Design and Optimization of Thermal Systems Episode 2 Part 3 pdf

272 Design and Optimization of Thermal Systems

where the increments ($R)i

and ($P)i

are calculated from the equations

t

t

¤

¦

¥

³

µ

´ 

t

t

¤

¦

¥

³

µ

´  F

R

R F

P

P FR P

i

i

i

i ii ( ) ( ) (,) $ $

t

t

¤

¦

¥

³

µ

´ 

t

t

¤

¦

¥

³

µ

´  G

R

R G

P

P GR P

i

i

i

i ii ( ) ( ) (, $ $ )

The four partial derivatives in the above equations are calculated for the R and

P values at the ith iteration, using analytical differentiation of the functions F and

G. The iterative process is continued until a convergence criterion of the form

F2  G2 a E is satisfied. Figure 4.32 shows the computer output for E  10–4 and

starting values of 2 and 100 for R and P, respectively. The results obtained are very

close to those obtained earlier by the successive substitution method. The program

is simpler to write for the successive substitution method. However, the Newton￾Raphson method converges at a faster rate, due to its second-order convergence.

It usually converges if the initial guessed values are not too far from the solution.

Nevertheless, if divergence occurs, the initial guessed values may be varied and

iteration repeated until convergence is achieved.

4.5.2 DYNAMIC SIMULATION OF LUMPED SYSTEMS

Dynamic simulation of thermal systems is used for studying the system character￾istics at start-up and shutdown, for investigating the system response to changes

in operating conditions, and for design and evaluation of a control scheme. We are

interested in ensuring that the system does not go beyond acceptable limits under

such transient conditions. For instance, at start-up, the cooling system of a furnace

may not be completely operational, resulting in temperature rise beyond safe lev￾els. This consideration is particularly important for electronic systems since their

performance is very sensitive to the operating temperature [see Figure 3.6(b)].

Similarly, at shutdown of a nuclear reactor, the heat removal subsystems must

remain effective until the temperature levels are sufficiently low. In many cases,

sudden fluctuations in the operating conditions occur due to, for instance, power

   

   

   

  



   

  



  

  

   

 





 



!   

! "  



  

FIGURE 4.32 The results by the Newton-Raphson method for Example 4.7.

Numerical Modeling and Simulation 273

surge, increase in thermal load, change in environmental conditions, change in

material flow, etc., and it is important to determine if the system exceeds safety

limits under these conditions.

Analytical Solution

If the various parts of the system can be treated as lumped, the resulting equations

are coupled ODEs. Modeling of a component as lumped was discussed in Chapter

3 and the resulting energy equations, such as Equation (3.7), Equation (3.10), and

Equation (3.11), were given. For a lumped body governed by the equation

R

T

CV dT

d qA hA T Ta  ( ) (4.41)

the temperature T(T) is given by

Q Q

R

  T ¤

¦

¥

³

µ

´

¤

¦

¥

³

µ

´ T T q

h

q

h

hA

CV a o exp (4.42)

where the symbols are the same as those employed for Equation (3.7) through

Equation (3.10). In the analytical solution given by Equation (4.42), the steady￾state temperature is q/h, obtained for time T l ∞. The initial temperature at T  0

is To, represented by Qo  To – Ta. This solution gives the basic characteristics of

many dynamic simulation results in which the steady-state behavior is achieved

at large time. If q  0, the convective transport case of Equation (3.7) is obtained,

with Equation (3.9) as the solution. The quantity RCV/hA is the response time in

that case, as given earlier in Equation (3.1). If convective heat loss is absent, only

qA is left on the right-hand side of Equation (4.41) and the solution is T – To 

(qA/RCV)T, indicating a linear increase if the heat input q is held constant.

The simulation of a system involves a set of ODEs, rather than a single ODE.

These equations may be linear or nonlinear. Most nonlinear equations, such as

Equation (3.11), require a numerical solution. Even with linear equations, the pres￾ence of several coupled ODEs makes it difficult to obtain an analytical solution.

As an example, let us consider two lumped bodies, denoted by subscripts 1 and 2,

exchanging energy through convection. The governing equations are

( ) R

T

CV dT

d

hA T T 1

1

1  12 1 ( ) (4.43a)

( ) R

T

CV dT

d

hA T T hA T T 2 a

2

1  2 2 2 11 2 ()() (4.43b)