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. Ĥ with change in P (at constant T and phase) (F&R 8.) Ĥ with change in T (at constant P and phase) (F&R 8.3) Ĥ 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 1 1
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) Since Uˆ and Ĥ 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 phase transition temperatures T m, T b (at which data are often listed, e.g. latent heats) latent heats heat of vapourization, heat of melting CHEE 1
Hypothetical Process Paths Calculate the enthalpy change as 1 kg of ice at 0C is transformed to superheated steam at 400C at 10 bar. H O (s) 0C, 1 atm ˆ? H H O (v) 400C,10 atm We could use the steam tables and find (Remember: final-initial): Ĥ 364 ( 334) 361 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 1 3
Hypothetical Process Path Hˆ Hˆ 1 m at the normal melting point ( T ) of H O (Table B.1) m H reference state 100 ˆ H O( l ) C p 0 dt ( C p expressions are found in Table B.) H O (s) (0C, 1 atm) ˆ? H Ĥ 1 Ĥ 5 H O (l) (0C, 1 atm) Ĥ H O (l) (100C, 1 atm) true path Ĥ 3 Hˆ Hˆ 3 v at the normal boiling point ( T ) of H O (Table B.1) b Hˆ Hˆ 0 Hˆ Hˆ Hˆ Hˆ Hˆ final 1 CHEE 1 4 H O (v) (400C, 10 atm) H O (v) (400C, 1 atm) 3 Ĥ 4 H H O (v) (100C, 1 atm) Hˆ 5 0 ( A reasonable assumption for small changes in P) ( C p expressions are found in Table B.) 4 400 ˆ H O( v) 4 C p 100 5 dt
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) H ˆ 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 Liquids and Solids: H with respect to U ˆ 0 pressure are small Hˆ PVˆ (but still very small) and will be neglected (except for steam tables). CHEE 1 5
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. H vapourization 1) Sensible heat of solid, H (T initial T melting ) ) Sensible heat of liquid, H (T melting T vapourization ) H melting 3) Sensible heat of gas, H (T vapourization T final ) T initial T final CHEE 1 6
Changes in U and H 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) C v ( T ) lim Uˆ T T 1 C T 0 v ( T ) dt Uˆ T Uˆ T V Slope = C v = heat capacity at constant volume Ideal gas: exact Solid or liquid: good approximation Nonideal gas: valid only if V constant CHEE 1 7
Changes in U and H 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) C p Hˆ ( T ) lim T T 1 C T 0 p Hˆ T Hˆ T P Ideal gas: exact Solid or liquid: good approximation Nonideal gas: exact only if P constant ( T ) dt C p = heat capacity at constant pressure Liquids and Solids: C p C v Ideal Gases: C p = C v + R CHEE 1 8
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 C p were constant, our job would be easy: H = C P (T -T 1 ) But, heat capacities are functions of temperature and are expressed in polynomial form: C p = a + bt + ct + dt 3 (Form 1 ) or, C p = a + bt+ ct - (Form ) Values of coefficients a, b, c, and d are given in Table B.. CHEE 1 9
Notes Regarding Table B. Be sure you use the correct functional form C p = a + bt + ct + dt 3 (Form 1) or C p = a + bt+ ct - (Form ) 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 10 3 = 13.0 a = 13.0 x 10-3 Be careful when you integrate! (T T 1 ) (T T 1 ) CHEE 1 10
Heat Capacity Calculations Integration C p kj 3 a bt ct dt mol C T kj Ĥ mol T 1 a(t a bt ct T 1 ) b (T dt 3 T 1 dt ) c (T 3 3 T 3 1 ) d 4 (T 4 T 4 1 ) CHEE 1 11
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 C p equation. Interpolation may be required. The reference state of these gases is: 1 atm and 5C. Use this table as you would for the steam tables, however, note that for H O, the units and reference state are different than the steam tables. CHEE 1 1
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). H vapourization ˆ H v ( T, P) : Hˆ m ( T, P) : liquid gas solid liquid H melting Table B.1 reports these two latent heats for substances at their normal melting and boiling points (i.e., at a pressure of 1 atm). CHEE 1 13
Hypothetical Process Path Hˆ Hˆ 1 m at the normal melting point ( T ) of H O (Table B.1) m H reference state 100 ˆ H O( l ) C p 0 dt ( C p expressions are found in Table B.) H O (s) (0C, 1 atm) ˆ? H Ĥ 1 Ĥ 5 H O (l) (0C, 1 atm) Ĥ H O (l) (100C, 1 atm) true path Ĥ 3 Hˆ Hˆ 3 v at the normal boiling point ( T ) of H O (Table B.1) b Hˆ Hˆ 0 Hˆ Hˆ Hˆ Hˆ Hˆ final 1 CHEE 1 14 H O (v) (400C, 10 atm) H O (v) (400C, 1 atm) 3 Ĥ 4 H H O (v) (100C, 1 atm) Hˆ 5 0 ( A reasonable assumption for small changes in P) ( C p expressions are found in Table B.) 4 400 ˆ H O( v) 4 C p 100 5 dt
Examples Week 9 Calculate the increase in specific enthalpy that occurs when acetone (v) is heated from 5 C to 100 C. A stream of nitrogen flowing at a rate of 1kg/min is heated from 50C to 00C. Calculate the heat that must be transferred. Fifteen kg/min of air is cooled from 400C to 30C. Calculate the required heat removal rate. Estimate the increase in specific enthalpy when H O (v) is heated from 300 C to 450 C. CHEE 1 15
Example 8.4- One hundred moles per second of liquid hexane at 5 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 1 16
Procedure for Energy Balance Calculations 1.Draw and completely label a process flow diagram.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 Û i or Ĥ i values Use a separate row for each phase of a species Identify unknowns with variables (e.g., Ĥ 1, Ĥ, etc.) CHEE 1 17
Procedure for Energy Balance Calculations 6.Calculate all required values of Û i or Ĥ i and insert values into table 7.Calculate U or H (e.g., H=m i Ĥ i - m i Ĥ i ) 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, E k, E p ) 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 1 18
Example 1 F&R 8.3-5 A stream containing 10% CH 4 and 90% air by volume is to be heated from 0C to 300C. Calculate the required rate of heat input in kilowatts if the flow rate of the gas is.00 x 10 3 litres (STP)/min. CHEE 1 19
Example 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 1 0
Example: Final Exam 006 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 C and 5 atm. Shaft work is delivered to the system at a rate of 30 kw to achieve the compression. 150 C 1 atm 5.760 mol/s 0.10 mol MeOH (v) /mol 0.90 mol air/mol 5.184 mol air/s 0.058 mol MeOH (v) /s 0 C 5 atm 0 C 5 atm Q W s 0.518 mol MeOH (l) / s 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 1 1
Interactive Tutorial Now is a good time to try Interactive Tutorial 5 CHEE 1