Mean Temperature Difference in the Field or Bavonet Tube J
NORMAS L. HURD The Texas Company, Xew York 17, N. Y .
T
HE essential mechanical characteristic of the Field or bayonet’tube construction (Figure 1)is the use of an inner tube to conduct the fluid to the far end. There the fluid turns hack through the annular space betiwen the inner and the sealed outside tube, exchanging heat with the shell side fluid through thc: outer tube wall. Freedom from expansion difficulties and the ease of replaring individual tubes of a nest are among the advantages claimed for t,his arrangement. It is sometimes used to minimize shell side friction loss-for example, in vacuum condensers-since a longitudinal approach to the free end of the tube bundle practirally eliminates the impact and turn losses required by the conventional tube sheet construction. The distinguishing feature from the standpoint of heat transfer is the interchange of heat bctxecn passes of the tube side fluid, since heat is exchanged b e h e e n the inner conducting tube and the annular space as n-ell as between t,he latter and the shell side fluid. I n the majority of applications where the tube side niatc:rial is at. essentially constant temperature-for example, in conden+ ing steam or vaporizing refrigerant,-this int’erchange is negligible; however, in applications where the tube side fluid undergoes no change in phase, the interchange cannot in general be neglected. The effect of this interchange is to lower the mean tempeizturc difference between the shell side and tube side fluids. The derivations which follow evaluate the true mean temperature difference Atnl between the shell side and annular space, so that the t,otal heat transferred betwecn media is given by
W C ( A - Tz) = H P L A L COURTESY, GRlSCOM-RUSSELL COMPAUY
Field Tubes in Vacuum Condenser Service; Bundle Entrance Losses Are Rlinimized
Derived equations and curves are given for the mean temperature difference across the outside tube wall for the four possible arrangements of longitudinal fluid flow in the Field or bayonet tube. A concept of equivalent greater and lesser ‘ temperature difference is introduced from which the true mean temperature difference may be obtained by taking the logarithmic mean through the usual chart or slide rule methods. This concept appears to have promise of general application to heat transfer problems involving varying surface arrangements for “mixed” streams in combinations of cocurrent and countercurrent flow.
(1)
Four cases were studied (Figure 2) representing the poxsiblt. combinations of tube side and shell side flow. These four arrangements of Field tube flow were studied prcviously by Gel’perin (4). His results corresponding t o those of cases I11 and 1 V of the present article can be reduced to the more simplified equations given here; however, in his cases corresponding to eases I and I1 an error occurs in the differential heat balance, as the effect of the tube side interchange on the temperature in the annular space is omitted. (This error occurs in tlic base case and in case variants I and I11 of the reference cited. A typographical error also appears in Equation 15 of variant 11, in which bl - a1 should replace bl f al.) I n terms of the derivation given in the present text, the middle term of Equation 2 is absent. NOMENCLATURE
I n the following definitions any set of consistent units may be used. 5“ = temperature of shellside fluid a t anyopoint, O F. TI = inlet temperature of shell side fluid, F. T2 = outlet temperature of shell side fluid, F. t X = temperature of tube side fluid a t any point in second pass, F. t2 = outlet temperature of tube side fluid, O F.
1266
December, ,1946
INDUSTRIAL AND ENGINEERING CHEMISTRY
1267
t' = teTperature of tube side fluid a t any point in first pass, F.
tl
= inlet temperature of tube side fluid F. H = heat transfer coefficient between shell fluid and tube side
P
=
h
=
p
=
z
=
fluid in the annular space based on the area a t perimeter P , B.t.u./hr./sq. ft./ F. perimeter of outer tube, sq. ft. of outer tube surface/ft. of length heat transfer coefficient between tube side fluid in annular space and that in inner tube, based on area a t perimeter p , B.t.u./hr./sq. ft./a F. perimeter of inner tube, sq. ft. of inner tube area/ft. of tube longitudinal distance along exchanger, measured from tube side inlet and outlet end, a t any point, ft. mass rate of flow of tube side fluid, lb./hr. specific heat of tube side fluid, B.t.u./lb./' F. mass rate of flow of shell side fluid, lb./hr. specific heat of shellside fluid, B.t.u./lb,/" F.
w = c =
W = C =
Ti
2
- T,
O F .
G = quantity defined such that 2 = eQx is a solution of Equation 10, f t . - 1 G I , ~= particular values of G as $veri by Equations 13 and 13A a, b = constants of integration, F. L = total longitudinal length of exchanger, ft. A
TUBE SIDE
SHELL SIC€ FLOW c -
OUT
.
-
= value of the radical expression in Equation 12, dimen-
sionless Af,,, = true mean temperature difference between shell side and
tube side fluids, F. R = - T i- T2, dimensionless T
- t' = -,dimensionless HP
bfd =
A =
B = D = E =
true mean temperature difference between shell side and tube side fluids per degree of total temperature change of tube side fluid, dimensionless equivalent greater temperature diffcrence per degree change of tube side fluid (Equation 27), dimensionless equivalent lesser temperature difference per degree change of tube side fluid (Equation 27), dimensionless equivalent arithmetic mean temperature difference per degree change of tube side fluid (Equation 22A), dimensionless "half range" of equivalent temperature differences per degree change of tube side fluid, A - D = D - B (Equations 30 and 31), dimensionless
The mathematical approach of the following derivation is adapted from that used by Underwood (7) for the case of the usual type of tube in multipass arrangement. The -assumptions made a r e that (a) the heat transfer coefficients h and H remain constant; ( b ) the rate of flow of each fluid is constant; (c) the heat capacity of each fluid is constant; ( d ) the flow of the shell side fluid is longitudinal; ( e ) the area of
DEVELOPMENT. In Figure 3 a differential heat balance on each tube side stream gives hp(t"
- t')&
= wcdt'
(3)
Adding,
HP(T
- t")dx
=
wc(dt'
- dt")
COURTESY, H L N R Y Y O 8 7 M A C H I N E COMPANY
Field Tubes in Suction Tank Heater, Designed for Heating Viscous Fluids by Means of Sream in the Tubes; Flow Arrangement Is That of Case I, Figure 2.
(4)
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 38, No. 12
sum of two linearly independent particular solutions, each multiplied by an arbitrary constant ( 5 ) . Therefore = TI
T
- T
=
+
aeQls
b&s
(14)
From the terminal condit,ionv that T = Ti when x = L, and = TZwhen x = 0, the constants are determined to be
DETERMINATION OF MEAN TEMPERATURE DIFFERENCE
Differentiating Equation 14 with respect t o x and substituting Equation 6, aGleQl5
which, evaluated a t x
X
Figure 3.
+ bG2eG*z = -
Temperature Diagram for Case I
=
(T
- t”)
(16)
0, kwonicts
P -H wc (7’2 - t 2 )
u G ~f bGz
(17)
Dotted lines indicate direction of heat flow.
Substituting Equations 13, 13A, and 15 in 17 and simplifying, By a heat balance on the right-hand side, Solving for eGIL/eGaL and taking logarit h i s , Differentiating Equation 5 and substituting in Equation 4,
dT
&
=
- G2)L = 1 (Tz - t 2 ) + ;i (T1 -
(GI
HP r c (T - t ” )
DifferentiatingEquation 6,
(TI
In
- td
(1
Y’2)
+ 2)* f x (Tl - Tz)
1
+ 21 (T1 - Tz) (1 + E)- 21 x (Tl - T2)
Differentiating Equation 5 with respect to r a n d rearranging, 12.0
_dt” _- _dxdt’ - -Wwc -C dT dx dx
10.0 go
From Equations 3 and 5,
a0 7.0
60 5.0
Substituting Equations 8 and 9 in Equation 7 and expressing temperature in terms of variable 2,
(10)
40
1.5 1.0
SOLUTION.Substituting G i n Equation 10 and factoring,
a5 QI
PO
Solving this quadratic equation,
0
I
0 2
Figure 4.
Designating the radical expression by X and letting GI and GZbe given by Equation 12 with the plus and minus signs, respectively,
--(l+z+x)
GI = 1 H P
2wc
(13)
1 HP Since Equation 10 is a reduced linear differential equation of the second degree, the general solution can be expressed as the
R-CASESIIH
I
2
3
4
5
3
4
5
6
7 R-CASESIIfm
Solution of Equations 30 and 31
Eliminating GI and G2 by Equation 13 and c,limiriating L by Equation 1, Equation 19 bwomcs dt, = (7‘1
(T2
- tz)
-
T2)X
+ 5 (T1- Ta) 1
(20)
Substituting the value of A and eliminating the heat capacity terms by the limits of Equation 5, this becomes At,,,
1269
INDUSTRIAL AND ENGINEERING CHEMISTRY
December, 1946
=
+ 4hP HP (tn - t P TZ- tz + TI- tl + d ( T 1 - T I + k + 4hP j p (h - ti)2 TZ- tz + TI- - TI - TZ+ k - tJ2 + 4hP HP (tp - td* d ( T 1 - TI
[
+
t2
-
ill2
.I
t1)l
where E is given by Equation 30 or 31, depending upon the case under consideration. Figure 4 provides a graphical solution for E for all cases. Atd is then obtained from Figure 5, which is a plot of Equation 26. At,,, is obtained by multiplying Atd by the tube side temperature change (Equation 22C). CONCEPT OF EQUIVALENT TEMPERATURE DIFFERENCE
The section which follows provides an alternate method for the solution of Equations 23 and 25 which will be particularly convenient for those accustomed to extracting the conven(21) tional log mean temperature difference. Equation 21 gives the mean temperature difference in terms of the It is also of interest because of its general application to other terminal temperatures. types of heat exchange problems.
In
tl
DEYELOPMENT OF DIMENSIONLESS EQUATIONS
To ensure a graphical presentation which will provide maximum coverage of practical cases, as well as to simplify the form and aid in the visualization of the effect of changes in the variables, Equation 21 is expressed in terms of dimensionless ratios. The following dimensionless ratios will be used:
R
-
T1 t2
- Tz - ratio of
-
tl
shell to tube side temperature changes (22)
D = ! Ti+Tz-ti-tz
(
)
= arithmetic mean temperature 2 t z - tl difference per unit change in tube side fluid temperature (22A)
&
r
Atd =
= ratio of conductances per unit of tube length
(22B)
A tm = -
true mean temperature difference per unit tl change in tube side fluid temperature (22C) t2
E/D
Using these ratios, Equation 21 can be written as follows: Atd =
Figure 5.
+ + + +
V'(R 112 4r 1n[2D v'(R 1Iz 20 d ( R 1)2
-
(23)
+ 4r
+
This equation expresses the mean temperature difference in terms
Equations 23 and 25 indicate a similarity of form when compared with the expression for the usual log mean temperature difference. Therefore may be expressed as f o l l o ~ s :
of dimensionless ratios.
Atd =
Similar derivations for cases 11,111,and I V show that Equations 21 and 23 apply t o case IV as bell as t o case I,whereas the corresponding equations for cases I1 and 111,also identical, are as follows:
=
Tz
+
+
- -
Atd =
+ +
-
- tz + -k
A- B In
A
(27)
where A and B will be evaluated by comparison with the above equations and will become, respectively, the equivalent greater and lesser temperature d ( T 1 T 2 k t1)2 H 4hP P (tn t l ) 2 difference, from which the value of Atd may be (24) obtained by taking the usual log mean. TI - tl d ( T 1 Tz - k t l ) Q 4hP HP (k - t , ) 2 (These and subsequent temperature differences mentioned in this section are underT~ tl Tr tz 612 m(ts 4hP tl)* stood to be on the dimensionless basis of unit temperature change of tube side fluid.) CASES I AND IV. Equating the corred ( R 1)' 4r sponding parts of Equations 23 and 27 and solving simul(25) taneously,
-
AL
Solution of Equation 26
-
-
+ + - - + +
-
dpl +
-
1
Equations 23 and 25 can be readily rearranged t o the form,
A - D + E
(28)
B - D - E
(29)
where, for these cases, (26) Comparison of Equations 28 and 29 reveals that the quantity
D lies midway between the equivalent greater and least tempera-
INDUSTRIAL AND ENGINEERING CHEMISTRY
1270
From A and B tho mean tcsmpornturc differoncc per unit change‘ in tulie side temperature, Atd, is readily obtained through thc: usual charts ( 6 ) and slide rule methods (2) for the solution of the log mean (Equation 27).
REFRIGERANT
rl
-.
REFRIGERANT
-.
CHARTS A S D EQUATIOSS
I n applying these methods to problems involving cond(~ni?at,ion, vaporization, or chemical reaction it should be notcd that basic assumption c limits the application to cases where thi, cornb i n d sensible and latent heat (or heat of reaction) p i ' unit, changc in tempcrature may be considered constant. Where the tube sidc medium, as distinguished from ihal on t hi. shell side, is a t constant tcmperaturc, the methods of this papcr are not necessary, since the iisual log mean tempereaturc tiif ence applies. General Field tub? calculation procedurt. can be summai,izcd i n the following steps:
MIX'ACID + HC
ACID t
I?-
,-
Vol. 38, No. 12
b.
1. Determine R ar:d I1 from the terminal temperatures by Equations 22 and 22A, and r from the transfer rates and tube dimensions by Equation 22B. H and h are not single film rates but, over-all rates. 2. From R and T read E in Figure 4 using the R scale corresponding to the case number, which is obtained from the flow arrangements of Figure 2. If D is equal to or great,er than 3 times E , proceed directly to step 4,using the value of D as that of Atd.
3. Calculate E / D and read Atd/D from Figure 5 . Multiply by D to obtain At,+ 4. Multiply Atd by ts - tl t'o obtain the true mean temporature difference Atm for use in Equation 1.
\i COMPOSITE
'
FEFD-+
i il ll I"\
STEAM
PUMP-OUT 3 + I
.
A
-
TURBINE
DRIVE
I n employing the equivalent temperature difference r o n i q ) t , the steps remain the same as those given, except that, step 3 is rcplaced by the following: To D, alternately add and subtract I>' (as in Equations 28 and 29) giving A and B. The log mran of these (Equation 27) is Ltd. ' ILLUSTRATIVE EXAMPLE. I t is required t o heat 1000 pounds per hour of fluid of 0.5 specific heat from 50" to 100" F. by means of 250 pounds per hour of shell side heating medium of specific hcat 1.0 ent,ering a t 200" I?. The heat transfer coefficimt from the shell side to annular spacr. H is estimated to be 60 B.t.u./tlr. F./sq. ft. of outer surface of the out,sidetube, whereas the cocifficient h, betv-een the annular space and the inside of the condui,!ing tube, is 120 based on thc o u t u surface of t,heinner tube. The. diameters of the tubes are 2 inches and 1 inch outer diamctw. The flow arrangement is to be that of case I11 of Figure 2. ( : a h late the length of the Field tube rcquired. By a heat balance the o u t M t,emperature of the shell side fluid is found to be 100' F. The dimensionless ratios then become, h y Equations 22h and 2213,
COURTESY, ITRATFORD E N O l N E E R l N G COMPANY AND ORISCOM-RUSSELL COMPANY
Figure 6.
R=
Field Tubes in an Alkylation Contactor
The tube side medium may be either vaporizing refrigerant or water. T h e ahell aide medium is an emuleion of acid and hydrocarbons.
200 - 100 = 2.0 100 - 50
D = 5 1( - 200
+ 100 - 50 - 100) 100 - 50
=
1.50
ture differences and, therefore, represents the equivalent arithmetic mean temperature difference as well as representing, by definition, the true arithmetic mean temperature difference. The quantity E is half of the difference between the equivalent greater and lesser temperature differences and may be termed the "half range" of the equivalent temperature differences. CASESI1 A X D 111. Equations 28 and 29 apply here as well; however, for these cases,
I n E'igure 4, for R = 2.0 on the scale for case I11 and for r = 1.0, the abscissa and ordinate intersect at E = 1.10. Therc4or.e E / D = 1.10/1.50 = 0.734 and, from Figure 5 , A t d / D = 0.788 and Atd = 0.783 X 1.50 = 1.17.5. nyEquation22C,
= ; d m
Atm = 1.175 X 50 = 55.8'F.
E
(31)
The solution of Equations 30 and 31 is given by the intersection of abscissa and ordinate in Flgure 4, using the abscissa scale corresponding to the case considered.
In using the equivalent temperature difference concept for ( 1 ~ termining the mean temperature difference, obtain A and H from Equations 28 and 29, facilitated by Figure 4:
INDUSTRIAL AND ENGINEERING CHEMISTRY
December, 1946
A = 1.50 f 1.10 = 2.60 B 1.50 - 1.10 0.40 By any of the usual chart (6) or slide rule (2) methods available for obtaining the log mean average, Equation 27 becomes Atd
=
2.60
- 0.40 =
and At,,, = 58.8’ F. as before. From the true mean tempertiture difference the outside tube area P L is now readily obtained from Equation 1 :
and since the perimeter P i s 21r = 0.525 ft. 12
1271
I n cases where D is equal to or greater than 3 times E as read from Figure 4,D gives directly the true mean temperature difference per unit change of tube side fluid temperature, with an error of 4% or less. Analysis shows that no change is required in either equations or nomenclature if the tube side material is the heating medium, with the cooling medium in the shell, nor are changes required in cases where the tubes are provided with fins, aa in Figure 6 . The concept of equivalent greater and lesser temperature differences appears t o have a general application t o heat transfcr problems where arbitrary arrangements of cocurrent and countercurrent flow of the heating and cooling media are specified, provided that each fluid may be considered as mixed (of uniform temperature) at any cross section of its flow. A preliminary investigation indicates that i t is applicable with benefit of simplicity to all such cases for which mean temperature difference equations are readily available-namely, the 1-2 and 2-4 arrangements of shell and tube passes in multipass heat exchangers as given by Underwood ( 7 ) , the 1-4 arrangement of Yendall ( I ) , and the unbalanced pass arrangement of Gardner (3).
the required length is ACKNOWLEDGMENT
“ 1
L
=
3 = 13.5 ft. 0.525
The writer wishes to acknowledge the helpful suggestions of
If the effect of the interchange of heat between tube side streams were neglected-that is, r = 0-the log mean of the terminal temperature differences in countercurrent flow woula apply, giving a mean temperature difference of 72” F., or over 22% greater than the true value.
J. A. Davies, J. J. Gabor, and L. P. Gaucher of The Texas Company, Particular acknowledgment is due to K. A. Gardner, of the Griscom-Russell Company, who reviewed the manuscript and kindly made available his own uncompleted article on the subject, in which the same basic equations are derived. LITERATURE CITED
DISCUSSION
Equation 26 and Figure 5 show that, other things being equal, smaller values of E give larger meaq temperature differences. The flow arrangements of cases I1 and 111 are thus superior to those of cases I and IV in this respect, as can be seen by comparing Equations 30 and 31. It is therefore desirable to have countercurrent flow between the shell side fluid and that in the annular space.
(1) Bowman, Mueller, and Nagle, Trans. Am. SOC.Mech. Engrs., 62, 283 (1940). (2) Dalton, T. N., IND. ENG.CHEM., 30,1081 (1938). (3) Gardner, K.A., Ibid., 33, 1215 (1941). (4) Gel’perin, N. I., Khim. Mashinostroenie, 4, 1 (1939). (5) Sherwood and Reed, “Applied Mathematics in Chemical Engineering’’, p. 94, New York, McGraw-Hill Book Co., Inc., 1939. (6) Tubular Exchanger Mfr. Assoc., Inc., “Standards of Tubular Exchanger Manufacturers Association”,p. 19 (1941). (7) Underwood, A. J. V., J . Inst. Petroleum Tech., 20, 145 (1934).
Calculation of Number of Theoretical
Plates for Rectifying Column A. E. STOPPEL University of Minnesota, Minneapolis, Minn.
By
extending the equilibrium curve and operating line of the typical McCabe-Thiele distillation diagram beyond their usual limits, the operating line will be found to cut the equilibrium curve at two.points, once on the chart proper, and again somewhere outside the chart. The number of theoretical plates required to operate the column may be expressed as a simple power function of these two points of intersection.
T
HE problem of the rapid calculation of the number of theoretical plates required to effect a given separation by distillation of a two-component system was studied by a number of investigators (1-4,7 , 8) who have approached the problem in a variety of ways. Some of the proposed methods make simplifying assumptions and must be regarded as approximate when
used over wide ranges of operating variables. Others are empirical in nature and involve the use of charts. A few are exact for infinite reflux but do not hold a t lower reflux ratios. A method which is exact for any reflux ratio, purity of overhead product or waste, or condition of feed, but which assumes constancy of the relative volatility, was proposed by Smoker ( 7 ) . More recently Harbert (6) developed equations designed to “jump over” a number of plates in a single step. The method described here leads to an expression algebraically identical to the equation proposed by Smoker ( 7 ) . By proper choice of constants picked from a modified McCabe-Thiele chart, the equation may be expressed rather simply and in a form resembling the expression obtained by Fenske (3) for conditions a t total reflux. Figure 1 is an expansion of the familiar McCabe-Thiele (6) diagram. Within square OPQR are represented equilibrium curve
