English translation
Application Examples for Analog Computers — Example 4: Heating of a Cable
This is an English translation of the original German document (Telefunken, 15 October 1963).
Example 4 — October 15, 1963
HEAT CONDUCTION
1. Problem Statement
The heating process in a cable is to be investigated. The conductor (r ≤ r₀) acts as a cylindrical heat source at constant temperature. The insulating sheath is assumed to be initially cold. The question is the temperature distribution in the insulating material as a function of time and location (Fig. 1).
Figure 1 — Cross-section through a pipe with cylindrical heat source. u = excess temperature, r = radius.
2. Equations
The governing partial differential equation is:
∂u/∂t = a(r) · (∂²u/∂r² + (1/r) · ∂u/∂r)
Boundary conditions:
u(r,t) = u₀ for r ≤ r₀ u(r,t) = 0 for r = R …(3)
Initial conditions:
u(r,0) = u₀ for r ≤ r₀ u(r,0) = 0 for r > r₀ …(3)
The wall thickness R − r₀ is divided into equidistant steps Δr.
The differential quotients are approximated by difference quotients [1], [2]:
∂u/∂r |r=rᵢ ≈ (uᵢ₊₁ − uᵢ₋₁) / (2Δr) …(4)
∂²u/∂r² |r=rᵢ ≈ (uᵢ₊₁ − 2uᵢ + uᵢ₋₁) / (Δr)² …(5)
1/r ≈ 1 / (r₀ + i·Δr) …(6)
From the partial differential equation one thereby obtains a system of ordinary differential equations:
duᵢ/dt = aᵢ · [(uᵢ₊₁ − 2uᵢ + uᵢ₋₁) / (Δr)² + (1 / (r₀ + i·Δr)) · (uᵢ₊₁ − uᵢ₋₁) / (2Δr)] …(7)
with initial values: u₀ = const. uᵢ(0) = 0 for i = 1, 2, … n …(8)
3. Normalization
With the normalized quantities:
Uᵢ = uᵢ / u_max , τ = λt …(9)
and dividing the wall thickness into 5 steps
(R = 5r₀, Δr = r₀/2; i = 1, 2, … 8)
the machine equation is:
dUᵢ/dτ = aᵢ / (λ·(Δr)²) · [Uᵢ₊₁ · (1 + 1/(2(2+i))) − 2Uᵢ + Uᵢ₋₁ · (1 − 1/(2(2+i)))] …(10)
Assuming constant values aᵢ = a, the time scale factor λ is chosen most conveniently so that the factor a / (λ·(Δr)²) = 0.5. This saves potentiometers. However, the constant a could equally well be a function of location.
The system of equations now reads:
U₀ = 1 U̇₁ = 7/12 U₂ − U₁ + 5/12 U₀ −U̇₂ = −9/16 U₃ + U₂ − 7/16 U₁ U̇₃ = 11/20 U₄ − U₃ + 9/20 U₂ … U̇₇ = 19/36 U₈ − U₇ + 17/36 U₆ −U̇₈ = 0 U̇ᵢ = dUᵢ/dτ …(11)
4. Computing Circuit
Figure 2 — Computing circuit for the solution of the system of equations Eq. (11).
The circuit shows a chain of integrators (summer-integrators with potentiometer inputs) implementing the finite-difference approximation. Each stage i corresponds to a radial node point in the insulating material, with potentiometer settings derived from Eq. (11).
5. Results
Figure 3 — Time course of temperature at locations r₀ to r₈ (normalized temperature Uᵢ(τ) vs. normalized time τ from 0 to 40).
The curves show that the temperature at the inner radius r₀ remains constant (held at 1), while the temperature at each successive outer ring rises progressively more slowly toward its steady-state value as τ → ∞.
Figure 4 — Spatial profile of temperature U_k(r) at specific times τ = 0 to τ = ∞.
From the functions Uᵢ(τ) with rᵢ as parameter (Fig. 3), the function U_k(r) with τ as parameter can be determined graphically (Fig. 4).
Literature
[1] L. Collatz, Numerische Behandlung von Differentialgleichungen, Springer-Verlag 1951.
[2] W. Giloi and R. Lauber, Analogrechnen, Springer-Verlag 1963.
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