Thermodynamics

Monday, February 12, 2007

HW #5, P12.27 - Volume Change of Mixing Two Liquids - 6 pts

The volume change of mixing (in cm3/mol) for the system ethanol(1) / methyl butyl ether (2) at 25oC is given by the equation:

Given that V1 = 58.63 cm3/mol and V2 = 118.46 cm3/mol, what volume of mixture is formed when 750 cm3 of pure species 1 is mixed with 1500 cm3 of pure species 2 at 25oC ? What would be the volume of the mixture if an ideal solution were formed ?

Wednesday, February 07, 2007

HW #4, P11.25 - Fugacity of a Mixture: Real vs. Ideal Solution - 8 pts

For the system ethlyene(1) / propylene(2) as a gas, estimate: f1^, f2^, phi1^ and phi2^ at T = 150 degC, P = 30 bar and y1 = 0.35.

a.) Through application of Eqs.(11.59) and (11.60).
(Actually, use the SRK EOS, like in class.)
b.) Assuming that the mixture is an ideal solution.

HW #4, P10.25+ - Bubble and Dew Point Calculations Using the SRK EOS - 16 pts

Assuming the validity of the DePriester Charts, make the following VLE calculations for the methane(1) / ethylene(2) / ethane(3) system:

a.) Pbub, given x1 = 0.10, x2 = 0.50 and t = -60oF.
b.) Pdew, given y1 = 0.50, y2 = 0.25 and t = -60oF.
c.) Tbub, given x1 = 0.12, x2 = 0.40 and P = 250 psia.
d.) Tdew, given y1 = 0.43, y2 = 0.36 and P = 250 psia.

Hints:
Do parts b and c only.
Use the results from my solution to this problem in HW#2, based on the DePriester Charts, as a starting point for SRK.

Use the SRK equations for mixtures to answer this question. The equations are the 2 SRK eqns in terms of Zliq and Zvap and three equilibrium eqns. The unknowns are The two Z's, two of the three yi's and the total pressure.

You will need to run Solver several times, tweaking your guesstimates of the unknown variables each time, in order to get Excel to converge to the correct answer. The correct answer is the one that yields the lowest value of the Σerror^2.

HW #4, P10.31+ - Equilibrium Flash Distillation Using the SRK EOS - 20 pts

A mixture consisting of 1 mol% ethane, 5 mol% propane, 44 mol% n-butane and 50 mol% isobutane is brought to a condition of 70oF at pressure P. If the molar fraction of the system that is vapor is 0.20, what is the pressure P, and what are the compositions of the vapor and liquid phases ?

Hints:
Use the results from my solution to this problem in HW#2, based on the DePriester Charts, as a starting point for SRK. Solve for P, the 4 xi's, the 4 yi's and the two Z's (compressibility of the vapor and liquid phases). That is a whopping 11 unks. I reduced the problem to 9 unks by forcing the Σxi = 1 and Σyi = 1. The 9 eqns include 3 independent material balance eqns, 4 equilibrium eqns and 2 SRK eqns in terms of two Z's, A's and B's. Remember that A, B and Z are not the same in the liquid phase as they are in the vapor phase.

HW #4, P11.28 - Excess Gibbs Free Energy of a Real Liquid Mixture - 16 pts

The excess Gibbs energy of a binary liquid mixture at T and P is given by:

a.) Find expressions for Ln g1 and Ln g2 at T and P.
b.) Show that when these expressions are combined in accord with Eq.(11.95) the given equation for GE/RT is recovered.
c.) Show that these expressions satisfy Eq.(11.96), the Gibbs/Duhem Equation.
d.) Show that (d Ln g1 / dx1)x1=1 = (d Ln g2 / dx1)x1=0 = 0.
e.) Plot GE/RT, Ln g1 and Ln g2 as calculated by the given equation for GE/RT and by the equations developed in part (a) vs. x1. Label points Ln g1∞ and Ln g2∞ and show their values.

HW #4, P11.36 - Excess Gibbs Free Energy of a Real Liquid Mixture - 12 pts

The data in the table below are experimental values of HE for binary liquid mixtures of 1,2-dichloroethane(1) and dimethyl carbonate(2) at 313.15 K and 1 atm.

a.) Determine from the data numerical values of parameters a, b and c in the correlating equation:

b.) Determine from the results of part (a) the minimum value of HE. At what value of x1 does this occur ?

c.) Determine from the results of part (a) expressions for : partial molar HE1 and partial molar HE1. Prepare a plot of these quantities vs. x1 and discuss its features.

x1 HE~ (J/mol)
0.0426 -23.3
0.0817 -45.7
0.1177 -66.5
0.1510 -86.6
0.2107 -118.2
0.2624 -144.6
0.3472 -176.6
0.4158 -195.7
0.5163 -204.2
0.6156 -191.7
0.6810 -174.1
0.7621 -141.0
0.8181 -116.8
0.8650 -85.6
0.9276 -43.5
0.9624 -22.6

Saturday, January 20, 2007

ChemE 326 - Test #1

If you have any questions related to the first test, please post them here instead of using email.

HW #3, P11.2 - Entropy Change of Mixing - 10 pts

A vessel divided into two parts by a partition, contains 4 moles of nitrogen gas at 75 degC and 30 bar on one side and 2.5 moles of argon gas at 130 degC and 20 bar on the other. If the partition is removed and the gases mix adiabatically and completely, what is the change in entropy ? Assume nitrogen to be an ideal gas with Cv = (5/2)R and argon to be an ideal gas with Cv = (3/2)R

Hints :

Calculate ΔS for each step of a hypothetical process path.
First convert N2 from the initial to the final state. Then, change Ar from the initial to the final state. Finally, mix the N2 and the Ar at the final T and P.

Ans.: ΔS ≈ 38 J/K

HW #3, WB-1 - A Perpetual Motion Machine of the Second Kind - 12 pts

A design for purifying helium consists of an adiabatic process that splits a helium stream containing 30 mole% methane into two product steams, one containing 97 mole% helium and the other containing 90 mole% methane. The feed enters the process at 10 bar and 117oC. The methane-rich product leaves at 1 bar and 27oC while the helium-rich product leaves at 50oC and 15 bar. The process produces work! Assuming He is an ideal gas with CP = (5/2) R and methane is an ideal gas with CP = (9/2) R, calculate the total entropy change of the universe for the process on the basis of 1 mole of feed in order to determine whether the process violates the 2nd Law.

Hints :

The first step in the analysis of this process is to solve the material and energy balance equations. The more subtle part is to calculate ΔS for the process. I broke the process into a hypothetical process path consisting of seven steps. Three of these steps consist of mixing pure He and CH4 at constant T and P while the remaining four steps involved changes in T or P for either pure He or CH4.

HW #3, P11.8 - Partial Molar Volumes of the Components of a Binary Mixture - 6 pts

If the molar density of a binary mixture is given by the empirical expression:

1/V~ = a0 + a1 x1 + a2 (x1)^2

find the corresponding expressions for Vbar1 and Vbar2.

Hints :
Use equations 11.15 and 11.16.
The xi's are mole fractions of species i.

HW #3, P11.13 - Partial Molar Volumes and the Gibbs-Duhem Equation - 10 pts

The molar volume in cm^3/mol of a binary liquid mixture at T and P is given by:

V~ = 120 x1 + 70 x2 + (15 x1 + 8 x2) x1 x2

a.) Find expressions for the partial molar volumes of species 1 and 2 at T and P.
b.) Show that when these expressions are combined in accord with Eqn 11.11 the given equation for V~ is recovered.
c.) Show that these expressions satisfy Eqn 11.14, the Gibbs-Duhem equation.
d.) Show that at constant T and P,

(dVbar1/dx1)@x1=1 = (dVbar2/dx1)@x1=0 = 0

e.) Plot values of V~, Vbar1, and Vbar2 calculated by the given equation for V~ and by the equations developed in part (a) vs. x1. Label points V~1, V~2, Vbar1 at infinite dilution and Vbar2 at infinite dilution.

Hints :
a.) This is similar to 11.8 Ans.: V2 = 14 x13 + x12 + 70
b.) Just a little bit of algebra here. Ans.: V = -7 x13 - x12 + 58 x1 + 70
c.) A bit of algebra with just a touch of differential calculus.
d.) Just a couple of derivatives and some plug-n-chug.
e.) A snap for you and your friend, Excel.

HW #3, P11.23b - Fugacity of Subcooled Liquid Isobutylene - 8 pts

Estimate the fugacity of isobutylene liquid at its normal boiling point temperature and 200 bar.

Use the SRK EOS to determine the fugacity coefficient of saturated liquid isobutylene at TNBP. Then, use the SRK EOS to estimate the molar volume of saturated liquid isobutylene at TNBP.

HW #3, P11.24b - Fugacity and Fugacity Coefficient of Isobutane as Functions of Pressure - 12 pts

Prepare plots of f vs. P and phi vs. P for isobutane at 40 degC. Use a pressure range 0 to 10 bar. At 40 degC, the vapor pressure of isobutane is 5.28 bar.

Use the SRK EOS to determine the fugacity coefficient of saturated vapor/liquid isobutane. You can also use SRK to calculate the fugacity coefficient of superheated vapors directly. Then, it is relatively easy to calculate the fugacity of the superheated vapors. Subcooled liquids require the use of SRK and the Poynting Factor. Then, use the SRK EOS to estimate the molar volume of saturated liquid isobutane.

Plot 21 data points from 0 to 10 bar, every 0.5 bar. This will entail using Solver for each data point. Remember to completely label your plots.

Monday, January 01, 2007

Welcome to the ChemE Thermo Blog !

For each HW assignment and each test, I will post a message to this blog.
You can come here and click the phrase "# comments" and post your questions on the appropriate HW problem. The "#" part is the number of comments that have already been posted on each Blog entry. Try it out. Post a comment on this Blog entry just for grins.

When you post a comment on a Blog entry, you must choose an identity. Please choose "Other" and make up a screen name. Please do NOT choose "anonymous". Nobody will know who you are from your screen name.

I hope you find the Blog as helpful as other students have. It is more efficient than me answering individual email questions.

Enjoy the quarter !

HW #1, 6.1 - Slope and Curvature of Isobars on a Mollier Diagram - 4 pts

Starting with equation 6.8, show that isobars in the vapor region of a Mollier (HS) Diagram must have positive slope and positive curvature.

HW #1, 6.2 - Exactness of Differentials of State Variables - 8 pts

a.) Making use of the fact that equation 6.20 is an exact differential expression, show that:

What is the result of application of this equation to an ideal gas ?
b.) Heat capacities CP and CV are defined as temperature derivatives respectively of U and H. Because these properties are realted, one expects the heat capacities also to be related. Show that the general expression connecting CP to CV is:

Show that equation B of example 6.2 is another form of this expression.

HW #1, 6.41 - Problem title here- 8 pts

Starting from Eqn (*) in the class handout from lecture #3,

derive the equation for the Helmholtz Free Energy in Eqn (**) of the same handout.

Show all your work and explain every step of the derivation.

HW #1, 6.49 - Enthalpy and Entropy Changes of Vaporization - 12 pts

From data in the steam tables:

a.) Determine values for Gliq and Gvap for saturated liquid and vapor at 150 psia. Should these be the same ?
b.) Determine values for DHvap / T and DSvap at 150 psia. Should these be the same ?
c.) Find values for VR, HR, and SR for saturated vapor at 150 psia.
d.) Estimate a value for dP*/dT at 150 psia and apply the Clapeyron Equation to evaluate DSvap at 150 psia. Does this result agree with the steam table value ?
e.) Use the SRK EOS to evaluate VR, HR, and SR for saturated vapor at 150 psia. Do these results agree with the values you found in part [c] ?

HW #1, 6.54 - Enthalpy, Entropy and Molar Volume of Ideal Gases - 12 pts

Estimate the molar volume, enthalpy and entropy for n-butane as a saturated vapor and as a saturated liquid at 370 K. The enthalpy and entropy are set equal to zero for the ideal gas state at 101.33 kPa and 273.15 K. The vapor pressure of n-butane at 370 K is 1435 kPa.

HW #1, WB.2 - Production of Acetylene from Calcium Carbide and Water - 15 pts

Five mole of calcium carbide is combined with 10 mol of liquid water in a closed, rigid, high-pressure vessel of volume 750 mL. Acetylene gas is produced by the reaction:

CaC2(s) + 2 H2O(l) => C2H2(g) + Ca(OH)2(s)

Initial conditions are 25oC and 1 bar, and the reaction goes to completion. For a final temperature of 125oC, determine

a.) The final pressure
b.) The heat transferred

At 125oC, the molar volume of Ca(OH)2 is 33.0 mL/mol. Ignore the effect of any gas present in the vessel initially.

HW #1, 6.62 - Real and Ideal Work from the Isentropic Expansion of Ethane in a Turbine - 12 pts

A stream of ethane gas at 220oC and 30 bar expands isentropically in a turbine to 2.6 bar. Determine the temperature of the expanded gas and the work produced if the properties of ethane are calculated by:
a.) The ideal gas EOS
b.) The SRK EOS

HW #1, WB3 - Molar Volume of a Real Gas Mixture - 12 pts

An equimolar mixture of methane and propane is discharged from a compressor at 5500 kPa and 90oC at the rate of 1.4 kg/s. If the velocity in the discharge line is not to exceed 30 m/s, what is the minimum diameter of the discharge line ?

Thursday, June 01, 2006

Baratuci's Final Exam on Monday

If you have any questions related to Baratuci's Final Exam, please post them here instead of using email.

I can answer Thermo questions for Castner's students, but I cannot tell you anything about your final exam. The answer is...I don't know.

It has been a pleasure working and blogging with you.
Best of luck on the final exam !

Adios,
Dr. B

Wednesday, May 10, 2006

HW #9, P#9 - Analysis of a Regenerative Gas Refrigeration System - 10 pts

A gas refrigeration system using air as the working fluid has a pressure ratio of 5. Air enters the compressor at 0°C. The high-pressure air is cooled to 35°C by rejecting heat to the surroundings. The refrigerant leaves the turbine at -80°C and then it absorbs heat from the refrigerated space before entering the regenerator. The mass flow rate of air is 0.4 kg/s. Assuming isentropic efficiencies of 80 percent for the compressor and 85 percent for the turbine and using constant specific heats at room temperature, determine...
a.) The effectiveness of the regenerator
b.) The refrigeration load
c.) The COP of the cycle
d.) The refrigeration load and the COP if this system operated on the simple gas refrigeration cycle. Use the same compressor inlet temperature as given, the same turbine inlet temperature as calculated, and the same compressor and turbine efficiencies.

HW #9, P#8 - Performance of a Two-Stage Cascade Refrigeration System - 8 pts

Consider a two-stage cascade refrigeration system operating between the pressure limits of 1.2 MPa and 200 kPa with refrigerant-134a as the working fluid. Heat rejection from the lower cycle to the upper cycle takes place in an adiabatic counterflow heat exchanger where the pressure in the upper and lower cycles are 0.4 and 0.5 MPa, respectively. In both cycles, the refrigerant is a saturated liquid at the condenser exit and a saturated vapor at the compressor inlet, and the isentropic efficiency of the compressor is 80 percent. If the mass flow rate of the refrigerant through the lower cycle is 0.15 kg/s, determine...
a.) The mass flow rate of the refrigerant through the upper cycle
b.) The rate of heat removal from the refrigerated space
c.) The COP of this refrigerator

HW #9, P#7 - Performance of a Geothermal Heat Pump - 6 pts

A heat pump with refrigerant-134a as the working fluid is used to keep a space at 25°C by absorbing heat from geothermal water that enters the evaporator at 50°C at a rate of 0.065 kg/s and leaves at 40°C. The refrigerant enters the evaporator at 20°C with a quality of 23 percent and leaves at the inlet pressure as saturated vapor. The refrigerant loses 300 W of heat to the surroundings as it flows through the compressor and the refrigerant leaves the compressor at 1.4 MPa at the same entropy as the inlet. Determine...
a.) The degrees of subcooling of the refrigerant in the condenser
b.) The mass flow rate of the refrigerant
c.) The heating load and the COP of the heat pump
d.) The theoretical minimum power input to the compressor for the same heating load

HW #9, P#6 - Ideal Brayton Cycle with Reheating and Multistage Compression - 8 pts

Consider an ideal Brayton cycle with two stages of compression and two stages of expansion. The pressure ratio across each stage of the compressor and turbine is 3. The air enters each stage of the compressor at 300 K and each stage of the turbine at 1200 K. Determine the back work ratio and the thermal efficiency of the cycle. Use variable specific heats. Solve this problem assuming...
a.) no regenerator is used
b.) a regenerator with 75% effectiveness is used

HW #9, P#5 - Effects of Reheating and Multistage Compression on the Efficiency of a Brayton Cycle - 2 pts

A simple ideal Brayton cycle without regeneration is modified to incorporate multistage compression with intercooling and multistage expansion with reheating, without changing the pressure or temperature limits of the cycle. As a result of these two modifications...
a.) Does the net work output increase, decrease, or remain the same?
b.) Does the back work ratio increase, decrease, or remain the same?
c.) Does the thermal efficiency increase, decrease, or remain the same?
d.) Does the heat rejected increase, decrease, or remain the same?

HW #9, P#4 - Thermal Efficiency of an Air-Standard Cycle - 6 pts

An air-standard cycle is executed in a closed system and is composed of the following four processes:
1-2 Isentropic compression from 100 kPa and 27°C to 1 MPa
2-3 P constant heat addition in amount of 2800 kJ/kg
3-4 Heat rejection at constant specific volume 100 kPa
4-1 Heat rejection at constant pressure back to the initial state

Assuming constant specific heats at room temperature...
a.) Show the cycle on PV and TS diagrams
b.) Calculate the maximum temperature in the cycle
c.) Determine the thermal efficiency.

HW #9, P#3 - NEW NEW NEW - Non-Ideal Binary Ammonia-Steam Rankine Power Cycle - 10 pts

A binary vapor power cycle consists of two Rankine Cyles with steam and ammonia as the working fluids. In the steam cycle, superheated vapor enters the turbine at 900 lbf/in2 and 1100°F. Saturated liquid leaves the condenser at 140°F. The heat rejected from the steam cycle is provided to the ammonia cycle. Saturated ammonia vapor enters the ammonia turbine at 120°F. Saturated liquid leaves the ammonia condenser at 75°F. Each turbine has an sentropic efficiency of 90% and the pumps operate isentropically. The net power output of the binary cycly is 7 x 107 Btu/h.

a.) Determine the quality of each turbine effluent.
b.) Determine the mass flow rate of each working fluid in lbm/h.
c.) Determine the thermal efficiency of the binary power cycle.

HW #9, P#2 - Ideal Reheat rankine Cycle - 10 pts

Steam enters the high-pressure turbine of a steam power plant that operates on the ideal reheat Rankine cycle at 800 psia and 900°F and leaves as saturated vapor. Steam is then reheated to 800°F before it expands to a pressure of 1psia. Heat is transferred to the steam in the boiler at a rate of 6 104 Btu/s. Steam is cooled in the condenser by the cooling water from a nearby river, which enters the condenser at 45°F. Show the cycle on a TS diagram with respect to saturation lines, and determine...
a.) The pressure at which reheating takes place
b.) The net power output and thermal efficiency
c.) The minimum mass flow rate of the cooling water required

HW #9, P#1 - Effect of Boiler Pressure on Rankine Cycle Performance - 3 pts

Consider a simple ideal Rankine Cycle with fixed condenser pressure. Assume that the boiler always produces saturated steam and the condenser always produces saturated liquid water. What is the effect of increasing the boiler pressure on...

a.) Pump work input
b.) Turbine work output
c.) Heat supplied
d.) Heat rejected
e.) Cycle efficiency
f.) Moisture content at turbine exit

Multiple choice: Increases, Decreases or Remains the Same

HW #8, P#8 - Entropy Generation and Lost Work in an Evaporator - 10 pts

Air enters the evaporator section of a window air conditioner at 100 kPa and 37°C with a volumetric flow rate of 10 m3/min. The refrigerant, R134a at 150 kPa with a quality of 0.3 enters the evaporator at a rate of 0.8 kg/min and leaves as saturated vapor at the same pressure. Determine the exit temperature of the air and the rate of entropy generation for this process ...
a.) Assuming the outer surfaces of the air conditioner are perfectly insulated
b.) Assuming heat is transferred to the evaporator of the air conditioner from the surrounding medium at 37 °C at a rate of 30 kJ/min.
c.) The lost work in kW, for both parts (a) and (b).

HW #8, P#7 - Shaft Work Requirment for Single-Stage and Two-Stage Compressors - 6 pts

Air enters a two-stage compressor at 100 kPa and 27°C and is compressed to 900 kPa. The pressure ratio across each stage is the same, and the air is cooled to the initial temperature between the two stages. Assuming the compression process to be isentropic, determine the power input to the compressor for a mass flow rate of 0.02 kg/s. What would your answer be if only one stage of compression were used?

HW #8, P#6 - Entropy Generation and Lost Work in a Mixing Chamber - 6 pts

Liquid water at 200 kPa and 20°C is heated in a chamber by mixing it with superheated steam at 200 kPa and 150°C. Liquid water enters the mixing chamber at a rate of 2.5 kg/s, and the chamber is estimated to lose heat to the surrounding air at 25°C at a rate of 1200 kJ/min. If the mixture leaves the mixing chamber at 200 kPa and 60°C, determine... (a) and (b) .
a.) The mass flow rate of the superheated steam
b.) The rate of entropy generation during this mixing process
c.) The lost work in kW

HW #8, P#5 - Entropy Generation and Lost Work Due to Heat Loss From an Iron - 2 pts

A 1000 W iron is left on the ironing board with its base exposed to the air which is at 20oC. If the surface of the iron is at 400oC, determine:
a.) The total rate of entropy generation during this process in steady operation
b.) How much of this entropy generation occurs within the iron ? (Sgen,int)
c.) The lost work in kW, assuming the temperature of the surroundings is 20oC

HW #8, P#4 - Entropy Generation and Lost Work in a Double-Pipe HEX - 6 pts

Cold water (cp = 4.18 kJ/kg · °C) leading to a shower enters a well-insulated, thin-walled, double-pipe, counter-flow heat exchanger at 15°C at a rate of 0.25 kg/s and is heated to 45°C by hot water (cp = 4.19 kJ/kg · °C) that enters at 100°C at a rate of 3 kg/s. Determine ...
a.) the rate of heat transfer
b.) the rate of entropy generation in the heat exchanger
c.) The lost work in kW, assuming the temperature of the surroundings is 20oC

HW #8, P#3 - Use of Isentropic Efficiency in the Analysis of a Real Turbine - 6 pts

Steam enters an adiabatic turbine at 7 MPa, 600°C, and 80 m/s and leaves at 50 kPa, 150°C, and 140 m/s. If the power output of the turbine is 6 MW, determine...
a.) the mass flow rate of the steam flowing through the turbine
b.)the isentropic efficiency of the turbine

HW #8, P#2 - Performance of a Reversible Compressor Along Different Process Paths - 8 pts

Nitrogen gas is compressed from 80 kPa and 27°C to 480 kPa by a 10-kW compressor. Determine the mass flow rate of nitrogen through the compressor, assuming the compression process to be ...
a.) isentropic
b.) polytropic with n = 1.3
c.) isothermal
d.) ideal two-stage polytropic with n = 1.3

HW #8, P#1 - Shaft Work for a Compressor vs. a Pump - 6 pts

Saturated refrigerant-134a vapor at 15 psia is compressed reversibly in an adiabatic compressor to 80 psia. Determine the work input to the compressor. What would your answer be if the refrigerant were first condensed at constant pressure before it was compressed?

Friday, April 14, 2006

HW #7 - P #10 - Compression of an Ideal Gas in an Open and a Closed System - 6 pts

Helium gas is compressed from 90 kPa and 30°C to 450 kPa in a reversible, adiabatic process. Determine the final temperature and the work done, assuming the process takes place ...
a.) in a piston and cylinder device
b.)in a steady-flow compressor

HW #7 - P #9 - Entropy Change for an Incompressible Substance - 5 pts

A 20-kg aluminum block initially at 200°C is brought into contact with a 20-kg block of iron at 100°C in an insulated enclosure. Determine the final equilibrium temperature and the total entropy change for this process.

HW #7 - P #8 - 1st & 2nd Laws Applied to a Reversible, Adiabatic Process - 6 pts

A piston-and-cylinder device contains 5 kg of steam at 100°C with a quality of 50 percent. This steam undergoes two processes as follows:
1-2 Heat is transferred to the steam in a reversible manner while the temperature is held constant until the steam exists as a saturated vapor.
2-3 The steam expands in an adiabatic, reversible process until the pressure is 15 kPa.
a.) Sketch these processes with respect to the saturation lines on a single TS diagram
b.) Determine the heat added to the steam in process 1-2, in kJ
c.) Determine the work done by the steam in process 2-3, in kJ

HW #7 - P #7 - Maximum Work output from an Adiabatic Turbine - 5 pts

Steam enters an adiabatic turbine at 800 psia and 900°F and leaves at a pressure of 40 psia. Determine the maximum amount of work that can be delivered by this turbine.

HW #7 - P #6 - Entropy Generation in an Evaporator - 3 pts

Refrigerant-134a enters the coils of the evaporator of a refrigeration system as a saturated liquid / saturated vapor mixture at a pressure of 160 kPa. The refrigerant absorbs 180 kJ of heat from the cooled space, which is maintained at -5°C, and leaves as saturated vapor at the same pressure. For this process, determine ...
a.) the entropy change of the refrigerant
b.) the entropy change of the cooled space
c.) the total entropy change

HW #7 - P #5 - Efficiency and Reservoir Temperatures for Reversible and Irreversible Cycles - 6 pts

Complete the following involving reversible and irreversible cycles.

a.) Reversible and irreversible power cycles each discharge QC to a cold reservoir at TC and receive energy QH from hot reservoirs at TH and T'H, respectively. There are no other heat transfers involved. Show that T'H> TH.

b.) Reversible and irreversible heat pumpcycles each receive QC from a cold reservoir at TC and discharge QH to hot reservoirs at TH and T'H, respectively. There are no other heat transfers involved. Show that T'H < TH.

HW #7 - P #4 - Thermal Efficiency of an Internally Reversible HE with Multiple Heat Transfers - 4 pts

A system executes a power cycle while receiving 750 kJ by heat transfer at its boundary where the temperature is 1500 K and discharges 100 kJ by heat transfer at another portion of its boundary where its temperature is 500 K. Another heat transfer from the system occurs at a portion of the system boundary where the temperature of the system is 1000 K. No other heat transfer crosses the boundary of the system. If no internal irreversibilities are present, determine the thermal efficiency of the power cycle.

HW #7 - P #3 - Entropy Change in an Adiabatic Nozzle - 1 pt

Steam is accelerated as it flows through an actual adiabatic nozzle. The entropy of the steam at the nozzle exit will be (greater than, equal to, less than) the entropy at the nozzle inlet.

HW #7 - P #2 - Entropy Change in a Reversible, Isothermal Heating Process - 1 pt

The entropy of the working fluid of the ideal Carnot cycle (increases, decreases, remains the same) during the isothermal heat addition process.

HW #7 - P #1 - Entropy Change for an Adiabatic Process - 1 pt

A piston-and-cylinder device contains superheated steam. During an actual adiabatic process, the entropy of the steam will (never, sometimes, always) increase.

HW #6 - P #8 - Cost of Opening the Refrigerator Door - 8 pts

It is often stated that the refrigerator door should be opened as few times as possible for the shortest duration of time to save energy. Consider a household refrigerator whose interior volume is 0.9 m3 and average internal temperature is 4oC. At any given time, one-third of the refrigerated space is occupied by food items, and the remaining 0.6 m3 is filled with air. The average temperature and pressure in the kitchen are 20oC and 95 kPa, respectively. Also, the absolute humidity of the air in the kitchen and the air inthe refrigerator are 0.010 and 0.004 kg water / kg dry air (BDA), respectively. Thus, 0.006 kg of water vapor is condensed and removed from the air in the refrigerator for each kg of BDA that enters the refrigerator.

The refrigerator door is opened an average of 8 times a day. Each time the door is opened, half of the air volume in the refrigerator is replaced by the warmer, more humid kitchen air. If the refrigerator has a COP of 1.4 and the cost of electricity is $0.075 / kW-h, determine :
a.) the cost of the energy wasted per year as a result of opening the refrigerator door.
b.) What would your answer be if the kitchen air were very dry and thus a negligible amount of water condensed in the refrigerator ?

HW #6 - P #7 - Analysis of a Carnot Heat Pump - 6 pts

A Carnot heat pump is to be used to heat a house and maintain it at 20°C in winter. On a day when the average outdoor temperature remains at about 2°C, the house is estimated to lose heat at a rate of 82,000 kJ/h. If the heat pump consumes 8 kW of power while operating, determine ...
a.) how long the heat pump ran on that day
b.) the total heating costs, assuming an average price of 8.5¢/kWh for electricity
c.) the heating cost for the same day if resistance heating is used instead of a heat pump

HW #6 - P #6 - Analysis of a Carnot Refrigerator - 6 pts

A Carnot refrigerator operates in a room in which the temperature is 25°C. The refrigerator consumes 500 W of power when operating and has a COP of 4.5. Determine ...
a.) The rate of heat removal from the refrigerated space
b.) The temperature of the refrigerated space

HW #6 - P #5 - Analysis of a Carnot Heat Engine - 6 pts

A heat engine is operating on a Carnot cycle and has a thermal efficiency of 55 percent. The waste heat from this engine is rejected to a nearby lake at 60°F at a rate of 800 Btu/min. Determine...
a.) The power output of the engine
b.) The temperature of the source.

HW #6 - P #4 - Effect of Source and Sink Temperatures on HE Efficiency - 8 pts

A heat engine operates between a source at TH and a sink at TC. Heat is supplied to the heat engine at a steady rate of 1200 kJ/min. Study the effects of TH and TC on the maximum power produced and the maximum cycle efficiency. For TC = 25oC, let TH vary from 300oC to 1000oC. Create plots of Wcycle and ηth as functions of TH. For TH = 550oC, let TC vary from 0oC to 50oC. Create plots of Wcycle and ηth as functions of TC. Discuss the results.

HW #6 - P #3 - A Reversible HE Used to Drive a Reversible Heat Pump - 6 pts

A reversible power cycle receives QH from a hot reservoir at TH and rejects Qsurr to the surroundings at T0. The work developed by the power cycle is used to drive a reversible refrigeration cycle that removes QC from a cold reservoir at TC and rejects heat to the same surroundings T0.
a.) Develop an expression for the ratio QC / QH in terms of the temperature ratios TH / T0 and TC / T0.
b.) Plot QC / QH versus TH / T0 for TC / T0 = 0.85, 0.90 and 0.95.
c.) Plot QC / QH versus TC / T0 for TH / T0 = 2, 3 and 4.

HW #6 - P #2 - Reversible, Irreversible and Impossible Power Cycles - 8 pts

A power cycle operating between two reservoirs receives QH from a hot reservoir at TH = 2000 K and rejects QC to a cold reservoir at TC = 400 K. For each of the following cases, determine whether the cycle is reversible, irreversible or impossible.
a.) QH = 1200 kJ and Wcycle = 1020 kJ
b.) QH = 1200 kJ and QC = 240 kJ
c.) QC = 600 kJ and Wcycle = 1400 kJ
d.) η = 40%

HW #6 - P #1 - "Show That" Problem Using the K-P Statement of the 2nd Law - 6 pts

Using the Kelvin-Planck Statement of the Second Law of Thermodynamics, demonstrate the following corollaries.
a.) The COP of an irreversible heat pump cycle is always less than the COP of a reversible heat pump cycle when both cycles exchange heat with the same two reservoirs.
b.) All reversible heat pump cycles operating between the same two reservoirs have the same COP.

HW #5 - P #10 - Filling a Balloon with Helium - 10 pts

A balloon initially contains 65 m3 of helium gas at atmospheric conditions of 100 kPa and 22oC. The balloon is connected by a valve to a large reservoir that supplies helium gas at 150 kPa and 25oC. Now, the valve is opened and helium is alowed to enter the balloon until pressure equilibrium with the helium at the supply line is reached. The material of the balloon is such that its volume increases linearly with pressure. If no heat transfer takes place during this process, determine the final temperature of the helium in the balloon.

HW #5 - P #9 - Operation of a Pressure Cooker - 8 pts

A 4 L pressure cooker has an operating pressure of 175 kPa. Initially, one-half of the volume is filled with liquid and the other half with vapor. If it is desired that the pressure cooker not run out of water for one hour, determine the highest rate of heat transfer allowed.

HW #5 - P #8 - Temperature Rise and Efficiency of a Pump - 6 pts

The pump of a water distribution system is powered by a 15-kW electric motor whose efficiency is 90 percent. The water flow rate through the pump is 50 L/s. The diameters of the inlet and outlet pipes are the same, and the elevation difference across the pump is negligible. If the pressures at the inlet and outlet of the pump are measured to be 100 kPa and 300 kPa (absolute), respectively, determine...
a.) The mechanical efficiency of the pump
b.) The temperature rise of water as it flows through the pump due to the mechanical inefficiency

HW #5 - P #7 - Steam Condensation Using Cooling Water - 6 pts

Steam enters the condenser of a steam power plant at 20 kPa and a quality of 95 percent with a mass flow rate of 20,000 kg/h. It is to be cooled by water from a nearby river by circulating the water through the tubes within the condenser. To prevent thermal pollution, the river water is not allowed to experience a temperature rise above 10°C. If the steam is to leave the condenser as saturated liquid at 20 kPa, determine the mass flow rate of the cooling water required.

HW #5 - P #6 - An Ideal, Adiabatic Open Feedwater Heater - 6 pts

In steam power plants, open feedwater heaters are frequently utilized to heat the feedwater by mixing it with steam bled off the turbine at some intermediate stage. Consider an open feedwater heater that operates at a pressure of 1000 kPa. Feedwater at 50°C and 1000 kPa is to be heated with superheated steam at 200°C and 1000 kPa. In an ideal feedwater heater, the mixture leaves the heater as saturated liquid at the feedwater pressure. Determine the ratio of the mass flow rates of the feedwater and the superheated vapor for this case.

HW #5 - P #4 - Steady Flow Through an Air Compressor - 6 pts

Air enters the compressor of a gas-turbine plant at ambient conditions of
100 kPa and 25°C with a low velocity and exits at 1 MPa and 347°C with a
velocity of 90 m/s. The compressor is cooled at a rate of 1500 kJ/min, and
the power input to the compressor is 250 kW. Determine the mass flow rate of
air through the compressor.

HW #5 - P #3 - Steady Flow Through an Adiabatic Steam Turbine - 6 pts

Steam flows steadily through an adiabatic turbine. The inlet conditions of
the steam are 10 MPa, 450°C, and 80 m/s, and the exit conditions are 10 kPa,
92 percent quality, and 50 m/s. The mass flow rate of the steam is 12 kg/s.
Determine...
a.) The change in kinetic energy
b.) The power output
c.) The turbine inlet area

HW #5 - P #2 - Outlet Temperature and Velocity in an Adiabatic Diffuser - 6 pts

Air at 13 psia and 20°F enters an adiabatic diffuser steadily with a
velocity of 600 ft/s and leaves with a low velocity at a pressure of 14.5
psia. The exit area of the diffuser is 5 times the inlet area. Determine...
a.) The exit temperature
b.) The exit velocity of the air

HW #5 - P #1 - Steady Flow Through a Nozzle - 6 pts

Steam at 5 MPa and 400°C enters a nozzle steadily with a velocity of 80 m/s,
and it leaves at 2 MPa and 300°C. The inlet area of the nozzle is 50 cm2,
and heat is being lost at a rate of 120 kJ/s. Determine...
a.) The mass flow rate of the steam
b.) The exit velocity of the steam
c.) The exit area of the nozzle

HW #5 - P #5 - Temperature Drop in an Adiabatic Throttling Valve - 6 pts

Refrigerant-134a is throttled from the saturated liquid state at 700 kPa to a pressure of 160 kPa. Determine the temperature drop during this process and the final specific volume of the refrigerant.

Tuesday, April 11, 2006

HW #4 - P #11 - Coefficient of Performance of a Household Heat Pump

A heat pump used to heat a house runs about one-third of the time. The house is losing heat at an average rate of 22,000 kJ/h. If the COP of the heat pump is 2.8, determine the power the heat pump draws when running.

HW #4 - P #10 - Coefficient of Performance of a Household Refrigerator

A household refrigerator with a COP of 1.2 removes heat from the refrigerated space at a rate of 60 kJ/min. Determine (a) the electric power consumed by the refrigerator and (b) the rate of heat transfer to the kitchen air.

HW #4 - P #9 - Thermal Efficiency of an Automobile Engine

An automobile engine consumes fuel at a rate of 28 L/h and delivers 60 kW of power to the wheels. If the fuel has a heating value of 44,000 kJ/kg and a density of 0.8 g/cm3, determine the thermal efficiency of this engine.

HW #4 - P #8 - Thermal Efficiency of a Steam Power Plant

A 600-MW steam power plant, which is cooled by a nearby river, has a thermal efficiency of 40 percent. Determine the rate of heat transfer to the river water. Will the actual heat transfer rate be higher or lower than this value? Why?

HW #4 - P #7 - 1st Law Applied to Cooking an Egg

An ordinary egg can be approximated as a 5.5-cm-diameter sphere. The egg is initially at a uniform temperature of 8°C and is dropped into boiling water at 97°C. Taking the properties of the egg to be = 1020 kg/m3 and cP = 3.32 kJ/kg · °C, determine how much heat is transferred to the egg by the time the average temperature of the egg rises to 80°C.

HW #4 - P #6 - Electrical Work and the 1st Law

A mass of 15 kg of air in a piston-and-cylinder device is heated from 25 to 77°C by passing current through a resistance heater inside the cylinder. The pressure inside the cylinder is held constant at 300 kPa during the process, and a heat loss of 60 kJ occurs. Determine the electric energy supplied, in kWh.

HW #4 - P #5 - Two Step Expansion with Stops

A piston-and-cylinder device initially contains 0.8 m3 of saturated water vapor at 250 kPa. At this state, the piston is resting on a set of stops, and the mass of the piston is such that a pressure of 300 kPa is required to move it. Heat is now slowly transferred to the steam until the volume doubles. Show the process on a PV diagram with respect to saturation lines and determine...
a.) The final temperature
b.) The work done during this process
c.) The total heat transfer

HW #4 - P #4 - Conduction and Convection Heat Transfer

A flat surface is covered with insulation with a thermal conductivity of 0.08 W/m-K. The temperature at the interface between the surface and the insulation is 300°C. The outside of the insulation is exposed to air at 30°C and the convective heat transfer coefficient between the insulation and the air is 10 W/m2-K. Ignoring radiation, determine the minimum thickness of insulation in meters such that the outside surface of the insulation is not hotter than 60°C at steady-state.

HW #4 - P #3 - Convection and Radiation Heat Transfer

A 5cm diameter spherical ball whose surface is maintained at a temperature of 70oC is suspended in the middle of a room at 20oC. If the convection heat transfer coefficient is 15 W/m2-oC and the emissivity of the surface is 0.8, determine the total rate of heat transfer from the ball.

HW #4 - P #2 - Two Step Expansion with a Linear Spring

A piston-and-cylinder device contains 50 kg of water at 250 kPa and 25°C.
The cross-sectional area of the piston is 0.1 m2. Heat is now transferred to
the water, causing part of it to evaporate and expand. When the volume reaches 0.2 m3, the piston reaches a linear spring whose spring constant is 100 kN/m. More heat is transferred to the water until the piston rises 20 cm more. Determine...
a.) The final pressure and temperature
b.) The work done during this process
c.) Show the process on a P-V diagram

HW #4 - P #1 - Boundary Work as a Gas Expands

During an expansion process, the pressure of a gas changes from 15 to 100 psia according to the relation P = aV + b, where a = 5 psia/ft3 and b is a constant. If the initial volume of the gas is 7 ft3, calculate the boundary work done during the process.

Monday, April 10, 2006

HW #3 - P7 - Hypothetical Process Paths and the Latent Heat of Vaporization

Use the hypothetical process path shown here to help you determine the change in enthalpy in Joules for 20.0 g of heptane (C7H16) as it changes from a saturated liquid at 300 K to a temperature of 370 K and a pressure of 58.7 kPa. Calculate the DH for each step in the path. Do not use tables of thermodynamic properties, except to check your answers. Instead, use the Antoine Equation to estimate the heat of vaporization of heptane at 300 K. Use the average heat capacity of heptane gas over the temperature range of interest. Assume heptane gas is an ideal gas at the relevant temperatures and pressures.

HW #3 - P6 - Vapor Pressure of Solids and the Clausius-Clapeyron Equation

Using the Clapeyron-Clausius equation and the triple- point data of water, estimate the sublimation pressure of water at -30°C and compare to the actual value of 38.02 Pa. ΔHsub = 2838.4 kJ/kg .

HW #3 - P5 - Using Vapor Pressure Data to Estimate ΔHvap

Using the Clapeyron equation, estimate the enthalpy of vaporization of steam at 300 kPa, and compare it to the tabulated value.

HW #3 - P4 - Determining Internal Energy Change Using Heat Capacity Polynomials

Determine the change in the specific internal energy of hydrogen (H2), in kJ/kg, as it is heated from 400 to 1000 K, using:
a.) the empirical specific heat equation (Shomate Equation) from the back of your book.
b.) The Cv(IG) value at the average temperature. (Use theat capacity polynomial to determine this CoV value.)
c.) The Cv(IG)value at room temperature, 25oC. (Use theat capacity polynomial to determine this Cv(IG)value.)

HW #3 - P3 - Determining Enthalpy Change Using Heat Capacity Polynomials

Determine the change in the specific enthalpy of nitrogen (N2), in kJ/kg, as it is heated from 600 to 1000 K, using:
a.) the empirical specific heat equation (Shomate Equation) from the back of your book.
b.) The Cp IG value at the average temperature. (Use the heat capacity polynomial to determine this Cp IG value.)
c.) The Cp IG value at room temperature, 25oC. (Use theat capacity polynomial to determine this Cp IG value.)

HW #3 - P2 - R-134a Table Fundamentals

Complete this table for refrigerant-134a.

T(oF) P (psia) H (Btu/lbm) x Phase Description
a.) 80 141.768
b.) 15 0.6
c.) 10 70
d.) 180 193.228
e.) 110 1.0

HW #3 - P1 - Steam Table Fundamentals

Complete the blank cells in the following table of properties of steam. In
the last column describe the condition of steam as compressed liquid,
saturated mixture, superheated vapor, or insufficient information; and, if
applicable, give the quality.

T(oC) P (kPa) V (m3/kg) U (kJ/kg) Quality
x Phase Description
a.) 30 200
b.) 130 270.28
c.) 400 1.5493
d.) 300 0.500
e.) 500 3084

Saturday, April 01, 2006

HW #2 - P #8 - Relative Humidity

A thermos bottle is half-filled with water and is left open to the atmospheric air at 70°F and 35% relative humidity. If heat transfer to the water through the thermos walls and the free surface is negligible, determine the temperature of water when phase equilibrium is established.

HW #2 - P #7 - Vapor Pressure of Water

During a hot summer day at the beach when the air temperature is 30°C, someone claims the vapor pressure in the air to be 5.2 kPa. Is this claim reasonable?

HW #2 - P #6 - An Application of Equations of State

A 0.016773 m3 tank contains 1 kg of R-134a at 110oC. Determine the pressure of the refrigerant using:
a.) the ideal gas EOS
b.) the generalized compressibility chart
c.) the refrigerant tables
d.) van der Waals EOS
e.) Soave-Redlich-Kwong EOS

HW #2 - P #5 - PVT Relationship Applied to an Automobile Tire

The pressure in an automobile tire depends on the temperature of the air in the tire. When the air temperature is 25°C, the pressure gage reads 210 kPa. If the volume of the tire is 0.025 m3, determine the pressure rise in the tire when the air temperature in the tire rises to 50°C. Also, determine the amount of air that must be bled off to restore pressure to its original value at this temperature. Assume atmospheric pressure is 100 kPa.

HW #2 - P #4 - Isochoric Condensation of Water Vapor

Superheated water vapor at 180 psia and 500°F is allowed to cool at constant volume until the temperature drops to 250°F. At the final state, determine...
a.) the pressure
b.) the quality
c.) the enthalpy
d.) Show the process on a TV diagram with respect to saturation curves

HW #2 - P #3 - Isobaric Heating of Water and Steam

A piston-and-cylinder device contains 0.1 m3 of liquid water and 0.9 m3 of water vapor in equilibrium at 800 kPa. Heat is transferred at constant pressure until the temperature reaches 350°C.
a.) What is the initial temperature of the water?
b.) Determine the total mass of the water.
c.) Calculate the final volume.
d.) Show the process on a PV diagram with respect to saturation curves.