Laminar flows of heavy-fuel oils through internally insulated pipelines

Applied Energy 15 (1983) 81-98

Laminar Flows of Heavy-Fuel Oils Through Internally Insulated Pipelines

S. D. Probert and C. Y. Chu School of Mechanical Engineering, Cranfield Institute of Technology, Bedford MK43 0AL (Great Britain)

SUMMARY For slow flows of hot oils through large diameter horizontal pipelines, natural convection currents within the oils affect the rates of heat transfer from the pipelines. This phenomenon is taken account of in the presented predictions of the optimal internal thicknesses of the pipes' thermally insulating liners, corresponding to the least rates offinancial expenditure upon energy. The recommended average temperatures for the transmission of the oil through short pipelines are also predicted jot a range of conditions commonly encountered.

NOMENCLATURE A, B a, b

Constants for viscosity of the hot oil, see eqn. (8). Coefficients representing the costs for heat and pumping energy respectively (arbitrary financial units per Ws expended). b' Effective cost of energy per Ws supplied to the pump, making due allowance that some of this energy re-appears as heat in the flowing hot oil (arbitrary financial units per Ws expended). C~ Specific heat of the considered pumped hot oil (J kg- 1 K - 1). D 1,D 2 Internal diameters respectively of the considered flow channel and the structurally strong, outer member of the pipeline (m). d Thickness of the outer, structurally strong pipe (m). (For the present analysis it is assumed that 2 d ~ D 2 - D1. ) 81

Applied Energy 0306-2619/83/$03"00 © Applied Science Publishers Ltd, England, 1983. Printed in Great Britain

82

S. D. Probert, C. Y. Chu

JEht, JE'vd

Steady-state rates of heat transfer and viscous dissipation respectively per unit length of pipeline (W m- 1).

Gr

Grashof number for the pumped hot oil[-flgp2DaL~Tt-Tw) 1 .

g

//i

hi ki

kt k~ L Nu

Local acceleration due to gravity (m s- 2). Friction head loss along the pipeline (m). Average coefficient for the steady-state transfer of heat from the conveyed hot oil to the insulant lining (W m- 2 K- 1). Mean effective thermal conductivity of the insulant lining which has been applied uniformly within the pipe (W m - 1 K - 1). Effective thermal conductivity of the hot oil at temperature Tt (Wm-1 K-l). Effective thermal conductivity of the soil surrounding the pipeline (W m- 1 K - 1). Length of the considered straight horizontal pipeline (m). Local Nusselt number for the heat flow from the moving hot oil to the pipe's insulating lining Vl -= h i~D 1 7-]"

Poxp

Steady-state total rate of financial expenditure resulting from the net rate of energy loss (arbitrary financial units per metre length of pipeline).

Pr

Prandtl number for the pumped hot oil

Ra

Rayleigh number for the pumped hot oil [= Gr. Pr].

Re

Reynolds number for the pumped hot oil F =

T

Absolute temperature (K). Effective pumping temperature of the pumped hot oil (K). Average temperature at the horizontal air/ground interface (K). Local temperature of the pumped hot oil (K). 'Inner-walr temperature of the pipe (K). Thickness of internally applied insulant { = (D a - D1)/2} (m). Mean speed of the hot oil flowing through the pipeline of circular bore (m s- 1). Steady-state volumetric flow-rate of the hot oil passing along the pipeline ( m 3 s - 1).

Tcpt rg T,

Tw

t

f,

[

c":7

= kt ]"

L

p'uD17. #t ]

Laminar flows of oils through internally insulatedpipelines

~p #l Pl P15

83

Specific volume of the hot oil, = 1/p (m 3 kg- ~). Depth of the axis of the horizontal pipeline beneath the horizontal surface of the ground (m). Thermal expansion coefficient for the flowing hot oil (K-~). Proportion of the viscous dissipation which is converted into heat in the flowing hot oil (0 < ~,< 1). Overall efficiency of the pump and driver set (0 < r/p < 1). Dynamic viscosity of the flowing hot oil at temperature T~ (Nsm-2). Density of the flowing hot oil (kg m-3). Density of the flowing hot oil at 15 °C (kg m-3).

Suffices ept g i l opt

At the effective pumping temperature of the hot oil. Of the horizontal surface of the earth. Of the insulant. Of the pumped hot oil. Optimal value of the parameter corresponding to which the least rate of financial expenditure upon energy occurs. Of the soil.

GLOSSARY

Effective pumping temperature is that constant temperature at which the oil suffers a flow resistance equal to that encountered by the oil in the considered cooling pipeline of equal bore. Heavy-fuel oil is the description commonly adopted for a range of highly viscous residual fuel oils (e.g. having Redwood No. 1 seconds > 3500 at 35 °C). Each particular oil is normally specified according to its viscosity (see BS 2869 or ASTM D396).

THE PROBLEM The conveyance of hot fuel oil along a buried pipeline results in heat losses and so a drop in the oil's temperature, and consequently an increase in its viscosity. The latter leads to a greater pressure drop per metre of pipe travelled. Insulating the pipe internally would reduce the rate of heat losses

84

s. D. Probert, C. Y. Chu

from the oil to the surrounding soil, but increase the necessary operating pumping pressure, because the effective bore of the pipe would then be smaller. These two conflicting factors result in the existence of an optimal flow diameter, D~ , corresponding to which the least total rate of financial expenditure upon heat and pumping energy occurs. The values of D~ op,are predicted in the presented analysis for the specified flow rates and temperatures assuming only laminar flow of the fuel oil occurs. •

opt

E N E R G Y DISSIPATIONS A steady flow of hot heavy-fuel oil, at temperature, Tt, through a horizontal pipe buried with its axis at a depth z, in homogeneous soil of effective thermal conductivity k~, is considered. The cylindrical internal lining of mean effective thermal conductivity k~, is applied to the pipe of length L, and constant internal and external diameters D 2 and (D 2 + 2d), respectively (see Fig. 1). The total rate of energy dissipation per unit length of pipe is the sum of the rate of heat loss/~ht, from a unit length of the pipe to the ground (whose horizontal surface is at a uniform temperature To), and that fraction of the rate of viscous losses/~od, per unit length accompanying the decay of the pressure head, which does not reappear as heat in the oil. HORIZONTAL AT

SURFACE

OF

TEMPERATURE

GROUND

Tcj

K

o

" ,

•

•

•

S01L

°

o '

.

•

,

.

~

,

.

4

4

. ,a

a

•

o41

..

"'''°

•

• ,.

" ~~

4

. "a

.

• "

~

" " .

' •

.

.

o

• D

•

"o

~" •

a

• .

•

.

b

"

"

~ o

q

'

°~

.

Eht

.

b ,°

.

"

~ " /

.

•

INTERNAL

,q

A-A

°

o

• ~/.,,~OIL

,

A

'o

'

•

a

a

A -

• t~

"

o

~'.

"

-

SOIL

o

=

.

~

D

•

.

•

•

"~

"e. o

o

.

.~

. .

.L •

VIEW

'

a

INSULANT"

•

. '

• "

.

o

-

~ A

o

~*~

.

LAMINAR FLOW OF

°,t.

', 4

•

-

a

~, o

°

•

A

.

.

CROSS-SECTIONAL VIEW

Fig. 1. Schematic sections through an internally insulated buried pipeline through which hot fuel oil is flowing. For the presented analysis, 2d ,4 D 2 - D 1, L ,> D 1.

85

Laminar flows of oils through internally insulated pipelines

STEADY-STATE HEAT LOSSES FROM THE PIPELINE The rate of heat loss per unit length F-~ht,through the homogeneous soil from the buried internally insulated pipe with its horizontal axis at a depth z below the surface of the ground can be described as follows:

f'ht

2n(T t - To) 1 in DD21+ ~ l n r ( 2 z ' ] + /(2z'~2

:

2

h,D~+~

(1) 1]

L\D2) ~it,D2)

In this expression, the thermal resistance of the pipewall (usually metal) is of relatively small magnitude compared with that of the insulant and so neglected. For laminar flow of the hot fuel oil, an empirically derived equation for the heat transfer coefficient between the oil film and inner wall of the pipe is regarded as most reliable and so preferred for design calculations. (The complication occurs due to the superimposition upon the forced convection of natural convection which always exists, and is especially apparent in large diameter pipes. It results in a higher heat transfer coefficient from the oil to the pipe.) From the results of large-scale tests for heated fuel oils being pumped through horizontal pipelines, with bores (i.e. internal diameters) varying from 0.08 to 0'40 m, a good correlation of Nusselt number, Nu, versus Rayleigh number, Ra, has been obtained--see Fig. 2 100 50

~2o ~, uJ

lO

1

10

102

104

105

7

100

109

RAYLEIfiH NUHBER,

Fig. 2. Correlation between the Nusselt number and the Rayleigh number for the

flows of the hot fuel oil through the horizontal pipeline(after Ford3).

86

S. D. Probert, C. Y. Chu

(Ford3). It can be seen that if Ra is less than 4000, the Nusselt number

tends to remain constant at 3.65, which is the theoretical prediction obtained from the true parabolic stream-profile. However, for a large pipeline conveying a very viscous fuel in which natural convection currents appear--see Fig. 3 - - R a becomes as large as 108, and the corresponding value of Nu can be up to 70. For Ra > 5.0 x 104, it is suggested (Gill and Russell 4) that: Nu = O"184(Ra) °'32

(2)

where: Ra =

Cpp2flgD31(T; - Tw)

(3)

Therefore: h 1=

0" 184kl(Ra)°'32 D1

(4)

The dependences of the specific heat, C v, and the thermal conductivity, k~, upon the oil temperature, can be calculated (Spiers 7) from: Cp

= v~pt (533 56.5 + 107.24(T, - 273)) 117"23

k, - - -

(1 - 0.000 54(T~ - 273))

(5) (6)

P15

where the density of the fuel oil at temperature, T~, is given by: pt = p15[1 -

fl(T~ -

288)]

(7)

and fl is its coefficient of volumetric expansion, which is usually assumed to be constant over the limited temperature range considered. The fuel oil viscosity, which depends markedly on the flow temperature but less so on its pressure, in the Newtonian region can be expressed simply by: ]g = 10 A(T'-zva)+B

(8)

where A and B are constants characteristic of the particular oil considered. It is assumed that all the heat losses are time invariant and occur radially. Therefore, the heat flux per unit length through each layer of the

Laminar flows of oils through internally insulated pipelines

87

composite pipeline remains constant with respect to time. Applying Fourier's law, we may write: Eht

= ztDlhl(Ti- T~)

2g(T w -

l,

~lntD-~~)+~

Tg)

-

ln[-fZz' + L\D / V\D

]

/

Therefore: hi=

2zt(Tw - T9)

DI(TI - Tw)I~ln (-D~I) + ~ln I (~) + ~/\D2, ] / ( 2z']2-

(9)

II]

From eqns (4) and (9), the convective heat transfer coefficient, h~, and average inner wall temperature, Tw, can be obtained by the iterative method which starts with an estimate for Tw (which is usually a few degrees less than T l for an insulated pipeline) in a computer algorithm. Viscous losses in the hot oil

When laminar flow occurs in an oil pipeline, the bulk oil temperature should not be used to calculate the rate of viscous losses as in the case of turbulent flow, because a considerable variation of temperature occurs radially across the pipe section. For the insulated pipeline, the contours of temperature in the oil become less radially asymmetric (see Fig. 3). The temperature difference between the bulk oil temperature and the inner

F

A ....

_PIPE WALL

A×IS #

,,

l'

s/ ~..."I~ORETICAL __~.......~-~ DISTRIBUTION --

' ~ " ~ " " ' ~ l

30 ]2

~

i

|

36 38

i

i

i

i

~0 4.2 4.4. 1,6 ~8

I

50

52

VIEW A-A

Fig. 3. Comparison of the measured and calculated temperature distributions (in °C) within a fuel oil flowing through a pipeline at a mean bulk oil temperature of 40.5 °C (after Gill and Russell4).

S. D. Probert, C. Y. Chu

88

1.0

0.8 Tept - Tw

T= -T.

0.6

0.t,

0.2

i

106

~

,

, 5, , , , 10I 7

~

A

. . . . .

5

I

I

100

2

,

,

I

. . . .

5

]

I

109

2

,

,

I

5

1 , , ,

101o

RAYLEIfiH NUMBER, P~

Fig. 4. Correlation betweenthe effectivepumping temperature, Tept,and the Rayleigh number for the flow of the hot oil, assumed to be at the corrected oil temperature for evaluating the viscous losses (after Ford3). wall temperature of the pipe becomes significant when the pipeline bore is large. Thus an equivalent effective pumping temperature, Tept, which is considered to be the uniformtemperature across the bore of the pipeline, was introduced to obtain an empirically correct oil viscosity. Using this, the equivalent viscous losses can then be evaluated accurately for the chosen bore of the pipeline (Ford3). This temperature, Tept, can be determined using Ford's correlation of effective pumping temperature with the Rayleigh number (see Fig. 4). Thus, the Reynolds Number, Re, corresponding to the temperature, T~pt, is obtained. If this value of Re is less than 2000, i.e. the flow is laminar, then the Hagen-Poiseuille equation: 128L#tl? H f - rcpgD~ (10) applies. The absolute viscosity, #~, is evaluated at the effective pumping temperature, T~p,. The rate of energy dissipation due to viscous losses,/~oa, per unit length of pipeline, namely:

Evd -

hg/-/f L

(ll)

So from eqns (10) and (11)"

/ vd-

128~t1172

(12)

Laminar flows of oils through internally insulated pipelines

89

ENERGY-CONSCIOUS DESIGNS FOR INTERNALLY INSULATED PIPELINES It has been shown (Probert et al. 6) that the rate of financial expenditure, Pexp, per metre length of a horizontal pipeline, resulting from heat and viscous losses, can be estimated by the sum of the contributory costs, i.e. : !

eexp

=

aEht +

"

b Eve

(13)

where"

The coefficients a and b are the unit energy financial costs (arbitrary units per Ws expended) for heating and pumping the hot oil respectively: r/p is the overall efficiency of the pumping set and ~ is the proportion of viscous dissipation which is converted into heat in the flowing oil (0 < 7 < 1). For each prescribed set of applied conditions (i.e. for given values ofD 2 and I2, the other influential parameters being kept constant), the minimum rate of financial expenditure due to the energy running costs occurs whenldPexp/dD ~ is set equal to zero. The value of Dlopt is then deduced, and it corresponds to the minimum of Pexp provided that d2Pexp/dD~ > 0for that value of D~opt (see Fig. 5). In this Figure, the Pexp being optimized is plotted against the internal diameter, D~, with the bulk oil temperature, Tz, being held at a constant value. Another value of T~can be selected and the process repeated. The locus of all these various minima (i.e. of the points for which dPexp/dD ~ = 0) indicates the value of the overall optimal oil temperature for a prescribed set of circumstances. For a given flow rate, I2, the Reynolds number can be expressed as: Re---

4pl)"

n#tD1

where the density and viscosity of the oil depend upon the flow temperature. Thus, for each prescribed flow temperature, there is a lower value of the flow diameter D'~, for which laminar flow persists, i.e. for Re to remain less than 2000. Therefore: 4pi2 - < 2000 n l~tD1 --

i.e.

_ D'l

pl)" 500n#t

(14)

S. D. Probert, C. Y. Chu

90

NUMERICAL PREDICTIONS (Figs 5-14) For the purpose of the present analysis, consider a short fuel-oil pipeline with D 2 = 0.40 m (except where otherwise stated), conveying 50 °C oil (for which the density p t = 9 7 0 k g m -3 at 15°C, viscosity: A = - 0 . 0 1 9 6 , B =0.4171; coefficient of volumetric expansion fl = 0-0007K -~) at a 120

i I

I I I

110

_ LAMINAR - FLOW

I I I I

5

100

90 1-

F--

70 6O

,,x,

N 3o 20

o.lo

~'2o

~3o'

oI~o

INTERNAL DIAMETER OF INSULANTUNING, gl, (m)

Fig. 5. Graphical representations of Pexv.(in arbitrary financial units per metre length of pipeline) and its components, aEht and b'E~a, for the flowing hot fuel oilwith a and b' in arbitrary financial units per Ws expended. For these predictions, it is assumed that f ' = 0 . 0 6 m a s -1, D 2 = 0 - 4 m , T g = 2 7 5 K , k i = 0 " 0 3 5 W m - l K -1, k s = l ' 5 6 W m - l K -z, z = l . l m , f l = 0 - 0 0 0 7 K -1 and p l s = 9 7 0 k g m - 3 at 15°C. It can be seen that D~opt increases from 0.257 m to 0.277 m, 0.287 m and 0.297 m as (b'/a) is changed from 1 to 2, 3 and 4 successively.

Laminar flows of oils through internally insulated pipelines

91

~7 C3

~, 06 ~o.s ~/=0.125m3sec-I

~s 0.4

_ /

~

_

~

_

_

~

V

*

=0.080m3set-'

=0.060 m3sec -'

•~ 0-3 ~0.2

%

0'.~

012

;s

d,

o'.s

0.6

d7

0'8

d9

~I0

1.10

OUTSIDE DIAMETEROF RPE, (Oz ÷2d),(rn)

Fig. 6. A family of design curves for selecting Dlop, for various values of the pipe's outside diameter (D 2 + 2d) at the stated flow rate of the fuel oil. It is assumed that d ,~ (O 2 - - D 1op)/2 and (b'/a) = l, the other applied conditions being as for Fig. 5.

steady rate ~" of 0"060m 3 s-1. The insulant liner has a mean effective thermal conductivity k i of 0.035 Wm - 1 K - 1 and its smooth surface finish is such that no asperities penetrate the boundary sub-layer. The thermal resistance of any contaminant or deposit between the oil film and the pipewall is neglected. Also, the effective thermal conductivity of the soil surrounding the pipe over the whole extent of the pipeline is taken to be invariant with a value of 1.56 W m - 1 K - 1. The average temperature of the horizontal surface of the ground is assumed to be constant at 2 °C and the depth of the centre-line of the horizontal pipeline below the horizontal ground surface is 1.10m. For the graphs presented in Fig. 5, the financial costs of energy expended for the systems are plotted as a family of curves. The rate of the heat loss,/~ht, per unit length of pipeline has been evaluated from eqn. (1) with an estimate for h 1, which was previously obtained iteratively from eqns (4) and (9) for each selected value of D 1. The rate of viscous dissipation,/;Svd, per unit length of pipeline was then calculated from eqn. (12). The rate of financial expenditure, Pexp can then be obtained from eqn. (13) for different values of the ratio (b'/a). It can be seen that the smaller the insulant thickness, the higher the rate of heat losses, but less

100

75

~ =1

~" 50 i-

V = 0.015m3s e c -1

z~

r~

7'5

16O

'~:~

1~0

1")5

26O

OPTIMAL THI[KNESS OF INTERNAL LINING, l'~t, {ram)

Fig. 7. This Figure should be read in conjunction witti Fig. 6. The least rates of financial expenditures upon energy are presented. For a particular pipe, of specified internal diameter, D2, and flow rate, I?, of the fuel oil, D1 optis first read from Fig. 6. With this value of Dlopt , topt{=(D2-Dtopt)/2} can be calculated, and hence from Fig. 7, the corresponding value of Pcxp can be read off or interpolated.

5

0.7 ~b'= 2

~

~ 0.5 z

~/=0-I~5m3sec-I

/~j__..___-

~ 0.4

v: oo6om3see-,

~ 0.3 ~

-- v: 0.015n~sec-~

~ 0.2 g

/

0.1

0

~1

01.2

0~3

01-~

135

01,6

1~-7

(~.8

O-9

OUTSIDE DIAMETEROF RPE, (D2 +2d}, (m)

Fig. 8. As for Fig. 6, except that

( b ' / a ) = 2.

1.'0

1.~0

Laminar flows of oils through internally insulated pipelines

93

100

b' 5-=2

=< 75

50 xuJ _z ~ ~ ~ - - ~ ~ ~

~

~/=0.125rn3secq

V: 0.030m3sec-~ - V = 0.015rnasec-~

OPTIMAL THICKNESS OF INTERNALLINING tom, (ram)

Fig. 9. This set of curves gives the least values of Pexp for the optimal geometry of the pipeline selected via Fig. 8, assuming that (b'/a) = 2. The volume rate of flow is stated with each curve.

:-0.7 8 c~ L~ Z 0,6

b'=3 cl

~ 0.5

~_....~...~._.T--

_z

J

~

-

~

:

~ 0-3

~

'V=0.1L~m3SiLk-1 V=0.080m3see-' V=0.060m3sec-'

,.12o. _-

0.2

~- 01

d~

d2

d~

ok

o's

o'~

~7

;8

;9

OUTSIDE DIAMETEROFPIPE, (O2 ÷2d),(m)

Fig. 10.

As for Fig. 6, except that (b'/a) = 3.

;o

~.io

S. D. Probert, C. Y. Chu

94 100

x

v Vo

'

OPTIMAL THICKNESS OF INTERNALLINING, tml, (ram)

Fig. I I.

As for Fig. 7, except that (b'/a) = 3. The volume rate of flow is stated with each curve.

A

E

c5~ 0.7 b' z

0.6

~

o.5

~/=0125 m3 sec-I 0 0 ~ ~sc~c-i

~S 0.a

0.3 ~ V = 0 0 1 5

z

rrP see-1

o2

~ 01 0

~'1

o'.2

o~3

o'.,,

&

&

d7

o~6

0'9

OUTSIDE DIAMETER OF PIPE, {Oz+2~) , (m)

Fig. 12.

As for Fig. 6, except that ( b ' / a ) = 4.

1~

1.~o

Laminar flows of oils through internally insulated pipelines

95

100

b'

i.-

~ so

w

t

25

OPTIHAL TH~KHESS OF INTERNAL LINING. topt, {mm)

Fig. 13.

As for Fig. 7, except t h a t

(b'/a)

= 4. T h e volume flow rate is stated with each

curve.

viscous losses are then incurred. So, for each value of the ratio (b'/a), there exists a clearly identifiable optimal diameter, DlopC The value of Dlopt increases with (b'/a), due to the greater pumping costs then being incurred to overcome the viscous dissipation. Graphs of the optimal diameter, D~opt, for various flow rates, 12, and different values of the external diameter of the pipeline (D 2 + 2d), by assuming a = 1, b' = 1, 2, 3 or 4, all in arbitrary financial units per Wattsecond expended, are shown in Figs 6, 8, 10 and 12. The corresponding financial costs of the rates of energy expenditure, Pexp (from eqn. (13)) are plotted against the optimal insulant thickness,/opt[ = ( D 2 - D~op)/2] and are presented in Figs 7, 9, 11 and 13. It can be seen for a given flow rate, 12, at high values of D2, only a slight increase of D~o,t results (see Figs 6 and 7). Also, Pexpchanges dramatically for low values of/opt, SO that a small increase of/opt in this region for small pipes results in a considerable reduction of P~xp. In Fig. 14, the value ofP~x v at d/5~xp/dD1 = 0, namely, P,Xpopt, is plotted

96

S. D. Probert, C. Y. Chu 90

~o

~

6O

•~

50

~ '=/+

-.---____

g z

------__.___

40

~.'=~ u-

N 2O LAMINAR FLOW

N i 030

~o

~

~

~

BULK OIL TEMPERATURE (*C)

Fig. 14. The variation of the least rate of financial expenditure with the bulk oil temperature, the other applied conditions being as for Fig. 5. It can be seen that Tjopt changes from 62-0 °C to 63.7 °C, 64.9 °C and 65.6 °C, respectively as a = 1 and b' increases from 1 to 2, 3 and 4, all in arbitrary financial units per Ws successively. The corresponding values of/Sexpopt are 29'8, 36.6, 41'6 and 45.8, all in arbitrary financial units per metre.

against the oil temperature, T~. It is clearly found that an individual minimum value of Pexpop~occurs at four different optimal values of T~opt according to whether the ratio (b'/a) equals 1, 2, 3 or 4. Each corresponding value of Dlopt is the overall optimal diameter for the prescribed set of conditions.

CONCLUSIONS The use of an optimal thickness of insulant for a pipeline appropriate to the applied conditions (e.g. flow rate and temperature distribution) is recommended. However, the optimal designs as predicted in the present analysis correspond solely to minimum rates of financial expenditure upon energy for pumping hot oil through the pipeline. No account of capital cost has been taken because, for instance, financial information

Laminar flows of oils through internally insulated pipelines

97

concerning internally lining pipes with thick layers of thermal insulants still remains rare and unsystematic. The economies of scale should, in future, significantly reduce the true costs of such pipelines. In addition, the initial capital cost of the pipeline is likely to be far less than the total energy running cost, as the pipeline will probably be operated over long periods ( > 30 years). The embarrassing uncertainties of inflation and interest rates for such a future period make net present value and discounted cash flow yield assessments in the present context almost guesswork. The presented analysis, based upon the ratio b/a of the unit fuel costs, to a large extent avoids the predictions being significantly affected by inflation. Thus, from such considerations, it is concluded that the predicted optima, provided the pipeline is operated for a long period, are likely to provide a worthy guide to the best designs. The temperature of the hot oil decays as it flows along the buried pipeline and this leads to the oil gradually increasing in viscosity if the pipeline is unheated. Thereby the rate of viscous losses per unit length increases the further along the pipeline one proceeds. Thus, an optimal temperature exists so that the least rate of energy expenditure ensues (see Fig. 14). The relationship between the pipe diameter, applied insulant thickness, rate of volumetric throughput and the chosen operating temperature for a short hot-oil pipeline has been investigated in this paper. The presented analysis can also be extended for optimal designs of long pipelines provided the variation of the effective pumping temperature with the position along the pipeline is known (Probert and ChuS).

ACKNOWLEDGEMENTS The authors wish to thank the Science and Engineering Research Council and BP Ltd. for their generous support of this project.

REFERENCES 1. ASTM. D396-75, Standard specification for juel oils, American Society for Testing and Materials, Committee D2, Philadelphia, USA, 1976. 2. BS2869, Petroleum Juels for oil engines and burners, British Standards Institution, London, 1970.

98

S. D. Probert, C. Y. Chu

3. P. E. Ford, Pipelines for viscous fuels. Proceedings of the Fourth Worm Petroleum Congress, Section VIII B, London, 1955, pp. 115-29. 4. F. Gill and R. J. Russell, Pumpability of residual fuel oils, Industrial and Engineering Chemistry, 46(6) (1954), pp. 1264-78. 5. S. D. Probert and C. Y. Chu, Optimal pipeline geometries and oil temperatures for least rates of energy expenditure during crude-oil transmission, Applied Energy, 14(1) (1983), pp. 1-31. 6. S.D. Probert, C. Y. Chu and C. M. Yeung, A possible alternative approach to the thermal insulation of pipelines, Applied Energy, 11 (1) (1982), pp. 15-34. 7. H. M. Spiers (Ed.), Technical data on fuel (Sixth edition), The British National Committee, World Power Conference, London, 1962, pp. 183-4 and p. 224.