Convective heat extraction from molten magma

Journal of Volcanology and Geothermal Research, 10 (1981)175--193 Elsevier Scientific Publishing Company, Amsterdam - - P r i n t e d in Belgium

CONVECTIVE

HEAT EXTRACTION

FROM MOLTEN

175

MAGMA*

H.C. HARDEE Sandia National Laboratories, Albuquerque, NM 8 7185 (U.S.A.) (Received March 10, 1980; revised and accepted August 2, 1980)

ABSTRACT Hardee, H.C., 1981. Convective heat extraction from molten magma. J. Volcanol. Geotherm. Res., 10: 175--193. The convective heat of molten magma in the upper 10 km of the continental crust represents a significant geothermal energy resource. Shallow basaltic magmas (< 10 kin) near the liquidus (1250°C) are predicted to offer heat extraction rates in the range of 15-50 kW/m 2 (20--80 MW/well). More accessible andesitic and wet rhyolitic magmas are predicted to offer heat extraction rates in the range of 5--25 kW/m 2 (8--40 MW/well). Convective heat transfer correlations are used which include corrections for high Prandtl number fluids and cylindrical boundary layers. The calculations based on these correlations agree well with laboratory tests using molten basalt at superliquidus temperatures (1450--1650°C). A t liciuidus and subliquidus temperatures an additional correction is developed for the thick solidified crust that forms on the heat exchanger. Non-Newtonian rheology is considered and shown to have a possible effect on the initiation of convection in the liquidus and subliquidus temperature range. In addition to heat extraction estimates, the analysis presented here is also relevant to convective heat loss to walls of magma bodies.

INTRODUCTION T h e c o n v e c t i v e h e a t o f m o l t e n m a g m a in t h e u p p e r 1 0 k m o f t h e c o n t i n e n tal crust represents a significant geothermal resource. In some instances a p o r t i o n o f t h e c o n v e c t i v e h e a t o f t h i s m a g m a is t r a n s f e r r e d t o h y d r o t h e r m a l s y s t e m s i n c o n t a c t w i t h t h e m a g m a a n d t h i s e n e r g y is t h e n a v a i l a b l e as c o n v e n t i o n a l g e o t h e r m a l e n e r g y . T h e e n e r g y a v a i l a b l e as c o n v e n t i o n a l g e o t h e r m a l e n e r g y , h o w e v e r , is a s m a l l f r a c t i o n o f t h e t o t a l c o n v e c t i v e e n e r g y in t h e magma. One way to effectively utilize the remaining energy would be to place a h e a t e x c h a n g e r d i r e c t l y in t h e m a g m a . C o l p a n d B r a n d v o l d ( 1 9 7 5 ) a n d Hardee and Larson (1977) have described the extraction of energy using a h e a t e x c h a n g e r d i r e c t l y in a s h a l l o w m a g m a b o d y . T h e a d v a n t a g e s o f d i r e c t energy extraction from magma are high source temperature, good heat extraction rates and a potentially enormous resource of thermal energy. Others have *This work was supported by the United States Department of Energy (DOE) under contract number DE-AC04-76DP00789.

0377-0273/81/0000--0000/$02.50 © 1981 Elsevier Scientific Publishing Company

176 also discussed the potential for energy extraction directly from magma. F e d o t o v et al. (1975) have discussed the potential for energy extraction from a magma b o d y in Kamchatka. Heffington et al. (1977) have examined the potential for energy extraction from magma in the main chamber of volcanoes. Bjornsson (1980) described the technique, which uses energy from a recent lava flow, to heat the town on the Island of Heimaey in Iceland. Bjornsson (1980) has also encouraged the direct tapping of magma energy, particularly at the shallow (3 km deep) magma chamber beneath the Krafla area in northern Iceland. A few heat extraction estimates are already available for magma: Hardee and Larson (1977) gave preliminary estimates on the order of 10 kW/m 2 for heat extraction rates in magma bodies. Hardee and Fewell (1975) reported laboratory heat extraction measurements on the order of 100--300 kW/m 2 for superliquidus basaltic lava and Hardee (1979a) measured transient heat extraction rates on the order of 10--200 kW/m 2 in a basaltic lava flow. Significant amounts of thermal energy exist in igneous-related systems in the upper 10 km of the crust of the western United States. Smith and Shaw (1979) estimate that 1023 J (10 s quads) exist in evaluated young igneousrelated systems and that the total in evaluated and unevaluated systems is on the order of 1024 J (106 quads). They estimated that about half of the evaluated systems contain magma chambers with a large molten fraction. There is evidence that some magma chambers exist at depths as shallow as 4--5 km in the continental United States (Eaton et al., 1975; Sanford et al., 1976; Chapin et al., 1979; Iyer et al., 1979). Basaltic magma chambers with their high temperatures, low viscosities, and potential convection offer the best prospect for efficient heat extraction. Although some shallow basaltic chambers are thought to exist in the continental United States at such places as San Francisco Peaks, Newberry, Medicine Lake, and others, (R. Decker, personal communication, 1979; Varnado and Colp, 1978), evidence for large quantities of shallow basaltic magma bodies in the continental United States is currently lacking. However, the more plentiful andesitic and wet rhyolitic (> 5% H20) magmas also appear to offer promising convective heat transfer rates. Although there are many ideas for open-flow heat exchanger systems and for enhancing the heat transfer rates in a viscous molten magma system (Hardee, 1979b; Bjornsson et al., 1980), the simplest initial approach for heat extraction is a closed cylindrical heat exchanger system. If a closed heat exchanger is placed in a magma b o d y , natural convection will be induced in the magma in the vicinity of the cool heat exchanger surface. This natural convection results from local density differences that occur when the magma in the vicinity of the heat exchanger surface is cooled. This natural convection process can have a significant and often controlling effect on the heat extraction rate for such heat exchangers. This paper examines heat extraction rates (energy transfer per unit area of heat exchanger surface) for conventional vertical cylindrical heat exchangers placed in a range of possible magma bodies. Convective heat transfer

177

equations are used which consider the high Prandtl number (Pr > 100) of magma and the effect of cylindrical heat exchanger geometry. These calculations are compared with some limited laboratory test data. The effect of a solidified crust around the heat exchanger is included in the steady-state heat extraction estimates and the estimates are compared with a few selected transient numerical calculations. Finally, some effects of non-Newtonian rheology on convective heat transfer rates are examined.

Fig. 1. Solidified c r u s t o n h e a t e x c h a n g e r surface a f t e r r e m o v a l f r o m m o l t e n lava. This p h o t o g r a p h s h o w s t h e c r u s t o f solidified lava t h a t f o r m s o n a cool h e a t e x c h a n g e r t u b e p l a c e d in m o l t e n basaltic lava w h e n c o n v e c t i o n is t h e d o m i n a n t h e a t t r a n s f e r m e c h a n i s m . A piece o f t h e c r u s t h a s b e e n b r o k e n away, a f t e r t h e h e a t e x c h a n g e r was r e m o v e d f r o m t h e m o l t e n lava, t o reveal t h e h e a t e x c h a n g e r surface u n d e r n e a t h . T h e i n n e r d a r k glassy z o n e f o r m e d d u r i n g t h e t r a n s i e n t f o l l o w i n g t h e i n s e r t i o n of t h e h e a t e x c h a n g e r i n t o t h e lava. T h e o u t e r d a r k glassy z o n e f o r m e d w h e n t h e h e a t e x c h a n g e r was r e m o v e d f r o m t h e lava a t t h e e n d o f t h e test.

178 HEAT E X T R A C T I O N T H E O R Y

When a heat exchanger is placed in a molten magma, a solid crust forms on the cooler heat exchanger surface. Fig. I shows a section of the crust that was broken away after the end of a test to show the heat exchanger Sube beneath. The crust grows in thickness until the conductive heat flux through the crust is equal to the convective heat flux in the magma at the edge of the crust as shown in Fig. 2. At this stage, the convective heat coming from the molten magma is equal to the heat being c o n d u c t e d through the crust and in turn this heat is equal to the heat being brought to the surface by the heat exchanger, which is termed the "heat extracted". If the temperature of the magma is well above the liquidus, the magma will tend to behave like a high Prandtl-number Newtonian fluid and the convective heat flux will be high and the crust that forms will be relatively thin as in Fig. 1. Magmas at liquidus or subliquidus temperatures will tend to behave like high Prandtln u m b e r fluids but will likely exhibit non-Newtonian fluid behavior (Kushiro et al., 1976). In this latter case, the crust that forms may be quite thick and the convective heat flux, although small, may still be significant in terms of heat extraction rates because of the increased surface area of the thick crust. If the crust is very thin, the heat transferred through the crust by conduction is adequately represented by the one-dimensional conduction equation

HEAT EXCHANGER WALL

SOL,O,.EOMAGMA CROST ,--"~

" :-~-~ ~ : J- .~ ~ ~.,~= ::~=,;:~-_~..-~_~_.~_~/~.-_

ro

MOLTEN MAGMAAT T~o

~

Fig. 2. Flow of heat from molten magma to the heat exchanger working fluid. In this diagram, the convective heat from the molten magma is equal to the conductive heat flowing through the solidified magma crust on the heat exchanger surface and, in turn, this heat is equal to the heat being carried to the surface by the heat exchanger coolant or working fluid.

179

(see Table 1 for explanation of symbols): K ( T o - - Te) qo -

(1)

l

The subscript " e " refers to conditions at the surface of the heat exchanger and the subscript " o " refers to conditions at the outer edge of the chilled crust that forms on the heat exchanger during early times when conduction dominates. In many cases the crust will be very thick and equation (1) is a poor approximation. A thick crust that forms around a cylindrical heat exchanger will have the geometry of a cylindrical annulus as shown in Fig. 2. The heat transferred through the cylindrical crust by conduction is given by Gebhart (1961): qo =

TABLE

K(To -- Te)

(2)

ro In (ro/re)

1

Thermal property data and nomenclature Magma heat exchanger area Acceleration due to gravity Thermal conductivity Test heat exchanger length Magma heat exchanger length Magma latent heat Magma h e a t exchanger radius Melt temperature--basalt, andesite A m b i e n t m a g m a temperature--basalt, andesite A m b i e n t m a g m a temperature--rhyolite Heat exchanger operating temperature Magma thermal diffusivity Magma volumetric thermal expansivity

A g K L L £ re To T® T~ Te

Magma density Magma Bingham yield stress Magma viscosity basalt at depth wet andesite at depth dry andesite, wet rhyolite (granite) basalt--variation with temperature

p % ~t

Magma Prandtl n u m b e r basalt at depth wet andesite at depth dry andesite, wet rhyolite

1571 m 2 9.8 m/s 2 2.93 W/mC 0.28 m 1000 m 4.186 × 105 J/kg 0.25 m 1050°C 1250°C (Kushiro et al., 1976; Marsh, 1978) 900°C 350°C 5 . 0 × 1 0 -7 m 2 / s

10 x 10 -s I/C (1050--1250°C) 5 × 10 -5 l/C (1250--1650°C) 2700 kg/m s 50 N / m 2 (1150°C) 10.7 kg/m s (Kushiro et al., 1976) 310.5 kg/m s (Kushiro et al., 1976) 2.7 × 104 kg/m s (Shaw, 1974) 1 × 1 0 - ' e x p [ 2 6 1 7 0 / ( T + 273)] k g / m s (Shaw et al., 1968) = ~lp~

7.9 X 103 2.3 X 10 ~ 2.0 X 10 7 pg~(T= --To)x 3

Rayleigh n u m b e r

Rax

Nusselt n u m b e r

Nu = hL /K

180 The limiting heat extraction rate at the heat exchanger surface is similarly given by: qe =

g(To --Te) re In (ro/re)

(3)

In order to find qe, equation (2) is first solved for r o and this result is substituted into equation (3). Equation (3) is then solved for qe which is the minimum heat extraction rate for the heat exchanger for a specified convective heat flux, qo, at the edge of the crust. By combining equations (2) and (3) it can be seen that:

qe = qo(ro/re)

(4)

This heat extraction rate at the heat exchanger surface, qe, may be several times larger than the convective heat flux, qo, in the magma at the edge of the crust because of the difference in surface area between the heat exchanger surface and the outer surface of the solidified crust. When the crust is very thin, the results of equations (2) and (3) reduce to the simple result of equation (1). The heat extraction rate in equations (3) or (4) is the heat extracted by the heat exchanger per unit area of heat exchanger surface. The total heat extracted b y the heat exchanger, QT, is simply qe times the surface area of the heat exchanger, or:

(5)

QT = Aqe

The heat flux, qo, entering the outer surface of the crust in Fig. 2 is the heat flux reaching the crust b y convection from the molten magma. This convective flow is caused by local density changes in the cooled magma in the vicinity of the heat exchanger surface. Convective heat flux correlations can be used to estimate the heat flux, qo, on the assumption that Newtonian properties can be used to estimate convection in magma for magmas near the liquidus (Shaw, 1974). Although the accuracy of these correlations near the liquidus is not known, they should be valid for magmas with appreciable superheat. Shaw (1974) used the correlation for natural convection with laminar flow given by Rohsenow and Choi (1961): Nu L =

0.56

Ra L

1~

(6)

This relationship was developed for fluids with Prandtl numbers near unity. For magmas and lava, the Prandtl n u m b e r is at least 100 and generally much larger than this, as noted in Table 1 and in Shaw (1974). Convective heat transfer correlations are available for the infinite Prandtl n u m b e r limit. Grober et al. (1961) point out that this infinite Prandtl number limit is effectively reached when the Prandti number exceeds 100. The infinite Prandtl n u m b e r convection correlations are therefore applicable for all practical lava and magma situations. The principal difference between a convection correlation for unit Prandtl number and one for infinite Prandtl

181 PRANDTL NUMBER EFFECT

(a)

(b)

\ \

THERMAL BOUNDARY LAYER

.ERMA BOUNDARY

/

i Om Pr=l

Pr

Nu=0.56 Ra 1/4 L

L

NUL= 0.67 RaL1/4

Fig. 3. T e m p e r a t u r e a n d v e l o c i t y d i s t r i b u t i o n in t h e c o n v e c t i v e b o u n d a r y layer for an o r d i n a r y fluid (a) a n d for a high P r a n d t l - n u m b e r fluid (b). In t h e o r d i n a r y fluid case t h e m a x i m u m c o n v e c t i v e velocity o c c u r s in t h e m i d d l e o f t h e t h e r m a l b o u n d a r y layer. In t h e high P r a n d t l - n u m b e r case, such as m a g m a , t h e m a x i m u m c o n v e c t i v e velocity occurs at t h e o u t e r edge o f t h e t h e r m a l b o u n d a r y layer.

n u m b e r is the relationship between the temperature and velocity boundary layers. The boundary layers are the regions where significant changes in temperature and velocity occur (Ozisik, 1968) as shown in Fig. 3. In a Prandtln u m b e r fluid near unity the temperature and velocity boundary layers will be of approximately equal thickness as shown in Fig. 3a. If the Prandtl number is very large, however, the velocity boundary layer will be thicker than the thermal boundary layer and the peak velocity will occur at a distance roughly equal to the thickness of the thermal boundary layer (Kuiken, 1962) as shown in Fig. 3b. Kuiken (1962) and Lin and Chao (1974) have solved for the convective heat flux on a vertical flat plate with infinite Prandtl number and give the result N u x = 0.50275 Raxll4

(7)

for the local Nusselt number at a vertical position x where the Rayleigh number, R a x , is defined in Table 1. The average Nusselt number can be obtained from the local value in equation (7) b y integrating the result over a length L. The effect o f this is to multiply the coefficient in equation (7) b y the factor 4/3 (Eckert and Drake, 1959) or: N u L = 0.670 RaL t/4

Equation (8) is developed for a vertical flat wall. The heat exchanger con-

(8)

182

sidered here is cylindrical and curvature effects are important since the convective velocity boundary layer can be very thick because of the high Prandtl number of lava and magma. Touloukian et al. (1948) developed a convective correlation for laminar natural convection on vertical cylinders at moderate Prandtl numbers (2 < Pr < 118) of the form: NuL = 0.726 RaL ‘I4

(9)

Since the coefficient in this correlation is valid for Prandtl numbers up to 118, the correlation should be accurate for typical lavas and magmas although the correlation may slightly underestimate the Nusselt number for magmas with very large Prandtl numbers. The result of equation (9) is consistent with work by Sparrow and Gregg (1956) who showed that curvature effects of the cylindrical geometry increased the Nusselt number, and the result is consistent with forced convection experiments by Mueller (1942) who found that the Nusselt number for parallel flow along small cylinders tended to increase as-much as 40% when the cylinder diameter became small relative to the boundary layer thickness. Touloukian et al. (1948) developed a correlation for turbulent natural convection on a vertical cylinder of the form: Nu, = 0.0674 RaL 1’3 (Pr”*2g)“3

(10)

This correlation has been verified for moderate Prandtl number fluids and turbulent flow with Rayleigh numbers up to 1012. LABORATORY

RESULTS

A laboratory heat extraction test was run with superliquidus molten basalt (Hardee and Fewell, 1975) at a high temperature (1450-1650°C) where convection was significant. The magma was molten degassed Hawaiian tholeiite lava and the heat exchanger consisted of a water cooled stainless steel tube shown in Fig. 1. The first test was run at 1450°C and the convective heat flux was determined by measuring steam generation rates. Steam was collected at a rate of 9.5 kW and the surface area of the heat exchanger in contact with the lava was 0.0529 m* resulting in a convective heat extraction rate of: q0 = 179 kW/m*

(11)

The solidified crust on the heat exchanger tube during the test was found to consist of two zones, an outer light-colored crystalline layer and an inner dark glass layer, each 0.6 cm thick. Differential thermal analysis (DTA) runs on the crust sample indicated that the temperature at the outside of the crystalline layer had been 1150°C and the temperature at the crystalline/ glassy interface had been 720°C. The convective heat flux from the magma was equal to the heat conducted across this zone of crust, and was determined using equation (2), the temperature drop AT = 1150-720°C and the

183 thermal conductivity from Table 1, giving: qo = 209 kW/m 2

(12)

The inner dark colored glassy zone of crust formed at temperatures below 720°C. The inner edge temperature of the dark glassy zone was estimated from thermocouple measurements on the heat exchanger surface to be 350°C and the convective heat flux was determined again knowing t h a t this convective heat flux must also equal the conductive heat flux through the glassy zone of crust. In this case AT = 7 2 0 - 3 5 0 ° C and: qo = 181 kW/m 2

(13)

Finally, the convective heat flux may be estimated using the high l~andtl number Newtonian correlation in equation (9). The Rayleigh number, using values from Table 1, is: pg{J(T RaL

=

-- To)L 3 p,o~ =

1.19 × 107

The convective heat flux, q o = h ( T - - T o ) , using the definition of the Nusselt number, N u = h L / K , and equation (9) is: qo = 0.726 R a 1 / 4 K (T.~ - T o ) Using T

(14)

= 1450 and values from Table 1 gives:

qo = 179 kW/m:

(15)

This heat extraction rate compares well with the measured result using (11) and the estimates given by (12) and (13). A second test was run at temperatures near 1650°C. The convective heat transfer rate based on steam generation measurements for this case was: qo = 314 kW/m 2

(16)

Using the convection correlation from equation (9) and properties from Table 1 gives: pg{3(T RaL

- - T O )L 3

=

=

4.35 X 107

P~

K qo = 0.726 R a L 1/4 - - ~ ( T

-- To) =

311 kW/m 2

(17)

which compares with the measured value of 314 kW/m 2 from equation (16). The tentative conclusion based on these limited measurements is t h a t the high Prandtl-number Newtonian correlation for natural convection in equation (9) works well for superliquidus lava or magma.

184 HEAT EXTRACTION FROM TYPICAL MAGMAS NEAR THE LIQUIDUS

The most favorable magmas for heat extraction are basaltic, andesitic or wet (> 5% H20) rhyolitic bodies because of their potential for convection (Shaw, 1974; Hardee and Larson, 1977). For the purpose of a heat extraction calculation, consider a 1000-m vertical cylindrical heat exchanger in contact with magma having the properties listed in Table 1. Further assume that the magma b o d y is at or slightly above the liquidus temperature at its equilibrium pressure in the 5--10 km depth range. This will place the temperature of a basaltic or andesitic magma b o d y at a b o u t 1250°C (Bowen, 1928; Yoder and Tilley, 1962; Hamilton et al., 1964; Kushiro et al., 1976; Marsh, 1978). This assumption is consistent with calculations by Marsh (1978) that indicate that the magma b o d y may even be superheated. It is also consistent with O'Hara's work (1978) which indicates that parental magma is as much as 250°C hotter than erupted lavas. The temperature difference across the convective boundary layer ( T . . - - T o ) will then be 200°C and this is consistent with previous convection calculations by Carmichael et al. (1976). Obviously there are many cases of erupted lavas at subliquidus temperatures which come up rapidly from magma sources at great depths. Such magmas have few important applications for the extraction of heat because of their short residence time at shallow depths in the crust. We are only interested here in magmas that tend to pool at shallow depths in the crust. For a basaltic magma the Rayleigh number is: p g{JL 3 Ran---

(T

- - T o ) = 9 . 9 X 1016

Then from equation (14): qo = 7.55 kW/m 2 This value is the expected convective heat flux in the magma under the conservative assumption of laminar flow. The actual heat flux at the heat exchanger surface is higher and is found by using equation (2). For an assumed heat exchanger radius of 25 cm: r_~o In -ro -= re re

g ( T o -- Te) reqo

Using the values in Table 1 and solving for ro/re gives: ro/r e = 1.82

and then from equation (4): ro

qe = qo - - = 7.55 (1.82) re qe = 1 3 . 7 k W / m 2

(18)

185 The total heat extracted by this heat exchanger, with surface area A = 1.57 × 103 m, is found using equation (5): QT = 21.5 MW The Rayleigh number, R a L = 9.9 × 1016, is large enough to consider the possibility of turbulent natural convection with its higher heat transfer rates. Normally natural convection will become turbulent when the Rayleigh number exceeds 109 (Rohsenow and Choi, 1961). The high erandtl number in this case, however, tends to raise this critical Rayleigh number turbulent transition point. Hieber and Gebhart (1971) studied this problem and their result for the critical Rayleigh number for the onset of turbulence can be put in this form: Racritical = 7.23 × 104 (Pr):

(19)

The critical Rayleigh number determined from equation (19) for the heat exchanger in basaltic magma is 4.5 × 1012. Since the actual Rayleigh number (9.9 X 1016) is much larger than the critical Rayleigh number (4.5 × 1012), the convection is turbulent. Assuming that the flow does become turbulent, we can use the standard turbulent natural convection correlation suggested by Touloukian et al. (1948) in equation (9) which in terms of the convective heat flux becomes: K qo = ~ - ( T

- - T o ) ( 0 . 0 6 7 4 ) R a L 1/3 (Pr°'29) 1/3

(20)

Using the values in Table 1, the convective heat flux at the edge of the solidified crust is: qo = 43.3 kW/m 2 and solving again for r o / r e using equation (18) gives: r o / r e = 1.18

The heat extraction rate at the heat exchanger surface is found again by using equation (4): qe = 51.1 kW/m 2 This result is slightly higher than would be obtained using the standard turbulent natural convection correlation suggested by Shaw (1974): N u L = 0.13 R a L 1/3

The total heat extracted by the heat exchanger is found again using equation

(5): QT = 80.3 MW These calculations are valid only after steady conditions have been reached and they neglect effects like phase change in the solidifying crust. A more

186

~G~ 20 qo= 10 kW/m 2

Z

qo= 5 k W / m 2

X W

i

re=25

cm

CONDUCTION

ONLY %= 0

, , ,,=,l

i

,

i i ,,,.I

,

• ,=,,,=1

........ 1

AY

I

.

.

,,

, IL

10

1 WEEK

T ME,XEARS

Fig. 4. Heat e x t r a c t i o n rates f o r a vertical cylindrical h e a t e x c h a n g e r e x p o s e d t o d i f f e r e n t m a g m a c o n v e c t i v e h e a t flux levels. S h o w n are the t r a n s i e n t h e a t e x t r a c t i o n rates o u t t o 30 y e a r s f o r a 25-cm-radius vertical cylindrical h e a t e x c h a n g e r i m m e r s e d in magma. TABLE 2 Heat e x t r a c t i o n rates qe a n d total h e a t e x t r a c t i o n QT for a 25-cm-radius, 1000-m vertical cylindrical h e a t e x c h a n g e r in m a g m a qo ( k W / m )

ro (cm)

qe (kW/m2) (numerical solution)

qe (kW/m2) (eq. 4)

QT (MW) (eq. 5)

0.1 0.5 1.0 1.06

562 220 138 122

-4.6 5.9 --

2.3 4.4 5.1 5.17

3.6 6.9 8.0 8.1

2.0 3.24 5.0 6.19

91 66 54 49

7.8 -12.2 --

7.3 8.52 10.8 12.1

11.5 13.3 17.0 19.0

7.55 10.0 19.4 20.0 43.3 50.0

45 43 34 34 30 29

-19.4 -30.0 -59.6

13.7 17.2 26.5 27.0 51.1 57.5

21.5 27.0 41.6 42.4 80.3 90.3

dry r h y o l i t e --dry a n d e s i t e or w e t r h y o l i t e (laminar) -w e t a n d e s i t e (laminar) -dry a n d e s i t e or wet rhyolite (turbulent) basalt (laminar) -wet andesite (turbulent) -basalt ( t u r b u l e n t ) --

187

detailed finite
TABLE Effect

3 of varying heat exchanger

qe (kW/m2) QT (MW)

radius, r e

r e = 1 cm

r e = 10 cm

r e = 25 cm

79

17.5

10.8

8.3

11.0

17.0

26.1

5.0

q o = 5 k W / m 2, L = 1 0 0 0

m.

r e = 50 cm

188 N O N - N E W T O N I A N F L O W A N D O T H E R EFFECTS

The previous analysis treated the convective b o u n d a r y layer as a Newtonian liquid, b u t the boundary layer in a magma below the liquidus will actually consist of a mixture of liquid and crystals. As the boundary layer thickness increases due to cooling, more crystals will form and the liquid will develop a Bingham, pseudo-plastic, or other nonlinear character (Bird et al., 1966), which will tend to increase the effective viscosity and reduce the heat transfer rate. The heat released by solidifying crystals in the boundary layer tends to improve the heat transfer rate and compensates for the increased viscosity. A Bingham fluid is one in which a minimum shear yield stress o0 must be exceeded before flow will begin. Field measurements b y Shaw et al. (1968) on Hawaiian tholeiite lava showed a Bingham characteristic behavior with shear yield stress 00 of 70--120 N/m 2 for 1135°C lava. Recent measurements by Pinkerton and Sparks (1978) on Mt. Etna lava at 1086°C also showed a Bingham characteristic behavior with a shear yield stress o0 of 370 N/m 2. Laboratory experiments by Murase and McBirney (1973) on samples of Columbia River basalt showed a transient Bingham t y p e yield stress which increased with time and was a strong function of temperature below the liquidus. Once the yield stress o0 is exceeded and flow has begun, the fluid in the b o u n d a r y layer behaves like a Newtonian fluid. The convection equations developed earlier for heat transfer will apply for the Bingham fluid once m o t i o n has begun. The basic question is whether induced natural convective flow will begin in a magma that behaves like a Bingham fluid. Using the high l>randtl number assumption shown in Fig. 3b and equating b u o y a n t forces in the b o u n d a r y layer to viscous shear at the wall gives: ~pg[j

(T

--To) 2

- ow

(21)

where o w is the shear stress at wall. This equation is valid whether the fluid is in motion or not. 6 is the thickness of the thermal layer and is the width of the zone where the temperature changes from the wall temperature to the ambient magma temperature (Ozisik, 1968). The thermal layer increases in thickness by conduction of heat from the fluid medium adjacent to the layer. The thickness of the thermal boundary layer can be estimated using the simple conduction solution for a step temperature change on a semi-infinite b o d y . For a suddenly imposed temperature change on a semi-infinite b o d y where the temperature at the edge of the boundary layer is within 10% of the ambient fluid temperature, the thermal boundary layer thickness is given b y (Carslaw and Jaeger, 1959): /4x/4-~ = 1.4

(22)

Substituting equation (22) into (21) gives the shear stress at the wall: O w = p g{3 ( T ~ - - T o )x/'-2-~

(23)

189

and this result is valid for a Bingham fluid as well as a Newtonian fluid. In order for convection to begin in a Bingham fluid uW > uo. Equation (23) however indicates that as time 7 increases, u, increases and a point will eventually be reached where uW > u. and flow will begin, provided that u. does not also increase with time. For example, substituting values from Table 1 into equation (23) results in: uw

=

0.054 6

N/m2

(24)

For the heat exchanger considered earlier with T,,= 105O”C, T_ = 125O”C, the average temperature in the boundary layer is 1150°C. From the field data of Pinkerton and Sparks (1978) and Shaw et al. (1968) the yield shear stress at 1150°C can be estimated as u. = 50 N/m2. Substituting this into equation (24) and solving for the time when motion will begin gives: 7 = 8.6 X lo5 seconds or about 10 days before convection begins if the magma behaves as a Bingham fluid. By referring to Fig. 4, it can be seen that this delay is not particularly noticeable in the full scale heat exchanger because the steady state convective heat extraction rates are not reached until about the first week anyway. This transient effect caused by the Bingham fluid character could have an effect on laboratory tests and could complicate the measurement of weak convective heat transfer rates below the liquidus. Laboratory tests are currently under way to measure convective heat transfer rates in basalt in the temperature range llOO-1300°C. A final consideration is the manner in which the buoyant convective force is treated. In natural convection calculations, it is common practice to represent the buoyant force by the product pgp(T_ - To) which is simply the density difference due to volumetric thermal expansion. Shaw (1974; personal communication, 1979) has noted that the diffusion of water into magma from country rock at the edge of a magma chamber may change the density more significantly than simple thermal expansion and in some cases may even reverse the direction of convective flow. While the diffusion of water into magma would not normally occur at the surface of a simple closed heat exchanger, it might be desirable in an advanced heat exchanger to inject water into the magma for the purpose of improving the heat transfer rates. Such a technique could increase the buoyant convective force and reduce the viscosity of the magma which in turn would improve the heat transfer rates (Shaw, 1974; Hardee and Larson, 1977). For the purpose of the present analysis of a simple closed heat exchanger it is sufficient to recognize that the magma density is a function of temperature, pressure, chemical composition, water content, partial solidification, etc. For a simple closed heat exchanger where there is no water injection and where pressure effects are not important, the density can be considered a function of temperature, p = f(T), for that particular magma composition, and the buoyant force can be represented by the term pgp(T_ - To). In this case p may well be an “effec-

190

tive” quantity which includes the effects of partial solidification, changes in water content or chemical composition where such changes are principally due to changes in temperature. The value of 0 in the temperature range 1050-1250% in Table 1 obviously includes some of these effects. CONCLUSIONS

Heat extraction from molten magma in the upper 10 km of the continental crust represents a significant potential energy resource. Laboratory tests with molten basalt at superliquidus temperatures (1450-1650°C) produced high heat extraction rates (179-314 kW/m’), which correlated well with simple Newtonian heat transfer relations. At temperatures for shallow magma bodies near the liquidus (125O”C), the convective heat transfer rates are lower. However, geometrical considerations caused by the thick crust that forms in this temperature range lead to reasonable heat extraction rates. Basaltic magmas offer heat extraction rates in the range of 15-50 kW/m2 (20-80 MW per well) and andesitic and wet rhyolitic magmas offer heat extraction rates in the range of 5-25 kW/m2 (8-40 MW per well). Non-Newtonian rheology is expected to cause early transient effects which may delay the onset of convection. A potential problem with non-Newtonian magma rheology is that it may complicate laboratory measurements of convective heat transfer rates at liquidus and subliquidus temperatures. The nonNewtonian rheology however does not appear to affect the long term heat extraction rates for full scale heat exchangers. Although evidence for shallow basaltic magma bodies in a continental crust environment is limited (e.g. Skaergaard -Wager and Deer, 1939; Norton and Taylor, 1979) the existence of andesitic or wet rhyolitic magma beneath a number of continental geothermal areas is likely. The convective heat flux values of q0 = 1.06 kW/m2 to g, = 3.24 kW/m2 for rhyolitic and andesitic magma bodies are reasonable. For instance, it has been estimated based on surface heat flow that the convection heat flux q0 for typical magma bodies like Yellowstone is on the order of 1 kW/m’ when side losses from the magma chamber are considered (Hardee and Larson, 1977) and this is in rough agreement with the expected convection rate. The estimates for a magma heat exchanger (S-80 MW thermal/well) are comparable to those for conventional geothermal applications. For instance, Kruger (1976) estimated extraction rates of 17 MW thermal/well for typical liquid-dominated geothermal plants. The Cerro Prieto geothermal field averages about 15 MW thermal/well (Guiza, 1975). In the vapor-dominated Geysers field, one new well is required per year for each 100 MW (Crow: 1979). Assuming a 30-year plant life and a 16% thermal efficiency (Colhe, 1978) gives an average extraction rate of 21 MW thermal/well for this field. The depths for conventional geothermal wells are shallower than for a magma well. At the Geysers field, the wells range from 1 to 3 km (Crow, 1979) and a magma well would be 4-10 km, or about four times as deep. The advan-

191

rage of magma energy, however, is that it may eventually turn out to be a much larger resource than the few existing natural geothermal fields, at comparable heat extraction rates. REFERENCES Bird, R.B., Stewart, W.E. and Lightfoot, E.N., 1966. Transport Phenomena. John Wiley and Sons, New Yor.k, N.Y., pp. 10--14. Bjornsson, S., 1980. Natural heat saves millions of barrels of oil. Icel. Rev., 18: 28--37. Bjornsson, H., Bjornsson, S. and Sigurgeirsson, T., 1980. Geothermal effects of water penetrating into h o t rock boundaries of magma bodies. Geotherm. Resour. Counc. Meet., Salt Lake City, Utah, September 1980. Bowen, N.L., 1928. The Evolution of the Igneous Rocks. Princeton University Press, Princeton, N.J. Carmichael, I.S.E., Nicholls, J., Spera, F.J., Wood, B.J. and Nelson, S.A., 1976. Hightemperature properties of silicate liquids: applications to the equilibration and ascent of basic magma. Lawrence Berkeley Lab. Rep., LBL-5238. Carslaw, H.S. afld Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford University Press, London, pp. 58--61. Chapin, C.E., Sanford, A.R., White, D.W., Chamberlin, R.M. and Osburn, G.R., 1979. Geological investigation of the Socorro geothermal area -- Final report. N.M. Inst. Mining Tech. Rep. NMEI-26. Collie, M.J., 1978. Geothermal Energy - - Recent Developments. Noyes Data Corporation, Park Ridge, N.J., pp. 146--147. Colp, J.L. and Brandvold, G.E., 1975. The Sandia Magma Energy Research Project. Proc. 2nd U.N. Syrup. on Development and Use of Geothermal Resources, San Francisco, Calif., May 1975. Crow, N.B., 1979. An environmental overview of geothermal development: The Geysers-Calistoga KGRA, 4. Environmental Geology. Lawrence Livermore Lab. Rep. UCRL52496, October 9, 1979. Eaton, G.P., Christiansen, R.L., Iyer, H.M., Pitt, A.M., Mabey, D.R., Blank, H.R., Jr., Zietz, I. and Gettings, M.E., 1975. Magma beneath Yellowstone Park. Science, 188: 787--796. Eckert, E.R.G. and Drake, R.M., 1959. Heat and Mass Transfer. McGraw-Hill, New York, N.Y., pp. 312--315. Fedotov, S.A., Balesta, S.T., Droznin, V.A., Masurenkov, Y.P. and Sugrobov, V.M., 1975. On a possibility of heat utilization of the Avachinsky volcanic chamber. Proc. 2nd U.N. Symp. on Development and Use of Geothermal Resources, San Francisco, Calif., May 1975, pp. 363--369. Gebhart, B., 1961. Heat Transfer. McGraw-Hill, New York, N.Y., pp. 20--21. Grober, H., Erk, S. and Grigull, U., 1961. Fundamentals of Heat Transfer. McGraw-Hill, New York, N.Y., 304 pp. Guiza, J.L., 1975. Power Generation at Cerro Prieto Geothermal Field. Proc. 2nd U.N. Symp. on Development and Use of Geothermal Resources, San Francisco, Calif., May 20--29, 1975, pp. 1976--1978. Hamilton, D.L., Burham, C.W. and Osborn, E.F., 1964. The solubility of water and effects of oxygen fugacity and water content on crystallization in mafic magmas. J. Petrol., 5: 21--39. Hardee, H.C., 1979a. Heat transfer measurements in the 1979 Kilauea lava flow, Hawaii. J. Geophys. Res., 84 (B13): 7485--7493. Hardee, H C., 1979b. Heat extraction from magma bodies. Paper presented at the Hawaii Syrup. on Intraplate Volcanism and Submarine Volcanism, Hilo, Hawaii, July 16--22, 1979.

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193 Wager, L.R. and Deer, W.A., 1939. Geological investigations in East Greenland, III. The petrology of the Skaergaard intrusion, Kangerdlugssuag, East Greenland. Medd. Gr~bnl., 105(4): 1--352. Yoder, H.S. and Tilley, C.E., 1962. Origin of basalt magmas: an experimental study of natural and synthetic rock systems. J. Petrol., 3: 342--532.