Heat conduction in one dimension
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Heat conduction in one dimension 3 Heat conduction in one dimension Consider a metal rod which for practical reasons we...
Problem 3 Using Fourier series expansion, solve the heat conduction equation in one dimension a?т ат...
Problem 3 Using Fourier series expansion, solve the heat conduction equation in one dimension a?т ат ax2 де with the Dirichlet boundary conditions: T T, if x 0, and T temperature distribution is given by: T(x, 0) -f(x) T, if x L. The initial 0 = *First find the steady state temperature distribution under the given boundary conditions. The steady form solution has the form (x)-C+C2x *Then write for the full solution T(x,t)=To(x)+u(x,t) with u(x,t) obeying the boundary contions U(0,t)...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
Question 7 When the conduction electrons in a metal are treated as an ideal gas of...
Question 7 When the conduction electrons in a metal are treated as an ideal gas of fermions, the mean number of electrons occupying a state with energy e is given by I+ (-2) (3)u (a) Define the Fermi energy [ marks] (b) Sketch n as a function of e for an electron gas: (i) at T 0 K, and to the Fermi temperature (ii) at a temperature above 0 K, but small compared Label and briefly justify each sketch [5...
Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection...
Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection as shown in Fig.1. L- 10cm Conductivity k = 100 w/m°C. Q(x)= 100.000W/㎡. At x = 0, q = 250 W/ m2. TL = 25 oC. Governing equation: _kdTeQ (0%L) Boundary condition: dT -k Figure 1 Heat conduction on a 1-D metal wire. a. Solve for T (x) with two linear elements (X1 = 0, x2-4cm, and X3 = 10cm) ; b. Compare with...
Cooling fins are used to increase the area available for heat transfer between metal walls and...
Cooling fins are used to increase the area available for heat
transfer between metal walls and poorly conducting fluids such as
gases. A rectangular fin is shown in the following figure.
To design a cooling fin and calculate the fin efficiency one
must first calculate the temperature profile in the fin.
If L>>B, no heat is lost from the end or from the edges,
and the heat flux at the surface is given by:
in which the convective heat transfer...
3. 3 points A metal rod has a length of 10cm, and has its left end at zn0 and right end at z = 5. The temperature T(z,1) of the rod satisfies the heat equation T,r(z,t)-m(r,t), its ends are kept...
3. 3 points A metal rod has a length of 10cm, and has its left end at zn0 and right end at z = 5. The temperature T(z,1) of the rod satisfies the heat equation T,r(z,t)-m(r,t), its ends are kept at 0 for all t, and its temperature distribution at time t0 is given by T(a,0)a. Find the temperature function T(,t) for the rod
3. 3 points A metal rod has a length of 10cm, and has its left end...
2. Solve the one-dimensional heat equation problem for a unit length bar with insulated ends with...
2. Solve the one-dimensional heat equation problem for a unit length bar with insulated ends with a prescribed initial linear temperature distribution: c2uxx = 111 , l4 (0,t)-14 (l,t)-0, 0 < x < 1 20-x) , Last Name A-M 3x, Last Name N -Z u(x,0) = The general solution to this problem is given in Example 4, page 563 in the text in terms of a Fourier Cosine Series. Write out the solution steps and evaluate the Fourier coefficients by...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
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...
4. Which one of the following is the correct expression for one-dimensional constant properties, heat conduction...
4. Which one of the following is the correct expression for one-dimensional constant properties, heat conduction equation for a cylinder with heat eady-e generation r dr qun 5. A 10 cm diameter sphere maintained at 30°C is buried in the earth at a place where the ·K. The depth to the centerline is 24 cm, and the thermal conductivity, k = 1.2 W/m earth surface temperature is 0°C. Calculate the heat loss from the sphere. (A) 25.3W (D)42.4W (E)20.0 w...
Problem 3 Using Fourier series expansion, solve the heat conduction equation in one dimension a?т ат ax2 де with the Dirichlet boundary conditions: T T, if x 0, and T temperature distribution is given by: T(x, 0) -f(x) T, if x L. The initial 0 = *First find the steady state temperature distribution under the given boundary conditions. The steady form solution has the form (x)-C+C2x *Then write for the full solution T(x,t)=To(x)+u(x,t) with u(x,t) obeying the boundary contions U(0,t)...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
Question 7 When the conduction electrons in a metal are treated as an ideal gas of fermions, the mean number of electrons occupying a state with energy e is given by I+ (-2) (3)u (a) Define the Fermi energy [ marks] (b) Sketch n as a function of e for an electron gas: (i) at T 0 K, and to the Fermi temperature (ii) at a temperature above 0 K, but small compared Label and briefly justify each sketch [5...
Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection as shown in Fig.1. L- 10cm Conductivity k = 100 w/m°C. Q(x)= 100.000W/㎡. At x = 0, q = 250 W/ m2. TL = 25 oC. Governing equation: _kdTeQ (0%L) Boundary condition: dT -k Figure 1 Heat conduction on a 1-D metal wire. a. Solve for T (x) with two linear elements (X1 = 0, x2-4cm, and X3 = 10cm) ; b. Compare with...
Cooling fins are used to increase the area available for heat
transfer between metal walls and poorly conducting fluids such as
gases. A rectangular fin is shown in the following figure.
To design a cooling fin and calculate the fin efficiency one
must first calculate the temperature profile in the fin.
If L>>B, no heat is lost from the end or from the edges,
and the heat flux at the surface is given by:
in which the convective heat transfer...
3. 3 points A metal rod has a length of 10cm, and has its left end at zn0 and right end at z = 5. The temperature T(z,1) of the rod satisfies the heat equation T,r(z,t)-m(r,t), its ends are kept at 0 for all t, and its temperature distribution at time t0 is given by T(a,0)a. Find the temperature function T(,t) for the rod
3. 3 points A metal rod has a length of 10cm, and has its left end...
2. Solve the one-dimensional heat equation problem for a unit length bar with insulated ends with a prescribed initial linear temperature distribution: c2uxx = 111 , l4 (0,t)-14 (l,t)-0, 0 < x < 1 20-x) , Last Name A-M 3x, Last Name N -Z u(x,0) = The general solution to this problem is given in Example 4, page 563 in the text in terms of a Fourier Cosine Series. Write out the solution steps and evaluate the Fourier coefficients by...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
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...
4. Which one of the following is the correct expression for one-dimensional constant properties, heat conduction equation for a cylinder with heat eady-e generation r dr qun 5. A 10 cm diameter sphere maintained at 30°C is buried in the earth at a place where the ·K. The depth to the centerline is 24 cm, and the thermal conductivity, k = 1.2 W/m earth surface temperature is 0°C. Calculate the heat loss from the sphere. (A) 25.3W (D)42.4W (E)20.0 w...
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