Lecture Objectives:

1
Lecture Objectives

• Discuss HW4
• Nonlinear equation solvers
• Continue with advance air systems
• Introduce control systems

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HW4 issues Nonlinear equation solvers
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Successive substitution method

• Iterative method
• Requires initial guess
• Requires equation in explicit form
• Can be used for solution of - steady-sate or
• -
  unsteady-state problems
• For unsteady-state problem we have to iterations
  for each time step

Solution for one time step
Simple Example
X-Y/2-1 X2-Y-3 X, Y are any physical variables
We know the solution By substitution method we
get X2-2x10 ? X1, Y4
Explicit form XY/2-1 YX23
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Successive substitution method
Simple Example
Explicit form XY/2-1 YX23
Initial guess ?
End of iteration ?
XY/2-1
YX23
Y
X
yes
Y2
no
X Y
initial guess 2
iteration 1 0 3
iteration 2 0.5 3.25
iteration 3 0.625 3.390625
iteration 4 0.695313 3.483459
iteration 5 0.74173 3.550163
--- ---- ---
Iteration 98 0.982059 3.96444
Solution
Solution for One time step
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Successive substitution method
When to stop with iterations?
We DO NOT know the exact solution!
Residual Difference in result between two
iterations RYY(n)-Y(n-1)lte ,

e is defined by requested accuracy

• For example eY0.0004
• Iteration 99 Y(99) 3.96132
• Iteration 100 Y(100)3.96169
  Y(100)-Y(99)0.00037lteY
• 
  stop the iterations

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Iterative method
Relaxation with iterative solvers When the
equations are highly nonlinear it can happen that
you get divergency in iterative procedure for
solving considered time step
divergence
variable
solution
convergence
Solution is Under-Relaxation YfY(n)(1-f)Y(n
-1) Y considered parameter , n iteration
, f relaxation factor For our example Yin
iteration 101fY(100)(1-f) Y(99) f 0-1
under-relaxation -stabilize the iteration f
1-2 over-relaxation - speed-up the
convergence
iteration
Value which is should be used for the next
iteration
Under-Relaxation is often required when you have
nonlinear equations!
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Newton-Raphson method

• Considerably better method than
• Successive substitution method
• Faster convergence
• Used in many professional tools (MathCAD, EES,
  MatLab, Mathematica, etc)
• More complex for programming
• Requires linear solver
• Based on Taylor-Series Expansion
• You need first derivative for each function to
  create the Jacobean matrix
• Equations in the form where all side are on one
  side of equality sign

Our simple example X-Y/2-1 ?
X-Y/210 X2-Y-3 ?
X2-Y30
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Newton-Raphson method(this is used in most
equation solvers)
Section 6.11 of handouts
Our simple example f1 X-Y/210 f2
X2-Y30 Steps 0) Find derivatives
d(f1)/dX 1 , d(f1)/dY -1/2
d(f2)/dX 2X ,
d(f2)/dY -1 1) Initial guess Y(0)2,
X(0)2 2) Find f1(Y(0),X(0))2-2/212
f2(Y(0),X(0))22-235 3
) Using derivatives and guess values find the
Jacobean matrix 4) Solve the matrix using linear
solver and find DX and DY 5) Find Y(1)Y(0) DY,
X(1)X(0) DX, Repeat step (2) with Y(1) and
X(1) .. Follow the procedure till
convergence
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Various Desiccant Systems
D. La, Y.J. Dai , Y. Li, R.Z. Wang, T.S. Ge
Technical development ofrotary desiccant
dehumidification and air conditioning A review
Renewable and Sustainable Energy
Reviews http//ac.els-cdn.com/S1364032109001737/1-
s2.0-S1364032109001737-main.pdf?_tid4e770220-bb3b
-11e3-a8b2-00000aacb361acdnat1396535094_db711889
daf20a355a7da4e49f4a3ab6
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Desiccant Enhanced HVAC
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Liquid Desiccant System
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Liquid Desiccant System
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Control
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The PID control algorithm
constants
time
e(t) difference between set point
and measured value
Position (x)
Differential
Proportional
Integral

• For our example of heating coil

Differential (how fast)
Proportional (how much)
Integral (for how long)
Position of the valve
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Proportional Controllers

• x is controller output
• A is controller output with no error
• (often A0)
• Kis proportional gain constant
• e is error
  (offset)

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Stable system
Unstable system
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Issues with P Controllers

• Always have an offset
• But, require less tuning than other controllers
• Very appropriate for things that change slowly
• i.e. building internal temperature

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Proportional Integral (PI)
K/Ti is integral gain
If controller is tuned properly, offset is
reduced to zero
Figure 2-18a
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Issues with PI Controllers

• Scheduling issues
• Require more tuning than for P
• But, no offset

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Proportional Integral Derivative (PID)

• Improvement over PI because of faster response
  and less deviation from offset
• Increases rate of error correction as errors get
  larger
• But
• HVAC controlled devices are too slow responding
• Requires setting three different gains

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Ref Kreider and Rabl.Figure 12.5
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The control in HVAC system only PI
Proportional
Integral
value
Set point
Proportional affect the slope
Integral affect the shape after the first bump
Set point
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The Real World

• 50 of US buildings have control problems
• 90 tuning and optimization
• 10 faults
• 25 energy savings from correcting control
  problems
• Commissioning is critically important

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HVAC Control

• Example
• Dew point control (Relative Humidity control)

fresh air
filter
cooling coil
heating coil
filter
damper
fan
mixing
T RH sensors
Heat gains
Humidity generation
We should supply air with lower humidity ratio
(w) and lower temperature
We either measure Dew Point directly or T RH
sensors substitute dew point sensor
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Relative humidity control by cooling coil
Cooling Coil
Mixture
Room
Supply
TDP
Heating coil
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Relative humidity control by cooling coil (CC)

• Cooling coil is controlled by TDP set-point
• if TDP measured gt TDP set-point ? send the
  signal to open more the CC valve
• if TDP measured lt TDP set-point ? send the
  signal to close more the CC valve

• Heating coil is controlled by Tair set-point
• if Tair lt Tair set-point ? send the signal
  to open more the heating coil valve
• if Tair gt Tair set-point ? send the signal
  to close more the heating coil valve

Control valves
Fresh air
mixing
cooling coil
heating coil
Tair TDP sensors
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Sequence of operation(PRC research facility)
Control logic Mixture in zone 1 IF ((
TMltTSP) (DPTMltDPTSP) ) heating and
humidifying Heater control IF (TSPgtTSA)
increase heating or IF (TSPltTSA) decrease
heating Humidifier IF (DPTSPgtDPTSA) increase
humidifying or IF (DPTSPltDPTSA) decrease humid.
Mixture in zone 2 IF ((TMgtTSP)
(DPTMltDPTSP) ) cooling and humidifying Cool.
coil cont. IF (TSPltTSA) increase cooling or IF
(TSPgtTSA) decrease cooling Humidifier IF
(DPTSPgtDPTSA) increase humidifying or IF
(DPTSPltDPTSA) decrease hum. Mixture in zone 3
IF ((DPTMgtDPTSP) ) cooling/dehumidifying and
reheatin Cool. coil cont. IF (DPTSPgtDPTSA)
increase cooling or IF (DPTSPltDPTSA) decrease
cooling Heater control IF (TSPgtTSA) increase
heating or IF (TSPltTSA) decrease heating