Finite Element Heat Transfer and Structural Analysis of a non-Gasketed Flange
RAMA SUBBA REDDY GORLA
Professor of Mechanical Engineering
ClevelandStateUniversity
Cleveland, OH44115U.S.A.
Sreekantha R Gorla
Anthony C Ianetti
Mehmet F Taskan
Department of Mechanical Engineering
ClevelandStateUniversity
Cleveland, OH44115U.S.A.
Abstract:
Stress analysis of a non-gasketed flange was computationally simulated by a finite element method and probabilistically evaluated in view of the several uncertainties in the performance parameters. Cumulative distribution functions and sensitivity factors were computed for overall heat transfer rate due to the structural and thermodynamic random variables. These results can be used to quickly identify the most critical design variables in order to optimize the design and make it cost effective. The analysis leads to the selection of the appropriate measurements to be used in structural and heat transfer analysis and to the identification of both the most critical measurements and parameters.
Key-Words:Heat transfer, probabilistic analysis, Finite element analysis, Flanges.
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1 Introduction
Taylor-Forgemethod[1, 2]is extensively used in many international design codes and standards for the design of flanges. Experimental evidence of leakage in bolt preload exists for the seating conditions in a flange joint. A knowledge of the value of the gasket load drop is essential in order to achieve leak tightness. The published literature [3-6] does not take into consideration of the nonlinearity of the gasket material and therefore yields erroneous results for high pressure and high temperature conditions. Abid and Nash [7] provided a parametric study of the behavior of metal-to-metal contact flanges that have different surface profiles. They recommended the dimensions for no leak conditions from the joint.
To cost effectively accomplish the design task, we need to formally quantify the effect of uncertainties (variables) in the design. Probabilistic design is one effective method to formally quantify the effect of uncertainties. In the present paper, a probabilistic analysis is presented for the influence of measurement accuracy and apriori fixed parameter variations on the random variables for thermal stressesfrom a non-gasketed flange. Small perturbation approach is used for the finite element methods to compute the sensitivity of the response to small fluctuations of the random variables present. The result is a parametric representation of the response in terms of a set of random variables with known statistical properties, which can be used to estimate the characteristics of the selected response variables such as heat transfer rate or temperature at a given point.
2 Analysis
2.1 Finite Element Solution
Let us consider a two-dimensional partial differential equation of the form
(1)
The above equation is valid over an area A. We assume that on a portion of the boundary L1, T= T0 (x,y).
On the remainder of the boundary, labeled L2, the general derivative boundary condition is specified in the form
nx+ny+ (2)
Here, nx and ny are direction cosines of the outward normal to L2. The form of the functional may be written as
I(T)=(3)
For a simplex two-dimensional element, we have extremized the above functional with respect to the unknown nodal temperatures. The resultant element matrices are then obtained from the following relation:
(4)
The element matrix and the element column may be written as
= = (5)
Where
Bii= ,
Bij =
Bik =
Bjj=
Bjk =
Bkk=
Ci=
Cj=
Ck =
The element matrices were then assembled into the global matrices and vectors. The prescribed boundary conditions were implemented at the appropriate nodal points. The algebraic equations in the global assembled form were solved by the Gauss elimination procedure. These details may be found in reference [8].
3 Perturbation of the heat transfer problem
The finite element solution for the heat transfer problem may be reduced to the following equation in the unperturbed state:
(6)
The perturbed problem involving small fluctuations of the random variables may be written as
=(7)
where
=
=
(8)
Therefore, we may write equation (6) as
(9)
In the last step in equation (9), we ignored the second order term d [B]. d [T]. Here, xi are the random variables. A simple form of the iterative algorithm is given by:
(10)
(11)
In order to start the iteration, we may use
The effect of temperature-dependent properties may be included in equation (10). From equation (10), we may write:
(12)
From equations (10) and (12) we may write
(13)
From equation (13), we may write
(14)
where is the amplification matrix. The iterative process will remain stable if the spectral radius of the amplification matrix [A] is less than unity. This will be true when the imposed perturbations on the original element matrix are small.
4 Results and Discussion
The history of the iterative algorithm is illustrated by means of an example involving pressure loading, conduction and convection in a non-gasketed flange as shown in Figure 1. The inner surface of the tank was exposed to an environment maintained at 1100°C, while the outer surface exposed to ambient temperature at 15°C. The tank was filled with a fluid at a pressure of 15.3 N/mm2. Table 1 shows the random variables and their mean values used. All random variables were assumed to be independent. A scatter of 10 % was specified for all the variables. This variation amounted to two standard deviations. Normal distribution was assumed for all random variable scatters.
Fig 1 Model of the Non-Gasketed Flange
Random Variable / Mean ValuePressure / 15.3 N /mm2
Temperature / 1100 oC
Inner Diameter / 174.6 mm
Length / 102.6 mm
Thickness / 13.5 mm
Thermal Conductivity / 0.043235 J/(s*mm*oc)
Thermal Co efficient of Expansion / 0.0000135 /oc
Modulus of Elasticity / 203395 N/mm2
Poisson's Ratio / 0.3
Heat Transfer Coefficient / 0.003 J/(s*mm2*oc)
Ambient Temperature / 15 oc
Table 1: Initial values for non-gasketed flange material properties and temperatures
Maximum stress location was determined from a pre-analysis of the flange. This location was used to evaluate the cumulative distribution functions (CDF) and the sensitivity factors for thermal stress response. Temperature distribution in the flange is shown in Figure 2. A typical stress distribution in the flange is shown in Figure 3. CDF for the stress is shown in Figure 4. The sensitivity factors for the stress versus the random variables are shown in Figure 5. We observe that the modulus of elasticity, Poisson’s ratio, coefficient of thermal expansion of the tank material and inside fluid temperature of the flange have a lot of influence on the stresses.
Fig. 2 Steady State Heat Transfer Results
Fig. 3 Thermal Stress Distribution
Fig.4 Cumulative Probability of Hoop Stress
Fig. 5 Sensitivity Factors Versus Random Variables
A robust design is one that has been created with a system of design tools that reduce product or process variability while guiding the performance toward an optimal setting. Robustness means achieving excellent performance under a wide range of operating conditions. All engineering systems function reasonably well under ideal conditions, but robust designs continue to function well when the conditions are non-ideal. Analytical robust design attempts to determine the values of design parameters, which maximize the reliability of the product without tightening the material or environmental tolerances. Probabilistic design and robust design go hand in hand. In order to determine the domains of stability, the system has to be analyzed probabilistically.
5 Concluding Remarks
In this paper, a non-deterministic method has been developed to support reliability-based design. The novelty in the paper is the probabilistic evaluation of the finite element solution for heat transfer. Cumulative distribution functions and sensitivity factors were computed for heat loss due to the random variables. The inside fluid temperature, modulus of elasticity, Poisson’s ratio and the coefficient of thermal expansion of the flange material have a lot of influence on the thermal stresses. Evaluating the probability of risk and sensitivity factors will enable the identification of the most critical design variable in order to optimize the design and make it cost effective.
References:
[1] Waters, E. and Taylor, J.H., The Strength of Pipe Flanges, Trans. ASME, 1927, Vol. 49, pp. 531-542.
[2] Waters, E., Wesstrom, D.B., Rossheim, D.B. and Williams, F.S.G., Formulas for Stress in Bolted Flanged Connections, Trans. ASME, 1937, Vol. 59, pp. 161-167.
[3] Waters, E., and Williams, F., Stress Conditions in Flanged Joints for Low Pressure Service, Trans. ASME, 1952, Vol. 74, pp. 135-156.
[4] Gill, S., The Stress Analysis of Pressure Vessels and Pressure Vessel Components, Pergamon Press, New York, 1970.
[5] Nagy, A., StressState Investigation of Integral Type Flange Joint, Dissertation, Budapest, 1991.
[6] Varga, L., and Baratosy, J., Optimal Prestressing of Bolted Flanges, Int. J. Pressure Vessels, 1995, Vol. 63, pp. 25-34.
[7] Abid, M. and Nash, D.H., A Parametric Study of Metal-to-Metal Contact Flanges with Optimized Geometry for Safe and No-Leak Conditions, Int. J. Pressure Vessels, 2004, Vol. 81, pp. 67-74.
[8] Allaire, P.E., Basics of The Finite Element Method, W.C. Brown Publishers, Dubuque, IA, 1985.
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