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Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection...
Solve the 1D heat conduction equation with a source term. The 1D heat conduction equation with a ...
Solve the 1D heat conduction equation with a source term.
The 1D heat conduction equation with a source term can be written
as:
dr dr Using the Finite Volume Method, we use this equation to solve for the temperature T across the thickness of a flat plate of thickness L-2 cm. The thermal conductivity is k-0.5 W/Km, and the temperatures at the two ends are held constant at 100°C and 200°C, respectively. An electric current creates aAL constant heat source...
1). Consider 1D heat conduction in a solid plate as shown. The temperatures at two boundaries...
1). Consider 1D heat conduction in a solid plate as shown. The temperatures at two boundaries are 20 K and 10 K, respectively. lm- 2 1 1 3 4 5 T20 K T = 10 K 0.25m 0.25m 0.25% 0.25 (a) Write down the governing equation for the temperature distribution inside the plate. Assume no heat source inside the entire plate. (6) The domain has been discretized using 5 equally spaced grids. Discretize the governing equation in (a) using finite...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where ?(?,?)...
Q2 Given the following heat conduction initial-boundary value
problem of a thin homogeneous rod, where ?(?,?) represents the
temperature. 9??? = ?? ; 0 < ? < 6; ? > 0; B. C. : ??
(0,?) = 0; ?? (6,?) = 0; ? > 0; I. C. : ?(?, 0) = 12 + 5??? ( ?
6 ?) − 4???(2??); 0 < ? < 6 (a) When ? = 0, what would be the
temperature at ? = 3? (Use...
The heat that is conducted through a body must frequently be removed by other heat transfer...
The heat that is conducted through a body must frequently be removed by other heat transfer processes. For example, the heat generated in an electronic device must be dissipated to the surroundings through convection by means of fins. Consider the one-dimensional aluminum fin (thickness t 3.0 mm, width 20 cm, length L) shown in Figure 1, that is exposed to a surrounding fluid at a temperature T. The conductivity of the aluminum fin (k) and coefficient of heat convection of...
Consider steady one-dimensional conduction in a slab having a thickness of L and a constant thermal...
Consider steady one-dimensional conduction in a slab having a thickness of L and a constant thermal conductivity of k. The two ends are maintained at temperatures T_0 (at x = 0), and T_L (at x = L). There is a heat source, with strength A(x/L)(1 - x/L) W/cm^3. a) Define a set of dimensionless independent variables (depending on position), and a dimensionless dependent variable. b) Obtain a differential equation in which each term is dimensionless. c) Define an appropriate dimensionless...
why m=1, and why n=pi/2, 3pi/2, ............ The problem of unsteady heat conduction in a metal...
why m=1, and why n=pi/2, 3pi/2, ............
The problem of unsteady heat conduction in a metal plate of length 1 meter is described by the equation: where u is the temperature, with initial/boundary conditions: a(2,0) = 1 (0,t) = 1 Ou
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t)...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Uti 0<x< 6; t> 0; B.C.: ux(0,t) = 0; ux(6,t) = 0; t> 0; 1. C.: u(x, 0) = 12 + scos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut 0<x< 6; t> 0; B.C.: 4x(0,t) = 0; uz (6,t) = 0; t> 0; 1.C. : u(x,0) = 12 + scos (x) – 4cos(2x); 0 < x < 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t)...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Ut; 0<x< 6; t> 0; B.C.: 4x(0,t) = 0; uz(6,t) = 0; t> 0; 1. C.: 4(x,0) = 12 + Scos (6x) – 4cos(27x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or...
Solve the 1D heat conduction equation with a source term.
The 1D heat conduction equation with a source term can be written
as:
dr dr Using the Finite Volume Method, we use this equation to solve for the temperature T across the thickness of a flat plate of thickness L-2 cm. The thermal conductivity is k-0.5 W/Km, and the temperatures at the two ends are held constant at 100°C and 200°C, respectively. An electric current creates aAL constant heat source...
1). Consider 1D heat conduction in a solid plate as shown. The temperatures at two boundaries are 20 K and 10 K, respectively. lm- 2 1 1 3 4 5 T20 K T = 10 K 0.25m 0.25m 0.25% 0.25 (a) Write down the governing equation for the temperature distribution inside the plate. Assume no heat source inside the entire plate. (6) The domain has been discretized using 5 equally spaced grids. Discretize the governing equation in (a) using finite...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
Q2 Given the following heat conduction initial-boundary value
problem of a thin homogeneous rod, where ?(?,?) represents the
temperature. 9??? = ?? ; 0 < ? < 6; ? > 0; B. C. : ??
(0,?) = 0; ?? (6,?) = 0; ? > 0; I. C. : ?(?, 0) = 12 + 5??? ( ?
6 ?) − 4???(2??); 0 < ? < 6 (a) When ? = 0, what would be the
temperature at ? = 3? (Use...
The heat that is conducted through a body must frequently be removed by other heat transfer processes. For example, the heat generated in an electronic device must be dissipated to the surroundings through convection by means of fins. Consider the one-dimensional aluminum fin (thickness t 3.0 mm, width 20 cm, length L) shown in Figure 1, that is exposed to a surrounding fluid at a temperature T. The conductivity of the aluminum fin (k) and coefficient of heat convection of...
Consider steady one-dimensional conduction in a slab having a thickness of L and a constant thermal conductivity of k. The two ends are maintained at temperatures T_0 (at x = 0), and T_L (at x = L). There is a heat source, with strength A(x/L)(1 - x/L) W/cm^3. a) Define a set of dimensionless independent variables (depending on position), and a dimensionless dependent variable. b) Obtain a differential equation in which each term is dimensionless. c) Define an appropriate dimensionless...
why m=1, and why n=pi/2, 3pi/2, ............
The problem of unsteady heat conduction in a metal plate of length 1 meter is described by the equation: where u is the temperature, with initial/boundary conditions: a(2,0) = 1 (0,t) = 1 Ou
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Uti 0<x< 6; t> 0; B.C.: ux(0,t) = 0; ux(6,t) = 0; t> 0; 1. C.: u(x, 0) = 12 + scos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut 0<x< 6; t> 0; B.C.: 4x(0,t) = 0; uz (6,t) = 0; t> 0; 1.C. : u(x,0) = 12 + scos (x) – 4cos(2x); 0 < x < 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Ut; 0<x< 6; t> 0; B.C.: 4x(0,t) = 0; uz(6,t) = 0; t> 0; 1. C.: 4(x,0) = 12 + Scos (6x) – 4cos(27x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or...
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