Assessing the Design Effect of Pressure Vessel Height and Radius on Reactor Stability and Safety

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  Control Theory and Informatics ISSN 2224-5774 (Paper) ISSN 2225-0492 (Online) Vol.4, No.1, 2014 19 Assessing the Design effect of Pressure Vessel Height and Radius on Reactor Stability and Safety A.I. Oludare 1 , M.N. Agu, 2  A.M. Umar, 3 S.O. Adedayo 4 , O. E. Omolara 5  and L.N. Okafor 6   1 Nigerian Defence Academy, Department of Physics, Kaduna 2 Nigeria Atomic Energy Commission, Abuja 3 Energy Commission of Nigeria, Department of Nuclear Science & Technology, Abuja 4  National Open University of Nigeria, Department of Information Technology, Abuja 5 Ahmadu Bello University, Department of Mathematics, Zaria, Nigeria 6 Nigerian Defence Academy, Department of Mathematics and Computer Science, Kaduna Corresponding author: email: ABSTRACT The Design of Reactor Pressure Vessel (RPV) should be that the height of pressure vessel is up to 16.0m and radius is up to 5.6m to ensure safe operation of nuclear reactor. The research conducted safety margin test on the design dimension of RPV in terms of the height and radius, secondly safety margin test was carried out on applied high temperature on the reactor graphite core and thirdly, safety margin test were perform on the cooling problem of the nuclear reactor core in relation to fuel temperature. By applying Linear Regression Analysis Techniques on some typical Reactor Pressure Vessel (RPV) models. The results of the statistical analysis on these types of nuclear reactor models reveals that the RPV models promises stability under application of pressure vessel up to 16.0m height and radius 5.6m. At this point the temperature seems at maximum and the reactor agrees to be more stable as the regression plot was optimized, that is the least squares method finds its optimum when the sum, S  , of squared residuals is at minimum. The safety margin prediction of 3.1% was validated for a typical RPV model as an advantage over the current 5.1% challenging problem for plant engineers to predict the safety margin limit. Keywords:  Reactor pressure vessel design, height and radius, high temperature effect, fuel element, risk and failure, reactor safety, safety factor, Ỳ , optimization, stability margin, reactor pressure vessel design models, selection of pressure vessel. INTRODUCTION The main drivers for reactor development are :   ã   Improved safety (for example by the incorporation of passive safety features) ã   Reduced capital cost ã   Reduced operating cost ã   Improved efficiency and utilization ã   Improved design effectiveness ã   Reduced build-time ã   Minimize the risk of failure These main drivers shall provide a good, novel approach and method for multi-objective decision-making in the development of the nuclear industry. The reactor pressure vessels are designed with great care to ensure safe operation when used within their pre-scribed temperature and pressure limits. The selection of pressure vessel must be the one which has the capability, pressure rating, corrosion resistance and design features that are suitable for its intended use. When pressure vessel could not function as to supply water or gas in cooling the reactor the phenomena can cause hydrogen built-up within the reactor and this can eventually melt down the reactor core, as in the case studies of nuclear accident in Japan when the pressure vessel fail to function, heat continue to built-up in the reactor and the reactor meltdown and was damaged [1]. Then Seawater was being pumped into the reactor in an attempt to cool down the radioactive core. Also, there was a recorded explosion which occurred, at the NDK Crystal manufacturing company in Belvidere, Illinois resulted from Stress Corrosion Cracking of High-Pressure Vessel [2], and fatal accident at Goodyear Tire and Rubber Plant in Houston , Texas , f ollowing Pressure Vessel Codes, the accident occurred, when an overpressure in a heat exchanger led to a violent rupture of the exchanger [3]. There was a safety concern because counterfeit reactor equipment of the reactor pressure vessel at Koodankulam   Nuclear Power Plant in the Tirunelveli district of the southern Indian state of Tamil Nadu [4]. Since these also include safety grade equipment, there is also a potential for accident.    Control Theory and Informatics ISSN 2224-5774 (Paper) ISSN 2225-0492 (Online) Vol.4, No.1, 2014 20 In exercising the responsibility for the selection of pressure equipment, the prospective user is often faced with a choice between over or under-designed equipment. The hazards introduced by under-designed pressure vessels are readily apparent, but the penalties that must be paid for over-designed apparatus are often overlooked[5]. The list of available nominal chemical composition of pressure vessel construction materials are highlighted in Table 1: Table I: Highlighted the nominal chemical composition of pressure vessel materials Major Elements (Percent) Material Typical Trade Name Fe Ni Cr Mo Mn Other T316/316 L Stainless Steel 65 12 17 2.5 2.0 Si 1.0 Alloy 20 Carpenter 20 35 34 20 2.5 2.0 Cu 3.5, Cb 1.0 max  Alloy 400 Monel 400 1.2 66 Cu 31.5  Alloy 600 Inconel 600 8 76 15.5 Alloy B-2 Hastelloy B-2 2 66 1 28 1 Co 1.0  Alloy C-276 Hastelloy C-276 6.5 53 15.5 16 1 W4.0, Co 2.5 Nickel 200 99 Titanium Grade 2 Commercially pure titanium Ti 99 min Titanium Grade 2 Commercially pure titanium Ti 99 min Titanium Grade 7 99 0.15 Pd Zirconium Grade 702 Zr + Hf 99.2 min, Hf 4.5 max Zirconium Grade 705 Zr + Hf 99.2 min, Hf 4.5 max Nb 2.5  A pressure vessel  is a closed, rigid container designed to hold gases or liquids at a pressure different from the ambient pressure. In addition to industrial compressed air receivers and domestic hot water storage tanks, other examples of pressure vessels are: diving cylinder, recompression chamber, distillation towers, autoclaves and many other vessels in mining or oil refineries and petrochemical plants, nuclear reactor vessel, habitat of a space ship, habitat of a submarine, pneumatic reservoir, hydraulic reservoir under pressure, rail vehicle airbrake reservoir, road vehicle airbrake reservoir and storage vessels for liquified gases such as ammonia, chlorine, propane, butane and LPG. DESIGN AND OPERATION STANDARDS In the nuclear industrial sector, pressure vessels are designed to operate safely at a specific pressure and temperature, technically referred to as the Design Pressure and Design Temperature . The pressure vessel is designed to a pressure, there is typically a safety valve or relief valve to ensure that this pressure is not exceeded in operation. A vessel that is inadequately designed to handle a high pressure constitutes a very significant safety hazard. Because of that, the design and certification of pressure vessels is governed by design codes such as the ASME Boiler and Pressure Vessel Code in North America[6], the Pressure Equipment Directive of the EU (PED), Japanese Industrial Standard (JIS), CSA B51 in Canada, AS1210 in Australia and other international standards like Lloyd's, Germanischer Lloyd, Det Norske Veritas, Stoomwezen and so on. SHAPE OF A PRESSURE VESSEL Theoretically a sphere would be the optimal shape of a pressure vessel. Most pressure vessels are made of steel. To manufacture a spherical pressure vessel, forged parts would have to be welded together. Some mechanical properties of steel are increased by forging, but welding can sometimes reduce these desirable properties. In case of welding, in order to make the pressure vessel meet international safety standards, carefully selected steel with a high impact resistance are be used. Most pressure vessels are arranged from a pipe and two covers. Disadvantage of these vessels is the fact that larger diameters make them relatively more expensive, so that for example the most economic shape of a 1000 litres, 250 bar (25,000 kPa) pressure vessel might be a diameter of 450 mm and a length of 6500 mm.  Control Theory and Informatics ISSN 2224-5774 (Paper) ISSN 2225-0492 (Online) Vol.4, No.1, 2014 21 No matter what shape it takes, the minimum mass of a pressure vessel scales with the pressure and volume it contains. For a sphere , the mass of a pressure vessel is ,…………………………………………Equation (1) Where :    M   is mass  p  is the pressure difference from ambient- the gauge pressure V   is volume ρ  is the density of the pressure vessel material σ  is the maximum working stress that material can tolerate. Other shapes besides a sphere have constants larger than 3/2 (infinite cylinders take 2), although some tanks, such as non-spherical wound composite tanks can approach this. CYLINDRICAL VESSEL WITH HEMISPHERICAL ENDS This is sometimes called a bullet for its shape, although in geometric terms it is a capsule.  For a cylinder with hemispherical ends, ,……………………………Equation (2) where ã   R is the radius, W is the middle cylinder width only, and the overall width is W + 2R CYLINDRICAL VESSEL WITH SEMI-ELLIPTICAL ENDS In a vessel with an aspect ratio of middle cylinder width to radius of 2:1, .…………………………………………Equation (3) GAS STORAGE In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus . ………………………………………Equation (4) The other factors are constant for a given vessel shape and material. So we can see that there is no theoretical efficiency of scale , in terms of the ratio of pressure vessel mass to pressurization energy, or of pressure vessel mass to stored gas mass. For storing gases, tankage efficiency is independent of pressure, at least for the same temperature. So, for example, a typical design for a minimum mass tank to hold helium (as a pressurant gas) on a rocket would use a spherical chamber for a minimum shape constant, carbon fiber for best possible , and very cold helium for best possible . STRESS IN THIN-WALLED PRESSURE VESSELS Stress in a shallow-walled pressure vessel in the shape of a sphere is ,………………………………….Equation (5) where is hoop stress, or stress in the circumferential direction, is stress in the longitudinal direction,  p  is internal gauge pressure, r   is the inner radius of the sphere, and t   is thickness of the cylinder wall. A vessel can  Control Theory and Informatics ISSN 2224-5774 (Paper) ISSN 2225-0492 (Online) Vol.4, No.1, 2014 22 be considered shallow-walled if the diameter is at least 10 times (sometimes cited as 20 times) greater than the wall depth. Stress in a shallow-walled pressure vessel in the shape of a cylinder is ,……………………………………………..Equation (6) ,…………………………………………Equation (7) where is hoop stress, or stress in the circumferential direction, is stress in the longitudinal direction,  p  is internal gauge pressure, r   is the inner radius of the cylinder, and t   is thickness of the cylinder wall. Almost all pressure vessel design standards contain variations of these two formulas with additional empirical terms to account for wall thickness tolerances, quality control of welds and in-service corrosion allowances. For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are: Spherical shells: …………………….Equation (8) Cylindrical shells: ………………………………Equation (9) ………………………….Equation (10) Where,  E   is the joint efficient, and all others variables as stated above. The factor of safety is often included in these formulas as well, in the case of the ASME BPVC this term is included in the material stress value when solving for Pressure or Thickness. LINEAR EXPANSION To a first approximation, the change in length measurements of an object ( linear dimension as opposed to, e.g., volumetric dimension) due to thermal expansion is related to temperature change by a linear expansion coefficient . It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write :   α L = (   )………………………………………………Equation (11) where L is a particular length measurement and is the rate of change of that linear dimension per unit change in temperature. The change in the linear dimension can be estimated to be: Δ = α L Δ T………………………………………………..Equation.(12) This equation works well as long as the linear-expansion coefficient does not change much over the change in temperature Δ T. If it does, the equation must be integrated. EFFECTS ON STRAIN For solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by and defined as :   ε thermal = (Lfinal – Linitial)/Linitial…………………….. Equation.(13)
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