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Condensation of vapours from gas—vapour mixtures. An approximate method of design
Condensation
of vapours from gas-vapour mixtures. method of design
An approximate
B. HULD$N Ekaio Ann&&ion for Power and Fuel Economy, S. Eeplanadgatan 14, Helalnki, Pinland (Iteeeioed 16 April 1957) Abetract-By using a hypothetical drlvlng force, the well known Co~~u~~~Houo~_[6] analyale may be applied graphleally, without trial and error, to solve condenser design problems. The method, ae compared with the orlglnal calculation in [tl], is rapid and the result agreea wlthln 1percent. The graphical method is approximate and includes the assumption [tl] of latent heat transfer overbalancing sensible heat tranefer ae pointed out by SMITH[lo]. It la believed that reasonable surface eathnatee may be obtainedeven for initially superheatedmixtures,w shown for the example of Baas [l]. The method haa the advantage of glving a clear overall view of the design procedure in analogy to the enthalpy driving force method& R6sum6-Par l’emploi d’un gradient flctlf, ll eat poasllM d’utlliaer l’analyae de Colbum-Hougen (6) aow forme graphique, aana i&&ion, pour r&oudre lea probli?meade ealcul d&r comknaeum. La m~,~m~aucalculoriginsl(8),estrapideetdonnedea~tateenrrccotdBIpour~tp~. La m&ode graphlque eat approchee, et suppose (6) que l’echange de chaleur latente l’emport& aur l*dchange de chaleur sensible, comme Pa indlque Smith (10). 11 eat admls qu’une &ahmtlon normaMde la surface peut 6tre obtenue, m&mepour dea m4angea wblement em&au&, comme le montm l’e-xemplede Bras (I). La mdthode B l’avantage de dormer une vue d’enaembleclaim, en comparaleon avec lea m&hodea utiliaant lea gradlente d’enthalpie. ZuanxumenfnsaunS-Dae bekannte Verfahren von COLBUBNund HOU~EN[f3]zur Berechnung von KONI~NSAZXXLIZN kann durch Eh&lhumg hypothetiecher treibender Kr&fte gmphlech ohne achrittwelae Verbeaserung verwendet werden. Diese Methode fOhrt achnell zu Ergebnieaen, die, vergllchen mit der Original-Methode [6], nicht mehr ale 1% Abweichung aufweieen. Die graphleche Methode let ein N6herungeverfahren und enth< die ANN-= [6], dasa die zu tlbertragenden latenten W&men gross slnd gegentiber den fOhlbarenWgrmen, wle es von SMITH [IO] r&her auegeftihrt wurde. Sogar fiir anf&nglich iiberhitzte Miachungen, wie bei dem Belepiel von Baas [l], erh< man vernunftige AbechlLtzungender OberfUche. Die Methode hat den Vorteil, elne klare %weicht Ober dae Berechnungsverfahren zu geben in Analogie zu jenen Methoden, die mit der Enthalpie ale treibender Kraft arbeiten.
THEORETICALLY,
the
simnhaneous
by BRAN [ 11. For the special case of water vaponr and air mixtures the simple enthalpy driving
transfer of
force method has been successfully applied (MICKLEY [8], BROWN [2]). An interesting extension of MICKLEY’S method has recently been
heat and mass across a phase boundary is an extremely complicated process. For the cooling of a saturated mixtnre of a condensing vaponr and an inert gas, COLBURN and HOUGEN [6] have presented a straighforward, stepwise, trialand-error method of calculation, later refined
published by CRIBB and NE-ON [7], who nse a corrected enthalpy transfer potential for the system moist coal gas-water. The aim of this paper is to show that by introducing a new hypothetical driving force concept, the basic COLBURN-HOUGEN analysis may be applied with little time-consumption and good accuracy. The writer has used the
by SMITE [lo]. CAIRNS [8] has presented experimental evidence for the validity of the uncorrected original COLBURN-HOUGEN method. However, the application of it is time-consuming. The same is true
for a general graphical method proposed 60
Condensation of vapours from gas-vapour mixtures.
The enthalpy of a condensate-free mixture may be expressed as
method successfully for some years in engineering practice.
i=c,M,,,(l
THEORY
+e)t+L,M;iz
(9)
or
By introducing a hypothetical potential of the form B=t-Fln(l--p/P) (1) Or 8 =t + Pln(l +z) (2)
i=L,M,
1 (10)
re(‘+~)t+il!
Using equations (2) and (8), the expression (10) may be written
for the total heat transferred, the rate of heat transfer at any point of a condensate surface will be proportional to the difference 8, - 0,. Assuming that diffusion and heat transfer occur independently of each other according to the rate equations
4A4
An approximate method of design
i-L -0 or
M 0 1+m(t’F)exp(6/F)-l exp (t/F) i = Lo Mu [u - exp (B/F) -
I l]
dq,/dA=L*M;k.ln
(13)
(5)
a =
1+ m @/F)
expU/F) and applying the semi-empirical relation suggested by C~ILTON and COLBURN[a].
;G (Pry
= &
m
+ 4&A
=
(SC)“/8 0)
dq/dA = h (8, -
0,)
(14)
Keeping the temperature factor F constant, the factor a is a function of the temperature t and the ratio m. If, however, (t/F) is small as compared with unity, the factor u, may be taken as approximately constant and ratios between enthalpy differences of the gas mixture may be approximated using only values of 8, i.e. . . 21 - 22 M exp(edF) - exp(e2/F) (15) is - i, exp (6/F) - exp 0%/F)
it follows that
4.W
(12)
with the notation
(8)
= h Vg- t,)
(11)
(7)
with
For small values takes the form . 21 i2-
(8) As can be seen from equations (l), (7) and (8) the potential 8 and the factor P have the dimension of temperature. Thus, 8 could be considered an “ effective ” temperature for a sensible heat transfer rate equivalent to the total heat transfer rate. Now, generally, the value of the factor F will vary with gas composition and temperature. For purposes of engineering design, however, good approximations may be obtained by using a mean, constant value of F throughout the whole condenser. The solution of a design problem requires knowledge of the relation between the potential 0 and the enthalpy i of the gas-vapour mixture.
of (O/F), the relation (15) . 22 w e, - e2 id e2 - e,
but the approximation have simultaneously
may be poor unless we
(17) As this is the case for a mixture of watervapour and air, the expression (16) thus confirms the
enthalpy
driving
force
approximation
for
small valties of (B/F), i.e. for low partial pressures
of steam. In a condenser with constant cross-section for gas flow, the sensible heat transfer coefficient h will usually decrease towards the cold end due 61
B. HIJI&N
.
to
a decreasing mass velocity. As the mass velocity is proportional to the factor (1 + ir), the relation between 0 and h may by the use of equation (2) be approximated using the proportionality
exp
4 h,=
491- t, F
exp [
1
( ) ~
$2
i
-
F
t2
1
lines from the operating line giving conditions [7], [8] and [9]. APPLICATION
The use of the method may best be shown by applying it to the carefully calculated example given in reference [6], as recommended by COLBURN[5]. The problem consisted in designing a counterflow condenser for a saturated mixture of steam and nitrogen at one atmosphere pressure and saturated at 95°C. The mixture is to be cooled to 40°C with water entering at 25’C and leaving at 60°C. An accurate calculation of the F-value is not justified, since even rough approximations give good accuracy. The mean value of the factor F for a steamnitrogen mixture will thus be taken as.
n
(18)
with IZ denoting the mass velocity exponent of h. If t is small as compared with 8, the proportionality may be approximated by the simpler expression
(19) This last expression will be used even when t is not small as compared with 8 because, to some extent it takes into account the temperature dependency of h. These approximations enable a rapid graphical evaluation of a sufficient number of condensate interface temperatures in a 8 - t - diagram for the saturated gas mixture in question, a subsequent integration giving the required condenser surface. The technique is analogous to that used in connection with the enthalpy potential, with tieTable 1. Point
F = 1500°C which is close to the arithmetic mean of the end values as is seen from Table 1. The value of the potential B can then be evaluated and plotted against t. The operating line, generally, will not be a straight line but an exponential curve, which is constructed as
Summary
1
3
of calculations 4
5
“C
(I-p,/p)
fmmPI
Mean F-value used ProperF-valueinherent in [S] 0 = tg- 15OOln(l - p,/P) exp (O/l~) 4u b from
PI
h P.c.u./(hr) (ft*)
Afrom [S]P.c.u./(hr)(ft2) tc tehm
lc - t
PI
LMT%
AA A A from [6] A A from [6]
OC OC “C
“C
(Z) PC) OC "C "C "C ft? fts ft2 ft2
95 0.165 1500 1855 2798 6.457 6O 80 84.5 84.5 _. 93.6 98.6 83.6 0 0
90
o*i305
1500 1470 1871 8481 4Q.5 42.8 28.8 28.0 84.4 84.8 48.9 88.4 288 226.4 288 226.4
62
85 0.430 1500 1505 1851 2461 88.8 85.8 19.3 18.0 72.9 78.0 89.1 41.5 76 81.0 309 807.4
75 0.622 1500 1550 787 1.690 28.8 30.5 15.4 14.5 53.5 54.4 24.7 81.4 82 85.4 891 892.8
9
7
--
---
5
interface
55 0.844 1500 1620 SO9 1.229 25.8 26.6 12d7 12.0 33.4 84.6 7.6 14.5
184 184.1 525 626.9
--49
0.927 1500 1680 154 1.108 25 25 12.0 11.7 27.7 27.8 2.7 4.8 165 168.1 690 696
Condensation of vapours from gas-vapour
mixtures.
follows (Fig. 1). With the end values given, the water temperature corresponding to e.g. 90°C gas temperature is calculated from a heat balance using the analogy (15), (neglecting condensate cooling)
60 - 40 _ exp 60 -
25
(2798/1500)
-
exp (2798/1500)
-
exp (1871/1500)
h (te -
An approximate method of design
tc) + L . AZ, *k * In
because, by definition, the left side of the equation (20) equals h (Bg - 8,) and consequently
eB-4-h_o
exp (154/15OO)
t, - t,
t, = 60 -
(60 -
25) @ii;
1
;:“;$
= 40.5%
(21)
h
Using the hot end values given in [S], viz. h0 - 81o ---= 34.5 h
9.0
a straight line from point 1 on the operating line with a slope of - 9.0 gives a condensate interface temperature intercept of 93.6’C. In the same way, the interface temperatures corresponding to points 3, 4, 5, 7 and 9 are calculated. For this purpose the relation (19) is used, with the mass velocity exponent ?z taken as 0.6. Thus, the value of h for e.g. point 3 is h = 34.5 (3*481/6-457)“‘6
= 23.8
(see Table 1). The neglecting of condensate cooling introduces an error in the position of the operating line and consequently also in the interface temperatures. A correction for this error could certainly be made, but considering that the error is usually comparatively small and the method approximate, a more accurate but time-consuming procedure seems not justified. With the interface temperatures thus calculated, the required surface between the points can be evaluated e.g. from the water-interface temperature differences. In this calculation the heat due to condensate cooling must not be neglected, or a considerable error will be introduced, because this amount of heat is clearly transferred with a smaller mean temperature difference than the rest of the heat. From the data given, the total heat of condensate cooling amounts to an additional 10 per cent. To simplify calculations, all condensate is assumed to be forme,d at the gas inlet point 1. The error thus introduced partly compensates the error made in approximating the operating line.
/ j i
FIG. 1. Approximate graphical solution of condenser problem given by COLBURNand HOUQEN[6]. Saturated mixture of steam and nitrogen at a total pressure P = 1 atmosphere. Temperature factor F = 1,500”C. Hypothetical temperature f? = t - 1,500 ln (1 - p#/P)“C. Similarily, the water temperatures corresponding to 85, 75 and 55% gas temperatures respectively are calculated, (see Table 1) and the resulting points of the operating line are plotted in the 0 -t -graph. Now, a “ tie-line ” from any point of the operating line having a negative slope equal to ratio of the heat transfer coefficients h,,/h gives an immediate graphical solution of the COLBURN-HOUGEN rate balance equation 63
The evaluation of the surface required between points 1 and 8 will accordingly run Total heat transferred without condensate cooling, from [6]
4,899,OOO
Heat from condensate cooling, [6]
498,000 P.c.u./hr
A summary of the calculations is- given in Table 1. Use of logarithmic mean values for the temperature differences is not theoretically justified but has been made since it gives slightly conservative results. In view of the many approximations made, _ the final result is in surprisingly good .agreement with the accurate calculation, the deviation being less than 1 per cent.
P.c.u./hr
Combined heat transfer coefficient, 810 P.c.u./(hr)
PI Exp (8/1500), point 1
6.457
Exp (d/MOO), point 9
1.108
Exp (8/1500), point 8
8.481
Temperature difference point 1, from graph
DISCUSSION
(ft”) (hr)
An analysis of errors shows that small relative errors in the position of the operating line and in the factor F should be divided by the factor E=V-S/h
886°C
Temperature difference point 8, from graph
48*9”C
Log. mean temperature difference, points l-8
88.4%
in order to obtain the resulting relative errors in the temperature difference. As the accuracy decreases with E, the mean value of F used should approach the proper value at the cold end in cases of a large variation in F between the hot and cold end. Preliminary checks of the method against the examples given in [9] and [a] show also good agreement with the calculations. Although not intended for initially superheated mixtures, the method is believed to give reasonable surface estimates even in such cases. It is then necessary ‘to assume saturation at the cold end. Depending upon the position of dew-point relative to intercept temperature, the method makes possible a quick determination of the onset of condensation, before which the surface calculation must be based on sensible heat transfer only. Using an F-value of 15OO”C, the method was applied to the example given by BRAS [l]. The problem was to design a condenser for moist converter gases at one atmosphere pressure entering at 136*4”C with a dew-point of 54*4’C and to be cooled to 29.8’C with water having an average temperature of 21’C throug&ut, the condenser. The terminal conditions indicated a constant average tie-line slope of about - 11.0. As the 0 - t - plot showed only a slight curvature in the equilibrium line and the operating curve was a straight vertical line, the required surface
Surface required between points l-8, without condensate cooling
Additional points i-a
surface
for
condensate
cooling
totalling 229 + 8.7 = 282.7 ft2 Similarily, surface is
between
dO/dt
points 8-4, the required
= 72.6 + 8.5 = = 76.1 ft2 As a result, the total surface between points 1 and 9 adds up to 690 ft2 as compared to the accurate figure of 695 ft’. 64
Condensation of vapours from gas-vapour mixtures. An
approximate
method of design
k = molar mass transfer coefficient L = Mtent heat of vaporization; Lo at enthalpy reference temperature M = molecular weight ; ill,,, mean molecular weight of gae mixture ; M, molecular weight of vapour m=factor n = mass velocity exponent of h p = partial pressure of vapour ; pg in bulk stream of gas mixture ; p, at condensate interface P = total pressure of gas mixture Pr = Prandtl number q = rate of total heat transfer ; q, sensible heat transfer rate ; qc heat transfer rate due to diffusion SC = Schmidt number t = temperature ; ts bulk temperature of gas stream ; tc temperature at condensate interface ; & cooling water temperature in = molar “ humidity ” : moles of condensing vapour per mole of inert gas 8 = hypothetical temperature for combined heat transfer
was evaluated using the end points only. The resulting surface was 14 per cent larger than that given by BRAS. With one intermediate point between the end points, an excess surface of only 7 per cent was obtained. The intercept tempertitures showed good agreement with those given, although the F-values inherent in the calculations at entrance and outlet were 1680 and 1600°C respectively. NOTATION A=sul&X? a=factor c = specific heat ; cmmean specific heat 6f.w mixture E=factor F = factor, temperature units G = molar mass rate h = coefficient of sensible heat transfer ; ho combined heat transfer coefficient i = enthalpy of gas mixture on inert gas basis
REFERENCE8
PI Bmis G. H. Chem. Engng. 1958, 228226, 288240. PI Brtowx G. Tram. Roy. Inst. Ttxk. 1954 No. 77 Stockholm. R. C. Chem. Engng. Sci. 1953 2 127-188. [al C!~UBNS PI PI
Cnrm-o~ T. H. and COL~URNA. P. Indwtr.
Engng. Chem. (Industr.) 1984 26 1188-1187.
PI PI
COLBURNA. P. and HOUGEN0. A. Industr. Engng. Ckem. (Industr.) 1934 26 1178-il82.
PI PI
McAnnaas W. H. Heat Trmwnision
COLSTJRN A. P. Inst. Mech. Engrs. Proc. Disc. Heat Transfer London 1951 James Clayton Lecture. CRIBBG. S. and NELSONE. T. Chewz.Engng. Sci. 1956 5 20-88. (2nd Ed.) 1942 pp. 285-292. McGraw-Hill, New York.
MICIUEYH. S. Chem. Engng. Progr. 1049 45 789-745.
[loI SMITHJ. C. Indust?. Engng. CM.
(Industr.) 1942 34 X248-1252.
65
of vapours from gas-vapour mixtures. method of design
An approximate
B. HULD$N Ekaio Ann&&ion for Power and Fuel Economy, S. Eeplanadgatan 14, Helalnki, Pinland (Iteeeioed 16 April 1957) Abetract-By using a hypothetical drlvlng force, the well known Co~~u~~~Houo~_[6] analyale may be applied graphleally, without trial and error, to solve condenser design problems. The method, ae compared with the orlglnal calculation in [tl], is rapid and the result agreea wlthln 1percent. The graphical method is approximate and includes the assumption [tl] of latent heat transfer overbalancing sensible heat tranefer ae pointed out by SMITH[lo]. It la believed that reasonable surface eathnatee may be obtainedeven for initially superheatedmixtures,w shown for the example of Baas [l]. The method haa the advantage of glving a clear overall view of the design procedure in analogy to the enthalpy driving force method& R6sum6-Par l’emploi d’un gradient flctlf, ll eat poasllM d’utlliaer l’analyae de Colbum-Hougen (6) aow forme graphique, aana i&&ion, pour r&oudre lea probli?meade ealcul d&r comknaeum. La m~,~m~aucalculoriginsl(8),estrapideetdonnedea~tateenrrccotdBIpour~tp~. La m&ode graphlque eat approchee, et suppose (6) que l’echange de chaleur latente l’emport& aur l*dchange de chaleur sensible, comme Pa indlque Smith (10). 11 eat admls qu’une &ahmtlon normaMde la surface peut 6tre obtenue, m&mepour dea m4angea wblement em&au&, comme le montm l’e-xemplede Bras (I). La mdthode B l’avantage de dormer une vue d’enaembleclaim, en comparaleon avec lea m&hodea utiliaant lea gradlente d’enthalpie. ZuanxumenfnsaunS-Dae bekannte Verfahren von COLBUBNund HOU~EN[f3]zur Berechnung von KONI~NSAZXXLIZN kann durch Eh&lhumg hypothetiecher treibender Kr&fte gmphlech ohne achrittwelae Verbeaserung verwendet werden. Diese Methode fOhrt achnell zu Ergebnieaen, die, vergllchen mit der Original-Methode [6], nicht mehr ale 1% Abweichung aufweieen. Die graphleche Methode let ein N6herungeverfahren und enth< die ANN-= [6], dasa die zu tlbertragenden latenten W&men gross slnd gegentiber den fOhlbarenWgrmen, wle es von SMITH [IO] r&her auegeftihrt wurde. Sogar fiir anf&nglich iiberhitzte Miachungen, wie bei dem Belepiel von Baas [l], erh< man vernunftige AbechlLtzungender OberfUche. Die Methode hat den Vorteil, elne klare %weicht Ober dae Berechnungsverfahren zu geben in Analogie zu jenen Methoden, die mit der Enthalpie ale treibender Kraft arbeiten.
THEORETICALLY,
the
simnhaneous
by BRAN [ 11. For the special case of water vaponr and air mixtures the simple enthalpy driving
transfer of
force method has been successfully applied (MICKLEY [8], BROWN [2]). An interesting extension of MICKLEY’S method has recently been
heat and mass across a phase boundary is an extremely complicated process. For the cooling of a saturated mixtnre of a condensing vaponr and an inert gas, COLBURN and HOUGEN [6] have presented a straighforward, stepwise, trialand-error method of calculation, later refined
published by CRIBB and NE-ON [7], who nse a corrected enthalpy transfer potential for the system moist coal gas-water. The aim of this paper is to show that by introducing a new hypothetical driving force concept, the basic COLBURN-HOUGEN analysis may be applied with little time-consumption and good accuracy. The writer has used the
by SMITE [lo]. CAIRNS [8] has presented experimental evidence for the validity of the uncorrected original COLBURN-HOUGEN method. However, the application of it is time-consuming. The same is true
for a general graphical method proposed 60
Condensation of vapours from gas-vapour mixtures.
The enthalpy of a condensate-free mixture may be expressed as
method successfully for some years in engineering practice.
i=c,M,,,(l
THEORY
+e)t+L,M;iz
(9)
or
By introducing a hypothetical potential of the form B=t-Fln(l--p/P) (1) Or 8 =t + Pln(l +z) (2)
i=L,M,
1 (10)
re(‘+~)t+il!
Using equations (2) and (8), the expression (10) may be written
for the total heat transferred, the rate of heat transfer at any point of a condensate surface will be proportional to the difference 8, - 0,. Assuming that diffusion and heat transfer occur independently of each other according to the rate equations
4A4
An approximate method of design
i-L -0 or
M 0 1+m(t’F)exp(6/F)-l exp (t/F) i = Lo Mu [u - exp (B/F) -
I l]
dq,/dA=L*M;k.ln
(13)
(5)
a =
1+ m @/F)
expU/F) and applying the semi-empirical relation suggested by C~ILTON and COLBURN[a].
;G (Pry
= &
m
+ 4&A
=
(SC)“/8 0)
dq/dA = h (8, -
0,)
(14)
Keeping the temperature factor F constant, the factor a is a function of the temperature t and the ratio m. If, however, (t/F) is small as compared with unity, the factor u, may be taken as approximately constant and ratios between enthalpy differences of the gas mixture may be approximated using only values of 8, i.e. . . 21 - 22 M exp(edF) - exp(e2/F) (15) is - i, exp (6/F) - exp 0%/F)
it follows that
4.W
(12)
with the notation
(8)
= h Vg- t,)
(11)
(7)
with
For small values takes the form . 21 i2-
(8) As can be seen from equations (l), (7) and (8) the potential 8 and the factor P have the dimension of temperature. Thus, 8 could be considered an “ effective ” temperature for a sensible heat transfer rate equivalent to the total heat transfer rate. Now, generally, the value of the factor F will vary with gas composition and temperature. For purposes of engineering design, however, good approximations may be obtained by using a mean, constant value of F throughout the whole condenser. The solution of a design problem requires knowledge of the relation between the potential 0 and the enthalpy i of the gas-vapour mixture.
of (O/F), the relation (15) . 22 w e, - e2 id e2 - e,
but the approximation have simultaneously
may be poor unless we
(17) As this is the case for a mixture of watervapour and air, the expression (16) thus confirms the
enthalpy
driving
force
approximation
for
small valties of (B/F), i.e. for low partial pressures
of steam. In a condenser with constant cross-section for gas flow, the sensible heat transfer coefficient h will usually decrease towards the cold end due 61
B. HIJI&N
.
to
a decreasing mass velocity. As the mass velocity is proportional to the factor (1 + ir), the relation between 0 and h may by the use of equation (2) be approximated using the proportionality
exp
4 h,=
491- t, F
exp [
1
( ) ~
$2
i
-
F
t2
1
lines from the operating line giving conditions [7], [8] and [9]. APPLICATION
The use of the method may best be shown by applying it to the carefully calculated example given in reference [6], as recommended by COLBURN[5]. The problem consisted in designing a counterflow condenser for a saturated mixture of steam and nitrogen at one atmosphere pressure and saturated at 95°C. The mixture is to be cooled to 40°C with water entering at 25’C and leaving at 60°C. An accurate calculation of the F-value is not justified, since even rough approximations give good accuracy. The mean value of the factor F for a steamnitrogen mixture will thus be taken as.
n
(18)
with IZ denoting the mass velocity exponent of h. If t is small as compared with 8, the proportionality may be approximated by the simpler expression
(19) This last expression will be used even when t is not small as compared with 8 because, to some extent it takes into account the temperature dependency of h. These approximations enable a rapid graphical evaluation of a sufficient number of condensate interface temperatures in a 8 - t - diagram for the saturated gas mixture in question, a subsequent integration giving the required condenser surface. The technique is analogous to that used in connection with the enthalpy potential, with tieTable 1. Point
F = 1500°C which is close to the arithmetic mean of the end values as is seen from Table 1. The value of the potential B can then be evaluated and plotted against t. The operating line, generally, will not be a straight line but an exponential curve, which is constructed as
Summary
1
3
of calculations 4
5
“C
(I-p,/p)
fmmPI
Mean F-value used ProperF-valueinherent in [S] 0 = tg- 15OOln(l - p,/P) exp (O/l~) 4u b from
PI
h P.c.u./(hr) (ft*)
Afrom [S]P.c.u./(hr)(ft2) tc tehm
lc - t
PI
LMT%
AA A A from [6] A A from [6]
OC OC “C
“C
(Z) PC) OC "C "C "C ft? fts ft2 ft2
95 0.165 1500 1855 2798 6.457 6O 80 84.5 84.5 _. 93.6 98.6 83.6 0 0
90
o*i305
1500 1470 1871 8481 4Q.5 42.8 28.8 28.0 84.4 84.8 48.9 88.4 288 226.4 288 226.4
62
85 0.430 1500 1505 1851 2461 88.8 85.8 19.3 18.0 72.9 78.0 89.1 41.5 76 81.0 309 807.4
75 0.622 1500 1550 787 1.690 28.8 30.5 15.4 14.5 53.5 54.4 24.7 81.4 82 85.4 891 892.8
9
7
--
---
5
interface
55 0.844 1500 1620 SO9 1.229 25.8 26.6 12d7 12.0 33.4 84.6 7.6 14.5
184 184.1 525 626.9
--49
0.927 1500 1680 154 1.108 25 25 12.0 11.7 27.7 27.8 2.7 4.8 165 168.1 690 696
Condensation of vapours from gas-vapour
mixtures.
follows (Fig. 1). With the end values given, the water temperature corresponding to e.g. 90°C gas temperature is calculated from a heat balance using the analogy (15), (neglecting condensate cooling)
60 - 40 _ exp 60 -
25
(2798/1500)
-
exp (2798/1500)
-
exp (1871/1500)
h (te -
An approximate method of design
tc) + L . AZ, *k * In
because, by definition, the left side of the equation (20) equals h (Bg - 8,) and consequently
eB-4-h_o
exp (154/15OO)
t, - t,
t, = 60 -
(60 -
25) @ii;
1
;:“;$
= 40.5%
(21)
h
Using the hot end values given in [S], viz. h0 - 81o ---= 34.5 h
9.0
a straight line from point 1 on the operating line with a slope of - 9.0 gives a condensate interface temperature intercept of 93.6’C. In the same way, the interface temperatures corresponding to points 3, 4, 5, 7 and 9 are calculated. For this purpose the relation (19) is used, with the mass velocity exponent ?z taken as 0.6. Thus, the value of h for e.g. point 3 is h = 34.5 (3*481/6-457)“‘6
= 23.8
(see Table 1). The neglecting of condensate cooling introduces an error in the position of the operating line and consequently also in the interface temperatures. A correction for this error could certainly be made, but considering that the error is usually comparatively small and the method approximate, a more accurate but time-consuming procedure seems not justified. With the interface temperatures thus calculated, the required surface between the points can be evaluated e.g. from the water-interface temperature differences. In this calculation the heat due to condensate cooling must not be neglected, or a considerable error will be introduced, because this amount of heat is clearly transferred with a smaller mean temperature difference than the rest of the heat. From the data given, the total heat of condensate cooling amounts to an additional 10 per cent. To simplify calculations, all condensate is assumed to be forme,d at the gas inlet point 1. The error thus introduced partly compensates the error made in approximating the operating line.
/ j i
FIG. 1. Approximate graphical solution of condenser problem given by COLBURNand HOUQEN[6]. Saturated mixture of steam and nitrogen at a total pressure P = 1 atmosphere. Temperature factor F = 1,500”C. Hypothetical temperature f? = t - 1,500 ln (1 - p#/P)“C. Similarily, the water temperatures corresponding to 85, 75 and 55% gas temperatures respectively are calculated, (see Table 1) and the resulting points of the operating line are plotted in the 0 -t -graph. Now, a “ tie-line ” from any point of the operating line having a negative slope equal to ratio of the heat transfer coefficients h,,/h gives an immediate graphical solution of the COLBURN-HOUGEN rate balance equation 63
The evaluation of the surface required between points 1 and 8 will accordingly run Total heat transferred without condensate cooling, from [6]
4,899,OOO
Heat from condensate cooling, [6]
498,000 P.c.u./hr
A summary of the calculations is- given in Table 1. Use of logarithmic mean values for the temperature differences is not theoretically justified but has been made since it gives slightly conservative results. In view of the many approximations made, _ the final result is in surprisingly good .agreement with the accurate calculation, the deviation being less than 1 per cent.
P.c.u./hr
Combined heat transfer coefficient, 810 P.c.u./(hr)
PI Exp (8/1500), point 1
6.457
Exp (d/MOO), point 9
1.108
Exp (8/1500), point 8
8.481
Temperature difference point 1, from graph
DISCUSSION
(ft”) (hr)
An analysis of errors shows that small relative errors in the position of the operating line and in the factor F should be divided by the factor E=V-S/h
886°C
Temperature difference point 8, from graph
48*9”C
Log. mean temperature difference, points l-8
88.4%
in order to obtain the resulting relative errors in the temperature difference. As the accuracy decreases with E, the mean value of F used should approach the proper value at the cold end in cases of a large variation in F between the hot and cold end. Preliminary checks of the method against the examples given in [9] and [a] show also good agreement with the calculations. Although not intended for initially superheated mixtures, the method is believed to give reasonable surface estimates even in such cases. It is then necessary ‘to assume saturation at the cold end. Depending upon the position of dew-point relative to intercept temperature, the method makes possible a quick determination of the onset of condensation, before which the surface calculation must be based on sensible heat transfer only. Using an F-value of 15OO”C, the method was applied to the example given by BRAS [l]. The problem was to design a condenser for moist converter gases at one atmosphere pressure entering at 136*4”C with a dew-point of 54*4’C and to be cooled to 29.8’C with water having an average temperature of 21’C throug&ut, the condenser. The terminal conditions indicated a constant average tie-line slope of about - 11.0. As the 0 - t - plot showed only a slight curvature in the equilibrium line and the operating curve was a straight vertical line, the required surface
Surface required between points l-8, without condensate cooling
Additional points i-a
surface
for
condensate
cooling
totalling 229 + 8.7 = 282.7 ft2 Similarily, surface is
between
dO/dt
points 8-4, the required
= 72.6 + 8.5 = = 76.1 ft2 As a result, the total surface between points 1 and 9 adds up to 690 ft2 as compared to the accurate figure of 695 ft’. 64
Condensation of vapours from gas-vapour mixtures. An
approximate
method of design
k = molar mass transfer coefficient L = Mtent heat of vaporization; Lo at enthalpy reference temperature M = molecular weight ; ill,,, mean molecular weight of gae mixture ; M, molecular weight of vapour m=factor n = mass velocity exponent of h p = partial pressure of vapour ; pg in bulk stream of gas mixture ; p, at condensate interface P = total pressure of gas mixture Pr = Prandtl number q = rate of total heat transfer ; q, sensible heat transfer rate ; qc heat transfer rate due to diffusion SC = Schmidt number t = temperature ; ts bulk temperature of gas stream ; tc temperature at condensate interface ; & cooling water temperature in = molar “ humidity ” : moles of condensing vapour per mole of inert gas 8 = hypothetical temperature for combined heat transfer
was evaluated using the end points only. The resulting surface was 14 per cent larger than that given by BRAS. With one intermediate point between the end points, an excess surface of only 7 per cent was obtained. The intercept tempertitures showed good agreement with those given, although the F-values inherent in the calculations at entrance and outlet were 1680 and 1600°C respectively. NOTATION A=sul&X? a=factor c = specific heat ; cmmean specific heat 6f.w mixture E=factor F = factor, temperature units G = molar mass rate h = coefficient of sensible heat transfer ; ho combined heat transfer coefficient i = enthalpy of gas mixture on inert gas basis
REFERENCE8
PI Bmis G. H. Chem. Engng. 1958, 228226, 288240. PI Brtowx G. Tram. Roy. Inst. Ttxk. 1954 No. 77 Stockholm. R. C. Chem. Engng. Sci. 1953 2 127-188. [al C!~UBNS PI PI
Cnrm-o~ T. H. and COL~URNA. P. Indwtr.
Engng. Chem. (Industr.) 1984 26 1188-1187.
PI PI
COLBURNA. P. and HOUGEN0. A. Industr. Engng. Ckem. (Industr.) 1934 26 1178-il82.
PI PI
McAnnaas W. H. Heat Trmwnision
COLSTJRN A. P. Inst. Mech. Engrs. Proc. Disc. Heat Transfer London 1951 James Clayton Lecture. CRIBBG. S. and NELSONE. T. Chewz.Engng. Sci. 1956 5 20-88. (2nd Ed.) 1942 pp. 285-292. McGraw-Hill, New York.
MICIUEYH. S. Chem. Engng. Progr. 1049 45 789-745.
[loI SMITHJ. C. Indust?. Engng. CM.
(Industr.) 1942 34 X248-1252.
65