insulation (Phetteplace and Meyer 1990). After simplification the following result is

obtained

(2-4)

where γ = *k*i/*k*s (dimensionless)

In this form it becomes easy to see how each factor affects this parameter. The

(4*H*p/*D*o)γ factor represents the contribution of the soil to the overall thermal

resistance. If γ << 1, that is, if the soil conductivity is much greater than the insulation

thermal conductivity, then this factor will be close to unity and the overall thermal

resistance reduces to the thermal resistance of the insulation alone.

To obtain a simpler form for the cost of heat loss, we make the following

assumptions:

1. That the soil temperature at the pipe depth varies sinusoidally over the yearly

cycle about a mean temperature.

2. That the cost of heat is constant over the yearly cycle. This does not limit us to

fixed heat cost over the life of the system, since escalation factors may be used.

3. That the outer surface temperature of the carrier pipe is equal to the tem-

perature of the carrier medium.

The result of these assumptions is the following form for the cost of heat loss

(2-5)

∫

where *I*1 = *PVF*h L 4π*k*i ( *C*h Tavg d*t * *A*t Ch Tm) ($)

Now let's consider the pumping costs. A cost is associated with the electrical

energy input to drive the pumps. The portion of this energy that results in frictional

heating of the fluid in the pipes is recovered as heat. In general the value of the heat

recovered will, of course, be less than the value of the electrical energy input to drive

the pumps. It can be significant, however, and therefore it has been included here.

Thus, we have the following for the pumping cost

∫

∫r ChPPf d*t*

(2-6)

where *PVF*e = present value factor for electrical energy (dimensionless)

inefficiencies (W)

6