- .. ----.-----------------------~

CBE20260: Thermo HW 6, #1: Solutions Author: Dane Grismer

4.2 Steam undergoes a state change from 450°C and.3.5MPa to 150°C and 0.3 MPa Determine Ml andMusing:

(a) steam table data(b) ideal gas assumptions (Be sure to use the ideal gas heat capacity for water).

b t\ = - 5~.1 f.J/J.:,h ~~ O. CT-!-/=I- Ic)Ik, -}c'

-r;: 45~~P, ='3.>~Ptr

ft, =. 33~o ~A,5,-=:f. CXJ1t IJ/J,-/(

(1) (p. -== 4-.0+1 (R) "'"35.' Icj/k~I-/( -- /J'. -:; /rJ/J.,-k~

AH-= lMCPAT~ (()(J.g:f){I50-~o) = [-StllcJ/", =lHl J~5-; CpJl'\ (~/1i)-RIt\(~;?):::1J.c I"" t 4'23;~23)- ~ I~ (o'~.s)

= 2.£/.' J:)1lc-!-k1/)5= 0.13.,. kJ/t.,.k]

CBE 20260: Thermo HW 6, #2: Solutions Author: Dane Grismer

4.3 The following problems involve one mole of an ideal monatomic gas, Cp = 5R/2, in a variable vol-ume piston/cylinder with a stirring paddle, an electric heater and a cooling coil through which refrig- C

erant can flow (see :figure). The piston is perfectly insulated. The piston contains 1gmole of gas.Unless specified, the initial conditions are: r = 25°C, pi = 5 bar.

Heater coil

Cooling coil

(a) Status: Heater on; cooler off; paddle off; piston fixed. Five kJ are added by the heater. Find AU,M, I:1P and sr.

(b) Status: Heater off; cooler off; paddle off; piston moveable. What reversible volume change willgive the same temperature rise as in part (a)? Also .find AU, AS andLlP.

(c) Status: Heater off; cooler off; paddle on; piston fixed. What shaft work will give the same AU,I:lS as part (a)?

(d) Status: Heater off; cooler off; paddle on; piston fixed. The stirring motor is consuming- 55 wattsand is 70% efficient.Wbat rate is the temperature changing? At what initial rates are U and Schanging?

(e) Status: Heater unknown; cooler unknown; paddle off; piston free. We wish to perform a revers-ible isothermal compression until the volume is half of the initial volume. If the volume isdecreasing at 2.0 cm3/s, at what rate should we heat or cool? Express your answer in terms of theinstantaneous volume. What is the total heat transfer necessary?

(h) PV:::••.RTJ ,,::: I ,"ole

V. - fl.T. - [1!'3./if3crt'~.kd/fflO1 ;J:H-{~t.IS)t2? ~ tr.-tJS=7 .82 ~.7 /1910 f =- 'f., qYf.ti2 GjoC\3,- -p,-~- ·~W I

E>'-,: do.seJ I I-Ie ~( ,4S

A 0 0LlU-=~ +-v£"(: +- y( {t1V!=0)

- use $ -I,II(/~ ..to f.tkJ. Va.-a s S t( IM-f! ~+w-p rt: Cadra bet~ <t- re.,.e".-s; bk)

Jt\ S -; ~ ( S I~ + ¢ff +- ~ re.ven:lk

0•••.pI"..,. 0 ."Jpt~

g5-=-OJ~CV (VI (Tz/1; ) + R ,~ (Vz./v,)

~ V'Z.. -= \/, (Tz./-r.) -<.,h = ff,er5'=t-.K2 <toO 3){''19.°f2ql.~)-"~ _ IJ '380. r-fc.~'3

b v:: - '3 S=l-(. q5Co••• J (Jeu-eas4d)

p.f~~ = nR12IV2. -== Lf; 2/ MPc(

LtJP:: 3. ?/ MRt 7D U-=Wcr p. c; lJT dos.eJ$yS. ,'1V= 5 IcJ/..-I]

,..--- hTs••"'. M L!. .I~ 'P4r*-t:-)

fPT; l.jO:>.lJt k](c.) part-~) wi ~JJf.e ~l hC4+ ..sa•••-e, /JlJ d-iJ 5 a$l f'1I(rf-(a)

do h~~-FF

/J U = W$ +- ;VI: +;r-0~ ~ -IS' k·)j~/.: ~ J{'rsf- .f!r~

(J) O. '7 (t;~ w) = 33. ~ w •.. 3? S J,&- = Ws j Saw-l! ~rJ(bHJ «s par+Cc.}

U= ~ =- c:v t: ~"";> r=.3.. ; '3~~)lj -= p. I ~/s--=fl[U:: 3t. ~ J/s I c:::v (2. Lf:}.,J"Ik - J

S ~ S~ ~ Cv ~ = Cf- (3.1 k"/s) .:l!:e+(3.1)- @. 13 J!k-J = S-]

(e) U 0;:: 0 :::Q.f- Wet: -:::"/ Q = - W.#' 'Et:.Li~o~,

. pdll trrWEe. -= - -;;ft : -V ~~ V= lfcl )7.6- 2.0-t i -t::C=S]

Q -:::K.'3/'t(ml52ttCfS!1.'-2-l::

CBE20260: Thermo HW 6, #3: Solutions Author: Dane Grismer

4.7 An isolated chamber is divided into two equal compartments, one containing gas and the other evac-uated. The partition between the two compartments ruptures. At the end of the process, the tempera-ture and pressure are uniform throughout the chamber.

(a) If the filled compartment initially contains an ideal gas at 25 MPa and 650 K, what is the finaltemperature and pressure in the chamber? What is Mfor the process? Assume a constant heatcapacity of Cp/R = 4.041.

(b) If the filled chamber initially contains steam at 25 MPa and 650 K, what is the fmal temperatureand pressure in the chamber? What is Mfor the process? (Use the steam.tables).

tt) ~I 91l$ '_ •. ~lQ J~.:O ty.t!~ da -= JW'e;:-0 .., r IS c.sr.dU-- a +a 'Ifj::. ) s )

A '" z: C It'\ (~)- R("&') :::q.,otl{1.1t't)/KCJ)-r. '3/'t/Ktf)-=: rs;1-( J/,,",o/./C-=-/J~?tUV':>9td P T, '1

II 1/__? I ~ ~~ =4 ~ [P. ~ /2.S' MP~f7;. :: t9:J!:J }J:ljlV,

(h) ~J. V =dQ +~c =0 5~e.e JQ";. d w~....oU. -::: uz, V

2=- 2 V, cP 2S'MPa d- (So/« '3-:ff°C))

ed,..••f'O~ ~ L.f'OJ d- '+~o°c) ~U. = 22'14' kJ/k, I ).-==..t-R3S /c.J/iz,-K) '" --E' roLf-53 vvt

3~

.: V~ --0. COqo6 w.3/~ J Uz ::.y.'2.t::tLf I:J/J,- be It-. 2-,.,kase. ~.~ "" .fa ~J.e.sJ a~S wey- LM I.( sf..h::At\~'··9

U -VL \t _VL.lI- _-2i5vViIP =: tt = ~

-ql4-es.s I, e..lc. tt ~ V;. c~k \4f~~ el q f...- U by "1\ ok,'eCh've ~. ft-r.- v - 'lr;...v)_ ~ ~ $o\ve &.ts,~,~Gel i3 +k -t"'~kert d- ~tesr MeH.od

T (0C) P (Mpa) Vl VV ul UVqlromV qlromU obj. fxn. s' « onV

330 12.858 0.001561 0.013 1505.8 2499.15 0.655564 0.793477 -0.13791 3.5518 5.4422 4.791079340 14.6 0.001638 0.0108 1570.62 2464.44 0.810085 0.809313 0.000772 3.6601 5.3356 5.017398

-Since the objective function changed signs, the final state is between the 2 T's, but closer to 340.-Assuming linear interpolation with respect to T, enter the linear interpolation formulas and solve with Solver or GoalSeek. .

T (uC) P (Mpa) V· V U· U qlromV qlromU obj. fxn. s· s.V onV

339.952 14.59165 0.001638 0.010811 1570.309 2464.606 0.809161 0.809228 -6.80E-05 3.659581 5.336111 5.016164

[It' 31fO.O DC1 ~ == 1'f.5LJNP41 1.= 1/%

~'Z -:::5:0162 JJ/~'K ="/ /:lS,.~ -.::O.IZZ~ j/9"~ :: [2. 20!( J//Mo/ ••/C""'bS"tl.S (

( /) S il less ~ ;f * Lcd ~ ene.-pk~ U li'lIA~d-li1ttt Jeere.ses atf,r,py)

CBE20260: Thermo HW 6, #4: Solutions Author: Dane Grismer

4.15 A common problem in the design of chemical processes is the steady-state compression of gasesfrom a low pressure PI to a much higher pressure P2' We can gain some insight about optimaldesign of this process by considering adiabatic reversible compression of ideal gases with stage-wise intercooling. If the compression is to be done in two stages, first compressing the gas from PIto P*, then cooling the gas at constant pressure down to the compressor inlet temperature Tl' andthen compressing the gas to P2' what should the value of the intermediate pressure be to accomplishthe compression with minimum work?

D

E-b~1 [~ I~-~f'res~): J1H-=;(+-W- WS-- bllC I (tIh I s~_~51Jr): .L1.s' =-0

.5 -1,4/ 9''Ve.s aaraJ""rl-,? ~ij,1e ideAl 94S => T'lctT, =: (P"1fOf?/Cp

£"-~( 9".,es w,: Lp (T*- T,) = ~ T,C{P"'/P, )~/~-I]

- T~ .:w ~S$Ot' 2 ~M~.sly aHd ~b,~e ...

V:;: \N, +W~ =- Cf -r;(izplt/P,J1?'Gv-I] +-[(t;/p·)1Z/~-DJ

- M~;~ic-e- & f;r-.. by ~~ +/..e dert~ J- s~ 10 0)

~ = r_T. C 1V~(.I:)RI'?_ 3&( P'2 ) ~Cp~ -:::0tip· .• I {71-p, r: f% '.f

(ptk)~= P, P'1 :;>Ii-;:m J

CBE20260: Thermo HW 6, #5: Solutions Author: Dane Grismer

4.23 Steam is used in the following adiabatic turbine system to generate electricity. 15% of the flow fromthe first turbine is diverted for other use.

P=8MPaT=_550°C•..rl P = 5 MPa

300~ P=O;8MPa. 15% withdrawn

for other use

(a) How much work (in kJlh) is generated by the first turbine which is 80% efficient?(b) How much work (in kJ/h) is generated by the second turbine which is 80% efficient?(c) Steam for the turbines is generated bya boiler. How much heat must be supplied to the boiler

(not shown) which has 300 kg/h of flow? The stream entering the boiler is T ~ 170°C, P = 8MFa. The stream exiting the boiler matches the inlet to the first turbine.

I ';;50? 3521. f"}., ~ '3161. 3''2 5 ~J

3 a1 2~~ ~ f;-. fY1>bbot

.s~

C4)~ 2 1,/1'\ 4£0 d-,ev o:J~ (f,2)-:=' 3J(l.3 I=J/J,H I~ '331*.2 +-& c.~'lt:t~. '821)/(c.q;ffl-(,f21~(3'+3Lf-.1- - 33. -

~ .!JPI'::: -I (P. 5''t-=7- ~/k,Ws:= o. g~ fI ' ):::: - J2f JeJ/'" ==? Hz ~ ~ IcJIt,~ ~1~gzl+ll~'3-331?-.1)1{3Lt3t::1-'33f?-.7)J(,.e'1-81-,.1?2I)~j'.q~ Icj/~.k'

~:::: 3tX> (- 12K) ==-1- 3~ LfOO kJ/ Ar = ¥sJ(b) ~ ~5 ~ .ftt,r~hej tnd-Iet is 5~ 511M!e 5>5$<tfV ~O,gMR,/

"",1\ be 1/r\ ~ ~ 250°Ctt'3'-;; 28~.:1-f-a,.c1Z2K-f>.fJ~)/{9.0lfOJ-'.3r:MJJ(21Sb. 'f-2t3!1.q.)~.w2 Icj~

Atl'-;::: -£'01 Ic)/~'W.s:::bH :: 0. g (-§OJ) = -tol 1c)11, ==") U~ <=J>~qcn k.j/~

~.::: 3CO(O.f5J(-401)=[-IOJ2Z' I=J/I\Y"=-J(s I--i)-o.IS)

(c) £-k.t: M (H"wt- _Htk) == ~flO\;: f\- +y,1P.:: Z?-Cg.5 -I- 1(8-0.8>:- 2-:;:?-' ~J//::?

g.:: 3a:::>('3~2I.t-1.~6J - 12.23) TlfO /cJ/AY" == ~l

CBE20260: Thermo HW 6, #6: Solutions Author: Dane Grismer

4.25 Liquid nitrogen is useful for medical purposes and for research laboratories. Determine the mini-mum shaft work needed to liquefy nitrogen initially at 298 K and 0.1013 MPa and ending with satu-rated liquid at the normal boiling point, 77.4 K and 0.1013 MPa. The heat of vaporization at thenormal boiling point is 5.577 kl/mol, and the surroundings are at 298 K. The constant pressure heatcapacity of gaseous nitrogen can be assumed to be independent of temperature at 7/2R for the pur-pose of this calculation. .

C4) ~ - s-lctJe. f/o..r sy~ . .e-I::-cl: O=.f.I '''';/rl _ Hocrt-.,QI.{f- +- ~~ +- ~s = -L1#ti -I- QH +-.!¥s5-ktJ: O=: 5''''t1'''-S~~At+--4- Q-II' •.-r; +~9et1 -= -II-4,,,•.•••p b5~+gH ~ =0 ~ ~;J,k

tt),,,,tI1' )

-~~ ~== 0 1* +he. ~1)1. ~k win k .(;rd 11I!!v, 'f'~SS'-,So\v-e 5-1,..1. Wr gl-f) ~ ~ ," F-1a/ ~ .(1..J .ws.-~-eeJ ~f-f) AsDH::: ):LfCpdr-~H""jep::= ~~

btl ~~8. '31,,)(":1'1.'t-'2!18) - £s-:FI-;# -, Iff! - 55+1 ~ -{/~t16J/IftoI

t~= i:F1.I#4.Ji- ~"":: +/~~/If)IY\IJ!J.)-..££!!.:::-/l1.2f3 J/•••.,,/·k~ T ~ ~(6.~ t~ ~t

gH = 1H,t'""", ss, = 2Qg(-flI,1g3){[) -=- -33f 16'2 J/~irlset 1;.,fUlf\l' = Zq~ TsUI'T hrr •...•r",·JM.&lJt( w~k .

F-L..cl: .!¥S = /J tf,; - gff ~ -II/I'll (I) of '33) ,,:2 ::.2/. r7 ~)I",f~(~)=[O. 35'3 kW= ~

(bl &.kk sl"~ ,- ~, c..._ p pr=ec,9[ ~ 2'181< d<><eJ sysfeoo

F-/"AI.: dU; dQ;.f-d~ + dWe-e ~ dQI-f +dk's -PJV>::.':> tlU-4-PdV+ VJP = dH

Fdr lit of. P~J dN==JQff+df,(r ->bH== QH+~

~- ~J: JS::.A9L&.~.s Wlft'f"~ Mt~tly -the- S4vJte =2:>&5: O.3S3kiiJ

(G-) W. . dre #.e ~e b~u ~ Jepe,.J ~ .s.f1tles .wf- ~s ~ TkiS Cd~· no ~/4tr a~-~t"Ilt\

(a) Consider nitrogen entering a flow device at 1 mol/min. Give shaft work in kW.(b) Consider nitrogen in a piston/cylinder device. Give the work in kJ per mole liquefied.(c) Compare the minimum shaft work for the two processes. Is one of the processes more advanta-

geous than the other on a molar basis? '

WJ-h61rl- ~ -hfJ,J.e.sdr C:~1

W€ C4kulale -H 1f..,J 5 4SSU~ I~(

l)tA s fa \41 is VilIitI.