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Laminar film boiling on a vertical fin is formulated as a ... related to film boiling phenomena. ... of Bromley (t950) was improved by Sparrow (1964) and his.
W~rmeundStoffiibertragung

Wfirme- und Stofftibertragung 24, 19-23 (1989)

© SpringerWerlag 1989

Laminar film boiling on a vertical fin P. K. Sarma, K. V. Sharma and V. Dharma Ran, Visakhapatnam, India

Abstract. Laminar film boiling on a vertical fin is formulated as a conjugate phenomenon and investigated for no slip and zero shear conditions at the vapor-liquid interface. The results indicate that the combined effects of thermal leakage at the ends of the fin and radiation from its lateral face have profound influence on the average Nusselt number. Further, from the formulation it can be shown that the isothermal condition can be deduced by suitably changing the boundary conditions of the fin at its extremities. The results of the investigation are rendered into dimensionless functional relationships between the average Nusselt number Nu,,, fin parameter M, radiation parameter NR and temperature ratio term g~. The proposed equation can be made use of in design calculations. Laminares Filmsieden an einer vertikalen Rippe

q

Ra T T÷ Tw,+L U X

y

Rayleigh number, (Gr Pr) temperature dimensionless temperature, (T~ - T y (Tw, o - ~) ( L , L - - T~)/(Tw,o -- Z) velocity direction along the test section direction perpendicular to the test section

Subscripts 1 S

Zusammenfassung. Laminares Filmsieden an einer vertikalen Rippe wird als ein konjugiert-komplexes Problem formuliert und ffir den Fall untersucht, dab entweder kein Schlupf oder keine Schubspannungen ander FKissigkeits-Dampf-Grenzfl/iche auftreten. Die Ergebnisse zeigen, dab die zusammengefat3ten Effekte yon thermischen Vertusten an den Enden der Rippe, sowie Strahlung yon der Seitenfl/iche, die mittleren Nusselt-Zahlen beeinflussen. Weiterhin kann gezeigt werden, dab bei zweckm/iBiger A.nderung der Randbedingungen an den Enden der Rippe die isotherme Randbedingung abgeleitet werden kann. Ftir die Ergebnisse der Untersuchung werden dimensionslose funktionelle Beziehungen zwischen der mittleren Nusselt-Zahl Nu,,, dem Rippenparameter M, dem Strahlungsparameter NR und dem Temperaturverh/iltnis ~ aufgestellt. Die aufgestellte Gleichung kann ffir Konstruktionsberechnungen benutzt werden.

heat flux

Qo, QL+ conduction rate at the base and tip of the fin respectively + Qo, QL -(dT+/&l) at the base and tip of the fin respectively

l) W

w, 0 w, L

liquid saturation vapor wall on the wall at x = 0 on the wall at x = L

Greek symbols 3 A e /~ v ~P

vapor layer thickness (6/L) (Ra Ku) lj4 emissivity of the thest surface x/L dynamic viscosity kinematic viscosity density Stefan-Boltzmann constant temperature ratio term, TJ(T~. o - T~)

Nomenclature A B C g Gr h, hm hsg k Ku L M NR Nu Nu,~ P Pr

cross sectional area of the fin constant in Eq. (5) specific heat at constant pressure acceleration due to gravity gL3(°~- ¢v)l(~% v2) local, average heat transfer coefficients respectively latent heat of vaporisation thermal conductivity hfg/C (T~, o - E) length of the fin fin parameter, (k vPL) (Ra Ku) 1/¢/(kwA) radiation parameter, et;(T~, o -- T 9 L/[kv(RaKu) 1/4] local Nusselt number, h L/k average Nusselt number perimeter of the fin Prandtl number of vapor

1 Introduction Film boiling studies on different geometries have been extensively reported in the literature inview of the practical importance of the phenomenon. Recent reviews of Jordan (1968) and K a l i n i n et al. (1975) summarise different aspects related to film boiling phenomena. The theoretical models consider different types of thermal conditions that can possibly prevail on the test surface. In general the analysis may relate to any one of the conditions viz., constant heat flux, constant wall temperature, variable heat flux and variable wall temperature o.f specified nature. However, from a realistic point of view the test surface m a y be dissipating heat

20

Warme- und Stoffi~bertragung 24 (1989)

to the vapor boundary layer under non-isothermal conditions which will not fall under any of the categories stated above. The temperature distribution in the test section is governed by the conjugate mechanism of radiant-convective heat transfer with simultaneous conduction in the test section arising due to thermal leakage into the terminals across which the test surface is held. The pioneering investigation of Bromley (1950) includes the order of magnitude analysis in assessing the contribution of radiation on the laminar film boiling heat transfer. Subsequently, the qualitative analysis of Bromley (t950) was improved by Sparrow (1964) and his analysis includes the following cases: - The radiant heat exchange between the test surface and vapor-liquid interface is considered in describing the phase transformation process treating vapor as a non-participating medium. - the radiant characteristics viz., the selective absorptivity and emissivity of the vapor are introduced in an appropriate manner into the equation of mass balance. However, it is observed that for low pressure of the system, the radiative properties of vapor do not significantly affect the heat transfer coefficients. The purpose of the present article is to make the approach more general at least to the extent of including the fin effects together with radiation from the test surface to the vaporliquid interface. These aspects if included in the analysis, might prove to be significant enough, reflecting on the magnitudes of film boiling heat transfer coefficients.

formulation of the problem the following assumptions of Bromley (1950) are also utilised. 1. The role of inertial force terms in the m o m e n t u m equation of vapor is of secondary importance. 2. The tempterature profile in the laminar vapor boundary layer is linear implying thermal conduction across the layer. 3. The possible dynamic conditions of flow of vapor at the interface can fall into any one of the categories viz., no slip or zero shear condition. 4. Further, based on the earlier observations of Sparrow (1964) the selective behaviour of the vapor towards the absorption and emission of thermal radiation is neglecte& This assumption implies that the vapor is a perfectly transmitting medium. With these considerations and treating the test section as a radiating fin facilitating evaporative mass transfer at the vapor-liquid interface the following differential equations are arrived at

k~,A ~2--~- k~P(Tw--T~)-~rP(T~-T~4)=O dx 2 6

(1)

Further, the evaporation at the interface is given by the balance

dx o G'hy°udy

k.(T w - T~)/6 + e a ( T ~ - T~'*).

(2)

Equation (2) is further simplified with the aid of possible velocity pofiles for the conditions of no slip and zero shear at the vapor-liquid interface

2 Formulation

Velocity profile: The physical model for a vertical fin immersed in a liquid medium at saturation temperature is shown in Fig. 1. The thermal conditions on the test section are however relaxed to include thermal conduction at the terminals of the test section. In addition, the radiation from the test surface is considered in the vapor boundary layer analysis. Further, in the

u --

g(0t -- G) 62 2#v g(ez - G) 62

u-

2#~

[(y/b) --

(y/3) 2]

[2 (y/d) -

No slip condition,

(3)

(y/6) 2] Zero shear condition. (4)

Equation (2) after further simplification can be written as

Test section

1 d ~g(~t-G:)5 3-] B dx L #v

T

....

Vapor-liquid interface

Temperature profite

I

u,v Velocities along(x,y)

(5)

where B = 12 B= 3

g

k~,(T,.,,,- T~) e.a(T~-- T~4) G h¢o 5 -1 G hso

for for

no slip condition, zero shear condition.

Equations (1) and (5) are rendered dimensionless and are respectively as follows

(gravi V

Physical model

Fig. 1. Physical configuration of the test section

d2T + d,ff dA ~ d~

MT+ II + N]~ [(T + + 7")4 _ 7' 4] ~ d- l = 0, A 4BT + I+NR[(T++7"p_7"*]~_, 3

(6)

(7)

P. K. Sarma et al.: Laminar film boiling on a vertical fin

21

4 Discussion of results

where T+=(T,-

T~)/(T,~,o- T~); q = x / L

tp = Tj(Tw, ° _ T~);

A = (5/L)(Ra Ku) u4

K u = hfo/C(T,~ ' o - T~) Ra = [g L3 (0z -- ~0~)/(0~ v~Z)][Pr]

(8)

M = (k~ PL) (Ra Ku)X/4/(k, A) NR = ~ a(Tw, o -- T~)3 L/[k~(Ra Ku)l/4]. L, P and A are length, perimeter and cross sectional area of the fin respectively.

3 Boundary conditions Equations (6) and (7) represent respectively the thermal conduction in the fin and the variation of vapor boundary layer thickness along the fin. The boundary condition for Eq. (6) at the leading edge is Tw=Tw, o

x=0;

or

r/=0;

T+=J.

(9)

Various boundary conditions can be stipulated for Eq. (6) at the other end i.e., at x = L depending on the actual working circumstances of the test section. i) Prescribed temperature at x = L Tw =T,~,L

where

Tw,L_>_Tw,0,

T,~+ = Tw+L

where

T~L>=I.

or (10)

ii) Prescribed heat flux at x = L dT~/dx=O.

(11)

The boundary condition for Eq. (7) is that at x=0,

6=0

or

t/=0,

A~=0.

It is evident from the equations that the inclusion of radiation effect and the thermal leakage at the ends of the test section in the analysis resulted in additional parameters such as the fin parameter M, the radiation parameter NR and the temperature ratio term ~P. The isothermal case of plane vertical surface solved by Bromley (1950) is a particular case of the present formulation in which M - - N R = 0. Further, from the present investigation the earlier analysis of Sparrow (1964) can also be generated by putting B = 1 2 and M = 0 and varying the radiation parameter NR. Test runs are carried out on H P 1000 to check the program and the numerical procedure adopted. It is observed that the results of both Bromley (1950) and Sparrow (1964) could be generated and the agreement is satisfactory. These results are not presented to conserve space. For further study in order to bring out the influence of the parameters (M, NR and ~) the two possible cases viz., no slip and zero shear conditions at the vapor-liquid interface have been undertaken. Figure 2 depicts the possible temperature distribution along the test section, when its terminals are held at equal temperatures i.e., T+o = Tw+L = 1 (i.e., at x = 0 and x = L) for 1 _