Economics of thermal insulation

Economics of thermal insulation

ECONOMICS OF T H E R M A L INSULATION S. D. PROBERT and S. GIANI School of Mechanical Engineering, Cranfield Institute of Technology, Cranfield, Bed...

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ECONOMICS OF T H E R M A L INSULATION

S. D. PROBERT and S. GIANI

School of Mechanical Engineering, Cranfield Institute of Technology, Cranfield, Bedford MK43 OAL (Great Britain)

S UMMA R Y

Thermal insulation can be seen, from current option-costing analyses, to be usually an excellent financial investment. Considerations of capital and revenue costs lead to simple design procedures for predicting the optimal economic thicknesses of insulants for plane walls and pipes, respectively. An energy utilisation inefficiency factor is proposed, together with a new profession of engineer--the energy accountant.

NOMENCLATURE

A C

Ch Ci Ctot f G

hi

ho J k L

m2 Area of plane surface Cost of applying and maintaining the insulation averaged £ (year)- 1 throughout the life of the system £ (year)Annual heating charge £ (year)- 1 Cost of insulation divided by life of the system £ (year)- 1 Annual cost of heating and insulation Inflation factor for the cost of fuel Rate of interest on capital if i t had not been invested in insulation, including allowances for inflation and plant depreciation Per cent (year)Average transfer coefficient for the heat passing between the contained fluid and the internal surface of the system W m - 2 K - 1 Average transfer coefficient for the heat passing between the external surface of the system and the surrounding air W m - 2 K - 1 Cost of a unit of heat £ j- 1 Thermal conductivity of the insulant W m- ~ K- 1 Length of the part of the insulated pipe considered m 189

Applied Energy (2) (1976)--© Applied Science Publishers Ltd, England, 1976 Printed in Great Britain

190 N P Q r1 r2 r2e rc T T~ U V x xe Z

s . D . PROBERT, S. GIANI

Total period of operation during a year s Energy Utilisation Inefficiency Factor £ (year)-2 Rate of loss of heat through the part of the plane or cylindrical system considered W Radius of the external surface of the pipe m Radius of the external surface of the insulant applied to the pipe of radius r 1 m Economically most favourable value of r2 m Critical external radius of the insulant m Mean temperature of the fluid contained within the insulated system K Ambient temperature of the air surrounding the insulated system K Overall thermal transmittance of the wall of the insulated system W i n - 2 K-X Volume of the insulant m3 Thickness of the insulant m Economic thickness of insulant m Cost of insulant per unit volume £ m-3

ENERGY THRIFT

Because of balance-of-payments trade deficits, uncertainties with respect to the security of future oil (and coal) supplies and the advent of nuclear-fusion power, as well as endeavours to reduce the rapid depletion rate of the world's limited reserves of fossil fuels, energy conservation is at last being vigorously advocated, although often only ineffectually practised. ' I f Britain could save 10 per cent of its present energy consumption by the 1980s, it would be worth £600 millions a year at current prices'--Eric Varley, Secretary of State for Energy, House of Commons, London, July, 1974. During 1975 the U K imported about £3500 million of oil. The rapacious appetite for power in the U K is about 6 kW per person (i.e. national consumption divided by population) whereas, for personal survival, one needs only about 80 W. A national plan for reducing, by at least 30 per cent, the annual rate of energy consumption relative to 1972 (i.e. the year prior to the OPEC cartel which imposed an oil boycott and subsequently a major j u m p in its price) could be achieved by 1980 without adversely affecting our quality of life or gross national product, 1 but unfortunately such an aim is rarely stated, let alone used as a reference standard with which to gauge our energy conservation performance. North Sea oil will provide a short temporary respite exceeding the lifetime of a Government, but will not resolve the U K energy quandary. Nevertheless, it should release sufficient capital for investment in a co-ordinated energy-conservation programme.

ECONOMICS OF THERMAL INSULATION

191

If the U K energy-consumption pattern were changed so that our industries and food-producing activities (from the smallest to the largest) were encouraged by governmental incentives to consume more effectively a larger proportion of a 30 per cent smaller national energy budget, then many of our present economic problems could be overcome. The UK would then be nearer self sufficiency with respect to energy and food, and more sophisticated artefacts could be exported, so reducing our balance-of-payments problem. We might even reach a stage when electrical power (often intermittently available in excess of demand) could be used for extracting (i) metals from ores which are at present too low grade to be economically viable, or (ii) in the extreme, minerals from our vast indigenous source--the sea. These goals could be achieved most economically and more permanently by redeploying some of the huge investments being made in energy release and conversion activities into energy conservation measures, such as thermal insulation, thereby permitting the use of energy for more productive purposes. To save energy by conservation is far more profitable than to prospect for oil, yet even at this time when their services are vitally needed, many thermal insulation firms are suffering severe cash-flow problems. Governments still prefer to foster 'lame ducks' such as the car industry, which encourages energy consumption in our energy profligate society, rather than give an incentive to all householders (not just owners) either by permitting all expenditure on thermal insulation to be deducted from gross salary before tax deductions, or by encouraging Building Societies tc provide mortgages for insulation, because such expenditure is in both the individual's and the national interest. A less attractive alternative would be for the Government to make a grant of, say, £100 per dwelling to encourage tenants or houseowners to insulate now, provided the occupier invested an equal amount. Irrespective of the exact method of financing, a national scheme is now needed by which whole streets of houses would have their thermal insulations upgraded and heat recovery systems installed, at one time and therefore at lower cost, as was achieved when the conversion to the use of North Sea gas occurred. During 1972, of the UK gross energy consumption, industrial and domestic customers used respectively 41 per cent and 25 per cent (four-fifths of the latter being for space heating).2 Heating domestic, commercial and industrial buildings for comfort purposes accounted for 42 per cent of the total consumption. By the implementation of reasonable thermal insulation and heat exchanger measures, this figure could be cut by two-thirds. Of the eighteen and a half million houses in the UK, since construction, only 4 million had by October, 1975 any degree of thermal insulation introduced. Thus a major means of conserving energy would be by thermal insulation--not merely of buildings, but also of industrial process plants. The £1000 million spent upon the production and application of thermal insulants within the UK during the 20 years to December, 1974 has led to fuel savings during 1975 alone of at least £720 million. By investing a further £150 million annually for the next decade in improving thermal insulation in the UK, it

S. D. PROBERT, S. GIANI

192

is estimated that by 1986 energy consumption would be reduced by more than an additional £ 1000 million annually at present f u e l prices. By 1 July, 1975, the average price index for fuel had risen to over 450, relative to a mid-1968 index of 100, whereas, during the same period, that for insulants rose by a factor of only 2.3, i.e. at just over half the inflation rate for fuel. This trend is likely to continue; within the next decade fuel prices will probably rise by at least a further factor of 2-5. 3 Thus the continuous and cumulative energy (and hence revenue) savings accruing from the installation of properly-designed, thermal insulation make it an increasingly attractive financial proposition.

ECONOMIC THICKNESSOF INSULANT The analysis presented in this paper is an extension of that of McMillan. 4 For each system, there is an economically most favourable shape, as well as an economic thickness of a particular insulant, the value of which' can be predicted, on the basis of several (sometimes unsubstantiated) assumptions, e.g. with respect to the rate of inflation during the proposed lifetime of the system. If no insulation is used, there would probably be an excessi~/e rate of heat exchange. However, the greater the thickness of insulant applied, the smaller this 'wild' thermal current (and hence running cost) would be, but the insulation cost (including installation charges and loss of interest on capital so invested) then increases. The economic thickness of insulant is that for which the sum of these two costs exhibits a minimum. Although the following derivations are presented as though for the insulation of heated systems, identical analyses can be used for refrigerated plant. The annual heating cost is given by: Ch = JQN (1)

• •

IZ~

PLANE~ WALL ~ -

/ I N S U L A N T OF ~/ C O N D U C T I V I T Y

:I I I:~::I

:::!: :7:i

k

/

?ii~,i!i;-.ii~iL !:: HEAT FLUX 0

•.. i,;.:..:'~/ 2,?::; ..,~....::..

,~MBIENT TEMPERATURE TQ

(:., ,i.ii'iS!

Fig. 1. Part of an insulated plane wall.

ECONOMICS OF THERMAL INSULATION

193

The annual insulation cost, Ci, depends upon the interest that would have otherwise accrued from such a capital investment, as well as upon depreciation, repair and insurance costs. It may be expressed as" G

Ci = 7--... Z V + C IUU

(2)

where C represents the total fixed cost of applying and ma&taining the insulation (including costs for such factors as protective materials, scaffolding and labour, which are almost independent of insulant thickness) divided by the expected lifetime (in years) of the system. The total, average, annual cost, C,ot, throughout the lifetime of the system is given by: C,o, = Ch + Ci (3) (i) For an insulated plane wall (see Fig. 1) The economic thickness, xe, of insulant is that for which C,o, is a minimum, and is obtained by differentiating Cto t with respect to insulant thickness, x, and equating

oJ x t~ t-

< o z

THICKNESS OF INSULATION.x {LINEAR SCALE1

Fig. 2.

Schematic plots of the mean insulation, heating and total costs for the insulated plane wall (shown in Fig. l) averaged t h r o u g h o u t the lifetime (in years) of the system.

194

S. D. PROBERT, S. GIANI

the differentiated expression to zero, viz.

~Ctot #x

(~0 G a V = J N-ff-ffx +'16-6 Z-~x = O

(4)

provided d2C,ot/dx2 is positive for x = xe. The rate of heat loss through the system 0 = A U(T'-

7".)

(5)

- T.)

(6)

Consequently, from eqn. (1): Ch = J A U N ( T

The thermal resistance (i.e. the reciprocal of the thermal transmittance, U) of the insulated plane wall can be obtained from the sum of the resistances in series of (i) the fluid boundary layer adjacent to the internal plane surface, (ii) the insulant and (iii) the air boundary layer on the external plane surface, i.e. 1

1

x

1

(7)

=/,. + Z- + ha t I I

II INSULATED CYLINDER

~\

mr,// I /

"INSULANT CONk~Fuc TiViTY

Fig. 3. Cutaway showing part of an insulated pipe.

ECONOMICS OF THERMAL INSULATION

195

In this equation it is assumed that the wall itself only provides a negligible resistance and that perfect thermal contact exists between the wall and the insulant. Therefore: JAN(T-

Ch =

Ta)

(8)

(l/h, + x/k + 1/ho)

Its dependence upon x can be seen in Fig. 2. Because the volume, V, of the insulant equals Ax, by substituting for Ch from eqn. (8) into eqn. (3) it can be seen that: JAN(TCt°t ~-

To)

G

(1/h.i -~ x/k di- 1/ho) "~- - ~ Z A x

--~ C

(9)

Hence: dCtot Ox - 0 =

J A N ( T - To) G k(l/h~ + x / k + 1/ho) 2 + -~-~ZA

(lO)

i.e.

Xe : 10 ,(JNk(T---GZ Ta!) ½ - k (~i + ~o)

(11)

(ii) For a thin, insulated, metal pipe of negligible thermal resistance (see Fig. 3) The rate of heat loss (2 in this instance is given by: (2 =

2rrL(T - Ta) I/(rlhl) + ( l / k ) I n (r2/ra) + l/(r2ho)

(12)

where both the internal and external radii of the pipe are taken as equal to r 1. The annual heating cost is therefore: 2rrLJN(T - Ta) Ch = l/(rlhi) + (l/k)In (r2/rl) + l/(r2ho)

(I 3)

The volume of the insulant in this instance is rtL(r22 - rl 2) and so:

Ci = ~GZL(r2 2 -

rl 2) + C

(14)

Thus, from the eqn. (3): Clot

7~ 2 n L J N ( T - To) + ~ GZL(r2 2 - rl 2) + C (15) 1/(rlhi) + (I/k)In (r2/rl) + 1/(r2ho)

For minimum cost, r2 = r2e ~Ctol - Or2 is positive,

i.e.

=

0

and

~2Ctol ar 2 2

196

S. D. PROBERT, S. GIANI

Oar

w -'n

t~

w x:

I

\

I

--r-

Ci

I

THICKNESS OF INSULATION,x(:r 2-fi) (LINEAR SCALE) Fig. 4.

Schematic plots of the mean annual insulation, heating and total costs for the insulated pipe shown in Fig. 3.

(kr2e hirl + r2eIn (r2et k )

= 10

(JNk(T__-Ta))½( I - k )½

(16)

where r2e is the optimal, external radius of the insulant. This equation has to be solved numerica!ly in order to determine the economically most favourable value, r2~, and hence the economic thickness of insulant. The significance of the critical thickness of insulant as shown in Fig. 4 is discussed in Appendix 1.

APPROXIMATE SOLUTIONS

By neglecting the thermal resistances of the fluid boundary layers within and outside the considered systems by putting hj = = oo) compared with the much larger thermal resistance of the insulant, it is possible to simplify eqns. (1 I) and (16) to the forms shown respectively in eqns. (17) and (18). These assumptions permit slightly exaggerated (by < 10 per Appendix 2) estimates of the

(i.e.

ho

cent--see

12 25 38 50 63 75 88 100 113 125 150

Outside diameter of pipe 2rl ram)

VALUES OF

0.542 0-915 1-187 1.385 1-563 1.703 1-836 1-944 2-050 ---

0-755 1-208 1.519 1.740 1-934 2-085 2-226 2.341 2"452

--

--

33.4

21.3

In (r2/ri) FOR

TABLE 1

0.403 0.711 0.945 1-122 1.283 1.412 1.536 1.637 1.737 1'821 --

48-3

0-335 0.604 0.816 0.978 1.128 1.249 1-366 1-463 1.558 1-638 1.788

60.3

0.284 0.522 0.714 0.863 1.003 1.117 1.227 1.319 1.410 1.487 1-631

73-0

VARIOUS COMMONLY-USED

0.239 0-446 0.618 0.754 0.883 0.989 1.092 1.179 1.265 1.338 1.476

88.9

0-191 0.363 0.510 0-629 0-743 0.838 0-932 1-012 1.091 1.159 1.288

114-3

0.136 0.265 0.379 0.474 0.567 0-646 0.726 0.794 0'862 0.922 1-036

165.1

0.104 0.206 0-298 0.376 0.454 0.522 0.590 0-649 0-709 0-761 0-863

219-1

0.074 0.148 0.217 0.276 0.337 0-390 0.445 0.493 0.542 0.585 0-670

314.3

PIPE SIZES AND INSULANT THICKNESSES

0.058 0-117 0.173 0.227 0.272 0'316 0.362 0-403 0-445 0.483 0.556

403.2

0-101 0-149 0'192 0.236 0.275 0-316 0.353 0.391 0'424 0.491

473.1

0.091 0-138 0-174 0"214 0.250 0-288 0.328 0.357 0-388 0.451

527-1

m

~d

Z

> ,d

r-

>

m

©

© z ©

198

S. D. PROBERT, S. G I A N I

economic thickness to be obtained as fuel costs rise. For plane surfaces:

rapidly:

the magnitude of the overestimate falls

(JNk(T - T,,)~½

x~ z 10\.

G-Z~

/

(1 7)

For cylindrical surfaces: r2e In (r2" t ~ 10 (JNk(T \rl / \ --~-~

Ta!)½

(18)

For a particular specification of insulated plane wall, values of all the parameters on the right-hand side of eqn. (17) would be either known or readily available and so x e could be calculated immediately. For the cylindrical system, the radius, r I, of the pipe which it is wished to insulate would also have been specified and so the following evaluation procedure is recommended. With the aid of Table 1, choose two realistic values for r 2 such that for one the numerical value of r 2 In (r2/rl) slightly exceeds that of lO[JNk(T - To)/GZ] ~, whereas, for the other, the converse is true. Interpolation between these higher and lower values for r 2 In (r2/rl), in order to satisfy eqn. (18) exactly, will then yield a good approximation for tee, from which the appropriate economic thickness ( = r2~ - rl) may be calculated.

PRACTICAL CONSIDERATIONS (i) Insulation cost This involves such parameters as the cost of the external cladding, as well as the transportation costs for bringing the labour to the site. Thus the cost of insulation per unit thickness decreases as the insulant thickness is increased. The magnitude of this marginal cost decrease is partially offset by several factors, such as for a cylindrical system, the increasing volume of insulant needed to achieve unit thickness as r2 increases. Also to apply a large economic thickness of low-strength insulant, it may become necessary to develop special, and hence expensive, fixing systems (which prevent the crushing of the insulant) rather than ordinary lacings and clips, as well as independently supported external claddings. Hence the straight lines presented in Figs. 2 and 4 for Ci are oversimplifications. Tliis emphasises that each new design should be analysed from first principles. TABLE

2

fFACTOR ASSUMING A STEADY 10 PER CENT INFLATION RATE

Plant life (years) Mid-life (years) f factor [f]½

0 2 4 6 8 I0 12 14 16 18 20 0 1 2 3 4 5 6 7 8 9 I0 1.00 1-10 1.21 1.33 1.46 1.61 1-77 1.95 2.14 2-36 2.59 1.00 1.05 1-10 1.15 1.21 1.27 1.33 1.40 1.46 1.54 1.61

200

S. D. PROBERT, S. GIANI

(vii) Heat transfer considerations The local heat transfer coefficient will vary over the surface of an insulated pipe. Thus a small variation of the economic thickness of insulant with radial position occurs, but adoption of this variation in practice would not be commercially feasible. Occasionally two layers--each of uniform thickness but with staggered joints--of different insulants are applied in series: this situation can be dealt with easily by the present analysis. The contact resistance between the insulant and the surface being insulated has been ignored in the analysis, and this leads to an overestimate of the economic thickness of the insulant. The economic thickness of insulant for the pertinent conditions is (i) often insufficient to prevent condensation on cold surfaces but (ii) usually more than adequate to ensure 'safe-touch' temperatures for personnel coming into temporary contact with the surface of the insulant. I° Nevertheless, insulants are used for many purposes, only some of which can be instantly recognised as having an immediate economic value. 1 x For example, insulation may inhibit the escalation of a fire, but the economic significance of this is difficult to quantify except possibly by lower insurance charges.

ECONOMIC OPINIONS

Since the start of the Industrial Revolution, some two hundred years ago, there has been a steady growth in national energy consumption to which we have all become accustomed. During the last twenty years, energy in the form of oil was so cheap that efficiency has often given way to convenience. Until recently, engineers had in general been constrained by accountants and customers to design systems which cost least, often with complete disregard to their subsequent, high runningcost, legacy. This 'down to a price', rather than 'up to a standard', design policy arises partly because energy is still cheap and usually invisible, and is therefore largely ignored. Governmental exhortation and market forces are failing to achieve the optimal use of energy because energy costs usually represent only a small part ( < 10 per cent) of total manufacturing costs or of the domestic budget. The price mechanism alone, at the present low level of energy prices ( ~ 1£ GJ-1), is inadequate to ensure only the thrifty use of energy. Too often precedence is still given to capital cost, rather than to energy consumption, when purchasing an artefact. If fuel costs equalled our monthly house mortgage repayments we would treat the fuel crisis more seriously. Consequently, legislation needs to be introduced so that a parameter such as the Energy Utilisation Inefficiency Factor, P, which equals the maximum permitted annual running cost divided by the guaranteed lifetime, is adopted nationally as a purchasing criterion. If the energy-thrift performance of a system is

ECONOMICS OF THERMAL INSULATION

201

found to be poorer than would be expected from the stipulated P factor for that class of goods, the buyer should have legal redress against the vendor. Statutory acceptable P values would probably vary initially between different types of system intended to satisfy the same basic function, because, for example, heating a flat in a row of terraced houses is in general less energy consuming than heating a detached house. Such P differences would, however, reveal immediately good and poor designs with respect to fuel consumption and so should lead eventually, by consensus pressures, to a single P factor being insisted upon for all artefacts attempting to satisfy the same purpose. An analysis of the economics of the insulation of buildings is much more complicated than those presented for the insulation of plane walls or pipes, because allowances must be made for the ventilation heat losses as well as for the building materials themselves providing some resistances to heat flow. The building may not have the optimal shape or location for minimum heat losses, 12 or the insulant may be rendered ineffective by a thermal current by-passing it via a 'thermal' bridge. Thus energy conservation measures must be considered at the design stage and not as afterthoughts when they will be probably less cost-effective. The larger the glazed area of the external walls of a building, in general, the greater will be the rate of heat loss. However, for psychological comfort we need to view our external environment and hence windows are desirable. Recent studies I 3 indicate that the best compromise in the UK is achieved with about 25 per cent of the external facade glazed, the windows (which should be non-opening) being predominantly on the southerly aspect to maximise solar gains; ventilation during the winter being achieved by a heat exchanger taking in air (preheated by energy that would otherwise be lost) from the loft space. Government should take the lead in improving design for enhanced energy conservation. For example, the introduction of a national grid for supplying hot water and steam (which is at present wasted in electric power-generating stations) via district-heating schemes is long overdue. At the other extreme, local authorities should be discouraged from installing low capital cost electric heating systems for (i) poorly insulated old-people's homes (which is leading to pensioners suffering from hypothermia, because they cannot afford the necessary electricity) and (ii) swimming pools, thereby resulting in high running costs and consequently increased admission charges, so that fewer people use the facility, thereby accelerating the inflationary cost spiral. Surely, in order to encourage the best use of resources, the Government should introduce now, but guarantee to apply for the next twenty years, a fixed relative pricing policy for coal, methane, oil and electricity so that cost-effective investments in energy-consuming equipment can be made. This would be compatible with the national policy of restricting the use of a high-grade fuel such as electricity to applications which are best accomplished with it, e.g. lighting. An energy budget, i.e. the maximum permitted annual energy consumption,

202

S. D. PROBERT, S. GIANI

needs to be established for each system. Typical values to permit a building to be habitable in the UK might be 500 MJ m- 2 of floor area for a house or bungalow, or 250 MJ m -2 for a flat, both with an upper limit of 200 m 2 floor area. (Alternatively, a 10 GJ annual ration might be allocated per person.) Any consumption exceeding these survival limits should be at a punitively high cost, e.g. ten times the basic rate per MJ. This is, in effect, the inversion of existing tariff policies which involve high standing charges (which it is proposed should be abolished) yet low cost per unit of energy consumed. The permitted ration should refer to gross energy consumption (i.e. including the energy dissipated in the release and distribution of the energy) so militating against the use of electricity (which is still derived predominantly from fossil fue!s) for inappropriate low-grade purposes, e.g. space heating. The P factor certification of plant or facilities should be undertaken by professionally responsible energy accountants in an analogous manner to that in which registered garages are permitted to issue Ministry of Transport certificates for road worthiness of vehicles. These private-practice energy accountants could be licensed by the Government to carry out annual energy audits and testing of equipment for individuals and organisations and would have the authority to negotiate with the Department of Energy (who would establish energy consumption norms for systems and appeals procedures) in a manner comparable with that in which financial accountants handle taxation claims, on behalf of clients, presented to Her Majesty's Inspectors of Taxes. Although still relatively rare compared with financial and stock-taking type audits, it is now increasingly realised that an energy audit can equally well serve to direct attention to those energy cost centres where savings can most easily be achieved or are most desirable. However, to implement such proposals nationally would require the compilation of a register of chartered energy accountants or energy engineers who are, for instance, capable of, and experienced in, monitoring fuel consumption, testing the combustion efficiency of boiler systems or surveying with an infra-red camera internally heated (or cooled) insulated systems to locate any thermal bridges which might be present. Whenever an energy accountant is consulted, his fees should be based not, as for an architect, on the capital cost of the building or equipment installed, but rather on the decreased revenue consequences obtained b~, implementing his proposals (e.g. with respect to the reduction in energy consumption). Our basic economic strategy during this period of energy uncertainty (e.g. with respect to the prospects for cheap nuclear-fusion power, exhaustion of readilyrecoverable fossil fuel reserves and political disruptions) should be to invest adequately in thermal insulation. This will involve ensuring that the appropriate technical and economic information is made available to those who will make decisions concerning the design and operation of equipment. Then radical changes in our life styles and enforced reductions in our quality of life within the next twenty years will prove unnecessary--i.e. 'Pay now, live later'.

203

E C O N O M I C S OF T H E R M A L I N S U L A T I O N

A P P E N D I X 1. C R I T I C A L T H I C K N E S S OF I N S U L A N T

For small diameter wires and pipes (e.g. hypodermic tubing), the application of a layer of thermal insulant may lead to an increase in the rate of loss of heat from the system to the surroundings (see Fig. 4) because the external surface area from which heat is lost is then increased. Thus the plastic sleeves on electric-current carrying wires should not only provide electrical insulation but also prevent overheating (and hence resistance rises and still further increased rates of ohmic heating, etc.) by being designed to rnaximise the rates of loss of heat. Assuming (i) the heat transfer coefficient, ho, for the outer surface of the insulant remains invariant irrespective of insulant thickness, (ii) the thickness of the pipe wall is infinitesimal so that its conductance is infinite and (iii) no boundary layer exists adjacent to the internal wall, then:

(2 =

2xL(T-

To)

(19)

(l/k) In (r2/rl) + l/(r2h o) Differentiating this with respect to r2 and equating the resultant to zero gives:

80 t~r2 2 - -

2nL r2

(T-

To)(l/(r2ho) - l / k )

{(l/k) In (r2/rl) q- l/(r2ho)} 2

= 0

(20)

i.e. =

k

(21)

The value of ~2Q/Sr22corresponding to this value of r2 is negative. Thus it may be concluded that the maximum value of ~ occurs for a 'critical' external radius, r~, of the insulant equal to k/ho. (The corresponding critical external radius for a spherical shell of insulant equals 2k/ho.) Typical insulants at 290 K and 1200 K have conductivities of 0.03 W m - 1 K - 1 and 0.13 W m - 1 K - 1, whereas ho for common geometries in air ranges from 5 to 45 W m - 2 K - 1. So the critical radii might be expected to vary from about 1 mm at normal ambient temperatures to about 5 mm for high temperature applications. Thus the critical thickness of insulant will be of importance only for very small systems, and so is usually of academic, rather than technological, interest to the thermal insulation engineer. In practice, the heat transfer coefficient, ho, is dependent upon the radius r 2 and on the temperature distribution.14 By taking these factors into consideration, it can be shown that the critical radii for pipes and spheres are approximately 0.6(k/ho) and 1.4(k/ho) respectively, i.e. even smaller than the above analysis suggests.

S. D. PROBERT, S. GIANI

204

APPENDIX 2. THE APPROXIMATION Typical values for the parameters involved in eqn. (11) are as follows: J = 2 x 10 - 3 £ M J - 1 ; N = 20 M s ; k = 4 x 10 - 2 W m -1 K - l ; T - Ta = 20K;G= 7.5percent;Z= 30£m-3;hi= 5Wm-2K -landho= 15Wm -2 K-1. Substituting these values into the e q u a t i o n shows that the neglect of the k ( l / h i + l/ho) term leads to the economic thickness of i n s u l a n t being overestimated by 8-9 per cent, i.e. 12 cm rather t h a n 10.9 cm.

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14. E. M. SPAgROW, Re-examination and correction of the critical radius for radial heat conduction, d. Am. I. Chem. Eng., 16(1) (1970) p. 149.