Here is the link to the slides from our presentation: https://wikis.mit.edu/confluence/download/attachments/62587827/lab3wiki_presentation.pdf
While we worked on the project together, Sandhya presented on our results from the tank experiment and Katie presented our atmospheric data analysis. Below you will find the main ideas for each slide.
Slide 1: Introduction
Slide 2: Experimental Setup
For our experiment a heating pad was placed at the bottom of a tank. Though not pictured in diagram, we placed temperature sensors at different heights along the side of the tank. We then filled the tank with water. We did two versions of the experiment. The first time, the water was of uniform temperature. The second time, the temperature profile of the water was linear in heat. After filling the tank, we turned on the heating pad and observed the rise in the convective layer.
Slide 3: Photo from the Linear Stratified Experiment
Notice the heating pad at the bottom and the temperature sensors along the left side of the tank. The pink is potassium permanganate which was added to better observe the convective layer.
Slide 4: Homogenous Fluid at Constant Depth
From the thermodynamic balance equation, we can derive a predictive relationship of temperature as a function of time.
Slide 5: Our Experiment
This slide gives the predicted temperature as a function of time, based on the values for our specific experiment.
Slide 6: Temperature vs. Time Plot
We had two temperature sensors, one at a height of 6.25 cm and one at 17.25cm and as expected, the temperature readings were nearly identical for both. The graph shows the temperature as a function of time as given by the temperature sensors. Superimposed is our predictive temperature. As you can see, our data matched our prediction relatively well. In fact, a linear regression of the data returns our predicted line as the best fit.
Slide 7: Linearly-Stratified Layer
From the thermodynamic balance equation, we can derive a predictive relationship of temperature as a function of time. Unlike the homogenous fluid of constant depth, we now assume that temperature is a linear function of height. As a result, the predictive temperature is now on the order of square root of time.
Slide 8: Our Experiment
This slide gives the predicted temperature as a function of time, based on the values for our specific experiment. To create approximately linear stratification, we attempted to increase the temperature of the water by 2K for every 3 cm, starting at 298K and ending with 317K. However, this process was difficult and introduced some amount of human error.
Slide 9: Photo of Experiment
A photograph taken a while after the heating pad was turned on. The height of the convective layer is approximately 7.5cm (each line is 3 cm of height). Potassium permanganate was added to enhance the visualization.
Slide 10: Temperature vs. Time
Because our initial temperature profile was not homogenous, we used 6 temperature sensors instead of 2. The graph shows the data, starting from when the heating pad was turned on. Superimposed is our prediction. A qualitative analysis shows that the data fits the predicted model reasonably well.
Slide 11: Temperature vs. Time
A zoom in of the previous graph to show more clearly how our data fits the prediction.
Slide 12: Atmospheric Data: Dry Convection
An introduction to the concept of dry convection in the atmosphere. The dry adiabatic lapse rate is the rate at which temperature changes with height in the atmosphere when no moisture is present. From this, we can derive the stability of a temperature profile.
Slide 13: Potential Temperature
The slide gives the definition of potential temperature in both words and equation form, and the stability of a temperature profile in terms of potential temperature. Because of the compressibility of the atmosphere potential temperature is analogous to temperature in the tank experiment.
Slide 14: Case Study: Yuma, Arizona
For the atmospheric analysis we used data from Yuma, AZ on June 18, 2007. As you can see from the picture, Yuma is in the middle of the desert.
Slide 15: Potential Temperature Profile
Plot of potential temperature versus height in the atmosphere over time. There are three distinct layers: the convective layer (0 to approximately 3000m), a stable layer (approximately 3000m to 11000m), and a very stable layer (approximately 11000m to 17000m).
Slide 16: Evolution of the Convective Layer
The previous plot zoomed in on the convective layer. As time progresses throughout the day, the height of the convective layer increases. There is a noticeable temperature inversion at 2139Z (2:39pm local time) around 2700m.
Slide 17: Making Connections
A diagram demonstrating the relationship between the temperature profile of the tank experiment as compared to the potential temperature profile from the atmospheric data. The tank is heated from below in the same way that solar radiation is reflected at the surface of the earth, giving rise to a convective layer that grows as the heating continues.
Slide 18: Energy Calculations
Here we try to quantify how the energy generated by solar radiation becomes energy in the convective layer. This energy is equal to the change in internal energy of the system. We use a discretized form of the integral to estimate heat.
Slide 19: Heat Supplied over time
A plot of heat over time. The blue line is data taken directly from radiosonde measurements, while the pink is our estimated heat from the calculation made using the equation from the previous slide. We performed the calculation over three two hour intervals which is why there are three distinct values for our right-hand-side.
Slide 20: Total heat supplied
A comparison of the total heat supplied over time based on the radiosonde data (RHS) and our calculated values (LHS). As you can see, our calculations are on the same order of magnitude.