Radial heat flow in two coaxial cylindrical discs

Radial heat flow in two coaxial cylindrical discs

Citation for published version (APA):

Daalder, J. E. (1977). Radial heat flow in two coaxial cylindrical discs. (EUT report. E, Fac. of Electrical Engineering; Vol. 77-E-76). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977

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RADIAL HEAT FLOW IN TWO COAXIAL CYLINDRICAL DISCS

TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

AFDELING DER ELEKTROTECHNIEK GROEP TECHNIEKEN VAN DE ENERGIE-VOORZIENING

EINDHOVEN UNIVERSITY OF TECHNOLOGY THE NETHERLANDS

DEPARTMENT OF ELECTRICAL ENGINEERING GROUP APPARATUS AND SYSTEMS FOR ELECTRICAL ENERGY SUPPLY

RADIAL HEAT FLOW IN TWO COAXIAL CYLINDRICAL DISCS

J.E. Daalder

TH Report 77-E-76 ISBN 90 6144 076 9

List of symbols thermal conductivity K thermal diffusity p specific mass c specific heat r radius

Q strength of a line source

T temperature -1 -1 [Watt m K ] 2 -1 [m sec ] -3 [Kgm ] [m]

I ,K modified Bessel functions of the first and second kind of zero order o 0

J

o,J1 Bessel functions of the first kind of zero and first order.

1. Introduction.

In this report an analysis will be given of the radial heat flow in two coaxial cylindrical disks which have different thermo-dynamical properties.

The heat flow is initiated by an instantaneous line source, situated at the

axis of the disks. Of particular interest is the temperature variation with time at the common boundary of the cylinders.

The problem has its origin in the determination of the energy transfer from an arc to an electrode and the ensuing heat flow to the surroundings via its support.

The enthalpy increase of an electrode (either cathode or anode) due to arcing is a quantity which is of relevance in the study of arc physics. If the amount of energy transferred to the electrode surface is known, con-clusions can be drawn regarding electrode surface temperature, electrode erosion and the energy balance (s) of the arc [ 1

J.

If no heat loss from the electrode to the surroundings is existent the enthalpy increase is easily obtained by dertermining the steady state tem-perature Too The enthalpy increase w:f,ll then be MCp'l'';;, where M is the mass of the electrode ,and C its specific heat.

p

However, losses due to conduction, convection and radiation are generally

unavoidable and for these reasons only an approximate value of the steady state temperature can be observed.

The purpose of this analysis is to establish conditions for which an

accurate value of the prospective steady state temperature of an electrode Can be obtained from its temperature variation with time. Only conduction of heat from an electrode to its support is taken into account. convection and radiation losses are neglected, a situation which is valid for e.g.

arcing in vacuum.

In chapter two a general approach will lead to a solution of the stated problem. In chapter three this result is discussed in relation with con-struction aspects of an experimental design which can be used for enthalpy

measurements.

The analysis of this report has been applied in the interpretation of experimental results and their accuracy as reported elsewhere [lJ,

2) Heat flow in two coaxial cylinders.

We consider two cylinders which have a coaxial arrangement as shown in

fig. 1.

~

fig: 1

The inner cylinder has a radius a, the width of the outer cylinder is 1. Symbols for the inner cylinder are denoted by subscript 1, for the outer cyHnder by subscript 2.

coincident with the common axis of the cylinders, r=O, an instantaneous

line source is situated having a strength Q and delivering a finite amount of heat equal to QPlcl joule per unit length at time t=O

[2].

At the

outer boundary r=a+l, the temperature is maintained at zero for all values of time: T2 (t, a+l) =0.

There is radial symmetry and only the radial dependency of the temperature variation is considered. Other limiting conditions are that the radius of the inner cylinder is much larger than the width of the outer cylinder: a»l and in general Al>A2 and K

1>K2• The temperature variation as a function of time at r=a: Tl (a,t) is required.

2a) Mathematical formulation of the probZem.

To start with we use Green's function of the temperature T in an infinite

o

medium at r and at time t due to an instantaneous line source at r=O:

T = o exp (-2 r 4K t) 1 (lal

For the region 0 < r < a we will now try [3] to find a solution of the form:

( lb) Here Tll must vanish for t=o and satisfy the diffusion equation, which is in cylindrical coordinates: 1 oT 1l

-+ r ar O<r<a (2)

.. /3

(Equation (la) is also a solution of (2) ).

The solution Tl must satisfy the boundary conditions:

at r=a (3)

and at r=a ( 4)

In the region a < r < a+l the temperature T2 is found by the solution of the diffusion equation:

a < r < a+l (5)

and the fulfilment of the boundary conditions (3) and (4) together with: T2 (a+l,t)=O

The initial conditions are for t=O

o

< r < a ( 7a)

a < r < a+l (7b)

2b) Lap~ace information.

The system of equations (1-7) will now be solved by the application of one-sided Laplace transformation: L { T (r,t)}= T (r,p).

p

After transformation the following set of formulas is found:

d2T dT Pll ₁ PH ₂ 0 2 + - - q T = dr r dr 1 PH

o

< r < a (8) d2T dT P2 + 1 P2 2 0 2

- - -

- q T = dr r dr 2 P2 a < r < a+l (9) Here _{ql =} 2 l? and _q2 2 l? Kl K2 (lOa,b) T _Q- Ko (ql r) Po 27TK 1 (11 ) dT dT Ai Pi _P2 - - = A dr 2 dr r=a ( 12) r=a ( 13)

o

r=a+l ( 14) (15)

Equations (8) and (9) are modified Bessel differential equations. The solutions are:

o

< r < a (16) a < r < a+l (17) Here I and o second kind

K are zero order modified Bessel functions of the first and o

respectively.

The solution (16) is bounded for r=o if the coefficient B is taken zero. Introducing (16) into (15) T can then be written as:

P1

Q

- 2 - K (q1 r)

o

< r < a ( 18)

TIK1 0

By using the boundary conditions (12-14) the coefficients A,C,D can be

determined. After inverse transformation the exact solution will be obtained.

A different approach will be pursued here by taking the condition ~«a into account. Using this condition an approximate expression is evaluated for the derivative of the boundary temperature at r=a:

dT P

( _ _ 2)

dr r=a

In combination with the boundary condition (12) the problem can then be

solved by application of the expansion theorem.

The constants C and D are determined from (13) and (14) together with (17). The derivative of T will then be:

dT P ( _ _ 2) dr r=a P2 (19)

. ./5

Here Z (q2) is given by: z KO (q2 (a+l)) Ii (q2 a ) ) + 10 (q2 (a+l)) Kl (q2 a ) K o(q2(a+l)) Io(q2 a ))- Io(q2(a+l)) Ko(q2 a ) (20) The functions K

o(q2(a+l)) and Io(q2(a+l») will be written as series. Using Taylor's formula for a single variable the expansion around r=a of these

functions can be set down as:

and

a

Here I' = ( 0

a

and K' 0 = ( 10 (q2 r )

a

(q2r) ) r=a K (q2r) o )

a

(q2r) r=a 2 (q21 ) (q a) + 10" (q2 a ) .j. •••• 2 · 2! etc.

Using these power series and by application of the Bessel recurrence formulas:

s K' (s) + v K (s) = -s K 1 (s)

v v v- v=O,lt2 •••• ~,

the fraction Z can be written as

2 3 1 (q2 1 ) (q2 1 ) - - + 2!q2 a _3! (q2 a ) 2 +

....

q2 a Z = 1 12 13 - - +

- - -

+

....

a 2!a 2 3!a 3

The derivative of the temperature T at the boundary r = a is now given by

P2 d T P2 dr ) _{r = a} - T (a) (1 + Pi 1

-.

. ..

) (21)

2

By introduction of q2 =

expression is

=

a

K2 (lOb) and after inverse transformation, the

+ ••••

(22)

...

}

The derivative of the temperature T2 at r = a is noW given as a series of Tl (al

and its derivatives.

In case the fraction of

~

is of the order of 10-1 the third term in the

denominator has a value a of 1,67.10 -3 . The error introduced by taking only the first two terms of the alternatering series in the denominator will be less than this value. This is acceptable for our purpose.

The expression (22) will be further approximated by neglecting the term

aT I

(---) and the higher derivatives. The influence on the accuraccy of the

at

r

=

a

results by neglecting these terms in the numerator of (22) will be analysed further on (see appendix).

The approximate form of (22) is after transformation:

dT p

(-2)

dr r = a = 1 l(I-!....} 2a

The boundary condition (12) can then be written as

dT 1.2 PI _{I T} d r = Al 1 PI (a) at r = a 1(1

-

- } 2a dT PI

_{_ r.}

_T (a) or _{( d r )} PI r = a a where A2 _I Y = Al 1 _{( I}

_-

1 2a a (23) (24) (25)

.. /7

Together with (16) the temperature T at the boundary r = a is found to be

Pl

1

(26)

2c)

Inverse transformation.

The inverse transformation of (26) can be performed by the application of the expansion theorem (cf. e.g. [5J).

Here the formal transformation can be given by

Q '"

1 T (t,a)

_Q- _~ = - - E

T (a) = _{27rK 1 n=l}

p 27rKl N(p)

where P are the roots of the denominator N(p) •

n P t n e Cd N(p» dp p = P n

To this purpose we will first establish the roots of the denominator of (26). The functions Io and Il are positive for real values of qla and as for

qla = 0 , Io(O) = 1 and Il (0) = 0, no real roots are existant.

Using now Im(X) as

-m

= j J (jx)

m m

=

0,1,2 .. , the denominator N(p) can be written

If the imaginary roots of N(p) = 0 are ~n we can write y Jo(~) - ~ Jl(~) = 0 and _~n

Introducing (cf lOa) qln

=\/::'

we find

where ~n are the roots of y Jo(~) - ~ J

1 (~)

=

0

The first six roots of this equation for varying values of yare tabulated

[ 4J.

The derivative of N(p) can be found by straightforward differentiation and the use of the Bessel recurrence relations. The result is:

d N

(~

p n 2 = -::---=a,-::- [y J 1 (~ ) + ~ J (~ )] 2K 1 ~n n n o n

Inserting these results in the formula of the expansion theorem we finally have as a general solution of (26)

T1 (t,a) =

..L

2 TTa Here T

=

n 2 a n

=

and ~ are the roots of

n tiT - n + ~ J (~ ) n o n (27) (28) (29)

The instantaneous line source has liberated an amount of heat equal to

Q P1c1 joule per meter. If a steady state temperature T~ is defined which is achieved by the inner cylinder if no heat conduction occurs at r

=

a, i.e.

a

T1 2

(ar)r=a == 0 we can write QPlcl = 1I'a PlclTCX)

and (27) will then be

tIT ~n e - n 00 T 1(t,a) - T 00 E _yJ1(~n) + ~ _J (~ ₎ n = 1 n o n

AS is shown by formula (24) the condition y = O. In case y = 0 ~1 = 0 and ~n(n>l) Thus T1 + 00 and (30) can be written as

T [1 + E 00 n = 2 tIT _n e -=-J --:('""~ """')- ] o n

a

T1 ( - - )

ar

r >

o.

(30) = a = 0 requires a value

For large values of time the second and higher terms of the series diminish and the temperature will indeed reach its steady state value T

00

The expression (30) is a series of rapidly converging exponential terms. The temperature dependence with time at the boundary r = a has the general appearance as shown in fig. 2.

t

_..

fig: 2

From an analysis of (28) and (29) i t appears that generally TI > T2 > T3 > •• ,

The first time constant TI is strongly dependant on y and is mainly related with the tail of the temperature curve if AI > A2 and KI > K

2. The time constants T

2,T3 etc. are largely independant of y. Due to their increasingly smaller

values their influence on the temperature curve is predominant in its front part and the contribution of the higher terms in the series will become negligible after a short time. If y is small the first time constant TI is large and mainly related with the heat flow over the boundary from the inner cylinder to the outer cylinder. The other time constants will then reflect the

characteristic relaxation times of temperature variation in the inner cylinder.

3. Heat flow in an electrode system.

The solution (30) as found in the previous chapter will now be used for the analysis of the temperature variation in an electrode geometry which is heated by arcing. In the geometry as shown in fig. I the inner cylinder depicts the electrode (cathode or anode), while the outer cylinder is the support necessary for reasons of mechanical strength and current conduction.

The heat supplied to an electrode by an arc is thought to be delivered in a small surface area and at the centre of the system. This will be particularly true in case of a vacuum arc where the dimensions of the region of heat supply

at e.g. the cathode are less than one mm2• The arc duration is generally much smaller than the thermal time constants involved and in a number of cases the power input by the arc can therefore suitably be described in terms of an

instantaneous point source situated at the electrode surface. For the particular

situation here the thickness of the electrode is much less than its diameter

(in the order of 5%), which means that the heat flow except for a small volume around the arc root will be in radial direction. The problem can therefore be analysed by using an instantaneous line source instead of a point source. Comparison of the time durations involved in heat propagation in radial and axial directions shows that this approach is justified.

The objective of a number of experiments performed with different electrode metals [1] was to measure the enthalpy increase by arcing and for this reason the temperature of an electrode was measured by placing thermocouples near its boundary. As (30) shows the temperature variation of the electrode is related with the steady state temperature T as previously defined. From an

00

experimental point of view the maximum temperature the boundary attains after arcing is a convenient quantity to measure for two reasons. Firstly if conditions are such that the difference between steady state temperature and the maximum attainable temperature is sufficiently small (cf. fig. 2) we can use the latter value as our experimental result. In addition the measurement of the temperature at its maximum value is advantageous with respect to experimental accuraccy as the fractional error in the result will then be a minimum.

The maximum temperature

(30) and, if we use the

Tea) _max where t = T 00

=

max "(1"(2 "(1 - "( 2 (The value of Jo(~2)

In

is

T (a)

max at the boundary is found by differentiation of

first two terms only, the result is

~2 expe (31) ---'---~ --2 2 ~ Y + ~1 Jo(~I)

[

_ (.2) 4 _{~~ Jo(~2)}

]

(32) ~1 2 Y + negative for y

"

0) •

In fig. 3 the variation of T (a) _T . is shown as a function of y. If the max 0:>

maximum temperature is taken as a measure for the steady state value it is clear that a low value of y is required for an accurate result. Low values of

y mean a high ratio of the first two time constants as fig. 4 shows. Even for a large difference as "(1

=

100 "(2 the error made by taking Tmax as the steady state value is still around 6,5%. If

Y should be of the order of 10-2 and

~_accuracy

_1 > 700.

"(2

is required' of around 1%,

A2

The quantity y is proportioned to

r-

(cf. 25) and a large ratio of thermal

1

conductivities is advantageous. StaLnless steel has a low thermal conductivity

-1 -1

(A = 14,7 Watt m K ) in comparison with other metals (see table 1) and it is therefore a proper metal for the construction of the support.

However for the investigation of a range of metals as electrode material having a wide range in A (see table 1) a sufficient low value of y will not always be ensured by the construction as given in fig. 1.

A construction which is better suited for this purpose is shown in fig. 5. The support is now consisting of three rectangular strips of length 1 and

fig: 5

aT 2 ( - )

ar r

from the inner cylinder over a the

crossection A. This construction limits the heat flow from the electrode disk and lowers the value of y. The derivation made in the previous chapter is also applicable here if certain conditions are met. The boundary condition (3) will now take a form which follows from the derivation given here.

The heat flow Q

1 at the boundary of the

inner cylinder is given by:

aT 1

Q - -21TadA ( - - ) (Watt).· Here

1 - 1 ar r = a

d is the thickness of the inner cylinder. The heat flow through the strips at r

₌

a (Watt). We now take Q

1

=

Q2 or the heat flow boundary r

=

a (the situation of fig. 1) _is equal to the heat flow along the strips at r

=

a. This will be true provided Tl » T2 which means that the relaxation time(s) for temperature equalization in the electrode is much less than the relaxation time of heat loss along the strips. The same condition is required if T

%

T •

max c;o

The boundary condition (3) will now be

= (33)

After transformation and introducing (23) we have dT I? 1 ( _ _ 1) ₌

_L

T . Pi (a) dr r

=

a a (34) 1 A2 3A 1 Here y = Ai 27Td _{1(1 - - )} 1 2a (35) •• /12

12

r--

J,:'ax

(0J0) 10 B 6 4

2

fig: 3

p.L

_1:2 10 2 1 10 -3 10 fig: 4

l

-'"

['\

i-'"

I--'V

-2

10

"'-1

/

II

1/

L

lL

Y

-~

1"-..

_r-..

l'..

"'-~

Y

-.. /13

(Compare (34) and (35) with (24) and (25». other conditions remain the same.

If the enthalpy increase by arcing is measured by a construction as given in fig. (5), the following values are feasible:

A = 10-6 m .2 d = 2.10- 3 m

1 = 2.10- 3 m

a = 18.10-3 m

1

If we use these values y can be calculated for different metals and the result is shown in table 2, second column. From (28). together with (29) the numerical values of the first three time constants are calculated and are also

T T(a)

' 0 0 _ max

shown in table 2. From (32) and (31) t and the fraction T

max _~

are obtained and found in the same table.

The first time constant T1 has a value of hundred seconds and more and differs considerably from the second time constant which is around one second or less.

As has already been observed the higher time constants are nearly independant

1 T2

of y and the fraction of -- is therefore.about constant

T3 ~2 3

~=

2 3,35 •

For the range of values as given in table 2 the contribution of the third term governed by T3 is negligible small and has therefore been neglected in the calculation of T (31). If therefore a small region around t = 0 is

max

excluded the temperature variation with time can be described as a process governed by the time-constants Tl and T

2•

From table 2 it appears that in most cases the maximum temperature is identical within some two percent with the steady state temperature. The accuracy of this model is sufficient as a basis for the interpretation of experimental results. If necessary a higher accuracy can be obtained by an additional determination of the time-constants T 1 and T 2 which

the front and tail slopes of the temperature curve.

1

experimental value of y can be calculated and thus

are readily obtained from From their r~io the

max the fraction ~ •

. ~

metals like Pb where a difference of 5% is applied.

observed, this procedure can For be

Table ·2. _{T - T}

~ max

T

Tl T2 T3 t ~

Metal y 1 max

sec sec sec sec

,

Ag 4,34 10-3 2,98 102 0,130 0,039 1,12 0,5

eu

4,65 10-3 4,36 102 0,190 0,057 1,65 0,5 Al 7,82 10 -3 2,31 102 0,226 0,068 1,77 0,8 Mo 1,33 10-2 2,05 102 0,356 0,106 2,59 1,4 Zn 1,61 10-2 2,56 102 0,533 0,159 3,79 1,7 Cd 1,99 10-2 1,68 102 0,457 0,136 3,13 2,0 Ni 2,07 10-2 3,52 102 0,957 0,286 6,55 2,0 Sn 2,89 10-2 1,46 102 0,559 0,167 3,64 2,7 Pb 5,37 10-2 1,31 102 0,940 0,280 5,54 4,7 •• /15

Conclusions

An analysis has shown that the enthalpy increase of an electrode which is heated by arcing can be determined with a high degree of accuracy. If a

proper design is chosen the observation of the maximum value of the temperature at the boundary of an electrode is sufficient to establish this quantity. The accuracy of this measurement increases with the increase in the ratio of the two dominant time constants by which the process of heat flow is governed in the electrode and its support system.

Appendix.

In the second chapter an approximate expression has been found for the derivative of the temperature T2 at the boundary r = a. In the

only the first term of (22) was used. Here we will analyse the

calculations

magni tude of the error made by neglecting the higher terms in relation with the solution found in the previous chapter.

The expression (22) can be written as

) r = a 1 - - - - = 1 - - [T 1 (t.,a) + 1 (1 2a) Here

s

=

Introducing (33) we can write

1

r

=

a - : [T1 (t,a) + (36)

The heat flow over the boundary r = a is given by Q

1 =

aT 1 ( - )

3r. r = a

The amount of heat which passes over the boundary at r

=

0 from t

=

0 until t

=

t i.e. the time the maximum temperature is reached, is:

max 3T 1 (-, - ) dt. or r = a

o

o

Formula (36) shows that the second and higher terms contribute to a higher

T1

value of (~) This means an increased heat flow over the boundary in

or r

=

a

comparison with the situation where only the first term of (36) is taken into

account. The contribution of the higher terms therefore means a decrease of the max. attainable value of the temperature T(a)max·

The additional heat flow introduced by the second term of (36) is

dt = 21TdA1 Y 1 S T (a)

max

If we compare this contribution of the second term in (36) with the total amount of energy increase of the electrode when no heat conduction over the

boundary occurs; i.e. an amount:

r

o

f

t max 1 llQldt

o

2K₁Y ST(a)

_______________ =

__

~m~a=x~ 2 T a ~

This fraction agrees with the fractional decrease of the previously calculated T(a) if the second term in (36) is taken into account. Introducing the

max

appropriate values i t appears that this decrease is of the order bf some promille for the range of materials as represented by table 2. Calculations show that the contribution of the third term is entirely negligible.

In (22) the series in the denominator has been cut off after the second term and the related error is for the numerical values used around 2 promille. The errors resulting from neglecting the second term in the numerator of (22) and the third term in the denominator are counteracting each other and the total result is an error of around one promille.

The conclusion therefore is that the approximations used in (22) are justified and does not influence the outcome of chapter 3.

Literature

[1] Daalder, J.E. to be published J. Phys. D: Appl. Phys. (1977).

[2] Carslaw, H.S.; Jaeger, J.C.

Conduction of heat in solids Oxford U.K., (1973) p.256.

[3] ib. p. 358

[4] ib. p. 493

[5] Fodor, G.

Laplace transforms in Engineering Akademiai Kiado Hungary (1965).

Table. 1.

Survey of thermodynamical parameters for a number of metals

A p c K

metal p

Watt m K -1 -1 Kgm -3 Joule kg K -1 -1 m sec 2 -1

Pb 3,46 101 11 ,34 103 1,30 f02 2,35 10-5 Cd 9,3 101 8,37 103 2,30 102 4,82 10-5 Sn 6,4 101 7,18 103 2,26 102 3,94 10-5 Zn 1,15 102 7,14 103 3,90 102 3,13 10-5 Al 2,37 102 2,70 103 9,01 102 9,74 10-5 Ag 4,27 102 10,5 103 2,39 102 1,70 10-4 Cu 3,98 102 8,89 103 3,85 102 1,16 10-4 Ni 8,99 101 8,8 103 4,44 102 2,30 10-5 Mo 1,40 102 9,01 103 2,51 102 6,18 10-5 Stainless 1,47 101 7,9 103 5,03 102 3,7 10-6 steel •• /20