I
S. L. SULLIVAN, Jr., and C. D. HOLLAND Department of Chemical Engineering, A. & M. College of Texas, College Station, Tex.
An Analytical Solution for a Double Pipe Heat Exchanger When gases are heated by condensing vapors, a highly accurate formula for the calculation of the area required to effect the heat transfer has been developed. This formula takes into account the variation of the physical properties of the system with .temperature
WHERE
GASES are heated over a wide range of temperature by condensing vapors, an accurate formula for the calculation of the area required to effect the transfer of heat has been developed. Only the development for the heating of gases within the tubes by condensing vapors on the outside is given because the formula obtained is applicable to the heating of gases on the outside of tubes by condensing vapors on the inside. A typical exchanger is shown below. The proposed equations take into account the variation of the physical properties of the system with temperature. Next to the method presented, perhaps the best is the one suggested by Colburn (3) in which the over-all coefficient of heat-transfcr was assumed to tary linearly with the difference between the bulk temperatures of the two streams. The development of the formula for the area required to effect the specified heat-transfer follows closely that shown previously (70) for the heating of liquids by condensing vapors. The principal difference lies in the treatment of the coefficient of heat-transfer from the tube wall to the gas. I t is assumed that this coefficienr varies linearly with the bulk temperature of the gas. Further approximations are as follows:
o The coefficient of heat-transfer from the tube wall to the gas is given by Equation 1. The specific heat of the gas varies linearly with the bulk temperature of the gas. The coefficienr of heat-transfer from the condensing vapor to the tube wall is a linear function of the bulk temperature of the gas inside of the tubes. Evaluation of this coefficient is by Equation 428 ( Z ) , which assumes film-type condensation and laminar flow of the condensate. The values of the coefficient at each end of the exchanger are used to determine the linear function. 0 The thermal conductivity of the tube wall is assumed to vary linearly with
hand side of Equation 1, which contains these variables. was evaluated for a variety of gases' at each of several temperatures (Figures 1 and 2). For all gases except methane, the data were taken from the collection given by Echert (5). For methane, the data given by Brown ( 2 ) were used. In view of the nature of the variation of the viscosity and thermal conductivity of gases with temperature, the well-behaved linear relationship of the coefficient of heattransfer with temperature is rather remarkable. These graphs do not include effects due to radiation. Apparently, the relatively small variation of the coefficient with temperature makes its experimental determination rather difficult. The efforts of numerous investigators have been summarized by McAdams (7). Based on the preceding assumptions the formula for the calculation of the area required to effect the heat-transfer is developed in the following manner. An enthalpy balance for a differential element of the exchanger leads to the well-known result [Equation 1 ( 70)] from which the following expression for the total area required to effect the heat-transfer is readily obtained.
the bulk temperature of the gas. Two values are calculated for the ends at the Of the tube These are used to determine the linear function of the bulk temperature. Application and Derivation
Since the film coefficient for the transfer of heat from the wall to the gas generally controls the over-all rate of heattransfer, it was investigated rather thoroughly. Under the same conditions stated previously (70), the following formula for the film coefficient was assumed to apply.
In the illustrative example shown herein, the following values for the constants were used, a = 0.8, b = 0.4, and d = 0.023. Both the viscosity and the thermal conductivity of a gas vary widely and in a nonlinear fashion with temperature. Although the Prandtl number for a gas remains nearly constant with temperature, it does go through a minimum for many gases. For gases such as air, this occurs in the general neighborhood of 400' F. The right
y-
CONDENSING VAPOR A T
TEMPERATURE
T
-------
GAS A T T E M P E R A T U R E
t
Diagram of a typical heat exchanger VOL. 53, NO. 9
SEPTEMBER 1961
699
, IOCC
z20
63
0 I80
i.;.
00
23c 300 400 TEMPERATURE (DEG R J
50c
A Figure 2. The variation of the coefficient of heat transfer i s linear over wide ranges at low temperatures
4
I
60
4?c
33c
6~-
ICCO
1200
6OC
L?i
530 I800
TEMPERATURE (DEG R)
+ Iz + 1 3
11
Ai
I;, 1,:and 1 3 represent the areas required
(2)
to effect the transfer of heat across the resistances given by Equation 2. In view of the assumptions stated previously, the formal integration of each of the expressions, Ii,12, and 13, is the same. For example, consider 11. The assumptions that both h and c, are linear functions of the bulk tempera-
(3) I? =
m ( D , - Di)Di 2 D,
k,(T - t )
r3 - W D ~ J " Do
cpdt ho(T - t )
Figure 1. Above 450" R., the coefficient OF heat 'transfer varies linearly over wide ranges of temperature
(4) (5)
ture mav be represented i n the follow ingmanner.
-h _- a i hi
= as
CPl
+ 616
(6)
+ bsQ
(7)
lvhere,
qi = t / t l
EXAMPLE Carbon dioxide i s to be heated from 8 0 " to 260" F. inside the tubes of a single pass exchanger. Saturated steam a t 2 8 0 " F. condenses on the outside of the tubes ( 1 inch, 18 B.W.G. brass). The mass velocity of the carbon dioxide i s 1.88 (Ib.)/(sec.) (sq. ft.) and 33 tubes are to be used. Find the total area required.
Evaluation of I , . Do = 1 .OOO inch, Dt = 0.902 inch. For a mass velocity of 1.88 (Ib.)/(sec.) (sq. ft.) and 33 tubes; w = 9 9 0 (lb.)/(hr.) Data used to solve this example were taken from ( 7 ) , (5), and ( 7 7 ) . For carbon dioxide: Temp.,
F.
CP
0.203 0.232
80 260
k 0.009575 0.01422
P
10.051 X 10" 12.98 X
The calculation of the inside coefficient, h, b y use of Equation 1 gives
hi h2
=
5.495
B.t.u. (hr.) (sq. ft.)
B.t.u. 6.540 ( h r . ) (sq. ft.)
k,
=
and
/3
are as
(B.t.u.) ( f t . ) -__ ( h r . ) (sq. f t . ) (OR. j
60.55
h,, = SO00
12
~~~~
(l3.t.u.) (hr.) (sq. ft.) (OR).
ho2 = 16000
~
(B.t.u.) (hr.) (sq. ft.) (OR.)
Since k, is essentially constant (k& = 60.55 and (kEA)nb reduces to k,(A),, where (A), i s the log-mean-temperature difference. Using Equations 10 and 11, the following values are obtained for 12 and 13.
rz = 0.03 sq. ft. r3 =
0.03 sq. f t .
The total area is the sum of /I,
13,
and
13.
A = 81.85 sq. ft.
at 80" F.
(OR.)
a t 260' F.
(OR.)
These values are used in Equation 8 to give Zi = 81.79 sq. ft.
Calculation of 12 and 13. The temperature drop across each resistance was calculated b y a trial and error procedure. The temperature drops across the tube wall and the steam film were so small that the values for the thermal conductivity of the wall and the physical properties o f the condensate were evaluated at 280" F.
700
The quantities needed to evaluate follows:
INDUSTRIAL AND ENGINEERING CHEMISTRY
This area i s 0.94% lower than value of 82.63 sq. ft. obtained b y the graphical integration of Equation 2 (70). This difference i s considerably below the percentage deviation quoted for the original data. Colburn's formula gave 80.15 sq. ft. which i s 3.ooyO lower than the value obtained b y graphical integration. Even better accuracy would have been obtained if the integral had been broken up into several parts, as cp i s not exactly linear from 8 0 " to 260" F. But the percentage deviation of the area given b y this method from that obtained b y graphical integration i s considerably less than the stated accuracy of the expression used for cpas a function of temperature.
DOUBLE P I P E H E A T E X C H A N G E R As shown previously (70), substitution of these expressions into Equation 3, followed by integration and rearrangement yields
where
and h, denotes the logarithmic mean of hz and h l . Similarly
The definitions for (kwA),w,(kw)m, (h0A)%, and (hJm are analogous to those for ( h a ) , and (h),,,, If two or more straight lines are required to describe any of the variables for a given problem, Equations 8, 10, and l l may be applied over the range corresponding to each straight line. ’ Expressions which differ from Equation l have been suggested by several investigators. Amoqg others, Knudsen and Katz (6) have collected and discussed many of these formulas. Colburn (4) and other workers proposed expressions with an exponent of the Prandtl number of l / 8 rather than 0.4. Because of the small variation of the Prandtl number with temperature, the particular power to which it is raised does not alter appreciably the linear relationship of h us. temperature. In some expressions for h, certain of the physical properties are to be evaluated at a film temperature. In this case, the terminal values, hl and h?, are computed according to the given expression. Then, when it is assumed that the film coefficient varies linearly between its terminal values with the bulk temperature, Equation 8 is applicable. Where the expression for h contains a ratio of viscosities as suggested by Sieder and Tate (9), it is again recommended that a linear variation of h between its terminal values with bulk temperature be employed. Thus, where expressions for h other than Equation 1 are available, the procedure recommended leads to the same formula for 11 as that given by Equation 8.
Heating of Gases on the Outside b y Condensing Vapors on the Inside of Tubes
For the flow of fluids in smooth annuli, formulas proposed for the calculation of the coefficient of heat-transfer have been summarized by Knudsen and Katz (6). Except for the use of an equivalent diameter and an additional factor consisting of the ratio of the diameters of the inner and outer boundaries of the annulus raised to a power less than unity, the expressions are of the same form as Equation 1. Formulas proposed for the calculation of the coefficients of heat-transfer for fluids flowing normal to tube banks have been summarized by McAdams (8). In most of these expressions, the exponent of the Reynolds number is of the order of 0.6. Except for the use of an exponent of the viscosity of 0.6 instead of 0.8, graphs similar to Figures 1 and 2 were constructed. For the same gases shown in Figures 1 and 2 and over the same temperature range, essentially straight lines were obtained for the film coefficient. Where physical properties are to be evaluated at film temperatures, the same procedure as that described previously is recommended. Thus, the formulas for Zl,Iz,and 13 are the same as those given by Equations 8: 10, and 11.
coefficient of heat transfer from the condensing vapor to the outside of the inner tube, (B.t.u.)/(hr.) (sq. ft.) (” R.) ( h ) , = logarithmic mean of the coefficient of heat-transfer, (B.t.u.)/(hr.) (sq. ft.) (” R.) ( h a ) , = logarithmic mean of (ha), defined by Equation 9, (B.t.u.) (” R.)/(hr,) (sq. ft.) (” R.) ZI,Iz,1 3 = used to denote certain integrals, defined by Equations 3, 4, and 5, respectively k = thermal conductivity of the gas, (B.t.u.)/(hr.) (sq. ft.) (” R. per ft.) k, = thermal conductivity of the wall of theinnertube. iB.t.u.)/ (hr.) (sq. ft.) (” R. pei ft.) Pr = Prandtl number = =
h,
I ,
~
(360;
Re t
T w
Gpp)
,
(dimensionless) ‘ = Reynolds number = ( D G / p ) , (dimensionless) = bulk temperature of the gas: t, denotes the temperature of the inner wall of the inner tube; to denotes the temperature of the outer wall of the inner tube; tl and t 2 denote the inlet and outlet temperature of the gas, respectively, ( ” R.) = temperature of the condensing vapor, (” R.) = maw rate of flow, (lb.)/(hr.)
Greek Letters A = denotes the temperature differt ; AI = T - t l ence, T and A, = T - t 2 , (” R.) p = viscosity of the liquid, (lb.)/ (sec.) (ft.) 6 = a variable employed in the derivations; C#J = t/tl, (dimensionless)
-
Acknowledgment
This work was supported in part by the Texas Engineering Experiment Station. Nomenclature a
=
al, a2 = Ai =
b
=
b l , b2 =
power to which the Reynolds number is raised; see Equation l , (dimensionless) constants used in the derivations heating surface of the inner tube, based on its inside diameter, (sq. ft.) power to which the Prandtl number is raised; see Equation l , (dimensionless) constants used in the deriva-
cp
tions .._~.~ = heat capacity of the gas, (B.t.u.)/
Di
= internal. diameter of the inner
D,,
= external diameter of the inner
(lb.) ( ” R.) tube, (ft.)
tube, (ft.) D,... = logarithmic mean of the diamk e r s Do and Q, (ft.) = mass velocity of the gas flowing G through the inner tube,, Ob.)/ . ,. (sec.)u(sq.ft.) h = coefficient of heat-transfer from the inner wall of the inner tube to the gas, (B.t.u.)/(hr.) (sq. ft.) ( ” R.)
literature Cited (1) Badger, W. L., Banchero, J. T., “Introduction to Chemical Engineering,” p. 733, McGraw-Hill, New York, 1955. (2) Brown. G. G.. and Associates, “Unit Operations,” pp. 440, 449, 584,’ Wiley, New York, 1950. (3) Colburn, A. P., IND.ENC. CHEM.25, 873 (1933). (4) Colburn, A. P., Trans. A.I.Ch.E. 29, ‘ ’174 (1933j. (5) Echert, E. R. G., Drake, R. M., Jr., “Heat and Mass Transfer,” pp. 504-7, McGraw-Hill, New York, 1959. (6) Knudsen, J. G., Katz, D. L., “Fluid Dynamics and Heat Transfer,” pp. 394, 403, McGraw-Hill, New York, 1958. (7) McAdams, W. H., “Heat Transmission,” 2nd ed., p. 173, McGraw-Hill, New York, 1942. (8 Zbid., 3rd ed., p. 272, 1954. ENC. (91 sieder, E. N., Tate, G. E., IND. CHEM.28, 1929 (1936). (10) Sullivan, S. L., Jr., Holland, C. D., Zbid., 53, 285 (1961). (11) Weber, H. C . , Meissner, H. P., \
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“Thermodynamics for Chemical Engineers,” 2nd ed., p. 493, Wiley, New York, 1957. RECEIVED for review February 27, 1961 ACCEPTED April 17, 1961
VOL. 53, NO. 9
SEPTEMBER 1961
701
