(3) heat conduction equation [compatibility mode] (1)

7/22/2019 (3) Heat Conduction Equation [Compatibility Mode] (1)

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THERMAL TECHNOLOGY LABORATORY

INSTITUTE OF THERMAL TECHNOLOGY

FACULTY OF ENERGYAND ENVIRONMENTAL ENGINEERING

SILESIAN UNIVERSITY OF TECHNOLOGY

LABORATORY INSTRUCTION

Theme of the exercise:

NATURAL CONVECTION


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Laboratory Instruction: Measuring the Emissivity of a Surface 2

2010 W Kostowski, Institute of Thermal Technology, Silesian UT Gliwice

1 IntroductionConvection is one of three basic mechanisms of heat transfer. The mechanism of convection

concerns fluids (gases and liquids) and is based both on heat conduction and on fluid motion.

Fluid motion enhances heat transfer; heat convection is thus much more intensive than pure

conduction [2]. Due to this intensity, in many engineering problems the temperaturedistribution in a fluid is assumed to be uniform and the mechanism of convenction is analyzed

only in a boundary layer between the fluid and the adjacent solid surface.

The phenomenon of convection is much more complex than conduction in solids. The rate of

convection heat transfer between a solid surface and the fluid flowing over that surface

depends both on fluid and flow properties, such as:

1. fluid thermal conductivity k (W/m-K); this parameter shows the role of heatconduction in the convection mechanism;

2. fluid dynamic viscosity (Pas),3. fluid density , (kg/m3),4. fluid specific heat Cp(J/kg-K),5. flow velocity w (m/s),6. flow area geometry identified by a characteristic dimensionL(m),7. type of flow (laminar, turbulent).

It is also essential to note that the rate of convection heat transfer is proportional to the

temperature difference between the surface and the fluid. This fact is known as the Newtons

law of cooling:

( )= TThAQ ss&

, (1)

where h is the convection heat transfer coefficient expressed in (W/m2K), As is the heat

transfer area, Ts is the temperature of the surface, and T is the temperature of the fluid

sufficiently far from the surface.

The convection heat transfer coefficent hdepends on the factors (17) listed above. Numerous

experimental studies were performed in the 20th

century to find correlations suitable for a

specific type of flow, fluid and geometry. Many such empirical correlations are given in heat

transfer handbooks. In this exercise, you will also follow this approach.

However, you should remember that your specific engineering problem usually differs from the

one investigated by the researcher, who gave an empirical formula that you wish to apply. If youneed a more detailed solution of your convection problem, you should either perform problem-specific measurements, or simulate the problem using Computational Fluid Dynamics (CFD).

For typical flow and geometry patterns, empirical formulae are available in literature. Most

of these formulae contain a dimensionless form of the convection heat transfer coefficient h,

i.e. theNusselt number, defined as:

k

hL=Nu , (2)

where L is the characteristic object length, and k is the thermal conductivity of fluid. The

Nusselt number compares the actual heat transfer by convection with heat transfer in the same

fluid by pure conduction. When no fluid motion is present, then Nu = 1.

Convection problems may be classified into two large groups:


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forced convection, when the fluid flow is forced by an external source such as a fan,pump or compressor;

natural convection, when the fluid flow is only due to differences in fluid density; it isless intensive than forced convection in the same medium.

Natural convection can occur due to the gravity and the buoyancy force. There is no naturalconvection inside a spacecraft, even if it is filled with atmospheric air [2].

In this exercise you will investigate a free convection problem using a test rig presented in

Fig. 1. An electric heater (2) is placed inside a copper pipe (1); its thermal power can be

adjusted using an autotransformer (7). High conductivity of copper ensures good temperature

equalization along pipe walls. The electric heater is placed in the central part of the pipe so

that both pipe ends are not heated, and heat losses through the ends can be neglected (their

area is less than 0.5% compared to that of the side walls). Heat transfer to the pipe interior is

thus not analyzed here. It can be assumed that the entire heat flux generated by the heater is

transferred to the surrounding air through the pipe side surface.

Fig. 1. The free convection test rig

1 pipe with air flow, 2 electric heater, 3 thermocouples for air inlet, 4 thermocouple

switch, 5 temperature indicator,6 wattmeter, 7 autotransformer

Pipe length: 830 mm, inner diameter: 13 mm

Thermocouples (3 Ni-CrNi) soldered to the pipe surface; their indications can be used to

verify if the wall temperature is equalized. Cold ends of the thermocouples are placed in the

ambient temperature, the thermoelectric force is thus proportional to the temperature

difference between the ambient and the pipe surface. The temperature indicator (5)

automatically converts the thermocouple voltage into temperature and adds the ambient

temperature measured with an additional sensor. The temperature you see on the display is

thus directly the temperature at the given point. The thermocouple switch (4) can operate in a

manual or automatic mode. The ambient temperature should be read from a common glass

thermometer.

The aim of the measurements is to determine the convection heat transfer coefficient h, which

can be done using 2 independent methods describe in the subsections below.


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1.1. Method 1: Heat transfer effectThis method is based on a direct observation of cooling effects. It should be stressed that the

pipe considered is cooled both by convection and by radiation. Radiative heat transfer can

usually be neglected in forced convection problems, but should be considered in natural

convection ones, since the latter are less intensive. In this exercise, it is possible to determine

the total rate of heat transfer, however, it is not possible to evaluate which part of heat transferis realized by convection or radiation, respectively.

The Newtons law of cooling (Eq. 1) can be modified to account for radiation:

( )

= TTAhQ sscombined& , (1)

where hcombined is the combined heat transfer coefficient accounting both for convection and

radiation. Mean values of the temperatures sT and T result directly from measuerements. The

areaAscan be calculated from pipe dimensions given to the Fig. 1.

1.2. Method 2: Detailed analysis of natural convectionWithin this analysis you will determine the convection heat transfer coefficient via Nusselt

number (Eq. 2). In natural convection problems, the Nusselt number depends on the product

of the Grashof and the Prandtl numbers:

( )nC PrGrNu = , (1)

The product Gr Pr is sometimes referred to as the Rayleigh number Ra.

As you know from the lectures, the Grashof numberdescribes the ratio of the buoyancy force

to the viscous force and is defined as:

( )2

3

Gr

cs

LTTg

= , (1)

where:

g = gravitational acceleration, m/s2

= coefficient of volume expansion 1/K (b = 1/T) for ideal gases

Ts = temperature of the surface, C

T = temperature of the fluid far from the surface, C

Lc = characteristic object length, m

= kinematic viscosity of the fluid, m2/s

The Prandtl number describes the ratio of the molecular diffusivity of momentum to the

molecular diffusivity of heat and is defined as:

k

Cp=Pr , (1)

where is the dynamic viscosity of the fluid, Pa s.

All fluid properties required for the calculation of the Grashof and Prandtl numbers should be

calculated for the mean film temperature Tf:

( )2

+=

TTT sf , (1)


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In order to do find the convection heat transfer coefficient, you should:

1. Find all the required fluid (air) properties for the average film temperature.2. Calculate the Grashof and the Prandtl number.3. Find a formula for the Nusselt number suitable for natural convection in air and for the

given geometry.

4. Calculate the Nusselt number and the convection heat transfer coefficient.2 Aim of exerciseThe aim of the exercise is to demonstrate the phenomenon of the natural convection and to

compare two ways of determining the convection heat transfer coefficient. The first method is

based on analyzing the heat transfer effects (measured in the experiment) and does not go into

details of the phenomenon. The second method uses the formulae given in the heat transfer

literature to determine the coefficient value theoretically.

From the first method you will obtain the combined heat transfer coefficient, accounting bothfor natural convection and for radiation. The second method allows you to find the convection

heat transfer coefficient, accounting for convection and excluding thermal radiation.

3 Measurement methodologyThe test rig is highly sensible to external impulses, such as flow of air from an opened

window or fluid motion caused by moving persons. Please try to ensure maximum stability of

air during the experiment.

During the experiment you should proceed as follows:

1. Check if the autotransformer power is 0.2. Switch on the test rig equipment.3. Set an electric heating power according to the tutors instructions.4. Write down thermocouple indications and the electric power in time intervals of 1

minute.

5. Wait for equilibrium: two recent records should be in an acceptable agreement.6. Check and write down the ambient (room) temperature.7. Repeat the items 26 for different power settings.


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4 Data sheet for measurement resultsPage 1

Time Nel T1a T1b T1c T2a T2b T2c T3a T3b T3c

min W C

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Indoor air temperature: Atmospheric pressure:

Caution:this page has to be confirmed by tutors signature!


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Data sheet Page 2


min W C

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Data sheet Page 3


min W C

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5 Guidelines for report preparationThe report should contain the following elements:

1. Calculation of the average combined heat transfer coefficient combinedh , performed foreach value of flow rate, based on measurement results.

2. Discussion of results (compare results between the series). Please prepare a graphshowing the dependency between the convection heat transfer coefficient and the

temperature difference between the fluid and the surface.

3. Detailed process analysis: please include calculations as well as the formulae youapplied. Ask the tutor for help if you encounter problems.

4. Comparison of results between the two methods of analysis.5. Final conclusions (you can comment on the results or on the exercise, discuss the

conditions in the laboratory room, find factors which might have influenced the

results, or give your proposal how to improve the exercise to make it more valuable ormore interesting).

References[1]A. Bejan, A. D. KrausHeat Transfer Handbook, J. Wiley & Sons, Hoboken NJ, 2003[2]Y. A. engelHeat Transfer A Practical Approach, McGraw-Hill, New York 2003[3]E. Kostowski Przepyw Ciepa(Heat Transfer, in Polish), Wydawnictwo Politechniki

lskiej, Gliwice 2006.

[4]R. Weber Lecture Notes in Heat Transfer, International Studies in Science andEngineering, IEVB TU Clausthal, 2008

[5]Laboratory instruction Konwekcja swobodna w powietrzu od rury (in Polish), ITCGliwice, 2004, available at www.itc.polsl.pl

(3) heat conduction equation [compatibility mode] (1)

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