

.............................(6)

Hence from known values of density, specific heat and thermal conductivity and assuming reasonable thickness, time duration t to warm up the cup can be calculated from eqn.(6)