Design and Optimization of Thermal Systems Episode 1 Part 9 pdf

172 Design and Optimization of Thermal Systems

local temperature. Here, ￾2  t2/tx2  t2/ty2. For the solid region, the energy equa￾tion is

( ) R

T

C T k T s s

t

t  2

where the subscript s denotes solid material properties. The Boussinesq approxi￾mations have been used for the buoyancy term. The pressure work and viscous

dissipation terms have been neglected. The boundary conditions on velocity are the

no-slip conditions, i.e., zero velocity at the solid boundaries. At the inlet and outlet,

the given velocities apply.

For the temperature field, at the inner surface of the enclosure, continuity of the

temperature and the heat flux gives

T T T

n k T

n s s

s

 t

t

¤

¦

¥

³

µ

´  t

t

¤

¦

¥

³

µ

´ and

where n is the coordinate normal to the surface. Also, at the left source, an energy

balance gives

QL k T

y

k T

y ss s  t

t 

t

t

¤

¦

¥

³

µ

´

where Qs is the energy dissipated by the source per unit width. Similar equations

may be written for other sources. At the outer surface of the enclosure walls, the

convective heat loss condition gives

t

t k  T

n hT T s i ( )

At the inlet, the temperature is uniform at Ti

and at the outlet developed tempera￾ture conditions, tT/ty  0, may be used.

Therefore, the governing equations and boundary conditions are written for this

coupled conduction-convection problem. The main characteristic quantities in the

problem are the conditions at the inlet and the energy input at the sources. The

energy input governs the heat transfer processes and the inlet conditions deter￾mine the forced airflow in the enclosure. Therefore, vi

, Hi

, Ti

, and Qs are taken as

the characteristic physical quantities. The various dimensions in the problem are

nondimensionalized by Hi

and the velocity V by vi

. Time T is nondimensionalized

by Hi

/vi

to give dimensionless time T`  T(vi

/Hi

). The nondimensional temperature

Q is defined as

Q   T T

T

T Q

k

i s

$ , where $

Here $T is taken as the temperature scale based on the energy input by a given

source. The energy input by other sources may be nondimensionalized by Qs

Modeling of Thermal Systems 173

The governing equations and the boundary conditions may now be nondimen￾sionalized to obtain the important dimensionless parameters in the problem. The

dimensionless equations for the convective flow are obtained as

• 

t

t  •     ¤

¦

¥

³

µ

´ 

V

V V V Gr

Re

0

2 T

( ) p e Q 1

1

2

2

Re

V

V

RePr

( )

.( ) ( )



t

t   

Q

T

Q Q

The dimensionless energy equation for the solid is

t

t  ¤

¦

¥

³

µ

´

¤

¦

¥

³

µ

´ 

Q

T

A

A

Q s 1 2

RePr ( )

where the asterisk denotes dimensionless quantities. The dimensionless pressure

p pvi *  R/ 2 and A is the thermal diffusivity. Therefore, the dimensionless parame￾ters that arise are the Reynolds number Re, the Grashof number Gr, and the Prandtl

number Pr, where these are defined as

Re Pr   vH g H T ii i

N

B

N

N

A

Gr = 3

2

$

In addition, the ratio of the thermal diffusivities As/A arises as a parameter. Here,

NM/R is the kinematic viscosity of the fluid. The Reynolds number determines

the characteristics of the flow, particularly whether it is laminar or turbulent, the

Grashof number determines the importance of buoyancy effects, and the Prandtl

number gives the effect of momentum diffusion as compared to thermal diffusion

and is fixed for a given fluid at a particular temperature.

Additional parameters arise from the boundary conditions. The conditions at

the inner and outer surfaces of the walls yield, respectively,

t

t

¤

¦

¥

³

µ

´  ¤

¦

¥

³

µ

´

t

t

¤

¦

¥

³

µ

´

Q Q t

n

k

k n

s

* *

fluid solid

Q Q t 

¤

¦

¥

³

µ

´ n

hH

k

i

s *

Therefore, the ratio of the thermal conductivities ks/k and the Biot number Bi 

hHi

/ks arise as parameters. A perfectly insulated condition at the outer surface is

achieved for Bi  0. In addition to these, several geometry parameters arise from

the dimensions of the enclosure (see Figure 3.20), such as di

/Hi

, Ho/Hi

, Ls/Hi

, etc.

Heat inputs at different sources lead to parameters such as (Qs

)2/Qs, (Qs

)3/Qs, etc.,

where (Qs

)2 and (Qs

)3 are the heat inputs by different electronic components.

The above considerations yield the dimensionless equations and boundary

conditions, along with all the dimensionless parameters that govern the thermal

transport process. Clearly, a large number of parameters are obtained. However,

if the geometry, fluid, and heat inputs at the sources are fixed, the main governing

parameters are Re, Gr, Bi, and the material property ratios As/A and ks/k. These