Development of a unified heat transfer model with special emphasis on thermal conduction

Int. Comtm HeatMass Transfer, Vol. 26, No. 5, pp. 657-667, 1999

Copyright© 1999ElsevierScienceLtd Printed in the USA. All rights reserved 0735-1933/99/S-see front matter

Pergamon

PII S 0 7 3 5 - 1 9 3 3 ( 9 9 ) 0 0 0 5 2 - 4

DEVELOPMENT OF A UNIFIED HEAT TRANSFER MODEL W I T H SPECIAL EMPHASIS ON T H E R M A L CONDUCTION

B.I. Kilkis Department of Mechanical Engineering Middle East Technical University Ankara, Turkey A.S.R. Suntur Stileymaniye Association Istanbul, Turkey

(Communicated by E. Hahne and K. Spindler) ABSTRACT It is postulated that energy fields in space and matter are quantized by an entity having coincident properties of wave and virtual matter. A unified link between thermal radiation and conduction was established as confirmed by Planck's quantum theory, and a unified heat transfer equation was developed. One-dimensional, steady-state thermal conduction was solved by using approximate scatter and mobility models in this equation. Coefficient of thermal conductivity of various materials were calculated and compared with the literature. Even with approximate scatter and mobility models, a good agreement was observed. © 1999 Elsevier Science Ltd

Theory Any seamless link between thermal radiation and conduction has not been recognized yet. For example, Klemens-Callaway theory for thermal conduction with lattice vibrations, phonon scattering, and dispersion is completely different from Planck's thermal radiation theory [1]. Quantum mechanics of thermal conduction in conjunction with other energy fields might be better understood by de Broglie analogy which states that if, light exhibits particle aspects then, particles of matter should exhibit coincident wave properties [2]. In fact, all descriptions of forces explain the presence of the force by the exchange of special particles between objects, and it is quite possible that there is a unique energy transfer and quantization entity which is common for all types of energy fields. In such a unified framework, it may be conceptualized that thermal conduction is quantized similar to thermal radiation. According to Unified Thermal Potential Theory, maximum transferable energy ofa quantization entity in a given host medium is simultaneous and identical functions of its particle (kinetic), wave, and thermal potential energies, as shown 657

658

B.I. Kilkis and A.S.R. Suntur

Vol. 26, No. 5

in Eq. (l-a) [3,4]. This equation applies to any energy field, in terms of equivalent thermal potential Qk and equivalent temperature P [3]. Along an isotherm of a thermal energy field, P is constant and equal to T. For energy transfer in space such as thermal radiation, vk is equal to c. This is called Photonie state.

(l-a)

ek ........ - l/2mkvk 2 =--Qk = mkCpP =- h f k = hve/';tk,,ax

If, energy transfer is in dense matter such as thermal conduction; vk << c, m k ~ mko, and the quantization entity virtually acquires material properties of the host medium. This is called Atonic state, and from Eq. (1-a) with Cp ~ 2 zcB 107/A, following equations are derived.

m k : 2h/(vk.,~kmax )

(l-b)

. ~ = ekmaffh = ll2mkoVk2/h =_ 3.62" 107p/A

(l-c)

2kmax = v d f k ~-- 6.34' IO'4(A/P) v2

(l-d)

A,tm~P = 2.764" 10-8VF1

(l-e)

ekm~~_= 4.554" 10-ZSVk2~ 2.4" lO'19p/A

(l-f)

From these Atonic state equations if, Eqs. (1-a)-(1-b) are simultaneously solved for Photonic state with the condition vg = ¢, Planck's maximum energy equation for thermal radiation is identically obtained:

ekm,~x = l/2mkc 2 = 1/2(4.42" l O'37 /.~kmax)C 2 ~ 1.99. l O-16/.~kmax .

(2)

This identity validates Eq. (l-a) and indicates a seamless link between Atonic and Photonie states. Since, Eq. (l-a) is valid for both states, Eqs. (l-b) to (l-f) must be also valid for Photonic state if, space has a matter property. In fact, when a virtual atomic weight o f A ' = 3.5.10 "4 is assigned to space, Eqs. (1-e)-(1-f) identically become Wien-Planck's thermal radiation equations, as shown in Eqs. (3)-(4) [3]. These results extend the Standard Model of the universe, and indicate that space is not a perfect vacuum which has already been evidenced by the latest findings that neutrinos have mass [5]. With A = A', space is a continuation of matter, and quantization in space and matter are similar and continuous. When Eq. (l-f) is applied to a unit volume, energy density E for all energy fields is given by Eqs. (5) and (8). 2k,,,,.rP = 2.764-108cA ' = 2.764.10 -8 2 9 9 7 9'101°.3.5.10-4-- 0.29 --- AkmaxT

{P=- T, vk=c}

(3)

ek,,,,x = 2.4-IOqgP/A = 2.4" 10"19p/3.5" 10.4 = 6.86" 10"16T ~ h]k

(4)

E = l/2ek,,axY k = 4.554" 10-28Vk2~ 1.2" IOq9ykP/A

(5)

q = 1/4Evg

{in a unit volume}

(6)

Vol. 26, No. 5

UNIFIED HEAT TRANSFER MODEL

659

Yk = bMST3

(7)

E ~ 1.2. IO-19bMST3p/A

(8)

Principle of Unification of Heat Transfer All modes of heat transfer are unified by the same quantization entity and governed by the same energy transfer equation namely, Eq. (8). Transition from one mode like radiation to another mode like conduction takes place by a change of state and thermal potential. For example, with b = 22. l(cm3.K3) -1, M = 1, S = 1, A = A' in thermal radiation, and P --- T, Eq. (8) identically becomes Stefan-Boltzmann law: E = 1.2"1019(22.1)T3T/3.5"10 4 = 7.57"1015T4 ---aT 4.

{thermal radiation}

(9-a)

From the similarity between Eqs. (l-e) and (3), E~ in Atonic state may be written as shown in Eq. (9-b). With vk = c and A = A', this equation reduces to Planck's equation for Ea.

& = 8~vflczA/&5(e -vkZA/('~kP)- l)

(9-b)

Derivation of One-Dimensional, Steady-State Heat Conduction Equation Any two isothermal and parallel planes having different thermal potentials Qkl and Qk2 in a finite volume of matter is shown in Fig. 1. Net thermal potential which drives a heat transfer and quantization entity from one plane to another is mk(Cp~T ~ - Cp2T2). If, the change o f Cp over the given temperature range is negligible then, P is T 1 - T2 or AT. In thermal convection, P is AT".

ekmax = 1/2mkvk2 = m k C p A T = h f k = hvk/2~ana x

{ A T = T 1 - 7"2, I"1 > 7"2}

(10)

vk = (2CpAT) 1/2 ~ I03(40~,BAT/A) 1/2

{vk<< c, AT> 0}

(11)

Since Eq. (11) does not cover the rate of deceleration gt, following simplistic equations for metals were developed. Deceleration factor Jtc is close to one if, A and Pe are small [3]. g,

oc A T / A [ . 2

Jtc = v j ( 2 C p A T ) t/2 = (I + A/5OO)q(l -p~. 106/273) ~

{AL > 0}

(12)

{Pe'106 _<200 ohm-cm}

(13)

i~/=eA/360-1

(14)

vl~ Z Jtc(2CpAT)I/2

(15)

660

B.I. Kilkis and A.S.R, Suntur

Vol. 26, No. 5

AL>0

d F k = mt~dvk/dt = mkgt

{mk ~_ mko }

dt =- dL/vk ~dFt3tL = ek = ~mkvkdvk = 1/2mkVk 2/Jtc z T2

T finite volume of matter (plane wall) A , p , 6p

plane 1

= Qk

plane 2

Tl > r z

Qkl > Ok2 --~L

rl

T1 P = A T = T1- T2

T~

Qk ~ .,kcpar

0K I Qkl FIG. 1 One-dimensional, steady-state thermal conduction model.

Intensity Ykand flux Ykvk/4 of the heat transfer and quantization entity are obtained from Eq. (7), and the numerical value ofb in thermal conduction is given in Eq. (16). b = 5" lO~6rp/A

(16)

Y!~= 5" I O16rpMST3 /A

(17)

Yk ~ 4 7r . 1023rlfft4S/A

{@ 293.15 K}

08)

-2 7rpoP/T

(19)

S=l-e

Mobility factor M depends on factors like lattice structure, thermal, magnetic, and electrical properties. For metals, an approximate expression for the numerical value of M w a s determined [3]: M _~4 ~ . 10-VA/LoSAT).

{AT > 0 K, Pc 106 < 200 ohm.cm}

(20)

Vol. 26, No. 5

UNIFIED HEAT TRANSFER MODEL

661

Heat flux q is obtained from Eqs. (l-f) and (6). It can be simplified by replacing vk, Yk at 293.15 K, and M with Eqs. (15), (18), and (20). Heat flow is towards the lower thermal potential. Factor (1/AL) represents the difference between Ye in a material volume of lxlxAL and a unit volume.

q = 1/4(4.554.10-28vk2/2)Ykvk(1/AL) q = 7.99"lO-l°rpCp3/2Jtc3/pe (AT/AL)

(21 ) {@ 293.15 K, S=- 1}

(22)

In Eqs. (21)-(22), coefficient of thermal conductivity is not an explicit parameter, as also claimed earlier [11]. However, Eq. (22) may be written in vector form to obtain the Fourier's law. (23)

q = -k t ( A T / A L ) k t = 7.99"lO-l°rpCp3/2Jtc3/pe

~ 3409rpJtc3/(peA3/2)

{@ 293.15 K, S ~ 1}

(24) (25)

ktp e = kt/k e -~ 3409rp/A3/2J~c3

Consider an Aluminum plate at an average temperature of 293.15 K AT is 20 K across the plate thickness of 2 cm. A = 26.98 g/g-atom, p = 2.7 g/cm 3, Cp= 0.9.107 erg/(g'K), Pe = 2'65'10"6 ohm'era, r = 1, J~c = 1. Using the new model; vk = 18,974 crrds, %,ox= 1.64-10 q9 erg, fk = 2.47"107 s -~, 2tkmax = 7.68"10 "4 cm, b = 5-1015 (em3'K3) -l, S = 1, M_- 8.99, Yk = 1.13" 1024 cm "3. From Eq. (22);

q = 2.2.108 erg/(s.cm z) = 2.2" 105 W/m 2, k s = 2.2" 107 erg/(s.cm.K) = 220 W/(m'K). The result is in good agreement with the literature i.e. for Aluminum 1100, k t is 221 W/(m'K). Table 1 shows k t of various metals as computed from Eq. (24), and compares them with the literature. Analytical prediction of electrical properties of metal alloys is very difficult. If, high-order statistical information are available, rigorous bounds can be obtained. Gaussian random field model also seems to be promising [12]. As shown in Eq. (26), an approximate solution which must be used with caution, was developed for alloys with two constituents [3]. Table 2 shows sample calculations for typical alloys where experimental values ofpe* were used in Eq. (24). An approximate correlation for r in dense matter is given in Eq. (29). For example, since A / p is 7.1 for Iron, r = 1.06 --~ 1. For metal alloys, ,4 * is used. In all tables, r is bracketed if, Eq. (29) does not agree with calculations.

joe* -~ Pebase( 1 + (tgealloy " Pebase)/Pebasey)l

{X_> 0.2, Pealloy > Pebase}

(26)

] = 1 + (0.25 - ( X - 0.5)2)'(£/3(,OealZoy/,Oebase) + lr)

(27)

A * = Abase(1 - X ) + AolloyX

(28)

r ~ In~A/(66p)[

(29)

662

B.I. Kilkis and A.S.R. Suntur

Vol. 26, No. 5

I

~

0

0

..

~,.o

0

~-

i

,~':~

~i ~~ ~

N

~

0

~

0

~

0

0

~

0

0

0

~

0

0

~

~

0

0

0

0

0

0 I

!

~

G

E

r,.)

• ,.-

0

r.~

m

N

Vol. 26, No. 5

UNIFIED HEAT TRANSFER MODEL

663

TABLE 2 Sample Calculations of Thermal Conductivity of Metal Alloys and Comparison with the Literature [9,10] Alloy

A*

Cp

P

g/g-atom erg/(g-K) g/cm 3

Pe* [9]

joe* Eq. (26) ohrn-cm

Brass 64.1 0.38x107 8.50 5.9-6.2x10 -6 6.29x10 -6 (70% Cu, 30% Zn) Bronze 77.3 0.34 8.55 15.2 20.30 (75% Cu, 25% Sn) Copper-Ni 61.6 0.42 8.92 15.7-37.5 8.30 (69.5% Cu, 29.5% Ni, 0.5% Fe)

kt(E q. (24) erg/(s'cm'K) @ 293.15K

r

k,

1.04-1.47x107 0.90x107

1

0.26-0.29

0.29

1

0.22-0.26

0.29-0.45

1

Although Pe may be as high as 1017 ohm.cm in non-metals, their effect on thermal conductivity are small above 200.10 -6 ohm.cm, and latent heat of fusion L H becomes a dominant factor. A simplistic expression for the numerical value of M w a s derived in terms o f LH. Table 3 shows sample calculations. M _~ 16.10-5w(A/AT) 3/2

(30)

w = B/xlA + [LH. 10-11(1.2 + ~/A/49)]3

(31)

k t = ,]A. 10-7rDCp3/2WJtc3/~

(32)

Discussion of Results and Conclusions

As far as this work is concerned, an acceptable agreement have already been obtained as shown in Figure 2 although, effects of lattice structure, magnetism, and other parameters were ignored. This model shows that thermal conductivity depends primarily upon atomic weight, density, and mobility which is related to electrical resistivity in metals. This conclusion is supported by empirical equations like Eq. (33). k t = 2.61' 106T/Pe - 8.37" 10"S(I'/,Oe)Z/(C~p)+ 2.32" 10-4/92C/(A D

(33)

From Eq. (33), k t of Aluminum at 293.15 K is 248 W/(m.K) versus 220 W/(m'K) from the new model, and 22 lW/(m'K) from the literature. Similar comparisons indicate that thermal conductivity may be predicted better by the new model, over a wider range. With this model, an alternative and unified insight to heat transfer was obtained, and thermal conduction was derived as a special case of thermal radiation. It is likely that, this model which identifies an Atonic state equivalent thermal potential for energy fields may explain from a new perspective the fundamental relationship between heat transfer and other energy fields like in thermoacoustics, thermomagnetics and thermoelectrics. Provided that scatter and mobility models are improved, coefficient of thermal conductivity of new a!loys or elements whose

664

B.I. Kilkis and A.S.R. Suntur

Vol. 26, No. 5

.o

~

o

oNo=-a

~

v

~

-

~

_

~

0

6

o

o

.o

..=

Z

Vol. 26, No. 5

UNIFIED HEAT TRANSFER MODEL

665

5xloT kt erg/(s.cm-K)

/ / ~

4 ~

tAu

5-

~ect Agreement

"! 2UJ

B,o+, I

1-

/

p , ~ : K

i0-|,o-~-

io-~,

p,,s '"'""1

iO'3

-I

k t erg/(s.em.K) ........ I ........ I

I0-a 10-I

I

'

'

'

I

2

'

'

'

I

3

'

'

'

I

4

'

'

'

5xld

Eqs. (24)-(32) FIG. 2 Comparison of calculations with experimental results from the literature [6,7,8,9,10].

thermal properties are not available yet may also be predicted whereas, new techniques may be developed for augmenting the heat transfer by using the effect of other energy fields. Equivalence of thermal potential among different energy fields may lead to analytical solutions for hybrid devices such as acoustic coolers. For example, with P equal to T/5 in sound propagation, the ideal thermoacoustic cycle for cooling a material may be defined from the seamless link between sound propagation and thermal conduction in that material [3]: (mkCpT/5)air =- (mkCpAT)materia 1. Then, AT < (Cpa,r/Cpmater,al)T/5. In air at room temperature, maximum AT for a brass fin (Cp = 0.38-107 erg/g'K) will be 154°C. In practice, due to scatter and other losses, AT will be lower, which is the case with AT,,o~ = 118°C in the literature [13].

666

B.I. Kilkis and A.S.R. Suntur

Vol. 26, No. 5

Acknowledgments This work was supported by Heatway Radiant Floors and Snow Melting, Springfield MO, USA. Authors are very thankful to Mr. Mike Chiles and Mr. Dan Chiles, who continuously encouraged and stood with them. Special thanks also go to members of Sildeymaniye Association.

Nomenclature A a B b c (~ E E~

atomic weight of the host medium, g/g-atom radiation constant: 7.565 7" 1015 erg/(cm3'K4) 4.186 8 proportionality factor for the intensity of energy transfer and quantization entity, (cm3K3) 1 velocity of light in space: 2.997 924 58.101° cnVs specific heat, erg/(g-K) energy density, erg/cm3 spectral distribution of energy density, erg/(cm3.cm)

ekmaxmaximum transferable energy, erg ]k

frequency, s -1

F

force, dyne

g~ h

Jtc

rate of deceleration, cm/s 2 Planck's constant: 6.626 075 5"10-27erg.s deceleration factor: Vk/(2CpAT)l/2, dimensionless

k kt k

Boltzmann's constant: 1.380 658'10 "16 erg/K coefficient of thermal conductivity, erg/(s'cm'K), W/(m'K) coefficient of electrical conductivity, (ohm-cm) "l

L

distance from an isothermal plane having higher equivalent thermal potential, cm

LH latent heat of fusion, erg/mole M mobility factor, dimensionless mk virtual mass of the energy transfer and quantization entity, g

rnko virtual rest mass: 9.109 389 7" 10-z8 g P equivalent thermal potential temperature, K QK equivalent thermal potential with respect to the cold point (0 K), erg q energy flux, erg/(s'cm2), W/m 2 r quantization number (integer), dimensionless S scatter factor, dimensionless T absolute temperature, K t time, s vk velocity of energy transfer and quantization entity, cm/s X alloying material content, dimensionless Yk intensity of energy transfer and quantization entity, cm -3 Z 2.764-10-8.4.965 11:1.37"10-7 K's e 2.718 281 8.. 2 k wavelength, cm

2kmaxwavelength for maximum energy, cm 7r p

Po

3.141 592 654.. density at average material temperature, g/cm 3 relative density: p/(1 g/cm3), dimensionless

Vol. 26, No. 5

p~ AL AT

UNIFIED HEAT TRANSFER MODEL

667

electrical resistivity, ohm'cm finite travel distance of the energy transfer and quantization entity (Kunnes), cm temperature difference: T1 - T2, K

Subscripts ~/oy alloying element base base metal in an alloy max maximum o relative variable, or rest state Superscripts n power for equivalent thermal potential temperature in convection (n is unity in thermal conduction) ' virtual material property * effective References

1, P.G. Klemens, Three-phonon Umklapp: A Two Step Process at Low Temperatures, in D. W. Yarbrough (ed.), Thermal Conductivi{y, 19, 39, Plenum Press, New York (1988). 2. L.V. de Broglie, Ondes et Quanta. C. Rendus de l 'Academies des Sciences, 177, 507, Paris (1923). 3. B.1. Kilkis and A.S.R. Suntur, Thermal Conduction, How is it Quantized ?, Proc. 2nd Trabzon lnt. Energy and Environment Symposium, Trabzon, pp. 53-58, Begell House, New York (1999). 4. B.I. Kilkis and A.S.R. Suntur, Application of the Unified Heat Transfer Equation to Thermal Conduction at Low Temperatures, Proc. Advances in Refrigeration Systems, Food Technologies and the Cold Chain, Sofia, in print (1999). 5. Y. Fukuda, T. Hayakava, and et al, Evidence for Oscillation of Atmospheric Neutrinos, SuperKamiokande Collaboration, Physics Review Letters, under review (1998). 6. L.S. Marks and T. Baumeister III (eds.), Mark's Standard Handbook for Mechanical Engineers, 9th ed., pp. 6.60-6.61, McGraw-Hill, New York, (1986). 7. DD. Hsu, Chemicool Periodic Table, http://www-tech.mit.edu/Chemicool/(1997). 8. M. Winter, Periodic Table, http://www.shef.ac.uk/chemweb-elements/(1997). 9. Materials Engineering 1989, Materials Selection, Penton Pub. Inc., Cleveland (1990). 10. S. Kakac and Y. Yener, Heat Conduction, 3rd ed., pp. 343-345, Taylor & Francis, Washington, D.C. (1993). 11. E.F. Auditori, The New Heat Transfer, 2nd ed., Ventuno Press, Ohio (1989). 12. A.P. Roberts and M.A. Knackstedt, Structure Property Correlation in Model Composite Materials, Physical Review E, 54, 2313 (•996). 13. L.S. Garrett, Development of a CFC-free Refrigerator to Help Reduce Depletion of the Ozone Layer, http ://www.rolexawards.eonffproject/projlau/d-garr-u.html (1993). Received January 15, 1999