Design and Optimization of Thermal Systems Episode 1 Part 8 doc

Modeling of Thermal Systems 147

a set of simultaneous ordinary differential equations arise. For instance, the tem￾peratures T1, T2, T3,z￾, Tn of n components of a given system are governed by a

system of equations of the form

dT

d FTT T T,

dT

d F T T ,T

n

1

2

T

T

T

1123

212 3

(, ,, , )

( ,

, , )

(, , , , ) 123

"

T ,

dT

d F T T T T,

n

n

n n

T

T  T

(3.15)

where the F’s are functions of the temperatures and thus couple the equations.

These equations can be solved numerically to yield the temperatures of the vari￾ous components as functions of time T (see Example 2.6).

Partial differential equations are obtained for distributed models. Thus, Equa￾tion (3.4) is the applicable energy equation for three-dimensional, steady conduc￾tion in a material with constant properties. Similarly, one-dimensional transient

conduction in a wall, which is large in the other two dimensions, is governed by

the equation

R

T

C T

x

k T

x

t

t  t

t

t

t

¤

¦

¥

³

µ

´ (3.16)

if the material properties are taken as variable. For constant properties, the equa￾tion becomes

t

t  t

t

T T

T x

A 2

2

(3.17)

where A  k/RC is the thermal diffusivity. Similarly, equations for two- and three￾dimensional cases may be written. For convective transport, the energy equation

is written for a two-dimensional, constant property, transient problem, with neg￾ligible viscous dissipation and pressure work, as

R

T

C T

u

T

x

v

T

y

k T

x

T

y p

t

t 

t

t 

t

t

¤

¦

¥

³

µ

´  t

t 

t

t

¤

¦

¥

2

2

2

2

³

µ

´

(3.18)

where Cp is the specific heat of the fluid at constant pressure and u and v are the

velocity components in the x and y directions, respectively. Partial differential

148 Design and Optimization of Thermal Systems

equations that govern most practical thermal systems are amenable to a solution

by analytical methods in very few cases and numerical methods are generally

necessary. Finite-difference and finite-element methods are the most commonly

employed techniques for partial differential equations. Ordinary differential

equations can often be solved analytically, particularly if the equation is linear.

The integral formulation is based on an integral statement of the conservation

laws and may be applied to a small finite region, from which the finite-element

and finite volume methods are derived, or to the entire domain. For instance,

conduction in a given region is governed by the integral equation

R

T

C T dV k T

n p dS q dV VS V

t

t  t

t  ``` ¯¯ ¯ (3.19)

where V is the volume of the region, A is its surface area, q``` is an energy source

per unit volume in the region, and n is the outward drawn normal to the surface.

This integral equation states that the rate of net energy generated in the region

plus the rate of net heat conducted in the region at the surfaces equals the rate of

increase in stored thermal energy in the region. Similar equations may be derived

for convection in a given domain. Radiative transport often leads to integral

equations because energy is absorbed over the volume of a participating fluid or

material. In addition, the total radiative transport, in general, involves integrals

over the area, wavelength, and solid angle. Figure 3.13 shows a few examples of

integral and differential formulations for the mathematical modeling of thermal

systems.

Further Simplification of Governing Equations

After the governing equations are assembled, along with the relevant boundary

conditions, employing the various approximations and idealizations outlined

here, further simplification can sometimes be obtained by a consideration of the

various terms in the equations to determine if any of them are negligible. This

is generally based on a nondimensionalization of the governing equations and

evaluation of the governing parameters, as given earlier in Equation (3.6). For

instance, cooling of an infinite heated rod moving continuously at speed U along

the axial direction x [Figure 1.10(d) and Figure 3.6(a)] is governed by the dimen￾sionless equation

t

t `

t

t   Q

T

Q Pe Q

X

2 (3.20)

where ￾2 is the Laplacian operator in cylindrical coordinates, nondimensional￾ized by the rod diameter D, and the Peclet number Pe is given by Pe  UD/A.

The dimensionless temperature Q is defined as Q  T/Tref, where the reference

temperature Tref may be the temperature at x  0. Also, dimensionless time T` is