- 2074 Bhadra · 6 marks
Steady vs unsteady heat transfer; three-layer cylinder
Short answer
In steady state temperature does not change with time; in unsteady state temperature varies with time. For a three-layer composite cylinder, series resistances give heat flow: Q = 2πL (T1 − T4) / [ln(r2/r1)/k1 + ln(r3/r2)/k2 + ln(r4/r3)/k3].
Differentiate between steady state and unsteady state heat transfer. Derive an expression for steady state heat transfer through a composite cylinder consisting of three layers.
Steady and unsteady heat transfer
| Point | Steady state | Unsteady (transient) state |
|---|---|---|
| Temperature with time | Does not change with time, T = f(x, y, z) | Changes with time, T = f(x, y, z, t) |
| Heat flow | Constant | Varies with time |
| Energy stored | Zero (inflow = outflow) | Non-zero; internal energy changes |
| Governing equation | d²T/dx² = 0 (no generation) | ∂T/∂t = α ∂²T/∂x² |
| Complexity | Simple | More complex |
| Example | Furnace wall during constant operation | Heating or cooling of a body, furnace start-up |
Heat transfer through a three-layer composite cylinder
A long cylinder has three concentric layers in perfect contact: radii r₁ < r₂ < r₃ < r₄ and conductivities k₁, k₂, k₃. The inner surface (r₁) is at T₁ and the outer surface (r₄) at T₄; the interface temperatures are T₂ and T₃. Heat flows radially and steadily through length L, with the same Q through each layer.
Cross-section (layers outward from the axis)
axis | layer 1 | layer 2 | layer 3 |
r1 r2 r3 r4
T1 --[R1]-- T2 --[R2]-- T3 --[R3]-- T4
Derivation
For a single cylindrical layer (Fourier's law with A = 2πrL, Q = −k·2πrL·dT/dr, integrated between the radii):
Q = 2πk L (T_in − T_out)/ln(r_out/r_in)
Apply to each layer:
T₁ − T₂ = Q ln(r₂/r₁)/(2πk₁L) T₂ − T₃ = Q ln(r₃/r₂)/(2πk₂L) T₃ − T₄ = Q ln(r₄/r₃)/(2πk₃L)
Adding, the interface temperatures cancel:
T₁ − T₄ = (Q/2πL) [ln(r₂/r₁)/k₁ + ln(r₃/r₂)/k₂ + ln(r₄/r₃)/k₃]
Q = 2πL (T₁ − T₄)/[ln(r₂/r₁)/k₁ + ln(r₃/r₂)/k₂ + ln(r₄/r₃)/k₃]
In resistance form, Q = (T₁ − T₄)/(R₁ + R₂ + R₃), with R_n = ln(r_{n+1}/r_n)/(2π k_n L) in series.
✓ Answer checked by Claude Opus 5.5 (high effort)
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