INDUSTRIAL AND ENGI*VEERING CHEMISTRY
September, 1928
889
Heat Transfer in Oils Flowing through Pipes' M. Garcia GULFREFINIWG C O M P A N Y , PORT
ARTHUR, T E X A S
T
HE transfer of heat in oils flowing through pipes is a which represents the law of temperature variation along the subject of primary importance to the petroleutri in- pipe. It has been well established from the work of McAdams dustry. Methods commonly used for the calculation of heat transfer rates are satisfactory for water and gases, and Frost,2 Rice,3 Morris and Whitman,4 and others that but in the case of oils and other highly viscous liquids the for turbulent flow the film resistance can be represented by variables involved require certain modifications that cannot an equation of the form be disregarded in the light of recent developments in the Ad r = theory of heat flow. (3) k The basis of design of heat transfer equipment is the overwhere d = inside diameter of all transfer coefficieit--that pipe is, the rate of heat flow per k = thermal c o n d u c unit surface per unit of mean tivity of liquid The logarithmic mean temperature difference equaz = absolute viscosity temperature difference betion is based on assumptions which are wholly unA , a , b = constants tween heating and cooling justified when dealing with oils, and may give misfluids. The reciprocal of leading results in heat transfer calculations. DifferThe dimensionless group this over-all coefficient is the ential equations have been derived, taking into account w / d z is derived f r o m t h e total thermal r e s i s t a n c e , the change in physical properties of the liquid with t u r b u l e n c e factor dus/r, n u m e r i c a l l y equal to the temperature, and the results obtained by integration where 1~ is the linear velqcity sum of the partial resistof these equations differ considerably from those given of the liquid and s the denances of the pipe mall and by the logarithmic mean formula and show a different sity, by substituting for v the fluid films on both sides law of temperature distribution along the exchanger. its value 4 w/d2s. The exof the wall. The mean temGraphical methods are given for the calculation of ponents a and b may be perature difference is taken mean liquid temperatures and mean temperature taken at 0.37 and 0.38, reas thr: logarithmic mean bedifferences,to be used in the correlation of experimental spectively. Very recent intween initial and final differdata or the design of exchangers for heavy viscous oils vestigations by Morris and ences-that is. involving large temperature ranges. JF'hitman' indicate that b is el e2 e, = not constant, b u t v a r i e s e In 2 with the degree of turbuez This expression for mean temperature differewe is based lence. For the purposes of the present derivation, honever, on the assumptions that the specific heat of the fluid and the the error introduced by taking b = 0 83 is comparatirely heat transfer coefficient remain constant during the heat small. For a given set of conditions, A , d, and zu are constants, interchange. The spc>cificheat of gases and liquids, however, is a function of the temperature, and the transfer coeffi- while c and z are functions of the liquid temperature. The cient depends on physical properties of the fluid that also thermal conductivity of oils has not been sufficiently investivary with the temperature. For petroleum oils these varia- gated, but its variation with temperature appears to be very tions are of such order that the assumption of constancy is slight, and consequently k may be taken as constant without entirely unjustified, arid a new expression for mean tempera- introducing any serious error. Equation (1) may therefore ture difference must be derired, taking this fact into account. be written in the form -A 'p(t)dt dS = ___ T-t
Derivation of Differential Heat Transfer Equations
The differential equation for heat transfer between a liquid in a pipe and the wall, when the wall temperature is constant, may be derived as follows: Let dS be an element of surface of the pipe wall and dt the differential change in temperature of the liquid flowing past this surface; then, by a heat balance, the total heat absorbed or lost by the liquid must equal the heat transmitted through the surface: where w c T t
r
= = = = =
weight of liquid flowing per unit of time specific heat pipe wall temperature liquid temperature thermal resistance of liquid film
Solving this equation for the surface S, we obtain ft P
s=
-w
1s
J
where and
Adb+lLll-b .4' = kl-L ( t ) = c l - - o z h - ~ ~=
(4
~
c U 6dZ0 46
(5)
The viscosity-temperature relation in petroleum oils fo11o~t-s approximately the equation z = at - p , plotting as a straight line of slope p on logarithmic paper. The constants CI and 9 vary with the specific gravity and baqe of the oil, ranging from 0.9 for Pennsylvania and light Oklahoma crudes to 9 or 9.5 for heavy Mexican fuel oils. The specific heat of oils has been shown by Fortsch and Whitmans to vary with the temperature, following a linear relation of the type c = -Ut S, t being expressed in degrees Fahrenheit and Jl and N depending on the specific gravity of the oil at 60" F. These physical properties are variously taken at the mean temperature of the liquid stream or at the mean film temperature, the latter being defined as the arithmetical mean between average stream temperature and temperature of the
+
tl
2
Presented before the Division of Petroleum Chemistry a t t h e 75th Meeting of t h e American Chemical Society, St. Louis, Mo., April 16 to 19, 1928.
6
1
Refvsvrgeratsng Eng , 10, 323 (1924). I N D ENG C H E M , 16, 460 (1924) I b t d , 2 0 , 234 (1928). Ibsd , 18, 79.5 (1926).
INDUSTRIAL AND ENGINEERING CHEMISTRY
890
inside wall surface. I n the present derivation, however, mean stream temperatures will be considered. The variation of film viscosity with temperature is of greater importance than the specific heat variation, since the former may cause the transfer coefficient to vary as the third or fourth power of the liquid temperature. If the specific heat is assumed constant over the temperature range, equation (4)may be simplified to
dS where and
B =
=
-Bdt t"fT--t)
- \Ad1.83~0.17(~/k)0.63Ly0.46
n = 0.46p
It is evident that the integration of equation (6) depends entirely on the value of the exponent n, which is determined by the logarithmic slope of the viscosity-temperature curve, and therefore no single equation can be derived to apply to all cases. When n equals zero, the thermal resistance becomes independent of liquid temperature, and the integrated equation is S = -Bwln-
T-t2 T-ti
(7)
from which the well-known logarithmic mean temperature difference can be derived. This is the ideal case of a liquid of constant specific heat and constant viscosity, the only one for which the logarithmic mean difference is theoretically correct. For petroleum oils, n may range from 0.41 to 4.4, and the integrated equations will differ widely. As an example, the integer values 1, 2, and 3 will be considered:
Vol. 20, No. 9
n = 2. At higher values the heating curve assumes the reverse curvature of the logarithmic mean curve (n = 0) given by most writers as typical for this case of heat interchange. These results emphasize the importance of establishing a rational criterion for mean liquid temperature in heat transfer calculations. All equations hitherto proposed for the film transfer coefficient, whether referred to the temperature of the main body of the liquid or of the boundary film, are based on the average liquid temperature. A difference of a few degrees in the temperature used may result in a five- or tenfold variation in viscosity, with a corresponding error in the film coefficient calculated. Graphical Integration
It is theoretically possible to derive the mean temperature difference from equation (2), expressing both the instantaneous thermal resistance and specific heat in terms of temperature. This would require an analytical derivation for each particular case, which obviously is impracticable. It is always possible, however, to determine the mean liquid temperature by graphical integration of a plot of temperature versus pipe surface passed over, when the approximate laws of change of thermal resistance and specific heat with temperature are known. 300
7502
200
(9)
a
I_
I
or approximately, dropping the term factored by 1/T as comparatively very small, -Bw =
s
-Bw -( T
T-(7, 1
1
-
ri)
$) (approximately)
200
300
400
iao
SO0
Morris and Whitman,4 in their extensive research work on film transfer coefficients, used a graphical integration based on the assumption that the transfer coefficient varies Application of Equations inversely with the square root of the viscosity, which agrees The difference in results yielded by these equations may be closely with the value 20'46 previously derived. The effect studied in their application to an actual case. Consider a of the variation of specific heat with temperature was not steam heater with a surface of 500 square feet, heating 10,000 taken into account. pounds of oil per hour from 100" to 300" F., the average It has been shown previously that the product of therspecific heat of the oil being 0.5 and the steam temperature mal resistance and specific heat varies inversely with some remaining constant at 350" F. The mean liquid temperature, function cp ( t ) of the liquid temperature. This function can mean temperature difference, and heat transfer coefficient be either obtained experimentally or calculated from the were calculated for values of n equal to 0, 1, 2, and 3. In viscosity-temperature and specific heat-temperature curves the last three cases the mean liquid temperature was found of the oil, by equation ( 5 ) , and a series of values of cp ( t ) by graphical integration of the curves plotted from equations At/(T-t) calculated and plotted for close temperature inter(8), (9), and (11); the mean temperature difference was then vals 4t over the required range; a graphical integration of obtained by subtracting this mean liquid temperature from this plot will give the liquid temperature curve. In practice the constant outside steam temperature. the plotting of the differential curve may be omitted and a The results are shown in the following tabulation: stepwise integration by Simpson's rule carried out simulMEANTEMPERAHEAT taneously, giving directly the integrated curve. MEANLIQUID TURE TRANSFER It may be questioned whether the difference in results BASIS TEMPERATURE DIFFERENCE COEFFICIENT F. F. obtained by this method, as compared with the logarithmic Constant transfer coefficient: mean value, is always such as to justify the additional work (n = 0 ) 226 0 124.0 16 1 Transfer coefficient varying required. This may be decided in any particular case by with liquid temperature. examination of the viscosity-temperature curve of the oil 207.6 142.4 14 1 n = l 165.9 184.1 10 9 n = 2 in question. Judging from the results shown in Figure 1, 152.6 197.4 9.9 n = 3 it may be said that the logarithmic mean is approximately Figure 1 gives the calculated heating curves. It must correct, probably within the limits of experimental error, be noticed that the over-all transfer coefficient decreases when the logarithmic slope of the curve is less than 2. For as the exponent n increases. For low values of n the curve slopes higher than 2-that is, for oils whose viscosity varie; flattens and approaches a straight line, the mean liquid tem- with the third, fourth, or higher power of the temperatureperature and mean temperature difference becoming identical the difference is undoubtedly large enough to merit considerawith the arithmetical mean somewhere between n = 1 and tion. n = 3,
=
t -
12
(11)
September, 1928
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Applications of Graphical I n t e g r a t i o n M e t h o d
The method above described was applied by the writer to a series of tests made by McCormick and Diederichs of Cornel1 University.6 These tests were made on a very viscous Mexican crude (about 7500 seconds Saybolt a t 100" F.) heated with steam in a commercial multitubular exchanger, the oil flowing outside of the tubes. The viscosity-temperature curve of the oil shows a logarithmic slope of !J.5. Heating curves were plotted and from them the mean liquid temperature and thermal resistances calculated.
ID0
80
-
and the linear velocity, 1.2 feet per second. The equations given by Morris and Whitman4 will be used for calculating r , taking P = f ( c z / k ) and N = F (dus/z) from their plots and using the value 0.078 for the thermal conductivity. The calculations are tabulated as follows: 2
1
80
0.44
13.0
0.0397
r c A f / T --t 0,00145
100
0.45
6.7
0,0374
0,00153
120
0.46
3.8
0.0320
0,00147
t
C
140
0.48
2.4
0.0203
0,00108
160
0.49
1.6
0.0144
0,00088
180
0.50
1.1
0,0108
0.00077
200
0.53
0 S
0.0090
0.00077
0.00149 0.00150
0.00127 0.00098 0.00083 0.00077
--0,00684
60
fi
Surface required, 533 X 0.00684 = 3.64 square feet per tube, or 436 square feet total. Tube length required, 3.64 X 1 2 / 0 . 6 2 ~ =22.5 feet.
:lr '0
u
s
891
20
IO
w
The method is also applicable to the case of two liquids changing temperatures from end to end, as in an oil-to-oil exchanger, in which case the working equations must be modified. The liquid film resistances on both sides of the mall are of such magnitude that the pipe wall resistance may be neglected in most cases. The differential equat'ion in this case is
For purposes of comparison, the obtained values of Twere plotted on logarithmic paper against Z U / ~ . The turbulence factor dvs/z cannot be computed in this case, since the flow took place in the baffled space around -(T-t)dS zlcdt __ = WCdT = the tubes and the velocity and equivalent value of d are una R + r known. The results are given in Figure 2, which shows a where W = pounds hoToil per second w = pounds cold oil per second line of much flatter slope than the average of the curves T = hot liquid temperature obtained by Morris and Whitman. This may be due to the t = cold liquid temperature flow being of the viscous or semiturbulent type. Typical R = thermal resistance of hot liquid film calculated heating curves are given in Figure 3. 7 = thermal resistance of cold liquid film The integration met>hodcan be used in the design of heat which integrated becomes Ptz transmission equipment, the actual plotting of the temperature curve not being necessary in this case. The general procedure is as follows: ( ~ 2 ) ~ ' ~ ~
Calculate rcAt/(T-t) for a series of temperatures :it 10- to 20degree intervals. Find the arithmetical averages between two consecutive values. The sum of these averages will give the surface required per pound of liquid flowing per hour.
The thermal resistances may be computed using actual values of the functions f ( c z / k ) and f(w/dz), obtained from plots of experimental data. When this information is available, all arbitrary assumptions as to the nature of these functions are avoided.
These integrations are only possible when the values of T - t can be computed over the entire integration range, since T is not constant. It is therefore necessary to know the relation between the cold liquid temperature a t any point and the corresponding hot liquid temperature across the tube wall. Let A and B be the specific heat equation constants of the hot liquid and a, b, those of the cold liquid, then
+
+
W(U bt)dt = - W ( A BT)dT and integrating B W A(Ti-Tz) 2 (Ti2-TzZ) = E [ ~ ( t i - t l ) b(tiz-t~2)l:= QlCV
+
+
Then Solving for TI,
This equation will enable the designer to find the hot liquid temperature T for any given value of t , from where the difference T - t and the heating curves for both liquids may be then calculated by the same method given before. As an example, consider the design of a steam heater to heat 64,000 pounds of oil per hour, from 80" to 200' F.?the steam temperature being 320' F. The heater is to contain 120 tubes of 0.62 inch inside diameter. Specific heat and viscosity data are available for the heating range, the average density of the oil being 59 pounds per cubic foot. The flow per tube is 64,000/120 = 533 pounds per tube, 6
Cornell University, IEng. Expt. Sta., Bull. 7 (February, 1927)
Spectrum of Ionized Hafnium-Analysis of the new descriptions of hafnium spectra by the Bureau of Standards has resulted in the classification of more than 200 lines characteristic of the ionized hafnium atoms. Because hafnium was not discovered until 1923, no supplementary data which have aided in analyzing other complex spectra were available, but the bureau's wavelength measurements have sufficed for the detection of the first regularities in the spectrum of this complicated new atom.
