I
S.
L.
SULLIVAN, Jr., and C. D. HOLLAND
Department of Chemical Engineering, A. & M. College of Texas, College Station, Tex.
Double Pipe Heat Exchangers When a liquid is to be heated by a condensing vapor, use this method to give the area required
A
H E A T TRANSFER problem encountered frequently is the heating of liquids inside of tubes by vapors condensing o n the outside of the tubes. Formulas available for the calculation of the area needed when the liquids are heated over a wide range of temperature are inaccurate in some cases. T h e best of the proposed methods is suggested by Colburn ( 4 ) in which the over-all coefficient of heat transfer was assumed to vary linearly with the difference between the bulk temperature of the two streams. T h e basic equation describing this type of heat transfer has been integrated with the aid of several assumptions; these are very good for many systems. 0
T h e formula presented applies for a single-pass exchanger where the temperature of the condensing vapor remains constant throughout the length of the exchanger. The operation of this exchanger is generally described by the following equation, w c
dt - = Uiai ( T - t ) dL
(1)
I n deriving this equation it was assumed that no heat was transferred in the longitudinal direction, and that the fluid was perfectly mixed in the radial direction. Separation of the variables vields
The local coefficient of heat transfer from the liquid to the tube wall divided by the specific heat is given by Equation 8. Specific heat of the liquid varies linearly with the bulk temperature of the liquid. Over a considerable temperature range, the specific heat of many substances approximates the linear relationship. Viscosity of the liquid varies with temperature in exponential fashion. For the liquid inside the tubes, the ratio of thermal conductivity k to the specific heat c, raised to the (1 - b ) power is a linear function of temperature. For many substances this is a good assumption (figure, page 286). T h e heat transfer coefficient for vapor condensing on the outside of the tubes is a linear function of the bulk temperature of the liquid or gas inside the tubes. The coefficient is calculated by Equation 428 ( Z ) , which assumes film-type condensation and laminar flow of the condensate. The coefficient is calculated a t both ends of the exchanger and these two values are used to determine the linear function.
0
Application and Derivation
The thermal conductivity of the tube wall is assumed to vary linearly with the bulk temperature. Two values are calculated for the two ends a t the actual temperature of the tube wall. These determine the linear function of the bulk temperature.
When c,/Ui is taken to be a constant, the well known formula, q = UiAiAt,,,, is readily obtained. I n many applications the variation of cp/ Ui is of the same order of magnitude as that of the temperature difference (2" - t ) . Generally, the largest change in U; results from the variation in viscosity of the fluid being heated. The viscosity of a liquid changes with temperature in a n exponential fashion. This relationship was employed in the integration of Equation 2. I n order to describe properly the integration of Equation 2, it is best to state l / U i in terms of the individual resistance to heat transfer
I n the integration of Equation 5, the general formula for h (Equation 409) ( 2 ) was assumed to apply. This equation involves the ratio of the length of a tube to its diameter raised to a power. I n the following treatment this power was taken to be zero.
T h e function, h/c,, appears in Equation 5. I n the example a = 0.8, b = 0.4, and d = 0.023. For many liquids the viscosity, p, varies exponentially with the recipro.ca1 of the absolute temperature as follows : p =
Ke
where ZI, Iz,and 1 3 represent the areas required to transfer the heat across each resistance given by Equation 3. More specifically, h( T
- t)
(5)
(9)
where K , B, and C are constants and t is the absolute temperature. Theoretical support for a n expression of this form has been discussed ( I ) . Equation 9 is a good approximation for the viscosity over a wide range of temperature. T h e details of the integration of Equation 5 are given later. T h e expression obtained for Ilfollows : 11 = r {e-c [Ei( 2 2 ) e-8
- Ei ( e ) ] -
[Ei( X Z ) - Ei (p)]) (10)
where,
Az = T - tz
x2=g-mml
Then Equation 2 may be stated as the following sum of integrals :
4 +c
zz=c-m
This symbol Ei is used to denote the following exponential integral.
Values of this integral are tabulated in many handbooks (5,6). I n the integration of the expression for ZZ,Equation 6, c, (for the stream undergoing heating), and the thermal conductivity, k,, of the metal wall were each assumed to vary linearly with the temperature, t , of the fluid. T h e following result was obtained :
Based on these assumptions, a highly accurate formula has been developed which gives the area required for heat transfer. VOL. 53, NO. 4
e
APRIL 1961
285
where,
I
0
I
U n i t s of k
=
(Btu) (hr)(sq f t ) ( D e g R per ft)
I
0.6
n '
Details of this integration are shown later. T h e primary approximation made in this integration consisted of taking the thermal conductivity, k,, of the w7all to be a linear function of the temperature of the fluid being heated. T h e approximation allows for the effect of the variation of k, with the mean-wall temperature. At the inlet of the exchanger, t - t ~ ti, = til, and to = t,l and k,l is evaluated a t t i l , which lies between til and tal. Similarly, kwz is evaluated at the corresponding value, tiz. I n the integration of the expression for 18, the heat capacity, cp, of the fluid and the heat transfer coefficient, h,, were each assumed to vary linearly with the temperature of the fluid. T h e approximation made in the estimation of the effect of temperature on h, is the same as that described previously for the thermal conductivity of the metal wall. Under these conditions :
T h e integration is the same as for I 2 which appears later. T h e definitions for (hoA)m and ( h J m are obtained by replacing k, in Equations 13 and 14 by h,. When h, (or k,) is independent of temperature, (hoA),,, = h,(A),, and (ho)m = h,l = h,z. When these values are substituted in Equation 15, the resulting formula is the same as the one obtained by carrying out the integration a t a constant value for ha. When resistances of the wall and steam film to heat transfer are negligible, Ai is equal to 11. Also, U may be based on a surface area other than the interior one, provided all of the formulas describing the system are so modified. I n the application of the formulas for 1 1 , 12, and 13,it is convenient to use the terminal values for variables such as ,c, k , and y . However, in some cases the accuracy may be improved by using the values lying on the best straight line through the data. Where the symbols tl, tz, and T appear in the formulas for 11,12, and I,, the absolute temperature must be used. This results from the use of absolute temperature in the approximations of the physical properties as a function of temperature. liquids Outside Condense Vapors Inside the Tubes
0.3
I 0.2 0
550
600
ABSOLUTE T E M P E R A T U R E
I 650
i
700
750
(DEG R)
For liquids, the ratio of the thermal conductivity to the specific raised to the power varies linearly with the absolute temperature
of the liquid film coefficient which is of the same form as the one given for fluids inside tubes, Equation 8. Thus, the expressions for 11:12, and 13 are applicable when the appropriate values for a, b and d are used. Methods of Integration In the integration of the expression for I i , Equation 5, it is convenient to make use of the variable + = t / t i 3 where ti is the temperature of the entering fluid. When stated in terms of 9, Equation 5 becomes
where @ = T / t l . The subscript 7 denotes the value of the given variable at the inlet of the exchanger, and 2 denotes the value of the variable at the outlet of the exchanger. In the derivations which follow, it is also convenient to restate Equation 8 as follows :
absolute temperature, where
and where a1 and bl are constants. Substitution of Equations 4A and 5.4 into 2A yields
When this expression is substituted in Equation IA, the following is obtained.
(7.4) In the integration of Equation 7A, it is convenient to express + in terms of u. Starting with Equation 4A, the following relationships are readily developed.
a = a - b /3
=
1 - b; see Equation 8
Similarly, Equation 9 may be written
Also, for convenience let and where,
INDUSTRIAL AND ENGINEERING CHEMISTRY
0.6
As shown previously, the function F may be represented by a linear function of the
where,
For this case, Brown and others (2) have given a correlation for computation
286
500
+ +
since by Equation 5A, a1 bl = 1. When the term d+/[u(al 614) ( @ - +)I of Equation 7A is replaced by the appropriate relationships shown in Equation 8A, then
Since (c
1 - Y ) (s -Y ) =
1 (C+)[FY
-
-1
1 (10A) c-Y and when the following changes of variable are made, z = c y , and x = g - y , Equation 9A becomes
Since the integrals enclosed by the brackets may be stated in terms of the exponential integral, Ei, defined by Equation 11,
-
where,
To demonstrate the use of the formulas developed here and to compare the results calculated b y several methods, consider the following example: Water is to b e heated from 70" F. to 200" F. inside the tubes o f an exchanger with a single tube pass. The heating medium consists of saturated steam at 220" F. Water enters the tubes (1 inch o.d., 1 8 B.W.G. brass) a t a velocity o f 4 feet per second. The heat transfer area required per tube is the problem.
Evaluation of 11. Do = 1.000 inch, D, = 0.902 inch. For water, the following data were taken from Brown (2) and Perry (3):
Evaluation of 1%and 13. The conventional trial and error procedure was employed for the calculation of the temperature drop across each resistance. After these had been computed, the thermal conductivity o f the wall, k,, and the outside film coefficient, h,, were calculated. The following temperatures were obtained: til =
673OR.,
ti2
200
cp
b
P
1.0 1.0
0.334 0.402
0.94 0.28
B.t.u. Units of c p = (,b.) (OR.), units of k = (B.t.u.) (ft.)
(hr.) (sq. f t . )
636'R.,
t,l
to2 =
kwl
= 58'6 (hr.) (sq. ft.) (OR).
, evaluated
at
(B.t.u.) (ft.) evaluated at 59'2 (hr.) (sq. ft.) (OR.)'
til
ti2
The steam film coefficients were computed as recommended b y Brown (2). The values obtained were as follows: B.t.u. hol = 1732 (hr.) (sq. ft.) (OR.)' evaluated at a film
and units of 1.1 = centipoises
1
= 642.5OR.
1
= 674.8'R.
- &I)
(T
(OR.)'
Calculation of the water-side coefficient, hl, a t the inlet conditions b y use o f Equation 8 gives B.t.u. hi = 834 (hr.) (sq. ft.) (OR.)
633OR.
til
674.3OR., t 2 z = 673.65"R.
(B.t.u.) (ft.)
Then,
kwz =
Temp., O F. 70
630°R.,
B.t.u.
at a
hot = 3120 (hr.) (sq. ft.) (OR.),
( T - tiz)
O n the basis of a velocity of 4 feet per second, lbs. w = 3,990 (hr.) (tube)
Cp1
which is needed in order to evaluate the quantity r. Before r may b e computed, it i s necessary to evaluate F2, where
-
cp2
(hoA), = 166,600
= 0
B't'u*
(hr.) (sq. ft.)
When these values are substituted in Equations 12 and 15, the following values for 12 and l3 are obtained: where AI = 1 5 0 and AZ = 2 0 " R. Other quantities needed in the computation o f I1 follow: m = ( a - 6 ) In
PZ
=
0.484
tz = 5.54, g = t2 = 0.543 t z - tl Fz T ( A i - A,) xz = g - m = 0.0581, z2 = c - m = 5.056 e-c 0.006374, e-Q = 0.5817 Ei (2%) = 41.9036, Ei (c) = 60.3200 Ei ( X Z ) = -2.2140, Ei (g) = 0.5915
e=----
Substitution of the above values into Equation 10 gives (sq. ft.)
11 = 6.690 __(tu be)
Because of the differences of numbers involved in the computation of 11, it is recommended that the exponentials and the exponential integrals b e stated accurately to several significant digits.
Then,
A,
=
-
ZZ = 0.5253
(G)
rs = 2.8065
(g)
11
+ I2 + z,
= 10.02
(tube.) sq. ft
This area is 1.7% lower than the value of 10.1 9 (square feet per tube) obtained b y graphical integration. This difference i s of the same order of magnitude as the computational accuracy. Colburn's formula gave 9.79 (square feet per tube), which i s 3.970 lower than the value obtained by graphical integration. Based on an average of Ui a t the inlet and U, at the outlet conditions, an area o f 10.42 (square feet per tube) was obtained. This value is 2.3Y0 higher than the one obtained b y graphical integration. It is better than can usually b e expected. The significant result i s that the area given b y the formulas presented here and the one obtained b y graphical integration agree to within the computational accuracy.
VOL. 53, NO. 4
APRIL 1961
287
Equation 10 is the solution of Equation 11A. The way the expressions after Equation 10 were obtained merits some elaboration. I n view of the definitions of y and n, z1
= c,zz = c
- a 1 nM cZ
and
Also, when a1 and bl are stated in terms of the terminal values of + and F , the result is
- FzAi - A2 - A i - Az
1
The expressions for n, c, and g beneath Equation 10 were similarly obtained. I n the integration of the expression for 12, Equation 6, t p of the liquid and the thermal conductivity, k,, of the wall are taken to be linear with respect to 4. Then
Equation 6 may be restated in terms of the variable 4 as follows : I2 =
[W
( D o - Di)Ai~p”.]s,:’ 2 A m L 1 b3
(E + .) (2 + .)(a
dc6
(14A)
- 4)
This integral is readily rearranged to the sum of twc integrals,
These integrals may be found in any standard set of integral tables. After the integrations have been carried out and after U Z , bz: u 3 , and ba have been evaluated in terms of c p and k , at the inlet and outlet conditions of the exchanger, Equation 12 is obtained. The integration of 13 is carried out in the same manner, where h, replaces k, in all of the expressions. Acknowledgment This work was supported by the E . I. d u Pont d e Nemours & Co., Inc. and the Texas Engineering Experiment Stations. T h e suggestions made by R. E. Basye are also appreciated. Nomenclature a
= power to which the Reyn-
old’s number is raised; see Equation 8, dimensionless al, a2, a 3 = constants used in the derivations
288
= internal heating surface of
the inner tube per foot of pipe, (sq. ft.),’(ft.) = heating surface of the inner tube, based on its outside diameter, (sq. ft.) = heating surface of the inner tube, based on its inside diameter, (sq. ft.) = heating surface of the inner tube, based on the logarithmic mean of the outside and inside diameters of the inner Tube, (sq. ft.) = power to which the Prandtl number is raised; see Equation 8 (dimensionless) = constant used in the expression for viscosity, Equation 9 = constant whose definition follows Equation 10 (dimensionless) = heat capacity of the liquid (B.t.u.)/(lb.) (OR.) = constant used in the expression for viscosity, , . Equation 9 = constant in Equation 8 (dimensionless? = internal diameter of the inner tube, (ft.) = external diameter of the inner tube, (ft.) = base for natural (or Naperian) logarithms = exponential integral defined by Equation 11 = a ratio; definition follows Eauation 10 idimensidnless) = a ratio: defined bv Eauation ’5A (dimenkoniess) = constant whose definition follows Equation 10 (dimensionless) = mass velocity of the liquid flowing through the inner tube, (lb.)/(hr.) (sq. ft.) = coefficient of heat transfer from the inner wall of the inner tube to the liquid, (B.t .u.) /(hr .) (sq. ft.) ( O R . ) = coefficient of heat transfer from the condensing vapor to the outside of the inner tube, iB.t.u.) /(hr.) (sq. ft.) ( O R . ) Zl, 12,13 = integrals defined by Equations 5, 6, and 7 k = thermal conductivitv of the liquid, (B.t.u.) /(hr.) (sq. ft.) (OR. per ft.) kw = thermal conductivity of the wall of the inner tube, (B.t.u.)/(hr.) (sq. ft.) (OR. per ft.) (,ktJm = logarithmic mean of the thermal conductivity of the wall: defined bv Equation ‘ 14, (B.t.u.)) (hr.) (sq. ft.) (OR.per ft.) (kwA)m = logarithmic mean of ( k A ) , defined by Equation 13, (B.t.u.) (‘R.)/(hr.) (sq. ft.) ( O R . per ft.) K = constant used for viscosity, Equation 9: (lb.)/’(hr.) (ft .I
INDUSTRIAL AND ENGINEERING CHEMISTRY
L, Lt
m n r
t
T
= length
of the exchanger measured from the inlet and the total length, respectively, (ft.) = a constant; definition follows Equation 10 (dimensionless) = a constant used in the derivations; definition follows Equation 4A = a constant defined after Equation 10, (sq. ft.)/ (tu be) = bulk temperature of liquid: ti denotes temperature of inner wall of inner tube, to denotes temperature of outer wall of inner tube; tl and t 2 denote inlet and outlet temperature of liquid, (“R.) = temperature of condensing vapor, (OR.)
U
= a symbol in the derivations,
ui
= over-all coefficient of heat
U
=
W
=
XZ
=
za
=
defined by Equation 4A transfer, based on D, (or Ai); defined by Equation 3, (B,t.u.)/(hr.) (sq. ft.) (OR.) a variable of integration used in the defincion of Ei, Equation 11 mass rate of flow, (lb.)/ (hr.) ( g - m) (dimensionless) as used in Equation 10 (G - m ) (dimensionless) as used in Equation 10
Greek Letters a:
= a constant used in the der-
s
= a constant used in the deri-
A
= temperature difference, T - t ; A, = T - t l and Az = T - t z , (OR.)
ivations vations
!J
9 @
viscosity of the liquid, (lb.)/ (hr.) (ft.) = a variable employed in the derivations; q = t / t ~(dimensionless) = a constant. @ = T,’tl (dimensionless) =
Literature Cited (1) Bird, R. B . Stewart, \V. E., Llghtfoot, E. N., “Transport Phenomcna,” p. 28, Wiley, New York, 1960. (2) Brown, G. G., others, “Unit Operations,” p. 440, 449, Wiley, New York, 1950. (3) Chemical Engineers’ Handbook (J. H. Perry, ed.), 3rd ed., p. 225. McGrawHill, New York, 1950. (4) Colburn, ‘4. P., Znd. Eng. Chenz. 25, 873 (1933). (5) (‘C.R.C. Standard Mathematical Tablcs” (C. D. Hodgman, ed.), p. 304, Chemical Rubber Publishing Co., Cleveland, Ohio, 1956. (6) Jahnke, Eugene, Emde, Fritz, “Tables of Functions TVith Formulae and Curves,” 4th ed., p. 6, Dover Publ., New York, 1945. RECEIVED for review August 1, 1960 ACCEPTED January 10, 1961
