Heat transfer from horizontal cylinders to liquids

HEAT TRANSFER

COMBINED

EFFECTS

FROM HORIZONTAL TO LIQUIDS OF NATURAL

CYLINDERS

AND FORCED

CONVECTION

A. K. A. JUMA and J. F. RICHARDSON* Department

of Chemical Engineering, University College, Swansea SA2 SPP, Wales (Received 10 February 1982; accepted 27 April 1982)

Abstract-Heat transfer rates have been measured from electrically heated cylinders to liquid flowing perpendicular to the axes of the cylinders. Results have been expressed as a Nusselt group as a function of Reynolds and Prandtl numbers. Natural convection exerts a significant influence when Grashof number divided by the square

of the Reynolds number exceeds about 0.2.

INTRODUCTION

transfer from cylinders of different diameters to liquids flowing perpendicular to their axes was studied for conditions where the effects of both natural and forced convection were significant. There is little previous work where the flow rate of the liquid over the surface is sufficiently low for natural convection to make an appreciable contribution to heat transfer and in this work particular attention has been paid to this area. The problem of heat transfer from a cylinder to a fluid flowing normally to its axis is complex because even when the flow in the boundary layer is streamline, separation may occur. However, many workers, including KramersIll, Piret et a1.[2], McAdams[3], Latif[4] and Davies[S] have published correlations for heat transfer coefficients for forced convection averaged over the whole surface of the cylinder; these correlations are applicable over limited ranges of Reynolds number, Prandtl number, and cylinder dimensions. The fluid velocities in natural convection currents are generally low, but the characteristics of the flow in the vicinity of the heat transfer surface are similar to those in forced convection. In natural convection, as in forced convection, the flow may be laminar or turbulent, depending on the fluid properties, the exter’it of the body, and the temperature difference between the surface and the fluid.’ Experimental results for natural convection heat transfer can most conveniently be correlated by an equation in which the Nusselt number NU is expressed as a function of the Grashof number Gr which accounts for the effect of buoyancy and viscous forces, and the Prandtl number Pr which accounts for the properties of the fluid. McAdams[3] has collected experimental data from various sources for natural convection from horizontal wires and cylinders and has recommended a generalised correlation from any fluid for Pr > 0.5 and for IO’< Gr < 109.

Heat

*Author to whom correspondence

should be addressed.

Of the few studies of heat transfer where both natural and forced convection are important, mention may be made of Kreith[6] who examined the boundary layer equation for one-dimensional forced flow and concluded that when the ratio GrlRe’ :, 1 natural convection effects cannot be neglected. Oosthwizen and Madan[7] and Jackson and Yen[S] studied that interaction of the two forms of convective heat transfer from horizontal cylinders to air flowing vertically upwards. They correlated their data in the forced convection region as a function of the Reynolds number Re, and for combined natural and forced convection as a function of the group Gr/Re*. Fand and Keswani[9] studied heat transfer from cylinders to water flowing upwards. They divided their data into four groups, according to the value of the ratio GtfRe’, for the purposes of correlation. In the first zone, forced convection predominates with the value of GrlRe2 < 0.5.

In the second zone, forced convection still dominates, but natural convection can contribute up to 10% of the total heat transfer. In zone three, defined by 2 < GtfRe’ (40, the behaviour is unstable and oscillates between that of zone two and zone four. In zone four, defined by GrlRe’>40, natural convection is predominant. APPARATUS ANDMETHOD OF CALCULATION A detailed description of the apparatus is given elsewhere[lO], and an account of the method of measuring heat transfer coefficients has been given in earlier publications[l2]. From the flow diagram Fig. 1, it is seen that the liquid was circulated by the centrifugal pump (3), to the test section, a column 104mm in diameter and 2.10 m tall (5), through the distributor section (7), after metering by one of the rotameters (4). The liquid, whose flowrate was regulated by the valves in the rotameter lines, was returned to the feed tank (2) and its temperature was controlled by adjusting the flow of cooling water fed to a submerged coil in the tank. In all the experiments, a paraffin oil of low volatility and high Rash 1681

A. K. A. JUMAand J. F. RICHARDSON

1682

Fig. 1. Flow diagram with the heating element in place

point was used; it had a viscosity of 1.06 x lo-* N.s/m* and a density of 834 kgm-” at 35°C. Two electrically heated elements, a hot wire element and a wound cylindrical element, were used. The hot wire element consisted of a tungsten wire, approximately 77 mm long and 0.13 mm dia. The cylindrical element consisted of a 2.45 m length of the same tungsten wire, wound on a tufnol cylinder of 5 mm diameter. The elements were supported horizontally and diametrically in the column by means of supporting stwctures as shown in Fig. 2. Direct current was supplied to each element through two contact tabs connected by means of thick copper leads to a Wheatstone bridge network, of which the element formed one arm. The temperature of the” wire TE was determined from its resistance RF T, = (RE - R”)/RO~

where ZR = the sum of all the resistances, including that of the leads, but excluding RE in that arm of the Wheatstone bridge which includes the element. In the steady state all the heat generated, qE, is transferred to the bulk of the liquid across the surface of the element so that YE = M&T, Combining

- TB).

(4)

eqns (3) and (4)

V,z

1

h=A,(Tr-

(5)

7’,)‘(R,+XR)2.RE’

The term RE CR, + XR)’

(1)

where R,(O) is the resistance of the element at 0°C. n (Km’) is the temperature coefficient of resistance. The rate of heat generation in the wire & may be exgressed as

was virtually constant because only small currents were passed and changes in resistance were therefore small; the average value was O.O477fI-’ for the cylindrical element for which A, = 0.8639 X 10m3 m*. Hence equation (5) becomes

Since

and for hot wire element

h( TE - TB) = 55.2 V,’

(R~ yzR); YE = (R, + PR)’ ’RE

(3)

= 0.0238 and AE = 0.2879 X 10m4m* h(TE - T,) = 826.7 V,‘.

(6)

> (6a)

Heat

transfer

from

horizontal

cylinders to liquids

1683

-

tom

I

i

I-

53mm

-I

Fig. 2. Cylindrical element support system

In all experiments, the maximum electrical energy supplied was only 0.17 W for the cylindrical element and 0.75 W for the wire element, and the temperature of the liquid was consequently not altered by a significant amount.

element C( TE - T&‘4 = 55.2 V,*.

(8)

By plotting (TE - TB)‘14 YS V,‘, a straight line was obtained, as shown in Fii. 3, whose slope (55,2/C) was found to be 0.642.

RESULTS

NaturaL convection The heat transfer coefficient for natural convection (zero liquid velocity) was determined for the wound cylindrical element, by supplying it with energy while it was submerged in stationary liquid. In this case, the heat generated in the element was transferred to the liquid and then dissipated to the surroundings. It took longer to reach steady state conditions than in the forced convection experiments in which the heat was carried away in the liquid stream. For natural convection

‘_

c =

55.2= &

Wlm2.

K-5”.

0.642

The heat transfer coefficient can then be calculated for any temperature difference. For this work on natural convection, therefore, in which AT was varied from zero when V, =0 to 10°C when VB = 6 volts, the corresponding value of (h) ranged from zero to 86 x lOIf = 153 W/m*. K. This value is close to 146 W/m2 K as calculated from McAdams correlation NU = 0.53( Gr x Pr)““.

h a (T, - TB)“4 = C(TE - Tg)“4 Substituting

from

eqn (7) into

eqn (6) for

(7) cylindrical

Forced convection Heat is transferred

from the surface

of the element

to

1684

A. K. A. JUMAand J. F. RICHARDSON

Fig. 3. (AT)S” YS voltage squared ( V.‘) for cylindrical element in paraffin oil-zero

velocity.

and of the wound cylinder, 52mm. while the tube diameter was 104mm, the average velocity around the element is higher than the average velocity in the tube, especially in streamline flow. In all cases the tube Reynolds number was calculated and for streamline flow, the mean velocity over the central portion of the tube across which the element in question was situated was used for calculating Re (element Reynolds number). Figure 5, which shows the effect of increasing the Reynolds number (or fluid velocity) on the Nusselt number (or heat transfer coefficient), also permits a comparison to be made with the experimental data obtained by other workers [I, 4,9]. Figure 5 shows the experimental data for various Iiquids (water, paraffin oils, transformer oils) obtained from the work of Davis [I l] and Piret et nl. [2] and from the present study, for both the wound cylindrical element (dia. 5 mm) and the hot wire element (dia. 0.13 mm); the range of Reynolds numbers covered was from 0.1 to 110, based on average Row rate around the element in the tube, and of Prandtl numbers from 2.3 to 1050. These experimental data were expressed and co;-

the moving fluid, first by conduction through the film of fluid in the immediate vicinity of the surface of the element where the fluid velocity approaches zero; and secondly by convection at extended distances from the surface in the region where the velocity increases with distance. As the fluid velocity increases the boundary layer becomes thinner, and natural convection becomes progressively less important. At high fluid velocities the boundary layer becomes turbulent and the flow may separate from the cylinder to give high heat transfer rates. For forced convection, the heat transfer coefficient will be independent of the temperature difference, and it will be seen from eqn (6) that a straight line should be obtained by plotting T, against V,‘. The slope of the line will be (55.2/h) for the wire-wound cyhfrical element; typical plots are shown in Fig. 4 for five different flow rates. The liquid temperature, obtained by extrapolation to zero voltage, lay between 35.0 and 35.2”C for all the experiments. Since the length of the hot wire element was 77 mm

-

=fJI

I

I

I

I

10

,

20

voltage

squared

cv:,vans

, 30

I

I

I

40

Fig. 4. Temperature of wound cylindrical element ( TE)vs voltage squared ( V,‘) at different liquid velocities

I 50

1685

Heat transfer from horizontal cylinders to liquids

-I

b

Paraffin oil

X

TransformerOil

v n 0 I

I,

25-37 79.3 187

>,

215

>I n

1050

635

Fig. 5. Nusselt number YS Reynolds number, including data from previous investigations.

1

t

*

Paraffin oil

X

Transformeroil

Fig. 6. Plot of NdPr”’ vs Reynolds number for different liquids.

25-37 79.3

1686

A. K. A. JUMAand I. F. RICHARDSON

related in terms of three dimensionless groups (Nusselt number, Nu, Reynolds number, Re, and Prandtl number, Pr). NulPr’ was plotted versus Re on a log-log scale for several values of the index (x) within the range from 0.28 to 0.36. It was found that the scatter of the data was minimised when x = 0.3. A plot of Nu/Pr” ’ vs Re is given in Fig. 6; from which it is found that the following relation gives a good representation of the data over the range of Reynolds numbers from 1 to 100. +

= 0.76 Re0-5.

This correlation underpredicts the value of Nu/Pr’-’ at Reynolds numbers less than 1. Kreith[6] suggested a correlation which was applicable within the range of Reynolds numbers between 3 to 100.

(11) Equation (11) fits the data plotted in Fig. 6 well at low Reynolds numbers (down to 1) but gives a poorer fit at high Reynolds numbers. Substituting into eqn (10) the value of Prandtl number (Pr = 170) for the liquid used in this work, the correlation becomes Nu = 3.55 ReO.S. The difference between the experimental obtained in the present work and the correlation

35 r0

I 5

I

10

1 1s

values was not

I

20

greater than 10% over the range I < Re < 100 except under those conditions where natural and forced convection mechanisms might both be expected to be important (Re- 10 for wound cylindrical element); this problem will now be examined. Combined natural and forced convection In the work on forced convection with the wound cylindrical element, the temperature difference (AT) between the element and the bulk liquid ranged from zero to 3.S”C. The heat transfer coefficient for natural convection would then range from zero to 86 (3.5)“4 = 118 W/m2 K (from eqn 7). For an average AT value of say l.S”C, the heat transfer coefficient at zero velocity would be expected to average about 100 W/m’. K; giving a Nusselt number of 4. In Fig. 5 where Nu is plotted vs Re, it is important to note that for cylindrical element at a Reynolds number of less than IO (corresponding to Gr!Re’ equal to about 0.2 at AT = 1.8”C) the graph deviates from the forced convection line. This value of GrlRe’ is of the same magnitude as that obtained by other workers for the interaction of natural and forced convection to be first observed. The graphs of temperature vs (voltage)* in Fig. 7 show a marked deviation from linearity at Reynolds numbers less than IO; this indicates that the heat transfer coefficient is not independent of AT and that natural convection is significant. All plots at low Reynolds number for the hot wire element give straight lines even at low Reynolds numbers, and there is therefore no evidence that natural convection is important. For the wound cylindrical element, it has been

I

I

I

25

30

35

Fig. 7. Temperature vs voltage squared ( V,‘) for cylindrical element

at low liquid

velocity.

1 40

J 45

Heat 6.0

,

,

I

I

transfer

I

4.0

from

horizontal

cylinders

I

I

I

I

I

6.0

w

100

1687

to liquids

I

I

I

I

I

tqo

1w

I

I

I

1 6.0

18.0

I

2QO

Gr/Re=

Fig. 8. Plot of (Nu-Nu,)

assumed that GtfRe’, i.e.

Nu-

NuF

is a function

of the

group

vs GtiReZ for paraffin oil.

from 1 to 100. Nu Pr “.j = 0.76 RE?~.

Nu - NuF = f(GrlRe’)

In eqn (13), NU is calculated for the measured value of the heat transfer coefficient and NQ is the corresponding value from eqn (IO). By plotting (Nu - Nu,) versus (GdRe’) (Fig. 8), it is seen that the relation might be expected to be of a modified logarithmic form. It was found that the following equation closely fitted the experimental results: NU - Nu,

= 1S7(GrlRe2)

w

(13)

“-75log [l + O.O%(GrlRe’)]

For cylinders of 5 mm diameter, natural convection exerts a significant influence at values of Gr Rem2 greater than about 0.2. This magnitude is consistent with the findings of several previous workers in the area[6, 91. Natural and forced convection effects are non-additive and natural convection makes a diminishing contribution as the Reynolds number is increased. For very small cylinders (dia. 0.13 mm), no effects of natural convection were observed even at very low Reynolds numbers; the minimum value of GrRC* then obtained was only 0.0064.

(14) From Fig. 8 it is seen that the effects of forced convection and of natural convection are non-additive, and that natural convection makes a diminishing contribution in absolute terms as the Reynolds number increases. The interaction is complicated and probably depends on both the geometry of the heating element and on the flow conditions. By the above method the additional contribution to heat transfer arising from natural convection can be estimated for cylinders of different diameters.

NOTATION

A, C C, de g Gr h b k

Nu CONCLUSIONS

Experimental results for forced convection from heated cylinders to a liquid flowing perpendicular to their axes are well correlated by equation (10) for values of Re ranging

NUF Pr qE

surface area of heat transfer element, m* constant in eqn (7), W/m2. K5’4 specific heat of fluid, J/kg. K diameter of the element, m gravitational acceleration, m/s2 Grashof number gp(AT)d,3 Y-2 heat transfer coefficient, W/m’. K current flowing through the heat transfer element, Amperes thermal conductivity of the fluid, W/m. K Nusselt number (=(h,dJk)) due to combined natural and forced convection Nusselt number (= h.d,lk)) in forced convection Prandtl Number ( = C&/r) heat generated within the element, W

I688

A. K. A. JUMA and J. F. RICHARDSON

R Rs Ro RI2 T TE 2-B u VB

resistance, fi resistance of heat transfer element, n resistance of the element at OT, R Reynolds number (&Up/p) temperature, “C temperature of the element temperature of the liquid in the bulk liquid velocity, m/s voltage applied across the Wheatstone

bridge,

Greek symbols

a p v Jo p

temperature coefficient of resistance, Km’ coefficient of volumetric expansion, Km’ kinematic viscosity ( = (p/p)), m2/s dynamic viscosity of the liquid, N. s/m2 density, kg/m’ REFERENCES

[l] Kamers H. Physica

1946 12(2-3) 61

V

121 Piret E. L., James W. and Stacy M., Ind. and Engng Chem. 1947 39 1098. .131. McAdams W. H.. Heat Transmission. McGraw-Hill. New York 1954. [4] Latif B. A. J. Variafion of porticte velocities and concentration in liquid-solid fluidized beds. Ph.D Thesis, Universitv Colleee of Swansea 1971. [5] Da& R., L&xl heat transfer in liquid-solid fluidized beds. Ph.D. Thesis, University College of Swansea 1975. [6] Kreith Frank, Principles of Heat Transfer, 3rd Edn, p. 326. Intext Press, New York 1977. [7] Oosthuizen P. H. and Madan S., ASME J. Heat Transfer (Feb. 1970) 194-196. [8] Jackson Thomas W. and Yen H. H., ASME J. of Heat Transfer (May 1971) 247-248. [91 Fand R. M. and Keswani K. K., Int. J Heat Moss Transjer 1973 16 1175. [IO] Juma A. K. A., Heal Transfer and segregation in fluid&d College of Swansea 1981. beds. Ph.D. Thesis, University [Ill Davis A. H., Phil. Mug. 1920 40 692; 1922 43 329; 1923 44 920. [12] Richardson J. F., Romani M. N. and Shakiri K. J., Chem. Engng Sci 1976 31619.