Final Design Project: Biodiesel Settling Tank Analysis

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1 MESSIAH COLLEGE ENGR 495 Finite Element Methods Tuesday, December 16, 2003 : Biodiesel Settling Tank Analysis Brandon Apple Jon Bitterman Becky Gast Kyle McNamara ENGR 495 Finite Element Methods 1

2 Abstract The purpose of this project is to analyze a 15 gallon conical bottom polyethylene tank under known loading conditions. The loading conditions for this project were determined by our Biodiesel Production senior project group. Our Biodiesel Production group requires a tank that will hold a maximum of 15 gallons of oil at a maximum temperature of 140 degrees Fahrenheit without failure. The results of the analysis should prove either that the tank will not fail or the reasons why failure occurred and the results of failure. Background Our Finite Element Method project analyzes a piece of equipment essential to our senior project of producing biodiesel fuel. The process consists of taking used cooking oil and through a chemical process making a diesel fuel substitute. We are using waste cooking oil from Dining Services on campus to create a product the campus is currently purchasing. We have designed a physical prototype of our system used to make the fuel. Within the system, the acidic oil is heated, mixed with basic chemicals, and then left to settle and separate. For our Finite Element Method project we are analyzing the reacting/settling/washing tank that the hot oil is fed into. The tank is a 15 gallon, conical bottom tank made of a linear low density polyethylene. Polyethylene is an inexpensive and versatile polymer and was chosen for the material of the tank due to its resistance to chemical erosion and price within that criterion. The used cooking oil is first heated to a minimum of 122 degrees Fahrenheit to a maximum of 140 degrees Fahrenheit in a metal tank. The used oil then enters the reacting/settling/washing tank to begin mixing with the chemicals. We are analyzing the reacting/settling/washing tank in I-DEAS to predict the effects of the temperature and pressure of the used oil on the integrity of the polyethylene tank. The manufacturer states that the deflection ENGR 495 Finite Element Methods 2

3 temperature of the polyethylene is 142 degrees Fahrenheit at 66 pounds per square inch (psi). Objectives 1. Model the reacting/settling/washing tank of the Biodiesel Manufacturing Senior Design Project Group. 2. Determine the temperature distribution through the wall of the polyethylene tank. 3. Establish whether the tank will fail due to yielding caused by stresses, induced by thermal and physical restraints. Theory and Procedure Due to the complex nature of analyzing the tank under thermal and physical restraints, both heat transfer and linear statics methods must be used. The analysis of the tank can be greatly reduced due to the axisymmetric symmetry of the tank. Heat transfer analysis is necessary because there are stresses produced due to a temperature gradient in the tank wall. The temperature gradient is defined as the difference between the temperature of the oil inside the tank and the ambient temperature outside the tank. As a result of this temperature gradient, heat is transferred across the thickness of the wall by conduction and then into the outside environment by convection. I-DEAS is capable of handling heat transfer when a temperature restraint is applied to the inside surface of the model and convection is applied to the outside surface of the model. Linear Statics analysis is necessary because there are stresses produced as a result of the pressure on the inside surface and the applied boundary conditions. The pressure varies in the tank and is dependent on the density of the fluid and the relative height of the fluid. This dependency is shown in the equation below, which can be used to find the pressure at various heights throughout the tank assuming uniform density oil. p = γ * h whe re γ = g * ρ ENGR 495 Finite Element Methods 3

4 The boundary conditions that are applied are a result of the brace that holds the tank in place and are assumed to be constant. I-DEAS is capable of handling linear statics analysis when a data edge is applied to the inside edge and appropriate boundary conditions are applied to the outside at the location of the brace. The material properties of polyethylene must also be determined and defined for an accurate I-DEAS analysis. Most of the properties are assumed to be constant with the exception being the modulus of elasticity. Since there is a temperature gradient in the wall, the modulus of elasticity will vary in the wall dependent upon the temperature. Since there is not a set standard for the variation of the modulus of elasticity with temperature, experimental data must be used to generate an approximate function. Once the material properties are defined, the heat transfer analysis is completed, and the linear statics analysis with respect to the heat transfer results is completed, the final results must be analyzed. The maximum stress that occurs in the model as a result of the stresses from thermal and physical restraints must be compared to the known 0.2% offset yield stress for polyethylene. Polyethylene is a ductile material so it can withstand deformation without assumed failure because the material will return to its original shape as long as the yield stress has not been reached. However, if the results show that the maximum stress is greater than the yield stress than failure has occurred because permanent deformation has occurred in the material. In this case, the permanent deformation may be a result of the plastic bending, stretching, or even melting. Assumptions In order to perform our analysis on the reacting/settling/washing tank we were required to make assumptions regarding our biodiesel production method and the properties of the specific polyethylene material in order to simulate and solve the problem in I-DEAS. We did acquire the specification sheet from the manufacturer regarding the exact polyethylene used for the tank, but there was still room for ENGR 495 Finite Element Methods 4

5 interpretation on some of the terms. Below we have stated the specific assumptions we have made throughout our analysis. 1. The inner surface of the polyethylene tank is constant at 140 F (60 C). 2. The ambient conditions are constant at 32 F (0 C). 3. There is a constant convection coefficient of 100 W/(m 2 K) from the outer surface. 4. Heat transfer due to radiation is negligible. 5. The thermal properties of the polyethylene tank are constant. 6. The analysis is at steady state. 7. The heat transfer analysis is two-dimensional. 8. The density of the used cooking oil is uniform at lbm / in The model of the tank is axisymmetric and therefore two-dimensional analysis can be used. Another assumption we made regards linear statics and the modulus of elasticity (E). This property is usually averaged as a constant value for a material, but in actuality is a temperature dependant property. Since our analysis and interpretation of results is dictated by temperature, we needed to use the temperature dependant modulus of elasticity. This information is experimental and sparse, but we found a small set of experimental data for low density polyethylene which is listed below. Temp ( C) E (MPa) Temp ( F) E (psi) Table 1. Experimental data for temperature dependence of the modulus of elasticity We compared the data with the averaged value for the modulus of elasticity from the manufacturer and found the data to be consistent. We graphed the data to obtain a function relating the temperature in degrees Fahrenheit to the modulus of elasticity in pounds per square inch. We chose a polynomial equation after comparing R 2 values of various types of functions. From the function we obtained the modulus of ENGR 495 Finite Element Methods 5

6 elasticity for the inner and outer surface temperatures of 140 F and 32 F. A graph of the modulus of elasticity versus the temperature is shown below. Modulus of Elasticity (psi) 1.E+05 9.E+04 8.E+04 7.E+04 6.E+04 5.E+04 4.E+04 3.E+04 Modulus of Elasticity vs Temperature E(exp) = T T R 2 = E(eqn) = T T R 2 = 1 2.E+04 1.E+04 0.E Temperature (F) Graph 1. Plot of the effects of temperature on the modulus of elasticity. Analysis Heat Transfer With the assumption of constant oil and ambient air temperatures, along with constant thermal conductivity of the tank (see Appendix A), there is no change in the temperature distribution of the cross section, regardless of location in the hoop direction. This allowed us to utilize axisymmetric elements for our analysis. Once the cross section was modeled, we revolved it 90 degrees, while only analyzing the cross-section. Having assumed a constant oil temperature, we applied a constant temperature restraint on the inner edge of the cross-section, excluding the roof of the ENGR 495 Finite Element Methods 6

7 tank. However, I-DEAS would not recognize this boundary condition when we attempted to mesh with axisymmetric elements. With the aid of Dr. Van Dyke, we found that I-DEAS required the temperature restraint to be applied to the entire inner surface, not just the edge of the cross-section of the tank. This allowed the boundary condition to be recognized when meshed with axisymmetric elements. To model the heat loss through the tank, we decided to apply convection over the outer surface. This simply required defining the surrounding air temperature and a convection coefficient. We chose to model the surrounding air at 32 degrees, and we selecting an average value of the convection coefficient to model free convection (Appendix A). I-DEAS, again proved reluctant to comply. After unsuccessfully attempting to apply the convection to the entire outer surface, as well as only the outer edge of the tank, we went to the tutorials and Dr. Van Dyke for help. He eventually determined that I-DEAS would calculate the convection only if it was applied directly to each element on the outer edge of the cross-section. The problem seemed to be that I-DEAS would not accept geometry based analysis for the convection, so it had to be applied to each element directly. This became a minor inconvenience since there were approximately 750 elements on which to apply the convection. With the temperature restraint and the convection applied, we meshed the cross-section with an element length of 0.06 inches, resulting in about four elements across the thickness of the tank. The model could then be solved for the temperature distribution throughout the cross section. For further analysis, I-DEAS has the capability to apply the temperature gradient as a boundary condition. Linear Statics We assumed constant physical properties of isotropic polyethylene, with the exclusion of the temperature dependence of the modulus of elasticity. I-DEAS allowed the modulus of elasticity to be input as a function, which was based on a best fit curve of experimental data that we found for polyethylene (Appendix B). Again, as in the heat transfer analysis, having assumed these properties, we were able to use axisymmetric elements for the analysis. Using the solution from the heat transfer ENGR 495 Finite Element Methods 7

8 analysis, we applied the temperature gradient to the cross-section, as well as a displacement restraint in the axial direction, on the fillet between the conical section and vertical wall on the outer edge of the cross-section of the tank (Appendix C). This restraint models the stand that will support the tank on the fillet. To model the pressure on the inner surface of the tank, we assumed that the tank was full of oil with uniform density. This allowed us to calculate a function for the pressure along the surface of the tank, which varied linearly with height (Appendix A). To apply, we had to define a Data Edge along the inner edge of the cross-section of the tank, with the function specified as the magnitude of the pressure varying with height. Then, to the edge a force could be applied as an axisymmetric intensity with a factor of one, since the Data Edge function accounted for the magnitude of the pressure at any point (Appendix C). This model could then be solved to determine the displacements and stresses in the tank. Results Figure 1. Temperature distribution through the cross section of the tank wall with inner wall surface of 140 o F and ambient temperature of 32 o F ENGR 495 Finite Element Methods 8

9 Figure 2. Displacement of the cross-section with inner wall surface of 140 o F and ambient temperature of 32 o F Figure 3. Von Mises Stress of the cross-section with inner wall surface of 140 o F and ambient temperature of 32 o F ENGR 495 Finite Element Methods 9

10 Figure 4. Temperature distribution through the cross section of the tank wall with inner wall surface of 140 o F and ambient temperature of 72 o F Figure 5. Displacement of the cross-section with inner wall surface of 140 o F and ambient temperature of 72 o F ENGR 495 Finite Element Methods 10

11 Figure 6. Von Mises Stress of the cross-section with inner wall surface of 140 o F and ambient temperature of 72 o F Figure 7. Displacement of the cross-section without wall temperature distribution ENGR 495 Finite Element Methods 11

12 Figure 8. Von Mises Stress of the cross-section without wall temperature distribution Discussion and Conclusions Heat Transfer Our I-DEAS results for the heat transfer analysis of the tank shows a linear temperature distribution through the thickness of the wall (See Figure 1). This temperature distribution is the same for all portions of the tank except the roof portion. The roof portion of the tank is at a constant temperature equal to the ambient temperature because a temperature restraint was not placed on the inner surface. With an inner surface temperature of 140 o F and an ambient temperature of 32 o F, our results show a linear temperature distribution from 140 o F on the inner surface to 59 o F on the outer surface. Linear Statics Our I-DEAS results for the linear statics analysis of the tank provided us with a maximum Von Mises stress and a displacement for the tank under the prescribed ENGR 495 Finite Element Methods 12

13 boundary conditions (see Figure 2 and Figure 3). We used the maximum Von Mises stress to compare with the 0.2% offset yield stress for polyethylene to determine whether or not the tank failed. Polyethylene is a ductile material so there are two possibilities for failure. One possibility for failure in this scenario refers to permanent deformation of the tank once the load is removed. Another possibility refers to indefinite deformation of the tank once the yield stress is reached and the load remains applied. Based on our analysis we determined that our tank will not fail since our maximum Von Mises stress is 1550 psi and the 0.2% offset yield stress is 2550 psi (See Appendix D). Since the maximum Von Mises stress will never reach the yield stress, the deformations incurred by the tank will be elastic. Being elastic, the tank will return to its original shape once the loads have been removed. The deformations of the tank will not hinder the production of biodiesel in any way and are therefore inconsequential in our failure analysis. Further Analysis To verify our results we also analyzed the tank under two different conditions. The first condition replaces the ambient temperature of 32 o F with 72 o F, and the second condition applies the physical restraints without the temperature and convection applied. The first condition simulated operating the biodiesel production in a warmer environment (See Figure 4). The results show that the maximum Von Mises stress is less than the results for the original condition (See Figure 6). This is due to the fact that modulus of elasticity is decreased since the average temperature is increased (See Appendix B). Therefore there will be no failure due to stresses in the tank. However, the displacement result is greater than in the original condition as a result of the decrease in the modulus of elasticity (See Figure 5). There needs to be some limit placed on the elasticity of our material for accurate failure analysis. As the modulus of elasticity decreases with increasing temperature, the material will cease to behave in an elastic manner. Without more knowledge of the materials behavior, this nonlinearity is impossible to predict and a physical analysis of the tank is required to ENGR 495 Finite Element Methods 13

14 predict its behavior. Based on Matt Steiman s 1 use of the tank at Wilson College under similar conditions, we can conclude that the linear assumption is still valid. The second condition simulated applying physical restraints without applying the temperature and convection. These results gave an estimation of the effects of the temperature by comparing them with the results from the original condition. An actual comparison of results for the displacement shows an approximate 30% decrease in the maximum deflection of the tank (See Figure 7). The results for the maximum Von Mises stress show a decrease from 1550 psi to 1150 psi (See Figure 8). This proves the significance of applying the temperature and convection to our model to analyze the tank properly. 1 Matt Steiman is the director of Fulton Farm for Sustainable Living at Wilson College in Chambersburg, PA. We had the opportunity to witness his working biodiesel processor with a similar settling tank to the one we are analyzing. ENGR 495 Finite Element Methods 14

15 Appendix A Properties of Polyethylene Poisson s Ratio.43 Modulus of Elasticity ( F) E(T) = (T 2 ) (T) (MPa) Mass Density 940 kg/m^3 Coefficient of Thermal Expansion /K Thermal Conductivity.3344 J/(m*K*s) Yield Stress 2550 psi Properties of Used Cooking Oil Mass Density lbm / in 3 Pressure Distribution f(z) = (N/m 2 ) Z* (N/m 3 ) Inner Surface Temperature 140 F Outer Surface Temperature 32 F Properties of Convection Convection Coefficient 100 W/m 2 *K Ambient Temperature 32 F ENGR 495 Finite Element Methods 15

16 Appendix B 1.E+05 Modulus of Elasticity vs Temperature Modulus of Elasticity (psi) 9.E+04 8.E+04 7.E+04 6.E+04 5.E+04 4.E+04 3.E+04 2.E+04 E(exp) = T T R 2 = E(eqn) = T T R 2 = 1 1.E+04 0.E Temperature (F) Figure B1. Plot used to develop a function for the change in the modulus of elasticity with temperature ENGR 495 Finite Element Methods 16

17 Appendix B continued Figure B2. I-DEAS plot used to verify the accuracy of the input function. ENGR 495 Finite Element Methods 17

18 Appendix C Figure C1. Heat Transfer Boundary Conditions Figure C2. Linear Statics data edge and force ENGR 495 Finite Element Methods 18

19 Figure C3. Linear Statics force Figure C4. Linear Statics displacement restraint and force ENGR 495 Finite Element Methods 19

20 Appendix D Figure D1. Von Mises Stress in Top Surface (Original Analysis) Figure D2. Von Mises Stress in Top Corner (Original Analysis) ENGR 495 Finite Element Methods 20

21 Figure D3. Von Mises Stress in Top -Right Edge (Original Analysis) Figure D4. Von Mises Stress in Beginning of Sloped Portion (Original Analysis) ENGR 495 Finite Element Methods 21

22 Figure D5. Von Mises Stress in Sloped Portion (Original Analysis) Figure D6. Von Mises Stress in Bottom Portion (Original Analysis) ENGR 495 Finite Element Methods 22