Download A Mathematical Analysis of Heat Transfer in an Earth

 Wesleyan University
The Honors College
Joop’s World: A Mathematical Analysis of Heat
Transfer in an Earth Analog Model
by
Greta Ramdin
Class of 2012
A thesis submitted to the
faculty of Wesleyan University
in partial fulfillment of the requirements for the
Degree of Bachelor of Arts
with Departmental Honors in Mathematics
Middletown, Connecticut
April, 2012
i Abstract The purpose of this experiment is to identify a greenhouse effect in the simple earth analog model: a uniformly heated sphere in a closed chamber. By filling the chamber with different gases, we can compare the effects of different gas properties on the overall heat transfer of the system. The goal is to determine the influence of each component of heat transfer on the overall system and investigate how this affects the average temperature of the sphere. Ultimately, we hope to be able to relate these results to the global climate system, specifically how changes in atmospheric gases affect average global temperature. We used a series of mathematical computations to determine the magnitude of the various methods of heat transfer and found convection to be an important component. We then concluded that the properties of each gas had a significant effect on convection in the chamber. From these results, we were also able to show the existence of the greenhouse effect when the chamber was filled with air, and carbon dioxide. ii Acknowledgements There are many people without whom this paper would not have been possible. First, I would like to acknowledge the wonderful Mathematics and Environmental Studies Departments at Wesleyan for preparing me well and equipping me with the tools to succeed in this endeavor. In particular, I would like to acknowledge my advisors, Johan (Joop) Varekamp and Michael Keane for working with me every step of the way. Without their encouragement and assistance, this paper would not be the triumph that it is. I would like to give an additional thanks to Joop for introducing me to this project and allowing me to make it mine for this year. Additionally, I would like to sincerely thank my housemate, Emily Pryor, Heidi Ransohoff, and Jane Nestler for supporting me through the good and the bad times, and for cutting me some slack in the house chores when I was too busy writing. I would also like to express my gratitude for my thesis buddies, Annie DeBoeur and Ally Wang, for giving me endless empathy and a quiet place to work. Lastly, a big thanks to my whole family, especially Edgar, the Scottish Terrier. A special thanks to mom for her encouragement, even when she thought I was crazy for pursuing this project and, most importantly, for reading and editing my entire paper despite not completely following all of the math. iii Table of Contents Title Page________________________________________________________________________________ i Abstract_________________________________________________________________________________ ii Acknowledgements____________________________________________________________________iii Table of Contents______________________________________________________________________ iv Table of Figures_________________________________________________________________________v 1. Introduction__________________________________________________________________________1 2. Background___________________________________________________________________________4 2.1 A Brief History______________________________________________________________4 2.2 Understanding Climate Change__________________________________________10 3. Theory_______________________________________________________________________________18 3.1 Radiative Cooling of the Sphere_________________________________________ 20 3.2 Radiative Heating from the Walls_______________________________________ 21 3.3 Heat from the Light Source______________________________________________ 21 3.4 Convective Cooling of the Sphere________________________________________22 3.5 Radiative Heating from Greenhouse Gases_____________________________ 27 4. Methodology________________________________________________________________________31 5. Results_______________________________________________________________________________33 5.1 Radiative Cooling of the Sphere_________________________________________ 33 5.2 Radiative Heating from the Walls_______________________________________ 34 5.3 Heat from the Light Source______________________________________________ 36 5.4 Convective Cooling of the Sphere________________________________________36 5.5 Radiative Heating from Greenhouse Gases_____________________________ 38 5.6 Tables______________________________________________________________________39 6. Discussion___________________________________________________________________________40 7. Conclusion__________________________________________________________________________ 44 Appendices____________________________________________________________________________ 47 Refereces______________________________________________________________________________ 68 iv Table of Figures 1. The Keeling Curve___________________________________________________________________ 2 2. IPCC Past and Predicted Temperature Rise_______________________________________ 3 3. Planck Curve for the Sun___________________________________________________________11 4. Vibrational States of CO2 Molecules_______________________________________________13 5. Atmospheric Convection Cells_____________________________________________________15 6. Coriolis Effect_______________________________________________________________________15 7. Heat Transfer To and From the Sphere___________________________________________20 8. Planck Curve________________________________________________________________________28 9. Greenhouse Gas Absorption Spectra______________________________________________29 10. Joop’s World_______________________________________________________________________31 11. Wall Partition Diagram___________________________________________________________34 12. Roof Partition Diagram___________________________________________________________34 v 1. Introduction Global climate change is becoming increasingly hard to ignore. Overall
changes in climate patterns as well as extreme weather events are becoming more
commonplace and are affecting larger amounts of people in more widespread areas.
This year has brought some of the strangest weather patterns I have personally ever
witnessed. Firstly, I have observed a general trend in warming in the Northeast
United States. This regional weather change is characterized by unusually high
winter temperatures and delayed winter storms. I also have experienced three events
of extreme weather in the last year alone.
The first of these experiences was in December of 2010 while I was studying
abroad in Dublin, Ireland. It is not typical for the capitol to experience snowfall, even
in the winter months. During my four-month stay, Dublin experienced the worst
snowfall in centuries and was forced to close bus and train routes, airplane travel,
schools, and businesses (Melia, 2010). The next two occurrences of extreme weather
were Hurricane Irene in August and the New England blizzard in October 2011. Both
of these storms left Wesleyan’s campus without power for multiple days causing
major disruptions in normal activities.
Of course, change in regional temperature trends and occurrences of extreme
weather events are not proof of global climate change but are arguably symptoms. To
see more conclusive evidence of climate change we look to the scientific community,
most notably the experiments of Charles David Keeling. Keeling measured the
amount of atmospheric CO2 over a period of forty-five years. His results showed a
general trend of increasing CO2 levels in the atmosphere (Varekamp, 2011), see fig. 1.
1 Figure 1- The Keeling Curve (Rafferty, 2012)
The International Panel on Climate Change (IPCC) has also verified evidence
of climate change. Using results provided from numerous scientific groups, the IPCC
has published four climate assessment reports to estimate the various impacts of
changing climate conditions. The most recent report, of 2007, predicted an average
three-degree warming for doubled CO2 levels (an increase from about 380 to 760
ppm) with a range from 1.5-6°C. It was also reported that this number could be
underestimated due to feedbacks of unknown factors. Figure 2 shows an IPCC graph
of observed temperature change and a number of scenarios for future predictions.
While the exact amount of warming is not agreed upon, it is fairly certain that some
warming would be expected if carbon dioxide levels continue to increase, especially
if this were to happen at current rates (Weart “General,” 2011).
2 Figure 2- IPCC Past and Predicted Temperature Rise (Solomon, ed., 2007)
In order to improve the existing results of climate modeling, we must continue
to investigate the properties that dictate climate mechanisms and how changes in the
fundamental properties will contribute to changes in global climate. We do this by
simplifying the terrestrial climate system to a more basic heat transfer problem: a
uniformly heated sphere surrounded by gases in a closed cylinder. The purpose of the
experiment is to investigate the role greenhouse gases play in the heat exchange in a
simulated atmosphere and the effect this has on the average temperature of the sphere.
From this model, we can eventually make inferences about how this would apply to
the real world climate and can possibly contribute valuable data to other existing
models. In this paper, I will investigate how greenhouse gases are contributing to
global climate change, particularly through atmospheric warming and the impacts that
these gases could have if left unchecked.
3 2. Background
2.1 A Brief History
Scientists have been investigating local and global weather patterns and how
humans affect overall climate over the last centuries. In the early twentieth century,
Vilhelm Bjerknes became the first scientist to model atmospheric motion. He
developed what were known as the seven “primitive equations” for heat, air motion,
and moisture in the atmosphere. Lewis Fry Richardson then used these “primitive
equations” to investigate a more complete prediction system for local climate, which
was published in 1922. In this method, Richardson used a grid system where he could
potentially input initial measured conditions, such as pressure and temperature, at one
location. He would then use a set of differential equations to predict the change in
weather over time. While these equations proved to be mostly accurate, they were
not practical for real life applications, as the calculations took much longer than the
time period for which they were valid. This means, for example, that it would take
two days to calculate weather patterns for a 24-hour period. Without a faster way to
perform the computations, Richardson’s equations were deemed useless (Weart
“General,” 2011).
In 1946, John Von Neumann began advocating the use of computers to predict
weather patterns. Von Neumann assembled a new meteorology group to investigate
modeling climate patterns with the ENIAC (Electronic Numerical Integrator and
Computer), a primitive computer. He appointed Jule Charney to head this group.
Charney simplified Richardson’s equations to analyze airflow only along a narrow
4 band since these early computers were slow. He was then able to compare the results
of this model with actual atmospheric changes. In 1950, Charney’s team ran a twodimensional model that consisted of a 270-point grid covering North America. The
points were spaced approximately 700 kilometers apart and a time step of three hours
was used. The model was able to predict many actual weather patterns. Though it left
much room for improvement, it was a promising step in numerical weather prediction
(Weart “General,” 2011).
General Circulation Models (GCMs) are computer models that use numerical
weather prediction to simulate long-term climate changes (Geerts, n.pag.). There are
five components in the climate system: hydrosphere (water), cryosphere (ice),
atmosphere (air), lithosphere (land), and biosphere (living). Because heat and energy
are transferred between each of these components, an ideal computer model will
incorporate them all. There are a number of types of computer-based models for
climate. The first, simplest type is an energy balance model, which is a onedimensional model that divides the Earth into latitude zones and then calculates
incoming and outgoing energy. Another type of model is an atmospheric general
circulation model (AGCM). These are three-dimensional models that use the
fundamental equations (conservation of momentum, mass, and energy, the equation
of state, and the fundamental equation of moisture) calculated at points on a grid to
show change from initial conditions. Two types of grids used are Cartesian grids,
which place a grid over the surface of the earth, and spectral grids, which represent
the fundamental equations as waveforms. Physical processes that affect energy
transfer, such as radiation, cloud coverage, and surface exchanges must also be
5 modeled and included with the fundamental equations. Another type of model is
oceanic GCMs (OGCM), which show heat intake, existence of sea-ice, and carbon
dioxide absorption by oceans (Hancock, 2000).
The first GCMs were produced in the 1950s. Following Charney’s ENIAC
model, Norman Phillips developed equations for a two-layer atmosphere, which could
more accurately simulate earth’s atmosphere. Building on this Joseph Smagorinsky
and Syukuro Manabe completed a three-dimensional model for a nine-layer
atmosphere in 1965. This was the first model to include radiation impediments of
greenhouse gases such as water (H2O), carbon dioxide (CO2) and ozone (O3). The
model also included heat exchange between the air and water, ice, and land, and parts
of the water cycle (rainfall, evaporation, and runoff). This model was a huge step
forward in climate modeling but still oversimplified some aspects, such as geography
and oceans, by use of a flat, moist surface representing a blended land/ocean (Weart
“General,” 2011).
Around the same time, Yale Mintz and Akio Arakawa were developing a twolayer atmosphere model with realistic geography. Julian Adem was also working on a
model that included many land and ocean processes. Additionally, in 1975, Manabe
with oceanographer Kirk Bryan published a coupled ocean/atmosphere model with
more realistic results but still not including ocean circulation. John Green contributed
work on short-term weather and large eddie actions and Jim Hansen included snow
albedo and snow layer melting in his model. Although all these models were
important advances in the overall understanding of global climate, better knowledge
of atmospheric processes such as cloud effects and vertical air motion is needed for
6 advancement (Weart “General,” 2011). Clouds are currently one of the most
uncertain aspects to predict. They are an example of what is known as a sub-scale
process, meaning it is smaller than the resolution of the grid. This makes it hard to
predict how larger areas will be affected. Currently, the best method to incorporate
cloud processes is to parameterize their large-scale effects, though this is not ideal
(Geerts, 1998).
Scientists then began using models not to predict a static climate but instead to
anticipate how changes in an initial condition, such as the amount of radiation the
planet receives, would change climate components, such as global temperature. In
1967, Manabe and Richard Wetherald created a one-dimensional model with the
purpose of testing changes in global temperature due to changes in carbon dioxide
(CO2) levels. They did this by first running the model at the current level then
repeating it but with a doubled CO2 concentration. The number they came up with for
the “climate sensitivity” (meaning the reaction to doubled CO2 levels) was a
temperature rise of approximately two degrees Celsius. In 1979, the National
Academy of Science appointed a panel led by Charney to compare existing CO2
climate sensitivity models and then develop its own. This panel concluded with high
confidence that if CO2 levels were to double the planet would see a warming of 1.54.5°C (Weart “General,” 2011).
Research on how infrared radiation is absorbed by the atmosphere began in
the 1820s when Joseph Fourier noted that energy in the form of visible light from the
sun heats up the earth’s surface but the energy does not easily escape from the
atmosphere. Simple calculations showed that without its atmosphere earth would
7 have a lower average temperature than is observed. This is due to the greenhouse
effect. In 1859, John Tyndall identified water and carbon dioxide as gases leading to
this effect, or greenhouse gases. In the late 1800s, Svante Arrhenius continued this
research by studying CO2 in past climates and concluding that it was a major factor in
determining the planet’s temperature. After, an intensive numerical calculation
Arrhenius showed that decreasing the atmospheric CO2 by half its current level would
result in cooling of four to five degrees Celsius in Europe. He also, surprisingly
accurately, showed that doubling CO2 would result in a 5-6°C rise in the global
average temperature. Together with Arvid Högbom, Arrhenius looked at the
anthropogenic effects on CO2 levels, concluding that humans were a significant
contributor of CO2, even comparable to geochemical processes (Weart “Carbon,”
2011).
Following these leading results, many scientists continued the research to add
their own outcomes to the overall theory. Critic Knut Ångström ran experiments to
show that CO2 was not a significant contributor of warming, as it is easily saturated to
its absorption limit and it has many absorption bands that overlap with water vapor.
Despite Arrhenius’ response to these criticisms, they were mostly accepted and
research slowed until the mid 1900s. At this time, Guy Stewart Callendar showed a
ten percent increase in CO2 levels since the nineteenth century and concluded that this
rise could explain warming trends that had occurred since this time. He also argued
that some parts of the CO2 absorption spectrum did not overlap with that of H2O.
Following these assertions, the United States military began radiation research in the
1950s by sending infrared radiation through gas filled tubes. The results showed that,
8 in fact, not all CO2 bands overlapped with those of H2O and that CO2 has a greater
effect in the upper atmosphere since H2O is not present there. This is because H2O
has a lower condensation point than CO2 and will condense and rain out before
reaching the upper atmosphere. Physicists Lewis Kaplan and Gilbert Plass then
verified that adding CO2 to the upper atmosphere would change the radiation balance
of the planet (Weart “Carbon,” 2011).
These results led to greater interest in the human influence on rising CO2
levels and the possible impacts here. Roger Revelle began looking into ocean uptake
of carbon and found that chemicals regulating ocean pH would prevent oceans from
absorbing the entirety of excess CO2. Following Revelle’s research, Bert Bolin and
Erik Eriksson published a paper asserting that it could take thousands of years for the
ocean to equilibrate to rising carbon levels. They also estimated that rising industry
was a major cause of increased CO2 levels and was continuing to grow. As mentioned
above, Charles David Keeling substantiated previous evidence with an extensive
study on atmospheric CO2, in which measurements of CO2 levels were taken over the
Hawaiian island of Mauna Loa beginning in 1960. His results showed clear seasonal
variations in CO2 levels caused by intake by the biosphere. He also reported an
increase in overall CO2 levels (Weart “Carbon,” 2011).
In the 1980s collaboration on GMCs began with the National Center for
Atmospheric Research (NCAR)’s Community Climate Models. In 1989, scientists
from the United States, Canada, France, the United Kingdom, Germany, China, and
Japan compared results and showed agreement in temperature and precipitation
changes, until cloud effects were added. Scientists continued to refine these models
9 through research into cloud effects, while at the same time looking into these effects
in past environments. By 2005, many models had accurately showed past climates,
such as the ice age climate, and the role that greenhouse gases had played, but future
climate predictions are still being refined (Weart “General,” 2011).
Currently, Earth System Models are being used to attempt to accurately
predict future climates. These models incorporate the effects of vegetation in terms of
reflectivity (albedo) and moisture transfer. They use more accurate data about the role
clouds play in absorbing radiation as well as increasing albedo. In 2007, a joint
archive was created to increase information sharing between groups working with
different models. The 2007 IPCC report, mentioned previously, incorporated
information from seventeen different groups to produce the most accurate results
possible. Scientists are continuing to improve these models by researching more exact
climate data and incorporating more aspects of climate interactions, to make
increasingly accurate circulation models. (Wert “General,” 2011).
2.2 Understanding Climate Change
In order to understand what role our experiment will play in climate modeling,
we must first look into the principles which determine global climate. There are many
aspects to climate but the one that we are mainly concerned about is global
temperature as this plays a large role in the other elements of climate (such as
precipitation and wind patterns). To calculate the expected average temperature we
first determine how much incoming radiation there is. This depends on the intensity
10 of the sun. To do this we look at the Planck curve for the sun, which gives us the
intensity of radiation as a function of wavelength (fig. 3).
Figure 3- Planck Curve for the Sun (Mihos, 2007)
By taking the derivative of this function at the peak, we get the temperature of
the sun. This process is also known as Wien’s Law. Next we use Boltzmann’s Law,
which entails integrating over the full Planck curve to get the energy radiated over the
whole spectrum of the sun. We then use the fact the energy dissipates at a rate of 1/R2
(where R is the distance between the earth and the sun). The resultant value is called
the solar constant and is the total amount of energy that arrives at the earth. This
value comes out to be 1367 W/m2. Because of the earth’s rotation we multiply S by
πRE2 to get the total amount of radiation that reaches the entire surface. We then
divide by the surface area of earth 4πRE2 because the energy dissipates over earth’s
11 surface. We get the final result of S*=SπRE2/(4πRE2)= S/4 for the total amount of
energy contributed to earth by the sun. However, because of the albedo, or reflectivity
of Earth’s surface, only about 67% of this energy reaches the surface. The earth is in a
steady state called dynamic equilibrium, meaning that the incoming radiation,
(0.67)S*, is equal to the outgoing radiation. Using the Stefan-Boltzmann law, εσT4, to
calculate the outgoing radiation, we have the final equation: (0.67)S*=εσT4, where ε
and σ are constants, ε= 1 and σ= 5.67x10-8 W/K4m2. By plugging in we can solve for
T, (0.67)342=(1)(5.67x10-8)T4. Thus, we find that the average temperature of the
Earth should be somewhere in the area of 255 K. However, from careful observation
we know that Earth’s actual average temperature is closer to 288 K. The reason for
this discrepancy can be attributed to the complex forms of heat transfer in the
atmosphere (Varekamp “Global Climate Change,” 2011).
Unlike the simplified model explained above, in the real climate system, heat
is trapped in the atmosphere as it attempts to escape. Greenhouse gases, such as water,
carbon dioxide, and nitrous oxide, cause warming by absorbing infrared heat, radiated
from the surface. Each greenhouse gas absorbs radiation at different wavelengths.
This is because each of the gases has more than two atoms per molecule; in most
cases they have three. Hence, the position of the atoms can change in relation to each
other by stretching, rotating, or bending, shown in figure 4. When a molecule absorbs
infrared heat its vibrational or rotational energy changes and each wavelength
changes the molecule in a different way. Hence, a molecule’s absorption spectrum
relates to its vibrational and rotational positions. Molecular absorption of infrared
radiation leads to planet surface. After the molecules absorb energy, it is reradiated in
12 all directions. Some of this energy travels back to the surface where it is absorbed. If
an absorbing gas particle collides with another particle before the heat is reradiated,
the energy will be transferred to other gas molecules. This causes atmospheric
temperature to rise, which in turn leads to a rise in surface temperature. (Varekamp
“Global Climate Change,” 2011).
Figure 4- Vibrational States of CO2 Molecules (Maurellis, 2003)
Additionally, the atmosphere allows for heat to be carried away from earth by
conduction and convection. Although this may have the immediate effect of lowering
the earth’s surface temperature, it also has more complex consequences in the
atmosphere, such as convection currents. Convection currents play a large role in
determining earth’s overall climate. Without these currents, we would see a much
13 more stratified change in latitudinal temperatures. This is because at the equator
sunlight enters at an angle nearly perpendicular to the surface. This means incoming
radiation is more intense here than at the poles. Additionally, the albedo is higher at
the poles due to the increased amount of snow and ice and because the angle of
incidence of the incoming light is lower. These two factors increase the amount of
light that is reflected from the surface. Because of the higher amount of radiation
received at the equator, the surface temperatures are higher here. The convection
currents, however, help spread out this heat so it is more evenly distributed over the
entire surface. Of course, we do still have higher temperatures at the equator than at
the poles but it is much less so than we would expect without these currents
(Varekamp “Global Climate Change,” 2011).
There are three main convection cells: the Hadley (tropical), mid-latitude and
polar cells. Figure 5, shows the flow of these cells and how they contribute to the
redistribution of heat described above. These cells are not completely accurate though,
as they do not show the complications of the Coriolis effect, which is shown in figure
6. The Coriolis effect is an apparent force caused by Earth’s rotation. This force
causes air to be deflected to the right in Northern Hemisphere and to the left in the
Southern Hemisphere. The Coriolis effect also causes wind to circulate around high
and low pressure systems rather than flowing directly from high to low pressure.
Hence, the Coriolis effect is the major cause of the complicated wind patterns we see
(NSIDC, 2010).
14 Figure 5- Atmospheric Convection Cells (Bloom, 2010)
Figure 6- Coriolis Effect (NSIDC, 2010)
15 The amount of conduction from the Earth’s surface and convection in the
atmosphere depend on the initial temperature at the surface. Hotter air will expand
more and therefore will rise higher in the atmosphere before it cools. This is because
the hotter air parcels in the lower atmosphere have more potential energy and must do
more work on the surrounding environment before cooling to the same temperature as
a cooler air parcel. This type of cooling is called adiabatic because heat is drawn from
the internal energy of the air parcel. Subsequently, the temperature of the upper
atmosphere increases because warmer air parcels are reaching higher levels (Cohen,
2011).
When the upper atmosphere warms, due to convection, radiation can more
easily escape. This is due to two properties of the upper atmosphere. The first is that
the gas is less dense is so the number density of greenhouse gas molecules is lower
and radiation encounters less absorbing material as it exits the atmosphere. The
second is that the distance to space is less so the radiation energy travels through a
smaller volume of gas to escape, again encountering less absorbing material. The
level in the atmosphere where all radiation can escape without being absorbed is
called the skin level and the temperature here is called the skin temperature. This
temperature is equivalent to the true average temperature that the Earth radiates at, in
other words, this is the temperature calculated in the example above (Varekamp,
“Global Climate Change,” 2011).
Because of the many complexities involved in the natural climate problem, it
can be advantageous to use a simplified model approach. By using our simplified
model of a uniformly lit stationary sphere, we can investigate the principles
16 underlying the real world climate problem. This setup, foremost, allows us to easily
interchange the gases in the chamber. By doing this we can examine how differences
in gas properties affect how heat is transferred in the system. For example, it is useful
to be able to investigate the problem where convection actually takes heat away from
the sphere rather than redistributing it over the surface. However, simplifying the
experiment to such a small scale does result in loss of important traits of the climate
system. The gravity force of our experiment is directed toward the floor rather than
being radially distributed around the sphere. This changes how heat is convected from
the sphere, meaning we cannot simulate convection currents as they occur in the
atmosphere. We also cannot replicate water patterns, which leaves out the possibility
of important contributing factors such as heat intake by clouds and water vapor, and
the greenhouse and albedo effects known to be caused by atmospheric water.
Additionally, we cannot estimate how ocean circulation and heat intake affects the
overall model. Conversely, the scale does make taking temperature measurements
easier and simplifies heat transfer equations. This method is good for getting to the
essentials of atmospheric heat transfer, which can then be applied to larger climate
problems. 17 3. Theory
Variables:
Q‐ heat [W] T‐ temperature [K] m‐ mass [kg] cP‐ specific heat [J/kgK] S‐ surface area [m] σ- Stefan-Boltzmann constant =
5.67x10-8 [W/K4m2]
t- time [s]
θ- angle [radians]
Subscripts S‐ sphere W‐ walls L‐ light G‐ greenhouse gas component C‐ convective component R‐ radiation from the sphere There are many equations required to determine how heat is transferred in our
model earth atmosphere. These equations are important for understanding what may
be causing changes in earth’s average temperature and in order to optimize the
experimental conditions. These equations are similar to the ones used by scientists
using computer models but are simplified due to the smaller scale and the more
simplistic model we are using. Consequently, it is possible for us to go into more
detail in calculating and analyzing these results.
The first important equation is the overall heat balance equation. This
equation helps us determine how heat enters and escapes the sphere and which
variables we will need to investigate further. A generic heat balance equation can be
written ΔQ= Qin-Qout. In this model, we have that the change in heat of the sphere is
equal to the heat received by the sphere minus the heat leaving the sphere. The
change in heat of the sphere is called the heat capacity. This is calculated by
multiplying the change in temperature, the specific heat of the sphere, and the mass
(Chenevert, 2011). So we have:
ΔQ=Qs=ΔTScSm => ΔQ= Qin-Qout= ΔTScSm (1)
18 In the Joop’s World model, there are five ways heat is transferred to or from
the sphere, as shown in figure 7. The most obvious way heat is added is by the light
source, which simulates the sun. Heat is also added by the walls and by greenhouse
gases. The walls radiate heat, which reaches the sphere. The cooling jacket helps
minimize this input but does not completely counteract it. Greenhouse gases add heat
to the sphere by reradiating absorbed heat it in all directions, including back to the
sphere. Heat leaves the sphere in two ways. The surface of the sphere radiates heat
and loses heat to convection. This provides the equations:
Qout = QR+QC
Qin = QL+QW+QG
(2)
Using this equation we can calculate the heat balance equations for a steady state
(ΔQ=0) and for a non-steady state:
Steady State: QL+QW+QG=QR+QC
Non-Steady State: QL+QW+QG-QR-QC= QS = ΔTScSm (3)
From here we want determine how we can calculate each of the components to be
able to analyze which are contributing the greatest effect.
19 Figure 7- Heat Transfer To and From the Sphere
3.1 Radiative Cooling of the Sphere
To calculate the heat radiated from the sphere we use the Stefan-Boltzmann
law. The Stefan-Boltzmann equation is:
QR=σTS4SΔt (4)
This equation is calculated by integrating over the Planck curve. This gives the
energy flux radiated from a surface at uniform brightness. All surfaces radiate heat.
When this energy is not in the form of visible light, as is the case with the sun,
infrared heat is radiated (Weisstein “Stefan-Boltzmann,” 2007).
20 3.2 Radiative Heating from the Walls
The process of calculating the incoming radiation from the walls is similar to
the example of calculating the amount of radiation earth receives from the sun shown
in the introduction. Since the walls are much bigger than the sphere, the radiation
does not all reach the sphere at an angle perpendicular to the surface. To adjust for
this we divide the roof and walls into different sections and use the angle of incoming
radiation to calculate the total radiation that reaches the sphere from each subsection.
We, again, use the Stefan-Boltzmann equation to sum the total heat given off by each
subsection of the walls and then multiply by the sine of the angle:
QW=σT4SΔt sinθ
(5)
We would then divide by the surface area of the sphere because this incoming
radiation is spread over the surface.
3.3 Heat from the Light Source
The heat from the light source is calculated using equation (3). We do this
using non-greenhouse gases before convection sets it so there is no convection or
greenhouse effect in a vacuum. Removing the convection and conduction terms from
equation (3) we end up with:
QL+QW-QR= ΔTScpm (6)
We then solve for QL and plug in from equations (4), (5), and (6) to get:
QL= ΔTScpm+σTs4SΔt- σT4SΔt sinθ
21 (7)
3.4 Convective Cooling of the Sphere
Varibles: v‐ velocity [m/s] x‐ distance [m] g‐gravitational acceleration [m/s2] h‐ heat transfer coefficient [W/m2K] L- characteristic length [m]
k- thermal conductivity [W/mK]
Vc- constant effective velocity [m/s]
q- heat flux [W/m2]
D- diameter [m]
T- temperature [K]
cP- specific heat [J/kgK]
t- time [s]
Q‐heat [W] T*- dimensionless temperature
n‐ constant power r, θ, ϕ- spherical coordinates Nu- Nusselt number
Ra- Rayleigh number = GrPr
Pr- Prandtl number
Gr- Grashof number = gα(TS-T∞)L3/ ν
ρ- density [kg/m3]
α- thermal expansion coefficient [1/K]
ν- kinematic viscosity [m2/s]
κ- thermal diffusivity [m2/s]
Subscripts
∞- points far from the surface,
or values at Pr>>1 (as in VC,∞)
0- values at Pr<<1 (as in VC,0)
S- surface/sphere
r, θ, ϕ- vector component In considering heat transfer in this experiment, we lump the conduction and
convection terms. This is because the convective transfer reflects the amount of heat
transferred by conduction from the sphere to the gases but prove to have a greater
overall influence on the heat transfer of the system. For free convection to occur, the
buoyant force must overcome the frictional and gravitational forces on the gas. We
determine the point when this happens using the Rayleigh number, a dimensionless
value calculated from properties of the gas to determine this value; we begin with
Stokes’ velocity equation (Weisstein “Rayleigh,” 2007):
v=
x 2 g!"!T
x 2 g!!T
=
(8)
3h
"
We find the Rayleigh number at the point where the transport time of heat rising in
the gas is smaller than or equal to the diffusion time across the surface of the gas
22 boundary layer (Weisstein “Rayleigh,” 2007). This means that the heat is not entirely
convected through the layer of gas to the next layer but instead the layer itself begins
to expand and rise. This can be expressed by:
2
!1 $
# x&
2x " 2 %
≤
(9)
!
v
x2
Substituting equation (8) for v we get: 2xκ ≤ 4
" r 2 g!!T %
$
' which then gives us
# 3" &
(Weisstein “Rayleigh,” 2007):
g!Tx 3!
≤ 1/6
"#
(10)
From this we define the Rayleigh number at the critical length to be (Weisstein
“Rayleigh,” 2007):
Ra=
g!TL3!
"#
(11)
From this equation we see the Rayleigh number depends on the gravitational
acceleration, the change in temperature between the sphere and the walls, and the
characteristic length, and properties specific to each of the gases. The characteristic
length is equal to the diameter of the sphere. The thermal expansion coefficient, the
kinematic viscosity, and the thermal diffusivity are properties, which are unique to
each gas and depend on the temperature of the gases (see table 3).
Free convection begins when the Rayleigh number reaches its critical value:
1700 (Weisstein “Rayleigh,” 2007). We would expect convection to begin earliest in
23 gases with a lower kinematic viscosity; however, how gases retain heat (and the
effects this has on the temperature gradient) is also a factor.
Once we have determined the time when convection begins to play a role in
the heat transfer of the system, we then want to calculate how much of a contributing
factor it is. To do this, we must use another dimensionless number, the Nusselt
number, which is used to calculate the coefficient of heat transfer. We have the
following general equation for the Nusselt number in a free convection system:
Nu=
hL
k
(12)
For free convection around a sphere the characteristic length is equal to the diameter
of the sphere (Bengtson, 2011). The Nusselt number can also be calculated
specifically for different geometries. To do this for a sphere we would start with the
constant property energy equation, a form of the Navier-Stokes equation, in spherical
coordinates (Jarfarpur, 1991):
! !T v" !T
v !T $
ρcp #vr
+
+ "
&=
" ! r r !" r sin " !# %
' 1 ! ! !T $ ! 1
!$
!T
1
! 2T *
k ) 2 #r2
+
sin
"
+
& # 2
&
, (13)
!" r 2 sin 2 " !# 2 +
( r ! r " ! r % " r sin " ! r %
The Navier-Stokes equations dictate movement in a fluid by relating velocity,
pressure, temperature, and density and are based on the principles of conservation of
mass, momentum, and energy. The equations were derived by G.G. Stokes and M.
Navier in the 1800s and are typically the basis for convection problems (Benson,
2008). Due to the symmetry of the system, equation (16) simplifies to (Jarfarpur,
1991):
24 ! "T v# "T $
! 1 ! $ ! 2 !T $
! c p # vr
+
& = k # 2 &# r
&
" "# r "# %
" r ! r %" ! r %
When the term
(14)
VC !T
is taken to approximate the left hand side of the equation, we
r !"
have the linearized energy equation (Jarfarpur, 1991):
! V "T $
! 1 ! $ ! 2 !T $
!c p # C
& = k # 2 &# r
& (15)
" r "# %
" r ! r %" ! r % Then plugging in the boundary conditions for thermal diffusivity, distance and time:
D!
k
x
D!
= κ, x=
and t= = and using the dimensionless temperature:
2
!c p
VC 2VC
T*=
T ! T"
we have the new equations (Jarfarpur, 1991):
TS ! T"
1 !T *
1 ! ! 2 !T * $
= 2
#r
&
! !t
r !r " !r %
(16)
We define the Nusselt number in this system as (Jarfarpur, 1991):
Nu(θ) =
qS D
(17)
k(TS ! T" )
And using the Fourier rate equation qS = -k(TS-T∞) δT*/δr and setting t = Dθ/2Vc we
get the equation for the Nusselt Number (Jarfarpur, 1991):
'! 2 $! DV $*
Nu(θ) = 2+ )# &# C &, (18)
(" ! %" !" %+
1
The area averaged Nusselt number can then be found by taking
2
resulting in:
Nu(θ) = 2 + 0.714 !
25 DVC
(19)
!
"
! Nu(! )sin! d!
0
To determine the effective velocity we use the Navier-Stokes equations for
conservation of momentum (Jarfarpur, 1991):
! v"
v! ! v"
! 2 v"
vr + =v
+gαsin(T-T∞)
!r
r !"
!r 2
(20)
We take the two cases where the Prandtl number, Pr, is either small (Pr<<1) or very
large (Pr>>1). For small Prandtl numbers we have the effective velocity (Jarfarpur,
1991):
VC,0 =
" g!!TD (21a)
And for very large Prandtl numbers we have (Jarfarpur, 1991):
VC,∞ =
"
g!!TD
(21b)
Pr
By substituting the term for another dimensionless value, the Grashof number, where
Gr=gβΔTL3/v2 we get (Jafarpur, 1991):
!v$
VC,0 = # & Gr1/2 (Pr<<1)
"D%
(22a)
!v$
1/2
VC,∞ = # & Gr1/2 ( 2 Pr ) (Pr>>1)
"D%
(22b)
We must now use the Churchill and Usagi blending technique, where: 1/(Vc)n =
1/( VC,0)n + 1/(VC,∞)n to obtain the following equation, which is valid for all
Prandtl numbers (Jarfarpur, 1991):
1/2
! v $! Gr $
# &#
&
" %"
%
VC = D 2 Pr 1/n (23) n/2
' ! 0.5 $ *
)1+ # & ,
)( " Pr % ,+
26 The value of n comes from a function of the Prandtl number. We use the value 9/8 as
recommended by Churchill and Chu. By substituting equation (23) into equation (18)
and using Ra=GrPr we get our final equation for the Nusselt number correlation
(Jafarpur, 1991):
Nu =
2 + 0.6Ra1/4
4/9
' ! 0.5 $9/16 *
)1+ # & ,
)( " Pr % ,+
(24)
Once we have calculated the correlation for the Nusselt and the coefficient of heat
transfer from equations (12) and (24) we are then able to calculate the value of heat
transfer using Newton’s Law of Cooling:
Q = hS(TS-T∞)
(25)
3.5 Radiative Heating from Greenhouse Gases
Varibles
T- temperature [K]
I- radiation intensity [W]
C- gas concentration [kg/m3]
z- length [m]
B(ν, T)- Plank’s law
k- Planck’s constant =
6.63x10-34 m2 [kg/s]
c- speed of light = 2.99x109 [m/s]
p- pressure [Pa]
S- line intensity
κ- absorption coefficient [m2/kg]
ρ- density [kg/m2]
τ- optical thickness
n- direction
θ- angle [radians]
ϒ- line width
Subscript o‐ intial G‐ for a particular gas C‐ at the center of the line ν- frequency [1/s]
To theoretically calculate the amount of radiation that is absorbed and then
radiated back to the surface of Earth is an intricate process. To do this we would start
with Beer’s equation (Varekamp, 2011):
I=IOe-Czκ (26)
27 The Schwartzchild equation then gives the change in radiation intensity as a function
of the optical thickness, direction, and frequency as the light travels through a thin
layer of gas (Pierrehumbert, 2010):
d
!1 ( !1 " h %+
I(τ, n, ν) = *tan $ '- [ I(τ, n, ν)-B(ν, T(τ))] (27)
# 11 &,
d!
cos! )
This function gives us the derivative of I with respect to τ, with two parts. The first
part I(τ, n, ν) is called the sink function. This is the amount of infrared radiation
absorbed by the gas. The second part B(ν, T(τ)) gives the radiation that the gas
radiates at a given temperature. We multiply by one over the cosine of the incoming
angle to account for the distance traveled. After having absorbed the radiation, the gas
will then re-radiated the energy. The function B(T) is Planck’s law and is given by
(Pierrehumbert, 2010):
B(T) =
2KTv 2
(28)
c2
It describes the amount of radiation given off by a black body. Figure 8 shows a
Planck curve for a number of bodies at different temperatures.
Figure 8- Planck Curve (Wittke)
28 Both the function B(T) and the optical thickness depend on the frequency (ie
the wavelength) of the light and the absorption spectrum of the gas. The optical
thickness is a function of the absorption coefficient, which is determined by how the
gas absorbs particular wavelengths of light and the Planck law is a function of the
temperature, which influences the wavelength of radiated light (Pierrehumbert, 2010).
Figure 9 depicts the absorption spectra of CO2, Methane, Nitrous Oxide, Oxygen and
Ozone, Water Vapor, and Air.
Figure 9- Greenhouse Gas Absorption Spectra (Johnson, 2009)
29 Generally, to calculate the amount of radiation absorbed and then re-emitted
by each greenhouse gas, we would have to calculate the optical thickness and Planck
function for each wavelength. The optical thickness of each gas depends on its
absorption spectrum, which cannot be modeled by a nice function. We originally
thought we could develop a method to integrate over the absorption spectrum of each
gas and calculate the absorption coefficients, but the functions are very difficult to
model. They are not as smooth as depicted above but instead have many sharp
oscillations. Additionally, collisions between molecules in the atmosphere can affect
the absorption spectrum of a gas. Individual spectral lines are defined by their
position (ie wavelength), shape, strength (ie intensity), and width given by the
expression:
κG(ν, p, T) =
S # v ! vC &
f%
( (29)
! $ " '
Therefore, the simplest way to try to integrate an absorption spectrum would be to
integrate each spectral line separately. This would only be slightly less tedious than
doing the calculations individually for each wavelength (Pierrehumbert, 2010).
30 4. Methodology
Figure 10- Joop’s World
Unlike the GCMs discussed earlier, Joop’s World is a real life tangible model.
It is shown in figure 10. Its purpose is to investigate heat transfer from a sphere on a
small scale. Joop’s World consists of an aluminum sphere placed within a cylindrical
chamber. The tank has a diameter of .26 m and a volume of 0.039 m3 and the earthsphere has a diameter of .025 m. The unit has three legs, and two gas ports. There are
four thermal couples: one located inside of the aluminum sphere, one on the roof
inside the chamber, another taped on the outside wall of the tank, and the last away
from the unit measuring ambient room temperature. The light is a 150-watt light bulb
located in a light box outside the chamber. The light is then transported into the
chamber through two optical fibers, which allow only light energy through and
prevent infrared radiation from entering the system. The light enters at ports 180
degrees apart to ensure equal heating of the world. The outer walls of the chamber
31 contain a water jacket for cooling. The water runs from an elevated container into the
roof of the unit. From the roof the water runs out and into the sidewalls of the
chamber and then through bottom and finally out into a bucket. The outside of the
chamber is covered with a foam-insulating jacket (not pictured above) to prevent
excess heat from entering the system this way.
Five different gases were used. They were Air, Argon (Ar), Nitrogen (N2),
Carbon Dioxide (CO2) and Helium (He). For each gas, I performed two trials: one
using room temperature water in the water-cooling jacket and the other with ice water.
For nitrogen and carbon dioxide, I also did a trial without running water through the
cooling jacket. I ran each condition a couple of times to ensure the best results. For
the run with no cooling, I began recording with Logger Pro® and turned the light on
immediately. For the runs using the water-cooling, I allowed the sphere to cool to a
steady state before switching the light on. This process ranged in time from 30
minutes to an hour and a half. Readings were taken every 12 seconds from each of the
four thermocouples until the system had reached a steady state (the sphere and walls
were no longer gaining heat). I then shut off the light and measurements were taken
as the system cooled. A full run took anywhere from three to five hours.
32 5. Results
From the experimental data, we can investigate how to apply the theoretical
calculations to the experimental conditions. Some important numbers to know for this
are in the following table:
Variable
Mass (m)
Specific Heat-Sphere (Cp)
Surface Area- Sphere (S)
Value
0.3044 kg
0.9 kJ/kg⋅K
.00435 m2
Table 1
5.1 Radiative Cooling of the Sphere
To calculate the heat radiated from the sphere we use equation (4):
QR=σTS4SΔt
When calculating this for the sphere, we use the temperature of the sphere at any
given time and the surface area of the sphere.
33 5.2 Radiative Heating from the Walls
Figure 11- Wall Partition Diagram
Figure 12- Roof Partition Diagram
To calculate the heat coming in from the walls, I first divided the wall, the
roof, and floor surfaces into smaller cross sections, as described earlier. These
partitions are shown in figures 11 and 12. We use equation (6) to sum the total heat
given off by each subsection of the walls. For the temperatures, we use the
34 temperature of the roof for each subsection of the roof and the temperature of the wall
for the subsections of the floor and walls. The surface area is the area of each
subsection. To get this angle we must take the inverse tangent of the opposite over the
adjacent lengths. The opposite measurement is the distance from the center of the
wall to the midpoint of the subsection. For the vertical walls this is: (r1+r2)/2, and for
the horizontal walls it is h or 2h. The adjacent measurement is the distance from the
sphere to the wall, 11 centimeters in both cases. We get the following equations:
(
" r + r %+
Horizontal Walls: Q = σT4roof(r22-r12)sin *tan !1 $ 1 2 '- Δt (30)
# 2(11) &,
)
(
" h %+
Vertical Walls: Q = σT4wall sin *tan !1 $ '- Δt (31)
# 11 &,
)
We have the following information for the horizontal wall cross sections:
R1 (cm)
2.25
5.5
8.25
Disk 1
Disk 2
Disk 3
Disk 4
R2
2.25
5.5
8.25
11
Angle
π/2
tan-1(0.375)=0.35877
tan-1(0.625)=0.558599
tan-1(0.875)=0.71883
Table 2
Surface Area (cm2)
π(2.25)2=23.7583
π(5.52-2.252)= 71.2749
π(8.252-5.52)=118.7914
π(112-8.252)=166.3081
For the vertical walls we have that r1=r2=4.4 giving us an area of 2π(11)(4.4), or
304.1062 cm2, for each cylindrical section. We have two angles that measure
tan-1(.4) = 0.38051 and tan-1(.8) = 0.67474, respectively.
To account for what percentage of the sphere was receiving the heat radiated
from the horizontal versus the vertical walls, I, first, made an estimate and then
checked this with the values before the light was turned on (such that radiation and
convection would not be a factor). I assessed that the radiation from the horizontal
walls was mostly being absorbed by the top and bottom quarters of the sphere and the
35 vertical walls were contributing heat mainly to the middle half of the sphere. These
numbers were fairly consistent with the experimental values with an average error of
4.11%.
5.3 Heat from the Light Source
We calculated the heat contributed from the light source using equation (7).
We assume that the light is fairly consistent for all the trials so we want to find a
single term to use for all the trials. Since this equation only works in non-greenhouse
gases before convection begins, we must average the values found within these
parameters to find a single value that we can use for all the gases and when
convection is occurring. To do this, we applied eq. (3) to nitrogen, helium, and argon
before the Rayleigh number reached the critical value, which happened in around 510 time steps (60-120 seconds). We then averaged these values so we had a mean
light value for each trial of helium, nitrogen, and argon; seven quantities in all. By
averaging these seven values, we found that the light contributed 6.83 J per time step.
5.4 Convective Cooling of the Sphere
To calculate the convection at each time step, we use equations 11, 24, and 25
for the Rayleigh and Nusselt numbers and to find the magnitude of heat transferred by
convection. For the Nusselt number, T∞ is equal to TR since this is the temperature
outside of the convection boundary layer and TS, the surface temperature, is same as
36 the temperature of the sphere, which is why the same variable is used. For the gas
properties used in these equations, each value must be looked up individually for each
gas at a specific temperature. It would have been difficult to get accurate numbers for
each temperature increment so I decided to round to the nearest five degrees, which
resulted in the following table of properties:
Thermal
Conductivity (k)
Thermal
Diffusivity (κ)
Kinematic
viscosity (ν)
Coeff. Thermal
Expansion (α)
Prandtl
2.52E-02
2.55E-02
2.59E-02
2.63E-02
2.05E-05
2.11E-05
2.17E-05
2.25E-05
1.47E-05
1.51E-05
1.56E-05
1.60E-05
3.47E-03
3.41E-03
3.35E-03
3.29E-03
7.17E-01
7.16E-01
7.15E-01
7.14E-01
1.58E-02
1.62E-02
1.66E-02
1.69E-02
1.03E-05
1.06E-05
1.10E-05
1.14E-05
7.83E-06
8.09E-06
8.34E-06
8.61E-06
3.47E-03
3.41E-03
3.35E-03
3.29E-03
7.64E-01
7.62E-01
7.59E-01
7.58E-01
2.51E-02
2.03E-05
1.46E-05
3.47E-03
7.18E-01
20˚
2.54E-02
2.10E-05
1.51E-05
3.41E-03
7.17E-01
25˚
30˚
Ar15˚
20˚
25˚
30˚
He15˚
20˚
25˚
30˚
2.58E-02
2.26E-02
2.17E-05
2.23E-05
1.55E-05
1.60E-05
3.35E-03
3.30E-03
7.17E-01
7.16E-0
1.71E-02
1.73E-02
1.76E-02
1.78E-01
1.94E-05
2.01E-05
2.07E-05
2.13E-04
1.30E-05
1.34E-05
1.39E-05
1.43E-04
3.47E-03
3.41E-03
3.35E-03
3.30E-03
7.17E-01
6.69E-01
6.69E-01
6.67E-01
1.46E-01
1.48E-01
1.49E-01
1.50E-01
1.66E-04
1.71E-04
1.76E-04
1.81E-04
1.14E-04
1.17E-04
1.21E-04
1.24E-04
3.47E-03
3.41E-03
3.35E-03
3.30E-03
6.86E-01
6.87E-01
6.88E-01
6.89E-01
Air15˚
20˚
25˚
30˚
CO215˚
20˚
25˚
30˚
N15˚
Table 3 (Fluid Properties, 2010).
37 5.5 Radiative Heating from the Gas
Joop's World was developed to help us study how greenhouse gases affect the
temperature of the earth, so being able to calculate how much of an effect the
greenhouse gases have on the sphere would be optimum; however, it is very difficult
to separate the actual radiative component of known greenhouse gases from the error
term seen in each experimental run. Because of this issue, I had to calculate this value
purely from experimental data. We start by manipulating eq. (3) to get a new equation.
This equation adds the term QX to express the excess heat in the system:
QX = QR+QC+QS-QL-QW-QG
(32)
For greenhouse gases, we can account for this term in part by the greenhouse effect,
which we are not yet able to calculate; however since we find that this term does not
equal zero in the non-greenhouse gas trials, we cannot assume that it is equal to only
the greenhouse term for these trials. To better determine the greenhouse component,
we first look at the non-greenhouse gas trials (Ar, N, and He) where the QG term will
equal zero. We average the QX terms for the non-greenhouse gases, to get Q X, this is
the average error term. This number should be valid over a wide range of gas
properties since we are averaging gases with diverse properties. By subtracting Q X
from the QX term found for each greenhouse gas we can get an idea of how much
radiation the gases are contributing (QG-exper.).
38 5.6 Tables
After completing all these calculations, we get a large spreadsheet from each
experimental run. The spreadsheets consist of calculations for each method of heat
transfer at each time step. It also takes a separate spreadsheet to calculate the heat
radiated from the walls in each experimental run due to the way the walls are split
into multiple sections. From these tables we get the following data for each run at a
steady state. Graphs for the full trial can be seen in appendix A and comparisons for
the steady-state temperature of each gas are in appendix E.
Ice Water Cooling:
Gas
Ar
Air
N
CO2
He
Time
223.0
110.6
136.6
185.6
143.0
Ts
307.3
305.4
302.3
301.7
292.9
Troof
292.4
293.4
287.9
287.3
286.6
Twall
292.9
286.4
285.2
284.0
QS
0.0
0.0
0.0
0.0
0.0
ΔT
14.9
13.0
14.4
14.4
6.3
QL
6.8
6.8
6.8
6.8
6.8
QR
26.4
25.8
24.7
24.5
21.8
QW
22.0
22.9
21.0
20.7
20.5
QC
148.2
141.3
20.5
435.2
3.0
Table 4
Room Temperature Cooling:
Gas
Ar
Air
N
CO2
He
Time
85.0
128.7
44.0
104.0
82.4
Ts
311.8
310.4
305.0
307.5
303.0
Troof
297.0
296.8
291.5
293.8
297.1
Twall
294.2
288.9
290.9
294.3
ΔT
14.8
14.6
13.5
13.7
5.9
QS
0.0
0.0
0.0
0.0
0.0
QL
6.8
6.8
6.8
6.8
6.8
QR
28.0
27.5
25.6
26.5
25.0
QW
23.3
24.0
23.2
22.6
23.6
QC
136.9
279.7
17.4
366.0
2.3
Ts
309.7
309.8
Troof
297.0
296.4
Twall
292.9
293.1
ΔT
12.7
13.4
QS
0.0
0.0
QL
6.8
6.8
QR
27.2
22.8
QW
23.3
23.3
QC
101.3
336.3
Table 5
No Cooling
Gas
N
CO2
Time
86.0
97.2
Table 6
39 6. Discussion
By observing the graphs for heat transfer, in appendix A, we find that the
convective term appears to be much larger than the other terms. This can likely be
attributed to the placement of the thermocouple. The thermocouple is currently placed
on the roof of the chamber in such a way that it is touching the metal walls as well as
the atmosphere. Since the walls are being cooled, the thermocouple reads a lower
temperature then would be shown if only the temperature of the atmosphere was
being expressed. Due to the colder temperature reflected here, the ΔT (|Ts-Tr|)
expression is much higher than it should be. The graphs in appendix B show that this
affects the convection term by a quadratic relationship.
We can investigate this by calculating the experimental value of convection.
Using eq. (3) and the fact that there is no greenhouse gas component, we write an
equation for the balance term, which equals the experimental value for conduction:
Balance = QC(exper.) = -QR-QS+QL+QW
(34)
After we have found the balance term for each trial, we can calculate the percent
error between the experimental value (balance) and the theoretical value (QC) with the
common equation: %error= (experiment value-theoretical value) x100
theoretical value
Gas
Ar- Ice Water
N- Ice Water
He-Ice Water
Ar-Room Temp
N- Room Temp
He- Room Temp
N-No Cooling
QS
0.0
0.0
0.0
0.0
0.0
0.0
0.0
QL
6.8
6.8
6.8
6.8
6.8
6.8
6.8
QR
26.4
24.7
21.8
28.0
25.6
25.0
27.2
QW
22.0
21.0
20.5
23.3
23.2
23.6
23.3
Table 7
40 QC
148.2
20.5
3.0
136.9
17.4
2.3
101.3
QX
145.8
17.4
-2.6
134.8
13.0
-3.1
98.4
Bal
2.4
3.1
5.6
2.2
4.4
5.5
2.9
%Error
-98.37
-84.80
86.87
-98.43
-74.60
134.19
-97.14
We also see the QX term in this table, which represents the excess heat after summing
all the terms. There is no clear explanation for this term in the non-greenhouse gases
but it helps us assess the greenhouse effect in the other trials. By using the method
described in the section above, we average QX in table 7, to get a value of 161.8, and
subtract this from the QX for air and CO2. This results are in the following table:
Gas
Air-Ice Water
CO2- Ice Water
Air-Room Temp
CO2- Room Temp
CO2-No Cooling
QS
0.0
0.0
0.0
0.0
0.0
QL
6.8
6.8
6.8
6.8
6.8
QR
25.8
24.5
27.5
26.5
22.8
QW
22.9
20.7
24.0
22.6
23.3
QC
141.3
435.2
279.7
366.0
336.3
QX
137.3
432.2
276.4
363.1
329.0
Bal.
4.0
3.0
3.4
2.9
7.3
Qx- Q X
-24.47
270.43
114.56
201.25
167.22
Table 8
The table is not a perfect representation of the radiative effects from the gases but it
does show that there is likely some contribution to the warming of the sphere.
To further investigate the convective and greenhouse components, I also
examined how the specific heat (the amount of energy it takes to increase the temperature of one gram of the gas by one Kelvin) affects the sphere temperature and the dimensionless numbers associated with convection. These
graphs are in appendix C. Specific heat was not a variable I used in earlier
calculations but since it dictates how the gas takes up heat it is probable that it would
have an effect on the way the gas convects heat. Table 9 and 10 shows the specific
heat, Rayleigh, and Nusselt number of each gas at the steady state.
Gas
Argon
Air
Nitrogen
Carbon
Dioxide
Helium
Specific Heat [kJ/kg⋅K]
0.52
1.01
1.04
0.85
Ts
307.3
305.4
302.3
301.7
Troof
292.4
293.4
287.9
287.3
Rayleigh
28936.34
21285.32
22288.97
94872.22
Nusselt
3304.77
2450.29
2566.45
10994.28
5.19
292.9
286.6
142.72
18.34
Table 9- Ice Water Cooling
41 Gas
Argon
Air
Nitrogen
Carbon
Dioxide
Helium
Specific Heat [kJ/kg⋅K]
0.52
1.01
1.04
Ts
311.8
310.4
305
Troof
297
296.8
291.5
Rayleigh
26455.48
22166.41
25765.24
Nusselt
3021.48
4251.22
2966.73
0.85
307.5
293.8
83332.73
9476.78
5.19
303
297.1
97.73
42.3
Table 10- Room Temperature
The first graph in appendix C shows the general trend relating specific heat
and sphere temperature at a steady state for non-greenhouse gases. The greenhouse
gases, notably air, are then shown as outliers off this curve. The trend shows that
gases with lower specific heat have a higher sphere temperature. This is this
consistent with what we know about specific heat. The gases with lower specific
heats require less heat from the sphere to raise their temperature enough for
convection to begin. This means if the sphere loses one watt of energy to the
surrounding gas, a gas with lower specific heat will show a greater rise in temperature
and convection will begin sooner. Hence, greater amounts of convection can occur
with less energy removed from the sphere. The greenhouse gas terms do not lie on
this curve because radiation from the gases also affects the overall temperature of the
sphere.
The graphs showing the relationships between specific heat and the Rayleigh
and Nusselt numbers at the steady state, also shown in appendix C, verify this finding,
except in the case of carbon dioxide. From these graphs, we see the general trend that
specific heat is inversely related to the Nusselt and Rayleigh numbers. That is to say
that gases with lower specific heats tend to have higher Rayleigh and Nusselt
numbers, which indicates that convection will occur sooner in gases with lower
specific heat. As shown above, this can be explained because these gases require less
42 energy to begin convection. The interesting part of these graphs is how large both
these numbers are for carbon dioxide. These numbers not only lie off the trend line,
but also are vastly above the others.
Lastly, to verify the results for the greenhouse gas component it was helpful to
plot the excess heat term at the steady state for each gas. This graph, in appendix D,
also showed evident of the greenhouse effect. This is seen because the value for QX is
higher for carbon dioxide than the other gases. We also see a large QX in the air term.
We would not automatically expect to see this effect on such a small scale because
CO2 concentrations are very small, around 380 ppm, in air. Another interesting aspect
of these graphs is that argon also had high QX values. Though not as significant as
carbon dioxide, this might be an interesting phenomenon to investigate in later
experiments.
43 8. Conclusion
Although we did not have as conclusive evidence as we had hoped, the
experiment was successful overall. In previous experiments done with this model, the
greenhouse term and convective term could not be separated through calculation.
Verifying how to theoretically calculate the convective component was crucial for
showing the existence of the radiation component in this model. Additionally, by
calculating the theoretical convective term, we were able to compute an experimental
value for the greenhouse effect that was not possible before. Although, we were not
able to directly calculate this term, we saw that the excess heat term was much greater
in the greenhouse gas trials than in the others. Even after accounting for the error seen
in the other trials, there was a significant excess heat term that could, at least in part,
be due to the greenhouse effect. It would be optimum if the theoretical calculations
concerning the greenhouse gas component could be worked out. Although, this would
be a complicated process, the results would be valuable for verifying the results found
in the experiment.
Additionally, when comparing specific heat with the steady-state sphere
temperature we saw that air was located significantly above the expected trend. This
suggested that the greenhouse effect is contributing to warming of the sphere causing
this temperature to be higher than expected. This is particularly interesting because
we do not see similar temperatures in nitrogen, the main constituent of air.
Additionally, because nitrogen accounts nearly 80% of the composition of air, these
gases have very similar properties, as seen in table 3. This means we cannot attribute
the unusual temperatures seen in the air trial to another property of the gas. Instead, it
44 is possible that air used in the experiment was moist air and not dry air as was
assumed. This would increase the greenhouse effect, as H2O is a powerful greenhouse
gas, and would explain why we see such high temperatures in these trials.
On the contrary, if the case explained above were in fact true, we would also
expect to see higher temperatures in the case of carbon dioxide; however, there is a
plausible explanation for why this result is not seen here. We see from the
experimental and theoretical values of convection, and from the graphs in appendix C
that CO2 has noticeably large values for convection in its pure form. This could be
attributed to its low kinematic viscosity and high Prandtl number. The large amount
of convection would take away excess heat from the sphere and could mask the
effects of greenhouse radiation we expect to see. This result shows how critical it is to
understand the convection term in our small-scale model.
In future experiments, refining the technique of calculating the convection
term and modifying our model to account for the error found could vastly improve the
results for the greenhouse term. To do this it would be important to relocate the
thermocouple inside the roof so that it is completely in the atmosphere and not
touching the cooled wall. It would also be helpful to have more thermocouples inside
the atmosphere, on the bottom of the chamber and possibly close to the horizontal
wall. This would give us more information about how the gases are flowing inside the
chamber. Furthermore, it would be ideal to have two thermocouples measuring the
influx and outflow of water. This would assist us in calculating the magnitude of
cooling done by the water system and could possibly contribute to more accurate
convection calculations.
45 Although convection proved to be very important in understanding the heat
transfer in this model, it does not provide much insight into global convection. The
convection patterns are much more complicated in the real-world climate, as they are
not only affected by gas properties but also factors such as high- and low-pressure
systems and stratified latitudinal temperatures. Additionally, gas concentrations
change by very little, on the order of parts per million, in the atmosphere. Although,
this has the potential to affect the greenhouse gas component, it is not likely that it
would have observable effects on convection, such as were seen in this experiment.
From the experiment, we have gained a basis for understanding the principles
underlying climate science and have made steps towards showing how greenhouse
gases are contributing to the heating of the earth. Once we have a more precise grasp
on the greenhouse gas terms, these results could be directly applied to the climate
crisis we are facing today. Understanding, how these feedbacks work on a small scale
will make a large impact on how we view earth’s climate system and will help us
make more accurate predictions as to the consequences if greenhouse gas levels
continue to rise as they are now.
46 Appendices Appendix A Temperature and Heat Transfer v. Time
Argon (Ice Water Cooling): 47 Air (Ice Water Cooling):
48 Nitrogen (Ice Water Cooling):
49 Carbon Dioxide (Ice Water Cooling):
50 Helium (Ice Water Cooling):
51 Argon (Room Temperature Cooling): 52 Air (Room Temperature Cooling):
53 Nitrogen (Room Temperature Cooling):
54 Carbon Dioxide (Room Temperature Cooling):
55 Helium (Room Temperature Cooling):
56 Nitrogen (No Cooling):
57 Carbon Dioxide (No Cooling):
58 Appendix B ΔT v. Convection
Ice Water Cooling:
59 60 Room Temperature Cooling: 61 62 No Cooling: 63 Appendix C Specific Heat at the Steady State
64 65 Appendix D Excess Heat at the Steady State
Appendix E Temperature at the Steady State
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