Design and Optimization of Thermal Systems Episode 2 Part 3 pdf

4.5.2 D YNAMIC S IMULATION OF L UMPED S YSTEMS Dynamic simulation of thermal systems is used for studying the system istics at start-up and shutdown, for investigating the system respons

where the increments ($R)i and ($P)i are calculated from the equations

t t

¤

¦¥

³ µ´

tt t

¤

¦¥

³ µ´

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 Raphson method converges at a faster rate, due to its second-order convergence

Newton-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 D YNAMIC S IMULATION OF L UMPED S YSTEMS

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

character-in operatcharacter-ing 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

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.

¤

¦¥

³µ´

h

q h

hA CV

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 T o, represented by Qo  T o – T a 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 – T o

(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

1

where T a is the temperature of the ambient with which body 2 exchanges energy

by convection The convective heat transfer coefficients h1 and h2 refer to the

inner and outer surfaces, as shown in Figure 4.33 The initial temperature is T o

at time T  0 Employing Q  T – Ta, these equations may be written, with Q1

QQ

2 1

Ambient medium,

T a

h2

h1

FIGURE 4.33 System consisting of two lumped bodies exchanging energy by convection

with each other and with the ambient medium.

ment The gradient dQ1/dT is zero at T  0 because Q2 Q1 Qo at the beginning

of the process Three first-order ODEs arise if three lumped bodies in energy exchange with each other are considered, four equations for four bodies, and so

on Analytical solutions may be obtained as given here or by using other cal techniques such as the Laplace transform method, for a few idealized cases, particularly if the equations are linear

analyti-Numerical Solution

If the ordinary differential equations are nonlinear, analytical solutions are erally not possible and numerical methods must be employed for the simulation The use of Runge-Kutta and predictor-corrector methods to solve a single ODE was discussed earlier For solving a system of equations, such as that given by Equation (4.44), let us consider two simultaneous first-order equations for depen-

gen-dent variables y and z:

values of y and z at the next step This procedure may be extended to a system of

three or more first-order differential equations, and thus also to higher-order tions All the conditions, in terms of the dependent variables and their derivatives, must be known at the starting point to use this method Therefore, the scheme,

equa-as given here, applies to initial-value problems If conditions at a different time must also be satisfied, a boundary-value problem arises and the shooting method, which employs a correction scheme to satisfy the boundary conditions, may be employed (Jalluria, 1996)

Finite difference methods may also be applied to solve a system of ODEs Algebraic equations are generated for each ODE by the finite difference approxi-mation and the combined set of equations is solved by the methods outlined ear-lier to obtain the desired simulation of the system Considering, again, Equation (4.46), we may write the finite difference equations as

where time T  i$T, subscripts i and i 1 represent values at time T and T

respectively, and the functions F and G may be linear or nonlinear Therefore,

values at T

formulation, which is particularly useful for nonlinear equations However, F and

ticularly at Tearlier for PDEs This is the implicit formulation that gives rise to a set of simul-taneous algebraic equations, linear ODEs generating linear algebraic equations, and nonlinear ODEs generating nonlinear ones Other, more accurate, finite-difference formulations are obviously possible for Equation (4.49) This set of equations is then solved to simulate the thermal system Higher-order equations arise in some cases, particularly in the analysis of dynamic stability of systems These may similarly be simulated using the finite difference method

Dynamic simulation is particularly valuable in areas such as materials cessing, which inevitably involve variations with time Lumping is commonly used in thermodynamic systems, such as energy conversion and refrigeration sys-tems, and the simulation outlined here helps in ensuring that the system behavior and performance are satisfactory under time-varying conditions The dynamic simulation of large systems such as power and steel plants is particularly impor-tant because of changes in demand and in the inputs to the systems Some of these aspects are considered in detail, employing examples, in Chapter 5 Let us consider a typical manufacturing system to illustrate these ideas.

pro-Example 4.8

Numerically simulate the casting of a metal plate of thickness L 0.2 m in a mold of

wall thickness W 0.05 m, assuming one-dimensional solidification, no energy age in the solid formed, uniform temperature in the mold, and initial liquid tempera-

stor-ture at the melting point T m 1200 K A convective loss at heat transfer coefficient

h 20 W/(m 2 ·K) occurs at the outer surface of the mold on both sides of the plate to

an ambient at temperature T a 20nC Find the total time needed for casting

Deter-mine the effect of varying h, using values of 10 and 40 W/(m2·K), and of varying W,

using values of 0.02 and 0.1 m Take density, specific heat, and thermal conductivity

of the cast material as 9000 kg/m 3 , 400 J/kg·K, and 50 W/m·K, respectively The corresponding values for the mold are 8000, 500, and 200, respectively The latent heat of fusion is 80 kJ/kg.

Solution

The problem concerns solidification of a molten material in an enclosed region,

as shown in Figure 1.3 However, a very simple, one-dimensional mathematical model is used, as sketched in Figure 4.35 The liquid is at the melting temperature

T m, a linear temperature distribution exists in the solid since energy storage in it is

neglected, and the mold is at uniform temperature T c(T), where T is time The erning equations are obtained from energy balance as

where the subscript c refers to the mold and s refers to the solid, D is the thickness

of the solid formed, and L f is the latent heat of fusion The first equation gives the energy balance for the mold, which gains energy from the solid and loses to the am- bient The second equation balances the energy removed by conduction in the solid

to the latent heat for phase change.

Therefore, two coupled ODEs are obtained for this dynamic problem, one for T c

and the other for D The material property values are substituted in the equations, which are then rewritten in the form of Equation (4.46) as

indepen-Figure 4.36 and indepen-Figure 4.37, in terms of the variation of T c and D with time Casting

is complete when D  0.1 m because heat removal occurs on both sides of the plate

It is found that a variation in the heat transfer coefficient has no significant effect

on the temperature or the solidification rate, over the range considered However,

the mold thickness W is an important design variable and substantially affects the

solidification time and the temperature of the mold From these results, the casting

time at h 20 W/(m 2·K) and W 0.05 m is 110 s A thicker mold removes energy faster and thus reduces the casting time Therefore, this example illustrates the use of dynamic simulation, which is particularly important for manufacturing pro- cesses, and for modeling the time-dependent behavior of the system.

4.5.3 D ISTRIBUTED S YSTEMS

In the preceding sections, we considered the relatively simple circumstances in which lumping may be employed for modeling the different parts of a given sys-tem This approximation leads to algebraic equations in the steady-state case and

to ODEs in the time-dependent or dynamic simulation case Even though the assumption of lumping or uniform conditions in each system part has been and still is very widely used because of the resulting simplicity, the easy availability

of powerful computers and versatile software has made it quite convenient to model and simulate the more general distributed circumstance in which the quan-tities vary with location and time Of course, if the lumped model is appropriate

for a given problem because of, say, the very low Biot number involved, there is

no reason to complicate the analysis and the results by using a distributed model However, there are many problems of practical interest in which large variations occur over the domain and the lumped approximation cannot be used Tempera-ture variation in the wall and in the insulation of a furnace is an example of this circumstance Similarly, the velocity and temperature fields in an electronic system, in the cylinder of an internal combustion engine, in the combustor of a

100 90 80 70 60 Time,
(s) (a)

Heat transfer coefficient = 10

= 20 = 40 W/m 2 K

Heat transfer coefficient = 10

= 20 = 40 W/m 2 K

50 40 30 20 10 300

gas turbine, and in the molten plastic in an injection mold are strong functions of location and time, making it essential to simulate these as distributed, dynamic systems for accurate results.

The governing equations for distributed systems are PDEs, which are quently nonlinear due to material property changes, coupling with fluid flow and the presence of radiative transport Several types of simple, linear PDEs, along

200 150

100 50

Time, (s) (b)

FIGURE 4.37 Calculated mold temperature T c and solid region thickness D as tions of time T for different values of the mold wall thickness W, at h  20 W/(m 2 ·K), in Example 4.8.

func-with the corresponding solution procedures, were discussed in Section 4.2.4 Finite difference, finite element, and other approaches to obtain simultaneous algebraic equations from the governing PDEs and to solve these were outlined Again, nonlinear PDEs lead to nonlinear algebraic equations and linear PDEs to linear algebraic equations Once the set of algebraic equations is derived, the solu-tion is obtained by the various methods for linear and nonlinear equations given earlier Nonlinear equations are often linearized, as discussed below, so that new values may be calculated using the known values from previous time steps or iterations In addition, commercially available software such as Fluent and Ansys

is generally employed in industry to simulate practical thermal systems

t

tt

tt

where the material properties are functions of temperature T If these are taken as

constant, the linear equation given by Equation (4.27) is obtained Then this tion may be solved conveniently by explicit or implicit finite difference methods

equa-if the geometry and boundary conditions are relatively simple For complicated domains and boundary conditions, finite element or boundary element methods may be used, as discussed earlier

If the properties are taken as variable due to material characteristics or perature range involved, the governing equation is nonlinear because the terms

tem-are nonlinear in T For instance, if the term on the right-hand side of Equation

k T x

T x

tt

tt

2

(4.51)

indicating the nonlinearity that exists in the equation There are several methods

of simulating systems in which such nonlinear equations arise In an iterative or time-marching process, the terms are commonly linearized by approximating the

coefficients, such as k(T) and ∂k(T)/∂x in the preceding equation, which cause the

nonlinearity in the terms, by the following three approaches:

1 Using the values of the coefficients from the previous iteration or time step

2 Using extrapolation to obtain an approximation of these coefficients

3 Starting with values at the previous time step and then iterating at the present time step to improve the approximation

If extrapolation is used, the value of k at the (n 1)th time or iteration step may be approximated as

4.5.4 S IMULATION OF L ARGE S YSTEMS

All the aspects considered in this chapter can easily be extended to large thermal systems that involve relatively large sets of algebraic and differential equations Such systems may range from a blast furnace for steel to an entire steel plant, from a cooling tower to a power plant, from the cooling system of a rocket to the entire rocket, and so on Though many of the examples considered here involved relatively small sets of equations for simplicity and convenience, the basic ideas presented here are equally applicable to large and more complicated systems.The two main features that distinguish large thermal systems from simpler ones are the presence of a large number of parts that leads to large sets of gov-erning equations and relatively independent subsystems that make up the overall system These aspects are treated by

1 Development of efficient approaches for solving large sets of equations

2 Better techniques for storing the relevant data

3 Subdivision of the system into subsystems that may be treated dently and then merged to obtain the simulation of the full system

indepen-All these considerations have been discussed earlier in this chapter and need not be repeated Methods such as Gauss-Seidel are particularly useful for handling large sets of algebraic equations while keeping the computer storage requirements small Similarly, modularization of the simulation process has been stressed at several places because this allows building up of the system simulation package while ensuring that each subsystem is treated satisfactorily Chapter 5 presents the overall design process for such large systems

Several specialized computer languages have been developed for the lation of engineering systems These are usually designed for certain types of systems and, as such, are more convenient to use than a general-purpose language such as C or Fortran Many of these simulation languages are particularly suited for manufacturing systems The general-purpose simulation system (GPSS) is

simu-a simulsimu-ation lsimu-angusimu-age suited for scheduling simu-and inventory control simu-applicsimu-ations dealing with different steps in a process Other languages that may be mentioned are SIMAN, SIMSCIPT, MAST, and MAP Each of these is particularly oriented

to a specific application, making it easier to enter the relevant data for simulation and to obtain the desired outputs for design, operation, and control of the system Other computational environments, such as those provided by specialized soft-ware for computer simulation and design (e.g., such as MATLAB MATHCAD, Maple, and other CAD programs) are also useful

4.5.5 N UMERICAL S IMULATION V ERSUS R EAL S YSTEM

It would be worthwhile to conclude the discussion on system simulation by ing the most important element, namely, that the simulation must accurately and closely predict the behavior of the actual system A satisfactory simulation of a system is achieved when the response of the simulated system to variations in operating conditions and to changes in the design hardware follows the expected physical trends and is a faithful representation of the given system Unfortu-nately, the real system is rarely available to check the predictions of the simula-tion because one of the main uses of simulation is to study system behavior for

stress-a vstress-ariety of designs without stress-actustress-ally fstress-abricstress-ating the system for these different designs Therefore, other methods must generally be employed to validate the models and to ensure that accurate predictions of system behavior are obtained from the simulation

As discussed in this and preceding chapters, the development of system lation involves several steps These include mathematical modeling, which gener-ally also contains the correlating equations representing the results from physical modeling, material property data, and component characteristics; numerical solu-tion to the governing equations; numerical modeling of different system parts; merging of separate models to yield an overall model for the system; variation of operating conditions to consider design and off-design conditions; and investiga-tion of system behavior for different design parameters Therefore, the validation

simu-of the simulation can be based on the validation simu-of these ingredients that lead

to overall system simulation and on any available results obtained from similar existing systems Finally, when a prototype is developed and fabricated based on the design obtained, the experimental results from the prototype can be employed

to provide a valuable check on the accuracy of the predictions

In conclusion, a validation of the numerical simulation of the system is carried out

to confirm a close representation of the real system by considering the following:

1 Validation of mathematical model

2 Validation of numerical schemes

3 Validation of the numerical models for system parts

4 Physical behavior of the simulated system

5 Comparison of results from simulation of simpler systems with able analytical and experimental results