full2011_inter.pdf - page 301

2011 International Conference on Alternative Energy in Developing Countries and Emerging Economies
- 301 -
ݍ ൌ σ ቀ݄௝ െ ݌൫ܳ௝ ൯ቁ ଶ ௡௝ୀଵ (13)
where,݌൫ܳ௝ ൯ ൌ ܽ଴ ൅ ܽଵ ܳ൅ Ǥ Ǥ ൅ܽ௠ܳ ௠ (14)
As the curve denoted as ݌ሺܳሻ should be fitted through
new sets of coordinates, a necessary condition for ݍ to be
minimum is to satisfy the following equations.
ݍǡ௔బ ൌ ݍǡ௔ భ ൌ ݍǡ௔మ ൌ ݍǡ௔ య ൌ Ͳ (15)
In matrix form, the normal equations are shown below.
ࢇ ൌ ࡭ࢀ Ǥ ࢈ (16)
As the elements of coefficient matrix are symmetric
which has a positive real eigenvalues then, Cholesky
decomposition method can be used in solving the
unknown vector parameter denoted as ࢇ.
E. Numerical calculation
The continuity equation, pump performance and
system characteristic curves can be expressed as
functional relations to be zeroed as shown in Eqn. (17).
݂௜ሺߜ ݌ǡ ߱ǡ ߱ଵǡ ߱ଶ ǡ Ǥ Ǥ ሻ ൌ Ͳ (17)
The independent variables are system pressure drop,
system flow rate and cooling flow rate requirements in
each ETS. The above equation can be rewritten as
ܨ௜ሺݏଵǡ ݏଶǡ Ǥ Ǥ ǡ ݏ௡ሻ ൌ Ͳ (18)
Considering the variables ࢙ be an entire vector denoted
as ݏ௜ and ࡲ denote the entire vector of functions ܨ௜ in
nonlinear form, the system of equations can be solved
using multivariable Newton-Raphson method where the
functions can be expanded in Taylor series of the form
ܨ௜ሺ࢙ ൅ ߜ࢙ሻ ൌ ܨ௜ሺ࢙ሻ ൅ σ డி೔
డ௦ೕ ݏߜ௝ ൅ ܱሺߜ ࢙ଶሻ
ே௝ୀଵ (19)
Neglecting higher order terms and by setting the left-
hand side equation equal to zero, a set of equations for
the corrections ߜ࢙ that move each function closer to zero
is simultaneously obtained where Eqn. (19) can be
reduced to an equation with Jacobian matrix ࡶ as shown
below.
ࡶǤ ߜ࢙ ൌ െࡲ (20)
As matrix Eqn. (20) is an over-determined system of
equations, Singular-Value Decomposition method is used
where the corrections are then added to the solution
vector and the process is iterated to convergence.
࢙௡௘௪ ൌ ࢙௢௟ௗ ൅ ߜ࢙ (21)
V. RESULT AND DISCUSSION
In this study, twelve (12) ETS with different cooling
loads are used as benchmarks for analysis as shown in
Tab. I. The total cooling flow rate requirement needed to
provide enough cooling to all buildings is 26,667 (gpm)
considering a design temperature difference of Δ T = 9°C
and diversity factor of 75%.
TABLEI
C OOLING LOAD REQUIREMENT IN EACH BUILDING
Building No. Cooling Load (Ton)
Tower A 3000
Tower B 2800
Tower C 2500
Tower D 2500
Tower E 2000
Tower F 1800
Tower G 1500
Tower H 1800
Tower I 1800
Tower J 1500
Tower K 1400
Tower L 1400
Prior to chilled water variable pumping system
calculation, pipe sizing is also deemed necessary to
reduce the head loss along the distribution network. The
piping network design criteria used are shown in Tab. II.
TABLEII
PIPING NETWORK DESIGN CRITERIA
Description Variable Value Unit
Velocity limit at primary line ݒ௣௟̴௟௜௠௜௧ 1.75 m/s
Velocity limit at secondary line ݒ ௦௟̴௟௜௠௜௧ 1.75 m/s
Velocity limit at plot take-off ݒ௣௧௢̴௟௜௠௜௧ 1.50 m/s
Pressure limit ݌௟௜௠௜௧ 100 Pa/m
The results of system simulation indicate that the
cooling flow rate requirement can be achieved even at
low pressure drop as shown in Fig. 1. The figure shows
the variable primary pumping system operating points
denoted as BEP1, BEP2 and BEP3 with their
corresponding pump curves as PC1 and PC2. BEP1 is
the initial pump operating point to satisfy the chilled
water system cooling flow rate requirement.