EXPERIMENTAL STUDY OF NATURAL CONVECTION HEAT TRANSFER CHARACTERISTICS OF AN

INCLINED CYLINDER

ABSTRACT

This study was concerned with the experimental

determination of free convective heat transfer from circular cylinders as the

cylinders were varied from the horizontal to the vertical at angles from the

horizontal. Cylinders in axially horizontal and vertical positions have

buoyancy forces only normal and parallel to their axes, respectively.

Cylinders inclined from the horizontal will have flow patterns that are three-

dimensional.

Many analytical studies have been done on cylinders

both in the horizontal and vertical positions. Typically cylinders in the

horizontal positions are assumed to be infinitely long in the axial direction

and hence the characteristic dimension is based upon the diameter. Cylinders

in the vertical position with small angles of curvature are treated essentially

as flat plates and the characteristic dimension is usually the length of the

cylinder. Though the need for data correlations for inclined cylinders for

applications, such as in design of inclined solar absorber tubing would seem

apparent, the topic has received less attention.

Bosworth performed several experiments on the

convective loss from fine wires in water and glycerine inclined at various

angles. Morgan and Koch performed similar fine wire experiments in air.

Stewart and Oosthiuzen performed experiments on aluminum cylinders in

air.

Chapter 1

Natural Convection

Convection is the study of conduction in a fluid as enhanced by its "convective

transport", that is, its velocity with respect to a solid surface. It thus combines the energy

equation, or first law of thermodynamics, with the continuity and momentum relations of

fluid mechanics.

In natural convection fluid motion is due solely to local buoyancy differences caused by

the presence of the hot or cold body surface. Most fluids near a hot wall, for example,

will have their density decreased, and an upward near-wall motion will be induced.

Natural convection velocities are relatively gentle and the resultant wall heat flux will

generally be less than in forced motion.

The prediction of heat transfer in buoyancy influenced flows is important for a number of

engineering applications, including cooling of electronics, heating and cooling of

buildings, process heat transfer (e.g., heat exchangers), and safety applications (e.g., heat

transfer from fires). In many of these applications, mixed convection exists in which both

forced and free convection contribute to the heat transfer. As a precursor to attempting

the complexities of mixed convection heat transfer, the current study will focus on heat

transfer in purely buoyant flows.

Natural Convection Equations

Necessary Dimensionless Numbers

Reynolds (Re) Grashof (Gr) Prandtl (Pr) Nusselt (Nu)

“Local”

“Average”

Properties with a subscript “f” just mean that the property has been obtained at the film

temperature, the average of the solid temperature (T0) and the fluid stream temperature

(T). If the volume expansion coefficient ( , where the derivative is

evaluated at the film temperature) isn’t tabulated, it can be found in one of two ways:

1. Equation of state for your fluid (e.g. ideal gas equation). If you assume your fluid

is an ideal gas, then f=1/Tf. Otherwise,

2. Use whatever density vs. temperature values you have tabulated and approximate

f with .

Although convective heat transfer problems can seem incredibly confusing given the

multitude of different equations available for different systems and flow regimes, it helps

to keep in mind that the whole goal of the problem is to find the overall heat transfer

coefficient, h, so that we can describe the heat transfer from object in the fluid medium.

Thus, finding NuL becomes the problem, and it does get a little ugly from time to time.

Notice that in certain cases (like flat plates) one can define a local dimensionless number

fx

xU

Re

20

3

fx

TTxgGr

f

f

Prf

xx k

xhNu

fL

LU

Re

20

3

fx

TTLgGr

f

L k

hLNu

dT

d

ff

1

Tf

1

by using x, the distance down the object in the direction of the flow. Finding the local

Nusselt number would allow one to then solve for hx, the “local” heat transfer coefficient.

We’re generally not interested in knowing what the heat transfer coefficient at one

particular point on the surface is, though—we want the average h (which is the one we’re

familiar with from all the heat transfer work we have done), and this is precisely the one

we get from NuL.

The problem is that the equations to find NuL are very problem-specific. Generally,

however, 2 distinguishing characteristics can be identified in the problem to find the

exact equation needed:

1. Turbulent or Laminar: - For forced convection problems Re determines whether

or not the flow is turbulent or laminar. For flow past flat plates, the transition

region from laminar to turbulent is about 105<Re<107. Go ahead and assume

turbulence if Re is much higher than 105. For convection problems with cylinders

and spheres, “L” (the length-scale) takes on slightly different meanings (and there

isn’t really an analog for the local dimensionless numbers). Nu is also called NuD

in these situations sometimes. Also, with cylinders and spheres it may not be as

cut and dry as saying “laminar” or “turbulent.” Certain equations have been

developed to be accurate over a specific (and sometimes relatively small) range.

For natural convection problems, the product Gr Pr is used to specify the flow

regime. The laminar natural convection equations we’re given hold from 104<Gr

Pr<109. The turbulent natural convection equations are valid from 109<Gr

Pr<1012.

2. High or Low Prandtl number:- After examining whether we have a laminar or

turbulent system, check the Prandtl number. There are typically different

equations for different ranges of Prandtl numbers. Unless otherwise specified, a

“high” Pr is one > 0.5 and a “low” Pr is < 0.05.

Horizontal Cylinder, Natural Convection (L=D/2)

104<Gr Pr<109

0.00835<Pr<1000

but if 0.6<Pr<10 use (8.18)

also, check special cases (7.45b,c)

109<Gr Pr<1012

works for most Pr

but if 0.6<Pr<10 use (8.21)

Inclined Cylinder, Natural Convection

NuD / (RD cos )0.25 = 0.53 + 0.555[(D/L cos )0.25 – (D/L)0.25]

25.0

5.0

4 Pr861.0

Pr902.0

25.0

L

L

Gr

Nu 5/23/215/75/2 Pr494.01Pr246.0

LL GrNu

Chapter 2

Literature Review

H.R.Nagendra, M.A.Tirunarayanan and A.Ramchandran (1970) worked on laminar free

convection from vertical cylinders with uniform heat flux, obtaining numerical solutions,

classifying cylinders as short, long or a wires and proposing heat transfer co-relations.

K.M.Krall, and E.R.G.Eckert (1973) studied local heat transfer around a cylinder at low

Reynolds number in transverse flow of air at low mach numbers (approximately 0.2)

P.H.Oosthuizen (1976) did an experimental study of free convective heat transfer from

inclined cylinders, proposing a co-relation

NuD / (GD cos )0.25 = 0.42[ 1 + ( 1.31 / (L/D tan )0.25)8]0.125

W.E.Stewart, Jr. (1976) carried out a study for the experimental determination of free

convective heat transfer from circular cylinders varying their angles of inclination,

proposing a correlation

NuD / (RD cos )0.25 = 0.53 + 0.555[(D/L cos )0.25 – (D/L)0.25]

B.Farouk and S.I.Guceri (1982) studied natural convection from a horizontal cylinder the

flow being in the turbulent region. The model involved studied turbulence through kinetic

energy and its volumetric rate of dissipation. The data was correlated using the co-

relation

A.V.Hassani(1992) studied the natural convective heat transfer from cylinders of

arbitrary cross section. The study aimed at predicting an expression natural convection

heat transfer from isothermal two-dimensional bodies of arbitrary cross section over a

wide range of Rayleigh and Prandtl numbers. The proposed correlation was

Nul = (2 l)/Pln(1 + (2L)/(((4Lzf/P2)0.25*C*Ra0.25)m + C*Ra1/3 )m)1/m)

M.S.Farid and C.K. Hsieh(1992) worked on measurement of free convective heat transfer

coefficient for a rough horizontal nonisothermal cylinder in ambient air using infrared

scanning. The flow was buoyancy driven and the cylinder was heated internally to a

steady state with nonuniform temperature on the surface. The effect of surface roughness

on the enhancement of heat transfer by free convection was studied.

S.B.Clemes ,K.G.T. Hollands and A.P.Brunger(1994) studied natural convection heat

transfer from long horizontal isothermal cylinders. They reported a new set of

measurements on natural convection heat transfer in air from isothermal long cylinders of

non circular cross sections at various orientations covering a Rayleigh number range from

about 104 to 109.The resulting correlation only requires the geometric specifications of the

body height and perimeter.

H.M.badr (1997) studied laminar natural convection from n elliptic tube with different

orientations. The effects of tube orientation ,axis ratio and Rayleigh numbers were

studied .The results showed that the Nusselt number increased with Rayleigh numbers

and that the maximum Nusselt numbers were obtained with the major axis vertical.

B. A/K Abu-Hijleh , M. Abu-Qudais and E. Abu Nada (1998)studied entropy generation

due to laminar natural convection from a horizontal isothermal cylinder. The entropy

generation was analyzed using computational fluid dynamics and the study was carried

out for different Rayleigh numbers and radii of the cylinders. Their main conclusion was

that entropy generation was lesser for larger cylinders at the same Rayleigh’s numbers.

D.T.Newport ,T.M.Dalton , M.R.D.davies , M.Whelan and C.Forno studied the thermal

interaction between an isothermal cylinder and its isothermal enclosure for cylinder

Rayleigh numbers of order 104 . An experimental investigation was made of the thermal

interaction between an isothermal cylinder and it’s water-cooled enclosure. This study is

of interest as the application of interest is the cooling of electronic systems.

Chapter 3

111Equation Chapter 1 Section 1Correlations for heat transfer from inclined cylinders:

The correlations developed by P.H.Oosthuizen and W.J.Stewart ,Jr. are presented

here.

In the first study, the correlation developed is presented in terms of Nusselt’s

number and Grashof or Rayleigh numbers. The fluid properties in these numbers were

evaluated using the mean model temperatures during the test and the ambient temperature

using the average of the two temperatures.

For the Grashof numbers considered, previous studies have indicated that for

Φ=00, the heat transfer rate is given by the equation of the form:

ND/GD0.25 = A(Pr) (1)

Since the Prandtl number remains almost constant in the range covered A is

constant the value is averaged to 0.42, using previous experimental data.

Further, for the range of Grashof numbers considered , previous studies have

indicated that for Φ=900 , the heat transfer rate is given by:

ND/GL0.25 = B(Pr) (2)

Previous studies indicate for a constant Pr B is 0.55 for the Grashof number

range.

Thus for an arbitrary angle of inclination Φ, the above equations indicate that ,

provided the boundary layer equations apply , which is so for the range of Grashof

numbers considered , then for a given Prandtl number we have:

ND/(GD cos Φ)0.25 = function(L/D tan Φ )

=function (L*) (3)213Equation

Chapter 3 Section 1

This result is obtained using the boundary later equations in terms of

dimensionless variables. The results for all of the models will be plotted for the given

variables.

For large values of L* , the component of buoyancy force parallel to the axis of

the cylinder will have a negligible effect on the flow , which is then effectively two

dimensional , and, therefore, the following will apply:

ND/GD0.25 = 0.42

Similarly, for small values of L*, the component of buoyancy force will have a

negligible effect on the flow which is then again effectively two dimensional; and in this

case then the equation that will apply is:

ND/(GL sin Φ)0.25 = 0.55 (5)

This can be rearranged as

ND/GD cos Φ)0.25 = 0.55/ (L*)0.25 (6)

The variations given by the above equations show that for L* greater than 10 the

equation number [4] applies , while for L* less than 1 equation number [6] applies. Thus

only for the region where L* lies between 1 and 10 are the three dimensional effects

important and the heat rate more than that predicted by either of the two equations.

In the intermediate range the variation of ND is represented by an equation of the

approximate form:

NDn = NDH

n + NDVn (7)

Where the terms on the right hand side are given by equations [4] and [6]

respectively. Using these two equations, equation [7] can be rearranged to give

ND/GD cos Φ)0.25 = 0.42 [1+ (1.31/(L*)0.25)n]1/n (8)

This equation correlates to the results for n = 8

ND/GD cos Φ)0.25 = 0.42 [1+ (1.31/(L*)0.25)8]0.125 (9)

This equation will apply in the Grashof number range of 104 to 109.

The second study presents a correlation in terms of Nusselt number , Rayleigh

number , angle of inclination and aspect ratio. Fluid properties were based on film

temperature as above.

The ratio of Nusselt number based upon cylinder diameter at an angle of

inclination to that determined at the horizontal position is plotted. The data

Morgan correlated the results of many investigations for both horizontal and

vertical cylinders . The suggested correlation for horizontal cylinders is the same as that

originally developed by McAdams as

ND = 0.53 (RaD)0.25 , (2.1)

A similar correlation for vertical cylinders for small aspect ratios has been given

as:

NL = 0.555 (RaL)0.25, (2.2)

The above equation can be modified by substitution of the aspect ratio to yield ND

for vertical cylinders,(Φ=900)

ND = 0.555 (RaD (D/L))0.25, (2.3)

The initial variation of ND follows that of a horizontal cylinder

ND (Φ small)= 0.53 (RaD cos Φ)0.25 (2.4)

While for large values of Φ , the ND should be given by equation [2.3].

considering the limiting cases of equations [2.1] and [2.3] an equation was developed to

approximate the variation in ND with inclination and aspect ratio.

ND(Φ,D/L) = 0.53 (RaD cos Φ )0.25 + 0.555(RaD D/L)0.25(1-cos Φ)0.25)

Which satisfies the equation [2.1] for Φ = 0 deg. And equation [2.3] for Φ = 90

deg. And provides for a linear correlation of ND between the vertical and horizontal

positions.

This can be rearranged as

ND/(RaD cos Φ )0.25 = 0.53 + 0.555(D/(L cos Φ )0.25 –(D/L)0.25

CHAPTER 4

EXPERIMENT

Experimental setup

Four cylindrical models were used in the above study; their main dimensions are

listed in the table below.

Model number Diameter, (D) mm Length, (L) ft L/D ratio

1 20 0.5 7.62

2 20 1 15.24

3 20 1.5 22.86

4 20 2 30.48

The cylinders were constructed out of hollow copper tubes. A nichrome wire

heating element was encapsulated in a fiber enclosure and placed along the center of the

tube .The gap between the heating element and the tube was filled with magnesium

powder. The two ends of the tube were sealed using staplon corks.

The surface temperature was measured using copper-constantan thermocouples.

The thermocouples were placed along the length of the cylinder. The thermocouples were

placed equidistantly on the surface. The number of thermocouples used was determined

by studies with similar models used in the study of heat transfer from vertical cylinders.

They have indicated that the number of thermocouples used was adequate in determining

the mean model temperature.

The thermocouples were made by fusing together copper and constantan wires of

0.5-millimeter diameter with the help of spark generation between the tips of two wires.

For placing the thermocouples on the surface first 2-millimeter diameter grooves were

punched on the surface to a depth of 1 millimeter and then the thermocouples were

inserted in these grooves. Then again by light punching the grooves were filled up and

the thermocouples were laid out axially for some length so as to minimize end correction.

The thermocouples were connected using a selector switch to a digital

millivoltmeter.

The cylinder was mounted on a movable frame to allow the reading to be taking at

various angles of inclination. The frame was supported using rubber pads.

Electric power was supplied using the laboratory main using a variac and was

measured using an ammeter and a voltmeter.

A thermo meter was used with the set-up to measure the ambient temperature.

A schematic of the experimental set-up is presented overleaf.

Experimental procedure:

The model was mounted on the frame and readings were taken after steady state

had been attained. Since the temperature range involved was small this was attained

quickly.

Also, since the thermocouples were linearly arranged on the cylinder, the cylinder

was given three rotations each of 900 degree to ensure that the average surface

temperature calculated from the readings truly approximated the mean surface

temperature.

With a given heat input, the surface temperatures were allowed to come to

equilibrium. The average of measured temperatures was then used as the average surface

temperature, nominally near 600C.

The heat transfer coefficient was then calculated by the equation:

(1)

Where A is the cylinder surface area

and δT is the difference between time averaged cylinder temperature and

ambient temperatures.

Heat losses other than by free convection are due to radiation from the cylinder

and end losses through the end pieces and the support structure.

Estimates of radiation losses from the cylinder to the chamber walls based upon

average surface temperatures come to about 5%. The necessary corrections were made to

the above equation to account for the radiation losses.

Staplon corks were used to close the ends and rubber pads were used to support

the cylinders. Temperature distributions within these were not measured and hence the

heat loss through these is difficult to estimate.

The ratio of thermal conductivities of copper to the materials used is very high.

Since the copper cylinders reach a steady-state condition, the members are assumed to

have attained steady state as well. Considering the large difference in conductivities, it

has been assumed that the heat loss through the supports and the end pieces was

negligible.