The Lumped Capacitance Method

dθ
′ =− adt
θ ′
Integrating from t=0 to any t and rearranging,
ln( θ′ ) = − at + C ⇒ θ′
= exp( − at + C )
1 1
θ′ = C exp( −at) or θ − b/ a = C exp( −at)
2 2
att = 0, T = T ⇒ C = ( T −T ) − b/
a
i
Hence, ( T −T ) − b/ a = [( T −T ) − b/ a] exp( −at)
∞
T −T∞
b/ a b/
a
= + [1 − ] exp( −at)
T −T T −T T −T
i ∞ i ∞ i ∞
i
∞
2
i
∞
T −T∞
b/
a
or, = exp ( − at) + ⎡1 − exp( −at)
⎤
T −T T − T
⎣
⎦ (5.25)
i
∞
i
To what does the foregoing equation reduce as steady state is approached?
How else may the steady-state solution be obtained?
∞