CHEE 221: Chemical Processes and Systems

Module 4. Energy Balances without Reaction

Part c: Calculating Changes in Enthalpy

(Felder & Rousseau Ch 8) Energy Balances. F&R Chapter 8

How do we calculate enthalpy (and internal energy) changes when we don’t have tabulated data (e.g., steam tables) for the process species? Basic procedures to calculate enthalpy (and internal energy changes) associated with the following processes are covered in Chapter 8 (no Reaction): Remember, 3 pieces of information set the thermodynamic state of matter: P, T and Phase. • Hˆ with change in P (at constant T and phase) (F&R 8.2)

• Hˆ with change in T (at constant P and phase) (F&R 8.3)

• Hˆ with change in phase (at constant T and P) (F&R 8.4)

To keep track of our calculations, we will summarize enthalpies in an Inlet‐Outlet Enthalpy Table.

Sections not covered in Ch 8 include 8.3c, 8.3e, 8.4b, 8.4d, 8.4e, 8.5

CHEE 221 2 Hypothetical Process Paths

 To calculate enthalpy changes, we need to construct a hypothetical process path, with: –A starting point: your defined reference state (phase, T and P) –An end point: the conditions of the stream of interest (inlet or outlet)

 SinceUˆ andHˆ are state properties (values are dependent only on the state of the species (phase, T and P) and not how they got there), any convenient process path from a reference state to a process state can be chosen

 The ideal process path will allow you to make use of: –sensible heats  heat capacities (Table B2)

– phase transition temperatures  Tm, Tb (at which data are often listed, e.g. latent heats) (Table B1) –latent heats  heat of vapourization, heat of melting (Table B1)

CHEE 221 F&R Ch 8.1b 3 Hypothetical Process Paths

Calculate the enthalpy change as 1 kg of ice at 0C is transformed to superheated steam at 400C at 10 bar.

ˆ H2O(s) H  ? H2O (v) 0C, 1 atm 400C, 10 atm

We could use the steam tables and find (Remember: final‐initial):

Hˆ  3264  (334)  3612 kJ/kg  - 334 kJ/kg is the enthalpy of ice

How would we calculate the enthalpy change if we didn’t have the steam tables?

CHEE 221 4 Construction of the Hypothetical Process Path

Hˆ  ? Our chosen H2O(s) H2O(v) reference state (0C, 1 atm) true path (400C, 10 atm)

Hˆ  Hˆ at the 1 m ˆ normal melting point Hˆ Hˆ H 5  0 1 5 ( A reasonable assumption (Tm ) of H2O (Table B.1) for small changes in P) H O H2O(l) 2 (v) (0C, 1 atm) (400C, 1 atm) 100 400 H2O(l ) Hˆ  C dT ˆ H2O(v) 2  p Hˆ Hˆ H 4   C p dT 0 2 4 100 (C expressions are p ˆ (C p expressions are found in Table B.2) H3 found in Table B.2) H2O(l) H2O(v) (100C, 1 atm) (100C, 1 atm) ˆ ˆ H3  Hv at the normal boiling point (Tb ) of H2O (Table B.1) ˆ ˆ ˆ ˆ ˆ ˆ ˆ H  H final0  H1  H2  H3  H4  H5

CHEE 221 5 Changes in H and U with P (constant T, phase)

Ideal Gases: –By definition, Uˆ  0(molecules don’t interact, so changing P doesn’t change the internal energy) – H = U + PV but PV = nRT = 0 (constant T)  Hˆ  0

Non‐Ideal Gases: – Changes in internal energy and enthalpy are small, provided P is small (< 5 atm)  Uˆ  Hˆ  0 –For steam, use tabulated values In a problem, state that changes in U and H with Liquids and Solids: respect to pressure are – Uˆ  0 small and will be –Hˆ  PVˆ (but still very small) neglected (except for steam tables).

CHEE 221 F&R Ch 8.2 6 Does H change with P? Steam Enthalpy Diagram

Use the steam tables to determine the state (liquid, vapour or mixture of the two; saturated or supersaturated) and approximate temperature (no need to use extrapolate) of 1 kg of water at 1 bar with the following enthalpies (relative to liquid water at the triple point)

a) 100 kJ b) 419 kJ c) 1500 kJ d) 2676 kJ e) 3000 kJ

Conclusion: Enthalpy is not a strong function of pressure below 10 bar CHEE 221 7 Phase Changes (at constant T and P): Latent Heat

Phase changes occur from the solid to the liquid phase, and from the liquid to the gas phase, and the reverse. The specific enthalpy change (heat) associated with the phase change at constant T and P is known as the latent heat of the phase change (i.e., latent heat of vapourization or simply heat of vapourization).

ˆ H v(T, P) : liquid  gas

ˆ Hvapourization H m(T, P) : solid  liquid

Table B.1 reports these two latent heats for substances Hmelting at their normal melting and boiling points (i.e., at a pressure of 1 atm).

CHEE 221 F&R Ch 8.4a 8 Changes in U and H with T (constant P, phase): Sensible Heat Sensible heat refers to heat that must be transferred to raise or lower the temperature of a substance without change in phase.

1) Sensible heat of solid, H

(Tinitial Tmelting) Hvapourization

2) Sensible heat of liquid, H

(Tmelting  Tvapourization)

Hmelting  3) Sensible heat of gas, H  (Tvapourization  Tfinal) 

Tinitial Tfinal

CHEE 221 F&R Ch 8.3a 9 Changes with T (constant P, phase): Sensible Heat

Closed System‐‐Find  U . The quantity of sensible heat required to produce a temperature change in a system can be determined from the appropriate form of the first law of thermodynamics: Q =  U (closed system; must be kept at constant volume)

 Uˆ   Uˆ  Cv (T )   lim     T 0 T   T     V T2 ˆ U  Cv (T)dT T1 Ideal gas: exact Solid or liquid: good approximation Slope = Cv = heat capacity at constant volume Nonideal gas: valid only if V constant

CHEE 221 10 Changes with T (constant P, phase): Sensible Heat

Open System‐‐Find  H. Enthalpy, like internal energy, also depends strongly on temperature. Q˙ =  H˙ (open system; calculate at constant pressure) Ideal gas: exact  Hˆ   Hˆ  C p (T )   lim     Solid or liquid: good approximation T 0 T   T     P Nonideal gas: exact only if P constant T2 Hˆ  C (T )dT  p Cp = heat capacity at constant pressure T1

Liquids and Solids: Cp  Cv

Ideal Gases: Cp = Cv + R

CHEE 221 11 Heat Capacity Formulas

Heat capacity – the amount of heat required to raise the temperature of one mole or one gram of a substance by one degree Celsius without change in phase units: J cal or mol K g C

If Cp were constant, our job would be easy: H = CP (T2‐T1) But, heat capacities are functions of temperature and are expressed in polynomial form: 2 3 Cp = a + bT + cT + dT (Form “1”) or, ‐2 Cp = a + bT+ cT (Form “2”)

Values of coefficients a, b, c, and d are given in Table B.2.

CHEE 221 F&R Ch 8.3b 12 Heat Capacity Calculations –Integration

C kJ  a bT  cT 2  dT 3 p mol C

T2 ˆ  kJ  2 3 H     a  bT  cT  dT dT  mol  T1 b 2 2 c 3 3 d 4 4  a(T 2T 1)  (T 2 T 1 )  (T 2 T 1 )  (T 2 T 1 ) 2 3 4

CHEE 221 13 Notes Regarding Table B.2

 Be sure you use the correct functional form 2 3 ‐2 – Cp = a + bT + cT + dT (Form 1) or Cp = a + bT+ cT (Form 2)

 Temperature units are sometimes K and sometimes C

 Positive exponent in table heading means you use negative exponent in the expression – E.g., if a x 103 = 123.0  a = 123.0 x 10‐3

2 2 2  Be careful when you integrate! (T2 –T1 )  (T2 –T1)

CHEE 221 14 Specific Enthalpies of Gases –Table B.8

Table B.8 lists specific enthalpies (kJ/mol) of selected gases (mainly combustion products, i.e. this table might be useful when solving energy balances for combustion problems) as a function of temperature.

•Can be used to estimate H changes as an alternative to integrating the Cp equation.

• Interpolation may be required.

•The reference state of these gases is: 1 atm and 25C.

•Use this table as you would for the steam tables, however, note that for H2O, the units and reference state are different than the steam tables.

CHEE 221 F&R Ch 8.3b 15 Putting it all together: Hypothetical Process Path

Hˆ  ? reference H2O(s) H2O(v) state (0C, 1 atm) true path (400C, 10 atm)

Hˆ  Hˆ at the 1 m ˆ normal melting point Hˆ Hˆ H 5  0 1 5 ( A reasonable assumption (Tm) of H2O (Table B.1) for small changes in P) H O H2O(l) 2 (v) (0C, 1 atm) (400C, 1 atm) 100 400 H2O(l ) Hˆ  C dT ˆ H2O(v) 2  p Hˆ Hˆ H 4   C p dT 0 2 4 100 (C expressions are p ˆ (C p expressions are found in Table B.2) H3 found in Table B.2) H2O(l) H2O(v) (100C, 1 atm) (100C, 1 atm) ˆ ˆ H3  Hv at the normal boiling point (Tb ) of H2O (Table B.1) ˆ ˆ ˆ ˆ ˆ ˆ ˆ H  H final0  H1  H2  H3  H4  H5

CHEE 221 16 Calculating enthalpy changes: Examples

1. Calculate the increase in specific enthalpy that occurs when

acetone(v) is heated from 25 C to 100 C. 2. A stream of nitrogen flowing at a rate of 1kg/min is heated from 50C to 200C. Calculate the heat that must be transferred. 3. Fifteen kg/min of air is cooled from 400Cto30C. Calculate the required heat removal rate.

4. Estimate the increase in specific enthalpy when H2O(v) is heated from 300 C to 450 C.

CHEE 221 17 Constructing a Process Pathway: Example 8.4‐2

One hundred moles per second of liquid hexane at 25 ºC and 7 bars pressure is vaporized and heated to 300 ºC at constant pressure. Estimate the rate at which that must be supplied.

CHEE 221 18 Procedure for Energy Balance Calculations

1. Draw and completely label a process flow diagram 2. Perform all material balance calculations 3. Choose a reference state (phase, T, P) for each species involved –If using enthalpy tables, use reference state used to generate table –If no tables are available, choose one inlet or outlet condition as the reference state for the species 4. Construct an inlet‐outlet enthalpy table – Columns for inlet and outlet amounts of each species along with their

corresponding Ûi or Ĥi values – Use a separate row for each phase of a species

– Identify unknowns with variables (e.g., Ĥ1, Ĥ2,etc.)

CHEE 221 F&R Ch 8.1c, 8.3d, 8.4c 19 Procedure for Energy Balance Calculations (cont’d)

6. Calculate all required values of Ûi or Ĥi and insert values into table

7. Calculate U or H (e.g., H=miĤi‐ miĤi) 8. Write the appropriate form of the energy balance equation, remove any negligible term, and calculate any other terms in the energy balance

equation (i.e., W, Ek, Ep) 9. Solve for the unknown quantity in the energy balance equation – Typically solve for Q but sometimes asked to solve for a mass (mole) flow or occasionally a T.

CHEE 221 F&R Ch 8.1c, 8.3d, 8.4c 20 Example 1: F&R 8.3‐5

A stream containing 10% CH4 and 90% air by volume is to be heated from 20C to 300C. Calculate the required rate of heat input in kilowatts if the flow rate of the gas is 2.00 x 103 litres (STP)/min.

CHEE 221 21 Example 2: F&R 8.1‐1

Acetone (denoted as Ac) is partially condensed out of a gas stream containing 66.9 mole% acetone vapour and the balance nitrogen. Process specifications and material balance calculations lead to the flowchart shown below.

The process operates at steady‐state. Calculate the required cooling rate.

CHEE 221 22 Example 3: Final Exam 2006

In the following process for condensing methanol vapour from air most of the entering methanol is liquefied in this steady‐state process, with the remaining fraction exiting with the air stream. Both exit streams are at 0 ºC and 5 atm. Shaft work is delivered to the system at a rate of 30 kW to achieve the compression. 5.184 mol air/s 0.058 mol MeOH /s 150 °C 1 atm (v)

5.760 mol/s 0 °C 5 atm

0.10 mol MeOH(v) /mol 0.90 mol air/mol 0 °C 5 atm

0.518 mol MeOH(l) / s Q Ws Construct an inlet‐outlet enthalpy table for the process, and calculate all unknown enthalpies. Identify the reference states selected for the components, and state all assumptions. What is the rate (kW) at which heat must be removed from the condenser?

CHEE 221 23