Unsteady Flow of Heat through Porous Media WILLIAM D. HARBERTI, D. C. CAIN2, AND R. L. HUNTINGTON University of Oklahoma, Norman, Okla.
ferent fluids such as air, water, and oil; and (c) the mechanism of the heat flow-i. e., by conduction or convection. It was fully realized that an answer to the above questions would not make possible the complete solution of the problem of heating an oil reservoir in view of the many other variables to be encountered, such as the presence of gas, oil, and water in one formation alone, to say nothing of the flow which would result from partial vaporization of the more volatile components in the crude oil.
The data presented in connection with this investigation will serve as a guide for the engineer who plans to heat porous media on a larger scale. The experimental work gives an idea of the relative rates of heat flow through the porous material, depending on the nature of the fluid filling the pore spaces. Furthermore, the results indicate that conduction controls to a certain degree for small size grains, while convection controls for spheres of larger size, if the interstices are filled with liquid. The presence of dissolved air in the water and oil may have been the cause for apparent conductivities higher than those for the pure materials.
Experimental Apparatus UNSTEADY FLOWOF BEAT INTO A CYLINDER.Since the production of oil or gas into a well in a reservoir involves the radial flow from a large area into an outlet whose dimensions are small compared to the radius of drainage in the reservoir itself, a steam-jacketed cylindrical apparatus was built (Figure 1) in which the flow of heat was to be likewise convergent and radial in nature. This setup was simpler to construct than one in which the heat would flow outward into a large space. The collection of condensate from the inside walls of the steam jacket made the apparatus substantially adiabatic, with the exception of a small heat loss through the upper wooden plug of the cylinder. The accurate placement of the copper-constantan thermocouples in the sand body was made possible through the use of a rigid spider made of wooden prongs and glass conduits. The couples were found to be accurate within *lo F. Temperature readings were taken more frequently during the early part of each run when rapid changes were occurring near the outer boundary of the sand body. The four thermocouples were located a t the following distances from the center of the cylinder:
I L fields which have reached the economic limit of their productivity by natural flow, gas lift, and various pumping methods, still contain, as a rule, a t least 25 per cent of their original petroleum hydrocarbons. At present such secondary recovery methods as water flooding, gas repressuring, etc., are being used in order to produce a part of this remaining crude oil. At some future date no doubt other methods will be used to recover still higher percentages of this crude oil from the reservoir. Among the new methods of recovery that have been suggested is one which involves the heating of the oil-bearing sand, and thereby sets up a tendency for flow to take place due to the vaporization of the more volatile hydrocarbons in the reservoir. Some experimental and field data have been reported from Russia (8) on the heating and vaporizing of crude oil in reservoir sands by combustion with preheated air. Many data are also available (1,S, 6, 7 , 9 )on the heating and cooling of solids such as bricks, glass, rubber, lumber, etc., since many industries require batch processes in which unsteady heat flow rates are involved. Although such solids as bricks and lumber might be considered porous in nature, they can hardly be put in the same class as a body of material consisting of unconsolidated sand grains. Then, too, the data reported on the heating and cooling of bricks and lumber probably include such variables as the change in moisture content during the period under investigation. The present work was undertaken therefore to determine: (a) the relative rates of heat transmission through porous media consisting of different size spheres; (b) the relative rates of heat flow through any one sand body filled with dif-
0
1
Distance, Foot Radius of cylinder Radius of location Thermocouple 1 Thermocoude 2 Thermocouple 3 Thermocouple 4
Ratio, R a d i u s of Location of Thermocouple/Radius of Cylinder
0.271 0.229
0.845
0.000
0.690 0.387 0 000
0.187 0.104
STEADYFLOW OF HEAT TBROUGH A CYLINDER. As a check against the apparent conductivity values which were obtained through the unsteady flow of heat into the cylinder, it was thought advisable to obtain the values of k (apparent conductivity) of the porous media (either gas or liquid iilled) under steady flow conditions. The apparatus is shown in Figure 2. To reduce the percentage of error to the minimum in the temperature measurements, the tube was made long (72 inches) and the annular thickness of the sand-filled space comparatively small (0.965 inch) so as to give the cooling water a chance to rise appreciably in temperature. Ample turbulence for the water was provided by constricting the flow near the thermometers, which indicated the incoming and exit water temperatures. An iron rod was also inserted in the water tube to give added turbulence and better mixing.
P r e s e n t address, University of Michigan, Ann Arbor, Mich. C o m p a n y , Oklahoma City, Okla.
* P r e s e n t address, Duncan-Mecklenburg
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I
1
WOODEN
ES ST
IN
SAND
- F LUI
2 I/Z'kTANDAR
FIGURE 1. APPARATUSFOR THE UNRADIALFLOWOF HEATIN A CYLINDER
STEADY
FIGURE 2. APPARATUS FOR RADIALFLOWOF HEATIN
Physical Properties of Materials Wilcox sand (from an outcrop near Ada, Okla.) consists of unconsolidated well-rounded grains of quartz. Its porosity is 38 per cent, its density 165 pounds per cubic foot, and its estimated specific heat 0.188. The screen analysis is as follows: Weight YoRetained Mesh (Tyler)
50 65 100 150 200
on Ecreen
3.42 13.90 57.90 18.68 6.10 -
100.00
Two sizes of commercial glass marbles were used-6/s-inch and 5/leinch diameter, respectively, their porosity is 38 per cent, their density 156 pounds per cubic foot, and their estimated specific heat 0.160. The lubricating oil has a viscosity at 210' F. of 7.0 centipoises, a t 130" F. of 28.6, and at 100' F. of 63.0; its density is 55.9 pounds per cubic foot at 60' F., and its estimated specific heat 0.5.
Three-Dimensional Flow of Heat from a Point Source into an Extensive Sand Body Since it may often be more convenient to apply heat at the center of a body than at its outer boundary, experimental data were obtained by releasing the latent heat of steam into a small hollow sphere located in the center of a body of sand quite extensive in volume compared to the dimensions of the heat source. On a commercial scale i t is conceivable that the heating of a partially depleted oil-sand reservoir might be conducted in somewhat similar fashion by injection of heat into wells whenever this method is resorted to in the secondary recovery of petroleum. The apparatus (Figure 3 ) consisted of a 50-gallon drum which contained a body of Wilcox sand, 22 inches in diameter and 28 inches in height. Saturated steam at atmospheric pressure was
T H E STEADY FIGURE3. APPARATUS FOR THE THREEA CYLINDER DIhlENSIOSAL FLOW O F HEATINTO A N EXTENSIVE BODYOF SAND
brought into the hollow steel sphere (2.07 inches i. d. and 2.37 inches 0. d.) located in the center of the sand body by means of a small seamless steel tube, l/g inch 0. d. The condensate was removed through a '/a-inch standard pipe nipple surrounding the '/a-inch 0. d. input tube. This '/a-inch pipe was well insulated in order to conhe the supply of heat to the sand through the central sphere only. The sixteen thermocouples were held in place by glass tubes (l/d inch 0. d.) supported by a wooden spider. The thermocouples were arranged in groups of four each, extending radially outward from the s here at directions of vertically upward, 45' from the vertical, Eorizontal, and vertically downward. The distances from the surface of the central ball to the thermocouples were 1, 2, 4 , and 8 inches, respectively, for each group of couples. For each experimental run, temperature readings were discontinued as soon as the thermocouple, 8 inches from the heat source, began t o show a rise in temperature, since beyond this point the sand body no longer functioned as a sphere.
Air-Sand System (for Radial Flow into Cylinder) The original data for the unsteady flow of heat from the saturated steam source to the cylinder filled with air and sand are presented in Figure 4,where a history of the temperature change for each thermocouple is plotted on Cartesian coordinates. These curves are similar in shape to those reported by McLean (6) on the heating of round timbers. It is easily seen that the heating takes effect earlier in the outer portion of the porous sand body. For example, a rise in temperature of thermocouple 1 at a distance of 0.042 foot from the outer boundary (the location of the heat source) took place within a minute's time, while for thermocouple 4 in the center of the sand body, 10 minutes were required to produce a noticeable rise. From Figure 4 a cross plot was made (Figure 5A) on which constant temperatures are shown in their relation to time and distance from the center of the sand body (initial sand temperature was 73" F.). For example, in 107 minutes the flow of heat brings a temperature of 160" F. a t the center of the cylinder, while in 15 minutes 75" F. is found in the center. Another cross plot of Figure 4 is shown in Figure 5B which presents a view of the temperature gradients at different time intervals. Since the right-hand boundary of this graph rep-
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INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1941
x 0
c% v
W
5: 2
22 03 0
0
4) 0
0
10
20
30
40
50
60
70
80
BO
100
110
120
I30
140
T I M E (MIN.)
us. TIME AT CONSTANT RADIIFOR UNSTEADY RADIAL HEAT FIGURE 4. TEMPERATURE FLOW IN A CYLINDER OF WILCOX SANDAND AIR Each curve labeled with the number of the thermocouple; o = experimental, x tion 3).
resents the center of the sand body, it is theoretically correct to assume that the temperature is a minimum a t this central line and then sweeps upward with the same curvature as the lines shown in the graph. The rate of change of temperature with respect to time was plotted against time in Figure 6A by taking the slopes of the four thermocouple lines in Figure 5B a t different times. The slopes of these curves in Figure 6.4 represent the second partial derivative of temperature with respect to time a t any one radius or
(b2T)/(bP)r. All of these curves pass through maxima a t elapsed times in the same order as the distances of the respective thermocouples from the heat source. This graph shows the rates a t which the temperature is rising a t various points within the sand body at different times; in
-
theoretioal (Equa-
other words, it presents the forward movement of the heat wave toward the center of the cylinder. I n Figure 6B the experimental and calculated rates (Equations l to 3) of heat flow us. time are given. The rate of flow of heat decreased as time increased. The experimental values were obtained by recovering the steam condensate from the outer walls of the sand-filled cylinder in a funnel m
c (D
2
so
; ?* V
m
cu
0
'v
2
5' m $0 71
t
FIGURE 5. RADIUS N us. TEMPERATURE AT CONSTANT TIME ( A )AND us. TIME AT 0 IO 20 30 40 50 60 70 80 90 100 ll0 CONSTANT TEMT I M E (MINI PERATURE ( B ) FOR FIGURE6. TIMEus. dT/& (GRAPHA ) AND us. dQ/& (GRAPH UNSTEADY HEAT B ) FOR UNSTEADY RADIALHEATFLOW IN A CYLINDER OP FLOW IN A CYLINWILCOXSANDAND AIR DER OF WILCOX SANDAND AIR Calculated point8 in B from Equation 1.
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I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
FIGURE7.
(Zi
T)/(Ti - To)
us.
kSTANT 0 / c ~ p R RADII 2 hT ‘ONFOR
uNsTEADY R~~~~~ HEAT FLOWIN C m I N D E R OF
A WIL-
cox SANDA N D AIR
placed directly below the cylinder. The agreement with the theoretical as shown by the dotted curve is very good in view of the fact that there was some beat loss from the cylinder: (a) heat to warm wooden plugs a t each end of cylinder, ( b ) heat loss from upper wooden plug, and (c) possibility of a small amount of condensation in drain line. McAdams (6) gives the following account: “In 1923, Gurney and Lurie (4) showed that the integrated equations for a number of bodies, such as the slab, sphere, long cylinder, etc., could be plotted in terms of four dimensionless groups:
Y
“A ‘relative time’ ratio:
Ice x=-pcnR2
vs. TIMEAT CONSTANT RADIIFOR UNSTEADY RADIAL HEATFLOW IN . . FIGURE 8. TEMPERATURE A. Wiloox s a n d a n d water B . Wilcox s a n d a n d SAE 20 lubricatifig oil
C.
D.
= TI - T
“A temperature difference ratio:
48-65 mesh s a n d and air, 100-150 mesh sand a n d air
E. F.
TI - To
A
CYLINDER
S/~e-inchmarbles and air E/s-inch marbles and air
26 1
INDUSTRIAL A N D ENGINEERING CHEMISTRY
February, 1941
l/hT R/k
“A position ratio:
k htR
jlf = __ =
“A thermal resistance ratio: a
N
=
$
(end of quotation)
These relations are shown in Figure 7. The shapes of the curves correspond somewhat to those reported by Gurney and Lurie (4).
Miscellaneous Porous Media Original data were obtained on the heating of Wilcox sand and water which are shown in Figure 8A. During the test the upper wooden plug was removed on several occasions, and no disturbance of the water in the sand could be observed such as flow due to convection or vaporization within the porous media. The data for the flow of heat through Wilcox sand, saturated with lubricating oil, are given in Figure 8B. These data bring out clearly that the nature of the fluid in the pore space has a profound effect upon the rate of heat transmission through the system. The variation of the rate of heat flow through several airfilled porous systems was studied by obtaining data on the following media : Graph of Figure 8 Wilcox sand 48-65 mesh Wilcox sand: 100-150 mesh Glass marbles, 5 / d n . diameter Glass marblea, 6/rin. diameter
c D
Apparent Conductivity. Air-Filled 0.27
E F
0.26 0.20 0.20
An unsuccessful attempt was made to measure the rate of heat flow through the s/lr and 6/s-inch marbles with waterfilled pore space. It was evident that convection currents were set up, both from the record of the temperature rise and from the observation of the flow of water over the surface of the marbles. Only 5 or 10 minutes were required to bring the whole vessel to a uniform temperature, substantially as high as that of the steam itself.
Theoretical Considerations I n calculating the heat flow into the cylindrical sand body the following equation was used:
Graphical integration was employed in obtaining theItheoretical values shown in Figure 6B. The experimental quantities were secured from measurements of the steam condensate m-hich was collected separately on the outer wall of the sandfilled cylinder. The theoretical points shown on Figure 4 were obtained through calculations based on the following equations. The differential equation for radial heat flow in a cylinder is :
The solving of this equation over the conditions of initial temperature constant and wall raised to constant temperature gives the equation (3):
0
20
40
60
80
100
120
140
160
BO
200
220
TIME CMIN)
FIGURE9. TEMPERATURE us. TIMEAT CONSTANT RADIIFOR
UNSTEADY RADIALHEATFLOW INTO A BODYOF WILCOX SAND AND WATER
For this equation the surface resistance is considered t o be zero, since the thermal resistances of the steam film and the iron are so low as to be practically negligible.
Curves are labeled with the distance in inches of the thermocouple from the heat souroe; direobion of flo& as follows: A . Vertioally upward C. Horizontal B . 4S0 from vertical D . Vertioally downward
INDUSTRIAL A N D ENGINEERING CHEMISTRY
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Vol. 33, No. 2
0
3 2
"p r w 3 0 b 4 -o
K O z m w
8 0
;
; 0
20
40
60
80
I20
IO0
I40
160
BO
ZW
220
T I M E (MIN)
FIGURE 11. TEMPERATURE us. TIMEAT CONSTAXT RADIIFOR UNSTEADY RADIALHEATFLOWAT THE SAMERATEIN ALL DIRECTIONS INTO A LARGE BODYOF WILCOXSAND AND AIR E a c h curve is labeled with t h e distance of t h e thermocouple from t h e h e a t source.
The values of k obtained by means of this equation for t h e various solid-fluid combinations are given in Table I. Three of the values are listed in Table 11,along with the corresponding steady state values calculated from original data obtained on apparatus shown in Figure 2. TABLEI. APPARENT CONDUCTIVITIESCALCULATED FROM UNSTEADY H E A T FLOW I N A CYLINDER Apparent Conductivity
System
B . t . u. h T . - ' f t . - '
F.-I
The conductivities of pure materials, according to McAdams ( 5 ) ,are as follows: k D r y sand Water Air Velocite Boil Sandstone
0.19 0.41 0.018 0.080 1.06
Temp.,
E'.
68 212 32 212 104
' F.
FIGURE 10. TEMPERATURE GRADIENTS AFTER VARIOUS TIMESOF HEATINJECTION AT VARIOUSDISTANCES FROM THE HEATSOURCE FOR WILCOXSAND AND WATER A , after 20 m i n u t e s ; B , a f t e r 60 m i n u t e s ; C, after 100 m i n u t e s
TEMP
*
F.
FIGURE12. TEMPERATURE GR.4DIENTS AFTER 125. MINUTES OF HEAT INJECTION FOR VARIOUS DIS-TANCES FROM THE HEATSOURCE
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1941
-
DATABASEDON STEADYTABLE 11. APPARENTCONDUCTIVITY HEATFLOW EXPERIMENTS (POROSITY, 38 PER CENT) Conductivity, B. t. u. Hr.-1 Ft.-’
Wiloox Sand with: Air
Water/Lb. Hr.
Water Oil
73.60 70.90 68.60 98.40 116.10 116.10 80.10 70.10 98.70
Of
70.73 69.97 71.35 69.96 70.89 71.96 72.10 71.61 71.02
Water* F.
’ F.-1
Steedy~nsteady flow flow
82.00 82.30 83.25 102.05 107.76 109.97 92.49 94.18 88.36
0.41 0.40 0.40 2.60 2.61 2.63 0.87 0.85 0.87
Acknowledgment The authors wish to express their appreciation for the help given by G. A. Van Lear and S. B. Townes in the mathematical analysis of the problem, and for the assistance of T. Whelan, J. Cheek, D. Brim, and H. M. Evans in constructing the apparatus and securing the original data.
0.26
Literature Cited Carslaw, “Mathematical Theory of Heat”, pp. 30, 38, New York, Macmillan Company, 1921. Frank and Mises, “Die Differential und Integralgleichungen des Mechanic und Physik”, Tiel 11, pp. 561-3, Braunschweig. Vieweg, 1935. Gurney, H. P., “Heating and Cooling of Solid Shapes”, unpub. monograph, Mass. Inst. Tech. library. Gurney and Lurie, IND.ENQ.CHEM.,15,1173 (1923). McAdams, “Heat Transmission”, Chap. 11, New York, McGrrtwHill Book Co., 1933. McLean, J. D., Proc. Am. wood-Preservers’ Assoc., 1930, 197. Newman, A. B., Trans. Am. I n s t . Chem. Engrs., 24,44 (1930). Bheinman, A. B., et al., Petroleum Engr., Dec., 1938. Shepherd, C. B., Hadlock, C.,and Brewer, R. C., IND. ENG. C H S M . , 30,388 (1938).
2.92 1.03
Obviously the size of the sand grains affected the value of k very little when the pores contained air, as shown by the values for the different size sand grains. The value of k for the marbles is approximate, since at no point do the true curves fit the pure conduction curves. Evidently the convection of the air has produced the peculiar shape.
Three-Dimensional Flow from a Point Source WATER-FILLED SANDBODY. The history of the temperature rise for the first three thermocouples, 1, 2, and 4 inches from the heat source, is shown in Figure 9 for the four direotions of heat flow. The upward vertical flow was more pronounced than it was in the other directions, the rate becoming progressively less toward the downward position. For example, after 240 minutes the temperature 1 inch from the sphere was 141” F. in an upward direction while it was only 128’ F. a t a distance of 1 inch downward. The other thermocouples, however, did not show such pronounced differences for the several directions from the heat source. The original data in Figure 9 are presented in a different manner in Figure 10 in order to give a better idea of the temperature gradient in various directions a t certain time intervals. It is evident that there are slight convection currents which are responsible for the more rapid rise of temperature in the upward direction. One of the water-sand runs was continued for several hours after the temperature rise had reached the thermocouples 8 inches from the heat source, a t which time the flow of heat reached a substantially steady state. SAND-AIR BODY. The original experimental data for the three-dimensional flow of heat through an air-filled sand body are given in Figure 11. Temperature records in each direction proved to be practically identical, which indicated that no appreciable upward convection currents were set up within the porous media as was the case with the water-filled sand body. A cross section of the data from Figure 11 is shown in Figure 12 a t 125 minutes from the start of the run and indicate the uniform flow of heat in each direction from the heat source, = time variable, hours
p
= density of material, Ib./cu. ft.
T
=
T
X
An error in presenting data occurred in the article in the June, The table gives the correct values for the saturated liquid and vapor density of propane, and the chart is a correct plot which should supersede Figure 4 in the article as published. 1940, issue, pages 837-8.
TABLE IV. SATURATED LIQUID AND VAPOR DENSITY OF PROPANE Density, G./Co. Density. G./Cc. Temp., e C.
30 35 40 45 50 65 60 66
Gas 0.0236 0.0270 0.0305 0.0343 0.0385 0.0440 0.0496 0.0562
Liquid
0.4858 0.4785 0.4715 0.4630 0.4543 0.4443 0.4340 0.4220
Temp., O C. 70 75 80 85 90 95 96.85
Gas
Liquid
0.0640 0.4080 0.0730 0.3923 0.0832 0.3760 0,0980 0.3562 0.1180 0.3320 0.1580 0.2930 0.2240
”
c v)
= =
O F . )
3
TEMPERATURE, DEGREES C
Q = integral heat entering cylinder, B. t. u. a = Ic/c*p E. = vth root of the equation J&) = 0 A, = W R v = number of root of equation J&) = 0 h = water film coefficient, B. t. u,/(” F. X t = average water temperature, F.
P- V-T Relations for Propane-Correction
>.
temperature variable, O F.
initial temperature, O F. radius variable, ft. R = radius of cylinder, ft. cp = specific heat of material, B. t. u./ (lb.)(O F.) k = conductivity (B. t. u. X ft. of thickness)/(hr. X sq. ft.
T
PRISIONTED before the meeting of the American Institute of Chemical Engineers, New Orleans, La.
t
Nomenclature
e
263
FIGURE 4
hr. X sq. ft.)
This error was called to our attention by W. V. Stearns of the Development Division, Sun Oil Company. GEORGE GRANGER BROWN
