http://www.me.umn.edu/

Main navigation | Main content

__Interferometry - Natural Convection around a Cylinder__

**Reading:** Before you begin, read the given material on the interferometer technique.

**Objective**

The objective of this experiment is to measure natural convection heat transfer from a horizontal heated cylinder. This will be accomplished with the aid of an interferometer. Through this we hope to gain experience studying heat transfer with optical devices.

**Background**

We begin with a description of the interferometer.

__Interferometer__

Figure 1 Mach-Zehnder Interferometer

A schematic representation of the interferometer to be used in this experiment is shown in Fig 1. It is known as a Mach-Zehnder interferometer. Monochromatic light enters through a lens and expands into a collimated beam of parallel light. This collimated beam is directed to a splitter plate; Half of the light passes straight through and the other half (the test beam) is reflected 90 degrees. The test beam of light is reflected 90 degrees by another a mirror and is directed through the test section. The other (reference) beam strikes a mirror and is directed to another splitter plate. At this second splitter plate, the two beams recombine and are directed to a screen or into a camera. The image that appears on the screen depends on what happens in the test section. If the light passing through the test section experiences the same conditions as the reference beam, then on the screen, there can be two possible scenarios. It will be either a uniform field, if the two beams are perfectly parallel upon recombination, or a set of parallel lines (wedge fringes) if the two beams are at some angle to each other. If the beam passing through the test section experiences a density difference relative to the reference beam, the wave length of this beam will change. Upon recombining with the reference beam, the two will be out of phase and will interfere with each other. In this case we will see a fringe pattern, such as the one shown in Fig 2.

Figure 2 Interferogram

In Figure 2, we see an interferogram showing the density (temperature) field in the air around a heated cylinder. This is very similar to what will be observed in this experiment. The dark circular section in the middle is the cylinder and each of the dark and light regions surrounding it are known as fringes. Each fringe represents a region of constant density (temperature). The temperature difference between each fringe is approximately equal to the temperature difference between the cylinder surface and the surrounding air divided by the number of fringes. For example, if there are 10 fringes and T_{cyl} - T_{ref} is equal to 30 K, then the temperature difference between fringes is approximately 3 K. The exact value for the temperature at each fringe can be found by applying the following equation for each fringe.

where,

ε - number of fringes from reference temperature (fringe) at edge of field

P - Ambient pressure(~100 kPa)

C - a constant (see below)

λ _{o} - Wavelength of the laser (632.8 nm)

T_{ref} - Ambient Temperature ( ~295 K)

R = 0.287 kJ/kg-K - Gas constant for air

L = Length of cylinder (along the light beam)

For example if L = 25 cm and ε = 7 at cylinder, then T_{cyl}-T_{ref} will be approximately 21^{o} C.

The Dale Gladstone constant (C) can be calculated using the following equation:

Where,

*n* = index of refraction

ρ= density of fluid.

For a laser of wavelength 632.8*nm* in air at 20^{o}C and 1 standard atmosphere of pressure, the value of *n-1* = 2.719X10^{-4}.

__Natural Convection__

In this experiment we will be studying natural (free) convection from a heated cylinder. As such, a brief discussion of natural (free) convection from a heated cylinder should be included. Natural convection occurs when flow is induced by density differences in a fluid contained in a body force (e.g., gravity) field. For a heated cylinder experiencing natural convection, we can describe the Nusselt number (Nu_{D}) as a function of the Rayleigh number (Ra_{D}). In the current experiment the Rayleigh number is in the range of 10^{4}-10^{7}, and for this range the Nusselt number averaged around the periphery of the cylinder is given by:

The Rayleigh number (Ra_{D}) is defined by:

The Rayleigh number may be a new term to some its terms are defined as follows:

g - Local acceleration due to gravity

β - Volumetric thermal expansion coefficient

T_{S} - Surface temperature

T_{∞} - Ambient temperature

D - Cylinder diameter

ν - Kinematic viscosity

α - Thermal diffusivity

For a more in depth discussion, the experimenter is referred to an undergraduate heat transfer text

**Procedure**

__Safety__

It is always important to exercise caution when performing experiments. This particular experiment involves use of a laser. When working with lasers, never allow the beam to come in contact with the naked eye. Avoid looking directly at the laser. A copy of the manual for our particular laser will be made available prior to and during the experiment. Familiarize yourself with the safe operation of the laser BEFORE performing the experiment. To avoid unexpected and unwanted reflections, remove your watch and other jewelry and put them in your pocket.

__Lab Procedure__

For steps 1 through 10, you should seek the assistance of the TA.

- Turn on the laser and ensure that the interferometer is aligned. The screen should show a uniform field.
- Check that the cylinder is horizontal and its axis is parallel to the beam.
- Turn on the power supply to heat the cylinder; set at about 23V (this value may need to be adjusted; try to get about 10 fringes).
- Monitor thermocouples until steady state conditions are achieved (approximately 1 hour).
- Record thermocouple readings for future comparison.
- Focus the digital camera onto the center of the cylinder through the optical lenses. Adjust the camera so that the interferogram as large as possible (can be done while cylinder is heating).
- Once a good fringe pattern is generated, take several photos. Also, block the reference beam and take a picture of the reference notch (without disturbing the camera). This is important as we need to have a reference (scale, i.e number of pixel = 1cm) for the digital images obtained.
- Convert the photos into Bitmap format using any imaging software. Improve the image to get a picture with very visible difference between the bright and dark fringes.
- Darken the cylinder portion using any imaging software and also clean the image to remove the dark portions outside the fringe pattern. Also find the center of the cylinder and note the scale of the image (1 cm = x pixels).
- Run Project software and give the coordinates of the center of the cylinder and the scale of the image. The program generates an Excel spread-sheet (output.csv) containing the required information about the variation of intensity with distance from the center of the cylinder.
- Using the output file, carry the analysis of locating the position of dark and bright fringes, temperature profile and finally the heat transfer coefficient at various angles on the cylinder surface.
- For a horizontal radius, identify on a printed picture of the interferogram the center of each fringe, measure the radial position of that center and compute the convective heat transfer coefficient at that location.
- Compare the results of (12) and the comparable results of (11).

__Deliverables__

- Calculate the average convective heat transfer coefficient for the cylinder based upon an appropriate correlation. (theoretical).
- Calculate the local heat transfer coefficient at the angles tested (every 30
^{°}) using the calculated average heat transfer coefficient (theoretical average value). See Figure 5 of the Kuehn & Goldstein paper (Refer to: T.H.Kuehn and R.J. Goldstein,*International Journal of Heat and Mass Transfer*vol.23 pp.971-979 and T.H.Kuehn and R.J. Goldstein,*International Journal of Heat and Mass Transfer*Vol. 19 No. 10 pp.1127-1134.). - Determine cylinder surface temperature using the fringe pattern data and compare it with the measured value. Explain any difference.
- Calculate the local heat transfer at each angle tested (every 30
^{°}) from the measured temperature gradient near the surface using the near-cylinder fringes.. - Plot the local experimental heat transfer coefficient on a radial plot. Compare to the results computed in step 2 above from the results in the Kuehn & Goldstein paper (Figures 5 or 10).
- Integrate local measured heat transfer coefficients to get total heat transfer from the cylinder. (NOTE: consider radiation heat transfer from cylinder) Compare to power input from the heater and explain difference, if any.
- Compare to the average heat transfer coefficients from the correlation (step 1 above).