To find the optimal diameter for a single pair of supply and return pipes, we need

to consider the costs involved and minimize their sum with respect to the pipe

diameter. The cost minimization is done for the life cycle of the system using a net

present value approach. Some types of heat distribution systems may have a salvage

value, while others will, in fact, have a disposal cost associated with the end of their

useful lifetime. Since these will in general be mild functions of the pipe diameter,

they will not significantly affect the optimal pipe diameter and thus will not be

treated here. With these limitations in mind, our objective function, the total life

cycle cost, becomes

min. *C*t = *C*hl + *C*pe + *C*pp + *C*m&r

(2-1)

where *C*t = total system owning and operating cost ($)

Now let's look at each of the costs in eq 2-1 in detail, starting with the cost of heat

losses.

The basic form of the heat loss cost is

(2-2)

where *C*h

= cost of heat ($/Wh)

= present value factor for heat (dimensionless)

= rate of heat loss (W)

= time of year (hr [0 ≤ *t *≤ 8760]).

In the most general case, the cost of heat *C*h can be a function of time because of

seasonal usage rates. The rate of heat loss *Q*hl will also be a function of time over the

yearly cycle. In fact, deterioration of the thermal insulation will result in increasing

heat losses as the system ages. This can not be incorporated directly into the

formulation as given above, but could be allowed for by using an appropriate

escalation factor in the present value factor for heat costs *PVF*h.

The only variable defined above that is dependent on our decision variable, the

pipe diameter *d, *is the heat loss rate itself *Q*hl. For a single buried pipe, the

relationship is

(2-3)

where *T*p = pipe outer surface temperature (C)

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