Handout3.pdf

CHE 205 Chemical Process Principles Section 3: F&R, Chapter 5

3-1 Copyright Richard M. Felder, Lisa G. Bullard, and Michael D. Dickey (2014)

EQUATIONS OF STATE FOR GASES Questions

A gas enters a reactor at a rate of 255 SCMH. What does that mean? An orifice meter mounted in a process gas line indicates a flow rate of 24 ft3/min. The gas

temperature is 195oF and the pressure is 62 psig. The gas is a mixture containing 70 mole% CO and the balance H2. What is the mass flow rate of the hydrogen in the gas?

A reactor feed stream consists of O2 flowing at 32 kg/s. The gas is to be compressed from 37oC and 2.8 atm absolute to 54oC and 284 atm. What are the volumetric flow rates at the inlet and outlet (needed to rate the compressor)?

A pitot tube indicates that the velocity of a stack gas is 5.0 m/s at 175oC. The stack diameter is 4.0 m. A continuous stack analyzer indicates an SO2 level of 2500 ppm (2500 moles SO2/106 moles gas). At what rate in kg/s is SO2 being discharged into the atmosphere?

A 70.0 m3 tank is rated at 2000 kPa. If 150 kg of helium is charged into the tank, what will the pressure be? How much more helium can be added before the rated pressure is attained?

Answers: Need an equation of state: relationship between temperature (T), pressure (P), volume (V), and number of moles (n) of a gas.

In Chapter 4, streams on flow charts labeled like this:

In this chapter, stream data just as likely to look like this:

For material balances, however, we still need moles and mole fractions. The job now becomes one of converting volumetric flow rates (or volumes) to molar flow rates (or moles), and (for gases) partial pressures to mole fractions. (Latter is easy: yA = pA/P )

Convert volumes to moles Solids & liquids: use tabulated densities (volume to mass) & molecular weights (mass to moles).

Mixtureseither look up mixture density data or assume volume additivity & calculate density from Eq. (5.1-1).

Gases, cant use tabulated densities. (Why not?) Instead, need an equation of state (EOS) a formula relating V, n, T, and P. Simplest is the ideal gas EOS.

Read Section 5.1. We wont lecture on it, but you need to know it.

100 mol/s

0.600 mol A/mol 0.400 mol B/mol

250 L/s @ 37oC, 800 mm Hg

pA = 420 mm Hg (partial pressure of A)



Ideal Gas Equation of State (Sections 5.2a & 5.2b)

(batch) or (continuous) or where / (or / ) is the of the gas

PV nRT PV nRT

PV RT V V n V n specific molar volume

T and P must be absolute temperature (K, oR) and absolute pressure (not gauge). R is the gas constantvalues given on inside back cover of text.

Convenientapplies regardless of what the gas is, & whether the gas has a single component or is a mixture. If you cant assume ideal gas behavior (i.e., if gas is nonideal or real),

Approximategreatest validity at low gas densities (high T, low P), when gas molecules are far enough apart for intermolecular forces to be negligible (behave like billiard balls). Usually ok for temperatures at or above 0oC & pressures at or below 1 atm. Rule of thumb for when to use it given in Eqs. (5.2-3) on p. 192.

Three will get you four. Given any three of the variables , (or ), , and (or )P V V T n n , calculate the fourth one.

Example

The volumetric flow rate of a stream of propane at 150oC and 70.0 atm being fed to a combustion furnace is measured and found to be 29.0 m3/h.

(a) Determine its molar flow rate in kmol/h.

Solution. We are given three of the four gas law variables (_____, ______, and ______) and so can determine the fourth one (________).

From the inside back cover, 0.08206 (L atm)/(mol K)R .

33 8 70.0 atm 29.0 m mol K kmol C H

h 0.08206 L atmh

kmol = 58.5 h

PVnRT

(b) Now, suppose an analysis of the combustion chamber products shows that the molar flow rate of the propane was 101 kmol/h. Think of four possible reasons for the discrepancy between the two stated values of the molar flow rate.

1. _______________________________________________________________________

2. _______________________________________________________________________

3. _______________________________________________________________________

4. _______________________________________________________________________



(c) Check the assumption of ideal gas behavior.

Solution:

0.08206 L atm L__________ , therefore from Eq. ____________, mol K mol

ideal gas beh

RTP

avior is a ___________ assumption.

For another example, work through Problem 5.10 in the workbook.

Standard temperature and pressure (STP): 0oC (273.16K, 491.67oR), 1 atm. At STP, 1 mol occupies 22.415 L, 1 kmol occupies 22.415 m3, & 1 lb-mole occupies 359.05 ft3. (Memorize)

While our text (and many other thermodynamic texts) use 0oC and 1 atm as STP, there are some other standards. In this course, please use the values on p. 194.

Suppose you are told that a gas flows at a rate of, say, 1280 SCFH [standard cubic feet per hour, or ft3(STP)/h]

(a) It does not mean that the gas is at standard temperature and pressure. It does mean that if you brought it from whatever its temperature and pressure really are to 0oC and 1 atm, its volumetric flow rate would be 1280 ft3/h. (See Example 5.2-4.)

(b) You can calculate the molar flow rate of the gas as 3

3

1280 ft (STP) 1 lb-mole lb-mole3.56hh 359.05 ft (STP)

n

(c) If you know that the actual temperature of the gas is 120oC (393K) and 0.800 atm, you can calculate its actual flow rate as

3 31280 ft (STP) 393 K 1 atm ft2300

h 273 K 0.800 atm hV

(See Example 5.2-3.)



Ideal Gas Mixtures, Partial Pressures, and Volume Percentages (Section 5.2c)

Suppose yA = mole fraction of a component A in a mixture of gases at pressure P and volume V. (Example: A mixture of gases at P = 1000 mm Hg with a volume of 200 liters contains 30 mole% CH4, 50 mole% C2H6, and 20 mole% C2H4.)

Partial pressure of a component of a gas: pA = yAP

Example: 4CH

0.30 1000 mm Hg = 300 mm Hgp The partial pressures of all components of a mixture add up to the total pressure (prove it).

Pure component volume and percentage by volume: The pure component volume of A is the volume A would occupy if it were by itself at the mixture temperature and pressure

DivideA A A AA

Pv n RT v n yPV nRT V n

(volume fraction = mole fraction)

The percentage by volume (% v/v) is 100 times the volume fraction. Thus,

% v/v = mole% for an ideal gas mixture

% v/v has no practical significance for a nonideal gas mixture

Three alternative ways of telling you the value of a mole fraction

From now on, any of these specifications may be given in material balance problems. You should immediately convert the first two to mole fractions when you label the flow chart, and if you label the partial pressure on the flow chart also label the mole fraction and count pA = yAP as another equation in the degree-of-freedom analysis.

MOLE FRACTIONS

yA

PARTIAL PRESSURE

pA = yAP

VOLUME

FRACTION % v/v = mole% for an ideal gas

mixture



Exercise. Liquid acetone (C3H6O) is fed at a rate of 400 L/min into a heated chamber where it evaporates into a nitrogen stream which enters at 27oC and 475 mm Hg (gauge). The gas leaving the heater is diluted by another nitrogen stream flowing at a measured rate of 419 m3(STP) at 25oC and 2.5 atm. The combined gases are then compressed to a total pressure P = 6.3 atm (gauge) at a temperature of 325oC. The partial pressure of acetone in this stream is pa = 501 torr (501 mm Hg). Atmospheric pressure is 763 torr.

(a) Write the complete set of equations you would solve to determine the molar composition of the product gas stream and the volumetric flow rate of the nitrogen entering the evaporator.

(b) How is it possible for the second nitrogen stream to be at standard temperature and pressure and at 25oC and 2.5 atm?

Solution. Verify that the flow chart is completely labeled.

(a) Degree-of-freedom analysis:

System equations:

(b) _________________________________________________________________________________

31

1 2

o

(m / min)

(mol N / min)

27 C, 475 torr

V

n

400 L/min C3H6O (l)

2 (mol/min)n

Evaporator

419 m3/min (STP) N2/min 3 2(mol N / min)n

25oC, 2.5 atm

Compressor 4 (mol/ min)n

y4 (mol C3H6O(v)/mol) (1y4) (mol N2/mol) 325oC, 6.3 atm (gauge) pa = 501 torr



Nonideal (Real) Gases (Section 5.3)

How would you measure P, V, T relationships in a laboratory?

Would the relationships be different for each gas species?

When you cant assume __________ behavior (low ___, high ____, outside criteria of Eqs. 5.2-3), rather than create massive tables of P, V, T data for each gas species, need to find a universal equation of state that incorporates properties of the gas species. Read p. 5-4 of the workbook.

It turns out that different gases behave similarly at the same _______________________ Section 5.3a. Each gas has a unique value for Tc and Pc (Table B.1). Lets review supercritical behavior.

We will discuss three approaches for non-ideal equations of state:

1. Virial equations of state (Section 5.3b).

1PV BPRT RT

(see p. 201 for how to calculate B)

2. Cubic equations of state and the SRK equation (Section 5.3c). Commonly used for single species (for mixtures, use compressibility factor EOS). The SRK equation of state (Eq. 5.3-7) may be the most commonly used EOS other than the ideal gas equation. Easy or hard to use, depending on which of the three variables ( , , )P V T is unknown.

Procedure:

For given species, look up Tc , Pc , and the Pitzer acentric factor (Table 5.3-1 for selected species). Calculate a, b, and m from Eqs. 5.3-8, 5.3-9, & 5.3-10.

If T and V are known, evaluate Tr from Eq. 5.3-11 and from Eq. 5.3-12, solve Eq. 5.3-7 for P. If T and P are known, enter Eq. 5.3-7 in Solver, enter all known values, and solve for V .

(Alternatively, use Solver in Excel, as in Example 5.3-3.)

If P and V are known, enter Eqs. 5.3-7, 5.3-11 for Tr, and 5.3-12 for into Excels Solver, enter all known values, and solve for T.



3. Compressibility factor equation of state (Section 5.4): PV = znRT , where z is the compressibility factor (a fudge factor the farther it is from 1, the farther the gas is from ideal).

Either you have tables where you can look up z for a given T and P (Section 5.4a, single species only), or you use the law of corresponding states (Section 5.4b) to estimate z.

Procedure: Given two of the variables T, P, and V (or V and n or and V n ) 1. Look up Tc and Pc (e.g. in Table B.1). Apply Newtons corrections for H2, He (p. 208).

2. Calculate two of the quantities reduced temperature. Tr = T/Tc , reduced pressure Pr = P/Pc , and

ideal reduced volume, ideal/r c c

VVRT P

, depending on which two of the variables T, P, and V are known.

3. Look up z on one of the generalized compressibility charts, Figs. 5.4-1 5.4-4.

4. Substitute known variables and z into the compressibility factor equation of state to determine the unknown variable.

Example: Example 5.4-2, p. 209. Note: If the gas were anything other than nitrogen, then Z will change because the critical constants are different. For example, Z = 1.4 for hydrogen.

Go through Test Yourself on p. 210. Kays rule: PVT calculations for nonideal gas mixtures using compressibility charts (Section 5.4c):

Calculate pseudocritical temperature and pseudocritical pressure by weighting Tci and Pci by mole fractions of ith component (Eqs. 5.4-9 and 5.4-10), then proceed as for single component. This is the only method we will present in this book for doing PVT calculations on nonideal gas mixtures.



Exercise. A natural gas (85 mole% CH4, 15% C2H6) at 20oC and 80 atm is burned completely with 30% excess air. natural gas 285 L/s.V The stack gas emerges at 280oC and 1 atm. What is stack gas ?V Solution.

(a) Draw and label the flow chart.

(b) What equations of state should you use to relate the volumetric flow rates of the fuel and stack gases to their molar flow rates?

Fuel gas: __________________________________________________

Stack gas: _________________________________________________

(c) Do the degree-of-freedom analysis (use atomic balances).

Furnace

CH4 + 2O2 CO2 + 2H2O C2H6 + 72 O2 2CO2 + 3H2O



(d) Write out the seven equations required to solve for the unknowns, letting (Tcm , Pcm) and (Tce , Pce) = critical temperatures of methane and ethane, respectively. (You could find their values in Table B1 but dont bother.)

Documents

Handout3.pdf