5 B Group B6 Optimization Problem Set

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The document is a problem statement asking to determine the optimum economic thickness of insulation for a 1" steel pipe carrying steam at 180°C. The pipe is 250m long and surrounded by air at 27°C. Insulation costs $200x per meter, where x is thickness in meters. The solution finds the optimum thickness as 10.442cm, with an insulating surface temperature of 39.9040C.

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Statement of the Problem ,[object Object]

Used: R Programming, Genetic Algorithms and Optimization Techniques Spring-Cleaning offers a wide variety of cleaning services. They provide carpet cleaning, tile and grout cleaning, upholstery cleaning, hardwood floor cleaning and air duct cleaning, etc. Broadly, they are categorised as residential and commercial cleaning. They have presence in 48 cities nationwide with various depots servicing specific regions.

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Presented in this short document is a description of what is called the (classic) “Pooling Optimization Problem” and was first described in Haverly (1978) where he modeled a small distillate blending problem with three component materials (A, B, C), one pool for mixing or blending of only two components, two products (P1, P2) and one property (sulfur, S) as well as only one time-period. The GAMS file of this exact same problem is found in Appendix A which describes all of the sets, lists, parameters, variables and constraints required to represent this problem. Related types of NLP sub-models can also be found in Kelly and Zyngier (2015) where they formulate other sub-types of continuous-processes such as blenders, splitters, separators, reactors, fractionators and black-boxes for adhoc or custom sub-models.

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Abstract Multi-objective formulations are realistic models for many complex engineering optimization problems. In many real-life problems considered objectives conflict with each other and optimization of one target solutions can lead to unacceptable results for other purposes. A reasonable solution of the multi-objective problem is to study the set of solutions, each of which satisfies the objectives at an acceptable level with no dominance of any of the solutions. The article provides a brief overview of multi-objective optimization methods (by Pareto criteria) and their improvement. Keywords: A fuzzy set, forecasting, risk, weakly formalized process for Pareto multi-objective optimization.

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Heat Conduction: In heat transfer, conduction (or heat conduction) is the transfer of heat energy by microscopic diffusion and collisions of particles or quasi-particles within a body due to a temperature gradient. The microscopically diffusing and colliding objects include molecules, electrons, atoms, and phonons. They transfer microscopically disorganized kinetic and potential energy, which are jointly known as internal energy. Conduction can only take place within an object or material, or between two objects that are in direct or indirect contact with each other. Conduction takes place in almost all forms of matter, such as solids, liquids, gases and plasmas. Thermal Conductivity of a metal: Thermal conductivity is a measure of the ability of a substance to conduct heat, determined by the rate of heat flow normally through an area in the substance divided by the area and by minus the component of the temperature gradient in the direction of flow, measured in watts per meter per Kelvin. Symbol: K is used to denote thermal conductivity.

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Tagliatela College of Engineering EASC1112 – Methods of Engineering Analysis TO: EASC 1112 Students FROM: EASC1112 Instructor Subject: Project 2, Optimum Pipe Insulation A long pipe is to be installed to transport steam from a boiler to another part of the plant. Insulation is needed on the pipe for both safety and economic reasons. You are to develop a spreadsheet to calculate the surface temperature of the insulated pipe and to model the heat loss to the surrounding air as a function of the thickness of insulation on the outside of the pipe. Your model should allow for variation in the key parameters to explore the effect of various changes. Using data generated by your model, select the best insulation thickness to maximize the present value of net savings in comparison to an un-insulated pipe. Provide appropriate plots and data tables to support your decision and to show the financial penalty for using a different insulation thickness. Heat Loss Calculation The steam pipe is to be made from schedule 40 steel with a diameter in the range of 2 to 3.5 inches (nominal pipe size). The pipe will be encased in fiberglass insulation with an aluminum sheet cladding to protect from weather. Heat loss for this case can be modeled using a combination of convection and conduction heat transfer rate equations. Heat from the steam is transferred to the inside wall of the pipe by forced convection, then through each of three layers by conduction (pipe wall, insulation, cladding) and finally from the outside of the cladding to the surrounding air by natural convection. The governing equations are shown below to calculate heat transferred per unit length of pipe: View of pipe looking along axis QS = rate of heat lost by steam to inside pipe wall Aluminum Cladding Q1, Q2, Q3 = rates of heat transferred through pipe wall, insulation, aluminum cladding, respectively r2 r1 r4 r3 Fiberglass Insulation Steel Pipe STEAM QA=rate of heat lost to air The heat transfer rate equations include constants for the thermal conductivity of the materials and heat transfer coefficients for the convective situations. Values for these will be fixed for the analysis. The temperatures of the steam and the air will be fixed values, but the temperatures at each surface will be dependent on the thickness of insulation and size of the pipe. The subscripts used for the temperatures correspond to radial distances from the center of the pipe. The radii values will be fixed for a particular case of pipe size and insulation thickness, but will be varied as part of the optimization work. The intermediate temperatures, to be found by simultaneous solution of the equation set, are: T1 = temperature of the inside wall of the pipe, at distance r1 from the pipe center axis T2 = temperature of the outside pipe wall and the inside of the insulation, distance r2 T3=temperature of the outside of the insulation and inside of the aluminum cladding, distance r3 T4 = tem ...

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