SUBSTITUTION OF ELECTRICAL DRYERS FOR NATURAL GAS DRYERS. Aditya Chintalapati, University of Michigan Nicholas E Myers, University of Michigan

Transcription

1 SUBSTITUTION OF ELECTRICAL DRYERS FOR NATURAL GAS DRYERS Aditya Chintalapati, University of Michigan Nicholas E Myers, University of Michigan Abstract In the U.S., there are numerous small and medium size manufacturing plants, where automated drying processes are utilized. Electric heaters are often used to preheat the air flowing with high velocities before it is blown onto a product in the drying process. One facility that uses such a process is the MIWI washers manufacturing plant in Jackson, Michigan. Our team looked at replacing the electric heaters with cheaper and potentially faster-heating natural gas burners. We determined how to control temperature, flame stability, and air quality changes in the dryer air stream due to the burning of natural gas. We also investigated the potential for cost savings, improvement in the production speed, and possible contamination from the products of combustion in the warm air stream due to the burning of natural gas. Keywords: Industrial Drying, Natural Gas Heater, Efficiency, Manufacturing Process Outline 1. Introduction 1. Overview of MIWI plant 2. Potential for cost savings 3. Plant Specifications 4. Technical Considerations 2. Fuel and Air Combustion 1. Stoichiometry 2. Flame Temperature 3. Mixing 4. Mass Flow Rates 3. Emissions 1. Soot 2. CO, NO X, and N 2 O 3. CO2 from energy use 4. Heat Transfer and Drying Process 1. Nozzle Jet velocity 2. Rate of transfer to plate and manufacturing process speed 3. Heat loss in pipe 5. Financial Analysis 6. Recommendations and Conclusions 1

2 1. Introduction In the U.S., there are numerous small and medium size manufacturing plants, where automated drying processes are utilized. Electric heaters are often used to preheat the air flowing with high velocities before it is blown onto a product in the drying process. In an automated production line, the speed at which a product can be manufactured is dependent on the speed at which the slowest process can be accomplished and in this project we are focusing on the drying process after a wash of the production items before a final surface treatment. This project builds off an initial assessment performed by the Industrial Assessment Center [1]. The purpose of this project was to investigate the potential for the MIWI facility in Jackson Michigan to switch their electrical dryers into natural gas dryers for the systems they manufacture. 1.1 Overview of the MIWI manufactured drying equipment The MIWI manufacturing plant is located in Jackson, MI. The plant designs and produces systems for drying of plastic and metallic components after a wash. Besides producing the drying systems for sale they also offer in house washing and drying services for external clients. In such a system (Fig 1), the electric heater heats the air after it is propelled by a blower, which is funneled into nozzles and is used to quickly dry the parts after the washing process is completed. The use of electricity to heat air is expensive and slow but easier to control than natural gas heaters. 1.2 Potential for cost savings Figure 1: Schematic of the the MIWI manufacturing plant s dryer The primary cost saving for the plant will come from switching the power source from electricity to burning of natural gas. Comparing the electricity used by the dryer with the energy equivalent cost of natural gas, we were able to determine the following cost savings potential (Fig 2). 2

3 Tout = 150 C Tout = 120 C Tout = 90 C Tout = 70 C Tout = 50 C Figure 2: Potential for cost saving with respect to current flow rates and temperatures At the current level of output temperature of 90 C and output volume of 11 m 3 /min, we could potentially save up to $3,000 annually per dryer by switching to natural gas. 1.3 Plant Specifications There were certain specifications that our system had to meet and specific conditions under which the system had to operate. These specifications (Table 1) were obtained from the Industrial Assessment Survey that was conducted prior to this project. Table 1: Showing the current plant specifications and operating conditions Parameters Value Ambient air temperature T AMB [K] Relative Humidity of air 70% RH AMB Outpute air temperature T OUT [K] Mass of exiting air m AIR [kg/s] System Power 15 Q EL [kw] Curent Elecrical Costs 3,750 AELC [$/yr] Outpute Relative Humidity 3.16% RH OUT Current Carbon Emissions 64.5 CO 2 [ton/yr] 1.4 Technical Considerations In order to switch use of electricity to natural gas we must meet the previous listed specifications plus a few other requirements must be addressed. A primary concern is that the emissions in the natural gas product gas flow should not impact the parts that were being dried, that is we are trying to use the diluted products from the combustion directly in the drying process. If that is 3

4 not possible then we must use a heat exchanger to heat air from the hot combustion products and use that air for the drying. This would introduce one additional step in the process and lower the overall efficiency of the heating process. There are also other concerns regarding the control of combustion to ensure that the dryer could be adjusted for different drying settings and that the temperature in the combustion chamber was at a safe level. We would also have to design a system such that it can easily interface with the existing air channels and does not require significant modification of the dryer itself in order to implement it. 2. Fuel and air combustion The primary purpose of this project was to investigate the feasibility of switching from electrical dryers to natural gas dryers and the challenges associated with it. In order to tackle these challenges, we conducted several theoretical and analytical tests to determine the ideal conditions to operate the system in along with addressing the concerns that the manufacturer brought up. The first thing that had to be done was to ensure that natural gas could be used as a viable fuel. This required our team to ensure that the required mass of natural gas could be safely combusted while minimizing flame temperatures and that a sufficient amount of secondary air could be added to cool down the system to the required 90 C output while ensuring that it is fully mixed. 2.1 Fuel Stoichiometry Natural Gas is made up of several hydrocarbons, however 95% of natural gas is Methane [2]. The combustion equation for natural gas is: CH 4 + 2(O N 2 ) CO 2 + 2H 2 O N 2 (Eq. 1) From a stoichiometric analysis of fuel combustion as shown in Appendix A, it was determined that in order to combust the required 1.08 kg of Natural Gas, we would need to add kg of primary air for stoichiometric combustion. 2.2 Flame Temperature One of the concerns that was brought up to us by the plant was the temperature in the combustion chamber itself. In the interest of safety, we had to ensure that the flame temperature had to be as low as possible. One way to reduce the flame temperature is to ensure that the combustion happens with excess air. Figure 3 [3] on page 5 shows the change in flame temperature with respect to the Fuel-Air ratio. 4

5 Figure 3: Flame Temperature vs. Equivalence ratio for Natural Gas An Equivalence ratio of 6.0 was determined to be ideal for this application [3] As shown in Fig 3 above, the minimum fuel to air ratio that still allows for combustion is 0.55 However, in order to stay above this limit and avoid a flame out scenario, the team concluded that a fuel to air ratio of 0.6 would be ideal. An equivalence ratio of 0.6 would reduce the flame temperature from 2100K at stoichiometric combustion ratio to a safer 1600K. As discussed in Section 3, this change also results in decreased emissions from the combustion of natural gas. To change the equivalence ratio, more primary air would be needed to added required recalculation of the primary and secondary air flows as shown below. 2.3 Mass Flow Rates In order to ensure that the air was cooled to 90 C before being blown on to the parts, secondary air would be mixed with the primary air to reduce the net combustion temperature. Enthalpy of combustion values for Natural Gas and the products were used to determine the necessary amount of secondary air required as seen in Appendix B. The necessary mass flow rates are: Fuel Flow Rate = 1.08 kg/hour Primary Air Flow = kg/hour Secondary Air flow = kg/hour 2.4 Mixing An important factor that had to be considered for this project was the mixing of the combustion product jet and the secondary air in the main pipe. Full mixing is the most desirable situation because having uniform temperature in the air blown onto the products would ensure that they would be dried evenly and thoroughly. The problem with this is that the proposed model has turbulent jet flow (see section 4.3 heat loss in pipes). Research in this area was completed by Insu Lee recently here at the University of Michigan [4]. In their study, recirculation zones were found on either side of the primary jet flow when its momentum increased in relation to the coflows (Fig. 4). As the relative momentum of the primary flow decreases, the size of the 5

6 recirculation zones decrease and they move farther away from the nozzles (very left of the pictures). Figure 4: Recirculation zones in confined flow. Larger C tnl values indicate lower ratios of primary jet flow to secondary co-flow [4] (These images are of half the width of the pipes) For the proposed system, the relative momentum of the primary jet is not very high compared to that of the secondary co-flow. From this, it can be assumed that recirculation zones would either not appear, or would be very small and towards the end of the pipe. Because the concentration only achieves a uniform profile after the center of the recirculation zone, it can be assumed that not enough mixing to produce a uniform temperature gradient in a one meter pipe. This means that some sort of mixing apparatus or solution must be implemented in the system. 3. EMISSIONS One of the main concerns with switching from an electric heater to a natural gas burner is the production of emissions in the combustion process. These pollutants have the possibility of contaminating the products being dried, interfering with any coating or future treatment. Three main forms of emissions from the combustion were taken into consideration in the creation of the natural gas drying system: soot, CO, and NOx. CO 2 emissions were considered as well but only for the energy usage. 3.1 Soot Soot is impure carbon particles that are a result of incomplete combustion. Primarily, soot is produced in diffusion flames where oxygen must diffuse into the flame and mix with the hydrocarbons. Because this mixing is limited by the rate of diffusion, typically not enough oxygen is incorporated into the combustion and it is not fully accomplished, resulting in excess impure carbon. In premixed flames, the oxidizer has been fully mixed with the fuel prior to combustion. For the premixed flames, the only issue resides in whether or not enough oxygen has been mixed with the fuel. In the combustion of hydrocarbons, the formation of soot should theoretically only occur when the C/O ratio exceeds 1[5]. From the stoichiometry mentioned in equation 1, a complete combustion of premixed flames occurs at an air/ch4 mole ratio of At any mixture above this ratio (more air), it can be assumed that minimal sooting will occur. Furthermore, the leaner (higher air/fuel ratio) the pre-mixture is, the more of a chance that the 6

7 oxygen combines with the hydrocarbons to form a complete combustion. Conversely, fuel rich mixtures will have less complete combustion (Fig. 5). With the proposed ɸ (air/fuel ratio/stoichiometric air/fuel ratio) of 0.6, it is believed that there will be more than enough air that sooting can be considered minimal. Figure 5.: On the left a rich fuel mixture with no premixed oxygen produces a yellow sooty diffusion flame, and on the right a full oxygen premixed flame produces no soot[6] 3.2 CO, N 2 O and NO X Because sooting is not likely with the use of a lean premixed flame, the major concern for combustion products is the CO that will be produced from the combustion. In an ideal combustion, all of the carbon in methane will react perfectly with the oxygen from the air and only the expected product of CO 2 that balances the stoichiometric equation will be produced. In reality, the combustion process is not completely efficient and some of the carbon will not become fully oxidized. This deviation from theory results in the production of CO. The amount of carbon monoxide produced is dependent on ɸ [2]. The more air is used, the more likely the carbon from the methane has sufficient oxygen to from CO 2 instead of CO. This effect is very pronounced, as the amount of CO produced from the combustion can be reduced by almost two orders of magnitude by using a ɸ of 0.6 instead of a ɸ of 1.0 (Fig. 6.). For ɸ = 0.6, the mole fraction of CO is just under 1/1000. This fraction of combustion products is almost negligible. Figure 6: Chemistry of Methane combustion as a function of distance through flame for multiple values of ɸ [2] 7

8 Because the oxidizer in the premixed flame is air and not pure oxygen, the nitrogen present in air must be accounted for. NO X is produced during combustion when nitrogen and oxygen are present. In this case, because air has a larger amount of nitrogen than oxygen, N 2 O can also be produced during the combustion. Unlike CO, this reaction will happen regardless of imperfect combustion as long as there is air in the mixture. Imperfect combustion does still factor in though because NO and N 2 O will be produced even when ɸ is greater than 1 (Fig. 7). A lean mixture is still the optimal choice when considering NO X and N 2 O pollution (Fig. 7) Since the N 2 O only is produced in small quantities (1 ppm for lean), the primary emission is NO, which is an order of magnitude smaller for the ɸ = 0.6 mixture than it is for ɸ 1. For this lean mixture, the total emission of NO and N 2 O are even smaller than that of CO and therefore can almost be ignored. Figure 7.: NO and N 2 O fractions as a function of distance through flame for multiple values of ɸ [2] With the large amount of secondary air that will be added to the combustion products (including soot), it can be assumed that the minimal amount of emissions that are produced will be so diluted they will have little to no effect on any further processes the dried parts might go through. 3.3 CO 2 from energy usage Since CO 2 is an expected product of methane combustion and remains fairly constant (on the same order of magnitude) regardless of ɸ, the CO 2 emissions were only considered as a part of determining which fuel source is more environmentally friendly. One of our objectives was also to reduce the net emissions from the system. It was determined form a life cycle analysis that the emissions from the system could be reduced by up to 67%, as shown in Appendix C. Reducing these emissions makes the plant more sustainable while reducing the environmental impact significantly. This reduction strongly adds value to our proposition since the facility becomes greener while becoming more profitable. 4 Heat Transfer and Drying Process The parts will be dried by a combination of blowing the water off the parts as well as through heat transfer to the part to evaporate the parts itself. Based on a benchmark for a similar drying system, it was determined that nozzle velocity would need to be 30 m/s [7] with the part being 8

9 placed 6 cm from the nozzle. There were also analysis done to determine how many parts that could be dried as well as well as the potential to lose heat through the system. 4.1 Nozzle Jet velocity To ensure that the parts are dried adequately, high-velocity jets were researched to find the speed at which a jet will shed a water droplet from a surface. This method adds a second dimension of drying on tip of the hot air that is being blown onto the products regardless of air speed. This values was not calculated but instead found to be 30 m/s [7]. After getting this value for the speed at which the air should hit the product, it had to ensured that the velocity coming out of the nozzles and the velocity hitting the pieces to be dried did not differ greatly. Equation 2 was used to find the maximum velocity of a turbulent jet flow as a function of nozzle diameter and distance from the nozzle [8]. While this does show that the velocity decays rapidly, due to the small proposed distance from the nozzle to the parts, the velocity decay is not a major concern. u = 5du 0 /x (u 0 = initial velocity, d = nozzle diameter, x = distance from nozzle) [Eq. 2] A second aspect of the nozzle jet velocity that had to be considered was the centerline jet decay. The effective area of drying had to be considered. It was discovered that the velocity decayed exponentially as the radial distance from the centerline of the jet increased (Eq. 3). Though this high level of velocity decay occurs, it was not considered to be a large issue due to the fact that even if the jet does not shed water droplets that aren t directly in line with the nozzle, the high temperature will have sufficient drying in and of itself. u(x, r) = u 0 exp(-50r 2 /x 2 ) (r = radial distance from centerline) [Eq. 3] 4.2 Rate of heat transfer to plate and speed of manufacturing process For this analysis, it was assumed that heat would be transferred to the plate via two nozzles that were to be placed at an interval of 0.25m and would blow hot air on to the conveyer at 30 m/s. Using heat transfer equations [9] shown below, [Eq. 4] [Eq. 5] We determined that heat could be transferred to a flat Aluminum sheet at a rate of 28.9 KJ/s as show in Appendix D. If we assume that a flat piece of Aluminum with dimensions 1m*0.5m*0.001m has a mass of 1.35 Kg, it takes 2.5 seconds to heat the part to 90C. Assuming it takes 10 seconds overall to dry each part, this translates to 900,000 such parts that can be dried by the plant in the year. For parts of larger mass that have to be dried, the plant would need to hold them for longer in front of the nozzles in order to transfer a greater amount of heat into these parts. 9

10 4.3 HEAT LOSS IN PIPES While the heat loss in the pipes leading from the combustion to the nozzle should theoretically be the same in both electrical and natural gas systems, it is an important component that needed to be calculated to be factored into the design of the system. To find the rate of heat transfer, Q u between the pipe and the air, the local surface-convection resistance R ku had to be calculated[9]. The first step in finding this value was to determine the kind of flow that would happen inside the system. Equation 6 was used to find the Reynolds number of the flow (where v is the mean velocity of the fluid, D h is the hydraulic diameter, and ν is the kinematic viscosity of air). This value was calculated to be 56,735, which was much larger than the critical Reynolds number of 2,300. This means the proposed flow would be turbulent. R e = ud h /ν [Eq. 6] After the kind of flow was determined, the normalized local surface-convection conductance (or Nusselt number) Nu had to be calculated for a turbulent flow (Eq. 7). < Nu 0.027R!.!! Pr!! (Pr air = 0.7) [Eq. 7] The Nusselt number was then used to calculate the number of transfer units NTU (Eq. 8) NTU = πl < Nu >! (M = mass flow rate, C p = specific heat) [Eq. 8]!!! This values was in turn used to calculate Ru (Eq. 9).! Ru = [Eq. 9]!!!!!!"#!!"# With Ru, and the local temperatures of the surface and air flow, the heat transfer rate was calculated to be J/s (assuming a 1m pipe) (Eq. 10). Q! =!!"#$%&'!!!"# [Eq. 10]!" The exit temperature of the air after the heat loss was also calculated to be Celsius (with original air temperature of 90 C) (Eq. 11). T!"#$ = T!"# (1 exp NTU )(T!"# T!"#$%&' ) [Eq. 11] This meant that the heat loss in the pipe only dropped the temperature a few degrees. This means that heat loss in the pipes was not a serious issue as it would be very easy to raise the original temperature of the air by a few degrees to offset this heat loss. This problem could be easily fixed by slightly decreasing the amount of secondary air added to the combustion products or by adding a layer of insulation to the pipe. 5 FINANCIAL ANALYSIS While the reduction of carbon emissions in the drying process is a great part of switching from electric to natural gas energy, an industrial drying plant would ultimately make its decision based on financial motivations. If switching to natural gas could save money, it would be the easy choice. The financial considerations for this kind of switch are ultimately two-fold. First, how much can using natural gas instead of electricity save in energy costs, and second, how much will the switch itself cost. According to the energy information administration [10], current average industrial retail prices in Michigan are $0.07 per kw-hr for electricity and $2.60 per MMBTU for natural gas. Working under the assumption that the MIWI plant runs for 3120 hours per year, the cost for using a 15 kw electrical heater is $3,276 per year for a single machine. For a natural gas burner using our mass flow rate of 1.08 kg/hr, the price of natural gas for a single burner would be $1,180 per 10

11 year. This means that for each natural gas burner the plant substitutes for an electrical one, they would save $2,097 per year. Assuming the plant operates 5 dryers, this saves the plant $10,483 per year. This is easily enough money that switching to natural gas would be a good choice for the MIWI plant. Energy savings alone are not the whole story. There would be initial costs in purchasing the natural gas burners to replace the electrical one. The proposed solution uses a Kromschroder BIC 50 industrial burner as a model (Fig 8) [11]. The BIC 50 burner costs $1,332 each. Average maintenance and replacement parts for these burners cost ~$544 a year (as quoted from Combustion 911 customer service representatives). This means that the first year costs for the burners would be $1,876 each and subsequent year costs would be $544 each ($9,380 and $2,720 for the 5 dryer assumption). Assuming that maintenance is similar for the electrical heaters and can be ignored, this is still $1,332 per burner, or $6,660 initial cost. While this seems like a large number, with the saving from the energy cost, the time it would take to cover this would be less than eight months. Provided the plant has the funds and would be operating for more than a year, this would be an easy decision to make. Figure 8: Kromschroder BIC 50 Burner used in the proposed solution [11] RECOMMENDATIONS AND CONCLUSIONS Natural Gas can be used to substitute for electrical drying and result in a cheaper, cleaner drying process for the company. If the plant operates 5 dryers, they can save up to $10,500 per year by switching. They can recover the initial cost of the dryers in savings in 8 months. The fuel flow rate in the dryers can be controlled with a maximum flame temperature of 1600K. The emissions from the burning of natural gas at an equivalence ratio of 0.6 would be very low and these would be further diluted by the addition of secondary air. It would reduce equivalent CO 2 emissions by 67%. The BIC 50 burner [11] that we chose has adjustable controls that the plant can use to control the flow of natural gas and primary air to change the temperature and magnitude of exiting air as required. The dryer is less than 0.5 cubic meters in volume; hence it is reasonable to assume that it could be used to replace the existing electrical heater without space being a constraint. 11

12 To have a uniform temperature for the output air, we would place a simple mixing apparatus or fan somewhere in the main pipe. This would alleviate the problem of recirculation zones. This would be a cheap and easy way to ensure total mixing. A recommendation that would help reduce emissions from the combustion would be to implement an exhaust gas recirculation system. This system would recirculate a portion of the exhaust gas back into the combustion chamber. This would effectively raise the temperature of the primary intake air and dilute its nitrogen. This would lower the NO X production of the system, further reducing overall emissions concerns. 12

13 REFERENCES [1] Substitution of Natural Gas for Electric Industrial Drying: A cost Saving Strategy. (2014). Industrial Assessment Center, University of Michigan. [2] Chemical Composition of Natural Gas-Union Gas. (n.d.). Retrieved April 24, 2015, from Gas [3] Belcadi, A., Assou, M., Affad, E. H., Chatri, E. H. (2012), CH 4 /NO X Reduced Mechanisms Used for Modeling Premixed Combustion. Energy and Power Engineering, 4, [4] Lee, I. A study of mixing in confined turbulent jets: Part I: Co-flowing streams. University of Michigan at Ann Arbor. [5] Haynes, B. S., & Wagner, H. G. (1981). Soot Formation. Progress in Energy and Combustion Science, Volume 7 (Issue 4), pp [6] Premixed flame. (n.d.). Retrieved April 23, 2015, from [7] Hot-Air Blow-Off Parts Drying. (n.d.). Retrieved April 24, 2015, from [8] Cushman-Roisin, B. (2014). Turbulent Jets. In Environmental Fluid Mechanics (pp ). New York, NY: John Wiley & Sons [9] Kaviany, M. (2011). Convection: Bounded Fluid Streams. In Essentials of heat transfer principles, materials, and applications. New York, NY: Cambridge University Press. [10] U.S. Energy Information Administration - EIA - Independent Statistics and Analysis. (n.d.). Retrieved April 23, 2015, from [11] Industrial Burner for Gas BIC. (n.d.). Retrieved April 23, 2015, from Appendix A: Fuel Stoichiometry m =!!!"#!$!"#!"#$ = 15 kw / kj/kg fuel = 1.08 kg/hour m(air) =!!.!"!".!"!".!" 1.08 = kg/hour We need at least Kg air an hour in order to get stoichiometric combustion Appendix B: Secondary Air As shown in equation 1, each mole of fuel requires 2 moles of primary air for combustion. For reducing the flame temperature to 90C, assume x times as much secondary air is added to the total gas flow which consists of CO 2 + 2H 2 O N 2 + x(3.76n 2 + O 2 ) per mole of fuel combusted. Heat generated from combustion of Methane: Lower heating value = kj/kg fuel 13

14 Molar Mass * LHV = kg/kmol fuel * kj/kg fuel = kj/kmol fuel This amount of energy gets transferred to the product gasses. Therefore the energy equation states, Energy = LHV*Molar Mass = kj/kmol fuel = energy in the products = h!"! + 2 h!!" h!! + x( h!! h!! ) per kmol fuel Assuming all the gas components are ideal gases, then at 90 C target we get : h!"! = kj/kmol ; h!! = kj/kmol ; h!!" = kj/kmol h!" = kj/kmol Solving the energy equation for x gives: x = Since for every 2 moles of primary air, x moles of secondary air are required: m(secondary, air) =! m(primary, air)! m secondary, air = Kg/hour If we are operating at an equivalence ratio of 0.6, you would require kg/hour of primary air. Hence the net secondary air that is required would be (additional air added in primary combustion) Kg/hour which results in a net secondary air requirement of Kg/hour. Appendix C: Emissions Comparison Net energy output of system (run for 1 hour) = 15 kw * 3600 = 54 MJ Combustion Efficiency: 94% Electrical Heater : ~100% [energy.gov] [1] Distribution : 94% [eia.gov] [2] Generation of electricity from Coal: 40% [highly efficient power plant, World Coal Association] [3] Net Energy needed = 54 MJ/(0.94*0.40) = 150 MJ Energy from 1 kg of coal = 30MJ [Anthracite, maximum energy content, University of Washington [4] Mass of coal needed = 150/30 = 5kg Emissions from 1kg of coal = 2.3 kg CO2/kg Coal [5] Net Emissions from running the system for 1 hour: 11.5 kg CO2/hour [1] [2] [3] [4] [5] 14

15 Appendix D: Heat Transfer Diameter of Nozzle = 2 * 10-3 m Length between nozzles = 0.25 m E = 1 πd 2 /(16L 2 ) E = 0.99 Reynold s number is calculated in equation Re D = R e = ud/ν = 57,464 (Reynolds number) ν = * 10-6 m 2 /s u = 30 m/s Pr = 0.69 The Nusselt s number is given by the equation below: <Nu> L = *0.986*0.031*0.81 = 7887 Where, ϵ = 0.99 Re D = ud h /ν = 5.74 * 10 4 T s = 300 K (Assumed) T f, = 360 K <A ku > L = 0.5 m 2 Hence, we obtain the net heat transfer rate to the Aluminum part as: <Q ku > L = 0.5 * 7887*0.0306/0.25*(60) = 28,961 J/s 15