Energy Balance
Change in T or P

Thus far, we have covered four energy topics. First, we covered some basic energy definitions (E_k, E_p, ΔH, and ΔU), then we learned to write energy balances on closed systems (those without streams, open systems (those with streams), and finally for mechanical systems (those where kinetic and potential energy dominate over internal energy like in for a waterfall).

I, myself am a global learner. In 201 and in thermodynamic courses (like ChEE 316 at Arizona), this expression is first derived from an energy balance before it is used extensively in the course material after where it's connection with energy balances fades. That is, we have learned a few energy balance expressions thus far, and for the most part, we will only apply this one reduced expression for the energy balances that will come in later classes. The topics to come deal with how to find ΔH (and thus Q) with energy balances on systems of change, that is, that involve chemical reactions, as well as changes in state, temperature, pressure. So, before we begin finding the enthalpies for these changes, I would like to solidify the connection between chapter 7 and this equation, resulting in a strong connection between the energy topics of the course.

If you have just said to yourself, this is obvious, the link is clear. Chemical engineering processes usually involve flows, resulting in the use of the open energy balance. And, these processes don't usually involve valves (or other components that change the velocity -> changes in kinetic energy), or pipes involving great height changes (no potential energy), and there is often no stirrer (no shaft work). Hence, the connection between the expressions learned in chapter 7 and the ΔH = Q expression that will usually be used from here on out, is obvious. If this was you response then bravo, and you probably skip on to the interrelations of this section, if your response was "hmm", or "huh?", then take a moment, and read on to solidify the connection.

Keeping in mind the basic equations we learned in chapter 7, and the energy balance equations for closed, open, and mechanical systems, let's consider a simple chemical process where methanol and water are separated in a condenser. What energy equations would you apply to the following system? How?

Here, the following expression is used to calculate the change in enthalpy for solids, liquids, or ideal gases.

In our problem solving, we would opt to find the enthalpy change for a change in temperature by consulting one of the tables or charts. If the tables do not have the information for our compounds or cover the temperature change we need, we would use the enthalpy change relation with heat capacity in it. The heat capacity could be given in the problem statement as a constant, or we may need to look it up in a table (Appendix B.2) where it is given as a simple polynominal in terms of temperature. Either way, whether constant or a polynominal, the intergal can be solved and we get ΔH for a temperature change.

Here, the interrelations to calculate enthalpy change are very straight- forward:

For liquids and solids, ΔĤ = Vhat*ΔP
For ideal gases, Δ Ĥ = 0

As you can see, the new interrelations introduced in this section are very simple.

All we do for a system in which there is a pressure change and a temperature change is simply calculate the ΔH for the temperature change, and then ΔH for the pressure change. Then we add up the two pieces to get the total ΔH. We can do this because enthalpy is a state function and we can use any pathway to add terms up in order to find the total.

The interrelations given here are not very complex. However, there are many different ways that we can choose to use when solving for these terms. The most important aspect of using them is clearly not math, it is reading the problem statement and recognizing that you have a temperature or pressure change between the inlet and outlet streams, and that you need to find ΔH. Then you need to be able to quickly choose a good method where you can find all the data you need.

In the next section, the definitions of ΔU and C_v are given, as are the relationships between ΔH and ΔU, and between C_v and C_p.

You should begin by realizing that ΔU can be found for an open system, and that ΔU, ΔH, C_p, and C_v are all related.

This should serve as a reminder that enthalpy and internal energy are related, as are the heat capacities. Thus, we use the open energy balance because it is convenient (tailor-made for flow problems). The reminder is that in problems where δH or δU is given or can be found, the other can be readily obtained using the following interrelations.

Relating ΔH to ΔU:

This applies to solids, liquids, and ideal gases. Simply combine this relation with the simple relations for ΔH given above to determine ΔH from ΔU and vice versa.

Relating C_p and C_v

where R is the ideal gas constant.

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