# Energy BalanceChange in T or P

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

• ΔH = Q, it's derivation and why it is so prevenlent from here on.
• Finding ΔH for changes in pressure and temperature
• By using Cp
By using tabulated enthalpy tables
• The relationship between ΔH and ΔU, and between Cv and Cp

## Let's Get a Perspective

 So far, we've covered many Energy balance topics. Where are we? Thus far, we have covered four energy topics. First, we covered some basic energy definitions (Ek, Ep, 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). For many of the rest of the problems in this class, ΔH = Q. Uh, what? 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. Make 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? As your experiences so far in the course tells you, the process is a fairly basic one: a single unit, two compontent process with one inlet stream and two outlet streams. So, consider your energy balance tool box: Which tool do you use, and how? Well, in the tool box we have a closed energy balance, an open one, and a mechanical system energy balance equation. There are streams in this problem (flowrates), thus the system is not closed. There are no valves or huge height changes so the system is not a strictly mechanical system using only a mechanical energy balance. Thus, it is clearly an open system where only the open energy balance equation applies. Finally, the equation is: ΔH+ΔEk+ΔEp=Q+Ws There clearly is no change in flow velocity (no valve), no real significate change in height, or a stirrer in the vessel. Therefore, respectively, the kinetic energy, potential energy, and shaft work can all be neglected. ΔH+ΔEk+ΔEp=Q+Ws And now we are left with a common reduced expression relating enthalpy to heat: ΔH=Q

## ΔH for Changes in Pressure and/or Temperature

 Now we understand the infamous expression ΔH = Q, where it comes from and why it recurs often in chemical engineering. Next, we will learn how to find ΔH (and thus Q) for various scenarios. On this webpage we will learn how to do this for changes in temperature and pressure. A later webpage will show us how to do this for changes in state and for chemical reactions. ΔH for a Temperature Change: Here, the following expression is used to calculate the change in enthalpy for solids, liquids, or ideal gases. or, ΔH = Hfinal - Hinitial Hinitial where, Hfinal and Hinitial for several different compounds are found directly from tables (such as table B8, B9), or for water from steam tables (such as tables B6 and B7). For the specific system of air and water vapor, the text offers a psychrometric plot (a cross-plot that relates several properties on on one plot) we can use to get Hinitial and Hfinal. For a tutorial on how to read a water vapor - air psychrometric chart, click here. 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. ΔH for a Pressure Change: Here, the interrelations to calculate enthalpy change are very straight- forward: For liquids and solids, ΔĤ = Vhat*ΔP For ideal gases, Δ Ĥ = 0As you can see, the new interrelations introduced in this section are very simple. ΔH for a Pressure and Temperature Change: 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. ΔHchange in T & P = Δ Hfrom T1 to T2 + ΔHfrom P1 to P2 Where the relations for ΔHfrom T1 to T2 and ΔHfrom P1 to P2 for solids, liquids, and gases have been given above. 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 Cv are given, as are the relationships between ΔH and ΔU, and between Cv and Cp.

## ΔU for Changes in Pressure and/or Temperature

 You should begin by realizing that ΔU can be found for an open system, and that ΔU, ΔH, Cp, and Cv 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: ΔH = ΔU + PΔV 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 Cp and CvCp = Cv (for solids and liquids)Cp = Cv + R (for ideal gases) where R is the ideal gas constant. Getting ΔU from Cv This interrelationship applies for a temperature change (at constant volume), and is analogous to the expression relating ΔH and Cp.ΔÛ = ∫ CV(T) dT, T1 TO T2 The relationships and equations were given here in this section so you can see how the variables are interrelated and can be interchanged. Also, remember that the open energy balance is used for a reason (flowing streams). Most of the systems in chemical engineering are open ones so an emphasis was placed on expressions of ΔH first.

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