McDonald and Bloomster (1977) discussed a model for laying out and sizing the
piping network for a city heated with geothermal water. Pipe diameter is deter-
mined using a "simple search" of feasible pipe sizes by minimizing the sum of the
annual capital cost, heat loss cost and pumping cost. They provided no information
on how to handle network constraints or consider annual load variations.
Bhm (1986) noted that, in the case of consumers directly connected to the
network, the "classical" approach of determining the optimal diameter by finding
the minimum of the sum of the capital, heat loss and pumping costs results in
pressures that are too high at the heating plant. He suggested the use of Munser's
(1980) method, which proportions the total available pressure loss in a network
using the equation
∆P1
L1
=
(1-1)
∆P0
n
1/3
∑ Li mi
˙
m1
˙
i =1
where ∆P1 = pressure loss in pipe number 1 (N/m2)
∆P0 = total pressure loss in the pipe network (N/m2)
L1 = length of pipe number 1 (m)
Li = length of pipe i (m)
m1 = mass flow rate in pipe number 1(kg/s)
˙
mi = mass flow rate in pipe i (kg/s)
˙
n = total number of pipes
i = pipe index.
Equation 1-1 is intended for use on "linear networks" that do not have branches.
Koskelainen (1980) developed a method that is able to solve for optimal diameters
in a branched network. His method consists of successively assuming that the
objective function and constraints locally are linear and repeatedly solving the
problem with a linear programming algorithm. He gives an example where his
"optimal" network has a cost that is 16.4% less than one sized using a head loss
design rule.
In this work we develop a rational design method that yields the optimal pipe
sizes for an application based on case-specific parameter values. This method allows
for the inclusion of all major costs and can account for such factors as escalation of
energy prices, seasonal energy costs, increases in heat losses over system life,
variation in seasonal heat demand, load management strategy, the effect of the heat
consumer, etc. Each of the major constraints on the design of a realistic district
heating network is derived and considered. This method is felt to be practical for
sizing much of the piping of a district heating system.
We begin our study in Chapter 2 by first finding a suitable method for deter-
mining the optimal size for a single pipe, independent of any others. In developing
this method, we endeavor to keep the formulation as simple as possible, yet
complete and accurate enough for design calculations. We make use of geometric
programming theory to identify a lower bounding problem that can be used to guide
us to our solution. At the end of Chapter 2 is an example that shows a 17% saving
in life cycle cost.
In Chapter 3 we study the heat consumer and the effect he has on the piping
system. We develop a new model for the consumer's heat exchanger, which uses the
geometric mean temperature difference as an approximation for the logarithmic
mean temperature difference, thus allowing for an explicit expression for return
temperature. We integrate this consumer model with our single pipe model of
Chapter 2 and show what effect the consumer has on the system.
In Chapter 4 we develop the constraints for systems with multiple pipes and
3
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