FINAL PROJECT REPORT – Team 10 Project Title: Open Hearth Surgery

Project Staff: Project Leader (Optimization): Marc Paré Project Member (Combustion): Christopher Beebe Project Member (Fluids): Nicholas Kretchmar Project Member (Heat Transfer): Matthew Redmond

TABLE OF CONTENTS

EXECUTIVE SUMMARY .....................................................................................................1

PROJECT OBJECTIVE .............................................................................................................. 1

PROJECT BACKGROUND ....................................................................................................... 1

DELIVERABLES ....................................................................................................................... 1

APPROACH ............................................................................................................................ 2

SCHEDULE ............................................................................................................................. 2

SUMMARY OF KEY CONCLUSIONS AND RECOMMENDATIONS ............................................ 3

VARIABLE LIST ..................................................................................................................4

DISCUSSION .....................................................................................................................5

BACKGROUND ...................................................................................................................... 5

SCHEMATIC ........................................................................................................................... 7

APPROACH ............................................................................................................................ 8

EQUATIONS ......................................................................................................................... 11

OPTIMIZATION .................................................................................................................... 21

CONCLUSION ...................................................................................................................... 25

REFERENCES ........................................................................................................................ 26

APPENDICES .................................................................................................................. 28

APPENDIX A ........................................................................................................................ 28

APPENDIX B......................................................................................................................... 31

APPENDIX C ......................................................................................................................... 34

APPENDIX D ........................................................................................................................ 36

APPENDIX E ......................................................................................................................... 37

APPENDIX F ......................................................................................................................... 38

APPENDIX G ........................................................................................................................ 39

1

EXECUTIVE SUMMARY

PROJECT OBJECTIVE

This project is centered upon the comprehensive design of an Improved Cook Stove (ICS), for implementation in developing nations to address the issues posed by the currently used three stone fire. Such a design is developed around three generic, measurable benchmarks: efficiency, particulate emissions, and economic sustainability.

The project objective includes specification of the following: design requirements and materials, efficiency and particulate emission improvement as compared to current benchmarks of the three stone fire, and an evaluation of economic sustainability in comparison to pre-existing initiative successes.

PROJECT BACKGROUND

Roughly half of the world’s population currently uses primitive methods of cooking that present major problems associated with inefficiency of fuel, particulate emissions and their impact on health, and economic sustainability. The design team has taken these specific problem criteria into direct consideration and aims to create a design to remedy all of them, as mentioned in the objective. The problem background and technical background of this project are detailed in subsequent sections of this report.

DELIVERABLES

Concept: March 5th The concept deliverable included critical areas of exploration for the stove design project.

The major areas of exploration included specific quantifiable parameters that were considered the most essential dependent variables. These major areas were centered upon the three criteria detailed in the aforementioned objective, and were combined to contribute to a successful ICS design.

Models: March 19th

Basic design of thermodynamic and fluid mechanics models were presented to address the questions posed by the concept, while satisfying the objective. The key to success for this deliverable was evaluating necessary simplifying assumptions to the applied engineering disciplines.

Equations: April 2nd

Detailed modeling work culminated in parameterized equations answering the questions posed by the concept document. This deliverable included explicit objective and constraint equations. The validity of these equations was tested against experimental results acquired from existing publications. Implementation of these equations in MATLAB was necessary to organize and aid in simulation generation.

Simulation: April 9th

Coupling the detailed thermodynamic and fluid mechanic equations presented simulations of overall stove performance, more specifically the required efficiency. This deliverable also involved the use of MATLAB. The results of the simulation, however, were not intended to quantitatively satisfy the objectives of this project.

2

Optimization: April 16th

An optimization scheme for a final design was devised by linking the coupled simulations and MATLAB equations. These optimizations presented the team with quantitative results satisfying the design objective, and detailing specific values for the chosen parameters. The result of the optimization was a final design suggestion.

Final Presentation: April 19th

A presentation was prepared to communicate our findings. Also included in this deliverable were the possible future steps to deploy the ICS design and business model. The focus of the presentation was the overall success of the design at addressing the needs of Improved Cook Stove technology in the Third World, and fulfillment of our objectives. Furthermore, feedback associated with the final presentation was taken into consideration and is reflected in this report.

APPROACH

While achieving an ICS design, a model was developed for use in assessing critical relationships present in cook stoves, as well as gaining insight into parameter relations and geometric optimization. This model specifically addressed the innovative coupling of a nozzle and skirt component of Improved Cook Stove design. The assessments were done via control volume analysis and implementation of modeling fluid dynamic, and heat transfer equations. The outcome of this comprehensive design model reaches beyond simple product development, as it will also be used in a complete business model. A detailed approach emphasizing assumptions can be found in the Discussion portion of this report.

SCHEDULE

The following schedule lists the major tasks for Open Hearth Surgery as well as the dates associated with initiation of the tasks, and the deadlines for their completion.

Date of Completion 3/5 3/12 3/19 3/26 4/2 4/9 4/16 4/19 4/30 5/3

Task Description (Date Begun)

1. Brainstorm ideas for concept document (2/12) X

2. Detailed research for concept ideas to validate their feasibility (2/12) X

3. Prepare concept document (2/12) O

4. Gather resources to aid in preparation of thermodynamic, heat transfer, and fluid dynamic models for the basic models report (3/5) X

5. Formulate basic models (3/12) X

7. Derive and validate equations for equations implementation (3/19) X

8. Prepare MATLAB implementations and Equations Report (3/19) O

3

9. Compile MATLAB implementations into final Simulation Report (4/2) O

10. Analyze simulation for necessary optimization scheme (4/9)

X

11. Implement optimization (4/9) X

12. Prepare results into Optimization Report (4/9) O

13. Prepare Final Presentation (4/17) O 14. Gather and organize all relevant

information and feedback (4/19)

X 15. Final Report (4/21)

O

X: Signifies a Completed/Submitted Task; O: Signifies a Completed/Submitted Milestone

SUMMARY OF KEY CONCLUSIONS AND RECOMMENDATIONS

Below is a condensed summary of the most important conclusions achieved as a result of this project. Also included are recommendations for future work to refine and continue the design process.

The ICS design achieved and the model developed provide valuable insight into the relations of fluid gas flow and heat transfer to efficiency and cook stove performance

The most sensitive parameter isolated for optimization included skirt gap width

The skirt gap width and stove height must be optimized together

The implementation of the internal nozzle, initially thought to improve heat transfer due to an increase in flue gas velocity, actually proved detrimental to stove performance according to the team’s analyses

Selection of a material with a high emissivity value could potentially be the most economically impactful engineering consideration for increased stove performance

Future work on this project might include: o More substantial qualitative analyses considering models with increased

complexity Not assuming perfect efficiency of all components by considering losses

through walls. Consider flame cooling due to excess air. This can be done by varying the fuel

to air ratio in models. Chemical composition of biomass used and the effect on combustion

efficiency o Prototyping of the optimized geometry to experimentally assess the theoretical

findings, as well as the effect of sustained human use on stove performance and longevity

o Implementing additional unique design components into the model for assessment of potential performance enhancement, such as fins to increase heat transfer.

4

VARIABLE LIST

Variables A area (m2) cp specific heat (J kg-1 K-1) d, D diameter (m) f friction factor F view factor g acceleration due to gravity (m s-2) G gap (m) h convective heat transfer coefficient

(W m-2 K-1)

H height (m) k thermal conductivity (W K-1 m-1) L length (m)

ṁ mass flow rate (kg s-1)

Nu Nusselt Number p pressure (N m-2) Pr Prandtl Number q heat flux (W m-2) Q energy released (J) R characteristic radius Re Reynolds Number T temperature (K) u, v velocity (m s-1) z gap between plate and nozzle (m)

diffusivity (m2 s)

bed voidage

sphericity

efficiency

viscosity(kg s-1 m-1)

density (kg m-3)

kinematic viscosity (m2 s)

Subscripts a annulus air air an where annulus and nozzle control volumes meet b bed bp from the bed to the pot c combustion chamber conv convective e exit f flame g gases h hydraulic n, N nozzle p pot ps pot sides p1 bottom pot surface p2 side pot surface rad radiative s pot bottom stagnation tot total w water Simplifying Variables A, B, C, D, E, S, Y, Z

5

DISCUSSION

BACKGROUND

Problem Background The problem addressed by this design project is one of global importance. Currently, half of

the world’s population–up to 90% of rural households in developing nations - relies on generating essential energy via inefficient combustion of biomass, such as wood, dung, and crop residue, in primitive cook stoves [1]. While these devices perform vital processes for the users, they also present issues that need to be addressed.

The first of these issues is inefficiency of the biomass consumed. This inefficiency exists pertaining to both fuel and time. In 1997, the Food and Agriculture Organization (FAO) estimated that people consume nearly 1,890 million cubic meters of wood per year. This means that 10% of all wood harvested from the world’s forests is used as fuel. The methods of combustion implemented compound inefficiency, as heat escapes into open air [2]. This inefficiency is a significant contributor to deforestation. Cook times are also substantially longer than necessary, further contributing to inefficient use of natural resources and time.

A second issue of great concern is Indoor Air Pollution (IAP) and safety. Cook stoves used rarely include flues or vents [1]. Consequently, people are exposed to high levels of emissions produced when burning solid biomass. These emissions include carbon monoxide, nitrous oxide and sulfur oxides, toxic hydrocarbons, and particulates. Inhalation of such emissions causes numerable adverse effects including, but not limited to, chronic obstructive pulmonary disease, acute respiratory infections, pulmonary tuberculosis, lung cancer, and birth defects. IAP is responsible for two million excess deaths in developing countries and 4% of global disease [3], making a resolution imperative. Additionally, open flames associated with the cook stoves result in devastating house fires, burns, and scalds.

Along with addressing these specific problems, other factors must be considered as well. Any ICS must be economically feasible and sustainable, and therefore must be capable of making a profit. Cultural, economic, and technical constrains must be taken into account for an ICS to be successful. A detailed discussion of these constraints and conditions is available in Appendix A.

In summary, an improvement in efficiency between 25 and 50 percent is desirable in stove design. Many times, designs perform differently in a controlled environment than they do in the real world due to variables beyond the designer’s control or insight. As a result, for a product to be valuable to the customer, it needs to yield improvements of at least 25 percent compared to the product the customer is accustomed to using, which is the three stone fire in this application. Also, it is important to consider that the user will most often be the woman in the household while the man of the household is the one purchasing the product. Both ‘customers’ have differing opinions on the most important aspects of a stove and both should be satisfied. As a result, stoves need to be marketed to women as a genuine improvement for their standard of living and to men as a low initial investment product that will yield benefits in the short term. Another cultural constraint that must be considered is the size and function of a stove. Different cultures have different cooking styles and capacities. In order for a stove to be effective and well received, cultural constraints must be carefully considered.

The ICS also needs to be inexpensive. When compared to the three stone fire, any stove is going to be more costly. Historically, stoves have been successful when manufactured for under 20 dollars. This cost level needs to be maintained in any new stove design. A for-profit company currently in business, StoveTec, manufactures biomass stoves for 20 dollars. The team’s design

6

needs to meet or exceed this benchmark to be competitive in the marketplace. This can be done through the selection of low cost, yet effective materials such as stainless steel and the selection of simple, yet effective designs. Furthermore, other factors such as the weather, altitude and environment of the region of stove implementation must be considered. In the past, stove designs have failed because they did not take these factors into account. This led to a low performance, low quality product.

In conjunction with crafting a model for a physical cook stove design, a comprehensive business model is also proposed. The business model first encompasses an educational campaign within the chosen region for implementation of the stove. The community must be educated about usage of the new ICS and its benefits. Specifically, these benefits include reduction of fuel usage, reduction of smoke and particulate emissions, decreased cooking times, and greater safety. Key members of societies will be chosen to be trained and educated with regard to the stove design, production, and maintenance. This education will be executed with hopes to establish an economically profitable industry within the culture, while dispersing and encouraging widespread use of the ICS design. The manipulative nature of the model created by Open Hearth Surgery allows for timely, effective alterations to be made to the stove design for customization specific to cultural and geographical constraints. Technical Background

General information and literature on the topic of biomass stoves is widely available. Perhaps the first and one of the most thorough technical analyses on biomass stoves is the book Biomass Stoves: Engineering Design, Development, and Dissemination written in 1988 by Samuel Baldwin [4]. This book establishes a foundation on which more recent research has been based. Baldwin’s work, along with white papers published by Aprovecho Research Center, a leading research group on biomass stove technology, has enabled the design team to compile a list of general biomass stove design recommendations [5][6]. These recommendations are included in Appendix C. Furthermore, the benchmarks of the three stone fire and existing ICS designs are detailed in Appendix B.

Although these existing benchmarks and general theory provide a solid foundation for improving biomass stove design, the actual process and analysis requires quantitative values and criterion for comparing multiple designs. Baldwin described measurable and theoretical quantities that have been the basis of improved stove performance. Of those, the design team intends to use the quantities of combustion efficiency, heat transfer efficiency, pot efficiency and control efficiency. Combustion efficiency is a measure of how well fuel is combusted; heat transfer efficiency is a measure of how well heat is transferred to the pot; pot efficiency is a measure of how well heat is transferred from the pot to its contents; and control efficiency is a measure of how much excess heat is generated [4]. This heat is the heat generated above and beyond what is required to cook food. These efficiencies are listed below in Equations 1-4.

(Eq. 1)

(Eq. 2)

(Eq.3)

7

Equations 1 and 2 can be combined for convenience and described as the thermal efficiency of the stove.

(Eq. 4)

Finally, the design team intends to consider a third efficiency, control efficiency, shown in

Equation 5 below.

(Eq. 5)

This theory and quantitative metrics form the basis of general knowledge and a starting

point to begin to make progress toward an improved cook stove design. In addition to the foundation of stove design presented by Balwin and Aprovecho, there also

exists the significantly difficult task of incorporating the highly coupled thermophysical properties that exist within a cook stove. Combustion and pyrolysis begin the complex sequence of events, as gasses emitted flow upward through the device while transferring heat to the cooking utensil and the environment. It is this intricate combustion process that required detailed assumptions for implementation into the team’s model. Literature published by Steward, entitled Prediction of Height of Turbulent Diffusion Buoyant Flames, in conjunction with the papers written by Bussman and Kausley provided Open Hearth Surgery with the necessary assumptions to manage this complexity[7][8][9]. These assumptions, detailed within the Discussion of this report, allowed for optimization results that illustrated valuable trends and relations of the thermophysical properties considered within the cook stove.

SCHEMATIC

In order to aid the reader in understanding the stove model presented in this report, an artist’s sketch of a typical cook stove is presented in Figure 1. While this sketch is not indicative of the specific design chosen for this design project, it serves as a basic reference. Fuel is fed from the bottom of the stove into an inner chamber where it undergoes combustion. A pot rests on the top of the stove. Heat is transferred to the pot by a combination of radiative and convective processes.

Figure 1. Artist’s Rendition of Cook Stove

8

Figure 2 below shows a more detailed, engineering schematic of the cook stove design

chosen. An air inlet at the bottom of the stove allows fresh air to be sucked through the grate and fuel bed. High temperatures in the fuel bed heat the air and cause pyrolysis. The gasses emitted from the fuel undergo combustion with air in the combustion chamber. The heat generated from combustion further heats up the hot gasses as well as providing energy for further pyrolysis of fuel, sustaining the combustion reaction. At this point, the hot gasses speed up as they pass through a nozzle and impinge upon a pot, heating it up. These gasses are redirected around the sides of the pot by a skirt through a channel gap, where they further contribute to heat transfer to the pot.

Figure 2. Engineering schematic of a cookstove

Figure 2 depicts the design that Open Hearth Surgery has chosen to analyze and optimize throughout the course of the design process. Essentially, it is consistent with the design of a Rocket Stove, which has already been proven more efficient than the three stone fire (described in Appendix B). However, the innovation lies within the unique combination of the aforementioned nozzle, and the skirt surrounding the pot. The implementation of these two design elements poses an interesting coupling of the thermophysical properties present within the cook stove, and will be analyzed to meet the design objective and criteria mentioned above.

APPROACH

The primary goal of this design is to explore how variations in the team’s unique cook stove geometry influence efficiencies, emissions, and cost. Because cost is primarily dependent on material and manufacturing factors, it can be reduced by using inexpensive materials and utilizing a simple design. Clever business models, as described in previous sections and in the

9

Appendices, also help lower cost so that efficient cook stoves can be available to anyone [6]. Emissions are difficult to model due to the complex chemical reactions that go on during combustion. As a result, the best way to approach this portion of the objective is a generalized approach that takes into account certain standards and rules of thumb of cook stove design that have been experimentally determined. This includes utilizing certain height to diameter ratios of the combustion chamber and burning fuel lean so as to reduce CO emissions. This leaves only one primary goal that can be readily quantified and optimized: efficiency. This became apparent early in the design process. The primary areas of focus should be the interaction of fluid flow, heat transfer, and combustion within the stove.

These theoretical areas of focus are highly coupled within a cook stove. They all are dependent upon one another. In fact, Kausley gives a theoretical relation that relates the fuel to air ratio with the flame temperature [7]. For any set fuel to air ratio, a certain flame temperature should result. This is very convenient and allows the complex combustion processes to be neglected. Instead, a certain flame temperature, set to be equal to the incoming air temperature, is assumed. With a known temperature of incoming air, relationships for fluid flow, heat transfer, and eventually, efficiency can be quantified and determined. This allows the optimization of a stove with varying geometrical parameters.

Inputs

The primary inputs into the stove system can be either variable or constant. Variable inputs are the geometries that are parameterized. By selecting a variety of geometrical parameters in a reasonable range for cook stoves, dimensions that yield optimum results can be selected for final design. Constants chosen are based on general knowledge and measured values common in most stoves. They do not vary across different stove designs.

The inputs shown in Table 1 below are variable and parameterized later in the optimization.

Table 1. Input Variables

Input Designation Description Units Lower Bound Upper Bound

Dn Nozzle Diameter cm 20 cm 40 cm

Gs Gap between pot wall and stove wall

mm 2 mm 12 mm

H Height of pot cm 20 cm 40 cm

Dp Diameter of pot cm 20 cm 40 cm

The inputs in Table 2 are those which are chosen based on available literature, general

recommendations for stove design, and common knowledge.

Table 2. Chosen Inputs and Values

Input Designation Description Units Chosen Value

Tf Flame Temperature K 860 K [7]

Tb Bed Temperature K 650 K [7]

Tw Water Temperature K 373.15 K

10

Control Volume Analysis The primary geometry that is intended to be explored concerns the relationship between

efficiency of a cook stove and changing nozzle diameter and skirt gap. Changing these parameters affects fluid flow, convective and radiative heat transfer, and efficiency. Other geometric relations to be explored include the effects of changing skirt length and stove diameter. Finding out how variations in these parameters affect the stove as well as their relative importance in stove performance should lead to better stove designs.

Parameterizing key variables to stove performance will allow equations and models to be created. This cannot be done by looking at the system as a whole but will instead be done by splitting up the stove into multiple control volumes and utilizing an energy balance on each control volume. This yields a system of equations that can be optimized and solved. Figure 3 below depicts the control volumes used to analyze the stove.

Figure 3. Control Volumes within the Stove

A couple of assumptions have been made for the generalized stove system. One major

assumption is that the flame fills the entire volume of the combustion chamber. This is a reasonable assumption stoves are often designed with an approximate height to diameter ratio within the combustion chamber of 3:1. A larger combustion chamber allows for a larger flame, giving more time for all the fuel and volatiles to be burnt to completion [5]. This reduces emissions and increases efficiency. But even more importantly, it implies that the combustion process takes place throughout the combustion chamber. If combustion takes place within the entire chamber, then the flame also fills the entire chamber. By making this assumption, it is possible to look up a literature value for the temperature of the flame, and thereby the inlet temperature of the hot flue gasses into the nozzle area.

Another assumption made in the overall model is that the double wall of the stove is considered adiabatic. It is commonly recommended to use sheet metal to make stoves due to its low thermal mass [5]. But it is still important to insulate the stove and reduce losses through its walls. As a result, a double wall made of sheet metal is chosen. The double wall, when made of a low emissivity material, serves as a radiation shield to prevent radiation heat transfer from

11

the bed and the flames to the environment. Polished stainless steel is chosen as the material because its emissivity is approximately .20 between 400 and 800K [10]. By placing a small gap in between the two metal walls, the air within can be treated as a static insulator because there is very little advection present. As a result, conduction within the fluid dominates. Air is a very good insulator with a fluid conductivity value of .03 W/mK. Glass fiber insulation has a conductivity value around .04 W/mK, depending on its density [10]. The walls of the stove are well insulated. Assuming adiabatic wall conditions is a very reasonable simplifying assumption that makes the problem much simpler to work with.

Outputs

Beginning with inputs we can apply control volume analyses, simplifying assumptions, and constraint equations. From these, useful outputs that help understand the stove system can be realized. The useful outputs that lend insight into the performance of the stove system are presented in Table 3 below.

Table 3. Output Variables

Variable Description Units

qconv,s Convective heat transfer to pot bottom W

qconv,ps Convective heat transfer to pot sides W

qrad Radiative heat transfer to pot bottom W

qtot Total heat transfer to pot W

Even so, the primary output that should be maximized is the efficiency of the stove system,

presented in Table 4. A higher efficiency stove ultimately creates the least waste. A high efficiency stove requires less fuel for a set task than a less efficient stove. This leads to less time spent gathering fuel as well as the conservation of vegetation in the surrounding environment. This is also reflected in emissions because less fuel is burned, therefore less particulate and chemical pollution is emitted. Efficiency is commonly defined as the ratio of useful work over the total amount of work done. In the case of a cook stove, that can be translated to be the total heat transfer to the pot’s contents over the total amount of energy released.

Table 4. Stove Efficiency Output

Variable Description Units

η Stove Efficiency dimensionless

EQUATIONS

Geometric Relations The design of the stove focuses on the alteration of geometries to optimize efficiency. These

geometries are shown in Figure 4, below. The dimensions of the skirt and nozzle were altered over a range and the heat transfer to the pot found. Due to the nature of the stove some of the dimensions are unique to the design. The flow goes vertically from the bottom to the top. After flowing through the nozzle the flow will be radial until the annulus.

12

Figure 4. Areas and Diameters within a Stove

The area between the nozzle and the bottom of the pot is designed to maintain a constant

velocity. To accomplish this, a continuity equation can be used and the gap is found to be inversely related to the diameter of the pot, as shown in Equation 6.

(Eq. 6)

Another important aspect is the annulus, as associated with the skirt of the pot. The several equations use values based on the dimensions of the annulus. An important value is the hydraulic diameter. Equation 7 shows the process of solving for the hydraulic diameter.

(Eq. 7)

The hydraulic diameter is a ratio of the cross sectional area, A, of the flow and the wetted perimeter, P. The solution shows that the hydraulic diameter is just the difference between the outer and inner diameters of the annulus.

Air Properties

Equations governing the fluid flow, heat transfer, and thermodynamic aspects of a cook stove are often dependent upon the thermophysical properties of the working fluid. Although these properties can easily be found in reference tables, they vary with respect to temperature. As a result, it is not realistic to constantly look up these values by hand. In order to avoid this, lines of best fit can be calculated for these values. Equations 8-13 represent the following curve fits for specific thermodynamic properties of air calculated for temperatures ranging from 300K to 1600K. Data has been gathered from Table B-2 of SFPE Handbook of Fire Protection Engineering [11] as well as Appendix A of Fundamentals of Heat and Mass Transfer [10] for density, specific heat, viscosity, kinematic viscosity, conductivity, and diffusivity of air. They most

POT

Ac, Dc

Fuel Bed

AN, DN

Ap, Dp

Aa, Dh

Dh=Dc-Dp

13

always yield errors of less than 1%. A sample calculation of this method is detailed in Appendix E of this report.

(Eq. 8)

(Eq.9)

(Eq. 10)

(Eq. 11)

(Eq. 12)

(Eq. 13) The working fluid throughout the stove is approximated to be air. In reality, the flue gasses

flowing through a stove have a different chemical composition than air. However, when running fuel lean so that CO emissions are reduced, the differences in the properties of air and flue gas become negligible. The gas composition of flue gasses for an excess air ratio of 162% is given by Kausley [7]. The calculation of flue gas density using partial pressures yields less than 1% error from tables giving the density of air. This calculation is detailed in Appendix E. Treating flue gasses as a gas with the same properties of air also is found in existing literature. Steward assumes that heat capacities of air, combustion products and fuel to be equal and independent of temperature in his paper on the height of buoyant diffusion flames [8].

Fluid Flow

Fluid flow in the stove is governed by a few equations: Bernoulli’s, continuity, and momentum. Figure 5 illustrates fluid gas flow throughout the stove design as well as the important variables associated with the velocities at key points within the flow. By observing properties of the air, as reported above, and gases at points in the stove, relations can be determined. In most cases the temperature of the air or gas determines density or pressure.

14

Figure 5. Velocities and Densities within a Stove

Equation 14 is the foremost equation for finding the flow rate: Bernoulli’s equation.

(Eq. 14)

Simplifying Equation 14 with the considerations applicable to the stove yields Equation 15 and Equation 16 below. This is almost identical to an equation for Stove Suction used by Kausley [7].

( ) ∑ (Eq. 15)

Coefficients Y and Z are introduced so that the entire fluid flow system can be represented by constants multiplied by the inlet fluid flow velocity, ui.

∑ (Eq. 16)

The unknowns are the inlet velocity and the sum of the pressure drops. Pressure drops

occur throughout the stove and are result of several phenomena. These pressure drops include pipe flow, flow through a packed bed, and flow through changes in geometry. Each pressure drop equation uses additional fluid laws to get common variables.

First, Equations 17 and 18 address the pressure drop from flow through a packed bed. Equation 17 is a function of the velocity. It is further simplified by setting coefficients A and B to the coefficients of ui, similar to the previous simplifications. This is shown in Equation 18.

(Eq. 17)

(Eq. 18)

POT

u,ρair

Fuel Bed

ug,ρgases

va,ρa

vn,ρf

vc,ρf

15

Next, flow through the skirt is considered. This flow uses an adapted version of common

pipe flow pressure drop, modeled by Equation 19.

(

) (

) (Eq. 19)

Because the skirt and pot system creates an annulus for the gases to flow through, the application requires some variable alteration. The diameter of a pipe becomes the Dh, the hydraulic diameter. Using continuity, the velocity in the annulus can be found to have a ratio to the inlet velocity. This value is substituted to gain a common variable, as shown in the progression from Equation 20 to 21.

(

) (

) (

) (Eq. 20)

(Eq. 21)

The nozzle of the stove also accounts for pressure drops. In order to model the nozzle, the

flow was assumed to suddenly go from a large pipe to a small pipe then from a small pipe to a large pipe. Because of the nearly instantaneous nature of this model, the pressure drop is an over estimation of the pressure drop associated with the stove nozzle. Equations 22 and 23 model this flow, which is based on ratios of the diameters of the two pipes.

*(

)

(

)+

(Eq. 22)

(

)

*

+

(Eq. 23)

Equation 22 is originally a function of the velocity for the large and small diameter pipe. The two velocities can be put in terms of the inlet velocity through continuity. Equation 23, the second nozzle equation, is originally based on the velocity of the small pipe and the diameters of the two pipes. Again, continuity is used to generate an equation in terms of the inlet velocity.

Now, the simplified Bernoulli’s equation can be rearranged to sum the different portions to zero. Next the different pressure drops must be added to get the final equation for finding velocity, as shown in Equations 24 and 25.

∑ (Eq. 24)

[ ] (Eq. 25)

Subsequently, Equation 25 is in the quadratic form, providing the ability to solve for the inlet air velocity, ui.

Throughout the fluid equations continuity is used to get common variables. Equation 26 balances to give Equation 27, which is equivalent to the desired velocity.

(Eq. 26)

(Eq. 27)

16

The velocity, cross sectional area, and density are used in the calculation of velocities. Once temperatures are determined the geometry is used to find the velocity anywhere in the stove.

Heat Transfer

Figure 6 below shows the Temperatures and Heat Transfer with a Stove. There are six unknown temperatures: Tn, Ta, Tan, Te, Tp1, and Tp2. The temperature of the flame, Tf, is given as 860K for actual flame temperature when 162% excess air available [7]. Bed temperature, Tb, can be set to 650K for wood undergoing pyrolysis [7]. Furthermore, the water can be assumed to be at boiling temperature. It is common knowledge that the temperature of boiling water is 100°C, or 373.15K. Ambient air temperature can be assumed to be 20°C, or 293.15K. The two mechanisms of heat transfer that are present in the diagram are radiation, which occurs from the fuel bed to the bottom of the pot, and convection, which occurs around all the surfaces of the pot. Heat losses through the exterior stove walls can be neglected due to adiabatic surface assumptions. The unknown temperatures are solved for primarily in two ways. Expressions can be set up relating the temperature drop of the fluid to the heat transfer from the fluid. Also, due to steady state conditions and assumptions, heat transfer rates into and out of pot surfaces can be made equivalent.

Figure 6.Temperatures and Heat Transfer within a Stove

In control volume analysis, it is customary to use average temperatures. Steward makes

such an assumption in his paper as well [8]. Applying this to the specific circumstances depicted above, we find that the average temperature within the nozzle control volume is equal to the average of the inlet and outlet temperatures. Likewise, the average temperature within the

17

annulus control volume is equal to the average of the inlet and outlet temperatures. This gives the following relations of Equation 28 and 29.

(Eq. 28)

(Eq. 29)

Radiation Radiation Heat Transfer is considered to occur solely from the fuel bed of the stove to the

pot. The effect of flame radiation is neglected. Steward makes this assumption [8], and Hottel shows that flame emissivity is quite low and only around 7% [12]. Even so, radiation from the fuel bed to the pot should be considered. Because the bottom of the pot is directly exposed to flames, it is expected to be highly sooted. Its surface can be considered to act as a blackbody with an emissivity of unity [4]. Vegetation has an emissivity between .92 and .96 [10]. Because the fuel for this stove is wood or biomass, the fuel bed is considered to have an emissivity of .94. Equation 30 is used for radiation from the fuel bed to the pot.

(

) (Eq. 30)

As the nozzle diameter changes, the view factor from the fuel bed to the pot will change as

well. It is assumed that the nozzle has a smaller diameter than the pot and that it is located close to the pot to increase convective heat transfer. Because the nozzle is made out of the same material as the stove walls, it has been chosen to have a low emissivity. The emissivity for polished stainless steel is approximately .20 between 400 and 800 K [10]. This means a smaller nozzle shields the pot from radiation. As a result, the view factor from the fuel bed to the pot is defined as all the radiation that is emitted from the fuel bed and passes through the nozzle. The view factor from the fuel bed to the pot, Fbp, can be calculated utilizing the view factor for coaxial parallel disks, detailed in Equations 31-34 [10]. Consider the disks represented by the fuel bed and nozzle. Their respective radii are and . They are separated by a distance of H, the height of the combustion chamber. R is a characteristic radius and S is a simplifying variable.

(Eq. 31) (Eq. 32)

(Eq. 33)

{ [ (

) ]

} (Eq. 34)

Convection at Pot Bottom Impingement of a gas traveling through a single circular nozzle where the nozzle is located

within one nozzle diameter from the surface is examined by Lytle and Webb [15]. The closeness of a nozzle to a surface is described as the ratio of the nozzle diameter to nozzle plate spacing, z/Dn. It can be generally stated that for a nozzle located closer to the impinging plate, or small z/Dn, the average heat transfer coefficient increases. However, for ratios of z/Dn< .25, significant acceleration of the fluid must occur between the nozzle-plate gap. As a result, a ratio of z/Dn equal to .25 is chosen so as not to invalidate the pressure drop equations previously described. They give the following empirical expression, Equation 35, for average Nusselt numbers over a plate radius approximately equal to nozzle diameter.

18

(Eq. 35)

The expression is valid for the range of nozzle-plate spacings of .1 <z/Dn< 1 and Reynolds

numbers between 3600 and 27600. It has an average error of 9%. The average convective heat transfer coefficient can be found using Equation 36.

(Eq. 36)

Once this value is known, convective heat transfer to the bottom of the pot can be

calculated using Equation 37. (Eq. 37)

Convection in Skirt Jaco Dirker and Josua Meyer have compiled a list of popular equations describing the

Nusselt number in a smooth annulus during forced convection [13]. They also have affirmed these equations, via experimentation and examination of previous literature, and recommended that the McAdams or Dittus-Boelter equations be used for annuli with annular diameter ratios of less than 2.5. McAdams describes the Nusselt Number as shown in Equation 38.

(Eq. 38)

Although this expression is for fully developed flow, the entrance region effects can be

neglected due to the low velocity of flow and the narrow entrance region of the annulus [4]. The average convective heat transfer coefficient can be found using Equation 39.

(Eq. 39)

These equations, along with the respective values for viscosity, conductivity of a fluid, and

geometrical constraints, yield Equation 40, an expression for the average heat transfer coefficient within the skirt section of the pot.

(Eq. 40)

Convection within Pot

The primary goal of a stove is not to heat up the pot, but actually to heat up the contents of the pot. As a result, the convection from the pot to the water must be considered. Because the pot is made of a high conductivity material, thermal resistance within the pot itself is neglected. It is assumed that a stove will most often be operating with liquid inside the stove at boiling temperature. Boiling occurs in multiple regimes: free convection boiling, nucleate boiling, transition boiling, and film boiling [10]. A pot with visible boiling water and bubbles that form at a hot surface and then rise is categorized under the nucleate boiling regime. The average convection coefficient for boiling water in nucleate boiling is experimentally approximated by Fritz in Equation 41[14].

(

)

(Eq. 41)

19

This approximation holds at approximately one atmosphere of pressure and under the

conditions

Although boiling mechanisms have been studied extensively, complete and reliable

mathematical models have yet to be developed [10]. As a result, a first order approximation is used to find a conservative value of 2500 W/m2K for the convection coefficient between the pot and the water. The details of this analysis are contained in Appendix F.

With this assumption made, we can further describe heat transfer from the pot bottom to

the water can with Equation 42. (Eq. 42)

Furthermore, heat transfer from the pot side to the water can be described by Equation 43.

(Eq. 43)

Heat Loss through the Stove

Although all of these equations and relations can readily yield expressions for heat transfer, they assume that the temperatures are known. However, that is not the case. It is necessary to develop a system of equations relating the temperatures present in the stove with the heat transfer various sections of the stove.

Recall from Figure 6 that although Tf, Tb, and Tw can be respectively set to 860K, 650K, and 373.15K, there are still six unknown temperatures: Tn, Ta, Tan, Te, Tp1, and Tp2. In order to eliminate unnecessary variables, we remember that the temperature within a control volume is defined as the average of the inlet and outlet temperatures of that volume, shown in Equations 44 and 45.

(Eq. 44)

(Eq. 45)

This eliminates Tn and Ta, leaving only four unknown temperatures. Next, an energy balance

may be written for the nozzle control volume at temperature Tn. The energy lost from convective heat transfer, qconv,s, results in a temperature drop from the fluid. A similar expression is presented by Kausley[7]. This is shown below in Equation 46.

∫

(Eq. 46)

A similar energy balance, Equation 47, may be written for the temperature drop that results

from convective heat transfer to the sides of the pot.

∫

(Eq. 47)

20

Next, an energy balance is written for the bottom of the pot. Because steady state conditions are assumed, the pot bottom temperature, Tp1, does not vary. As a result, heat transfer to the pot bottom must be equal to heat transfer from the pot bottom. This is shown in Equations 48 and 49.

(Eq. 48)

(

)

(Eq. 49)

Finally, a similar energy balance can be written for the sides of the pot, as shown in Equations 50 and 51. It should be realized that heat transfer to the pot side must be equal to heat transfer from the pot side.

(Eq. 50)

(Eq. 51)

These four equations create a nonlinear system of equations that have only four unknowns.

It can be solved by Newton-Raphson method. The method used to solve these equations is detailed in the optimization section of this report.

Stove Efficiency

It is possible to solve for all the geometries, fluid flow conditions, heat transfer rates, and temperatures for a variety of temperatures and still not know the optimum stove. Remember, the primary consideration of stove design is to maximize efficiency so that meals may be cooked with the lowest possible use of fuel. This both reduces time spent gathering fuel, as well as the overall amount of emissions released during the cooking process. This also means that we must come up for an expression for efficiency. Efficiency is commonly defined as the ratio of useful work over the total amount of work done. In the case of a cook stove, that can be translated to be the total heat transfer to the pot’s contents over the total amount of energy released.

Employing the same assumptions that have been made throughout this analysis, the total heat transfer to the pot can be described by Equation 52.

(Eq. 52)

Also, the total amount of energy released from the combustion chamber is equal to the

total amount of radiation from the bed plus the energy needed to heat ambient air to the flame temperature. This relationship is shown below in Equation 53.

∫

(Eq. 53)

Finally, an expression for efficiency of the stove can be obtained.

∫

(Eq. 54)

Objective Equation As noted in earlier sections of this report, because cost and emissions can be best

manipulated using qualitative methods, only the efficiency of the stove is effectively evaluated

21

in a quantitative way. Therefore, the stove efficiency should be maximized. This yields the objective shown below as Equation 55.

∫

(Eq. 55)

where y is to be maximized. Because is an efficiency, it should always be between zero and one. Any other solution will indicate an error in some step of the process.

Constraint Equations

Constrain Equations for this optimization problem consists of all the equations, relationships, and limitations discussed previously in the modeling section of this report. These include but are not limited to constraints on the following:

1. Conservation of energy 2. Conservation of mass 3. Geometric relationships 4. Limitations on Temperature 5. Limitations on dimensions

OPTIMIZATION

Introduction As detailed in previous sections, the simulation of an improved cook stove involved a

number of tightly coupled thermodynamic and heat transfer processes. For instance, the temperatures at various points in the stove greatly affected the fluid properties of the air medium transporting heat energy. The approach used to deal with this complexity was based on the Newton-Raphson method.

The simulation began with an initial guess for temperatures at key points in the stove. These temperatures were used to calculate relevant fluid mechanics properties such as density and heat transfer coefficients. Using these calculated values, it was possible to set up control volume energy balances for different regions of the stove. These control volume energy balances were used to calculate the error term for the Newton-Raphson method. Using MATLAB’s built-in solving functions, temperatures that satisfied the control volume energy balances were found. It was prohibitively computationally expensive to exhaustively search the solution space for a global optimum of gap size, pot height, pot diameter, and gap width. Also, the method of Lagrange multipliers did not fit the formulation of the problem. Instead, a study of the effect of varying various geometric parameters was performed, and a candidate design for a stove was constructed using insight gained from this process.

For reference to the equations and implementation of the numerous mathematical models, input and output variables, and optimization methods utilized, please see the MATLAB code provided in Appendix G.

Parametric Analysis

The first study of stove parameters performed was a coarse search of the solution space varying the nozzle diameter, gap size, pot height, and pot diameter. The results of this study are presented in Figure 7.

22

Figure 7. Coarse Search Results

There appeared to be candidate stove designs for any combination of desired heat transfer

and efficiency. It turned out that the height of the stove should always be set at its maximum value of 40 cm. This is shown by Figure 8, where stoves with a height of 40 cm are circled in red.

Figure 8. Candidate Stove Designs of Maximum Stove Height

Another parameter that could be set to its upper bound was the nozzle diameter. Figure 9

and Figure 10 show the results simulating the performance of a stove at a fixed diameter of 20

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Heat Transfer (kW)

Eff

icie

ncy

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Heat Transfer (kW)

Eff

icie

ncy

23

cm and a fixed height of 40 cm. Notice that the optimal efficiency and heat transfer occurred at the maximum nozzle diameter.

Figure 9. Efficiency of Fixed Dimension Stove

Figure 10. Heat Transfer of Fixed Dimension Stove

Based on the previous two results, it was possible to explore a narrower subset of the

solution space more exhaustively. Stove diameter was varied from 20cm to 40cm, and more data points were considered for the gap size.

5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Nozzle Diameter (cm)

Eff

icie

ncy

5 10 15 200

0.5

1

1.5

2

2.5

3

Nozzle Diameter (cm)

Heat

Tra

nsfe

r (k

W)

24

Figure 11. Optimized Stove Design Candidates

The stove configurations circled in red within Figure 11 were for a diameter of 20 cm, and

the stove configurations circled in black were a diameter of 40 cm. The stove that displayed the maximum efficiency in the above plot delivered enough power for a family’s cooking needs. Therefore, the optimal stove geometry for our design problem was selected and is presented in Table 5 below.

Table 5. Optimized Stove Geometry and Outputs

Geometry Value

Gap Size 9.4 mm

Nozzle Diameter 20.0 cm

Stove Diameter 20.0 cm

Stove Height 40.0 cm

Efficiency 0.72

Heat Transfer 1.8 kW

Final Product Drawing

Again, these results indicate that the stove height should be set to its maximum and the stove diameter to its minimum. Furthermore, the nozzle should be set to its maximum diameter, effectively eliminating any of its effects. At these geometries, the ideal gap size is 9.4 m. Figure 12 shows the stove and its dimensions.

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Heat Transfer (kW)

Eff

icie

ncy

25

Figure 12.Final Stove and Dimensions

CONCLUSION

From an engineering point of view, the modeling of an improved cook stove undertaken by this team led to a number of insights. It was found that it was possible to capture the heat transfer processes of a stove qualitatively and optimize its geometry. Through analysis, sensitive parameters were isolated. Among the most sensitive for rocket-type stoves with skirts were the gap width and presence of internal features. In fact, the sensitivity of the performance of stoves to internal features caused the potentially innovative design idea of a nozzle to actually decrease stove performance. A final engineering insight was that a high emissivity material inside the stove could greatly improve its performance.

There is still more work that could be done to transform this initial contribution of stove models into a real product. First, testing of human factors, such as how cooks’ existing habits impact the real-world performance of the stove, should be undertaken. Also, experiments to establish quantitatively accurate correlations for heat transfer and efficiency numbers could be performed. There were many simplifying models used, and experiments on real rocket stoves could validate or improve their accuracy. Effects that should be considered in more detail include the fuel-air ratio for combustion, the effect of non-adiabatic walls, and the heat transfer coefficient of the liquid in the stove. Despite the modeling challenges, it was possible to design a cook stove for the developing world that exhibited substantial improvements over native technology. Deploying such a technology would improve the quality of life for millions across the globe and significantly contribute to a reduction of emissions of greenhouse gases.

26

REFERENCES

[1]Smith, K.A. "Indoor air pollution in developing countries: recommendations for research."Indoor Air. 12 (2002): 198-207.

[2]Achard F, Eva H.D., Stibig H.J., Mayaux P., Gallego J., Richards T., Malingreau J.P., “Determination of deforestation rates of the world's humid tropical forests.” Science.297(2002):999-1002.

[3]Bruce, N., Perez-Padilla, R., Albalak, R. "Indoor air pollution in developing countries: a major environmental and public health challenge."Bulletin of the World Health Organization. 78 (2000): 1078-1092.

[4] Baldwin, Samuel F.. Biomass Stoves: Engineering Design Development and Dissemination. New York: Vita Pubns, 1988.

[5] Still, Dean, Nordica MacCarty.“Cooking with Less Fuel: Breathing Less Smoke.”Aprovecho Research Center Library. 2009. Sponsoring Foundations: Aprovecho Research Center, World Food Programme, School Feeding Service (PDPF), Partnership for Clean Indoor Air, Shell Foundation, EPA. <http://www.aprovecho.org/lab/pubs/researchlib/category/1/design>.

[6] Associated Press. "Cottage Grove company sends 'rocket stoves' to Haiti." 24 Jan. 2010. <http://www.oregonlive.com/news/index.ssf/2010/01/cottage_grove_company_sends_ro.html>

[7] Kausley, Shankar B., Aniruddha B. Pandit.Modelling of solid fuel stoves. Fuel 89 (2010) 782-791.

[8] Steward, F.R. Prediction of Height of Turbulent Diffusion Buoyant Flames. Combustion Science and Technology, Vol. 2, pp. 203-212, 1970

[9] Bussmann, Paulus. Woodstoves: Theory and Applications in developing countries. Thesis (Ph. D.)--TechnischeUniversiteit Eindhoven, 1988.

[10]Bergman, Theodore L., David P. Dewitt, Frank P. Incropera, and Adrienne S. Lavine.Fundamentals of Heat and Mass Transfer. 6 ed. New York, NY: Wiley, 2006. Print.

[11] SFPE Handbook of Fire Protection Engineering 2nd Edition. [12]Hottel, H., 1954, in Heat Transmission, W.H. McAdams (ed.), 3rdedition, McGraw-Hill, New

York. [13]Dirker, J. and Meyer, J. P.(2005)Convective Heat Transfer Coefficients in Concentric Annuli,

HeatTransfer Engineering, 26: 2, 38 — 44 [14] VDI-Wärmeatlas, 7th edition, Düsseldorf 1994. [15] Lytle, D. and Webb, B.W. Air jet impingement heat transfer at low nozzle-plate spacings,

International Journal of Heat and Mass Transfer. Vol. 37, No. 12, pp. 1687-1697, 1994. [16]Winiarski, Larry.“Ten Design Principles for Wood Burning Stoves.”Aprovecho Research

Center Library.2009.Aprovecho Research Center.<http://www.aprovecho.org/lab/pubs/researchlib/category/1/design>.

[17]Bryden, Mark, Still, Dean, Scott, Peter, Hoffa, Geoff, Ogle, Damon, Bailis, Rob, Goyer, Ken. “Design Principles for Wood Burning Cook Stoves.”Aprovecho Research Center Library. 2005. Sponsoring Foundations: Aprovecho Research Center, Shell Foundation, EPA. <http://www.aprovecho.org/lab/pubs/researchlib/category/1/design>.

[18] "Official Home Page of the Best Rocket Stoves on the Planet - Welcome to StoveTec." Official Home Page of the Best Rocket Stoves on the Planet - Welcome to StoveTec. N.p., n.d. Web. 29 Jan. 2010. <http://www.stovetec.net/us/index.php>.

27

[19]Hudelson, Nordica A., Bryden, K.M. and Still, Dean. “Global Modeling and Testing of Rocket Stove Operating Variations.” Department of Mechanical Engineering, Iowa State University, Ames, IA 50011-2161.

[20] An Improved Wood Cookstove: Harnessing Fan Driven Forced Draft for Cleaner Combustion <http://www.bioenergylists.org/stovesdoc/apro/witt/Final%20Cookstove%20Report%22-%20May05-3.pdf>.

[21] Barnes, Douglas F, Openshaw, Keith, Smith, Kirk R, van der Plas, Robert. “The design and diffusion of improved cooking stoves.” The World BankResearch Observer. Cary: Jul 1993. Vol. 8, Iss.2; pg. 119.

[22] Smith K.R., Shuhua G., KUN H. and Daxiong Q. “One Hundred Million Improved Cookstoves in China: How Was It Done?” World Development, Vol. 21, No. 6, pp. 941-961, 1993.

28

APPENDICES

APPENDIX A

Discussion on Technical and Cultural Constraints Many factors play a role into the success of a biomass fuel stove. These factors and

constraints can be broadly categorized into technical, cultural, and economic constraints. Any design should pay careful consideration to these constraints as well as seek to learn from the success and mistakes of stove designs in the past.

The result of experiments and research done on stoves often shows much promise with regard to improvements of new stoves over current ones giving their developers confidence in the stove design. The problem lies in the fact that studies done on stoves in labs are under ideal conditions. Once the stove is produced and installed, whatever improvements that have been made are only a percentage of the ideal performance. This can be due to multiple factors. The production quality of the stove is the first hit to performance. There is a difference in quality if the stoves are produced and shipped or if local populations are intended to build and distribute the stove. While having the local population assemble the stoves could improve the economy of the area, the production quality may suffer. This could be due to lack of materials available, unknown labor quality, and unknown production quality. Another aspect of performance reduction is the location that the stove is being operated in. Stove tests within labs are done under ideal conditions and with many variables held constant while usage in a house trusts that the user uses a similar quality of fuel and operates the stove properly. The last cause of over confidence in improved performance is in the fact that traditional stoves are not always as inefficient as assumed [22]. When all of these factors are considered, the improvements of a new stove can be greatly diminished. Douglas suggests an improvement efficiency between 25 and 50 percent is desirable to ensure an improvement once applied to real world conditions [21]. For a product to be valuable to the customer, it needs to yield improvements of at least 25 percent compared to the product the customer is accustomed to using.

In order to fully understand the market, it is important to observe and attempt to understand the dynamic of stoves within the household. Generally, women handle most of the cooking duties; a fact that designers should be cognizant of. A wooden stove designed for Kenya met failure when the design required that wood be cut into small pieces, and the women did not have the time or the tools necessary to accomplish this task. The stove faced harsh disapproval and some were subsequently retrofitted to hold larger wood [21]. Furthermore, although women act as the primary operators of the stove, men generally purchase stoves. Consideration must be made to ensure that the customer, and in general the men, are convinced that the benefits of the stove warrant the initial investment and financial costs associated with that investment. In the case of stoves, the customer is not so simplistic of a term as usual; there are actually two separate but distinct customers that both must be satisfied. The user, generally the woman in the household, must be convinced that the stove is useful and beneficial to her lifestyle. The purchaser, generally the man in the household, must be convinced that the stove is a worthy investment. This makes marketing the stove especially challenging because the purchaser of the stove does not give reliability and efficiency a significant weight of importance while the user of the stove is hesitant to change their ways no matter how inexpensive the stove actually is. As a result, stoves need to be marketed to women as a genuine improvement for their standard of living and to men as a low initial investment product that will yield benefits in the short term. A marketing campaign that highlights the benefits of such a stove will be valuable in gaining widespread acceptance and market share.

29

Another aspect of stove design that is influenced by culture is the size that a stove needs to be. Different cultures have different cooking methods. Some cook in very large pots and cook meals that take a lot of time to complete, others cook very small portions that are done cooking quite quickly. This variable will affect both the overall design of the stove as well as the cultural compatibility of the stove. Depending on the use of the stove, different variables will be more or less important in the optimization of its dimensions and features. Also, the cost of a stove will be affected by its size. But more importantly, any new design needs to fit within the cultural norms of the target consumer and not require significant change or adaptation on the part of the user. Change and adaptation are not well adopted by consumers, and more often than not lead to product rejection even with proper marketing. In order for a stove to be effective, subtle cultural constraints such as cooking style and pot size must be researched and carefully considered.

Currently, companies such as StoveTec offer high efficiency and low emission stoves to both domestic and international customers. Their low cost rocket stove has a manufacturing cost of approximately $20 [6]. They sell the stoves to consumers in the U.S. at a retail price of $40 and use their profits, along with some assistance from carbon credit programs, to sell the stoves to people in the developing world at a cost of $8 [6]. Their business model has resulted in a profit earning entity that is able to provide both a service to customers in the U.S. as well as those abroad. Currently, StoveTec is taking a leading role in mobile stove dissemination in the earthquake torn country of Haiti, where thousands are living in tents and using whatever they can scrounge for fuel. Their business model is a prime example of how business can be merged with social responsibility. Another stove distribution taking place in China in the 1980’s was successful with a price of approximately $9 per stove [22][21]. Adjusted for inflation and assuming the program took place in 1985, this translates into a price of $18 in 2009. A low price, high efficiency biomass stove can be a valuable necessity to any of the 3 billion people on the planet who cook over an open fire or use unimproved stoves. Any new stove should have a total cost of $20 or better in order for a business model similar to this to be leveraged. Costs greater than $8 to those in the developing world prove to be too large an initial investment to overcome.

Geographic conditions must also be a factor in design. Differences in weather conditions such as humidity, elevation, and temperature can impact the performance of a stove. One particular stove program in Nepal had great initial success with distribution of stoves to the population. However, once stoves were installed at different locations, problems began to arise. This is due to the mountainous nature and varying elevation of Nepal’s terrain. The prescribed stove was not suited to operating at different altitudes [21]. Conditions in different locales and geographic locations can further vary when multiple countries are considered. Designs should take into account variances that may arise in the target area of distribution and be prepared and designed to perform well within an allotted range.

Previous stove projects have successes and failures that can be learned from. As mentioned, during the 1980's, China installed most of the world’s biomass fuel stoves [22]. A key aspect of China's success was on concentrating and limiting initial efforts [22]. This means that they initially released a limited about of stoves so that the development and design of the stoves could be refined, costs could be lowered, and overall effectiveness increased. This program also made use of education and regulation of the program; 10% of the program costs were dedicated to ensuring that public adoption of the stoves was based on fact and not fiction. Through this program, they were able to create a niche in the marketplace for the stoves, providing a product along with jobs and creating a broader based economy. The success of this initiative in China is an invaluable model for future biomass stove designs. Any new design should be thoroughly

30

tested through both alpha and beta testing, where feedback from the consumer themselves is aggregated and used in the next iteration of design. Also, an educational campaign should be implemented so that the users of the stove understand more about it and so that the full potential and benefits of the stove may be realized. Finally, it is important that a new stove be accepted as part of the local economy, giving back to the people and becoming a permanent fixture there. Doing so will ensure long term acceptance and continued sales.

31

APPENDIX B

EXISTING ICS TECHNOLOGIES Significant effort has been put into experimentation and design of Improved Cook Stove

(ICS) technology. There are too many specific stove designs to consider individually, but the major classes of stove and design principles for ICS are important to review. The most rudimentary form of cook stove is the three-stone fire.This stove is widely used in developing countries and serves as a baseline for evaluating ICS technology. The Aprovecho Research Center, one of the leading research groups on biomass stove technology, regularly publishes benchmarks for various stove technologies. These benchmarks are illustrated in Figures 1B-4B[17].

Figure 1B. Fuel Consumption of Various Stove Designs vs. Benchmark in the Water Boiling

Test (WBT)

32

Figure 2B. Energy Use of Various Stove Designs vs. Benchmark in the WBT

Figure 3B. Carbon Monoxide Emission of Various Stove Designs vs. Benchmark in the WBT

33

Figure 4B. Particulate Matter (PM) Emissions of Various Stove Designs vs. Benchmark in the

WBT There are several observations of interest to note from these benchmarks. From Figures

1 and 2, it is clear that nearly every ICS design outperforms the three-stone fire. Also noteworthy is that ICS solutions have much more consistent performance than the three-stone fire. Figures 3 and 4 depict very impressive reduction in carbon monoxide and particulate matter (PM) emissions from Rocket and Fan stoves. However, stoves based on charcoal show an increase in CO emission. These findings and generalities relating to the classes of stove designs found above are consistent with the broad pool of information on the topic.

Rocket Stoves are a popular ICS option due to low cost and ease of installation. Rocket Stoves combine an air-intake and a combustion chamber to improve fuel efficiency at minimal construction cost. There has been significant work in improving Rocket Stove design leading to commercialization under the name StoveTec[18]. Nordica Hudelson, K.M. Bryden, and Dean Still considered design parameters for optimizing Rocket Stoves in the paper Global Modeling and Testing of Rocket Stove Operating Variations[19]. For Rocket Stoves, losses “can be quantified in several main areas: energy lost in the combustion gases, convection and radiation losses from the stove, convection and radiation losses from the pan, energy stored in the stove, and energy stored in the pan.” [19] One potential avenue for improvement in Rocket Stoves was explored by M. Benjamin Witt in An Improved Cookstove: Harnessing Fan Driven Forced Draft for Cleaner Combustion[20]. Witt considered adding a fan to aid in combustion efficiency. His findings suggested that adding fans to Rocket Stoves resulted in a much cleaner burn but insignificant gains in fuel efficiency.

34

APPENDIX C

Design Recommendations

Currently accepted recommendations for biomass fuel stoves are as follows:

1. Insulated material around the fire and flow path prevents heat loss through the walls of the stove.

2. A combustion chamber and/or chimney limits the release of both particulate and chemical pollutants into the air (smoke and pollutants).

3. Drafting air intake from below the fire and through the coals as opposed to through the fuel door or directly into the combustion chamber maintains a high combustion temperature, limiting particulate and chemical pollutants.

4. Unrestricted, unidirectional airflow prevents incompletely combusted fuel such as particulates and smoke and incompletely combusted gasses such as carbon monoxide from exiting the stove.

5. A grate prevents large incompletely combusted pieces of fuel from clogging air intakes. 6. Redirecting fluid flow close to the fire maximizes heat transfer from the fire to the fluid.

However, a fluid flow that is too rapid can quench the fire. 7. A higher velocity fluid flow near the pot will maximize heat transfer to the pot. However,

too small a cross section will create a large pressure drop and restrict fluid flow. 8. Any insulation materials used should have low thermal masses. Having a high thermal mass

near the fire can absorb heat, increase cooking time, and ultimately increase both the amount of fuel used as well as amount of pollutants created. Although these design recommendations are valuable, some other, less clear relationships

have been revealed by previous researchers. One interesting aspects of stove design is chimney length. A longer chimney increases combustion efficiency but also increases heat loss to the insulation materials in the chimney and environment. A 3:1 ratio of chimney height to diameter is generally recommended as a compromise between these two desired characteristics [5]. Study of this design variable could lead to valuable insight. Another important aspect of stove design is the fuel feeding process. Baldwin recommends that the stove door be closed to increase stove temperature and therefore increase efficiency while decreasing pollutants. But Aprovecho Research Center recommends that the door be left open while only the ends of biomass are burned. This reduces heat waste via conduction to the rest of the fuel and preserves fuel for future use. This variable will be more difficult to isolate and the design team believes this discrepancy really depends on which design consideration (fuel use or pollutant emission) is given more weight, as well as the type of cooking that a stove is more frequently used for. It also depends on cooking style. The local culture may traditionally prepare faster cooking meals like would be found in frying or slower cooking meals with a cooking method more similar to roasting. Cooking style plays an important role in consideration of the type and amount insulation that will be used on the stove, but it also may influence the choice of a fuel feeder. Baldwin also mentions one design consideration that seems to be neglected in other bodies of work. He recommends that the cooking pot be placed as close to the fire as possible so that a large portion of radiant heat from the fire will be absorbed by the pot. He cites an example of this by pointing out that a simple pot positioned above a three stone fire can at times be more efficient than some stoves partially due to this close proximity and high radiant heat transfer from the fire to the pot [4]. Lastly, based on Baldwin’s recommendations, the cross

35

sectional area of the hot gasses near the pot should be reduced slightly as to increase the velocity of the air and consequently the convective heat transfer coefficient. Even so, he admits that this design variable is sensitive on the order of millimeters; a gap too large will yield little or no improvement and a gap too small will restrict air flow too much [4]. This is most likely why Aprovecho recommends that the cross sectional area of flow remain constant. These recommendations and guidelines must all be considered in the design of a biomass stove with the knowledge that conflicting engineering requirements will require compromise on the final design choice.

Although biomass fuel stoves serve their purpose, they are of no use without a cooking container, namely a pot. Although heat transfer from the hot air to the pot itself is considered an aspect of stove design, various pots and their contents also can affect stove performance. The main variation in pot performance is material. Although variables such as surface roughness may affect the convection coefficients of pot-fluid interfaces, this too can be somewhat attributed to the material chosen; aluminum is a smooth metal while cast iron and clay pots generally exhibit multiple surface defects. Even though clay pots were once considered to be the cultural norm, aluminum has been widely proliferated amongst developing countries. Aluminum is considered a good pot material because of its high thermal conductivity as well as its relatively low thermal mass. Clay pots, on the other hand, result in poor performance due to their decreased conductivity and increased thermal mass. Fuel savings associated with switching from clay pots to aluminum pots are estimated to be upwards of 40% [4]. When that consideration is combined with the improved durability and decreased weight of aluminum pots, the marginal cost of aluminum is outweighed by its beneficial properties. Not to say that aluminum pots do not need to be well maintained, a pot that is plagued by soot build up will inhibit heat transfer by adding extra thickness as well as a contact resistance between the aluminum and the soot. As a result, it is important to always keep cookware clean and shiny.

36

APPENDIX D

The calculation of a curve fit for the viscosity of air is shown below. The following values have been taken for the viscosity of air [10]. They are plotted and a polynomial line of best fit is found using Microsoft Excel.

Table 1D. Temperature and Corresponding Viscosity of Air

Temperature (K) Viscosity (Ns/m2)

250 0.00001596

300 0.00001846

350 0.00002082

400 0.00002301

450 0.00002507

500 0.00002701

550 0.00002884

600 0.00003058

650 0.00003225

700 0.00003388

750 0.00003546

800 0.00003698

850 0.00003843

900 0.00003981

950 0.00004113

1000 0.00004244

Figure 1D. Polynomial Regression for Viscosity of Air Calculation

y = -1.8387E-17x4 + 5.7800E-14x3 - 7.8696E-11x2 + 8.2231E-08x - 5.2067E-07

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0.000035

0.00004

0.000045

0 200 400 600 800 1000 1200

Vis

cosi

ty (

Ns/

m^2

)

Temperature (K)

Viscosity of Air

37

APPENDIX E

The calculation of Flue Gas Density using Partial Fractions is shown below.

Table 1E. Flue Gas Density Given the composition of flue gas at 162% excess air [7]

CO2 H2O O2 N2

Flue Gas Fractions 7.35% 5.25% 12.34% 75.06%

Flue Gas Density (Kg/m3) 1.842 .804 1.331 1.165

Flue Gas Fraction *Density (Kg/m3) 0.135387 0.04221 0.164245 0.874449

Using the theory of partial fractions for gasses, it is shown that at 293.15 K

The density of air at 239.15 K is 1.2074 Kg/m3[11].

Percent Error =

=

= 0.73 %

38

APPENDIX F

The calculation of the convective heat transfer coefficient for boiling water is shown below. Given the following empirical relationship for nucleate pool boiling by Fritz[14].

(

)

Where

For a 30 cm diameter pot with pot sides of height 20cm

Previous literature leads us to believe that a small cookstove has a average power delivery of 5.6KW [7]. It also confirms that the skirt mechanism on a pot is generally the most important aspect of a stove[4]. Because of this, we pick the following values for heat transfer through these surfaces.

From this, we can solve for the convection coefficient of water at the stagnation point

(

)

And at the pot sides

(

)

As a conservative estimate, we designate the convection coefficient for boiling water within the pot to be equal to

39

APPENDIX G

Simulation.m—runs stove simulations to find optimal configurations. close all clear clc

% Calculate ideal gap (in terms of eff) heights = linspace(0.2, 0.4, 6); diameters = linspace(0.2, 0.4, 6);

hs = []; ds = []; gs = []; hts = []; effs = [];

for height = heights for diameter = diameters

mid_gap = diameter / 100.0; gaps = linspace(mid_gap / 2.0, mid_gap * 10.0, 20);

for gap = gaps ss = StoveSystem(gap, diameter, diameter, height);

T0 = [900; 900; 900; 900]; % Make a starting guess

at the solution options=optimset('Display','none'); % Option to display

output [x,fval] = fsolve(@ss.validate,T0,options); % Call solver

[qn qi] = ss.netHT(x);

if qn > 0 qn = real(qn) qi = real(qi)

hts = [hts qn]; effs = [effs qn/qi];

hs = [hs height]; gs = [gs gap]; ds = [ds diameter]; end

end end end

ii = (hs == 0.4); plot(hts./1000.0, effs, 'bx') hold on

40

plot(hts(ii)./1000.0, effs(ii), 'ro') xlabel('Heat Transfer (kW)') ylabel('Efficiency')

StoveSystem.m—MATLAB class used to capture the stove simulation. % Subscript Labels % _n = nozzle % _p = pot % _a = annulus % _f = flame % _c = combustion chamber % _w1 = inner wall diameter % _w2 = outer wall diameter % _b = bed % _water = water % _air = air

classdef StoveSystem properties %---------------------------- %GEOMETRY %---------------------------- gap D_n H_p % One Dimensional H % Height of combustion chamber (m) D % Diameter of combustion chamber (m) H_b % Height of fuel bed (m) D_b % Diameter of fuel bed (m) D_c % Diameter of combustion chamber (m) D_w1 % Diameter of inner wall (m) D_w2 % Diameter of outer wall (m) *note: 1 cm in between inner

and out wall D_p % Diameter of pot (m)

% Two Dimensional A_b % (m^2) Area of the bed (inlet area) A_i A_n % (m^2) Area of the nozzle A_c % (m^2) Area of the combustion chamber A_a % (m^2) Area of the skirt (annulus) and outlet area A_s % (m^2) Area of the stagnation point (bottom of pot) A_ps % (m^2) Area of the sides of the pot P_a % (m) Perimeter of skirt (annulus) and outlet area

% Hydraulic Diameters D_h % (m) Hydraulic Diameter of the skirt (annulus) and outlet

area

%---------------------------- %TEMPERATURE %---------------------------- % The following correspond to 162% excess air adn the

theoretical limit to

41

% flame temperature Taken from [Modelling of solid fuel stoves]

by Kausley, Pandit

T_amb = 293.15; % (K) Ambient air temperature T_f = 860 + 273.15; % (K) Flame Temperature T_b = 650 ; % (K) Bed Temperature - Ignition temperature of

wood. pyrolysis of fuel occurs with release of volatils at this

temperature T_b_fluid T_water = 80.0 + 273.0 ;% (K) about boiling

% ------------------------- % Constants % -------------------------

% universal constants g = 9.81; % (m/s^2) gravitational constant sigma = 5.67*10^-8 ; %(W/m^2K^4) Stephan Boltzman Constant

% friction factors ff = .024; % a reasonable assumption for D_h=.02, roughness of

steel = 4.57x10^-5 m, so relative roughness = .002. from moody chart,

this is close to f=.02. we also could choose f=.024 to be more precise,

but dr. orloff uses f=.02 in his notes

% fuel characteristics bed_voidage = 0.69 ;% sigma_v d_sphere = 0.015 ;% sphere diameter (m) sphericity = .874 ;% phi P_atm = 101325.0 ;% Pa (N/m^2)

% radiation emissivities emissivity_b = .94; %(unitless)

% Water properties mu_water = 0.000286; % kg / (m * s)

res_q_conv_stag res_q_conv_side res_q_rad end

methods (Static) function ret = cp_fit(T_1, T_2) aa = 1.9327e-10; bb = -7.9999e-7; cc = 1.1407e-3; dd = -4.489e-1; ee = 1.0575e3;

coeffs = [aa, bb, cc, dd, ee]; divs = 5:-1:1;

t1s = ones(1, 5) * T_1;

42

t2s = ones(1, 5) * T_2; dts = t1s.^divs - t2s.^divs;

ret = sum(coeffs ./ divs .* dts); end end

methods function f = StoveSystem(gap, D, D_n, H_p) f.gap = gap; f.D_n = D_n; f.H_p = H_p;

% One Dimensional f.H = .5;

f.D = D; % f.D = 0.3;

f.H_b = 0.05; f.D_b = f.D; f.D_c = f.D; f.D_w1 = f.D; f.D_w2 = f.D + .02; f.D_p = f.D - 2.*(gap);

% Two Dimensional f.A_b = pi./4.*f.D_b.^2; f.A_i = f.A_b/4; f.A_n = pi./4.*D_n.^2; f.A_c = pi./4.*D_n.^2; f.A_a = pi./4.*(f.D_w1.^2 - f.D_p.^2); f.A_s = pi./4.*f.D_p.^2; f.A_ps = pi.*f.D_p*H_p; f.P_a = pi.*(f.D_w1 + f.D_p);

% Hydraulic Diameters f.D_h = 4.*f.A_a./f.P_a;

f.T_b_fluid = (f.T_amb + f.T_b)/2; % (K) Temperature of the

fluid as it travels through the bed end

function F = validate(self, T) % Temperatures T_p1 = T(1); T_an = T(2); T_p2 = T(3); T_exit = T(4);

T_n = mean([self.T_f, T_an]); T_a = mean([T_an, T_exit]);

43

% Gas Thermophysical Properties *thermophysical properties

of air % calculated using air.m % % Density p_air = air('p',293.15); % density of air at atmospheric

temperature kg/m^3) p_f = air('p',self.T_f); % density of flue gas in the flame

(kg/m^3) p_a = air('p',T_a); % density of flue gas in annulus

(kg/m^3) p_n = air('p',T_n); % density of flue gas around bottom of

pot (by nozzle) p_b = air('p',self.T_b_fluid); % density of air gas in

traveling thru fuel bed (kg/m^3)

% Viscosity of Air (HT book) mu_b = air('mu',self.T_b_fluid); %(Ns/m) viscosity of air

traveling thru fuel bed (@ avg of ambient and fuel bed temp); mu_a = air('mu',T_a) ; % (Ns/m) viscosity of air traveling

thru annulus (@ approx temp in annulus)

% ------------------------- % PRESSURE DROP COEFFICIENTS % -------------------------

% Pressure drop for the fuel bed A = (150*mu_b*self.H_b*(1-

self.bed_voidage)^2)/(self.d_sphere^2*self.bed_voidage^3*self.sphericit

y^2); % multiplied times u B = (1.75*p_b*self.H_b*(1-

self.bed_voidage))/(self.d_sphere*self.bed_voidage^3*self.sphericity);

% multiplied times u^2

% Pressure drop for the annulus C =

self.ff*self.H_p/self.D_h*p_air/2*((p_air*self.A_b)/(p_a*self.A_a))^2;

% multiplied times u^2

% Pressure drop for the nozzle (assuming theta = 90 for

entrance, and using velocity of the chamber, not the nozzle %Big to Small E = p_f/2*(((p_air*self.A_b)/(p_f*self.A_n))^2-

((p_air*self.A_b)/(p_f*self.A_c))); % multiplied times u^2 %Small to Big F = .5.*((p_air*self.A_b)/(p_f*self.A_n))^2*p_f*(1-

(self.D_n/self.D_c)^2)^2; % multiplied times u^2

% Buoyant forces from change in density of air and flue

gasses due to temperature rise Y = p_air/2; Z = self.g*self.H*(p_air-p_a);

% Solve function for u, inlet air velocity quad_c = -Z; % pressure dro pcoefficient for u^0

44

quad_b = A; % pressure drop coefficient for u^1 quad_a = Y + B + C + E + F; % pressure drop coefficient for

u^2

% use quadratic equation to solve u_i = (-quad_b + (quad_b^2 -

4*quad_a*quad_c)^.5)./(2*quad_a); % (m/s) inlet velocity of air into

the stove

%Knowing input velocity, u , we can solve for u all over

the stove u_c = (u_i*self.A_i.*p_air)/(self.A_c*p_f); % (m/s)

velocity of flue gas through combustion chamber u_n = (u_i*self.A_i.*p_air)/(self.A_n*p_f); % (m/s)

velocity of flue gas through nozzle u_a = (u_i*self.A_i.*p_air)/(self.A_a*p_a); % (m/s)

velocity of flue gas through annulus (skirt)

% ------------------------- % HEAT TRANSFER CALCULATIONS % -------------------------

% Radiative Heat Transfer to the Pot**need to check this

against book...i % think its right but some reason does not come out this

way by hand... R_b = self.D_b/(2*self.H); R_p = self.D_n/(2*self.H); S = 1 + ((1 + R_p^2)/(R_b^2)); F_bp = .5*(S - (S^2 - 4*(self.D_n/self.D_b)^2)^.5); % View

Factor from the Bed to the Pot (through nozzle opening) q_rad =

self.sigma*self.emissivity_b*F_bp*self.A_b*(self.T_b^4 - T_p1^4);

% Convective Heat Transfer to the Pot at the Stagnation

Point % already defined p_n, density of flue gas around

stagnation point for air at T_n k_n = air('k',T_n); % (W/mK) fluid conductivity of flue gas

around stagnation point for air at T_n Cp_n = air('cp',T_n); % (J/KgK) specific heat of flue gas

around stagnation point for air at T_n mu_n = air('mu',T_n); % (Ns/m^2) viscosity of flue gas

around stagnation point for air at T_n Pr_n = Cp_n.*mu_n./k_n; % Prandtl Number of flue gas around

bottom of pot Z_over_D = .25; % ratio of distance that nozzle is from pan

over diameter of nozzle

% h_conv_stag =

k_n./D_n.*1.26.*(Pr_n).^.42.*(p_n.*u_n.*D_n./mu_n).*(D_p./D_n).^(-.5) %

(W/m^2K) convection coefficient at bottom of pot h_conv_stag =

self.D_p/k_n*.424*(p_n*u_n*self.D_n/mu_n)^.57*Z_over_D^(-.33); %

(W/m^2K) convection coefficient at bottom of pot

45

% Convective Heat Transfer to the Pot in the Skirt Section % already defined p_a, density of flue gas around annulus

(skirt) k_a = air('k',T_a); % (W/mK) fluid conductivity of flue gas

around annulus (skirt) for air at T_a Cp_a = air('cp',T_a); % (J/KgK) specific heat of flue gas

around annulus (skirt) for air at T_a mu_a = air('mu',T_a); % (Ns/m^2) viscosity of flue gas

around annulus (skirt) for air at T_a Pr_a = Cp_a.*mu_a./k_a; % Prandtl Number of flue gas around

annulus (skirt) for air at T_a

h_conv_side =

k_a/self.D_h*.23*(p_a*u_a*self.D_h/mu_a)^.8*Pr_a^(1/3)*(mu_a/self.mu_wa

ter).^.14; % (W) Convection Coefficient at sides (skirt) around pot

m_dot_a = p_a.*u_a.*self.A_a; % (Kg/s) mass flow rate of

air out of the stove (annulus)

q_stag = h_conv_stag * self.A_s * (T_p1 - self.T_f/2.0 -

T_an/2.0);

StoveSystem.cp_fit(self.T_f, T_an) eq_1 = m_dot_a * StoveSystem.cp_fit(self.T_f, T_an) + ... q_stag; eq_2 = m_dot_a * StoveSystem.cp_fit(T_an, T_exit) + ... h_conv_side * self.A_ps * (T_p2 - T_exit/2.0 -

T_an/2.0);

h_w = 2500; % (W/m^2K)

eq_3 = h_w*self.A_s*(T_p1-self.T_water) + q_rad + q_stag;

eq_4 = (h_w+h_conv_side)*T_p2 - (h_conv_side/2.0)*T_an -

... (h_conv_side/2.0)*T_exit - h_w*self.T_water;

F = [eq_1; eq_2; eq_3; eq_4]; end

function [qNet, qIn] = netHT(self, T) % Temperatures T_p1 = T(1); T_an = T(2); T_p2 = T(3); T_exit = T(4);

T_n = mean([self.T_f, T_an]); T_a = mean([T_an, T_exit]);

46

% Gas Thermophysical Properties *thermophysical properties

of air % calculated using air.m % % Density p_air = air('p',293.15); % density of air at atmospheric

temperature kg/m^3) p_f = air('p',self.T_f); % density of flue gas in the flame

(kg/m^3) p_a = air('p',T_a); % density of flue gas in annulus

(kg/m^3) p_n = air('p',T_n); % density of flue gas around bottom of

pot (by nozzle) p_b = air('p',self.T_b_fluid); % density of air gas in

traveling thru fuel bed (kg/m^3)

% Viscosity of Air (HT book) mu_b = air('mu',self.T_b_fluid); %(Ns/m) viscosity of air

traveling thru fuel bed (@ avg of ambient and fuel bed temp); mu_a = air('mu',T_a) ; % (Ns/m) viscosity of air traveling

thru annulus (@ approx temp in annulus)

% ------------------------- % PRESSURE DROP COEFFICIENTS % -------------------------

% Pressure drop for the fuel bed A = (150*mu_b*self.H_b*(1-

self.bed_voidage)^2)/(self.d_sphere^2*self.bed_voidage^3*self.sphericit

y^2); % multiplied times u B = (1.75*p_b*self.H_b*(1-

self.bed_voidage))/(self.d_sphere*self.bed_voidage^3*self.sphericity);

% multiplied times u^2

% Pressure drop for the annulus C =

self.ff*self.H_p/self.D_h*p_air/2*((p_air*self.A_b)/(p_a*self.A_a))^2;

% multiplied times u^2

% Pressure drop for the nozzle (assuming theta = 90 for

entrance, and using velocity of the chamber, not the nozzle %Big to Small E = p_f/2*(((p_air*self.A_b)/(p_f*self.A_n))^2-

((p_air*self.A_b)/(p_f*self.A_c))); % multiplied times u^2 %Small to Big F = .5.*((p_air*self.A_b)/(p_f*self.A_n))^2*p_f*(1-

(self.D_n/self.D_c)^2)^2; % multiplied times u^2

% Buoyant forces from change in density of air and flue

gasses due to temperature rise Y = p_air/2; Z = self.g*self.H*(p_air-p_a);

% Solve function for u, inlet air velocity quad_c = -Z; % pressure dro pcoefficient for u^0

47

quad_b = A; % pressure drop coefficient for u^1 quad_a = Y + B + C + E + F; % pressure drop coefficient for

u^2

% use quadratic equation to solve u_i = (-quad_b + (quad_b^2 -

4*quad_a*quad_c)^.5)./(2*quad_a); % (m/s) inlet velocity of air into

the stove

%Knowing input velocity, u , we can solve for u all over

the stove u_c = (u_i*self.A_i.*p_air)/(self.A_c*p_f); % (m/s)

velocity of flue gas through combustion chamber u_n = (u_i*self.A_i.*p_air)/(self.A_n*p_f); % (m/s)

velocity of flue gas through nozzle u_a = (u_i*self.A_i.*p_air)/(self.A_a*p_a); % (m/s)

velocity of flue gas through annulus (skirt)

% ------------------------- % HEAT TRANSFER CALCULATIONS % -------------------------

% Radiative Heat Transfer to the Pot**need to check this

against book...i % think its right but some reason does not come out this

way by hand... R_b = self.D_b/(2*self.H); R_p = self.D_n/(2*self.H); S = 1 + ((1 + R_p^2)/(R_b^2)); F_bp = .5*(S - (S^2 - 4*(self.D_n/self.D_b)^2)^.5); % View

Factor from the Bed to the Pot (through nozzle opening) q_rad =

self.sigma*self.emissivity_b*F_bp*self.A_b*(self.T_b^4 - T_p1^4);

% Convective Heat Transfer to the Pot at the Stagnation

Point % already defined p_n, density of flue gas around

stagnation point for air at T_n k_n = air('k',T_n); % (W/mK) fluid conductivity of flue gas

around stagnation point for air at T_n Cp_n = air('cp',T_n); % (J/KgK) specific heat of flue gas

around stagnation point for air at T_n mu_n = air('mu',T_n); % (Ns/m^2) viscosity of flue gas

around stagnation point for air at T_n Pr_n = Cp_n.*mu_n./k_n; % Prandtl Number of flue gas around

bottom of pot Z_over_D = .25; % ratio of distance that nozzle is from pan

over diameter of nozzle

% h_conv_stag =

k_n./D_n.*1.26.*(Pr_n).^.42.*(p_n.*u_n.*D_n./mu_n).*(D_p./D_n).^(-.5) %

(W/m^2K) convection coefficient at bottom of pot h_conv_stag =

self.D_p/k_n*.15*(p_n*u_n*self.D_n/mu_n)^.57*Z_over_D^(-.33); %

(W/m^2K) convection coefficient at bottom of pot

48

% Convective Heat Transfer to the Pot in the Skirt Section % already defined p_a, density of flue gas around annulus

(skirt) k_a = air('k',T_a); % (W/mK) fluid conductivity of flue gas

around annulus (skirt) for air at T_a Cp_a = air('cp',T_a); % (J/KgK) specific heat of flue gas

around annulus (skirt) for air at T_a mu_a = air('mu',T_a); % (Ns/m^2) viscosity of flue gas

around annulus (skirt) for air at T_a Pr_a = Cp_a.*mu_a./k_a; % Prandtl Number of flue gas around

annulus (skirt) for air at T_a

h_conv_side =

k_a/self.D_h*.23*(p_a*u_a*self.D_h/mu_a)^.8*Pr_a^(1/3)*(mu_a/self.mu_wa

ter).^.14; % (W) Convection Coefficient at sides (skirt) around pot

m_dot_a = p_a.*u_a.*self.A_a; % (Kg/s) mass flow rate of

air out of the stove (annulus)

q_conv_stag = h_conv_stag*self.A_s*(T_n - T_p1); % (W)

Convection Heat Transfer at Stagnation point to the pot q_conv_side = h_conv_side*self.A_ps*(T_a - T_p2); q_rad = q_rad;

% q_in = radiation (VF=1.0) + heat storage by air qIn = self.sigma*self.emissivity_b*self.A_b*(self.T_b^4-

T_p1^4) + ... m_dot_a * StoveSystem.cp_fit(self.T_f, 24+273); [q_conv_stag, q_conv_side, q_rad] qNet = sum([q_conv_stag, q_conv_side, q_rad]); end end end