equilibrium with the electrodes and its surroundings. Suddenly,
an electrical current is passed through the rod.
a. Using a first law analysis, what is the energy balance on the
rod? (
)
b. Using A c = (πD 2 /4), P = πD , derive the explicit finitedifference expression for node (n). Recall that heat generation
follows the I 2 R e law and that R e is defined as R e = ρ e Δx/A c .
Express your answer using the Fourier number in the explicit
finite-difference form.
c. What are the stability criteria at node (n)?
5.6. Refer to Figure 5.4, use the finite-difference explicit method to
derive the energy balance during the transient for the following
grid distributions:
(1) m = 1, 2, 3, 4, 5
(2) m = 1, 2, 3, 4
(3) m = 1, 2, 3
n = 1, 2, 3, 4, 5
n = 1, 2, 3, 4
n = 1, 2, 3
5.7 Refer to Figure 5.4, use the finite-difference implicit method to
derive the energy balance during the transient for the following
grid distributions:
(1) m = 1, 2, 3, 4, 5
(2) m = 1, 2, 3, 4
(3) m = 1, 2, 3
n = 1, 2, 3, 4, 5
n = 1, 2, 3, 4
n = 1, 2, 3
5.8. Use the finite-difference explicit method, and derive finitedifference energy balance equations for a 1-D hollow cylinder
during the transient with the following BCs:
(1) r = r 1 , T = T 1
(2) r = r 1 , −k(∂T/∂r) = h 1 (T ∞1 − T 1 )
r = r N , T = T N
r = r N , −k(∂T/∂r) = h N (T N − T ∞N )
(3) r = r 1 , −k(∂T/∂r)
= h 1 (T ∞ 1 − T 1 )
""
r = r N , −k(∂T/∂r) = q s
5.9. Use the finite-difference implicit method, and derive finitedifference energy balance equations for a 1-D hollow cylinder
during the transient with the following BCs:
(1) r = r 1 , T = T 1
(2) r = r 1 , −k(∂T/∂r) = h 1 (T ∞1 − T 1 )
r = r N , T = T N
r = r N , −k(∂T/∂r) = h N (T N − T ∞N )
(3) r = r 1 , −k(∂T/∂r)
= h 1 (T ∞ 1 − T 1 )
""
r = r N , −k(∂T/∂r) = q s
5.10. Derive Equations 5.23 and 5.25 for 3-D transient heat conduction
problems.
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Numerical Analysis in Heat Conduction
an electrical current is passed through the rod.
a. Using a first law analysis, what is the energy balance on the
rod? (
)
b. Using A c = (πD 2 /4), P = πD , derive the explicit finitedifference expression for node (n). Recall that heat generation
follows the I 2 R e law and that R e is defined as R e = ρ e Δx/A c .
Express your answer using the Fourier number in the explicit
finite-difference form.
c. What are the stability criteria at node (n)?
5.6. Refer to Figure 5.4, use the finite-difference explicit method to
derive the energy balance during the transient for the following
grid distributions:
(1) m = 1, 2, 3, 4, 5
(2) m = 1, 2, 3, 4
(3) m = 1, 2, 3
n = 1, 2, 3, 4, 5
n = 1, 2, 3, 4
n = 1, 2, 3
5.7 Refer to Figure 5.4, use the finite-difference implicit method to
derive the energy balance during the transient for the following
grid distributions:
(1) m = 1, 2, 3, 4, 5
(2) m = 1, 2, 3, 4
(3) m = 1, 2, 3
n = 1, 2, 3, 4, 5
n = 1, 2, 3, 4
n = 1, 2, 3
5.8. Use the finite-difference explicit method, and derive finitedifference energy balance equations for a 1-D hollow cylinder
during the transient with the following BCs:
(1) r = r 1 , T = T 1
(2) r = r 1 , −k(∂T/∂r) = h 1 (T ∞1 − T 1 )
r = r N , T = T N
r = r N , −k(∂T/∂r) = h N (T N − T ∞N )
(3) r = r 1 , −k(∂T/∂r)
= h 1 (T ∞ 1 − T 1 )
""
r = r N , −k(∂T/∂r) = q s
5.9. Use the finite-difference implicit method, and derive finitedifference energy balance equations for a 1-D hollow cylinder
during the transient with the following BCs:
(1) r = r 1 , T = T 1
(2) r = r 1 , −k(∂T/∂r) = h 1 (T ∞1 − T 1 )
r = r N , T = T N
r = r N , −k(∂T/∂r) = h N (T N − T ∞N )
(3) r = r 1 , −k(∂T/∂r)
= h 1 (T ∞ 1 − T 1 )
""
r = r N , −k(∂T/∂r) = q s
5.10. Derive Equations 5.23 and 5.25 for 3-D transient heat conduction
problems.
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Numerical Analysis in Heat Conduction