Numerical Methods of Mechanics and Computer Mechanics Lecture 07 PDF

FACULTY OF MECHANICAL ENGINEERING TECHNICAL UNIVERSITY OF KOŠICE Department of applied mechanics and mechanical engineering NUMERICAL METHODS OF MECHANICS AND COMPUTER MECHANICS www.kamasi.sk | ww...

FACULTY OF MECHANICAL ENGINEERING TECHNICAL UNIVERSITY OF KOŠICE Department of applied mechanics and mechanical engineering NUMERICAL METHODS OF MECHANICS AND COMPUTER MECHANICS www.kamasi.sk | www.strojarina.eu 1  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7. THERMAL ANALYSIS www.kamasi.sk | www.strojarina.eu 2  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction Heat transfer: conduction, convection, radiation. A temperature difference must exist for heat transfer to occur. Heat is always transferred in the direction of decreasing temperature. www.kamasi.sk | www.strojarina.eu 3  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction Thermal Analysis Item, [units], symbol Structural Analysis Item, [units], symbol Unknown: Temperature [K], T Unknown: Displacements [m], u Gradient: Temperature Gradient [K/m], ∇T Gradient: Strains [m/m], ε Flux: Heat flux [W/m2], q Flux: Stresses [N/m2], σ Source: Heat Source for point, line, surface, Source: Force for point, line, surface, volume [W], [W/m], [W/m2], [W/m3], Q volume [N], [N/m], [N/m2], [N/m3], g Indirect restraint: Convection Indirect restraint: Elastic support Restraint: Prescribed temperature [K], T Restraint: Prescribed displacement [m], u Reaction: Heat flow resultant [W], H Reaction: Force component [N], F Material Property: Thermal Material Property: Elastic modulus conductivity [W/(mK)], k [N/m2], E Material Law: Fourier’s law Material Law: Hooke’s Law www.kamasi.sk | www.strojarina.eu 4  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction Conduction takes place within the boundaries of a body by the diffusion of its internal energy. The temperature within the body, T, is given in units of degrees Celsius [C], Fahrenheit [F], Kelvin [K], or Rankin [R]. Its variation in space defines the temperature gradient vector ∇T, with units of [K/m] say. The heat flux vector q, per unit area is define by Fourier’s Conduction Law, as the thermal conductivity matrix k, times the negative of the temperature gradient, q = - k∇ T. The integral of the heat flux over an area yields the total heat flow for that area. q   k T www.kamasi.sk | www.strojarina.eu 5  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction Thermal conductivity has the units of [W/(mK)] while the heat flux has units of [W/m2]. The conductivity, k, is usually only known to three or four significant figures. For solids it ranges from about 417 W/(mK) for silver down to 0.76 W/(mK) for glass. A perfect insulator material (k ≡ 0) will not conduct heat; therefore the heat flux vector must be parallel to the insulator surface. A plane of symmetry (where the geometry, k values, and heat sources are mirror images) acts as a perfect insulator. In finite element analysis, all surfaces default to perfect insulators unless you give a specified temperature, a known heat influx, a convection condition, or a radiation condition. www.kamasi.sk | www.strojarina.eu 6  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction Convection occurs in a fluid by mixing. Here we will consider only free convection from the surface of a body to the surrounding fluid. Forced convection, which requires a coupled mass transfer, will not be considered. The magnitude of the heat flux normal to a solid surface by free convection is qn = hAh(Th – Tf), where h is the convection coefficient, Ah is the surface area contacting the fluid, Th is the convecting surface temperature, and Tf is the surrounding fluid temperature, respectively. The units of h are [W/(m2K)]. Its value varies widely and is usually known only from one to four significant figures. Typical values for convection to air and water are 5-25 and 500-1000 W/(m2K), respectively. www.kamasi.sk | www.strojarina.eu 7  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction www.kamasi.sk | www.strojarina.eu 8  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction www.kamasi.sk | www.strojarina.eu 9  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction Radiation heat transfer occurs by electromagnetic radiation between the surfaces of a body and the surrounding medium. It is a highly nonlinear function of the absolute temperatures of the body and medium. The magnitude of the heat flux normal to a solid surface by radiation is qr =εσAr (Tr4 – Tm4). Here Tr is the absolute temperature of the body surface, Tm is the absolute temperature of the surrounding medium, Ar is the body surface area subjected to radiation, σ = 5.67x108 W/(m2K4) is the Stefan-Boltzmann constant, and ε is a surface factor (ε = 1 for a perfect black body). www.kamasi.sk | www.strojarina.eu 10  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction Transient or unsteady, heat transfer in time also requires the material properties of specific heat at constant pressure, cp in [k J/(kgK)], and the mass density, ρ in [kg/m3]. The specific heat is typically known to 2 or 3 significant figures, while the mass density is probably the most accurately known material property with 4 to 5 significant figures. The one-dimensional governing differential equation for transient heat transfer through an area A, of conductivity kx, density ρ, specific heat cp with a volumetric rate of heat generation, Q, for the temperature T at time t is ∂(kx∂T/∂x)/∂x + Q(x) = ρ cp ∂T/∂t, for 0 ≤ x ≤ L and time t ≥ 0. It requires initial conditions to describe the beginning state, and boundary conditions for later times. For a steady state condition (∂T/∂t = 0) the typical boundary conditions of one of the following: www.kamasi.sk | www.strojarina.eu 11  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.1 Introduction 1. T prescribed at 0 and L, or 2. T prescribed at one end and a heat source at the other, or 3. T prescribed at one end and a convection condition at the other, or 4. A convection condition at one end and a heat source at the other, or 5. A convection condition at both ends. www.kamasi.sk | www.strojarina.eu 12  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.2 Heat transfer - described by the Fourier equation   T    T    T  T  k    k    k   Q   c x  x x  y  y y  z  z z  t where: Q is the internal heat source (heat per unit time per unit volume is positive), in kW/m3) ϱ is the mass density in kg/m3 c is the specific heat in J.kg-1.K-1 T   k T   Q   c Heat conduction equation t www.kamasi.sk | www.strojarina.eu 13  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.2 Heat transfer - described by the Fourier equation   T    T    T  T  k    k    k   Q   c x  x x  y  y y  z  z z  t - for isotropic material: k  kx  k y  kz   2T    2T    2T  T k  2    2    2   Q  c  x   y   z  t - stady state, any differentiation with respect to time is equal to zero, so Eq. becomes   2T    2T    2T  k 2  2  2 Q  0  x   y   z  www.kamasi.sk | www.strojarina.eu 14  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.2 Heat transfer - initial conditions: - boundary conditions (BC): BC 1 T = TB on S1 where TB represents a known boundary temperature and S1 is a surface where the temperature is known BC 2 𝑑𝑇 𝑞𝑥∗ = −𝑘𝑥𝑥 𝑑𝑥 on S2 where S2 is a surface where the prescribed heat flux 𝑞𝑥∗ or temperature gradient is known. On an insulated boundary, 𝑞𝑥∗ = 0. These different boundary conditions are shown in Figure, where by sign convention, positive 𝑞𝑥∗ occurs when heat is flowing into the body, and negative 𝑞𝑥∗ when heat is flowing out of the body. www.kamasi.sk | www.strojarina.eu 15  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.2 Heat transfer The three-dimensional heat conduction problem can be stated in an equivalent variational form as follows: Find the temperature distribution T(x, y, z, t) inside the solid body that minimizes the Integral www.kamasi.sk | www.strojarina.eu 16  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.3 Derivation of Finite Element Equations The step-by-step procedure involved in the derivation of finite element equations is given below: Step 1: Divide the domain V into E finite elements of p nodes each. Step 2: Assume a suitable form of variation of T in each finite element and express T(e)(x, y, z, t) in element e as where Ti(t) is the temperature of node i, and Ni(x, y, z) is the interpolation function corresponding to node i of element e. www.kamasi.sk | www.strojarina.eu 17  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.3 Derivation of Finite Element Equations Step 3: Express the functional I as a sum of E elemental quantities I(e) as where www.kamasi.sk | www.strojarina.eu 18  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.3 Derivation of Finite Element Equations For the minimization of the functional I, use the necessary conditions where M is the total number of nodal temperature unknowns. We have www.kamasi.sk | www.strojarina.eu 19  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.3 Derivation of Finite Element Equations Note that the surface integrals do not appear in Eq. if node i does not lie on S2 and S3. The first equation gives where www.kamasi.sk | www.strojarina.eu 20  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.3 Derivation of Finite Element Equations Thus, Eq. can be expressed as www.kamasi.sk | www.strojarina.eu 21  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.3 Derivation of Finite Element Equations and Step 4: Rewrite Eqs. in matrix form as Where is he vector of nodal temperature unknowns of the system: www.kamasi.sk | www.strojarina.eu 22  COMPUTER MECHANICS Lecture 7 7. Thermal Analysis 7.3 Derivation of Finite Element Equations By using the familiar assembly process, Eq. can be expressed as where and Step 5: The first equations are the desired equations that have to be solved after incorporating the boundary conditions specified over S1. www.kamasi.sk | www.strojarina.eu 23  FACULTY OF MECHANICAL ENGINEERING TECHNICAL UNIVERSITY OF KOŠICE Department of applied mechanics and mechanical engineering THANK YOU FOR YOUR ATTENTION www.kamasi.sk | www.strojarina.eu 24