User:EspadaNoches/Ice Cream

Ice cream is generally made freezing a milk or cream mixture containing sugar and various flavorings to a desired temperature and consistency. The creation process can be analyzed using the 1st and 2nd laws of thermodynamics.

Introduction
A classic method of making homemade ice cream is to place a bag of ice cream mixture into an insulated container of ice and salt. The salt inhibits the re-freezing of water during the ice's melting process as well as "lowers" the freezing/melting point of water which in turn limits the amount of heat transfer into the mixture. As the ice melts around the cream it absorbs energy, but since the ice container is insulated it must draw the thermal energy from the cream mixture. In other words, the cream freezes as the ice melts.

For analysis purposes, the system is the cream and the surroundings are the ice mixture. State 1 is a liquid mixture and state 2 is a completely solid ice cream

Analysis using the First Law
Using the first law of thermodynamics to get an equation for energy balance

ΔE=E2-E1 where ΔE=Q+W gives the equation

(Qin - Qout) + (Win - Wout) = E2 - E1

E1 + Qin + Win = E2 + Qout + Wout

Where E is the internal energy of the system, Q is the heat transfer, Wout is the work done by the system and Win is the work done on the system.

Assumptions

 * Volume and pressure changes are minimal and therefore the total change is equal to zero.
 * The work input contribution is small enough to be neglected as very little energy is absorbed by shaking.
 * There is no mass transfer between the mixture and ice.
 * The bag containing the cream is perfectly conductive.
 * The bag contains enough ice that the total temperature drop is insignificant

Final Equation
As there is no movement of the system, no elasticity, and no chemical change, the only energy contribution is from the change in enthalpy.

mh1 = mh2 + mhfs + Qloss

m is the mass of the mixture, h is the enthalpy, hfs is the heat of fusion, and Qloss is the heat lost by the cream to the ice. Re-arranging the equation and using the properties of enthalpy give a new set of equations.

m(h1 - h2) - mhfs = Qloss

mCpavg(T1 - T2) - mhfs = Qloss

Cpavg is the average heat capacity of the cream at constant pressure and T is the temperature of the mixture.

Analysis Using the Second Law
Using the entropy balance equation

ΔS = Sin - Sout + Sgen

Assumptions

 * Volume and pressure changes are minimal and therefore the total change is equal to zero.
 * The work input contribution is small enough to be neglected as very little energy is absorbed by shaking.
 * There is no mass transfer between the mixture and ice.
 * The bag containing the cream is perfectly conductive.
 * The bag contains enough ice that the total temperature drop is insignificant

Final Equations
Because there is no mass flowing through the boundary, the mass transfer contribution becomes 0. As there is also no heat flow into the system, Sin becomes 0.

m(s2 - s1) = Sgen - Qloss/Tsurr

Where s is the specific entropy and Sgen is the total entropy generated by the process. Now using a derivation of the second law, this equation becomes

mCpavgln(T2/T1) = Sgen - Qloss/Tsurr