Alex

Heat Transfer By Natural Convection

Introduction:

In 1915, german engineer and teacher Wilhelm Nusselt wrote a paper titled “The Basic Law of Heat Transfer.” In this paper, he discusses the “Nusselt Number” (Nu) which is a ratio of convective to conductive heat in a fluid/solid system. This number is extremely important in order to determine how actively heat is transferred through a system. In other words, the larger the Nusselt number, the more efficient a certain metal is in aiding with heat convection. Similar to the Nusselt number, the Convective Heat Transfer Coefficient helps in determining the efficiency of convection as it finds the rate of heat transfer between a solid surface and a fluid. This is significant for my DYO specifically because my goal is to find which hollow metal rod- either copper, aluminum, iron, steel, and silver- transfers heat most efficiently in convection. In this simulation, an apparatus is set up which first conducts heat from a heater to a certain metal, and then that heat is then convected from the surface of the metal to the outside air. Specifically, through conduction from the heater to the hollow metal rod, the warm particles touch the cooler particles on the metal surface and start to heat it up. This is unique in this experiment because the warm particles never move from the heater to the metal, they just bump into each other and attempt to find equilibrium in temperatures by heating the cooler body. Then, when this metal is heating up, warm particles disperse all over the metal, including to the surface. At the surface of the metal, the particles are warmer than the particles in the air, so the same thing happens where the area around the warmer particles start to heat up as well. Except now, these particles are in the air and move in a very specific direction. As these particles heat up, they gain energy and start to move faster. This increase in energy causes the area around the heat source to become less dense, and thus the particles rise. When the warm particles rise, they push the cooler particles in the air back down to the heat source, where the process will begin once again. This is what is referred to as a convection current, and is vital in understanding how the apparatus in my simulation works.

Background:

The driving force behind my specific experiment is the heat transfer theory, which is basically a prediction of how energy is transferred as a result of temperature difference. I wanted to take this theory one step further in my DYO however. Instead of simply understanding how natural convection works in my apparatus, I wanted to find the most active and efficient way to transfer heat in this environment.

• Newton’s Law of Cooling

  • Newton’s Law of Cooling yields the following equation- q = hAΔT- where A is the surface area, ΔT is the temperature difference between the surface and surrounding fluid, and h is the convective heat transfer coefficient, which was briefly discussed previously. What makes Newton’s Law of Cooling so important to my experiment is it shows the relationship between the rate of heat transfer and the convective heat transfer coefficient.

• Convective Heat Transfer Coefficient

  • The convective heat transfer coefficient cannot easily be calculated, as it requires an empirical approach of a large number of experiments, so for the sake of ease in this experiment and to calculate the rate of heat transfer, this value will be automatically calculated by the simulation.

• Nusselt Number

  • As introduced earlier, the Nusselt number is the ratio of convective to conductive heat transfer and is very useful in determining the rate of convective heat transfer. Because this number is a ratio, its formula is the following:

After this formula is obtained, the formulas for net heat transferred via convection and conduction can be substituted in yielding the formula to find Nu:

Methods:

Procedure:

1. Open the web simulation at https://vlab.amrita.edu/?sub=1&brch=194&sim=791&cnt=4

2. Set all control variables to begin with. When in fullscreen on the simulation, sliders will appear on the right side of the screen. Once these are set once, they should stay constant throughout the entire experiment, regardless of which metal is being tested. I have attached my list of control variables below.

3. One more control variable needed to set up is the voltage and current of the system. This variable is extremely important to be controlled because each metal must receive the same exact amount of heat in order to determine which one transfers heat energy the best. For my trials, I used a voltage of 100. volts, and a current of 0.20 amps.

4. To begin taking measurements, power the apparatus on until the stopwatch hits 5 minutes. Then, by clicking the arrows on the temperature indicator, record values for T₁, T₂, T₃, T₄, T₅, and T₆. Also, in the bottom right corner, record the heat transfer coefficient that has been automatically calculated.

5. Repeat this process for all 5 metals and create a neat table to display the measurements.

6.After all the data is collected, use the formula provided in the “Nusselt Number” section of the “Background” to calculate the Nusselt number of the apparatus for each specific metal.

Results:

This table is noteworthy for a multitude of reasons. The first thing to stick out is the similarity between the corresponding points on the rod. The reason this makes sense will be covered in the next section. Also, from this table, one is able to easily find which variables are controlled throughout the experiment, which is vital in determining the relationship between a specific metal and its Nusselt number. And finally, boarded by a thick box, is the heat transfer coefficient which is automatically calculated by the simulation. The Nusselt Number is the value that I calculated by hand and is the number that corresponds to the rate of heat transfer. To calculate the Nusselt Number for copper for instance:

Since, as seen through the table, each point is the same temperature for each metal, the graphs for each metal resemble this:

(where point 1 represents T₂, point 2 represents T₃, point 3 represents T₄, and point 4 represents T₅ on the x-axis)

Discussion:

One of the most interesting parts of the results section in my opinion is the “Temperature on Different Points of the Silver Rod” graph. As seen through this graph and the table, each point is the same temperature for each metal. And because of that, the slope of each graph for temperature vs. point on rod is the exact same. This is not what I had initially expected, as I thought each metal would have its own unique rate of heat transfer up the rod. After thinking about what, physically, is going on in this apparatus, the lines started to connect as to why I wasn’t seeing any variation in slope of this graph. As stated earlier, the Nusselt Number is a ratio between the convective and conductive heat in a fluid/solid system. And because I am looking specifically at which metal has the highest rate of convective heat transfer, it makes sense for the thermal conductivity to remain the same for each metal. This also makes sense computationally because as seen in the “Background” section, when setting up the ratio of q_convection to q_conduction to find the Nusselt Number, you are left with the equation Nu = (hΔT)/(k(ΔT/L). Then, by simplifying the equation, you can cancel out ΔT from the top and bottom of the fraction. By being able to cancel out this value, it means that the change in temperature on different points on the rod is immaterial. Now that we know that ΔT is not important in finding the Nusselt number, Nu can be calculated. As seen in the results section, silver has the highest Nu at 78.491, copper is the second highest at 78.485, aluminum comes right behind copper at 78.417, iron is the fourth highest at 78.095, and steel has the lowest Nusselt number at a drastically low 76.656. With the Nusselt number for each metal and an understanding of the direct relationship between Nusselt number and rate of convective heat transfer through Newton’s Law of Cooling, it is clear that the silver rod was most efficient in convective heat transfer.

Sources:

• Simulation: https://vlab.amrita.edu/?sub=1&brch=194&sim=791&cnt=1

• Information of Nusselt: https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/nusselt-ernst-kraft-wilhelm

• Nusselt Number Calculation: https://www.youtube.com/watch?v=Tx9iV1PffMg

• Heat Transfer Theory: https://www.alfalaval.us/globalassets/documents/local/united-states/hvac/the-theory-behind-heat-transfer.pdf

• Rate of Conductive Heat Transfer/Newton’s Law of Cooling: https://www.youtube.com/watch?v=Cm_Ggi9_FtY

• Bar Graph: https://www.rapidtables.com/tools/bar-graph.html