Energy Balance
Change in State

Upon studying this section, you should be familiar with the following:

ΔH for a change in state

We'll learn now to calculate the ΔH for a change in state:

A welders torch melts a metal, gaseous water molecules in the air condense on the outside of your cup, dry ice (frozen carbon dioxide) is seen to sublime. We will learn what the change in enthalpy is for each of these changes in state (which gets us to the heat released or absorbed).

The interrelation:

Calculating the phase change of a substance is fairly straight forward. The ΔH associated with a phase change is refered to as the latent heat (the energy needed to be added or removed to make the phase change happen). More specifically, there are two major types of latent heats: one gives ΔH for a change between a solid and a liquid, and the other gives the ΔH for a change between the gas and liquid phase.

ΔHvap or ΔHv:  heat of vaporization  (liquid that vaporizes)
ΔHfus or ΔHm:  heat of fusion        (solid that melts)

These values can be simply looked up in Appendix B.1.

So now, with ΔHv and ΔHm we can calculate ΔH for any phase change:

Solidification    ΔHl->s = -ΔHm
Melting           ΔHs->l =  ΔHm
Condensation      ΔHg->l = -ΔHv
Vaporization      ΔHl->g =  ΔHv
Sublimation       ΔHs->g =  ΔHm + ΔHv
Deposition        ΔHg->s = -(ΔHv + ΔHm)

notes:

There are two points I would now like to point out in doing change in state calculations:
1- Be careful with the signs. That is, ask yourself in using ΔHm, should you use (+)ΔHm (for melting because you are adding energy), or (-)ΔHm (for solidification because you are removing energy).
2- The tabulated latent heats in table B.1 correspond to specific pressures and temperatures where they are valid. For example, ΔHm, water, for the melting of ice, is listed at P=1atm and T=273K. Therefore, if asked to find conditions of melting ice from 250K and 5 atm to a final state of 1 atm and 350 K, then changes in temperature and pressure would also have to be taken in to account. That is, the problem only asked for the energy change of two states in which a melting occurs. But the calculation process would have to take into account energy release due to changes in pressure, temperature, and state. We can calculate the ΔH for each process, and then simply add them together to find the total . This second note is further developed in the following section.

Typical Calculation Process

As just mentioned, energy calculations that involve a change in state often involve additional changes in pressure and temperature.
The algorithm for doing a change in state calculation which involve initial and final temperatures and pressures that are different from the change of state temperature and pressure are as follows:

(Generic) Problem
Given initial conditions P1, T1, and state A, the substance in state A undergoes a change in state followed by an additional change in temperature and pressure, giving final conditions of P2, T2 and state B. Find ΔH of this process. Pictorially, the problem could be represented the following way:



The solution process is algorithmically as follows: It requires finding the change in enthalpy "in pieces." In each peice, we hold everything constant except one variable and calculate ΔH for that step. Then, we sum the ΔH for each step to get the total ΔH. Two points before we set up the algorithm:
1- A very important aspect in this calculation that we have to go to the exact temperature and pressure of the phase change. 2- And of course, in calculating ΔH for each change in T and P, we need to pay careful attention to the phase we are in.

Game Plan:



Notes:

1) P ref is almost always 1 atm, and Tref is the temperature for the tabulated phase change (the boiling point temperature for a vaporization and the melting point temperature for a solidification).
2) For a process, we often have multiple components, so we have to do this calculation for each component that undergoes a phase change, then sum up the ΔH of each component.
3) Often, there may just be one component that undergoes a phase change. In this case, this calculation process gives us the change of enthalpy for that component. Other components that undergo only a change in temperature and pressure and no phase change are calculated with integrals or enthalpy tables as before. The total enthalpy of the system is the sum of the enthalpies of each component added together.
4) The principle here is fundamental and intuitive: we take everything we start with, and account for any phenomena that effects molecular motion, whether a pressure, temperature, or a phase change. After all this is accounted for, we sum up the ΔH's for each step for each component to get the enthalpy change of the process. This is a similar algorithm used to determi

Here are two examples that are similar to those in Chapter 8 that require an energy balance on a system where a phase change is accompanied by a temperature change. example one, and example two.


Examples

Example 1:

An equimolar liquid mixture of benzene (B) and toluene (T) at 10 °C is fed continuously to a vessel in which the mixture is heated to 50 °C. The liquid product is 40.0 mole% B, and the vapor product is 68.4 mole % B. How much heat must be transferred to the mixture per g-mole of feed?

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Example 2:

A stream of pure cyclopentane vapor flowing at a rate of 1550 L/s at 150 °C and 1 atm enters a cooler in which 55% of the feed is condensed at constant pressure. What is the temperature at the condenser outlet?

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Example 3:

A liquid stream containing 50.0 mole% benzene and the balance toluene at 25 °C is fed to a continuous single-stage evaporator at a rate of 1320 mol/s. The liquid and vapor streams leaving the evaporator are both at 95.0 °C. The liquid contains 42.5 mole% benzene and the vapor contains 73.5 mole% benzene. Calculate the heating requirement for this process in kW.

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Example 4:

An air conditioner cools 226 m3/min of humid air at 36 °C and 98% relative humidity to 10 °C. Determine the heat required for this process.

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