Chapter 15 - Section A - Mathcad Solutions 15.1 Initial state: Liquid water at 70 degF. H 1 38.05 BTU lb m ⋅ := S 1 0.0745 BTU lb m rankine ⋅ ⋅ := (Table F.3) Final state: Ice at 32 degF. H 2 0.02 − 143.3 − ( ) BTU lb m ⋅ := S 2 0.0 143.3 491.67 − ⎛ ⎜ ⎝ ⎞ ⎠ BTU lb m rankine ⋅ ⋅ := T σ 70 459.67 + ( ) rankine ⋅ := (a) Point A: sat. vapor at 32 degF. Point C: sat. liquid at 70 degF. P = 85.79(psia). Point D: Mix of sat. liq. & sat. vapor at 32 degF with the enthalpy of Point C. Point B: Superheated vapor at 85.79(psia) and the entropy of Point A. Data for Points A, C, & D from Table 9.1. Data for Point B from Fig. G.2. 595
Introduction to chemical engineering thermodynamics
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Chapter 15 - Section A - Mathcad Solutions
15.1 Initial state: Liquid water at 70 degF.
H1 38.05BTUlbm
⋅:= S1 0.0745BTU
lbm rankine⋅⋅:= (Table F.3)
Final state: Ice at 32 degF.
H2 0.02− 143.3−( )BTUlbm
⋅:= S2 0.0143.3491.67
−⎛⎜⎝
⎞⎠
BTUlbm rankine⋅⋅:=
Tσ 70 459.67+( ) rankine⋅:=
(a)
Point A: sat. vapor at 32 degF.
Point C: sat. liquid at 70 degF. P = 85.79(psia).
Point D: Mix of sat. liq. & sat. vapor at 32 degF with the enthalpy of Point C.
Point B: Superheated vapor at 85.79(psia) and the entropy of Point A.
Data for Points A, C, & D from Table 9.1. Data for Point B from Fig. G.2.
595
HB 114BTUlbm
⋅:=
For superheated vapor at 85.79(psia) and S = 0.2223:
HD HC:=HC 34.58BTUlbm
⋅:=
For sat. liquid at 70 degF:
SA 0.2223BTU
lbm rankine⋅⋅:=HA 107.60
BTUlbm
⋅:=
For sat. liquid and vapor at 32 degF, by interpolation in the table:
Conventional refrigeration cycle under ideal conditions of operation:Isentropic compression, infinite flow rate of cooling water, &minimum temp. difference for heat transfer = 0.
(c)
The only irreversibility is the transfer of heat from the water as it coolsfrom 70 to 32 degF to the cold reservoir of the Carnot heat pump at 70degF.
Ans.η t 0.889=η tWdotideal
Wdot:=
Ans.
Wideal H2 H1− Tσ S2 S1−( )⋅−:=
Wideal 12.466BTUlbm
= mdot 1lbm
sec⋅:=
Wdotideal mdot Wideal⋅:= Wdotideal 13.15kW= Ans.
(b) For the Carnot heat pump, heat equal to the enthalpy change of thewater is extracted from a cold reservoir at 32 degF, with heatrejection to the surroundings at 70 degF.
(Note that minimum temp. diff. is not at end of condenser, but it is notpractical to base design on 8-degF temp. diff. at pinch. See sketch.)
Point C: Sat. Liquid at 98 degF.
Point D: Mix of sat. liq. and sat. vapor at 24 degF with H of point C,
Point B: Superheated vapor at 134.75(psia).
Point A: Sat. vapor at 24 degF.
η 0.75:=Practical cycle.(d)
The irreversibilities are in the throttling process and in heat transfer inboth the condenser and evaporator, where there are finite temperaturedifferences.
For the cooling process from this temperature to the final temperature of298.15 K, the entropy change is calculated by
602
mdot1
mdot2
mdot3
H3
S3
⎛⎜⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟⎟
⎠
Find mdot1 mdot2, mdot3, H3, S3,( ):=
S3 SliqH3 Hliq−
Tsat+=
H3 Hliq−( ) mdot3⋅ 300kJs
=mdot3 mdot1 mdot2+=
S3 mdot3⋅ S1 mdot1⋅− S2 mdot2⋅− 0kJs K⋅
=
H3 mdot3⋅ H1 mdot1⋅− H2 mdot2⋅− 0kJs
=
Given
S3 SliqH3 Hliq−
Tsat+:=H3
H1 H2+
2:=
mdot3 mdot1 mdot2+:=mdot2 mdot1:=mdot1 0.1kgs
⋅:=Guesses:
We can also write a material balance, a quantity requirement, and relationbetween H3 and S3 which assumes wet steam at point 3.The five equations (in 5 unknowns) are as follows:
∆fs S mdot⋅( ) 0=∆fs H mdot⋅( ) 0=
Assume no heat losses, no shaft work, and negligible changes in kineticand potential energy. Then by Eqs. (2.30) and (5.22) for a completelyreversible process:
(a)
Tsat 453.03K:=Svap 6.5828kJ
kg K⋅:=Hvap 2776.2
kJkg
:=
Sliq 2.1382kJ
kg K⋅⋅:=Hliq 762.6
kJkg⋅:=
For sat. liquid and vapor at 1000 kPa, Table F.2:
603
Scomp 7.1803kJ
kg K⋅:=By interpolation:
Hcomp 3084.4kJkg
=Hcomp H2H'comp H2−
ηcomp
⎛⎜⎝
⎞
⎠+:=
ηcomp 0.75:=H'comp 2993.5kJkg
:=
Compressor: Constant-S compression of steam from Point 2 to 1000 kParesults in superheated steam. Interpolation in Table F.2 yields
By Eq. (5.25), with the enthalpy term equal to zero:
(assumed)Tσ 300K:=THERMODYNAMIC ANALYSIS
S3 6.5876kJ
kg K⋅:=By interpolation,
Steam at Point 3 is slightly superheated.
H3 2.77844 103×kJkg
=mdot3 0.14882kgs
=
Ans.
mdot2 0.04274kgs
=mdot1 0.10608kgs
=
mdot1
mdot2
mdot3
H3
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟
⎠
Find mdot1 mdot2, mdot3, H3,( ):=
H3 Hliq−( ) mdot3⋅ 300kJs
=mdot3 mdot1 mdot2+=
H3 mdot3⋅ H1 mdot1⋅− H2 mdot2⋅− 0kJs
=
Hcomp H2−( ) mdot2⋅ Hturb H1−( )− mdot1⋅=
Given
H3 2770.kJkg
:=mdot3 0.15kgs
:=
mdot2 0.064kgs
:=mdot1 0.086kgs
:=Guesses:
The energy balance, mass balance, and quantity requirement equations ofPart (a) are still valid. In addition, The work output of the turbine equalsthe work input of the compressor. Thus we have 4 equations (in 4unknowns):
605
x1 0.285= S1 0.084BTU
lbm rankine⋅=
S1 Sliq x1 Svap Sliq−( )⋅+:=x1H1 Hliq−
Hvap Hliq−:=
H1 H4:=S2 Svap:=H2 Hvap:=
S4 0.07892BTU
lbm rankine⋅⋅:=H4 37.978
BTUlbm
⋅:=
For sat. liquid at the condenser outlet temperature of 80 degF:
Sliq 0.02744BTU
lbm rankine⋅:=Svap 0.22525
BTUlbm rankine⋅⋅:=
Hvap 103.015BTUlbm
⋅:=Hliq 12.090BTUlbm
:=
For sat. liquid and vapor at the evaporator temperature of 0 degF:
Some property values with reference to Fig. 9.1 are given in Example 9.1.Others come from Table 9.1 or Fig. G.2.
15.4
The percent values above express each quantity as a percentage of theabsolute value of the ideal work, to which the quantities sum.
The purpose of the condenser is to transfer heat to the surroundings. Thusthe heat transferred in the condenser is Q in the sense of Chapter 15; i.e.,it is heat transfer to the SURROUNDINGS, taken here to be at atemperature of 70 degF. Internal heat transfer (within the system) is not Q. The heat transferred inthe evaporator comes from a space maintained at 10 degF, which is part ofthe system, and is treated as an internal heat reservoir. The ideal work of the process is that of a Carnot engine operating betweenthe temperature of the refrigerated space and the temperature of thesurroundings.
Wdot 3.225 104×BTU
hr=Wdot mdot ∆H⋅:=
mdot 1845.1lbm
hr⋅:=S3 0.231
BTUlbm rankine⋅⋅:=
From Fig. G.2 at H3 and P = 101.37(psia):
H3 120.5BTUlbm
=H3 H2 ∆H+:=∆H 17.48BTUlbm
⋅:=
From Example 9.1(b) for the compression step:
607
QdotC
600
500
400
300
200
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
−BTUsec
⋅:=tC
40
30
20
10
0
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
:=
The following vectors refer to Parts (a)-(e):
TH Tσ:=Tσ 70 459.67+( ) rankine⋅:=
The discussion at the top of the second page of the solution to thepreceding problem applies equally here. In each case,
15.5
Wdot 32252.3BTU
hr=
The percent values above express each quantity as a percentage of theactual work, to which they sum:
9.14%Wdotlost.evap 2947.6BTU
hr=
14.67%Wdotlost.throttle 4730.2BTU
hr=
11.24%Wdotlost.cond 3625.2BTU
hr=
17.42%Wdotlost.comp 5619.4BTU
hr=
47.53%Wdotideal 15329.9BTU
hr=
The final term accounts for the entropy change of the refrigerated space (aninternal heat reservoir).
Wdotlost.evap Tσ mdot⋅ S2 S1−( )⋅
TσH1 H2−
TC⋅ mdot⋅+
...:=
608
H3
117.7
118.9
120.1
121.7
123.4
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
BTUlbm
⋅:=
From these values we must find thecorresponding entropies from Fig. G.2.They are read at the vapor pressure for80 degF of 101.37 kPa. The flow rates come from Problem 9.9:
From the results of Pb. 9.9, we find:
S1 Sliq x1 Svap Sliq−( )⋅+⎡⎣ ⎤⎦→⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
:=x1H1 Hliq−
Hvap Hliq−
→⎯⎯⎯⎯⎯
:=
H1 H4:=S4 0.07892BTU
lbm rankine⋅⋅:=H4 37.978
BTUlbm
⋅:=
For sat. liquid at the condenser temperature:
S2 Svap:=Svap
0.22244
0.22325
0.22418
0.22525
0.22647
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
BTUlbm rankine⋅⋅:=Sliq
0.04715
0.04065
0.03408
0.02744
0.02073
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
BTUlbm rankine⋅⋅:=
H2 Hvap:=Hvap
107.320
105.907
104.471
103.015
101.542
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
BTUlbm
⋅:=Hliq
21.486
18.318
15.187
12.090
9.026
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
BTUlbm
⋅:=
For sat. liquid and vapor at the evaporator temperature, Table 9.1:
The final term accounts for the entropy change of the refrigerated space (aninternal heat reservoir).
Wdotlost.evap Tσ mdot⋅ S2 S1−( )⋅
TσH1 H2−
TC⋅
⎛⎜⎝
⎞
⎠mdot⋅+
...:=
Wdotlost.throttle Tσ mdot⋅ S1 S4A−( )⋅:=
Wdotlost.cond Tσ mdot⋅ S4 S3−( )⋅ Qdot−:=
Qdot H4 H3−( ) mdot⋅:=
Wdotlost.comp Tσ mdot⋅ S3 S2A−( )⋅:=
613
H4 500.4kJkg
=H4 H1 H3+ H2−:=
By an energy balance, assuming the slurry passes through unchanged,
S3 7.4048kJ
kg K⋅⋅:=
By more double interpolation in Table F.2 at 143.27 kPa,
H3 2757.3kJkg
=H3 H2H'3 H2−
ηcomp+:=H'3 2737.0
kJkg⋅:=
For isentropic compression to 143.27 kPa, we find by double interpolation inTable F.2:
(sat. vapor)S2 7.3554kJ
kg K⋅⋅:=H2 2676.0
kJkg⋅:=
(sat. liquid)S1 1.3069kJ
kg K⋅⋅:=H1 419.1
kJkg⋅:=
ηcomp 0.75:=
Compression to a pressure at which condensation in coils occurs at110 degC. Table F.1 gives this sat. pressure as 143.27 kPa
15.7
614
A thermodynamic analysis requires an exact definition of the overallprocess considered, and in this case we must therefore specify the sourceof the heat transferred to the boiler.Since steam leaves the boiler at 900 degF, the heat source may beconsidered a heat reservoir at some higher temperature. We assume inthe following that this temperature is 950 degF.The assumption of a different temperature would provide a variation in thesolution.
15.8
Wdot 40.667kW= The figures on the right are percentages of theactual work, to which the terms sum.
18.22%Wdotlost.comp 7.41kW=
60.62%Wdotlost.evap 24.651kW=
21.16%Wdotideal 8.606kW=
Wdot mdot H3 H2−( )⋅:=
Wdotlost.comp mdot Tσ⋅ S3 S2−( )⋅:=
Wdotlost.evap mdot Tσ⋅ S4 S3− S2+ S1−( )⋅:=
Wdotideal mdot H4 H1− Tσ S4 S1−( )⋅−⎡⎣ ⎤⎦⋅:=
Tσ 300 K⋅:=
mdot 0.5kgsec⋅:=S4 1.5206
kJkg K⋅
=S4 Sliq x4 Slv⋅+:=
x4 0.018=x4H4 Hliq−
Hlv:=Slv 5.8203
kJkg K⋅⋅:=
Sliq 1.4185kJ
kg K⋅⋅:=Hlv 2230.0
kJkg⋅:=Hliq 461.3
kJkg⋅:=
This enthalpy is a bit larger than that of sat. liquid at 110 degC; find qualityand then the entropy:
615
Q 829.045− BTU=
Q 1 lbm⋅ H5⋅ 1 lbm⋅ m−( ) H4⋅− m H7⋅−:=
The purpose of the condenser is to transfer heat to the surroundings. Theamount of heat is
We take as a basis 1 lbm of H2O passing through the boiler. Requiredproperty values come from Pb. 8.8.
The ideal work of the process in this case is given by a Carnot engineoperating between this temperature and that of the surroundings, herespecified to be 80 degF.
616
S10 9.521kJ
kg K⋅⋅:=H10 796.9
kJkg⋅:=
S9 4.928kJ
kg K⋅⋅:=H9 285.4
kJkg⋅:=
S7 7.544kJ
kg K⋅⋅:=H7 719.8
kJkg⋅:=
S5 8.894kJ
kg K⋅⋅:=H5 1009.7
kJkg⋅:=
S4 9.359kJ
kg K⋅⋅:=H4 1140.0
kJkg⋅:=
Property values:
Refer to Figure 9.7, page 330 The analysis presented here is for the liquefaction section to the right of the dashed line. Enthalpy and entropy values are those given in Ex. 9.3 plus additional values fromthe reference cited on page 331 at conditions given in Ex. 9.3.
15.9
(absolute value)
The numbers on the right are percentages of theabsolute value of the ideal work, to which theysum.
Wideal 742.82BTU=
1.13%Wlost.pump.heater 8.36BTU=
4.90%Wlost.cond.valve 36.44BTU=
13.30%Wlost.turbine 98.81BTU=
30.24%Wlost.boiler.reservoir 224.66BTU=
50.43%Wabs.value 374.61 BTU⋅=
The absolute value of the actual work comes from Pb. 8.8:
The basis for all calculations is 1 kg of methane entering at point 4. Allwork quantities are in kJ. Results given in Ex. 9.3 on this basis are:
Fraction of entering methane that is liquefied:Fraction of entering methane passing through the expander:On this basis also Eq. (5.26) for Ideal Work, Eq. (5.33) for EntropyGeneration,and Eq. (5.34) for Lost Work can be written: