Engineering Problem #$3$: Soil Temperature Modeling ${?}^2$

Background

In the construction of new urban systems, one of the

important considerations is the depth at which water pipes

must be buried in order to ensure they do not freeze. It is

possible to model this situation by considering how the

temperature within the soil at a depth $x$ relates to the

surface temperature, $T_S,$ and the initial soil temperature, $T_i,$

over time $t($in seconds$).$ It has been found that this can be

approximately represented by the equation:

$\frac{T(x,t)-T_S}{T_i-T_S}=$erf$(\frac{x}{2\sqrt[\phantom{2}]{t}})=\frac{2}{\sqrt[\phantom{2}]{}}{\displaystyle \underset{0}{\overset{\frac{x}{2\sqrt[\phantom{2}]{t}}}{}}}{e}^{-u^2}du$

Here, erf, refers to a famous $($and very useful$)$ function in mathematics call the Gauss Error

Function, or simply the error function. It has the integral form shown above. The parameter $$

represents the thermal diffusivity of the soil.

Problem Statement

Prepare a report that summarizes how the soil temperature within Toronto would change with time

and depth over the course of a six$-$month period $($December $1-$ May $31).$ Most of the soil in

Toronto can be characterized as clay, and typically the initial soil temperature for all depths at the

start of December is $T_i=14C.$ You have been asked to consider the following specific cases:

a$)$ Demonstrate how the temperature changes with time at a $0\mathrm{.}75m$ depth over the course of

the first $31$ days $($use increments of $1$ day$).$

b$)$ Show how the temperature changes with time $($using $1$ day increments$)$ over the six$-$month

period for the range of depths of $0\mathrm{.}05m$ to $5m.$

c$)$ Determine:

a$.$ How deep the water lines need to be buried to ensure that the temperature does not

drop below $5C.$

b$.$ How deep the water lines need to be buried in order for the surrounding soil

temperature to change by less than $1\%$ of the initial soil temperature $($i$.$e$.,14C).$