Utilities must avoid freezing water pipes. If we assume uniform soil conditions, the temperature T(1,t) at a distance r below the surface and time t after the beginning of a cold snap is given approximately by T(1,t)- T. T- T, 2 %3D dt. Here T, is the constant surface temperature during the cold period, T, is the initial soil tem- perature before the cold snap, and a is the thermal conductivity of the soil. If z is measured in meters and t in seconds, then a = 0.138 10-6m2/s. Let T; = 20°C, and T, = -20°C, and recall that water freezes at 0°C. Determine how deep a water pipe should be buried so that it will not freeze until at least 60 days exposure under these conditions.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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 Utilities must avoid freezing water pipes. If we assume uniform soil conditions,
the temperature T(x, t) at a distance x below the surface and time t after the beginning of
a cold snap is given approximately by
T(x, t) − Ts
Ti − Ts
=
2

π
x
2

Z αt
0
e
−t
2
dt .
Here Ts is the constant surface temperature during the cold period, Ti
is the initial soil temperature before the cold snap, and α is the thermal conductivity of the soil. If x is measured
in meters and t in seconds, then α = 0.138 · 10−6m2/s. Let Ti = 20◦C, and Ts = −20◦C,
and recall that water freezes at 0◦C. Determine how deep a water pipe should be buried so
that it will not freeze until at least 60 days exposure under these conditions.

Utilities must avoid freezing water pipes. If we assume uniform soil conditions,
the temperature T(r,t) at a distance z below the surface and time t after the beginning of
a cold snap is given approximately by
T(r, t) - T.
T; - T,
dt.
Here T, is the constant surface temperature during the cold period, T, is the initial soil tem-
perature before the cold snap, and a is the thermal conductivity of the soil. If r is measured
in meters and t in seconds, then a = 0.138 10-6m²/s. Let T, = 20°C, and T, = -20°C,
and recall that water freezes at 0°C. Determine how deep a water pipe should be buried so
that it will not freeze until at least 60 days exposure under these conditions.
Transcribed Image Text:Utilities must avoid freezing water pipes. If we assume uniform soil conditions, the temperature T(r,t) at a distance z below the surface and time t after the beginning of a cold snap is given approximately by T(r, t) - T. T; - T, dt. Here T, is the constant surface temperature during the cold period, T, is the initial soil tem- perature before the cold snap, and a is the thermal conductivity of the soil. If r is measured in meters and t in seconds, then a = 0.138 10-6m²/s. Let T, = 20°C, and T, = -20°C, and recall that water freezes at 0°C. Determine how deep a water pipe should be buried so that it will not freeze until at least 60 days exposure under these conditions.
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