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 1^st and 2^nd 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

(Q_in - Q_out) + (W_in - W_out) = E₂ - E₁

E₁ + Q_in + W_in = E₂ + Q_out + W_out

Where E is the internal energy of the system, Q is the heat transfer, W_out is the work done by the system and W_in 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 + mh_fs + Q_loss

m is the mass of the mixture, h is the enthalpy, h_fs is the heat of fusion, and Q_loss 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) - mh_fs = Q_loss

mC_p^avg(T₁ - T₂) - mh_fs = Q_loss

C_p^avg 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 = S_in - S_out + S_gen

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, S_in becomes 0.

m(s₂ - s₁) = S_gen - Q_loss/T_surr

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

mC_p^avgln(T₂/T₁) = S_gen - Q_loss/T_surr