User:Mech425queens

Cultural Issues

As always with Appropriate Technology, close attention to cultural issues is extraordinarily important. Several authors have commented on cultural issues related to solar cooker technology transfer. Ramon Coyle pointedly begins his article by discussing a topic often overlooked in discussions of cultural issues related to technology transfer: the culture of the Western promoters themselves. Ramon observes that, especially in the United States, the dominant culture encourages great optimism and a narrow worldview and does not value humility and patience. According to Ramon, optimism "may tempt US-based solar cooker promoters to launch grand schemes with short time frames and too little research, planning and resources."

Local cultural appreciation is of course also important.

Solar Cooker Designs

There are a large number of solar cookers designs available freely on the internet. Some of these can be found on Appropedia. The Solar Cooking Wiki, run by Solar Cookers International, is another repository.

Existing Cookers

Solar cooker designs can be broadly grouped into three categories

• Box cookers
• Curved concentrator cookers
• Panel cookers

In light of the availability of these designs, the goal of the project was set not to provide yet another viable solar cooker design, but rather to study a particular aspect of solar cooker design to determine whether a more optimal design was available.

Assumptions

The following table contains the assumptions made in producing this project and their potential implications on model predictions.

Assumption	Potential effect
One dimensional heat transfer	Inaccuracy due to edge effects
No higher order Fresnel reflections	Model shows lower power transmitted into cooker

Thermodynamic Analysis

The solar cooker was modeled using a one dimensional thermodynamic analysis that accounted for conduction and convection.

Radiation

In designing the model, the effects of radiation were largely ignored. Radiation effects can be attributed to the Stefan-Boltzmann law given below. The Stefan-Boltzmann law relates the total power radiated by a surface to its temperature. The radiative power is also related to the emissivity of the material (the extent to which it deviates from an ideal blackbody) and to the surface area of the material. Emissivities of common construction materials (brick, wood, glass, etc.) range from about 0.8 to 0.94.

$\,P=A\epsilon \sigma T^{4}\,,$

$\,\sigma =5.670400\times 10^{-8}Js^{-1}m^{-2}K^{-4}\,,$

Given an outer surface temperature of 80 °C, the effects of radiative transfer are relatively small.

Convection

The general problem of convection is to understand how heat is transferred between a (generally) moving fluid and another body. The rate of heat transfer is proportional to the difference in temperature between the two bodies in question, as well as a proportionality h, known as the convection coefficient.

$q={\bar {h}}A_{s}(T_{s}-T_{\infty })$

Air flowing over a flat plate forms a boundary layer that grows thicker with distance downstream from the plate edge. Fluid in the boundary flows at a reduced velocity relative to the plate compared with the free stream. Also, temperature gradients exist in the boundary layer, allowing for the emergence of a steady-state temperature T_s at the surface of the plate and a different steady state temperature $T_{\infty }$ in the free stream. The rate of heat transfer via conduction is heavily dependent on the temperature gradient at the surface of the plate, as shown below.

$h={\frac {-k_{f}\partial {T}/\partial {y}|_{y=0}}{T_{s}-T_{\infty }}}$

Since the temperature gradient varies strongly with position along the plate, an average value must be calculated in order to calculate the total heat flow rate. In combination with a number of nondimensional relations, the above allows the calculation of the dependence of the mean convection coefficient on properties of the flow. Incropera et al. determined that the convection coefficient could be calculated as follows:

${\bar {h}}_{L}=0.664{\frac {k}{L}}Re_{x}^{1/2}Pr^{1/3}$

Thermal Resistance Method

Making the simplifying assumption that all conduction and convection processes are one dimensional, it is possible to apply an analogy with electric circuits to solve for the steady-state temperature in the cooker. In the analogy, heat flow corresponds to electric current, temperature corresponds to electric potential and the thermal resistance corresponds to electric resistance. The thermal resistance of an element depends on the type of heat transfer process (conductive, convective, radiative) and geometric factors. Thermal resistances for conduction and convection are given below.

Circuit analogy

Thermal resistance circuit

$R_{cond}={\frac {L}{kA}}$

$R_{conv}={\frac {1}{hA}}$

The circuit describes three parallel paths between the inside of the cooker (assumed to be at thermal equilibrium temperature T_b) and the outside (assumed to be at equilibrium temperature $T_{\infty }$). The path farthest to the left (path A) represents conduction through the insulating walls of the cooker and radiation to the surrounding air. The factor of two is present as this path represents conduction through two separate sides of the cooker. The center path (path B) represents conduction through the insulating wall directly to the ground. The right hand path (path C) represents conduction through the glass surface and convection from this surface.

The total resistance of the network is given below in terms of the resistances of paths A, B and C, and the temperature of the cooker is given in terms of the total resistance, q_rad and the outside temperature.

$R_{T}={\frac {ABC}{AB+AC+BC}}$

$T_{b}=q_{rad}R_{T}+T_{\infty }$

It is acknowledged that the aforementioned analysis presents a simplified view of the heat transfer occuring in the system. The assumption of one-dimensional heat transfer, while likely valid at the center of each surface, breaks down where the surfaces intersect. Also, complexities in the geometry of the device make it difficult to apply even this simplified model. In reality, the heat transfer characteristics would depend on the orientation of the cooker with respect to the mean flow direction. However, this characteristic was considered unimportant and an average taken in order to eliminate dependence on the orientation of the cooker. Despite its drawbacks, this model still allows for an estimation of the temperatures achieved in the cooker under different conditions.

Sun Elevation

In order to determine cooker temperatures at different times of day and at different locations on the Earth, it was necessary to construct a model of the sun's position as it varies with these variables. Of particular importance was the solar elevation angle, which gives the angle between the straight line connecting the sun to the observer and the flat plane of the Earth. The elevation angle has a strong impact on cooking temperatures. The solar elevation angle $\phi$ can be calculated as shown below

$sin(\theta )=cos{h}cos{\delta }cos{\phi }$

Fresnel Reflection

Side view with ray

Incident radiation partially reflected according to Fresnel reflection

The effect of the sun's rays striking the solar cooker at different angles is extremely important to this modeling exercise. The reflective behaviour of these rays was modeled using the Fresnel equations shown below.

$R_{s}=\left[{\frac {n_{1}\cos \theta _{i}-n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{i}\right)^{2}}}}{n_{1}\cos \theta _{i}+n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{i}\right)^{2}}}}}\right]^{2}$

$R_{p}=\left[{\frac {n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{i}\right)^{2}}}-n_{2}\cos \theta _{i}}{n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{i}\right)^{2}}}+n_{2}\cos \theta _{i}}}\right]^{2}$

$R={\frac {\left(R_{s}+R_{p}\right)}{2}}$

The above equations give reflection coefficients for light crossing a boundary between two media. R_s gives the coefficient for light polarized perpendicular to the interface and R_p gives the coefficient for light polarized parallel to the interface. R gives the coefficient for unpolarized light, where n₁ and n₂ are the indices of refraction of the first and second media, respectively and $\theta {}_{i}$ is the angle of incidence. Wikipedia provides a useful diagram of the variables here.

In order to deal with the two boundaries (air - glass and glass - air), the Fresnel equations were simply applied twice. The angle of refraction at the first interface was calculated from Snell's law and this angle was accounted for when calculating R at the second interface. No second-order reflections were taken into account in calculating the transmitted intensity, which was given by the equation below.

Simulation software

Simulation software was written in Java. The source code archive and runnable JAR file can be found here. <-- Wow! what a bunch of crap :) btw where's the link? as if it serves some purpose.

Results

Plot of temperature (°C) as time of day and window angle vary

Plot shows the temperature attained by the cooker with various window angles at various points during the day of April 14 2010 at 12°N, 1°W

The simulation code was used to predict the temperature of the solar cooker under a variety of conditions.

Future Work

As has been stated, this model is not an accurate representation of a solar simulator. Temperatures predicted by the model are much too high to reflect physical conditions. As was also mentioned, the model is notably lacking in its neglect to treat radiative losses from the cooker. This shortcoming needs to be addressed. Additionally, the model needs to be compared with several different box cooker designs operating under different conditions in order to make more reliable predictions.

According to the model, the design space is specified by 5 variables: Latitude, Time of day, Window angle, Cooker temperature, and Day of the year. A comprehensive implementation of the model would allow the user to specify design constraints in terms of any of the variables and assess the impact of these constraints on the other variables. It would also allow the user to optimize the window angle for these different design constraints. In theory, this is not difficult to do. Practically, there are many challenges. There is a computational challenge in calculating the temperature at every point in time during the year with reasonable resolution, at every latitude, and with every possible window angle. There is a challenge in terms of how to display data with so many free variables in a useful way. A potential implementation might allow the user to specify a cutoff temperature (i.e. greater than 100°C), a time of day over which the cooker must be functioning, and a particular day of interest and then obtain a plot similar to the one below. Such a plot would be highly useful in making a design tradeoff between optimal power at a particular location and effectiveness over a large region.