1. Lecture Objectives: Discuss HW4 Continue with advance air systems
Nonlinear equation solvers
Continue with advance air systems
Introduce control systems

2. HW4 issues Nonlinear equation solvers

3. 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
Explicit form:
X=Y/2-1
Y=X2+3
We know the solution:
By substitution method we get:
X2-2x+1=0 → X=1, Y=4

4. Successive substitution method
Simple Example:
Explicit form:
X=Y/2-1
Y=X2+3
Initial guess ?
End of iteration ?
X=Y/2-1
Y=X2+3
X=…
Y=…
yes
Y=2 no
Solution: X Y
initial guess 2
iteration 1 3
iteration 2
0.5
3.25
iteration 3
0.625
iteration 4
iteration 5
---
----
Iteration 98
Solution for
One time step

5. Successive substitution method
When to stop with iterations?
n-1 n RY
We DO NOT know the exact solution!
Residual:
Difference in result between two iterations RY=|Y(n)-Y(n-1)|<e ,
e is defined by requested accuracy
For example: eY=0.0004
Iteration Y(99) =
Iteration Y(100)= |Y(100)-Y(99)|= <eY
stop the iterations

6. 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:
Y*=f·Y(n)+(1-f)·Y(n-1) Y – considered parameter , n –iteration , f – relaxation factor
For our example Y*in iteration 101=f·Y(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!

7. 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= → X-Y/2+1=0
X2-Y= → X2-Y+3=0

8. Newton-Raphson method (this is used in most equation solvers)
Section 6.11 of handouts
Our simple example:
f1 = X-Y/2+1=0
f2 = X2-Y+3=0
Steps:
0) Find derivatives d(f1)/dX = , 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))=2-2/2+1=2
f2(Y(0),X(0))=22-2+3=5
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
Unknowns
(correction Dxi)
Jacobean matrix
Function values
for guessed
variables

9. 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

10. Desiccant Enhanced HVAC

11. Liquid Desiccant System

12. Liquid Desiccant System

13. Control

14. The PID control algorithm
constants
time
e(t) – difference between
set point and
measured value
Position (x)
Proportional
Integral
Differential
For our example of heating coil:
Differential
(how fast)
Proportional
(how much)
Integral
(for how long)
Position of the valve

15. Proportional Controllers
x is controller output
A is controller output with no error
(often A=0)
Kis proportional gain constant
e = is error (offset)

16. Unstable system
Stable system

17. 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

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

20. Issues with PI Controllers
Scheduling issues
Require more tuning than for P
But, no offset

21. 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

22. Ref: Kreider and Rabl.Figure 12.5

23. The control in HVAC system – only PI
Proportional
Integral
value
Set point
Proportional
affect the slope
Set point
Integral
affect the shape after
the first “bump”

24. 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

25. HVAC Control Example : Dew point control (Relative Humidity control)
fresh
air
damper
filter
cooling
coil
heating
coil
filter
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

26. Relative humidity control by cooling coil
Mixture
Room
Supply
TDP
Heating coil

27. Relative humidity control by cooling coil (CC)
Cooling coil is controlled by TDP set-point
if TDP measured > TDP set-point → send the signal to open more the CC valve
if TDP measured < TDP set-point → send the signal to close more the CC valve
Heating coil is controlled by Tair set-point
if Tair < Tair set-point → send the signal to open more the heating coil valve
if Tair > 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

28. Sequence of operation (PRC research facility)
Set Point (SP)
Mixture 2
Mixture 3
Mixture 1
DBTSP
DPTSP
Control logic:
Mixture in zone 1: IF (( TM<TSP) & (DPTM<DPTSP) ) heating and humidifying
Heater control: IF (TSP>TSA) increase heating or IF (TSP<TSA) decrease heating
Humidifier: IF (DPTSP>DPTSA) increase humidifying or IF (DPTSP<DPTSA) decrease humid.
Mixture in zone 2: IF ((TM>TSP) & (DPTM<DPTSP) ) cooling and humidifying
Cool. coil cont.: IF (TSP<TSA) increase cooling or IF (TSP>TSA) decrease cooling
Humidifier: IF (DPTSP>DPTSA) increase humidifying or IF (DPTSP<DPTSA) decrease hum.
Mixture in zone 3: IF ((DPTM>DPTSP) ) cooling/dehumidifying and reheatin
Cool. coil cont.: IF (DPTSP>DPTSA) increase cooling or IF (DPTSP<DPTSA) decrease cooling