Steady-state conduction through a plane wall without heat generation...
Description
Question 5 Measurements show that steady-state conduction through a plane wall without heat generation produced a convex temperature distribution such that the midpoint temperature was T0 higher than expected for a linear temperature distribution.
Assuming that the thermal conductivity has a linear dependence on the temperature k = k0(1 + αT) where α is a constant, develop a relationship to evaluate α in terms of T0, T1, and T2. Solution: Write Temperature Balance Equation : Since the system is in steady state and there is no accumulation or generation the heat balance equation reduces to:
∂ ∂T ×k ∂x ∂x
= 0 (Eq. 5.1)
Heat conductivity is changing linearly and is given as:
k ( T )=k 0 ( 1+αT ) ( Eq . 5.2) Integrate Eq. 5.1
k (T )
∂T =C 1 (Eq. 5.2) ∂x
Separate variables.
k ( T ) ∂ T =C 1 ∂ x
(Eq. 5.3)
Substitute k(T) and integrate both sides
∫ k 0 ( 1+αT ) ∂ T =C1 x+ C2 αT 2 k0 ¿
(Eq.5.4)
2
T+
)= C x 1+C 2
(Eq.5.5)
α T 12 T 1+ 2 )= C 1 x (0)+ C2 T 0 =T 1 =k 0 ¿
(Eq.5.6)
2
αT2 T2+ 2 )= C 1 x ( L)+C 2 (Eq. 5.7) T L =T 2 =k 0 ¿ Subtract T1 from T2: 2
α T2 α T1 T 2+ T1+ 2 2 )T 2 −T 1=k 0 ¿ k 0¿
2
) = C1 x ( L )−C 1 x (0)
(Eq.5.8)
Then C1 can be calculated as follows:
C1 =
T 2−T 1 L
2
2
αT2 α T1 k 0 (T 2+ )−k 0 (T 1 + ) 2 2 = L
(Eq.5.9)
Then C2 can be calculated by substituting C1 in Eq. 5.7 with Eq. 5.9: 2
α T2 T2+ 2 C 2=k 0 ¿
)-
(
(k 0 T 2 +
α T 22 αT 2 −k 0 T 1 + 1 )L 2 2 α T 12 =k 0 T 1+ L 2
) (
)
(
)
(Eq.5.10)
Then from Eq.5.4 Temperature Profile T(x) can be derived as:
k 2
T ( x )=¿
(¿ ¿ 0(T 2 +
α T2 α T 12 )−k 0 (T 1+ ) )x 2 2 L C1 x+C 2=¿
+
(
k0 T 1 +
α T1 2
2
)
(Eq.5.11)
Re-arrange to find α:
k (¿¿ 0 x)(T −T 21−T 12 xL) (Eq.5.12) 2 ( T ( x ) × L ) −(k 0 x )(2T 2 +2T 1−2 T 1 xL) α= ¿ 2 2
We know that:
T L= 2
T 1+ T 2 +∆T0 2
(Eq.5.13)
Therefore either T1 or T2 can be expressed though equation to compute α
Thank you for interesting in our services. We are a non-profit group that run this website to share documents. We need your help to maintenance this website.