Farbo, Jones Determining an Effective Method to Cool Soda Using Household Items James Farbo & Ryan Jones Department of Chemical Engineering, The Pennsylvania State University, University Park, PA, 16802 CH E 350, Section 1, Dr. Costas Maranas Prompt Using household items, design a way to cool a can of your favorite carbonated beverage from room temperature to a suitable drinking temperature of 3°C. Background/Hypothesis Our task was to cool an aluminum can of our favorite carbonated beverage, in this case, light beer, from room temperature to a suitable drinking temperature of 3°C. The general idea that comes to mind is to cool the can with an ice bath, however this takes far too long. Consequently, we considered various heat transfer methods to increase the rate at which heat is transferred from the can to the surroundings. Some 1
Transcript
Farbo, Jones
Determining an Effective Method to Cool Soda Using Household Items
James Farbo & Ryan Jones
Department of Chemical Engineering, The Pennsylvania State University, University Park, PA, 16802
CH E 350, Section 1, Dr. Costas Maranas
Prompt
Using household items, design a way to cool a can of your favorite carbonated beverage
from room temperature to a suitable drinking temperature of 3°C.
Background/Hypothesis
Our task was to cool an aluminum can of our favorite carbonated beverage, in this case,
light beer, from room temperature to a suitable drinking temperature of 3°C. The general idea
that comes to mind is to cool the can with an ice bath, however this takes far too long.
Consequently, we considered various heat transfer methods to increase the rate at which heat is
transferred from the can to the surroundings. Some brainstormed ideas included putting the can
in an ice bath and creating forced convection by introducing an outside force, creating a packed
and/or fluidized bed tightly surrounding the can, and implementing highly conductive annular
fins on the surface of the can to increase the convective heat transfer coefficient. While
theoretically possible, these ideas weren’t feasible using strictly household items.
Drawing from our knowledge of chemical engineering thermodynamics, we considered
ways in which to drop the temperature of an ice bath by freezing point depression. We
hypothesized that by adding table salt, a common household item, to an ice bath, we could drop
the freezing point of water significantly below its pure liquid value of 0°C at 1 atm, thus cooling
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the can more quickly. In theory, by adding salt (a solute) to an ice bath, the ice-solute mixture
becomes liquid at temperatures where pure water would be a solid.1 On a molecular level, this
phenomenon occurs because the addition of salt disrupts the dynamic equilibrium between water
and ice.2 Salt’s solubility in water prevents some water molecules from freezing; contrarily, the
rate at which ice melts is unaffected. Because melting occurs more rapidly than freezing, it will
continue to occur unless the temperature of the mixture drops low enough to re-establish solid-
liquid equilibrium.
This thermodynamic argument prompted us to design an experiment that tested the
effectiveness of cooling a can of light beer a in a salt-ice-water mixture versus a pure ice-water
mixture. Considering the phase map of salt-ice-water, shown below in Figure 1, we chose to cool
the can at 10% and 20% salt concentrations (by weight) and compare results to a pure ice-water
solution.2
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Figure 1. Phase map of a salt-ice-water solution.2 Salt becomes saturated in ice-water solution at 23.3% NaCl (by weight); the lowest possible mixture temperature is -21.1°C.
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These concentrations were selected because they promote significant freezing point
depression, and are close to but not exceeding the solubility limit of 23.3% salt (by weight).
To test the validity of our experimental setup, theoretical proof of concept is essential.
Arguably, the most important factor necessary for correctly modeling our experiment is
determining the value of the convective heat transfer coefficient, h, of the ice water bath. Due to
the absence of an external driving force, the liquid experiences no motion; thus, convection is
assumed to be free. Furthermore, the can is cylindrical and can therefore be modeled as a long
horizontal cylinder. Equations 9.33 and 9.34, shown below, have been empirically derived using
dimensionless variables and can be used to model external free convection for a long horizontal
cylinder at the film temperature.3
(9.33)
(9.34)
In Eqs. 9.33 and 9.34, RaD represents the Rayleigh number, a dimensionless parameter
that characterizes buoyancy-driven flow (i.e. free convection) involving both laminar and
turbulent flow, shown below in Eq. 9.25.3
(9.25)
The properties necessary to calculate the Rayleigh number (i.e. the fluid’s kinematic
viscosity, thermal diffusivity, and thermal expansion coefficient) are often tabulated for easy
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access.3 If they are not tabulated for a specific species, they can often be calculated by utilizing
common correlations (listed below) and linear interpolation. For the saltwater solutions,
obtaining the necessary constants requires referencing the literature for thermal properties
dependent on both temperature and percent salinity4 in addition to both linear interpolation and
extrapolation as necessary. By far the most troublesome constant to calculate is the thermal
expansion coefficient. This can be found by plotting the density of the fluid as a function of
temperature and dividing the negative of the slope of this curve by the density of the fluid
(equation 3) (Figures S2 and S3).
(1)
(2)
(3)
At this point, the Rayleigh number can be used to calculate the averaged Nusselt number
for a large range of Rayleigh numbers in both the laminar and turbulent flow regimes via
equation 9.34, and the average convective heat transfer coefficient can be subsequently found by
equation 9.33.
With the convective heat transfer coefficient of the ice-water bath in hand, the most
logical approach for assessing the temperature distribution inside the aluminum can is to
consider an effective heat transfer coefficient, U, which incorporates the convective loss from the
bath and the conductive heat loss from the aluminum can. Ultimately, the rate of heat loss is
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dependent on the least resistive element of the thermal circuit. Therefore, if the thermal
resistance of the aluminum can is negligible compared to that of the liquid inside the can, it may
be neglected.
R = {l} over {k} + {1} over {h} (4)
U= 1R } ¿(5)
The theoretical cooling time can then be calculated in one of two ways: from the
approximate solutions to the one-dimensional transient conduction temperature distribution for
an infinite cylinder or by numerical integration of the one-dimensional partial differential heat
equation in cylindrical coordinates.
(6)
The latter approach is more user friendly as it doesn’t require the copious calculations
associated with analytically solving transient systems. In short, this method analyzes one radial
slice of the can at a time, ultimately mapping out the temperature distribution within the can at
several different positions; these temperatures can then be averaged so that the liquid can be
treated as having a uniform temperature at all times. Additionally, the time and radius increments
can be adjusted accordingly to increase the accuracies of the local derivatives and provide similar
answers to those that would be obtained analytically.
Experimental
Cooling a can of light beer—pure ice bath
Firstly, we obtained three cans of light Busch beer and allowed them to equilibrate to
room temperature for 30 minutes. Upon obtaining laboratory access from Professor Wayne
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Curtis, we then created an ice bath by adding 4.5 Kg of water and 1.5Kg of ice to a rectangular
container and allowed it to equilibrate for 5 minutes. Using a thermocouple, we measured the
initial temperatures of both the ice bath and the liquid beer. Subsequently, we suspended and
submerged the can of beer into the ice bath using a ring stand and clasp (Figure S1). Upon
complete submersion, we immediately started recording time using a stopwatch while
simultaneously measuring the temperature of the beer until it reached a refreshing drinkable
temperature of 3°C.
Cooling a can of light beer—10 and 20 weight % salt solutions in a water/ice bath
Initially, 5 L of pure water and 4L of pure ice were added to a new rectangular container
(8.666 kg of H2O by weight) and allowed to equilibrate for five minutes. Using a balance, 0.962
kg of salt was weighed and added to the ice bath, creating a 10% salt solution. The solution was
then stirred using a spatula by hand for two minutes to promote the dissociation of salt. Next, the
initial temperatures of both the bath and the second room temperature beer were recorded. Note
that we ensured the initial temperatures of the beers were the same for each trial to ensure a
concise comparison. Finally, the beer was submerged and tested similarly to that of the pure ice
bath.
Analogously, mixing 5 L of pure water, 5.5 L of pure ice, and 2.501 kg of salt created a
20% salt-ice-water solution. Subsequently, the procedure was conducted identically to that of the
previous two trials.
Results and Discussion
Completion of the three experimental trials provided data that supported our original
hypothesis. Cooling the cans of beer to drinkable temperatures occurred much faster in ice water
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with increased salt concentrations relative to pure ice water, shown below in Table 1 (All
theoretical times were found using the cylindrical PDE solver depicted in Figure S4).
Pure Ice/Water 10% Salt in Ice/Water 20% Salt in Ice/WaterTi,can (°C) 25.3 25.3 25.3Tf,can (°C) 3.0 3.0 3.0Tbath (°C) 0 -7.7 -13.5ExperimentalTime (s)
Notice that the time required to cool a can of beer in 10% Salt in Ice/Water was less than
half of the time needed for a can of beer in Pure Ice/Water, despite the bath temperature
difference only being approximately -8°C. This indicates that there must be a strong driving
force, other than the increase in temperature difference, drastically improving the ability for the
bath to absorb heat from the can. Upon completion of theoretical modeling, a couple noteworthy
causes stand out.
Firstly, significant discrepancies appeared when the mean free convective heat transfer
coefficient was calculated from Eq.’s 9.25, 9.33, and 9.34 for each solution, as seen above in
Table 1. The large difference between h for pure ice/water and the two salt solutions helps
account for the significant difference in cooling times--the higher average h in the salt solutions,
relative to the pure ice/water, facilitated much quicker heat transfer from the can. From equations
9.33, 9.34, and 9.25, is it evident that h is directly proportional to the temperature difference
between the initial can temperature and the bath temperature. Inherently this makes sense: the
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Table 1. Experimental and theoretical data for cooling a soda can at various mixtures of water, ice, and salt
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colder the bath temperature the faster it will cool the can. However, it is notable that h decreases
by ~5% when increasing the salt concentration from 10 to 20%. This may seem counterintuitive,
but h as the salt concentration increases, other properties such as the density, viscosity, and
Prandtl number also increase and the thermal conductivity decreases, negatively affecting the
convective heat transfer coefficient. Despite this decrease in h, the drastically lower temperature
of the 20% salt bath dominates and drives the can to cool faster.
The net result of this experiment is that salt depresses the freezing point of water and thus
accordingly cools the room temperature can more quickly than an ice/water bath alone.
However, there is clearly diminishing returns associated with the amount of salt used. The 10%
and 20% salt solutions cooled the fluid in the can in approximately 8.7 minutes and 6.9 minutes,
respectively. Thus, as the saturation concentration of salt in water (23.3 weight %) is approached,
the cooling effects are diminished as the solution becomes denser, more viscous, and less
conductive. Ultimately, the saltwater solution drastically expedited the cooling process compared
to the ice/water bath which took nearly 22 minutes. The large discrepancies between the
experimental times mentioned above and the theoretical calculations (each yielding an
approximate percent error of 25-30%) is indicative of the many assumptions and approximations
mentioned in Table S1.
Conclusion
Through both experimental and theoretical verification, a beer can is cooled significantly
quicker in 10% and 20% salt in ice/water solution compared to a pure ice/water solution (see
Table 1). Furthermore, our theoretical model of each system proved to be fairly accurate,
producing reasonable percent errors, with respect to the experimental times (see Table 1). Given
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the large amount of assumptions (Table S1), a large error was expected, however all calculations
proved to be reasonable. In future beer can cooling experiments, we would use a larger scale
ice/water bath to decrease the dependency of T∞ on time. Furthermore, we would insulate the
bath to minimize heat loss and stir the salt solutions more effectively to facilitate the dissociation
of sodium chloride.
Supporting Information
Assumptions Reason for Assumption
Error Associated with Assumption
Can is an infinite cylinder; assume adiabatic tip (i.e. L>>D) for all trials. Furthermore, assume that a vertical cylinder produces the same h as a horizontal cylinder.
Modeling convective heat transfer coefficient for free convection of a cylinder
L is not that much greater than D; We felt that a horizontal infinite cylinder approximation would be more accurate than a vertical plane wall approximation. Because the can is vertical in the experiment (not horizontal, as requested by the infinite cylinder approximation formula) and horizontal in theoretical calculations, there is a slight error produced.
Beer and water are thermally equivalent
No strong temperature dependent thermal properties of beer (but the major component is water)
Slight discrepancies in thermal conductivity, density, and heat capacity due to the presence of syrups, dissolved carbon dioxide and alcohol
Treat water and ice in mixture as purely water at Tfilm
Majority of ice in bath had melted by the time the can was submerged
There was some ice remaining and we couldn’t quantitatively account for its properties in the bath
T∞ is not a function of time and the ice-water bath can be treated as infinite
We couldn’t solve the PDE if T∞ varied with time.
In practice, T∞ indeed changed as a function of time because ice melted in the bath
Neglect soda can resistance when calculating h
hcan accounts for less than .15% of the total thermal resistance
Negligible
Solubility of salt in ice/water bath is Allows us to assume Not all of the salt dissolved causing
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Table S1. Assumptions made to carry out theoretical calculations
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100% (applies only to salt solutions) 100% of salt added is in solution i.e. it’s a uniform mixture
error in calculations involving density
Can temperature is initially uniform This assumption allows for the calculation of the Rayleigh number (Ts=Ti) and is fairly accurate for cans that have been stored at room temperature.
It is likely that the temperature at the centerline of the can was slightly less than the temperature at the surface due to aluminum’s high conductivity and radiation effects in the room
Can is 100% aluminum Allows for simple calculation of resistance for can and ultimately, allows us to neglect it.
Negligible
Fluid in can is not moving We can now treat the fluid as a solid object, thus simplifying calculations
At any given time, the fluid is going to want to move around and equilibrate due to inherent buoyancy forces
Supplemental Figures
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Figure S1. Experimental setup for cooling soda can in ice/water/salt bath
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Figure S3. Plot of density as a function of temperature (20% salt solution)
Figure S2. Plot of density as a function of temperature (10% salt solution)
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Sample Calculations
Provided as supplementary data
References
1.) Matsoukas, T. Fundamentals of Chemical Engineering Thermodynamics, 1st ed.;
