化工热力学英文习题

Michigan State University

DEPARTMENT OF CHEMICAL ENGINEERING AND MATERIAL SCIENCE

ChE 821: Advanced Thermodynamics Fall 2008

1. (30) A thermodynamicist is attempting to model the process of balloon inflation by assuming

that the elastic casing behaves like a spring opposing the expansion (see below). The model

assumes that the piston/cylinder is adiabatic. As air (following the ideal gas law) is admitted, the

spring is compressed. The pressure on the spring side of the piston is zero, so that the spring

provides the only force opposing movement of the piston. The pressure in the tank is related to

the gas volume by Hooke’s law

P ? P i = k (V – V i )

where k = 1E-5 MPa/cm 3, P i = 0.1 MPa, T i = 300K, and V i = 3000 cm 3, Cv = 20.9 J/mol K,

independent of temperature, and the reservoir is at 0.7 MPa and 300K.

Provide the balances needed to determine the gas temperature in the cylinder at volume V =

4000cm 3. Perform all integrations. Do not calculate the gas temperature, but provide all

equations and parameter values to demonstrate that you could determine the gas temperature.

2. (30) Consider two air tanks at the initial conditions shown below. We wish to obtain work

from them by exchanging heat and mass between the tanks. No gas may be vented to the

atmosphere, and no heat may be exchanged with the atmosphere. Reversible devices may be

used to connect the two tanks.

Provide the balances necessary to calculate the maximum work that may be obtained. Perform all

integrations. Do not calculate the work value, but provide all equations and parameter values to

demonstrate that you could determine the work value. C p = 29.3 J/molK. Use the ideal gas law.

Tank A 400 K 5 bar 6 m 3 Tank B 200 K 0.1 bar 10 m 3

3. (a) (15) It is desired to express the derivative ()

T

V ?, which is related to isothermal compressibility in terms of ()

S V P ??, which is related to adiabatic compressibility. Derive a relation by starting with ()T V ? and interposing P and S using the Jacobian method. Leave the answer in terms of derivatives involving S .

(b) (15) Express ((()S P P T V S ??? in terms of measureable properties.

4. (10) Show ,,T V T N V μμ??????=?????????.

Equations

R = 8.3143 J/(molK) = 8.3143 cm 3MPa/(molK) = 83.143 cm 3bar/(molK)

dU = TdS --- PdV + μdN ? -(?P/?S)V = (?T/?V)S dH = TdS + VdP + μdN ? (?V/?S)P = (?T/?P)S dA = -SdT - PdV + μdN ? (?P/?T)V = (?S/?V)T dG = -SdT + VdP + μdN ? -(?V/?T)P = (?S/?P)T

Jacobian Formula

(),(,)Y X Y X X Y Y X

K K X Y K L K L K L X Y X Y Y X L L X Y ??ê?ê?á?á??ˉ?ˉ???????ê?ê?ê?ê?=-=á?á?á?á??ˉ?ˉ?ˉ?ˉ???????ê?ê?á?á??ˉ?ˉ??

e e d c b c b ????????=????????, , c b c b e d =??????????, , ,d e e e b b b d N c c d c ???????????=???????????????????? , ,,,b e d e c e

d c b d c b ????????????=?????????????

Michigan State University

DEPARTMENT OF CHEMICAL ENGINEERING AND MATERIALS SCIENCE

ChE 821: Advanced Thermodynamics Fall 2007

1.

The mass flow controllers are set to maintain constant molar of a gas in and out of the perfectly

insulated tank. The initial conditions are specified and at the start of operation, the mass flow

controllers are simultaneously and instantaneously put into operation at 1.5 mol/h. Conditions

of stream A are constant with time. The gas may be assumed to be an ideal gas with Cp = 29

J/mol-K.

(a) (10 pt) Write the energy balance for the tank in the most simplified form.

(b) (20 pt) Rearrange the energy balance to solve for the conditions in the tank as a function of

time. Perform all integrations. Do not calculate the values, but provide all equations and

parameter values to demonstrate that you could calculate the conditions.

(c) (20 pt) Write the entropy balance for the tank in the most simplified form for a boundary

including both valves and the tank.

(d) (10 pt) Provide formulas to calculate the entropy in the tank at any specified time. Show

that enough information is available to calculate all necessary values. Demonstrate how you

can prove if the process is reversible or not.

(e) (10 pt) Without performing calculations, do you expect the process to be reversible?

Explain.

2. (10) Using stability, show what is known about the sign of the adiabatic compressibility,

S

S P V V ?????????=1κ 3. (20) Use the method of Jacobians to express U

S P ????????using T, P as independent variables. mass flow controller mass flow controller stream A 1.5 mol/h

0.5L insulated tank initial conditions 1 bar, 25 C

Equations

R = 8.3143 J/(molK) = 8.3143 cm 3MPa/(molK) = 83.143 cm 3bar/(molK)

dU = TdS --- PdV + μdN ? -(?P/?S)V = (?T/?V)S dH = TdS + VdP + μdN ? (?V/?S)P = (?T/?P)S dA = -SdT - PdV + μdN ? (?P/?T)V = (?S/?V)T dG = -SdT + VdP + μdN ? -(?V/?T)

P = (?S/?P)T

Jacobian Formula

(),(,)Y X

Y X X Y Y X

K K X Y K L K L K L X Y X Y Y X L L X Y ??ê

?ê

?á?á??ˉ?ˉ???????ê?ê?ê?ê

?=-=á?á?á?á??ˉ?ˉ?ˉ?ˉ???????ê?ê?á?á??ˉ?ˉ??

e e d c b c b ????????=????????, , c b c b e d =??????

????, , ,d e e e b b b d N c c d c ??

?????

????=???????????????????? ,

,,,b e d e c e d c b d c b ????????????=?????????????

Michigan State University

DEPARTMENT OF CHEMICAL ENGINEERING AND MATERIALS SCIENCE

ChE 821: Advanced Thermodynamics Fall 2005

Exam 1, closed book, closed notes, equation sheet provided

1. An insulated rigid tank is connected to a reservoir as shown in the illustration below. The

tank and reservoir contain fluids that can be modeled using the ideal gas law. The valve is opened to increase the tank pressure to 1 MPa. The pressurization is assumed to be rapid, so that no heat transfer occurs. (Cp = 29.1 J/molK)

(a) (20) Set forth the simplified energy balance to be used to model the pressurization. It is imperative that you clearly indicate the boundary or boundaries used for your answer(s).

(b) (20) Rearrange the balance as necessary and provide an integrated energy balance. Supplement this result using constitutive properties of an ideal gas to derive equations that permit determination of the final temperature of the tank. Demonstrate that there are enough equations to find all unknowns, but you do not need to find the numerical answer.

(c) (10) Comment on whether the process is reversible or irreversible, and provide the reasoning for your answer without performing computations.

(d) (10) The entropy of an ideal gas can be calculated using S = Cp ln(T/T R ) – R ln(P/P R ) + S R . Set forth the equations and analysis that would support answer for part (c). Provide sufficient equations to determine all unknowns. It is imperative that you clearly indicate the boundary or boundaries used for your answer(s).

2. (20) Express (?P/?V)U in terms of measurable properties using V, T as independent

variables by using the Jacobian Method. Note: dU = TdS – PdV.

3. (20) Stability criteria for a pure fluid is that y (1)22 > 0. Starting with y (0) = U, provide any three of the six resulting stability criteria expressed in terms of variables from the set {S, T, P, V, μ, N}. U is a natural function of {S, V, N}, and the ordering of the natural variables is arbitrary.

initial conditions P = 0.1 MPa T = 380 K

R = 8.3143 J/(molK) = 8.3143 cm 3MPa/(molK) = 83.143 cm 3bar/(molK)

dU = TdS --- PdV + μdN ? -(?P/ ?S)V = (?T/ ?V)S dH = TdS + VdP + μdN ? (?V/ ?S)P = (?T/ ?P)S dA = -SdT - PdV + μdN ? (?P/ ?T)V = (?S/ ?V)T dG = -SdT + VdP + μdN ? -(?V/ ?T)P = (?S/ ?P)T

Jacobian Formula

(),(,)Y

X Y X X Y Y

X

K K X Y K L K L K L X Y X Y Y X L L X Y ??ê

?

ê?á?á??ˉ?ˉ???????ê?

ê

?ê?ê?=-=á?á?á?á??ˉ?ˉ?ˉ?ˉ???????ê?

ê?á?á??ˉ?ˉ??

Michigan State University

DEPARTMENT OF CHEMICAL ENGINEERING AND MATERIALS SCIENCE

ChE821: Advanced Thermodynamics Fall 2004

October 22, 2004, closed book with furnished equation sheet and inside front cover of Elliott and Lira textbook.

1. A gas tank of 1 m 3 volume contains air at 3 MPa and 440K. The tank is to be exhausted

through a special device from Carnaco engine works. The device is known to be reversible, but because of the configuration does not permit heat transfer to the tank, and there is no heat

transfer between the surroundings and the tank. The gas outlet from the device is the same T and P as the surroundings (292K, 0.1 MPa). The gas may be modeled as an ideal gas with Cp = 29.3 J/molK.

(a) (20) Set forth the simplified energy and entropy balances to be used to find the maximum work obtainable from the device subject to the constraints mentioned above. It is

imperative that you clearly indicate the boundary or boundaries used for your answer(s). (b) (20) Use the constitutive properties of an ideal gas to derive equations that permit

determination of the work. Demonstrate that there are enough equations to find all

unknowns, but you do not need to find the numerical answer.

2. (20) A fluid follows and equation of state where

()

(/)ig H H P b a T -=- and 2()/ig S S aP T -=- where a and b are constants. The fluid enters an adiabatic reversible turbine at (P 1,T 1). The fluid exits at P 2. Assuming that the outlet is one phase, set forth the step-by-step procedure to find the outlet T using the given departure functions. Clearly indicate how you would determine all

variables, providing the necessary equations; however you may assume that the equation of state is available for determining molar volume, and you do not need to give that equation.

3. The Joule-Thomson coefficient, H

T P ?ê?á?ˉ?, indicates how the temperature changes during a throttling process. For real fluids, the Joule-Thompson coefficient is often positive, meaning that temperature drops through a throttling process. However, for real fluids, the Joule-Thomson coefficient can be negative. For a given pressure, there is temperature, known as the inversion temperature, where the Joule-Thomson coefficient goes to zero. Above the inversion temperature throttling will cause a temperature increase; below the inversion temperature throttling will cause a temperature decrease.

(a) (10) Express the Joule-Thomson coefficient in terms of measurable derivatives, and find the constraint that must be satisfied at the inversion temperature. (Note: Cp will not be infinite at the inversion temperature).

(b) (15) Evaluate the derivative(s) for the van der Waals equation of state:

2/()/P RT V b a V =--

(c) (15) Set forth a step-by-step procedure that would provide the inversion temperature for a

given pressure as predicted by the van der Waals EOS.

Michigan State University

DEPARTMENT OF CHEMICAL ENGINEERING AND MATERIAL SCIENCE

ChE 821: Advanced Thermodynamics Fall 2003

Closed Book, with provided equation sheet, R = 8.314 J/molK

1. During emergency launch of a missile, the fuel is injected from a well-insulated holding tank as shown below.

controller ‘B’ is opened to permit a flow of 0.1 m

3/min. Device ‘A’ is a downstream pressure regulator that maintains the headspace pressure at 1 MPa. The headspace can be considered to be well mixed during the process. Heat transfer between the headspace gas and the liquid should be neglected. The C p of air is 29.3 J/molK. Use the ideal gas law.

(a) (10) Write the energy balance for the tank in the most simplified form while clearly indicating the boundary used for the balance.

(b) (30) Develop the energy balance and solve for the headspace temperature as a

function of time. Perform all integrations. Do not calculate the value, but provide all

equations and parameter values to demonstrate that you could determine the temperature.

2. (30) Consider two air tanks at the initial conditions shown below. We wish to obtain work from them by exchanging heat and mass between the tanks. No gas may be vented to the atmosphere, and no heat may be exchanged with the atmosphere. Reversible devices may be used to connect the two tanks.

Provide the balances necessary to calculate the maximum work that may be obtained. Perform all integrations. Do not calculate the value, but provide all equations and parameter values to demonstrate that you could determine the work. C p = 29.3 J/molK. Use the ideal gas law. Tank A 700 K 10 bar 5 m 3 Tank B 200 K 0.1 bar 15 m 3

3. (30) Consider a piston/cylinder device. The gas in the cylinder is reversibly and adiabatically compressed from an initial state of 0.1 MPa and 280K to a pressure of 5 MPa. The PVT properties of the fluid can be modeled by an equation of state of the form PV = RT + (b – a/T)P where a and b are known constants. C p = 29.3 J/molK.

Set up the problem to determine the final temperature for this non-ideal gas. Do not calculate a final value, but provide all integrations and parameter values and demonstrate that sufficient equations are available for all unknowns.

Michigan State University

DEPARTMENT OF CHEMICAL ENGINEERING AND MATERIALS SCIENCE

ChE821: Advanced Thermodynamics Fall 2003

Open Book, Closed Notes

1. (35) Express the derivative S

H T ? ? in terms of measurable quantities.

2. (35) Use the method of Jacobians, along with any other helpful techniques to express H

P V ? ? using T and P as independent variables.

3. (30) Consider y (0) = G(T, P, N 1, …N n ) and y (1) = H(S,P,N 1,…N n ). Use Table 5.3 (Table 5.1 in 2nd edition) to find y (1)12. (Note: I believe that the equation might be missing a minus sign).

Michigan State University

Department of Chemical Engineering and Materials Science

Fall 2002

1. Bottles of compressed gases are common in laboratories. Oxygen cylinders are particularly dangerous and the pressure regulators are frequently labeled “Oxygen – do not oil”. The rationale for this rule comes from the fact that if the regulator were to get hot with oxygen present, it could exceed the flash point of the oil, and a spontaneous explosion would result.

To investigate the possibility, consider an oxygen cylinder that is 40L in volume (represented by V T in the diagram below), initially at 15.17 MPa and 311 K. The tank is connected to a regulator through a valve. The connecting tubing and internal voids of the regulator are about 10 mL represented by variable V R in the diagram below. When V R is filled rapidly, the process can be considered adiabatic, and the pressure and temperature in the tank remain constant (act as a reservoir).

Assume that the gas initially in V R mixes completely with the entering gas. Taking the system as V R , provide equations that would lead to the final temperature in V R . Your answer should be simplified (fully integrated) and there should be sufficient equations to find the final temperature. Do not calculate a final temperature, but clearly indicate how the equations would be used.

2. A gas tank of volume V T and temperature T i and pressure P i is to be depressurized through a

new Carnoco company device that is reported to obtain the maximum work possible. The device is capable of transferring heat from the surroundings to the tank, as well as transferring heat to/from the exiting gas stream before it enters the surroundings. If the gas tank is initially filled with air, Cp = 29.3 J/molK, provide sufficient formulas that lead to the maximum work. Do not calculate the final value, but provide simplified equations and clearly indicate how the equations would be used. The surrounding temperature and pressure are T a and P a .

2. Use the Jacobian method to express the derivative S

P V ?? using P, T as independent variables. 3. According to the square-well potential, the second virial coefficient is given by

??=kT R R boR B εexp 11333 and the virial equation of state is Z = 1 + BP/RT. Derive the integrated enthalpy departure for

a fluid which follows the square-well potential. (Note: d[exp(f(x))]/dx = exp(f(x))d[f(x)]/dx)