Transfert Par Radiation Dans Un PFR

7/29/2019 Transfert Par Radiation Dans Un PFR

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Joint Meeting of The Scandinavian-Nordic and Italian Sections of The Combustion Institute

9.1.1

Validation of a Flux Method for Radiative Heat Transfer

in Plug Flow Reactors

M. Filla

Dipartimento di Processi Chimici dellIngegneria - Universit di Padova (Italy)

INTRODUCTION

Flux methods are conceptually approximate methods for the calculation of the rate of radiative heattransfer with remarkable computational advantages with respect to conceptually exactmethods likeHottels zone method and the MonteCarlo method. The attention has been focused here on the

quantitativeaspects of the flux method approximation in order to distinguish cases whereRoeslers one-dimensional flux method can be applied safely from those where caution is required.

ANALYSIS

The steady state temperature profile of a medium along a PFR is governed by the energy balance:

0]qdx

dTk)HVYTc(u[

dx

dRefffp =++ (1)

where the contributions of sensible and chemical enthalpy, effective diffusion and radiation appear

in this order, to be solved together with the auxiliary equations for the mass velocity u = u(x),

the equivalent unburned fuel mass fraction Yf=Yf(x), the effective thermal conductivity keff,and the absorption coefficient of the reacting medium ka. The temperature profile of the bounding

walls where the heat flux is assigned, and the heat flux profile where the temperature is specifiedare then also determined.

With reference to the elementary volume located by point P along the duct of fig. 1, where a beam

of incident radiation of intensity Ii(P, ) is visualized together with the relevant elementary solid angled, the radiative contribution to the energy balance eq. (1) is

]),()(4[/4

4 = dxIxTkdxdq iaR (2)

The intensity of the incident radiation Ii (x, ) being governed by the radiative transport equation,

the radiative contribution confers non-linear integro-differential form to the energy balance eq. (1).

In the zone method approach [1] the volume and the bounding walls are subdivided in N v and Ns zones,

and the integral contribution of absorption in eq. (2) is obtained by the sum of the contributions of eachzone, so that with reference to the volume zone Vn

na

4

pnpp

N

1

4

jnJj

N

14

i Vk/]TGGTGS[d),x(Ivs

+= (3)

where the blackbody emissive power of each zone is weighted by the relevant total exchange area [1].

The radiative contribution expressed by eq. (3) is conceptually exact. However, this is a form that

adds a considerable computational burden to the solution of the set of algebraic equations generated


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9.1.2

by the discretization of eq. (1), because all elements of the matrix of the coefficients, which is

tridiagonal in a non-radiating medium, become non-zero if the medium absorbs/emits thermalradiation. The radiative contribution can, however, be brought to yield a tridiagonal set of algebraic

equations by discretization if expressed by the flux method approach. The fluxmethod approximation

consists in the assumption of the angular distribution of incident intensity Ii(x, ). The discrete

ordinate one-dimensional flux method originated by Roesler [2] is characterized by a constant intensity

profile within each 2 solid angle centred about the positive and negative direction of the axis,as shown in fig. 2.

A function G(x) such that

d),x(I4

1)x(G

4

i=

can then be introduced, whereby the radiative contribution to the energy balance eq. (1) takes the form

)]x(G)x(T[k4dx/dq 4aR =

G(x) is governed by the ordinary differential equation

)cGc(cdx/Gd 21322 += (4)

so that the matrix of the coefficients of the set of algebraic equations generated by the discretization

of eq. (1) is now tridiagonal for radiating as well as for non-radiating media.

The heat fluxes absorbed at the side and end walls q sw(x) and qew,i are related to G(x) by the

relationships:

qsw(x) = sw[G(x) - Tsw4] , qew,i = cew,i[G)ew,i - Tew,i4] i = 1, 2

The explicit dependence of the coefficients c1, c2, c3, cew,i on the absorption thickness of the medium

and on the nature, temperature and emissivity of the side and end walls, as well as the boundary

conditions of eq. (4), can be found in Ref. 2..

VALIDATION

PROCEDURE - The accuracy of the flux method above has been determined by comparison of

the heat flux profile at the absorbing walls of a cylindrical PFR combustor obtained for assigned

temperature profile of the medium with the corresponding exact profile. The differences between

approximate and exact heat fluxes are functions of the emissivity of the absorbing walls (sw, ew,i),absorption thickness of the medium (kaD), and aspect ratio of the chamber (L/D), namely:

qi(x) = f(x, sw, ew,i, kaD, L/D)

For any given pair of values of kaD and of L/D the discrepancy qi,max is a maximum when the wallsare black. The is reported in figs. 3 and 4 in function of the absorption thickness of the mediumfor selected values of the aspect ratio are these maximum relative differences.

In the case of isothermal medium the PFR combustor is also a CSTR, a well known case for whichthe flux absorbed by the bounding walls is given by Hottel [1] in the form

)TT(Fq 4s

4

GsG

s

=


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9.1.3

which is exact as long as the total exchange factor sGF is determined using the exact emissivity of the

medium, i.e. that determined by the definition

sG A/gs= (5)

where gs is the gas-surface direct exchange area of the enclosure [1].

The validity of the results obtained being independent of the shape of the temperature profile, as shownin Ref. 3, this elementary case is as good as any other for the purpose of validation of the flux method

in object.

RESULTS The percentage differences between the rates of heat transfer to the walls obtained by theflux method and those obtained by Hottels method for black wall cylindrical chambers of aspect ratio

L/D =1 and L/D = 10 confining an isothermal medium of absorption thickness kaD are shown in fig. 3.The flux method underpredicts the rate of heat transfer with a maximum difference in correspondence

of absorption thicknesses of the order of unity, and tends to become exact in both the transparent andthick medium limits. A more complex behaviour becomes apparent as the difference of the rates of heat

transfer to the side wall are considered separately from those to the end walls, indicating that the result

above is due to a compensation between an underestimation of the former and an overestimation of the

latter. Moreover, the approximation of the heat flux to the side wall increases with increasing aspectratio, i.e. as the weight of the rate of heat transfer to the end walls decreases.

A very significant measure of the magnitude of the discrepancies introduced by the flux method

approximation is now obtained by the comparison with that introduced in the exact zone method

by the widespread practical approximation by which the emissivity of a grey medium is calculatedusing the estimate of the mean beam length Le = 3.5V/A in the relationship:

G = 1 - exp(-kaLe)

as an alternative to the exact but laborious determination by eq. (5).

The discrepancies in the heat flux profiles at the side wall of a cylindrical chamber with radiatively

adiabatic end walls are shown in fig. 4 for the two values of the aspect ratio L/D = 1 and L/D = 10 .

The rate of heat transfer is underestimated by both approximations with a maximum of about the same

magnitude in correspondence of kaD 1. The two differences differ by amounts of the order of 1%in media of increasing absorption thickness (kaD > 1), whereas in more transparent media (kaD < 1)the flux method tends to become exact, while the underestimation of the rate of heat transfer due to

the Le = 3.5V/A approximation on G reaches the order of 10%.

CONCLUSIONS

The rate of heat transfer to the end walls of a one-dimensional chamber is overestimated by Roeslers

flux method, and that to the side walls is underestimated.

The approximation to the exact rate of heat transfer to the side walls can be even better thanthat of the zone method when the Le = 3.5V/A approximation of the mean beam length is used for

the determination of the gas emissivity, e.g. as the rate of heat transfer to the end walls tends

to become negligible and/or the aspect ratio of the chamber exceeds the order of unity.

REFERENCES1. Hottel, H.C. and Sarofim, A.F.,Radiative Transfer, McGraw-Hill, New York (1967)

2. Filla, M., Chem. Eng. Sci.,39:159 (1984)

3. Filla, M., Combustion and Sustainable Development, Roma, Italy, June, p. II.5 (2002).


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Fig. 1 The intensity of incidentradiation Ii(P, ) on the elementary

volume along a 1-D duct is a function

of the point P(x) and of the direction

of the impinging beam .

Fig. 2 The incident intensity

rofile on the elementary volumein the discrete ordinate 1-D flux

method approximation.

-20

-15

-10

-5

0

0.01 0.1 1 10kaD

t%-30

-20

-10

0

0.01 0.1 1 10kaD

sw%

-10

0

10

20

30

40

50

0.01 0.1 1 10kaD

ew%

Fig. 3 Percentage differences between the

lux method and the exact rates of heat

transfer to the total (t%), side wall (sw%)

and end walls (ew%) of a black cylindricalenclosure of aspect ratio L/D=1 (full line)

and L/D=10 (dotted line).

-20

-10

0

10

20

0.01 0.1 1 10kaD

sw%

-30

-20

-10

0

10

20

0.01 0.1 1 10kaD

sw%

Fig. 4 - Percentage differences with respect to the exact rates of heat transfer to the

side wall (sw%) of a black cylindrical enclosure due to the flux method approximation

(full line) and to the Le=3.5V/A approximation on the emissivity G in the zone method(dotted line) for the two key values of the aspect ratio.

FIGURES

L/D = 1 L/D = 10

9.1.4


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