full2011_inter.pdf - page 300

2011 International Conference on Alternative Energy in Developing Countries and Emerging Economies
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IV. NUMERICAL METHOD
The theoretical system pressure drop and system flow
rate of distribution network are calculated using the
governing equations of mass conservation and energy
equations considering that sizing criteria of chilled water
cooling system and control compatibility of chillers are
met. The nonlinear equations which are formed consist
of pump performance curve, mass balance and system
characteristic curves.
A. Pump performance curve
The pump performance curve is taken from the
technical data sheet of pump model that will satisfy the
system cooling flow rate requirement based on the
desired chilled water temperature difference. The pump-
flow curve can be generally expressed as a function of
flow rate as shown in Eqn. (1).
ߜ ݌ଵ ൌ ݃ሺ߱ሻ (1)
where , ߜ ݌ଵ represents the pressure drop.
B. System characteristic curve
The system characteristic curves are derived based on
the piping configuration of distribution network from
central cooling plant to respective ETS of each building
which can be expressed as
ߜ ݌௝ ൌ σ ܿ௜߱௜ ଶ ௡௜ୀଵ (2)
where,݅ indicates the number of pipe components ݆ the number of ETS in the distribution network
(i.e., ݆ ൌ ʹǡ ͵ǡ Ǥ ǡ݉)
The ܿ௜ and ߱௜ denote the frictional coefficients of each
component and flow rate along the distribution network,
respectively. The friction factor of chilled water denoted
as ߣ can be expressed as a function of Reynolds number
and relative roughness which can be in the form of
ܿ ൌ ݇ ቀߣǡ ௅ ׎ቁ (3a)
ߣ ൌ ݄ ቀܴ݁ǡ ఌ ׎ቁ (3b)
The fluid thermal properties such as mass density,
absolute viscosity and specific heat are function of inlet
fluid temperature.
C. Dimensional analysis
As variable primary pumping system is used, the
pumps are modulated at its best efficiency point.
Dimensional analysis with a generalized approach
commonly known as Buckingham Π Theorem is adapted
to derive pertinent parameters to predict the performance
of pump model under different conditions of operation.
The variables used considering that the length scale is
neglected are shown below.
݂መ ൫݄ሗ ǡ ܳǡ ݊ǡ ܦǡ ߝǡߩ ǡ ߤǡ ܶǡ ߟ ǡ ݃ ൯ ൌ Ͳ (4)
The dimensionless parameters can be expressed as
׎ ቀ௛ሗ ஽ǡ ் ఘ௡
మ஽ఱ ǡ ொ ௡஽య ǡఘ௡஽మ
ఓ ǡ ఌ ஽ǡߟ ǡ ௚ ௡
మ஽ቁ ൌ Ͳ (5)
The new functional relations yield
௚௛ሗ ௡మ஽మ ൌ ׎ଵ ቂ ொ ௡஽య ǡ ఘ௡஽మ
ఓ ǡ ఌ ஽ቃ (6)
் ఘ௡
మ஽ఱ ൌ ׎ ଶ ቂ ொ ௡஽య ǡ ఘ௡஽మ
ఓ ǡ ఌ ஽ቃ (7)
ߟ ൌ ׎ ଷ ቂ ொ ௡஽య ǡ ఘ௡஽మ
ఓ ǡ ఌ ஽ቃ (8)
However, the effects of Reynolds number and relative
roughness can be neglected for fluids at high velocity.
Hence, the above equations which consist of head rise
coefficient, power coefficient and efficiency can be a
function of flow coefficient in the form of
ߟ ൌ ܥு ܥொ ܥ௉ Τ (9)
Based on these equations, the Affinity Laws for
geometrically similar pumps can be generally expressed
as follows.
ቂ ௚௛ሗ ௡
మ஽మቃଵ ൌ ቂ ௚௛ሗ ௡మ஽మቃଶ (10)
ቂ ் ఘ௡
మ஽ఱቃଵ ൌ ቂ ் ఘ௡మ஽ఱቃଶ (11)
ቂ ொ ௡஽యቃଵ ൌ ቂ ொ ௡஽య ቃଶ (12)
D. Cholesky decomposition
As system equilibrium occurs after pumps modulation
which is operating at its best efficiency point, the new
pump’s rotative speed can be identified using the Affinity
Laws. With the new rotative speed, new sets of flow rates
and total dynamic heads can be generated to identify the
pump characteristic curve at its modulated condition
using the Method of Least Squares as shown in Eqn. (13).