Design and Optimization of Thermal Systems Episode 1 Part 10 pdf

Modeling of Thermal Systems 197

PROBLEMS

Note: In all problems dealing with model development, list the assumptions,

approximations, and idealizations employed; give the resulting governing equa￾tions; and, whenever possible, give the analytical solution. Symbols may be used

for the appropriate physical quantities.

3.1. An energy storage system consists of concentric cylinders, the inner

being of radius R1, the outer of radius R2 and both being of length L,

as shown in Figure P3.1. The inner cylinder is heated electrically and

supplies a constant heat flux q to the material in the outer cylinder, as

shown. The annulus is packed with high conductivity metal pieces.

Assuming that the system is well insulated from the environment and

that the annular region containing the metal pieces may be taken as

isothermal,

(a) Obtain a mathematical model for the system.

(b) If the maximum temperature is given as Tmax, obtain the time for

which heating may be allowed to occur, employing the usual

symbols for properties

3.2. Solid plastic cylinders of diameter 1 cm and length 30 cm are heat

treated by moving them at constant speed U through an electric oven

of length L, as shown in Figure P3.2. The temperature at the oven walls

is Ts and the air in the oven is at temperature Ta. The convective heat

transfer coefficient at the plastic surface is given as h and the surface

Metal

pieces

R2

R1

Insulation

q

FIGURE P3.1

FIGURE P3.2

L

U

Ts

h, Ta

198 Design and Optimization of Thermal Systems

emissivity as E. The cylinders are placed perpendicular to the direc￾tion of motion and are rotated as they move across the oven. Develop a

simple mathematical model for obtaining the temperature in the plastic

cylinders as a function of the temperatures Ts and Ta, h, L, and U, for

design of the system. Clearly indicate the assumptions and approxima￾tions made.

3.3. A chemical industry needs hot water at temperature Tc o $Tc for a

chemical process. For this purpose, a storage tank of volume V and

surface area A is employed. Whenever hot water is withdrawn from

the tank, cold water at temperature Ta, where Ta is the ambient tem￾perature, flows into the tank. A heater supplying energy at the rate of

Q turns on whenever the temperature reaches Tc – $Tc and turns off

when it is reaches Tc  $Tc. The heater is submerged in the water con￾tained in the tank. Assuming uniform temperature in the tank and a

convective loss to the environment at the surface, with a heat transfer

coefficient h, obtain a mathematical model for this system. Sketch the

expected temperature T of water in the tank as a function of time for

a given flow rate m of hot water and also for the case when there is no

outflow, m  0.

3.4. Consider a cylindrical rod of diameter D undergoing thermal process￾ing and moving at a speed U as shown in Figure P3.4. The rod may

be assumed to be infinite in the direction of motion. Energy transfer

occurs at the outer surface, with a constant heat flux input q and convec￾tive loss to the ambient at temperature Ta and heat transfer coefficient

h. Assuming one-dimensional, steady transport, obtain the governing

equation and the relevant boundary conditions. By nondimensional￾ization, determine the governing dimensionless parameters. Finally,

obtain T(x) for (a) h  0 and (b) q  0.

3.5. Give the governing equations and boundary conditions for the steady￾state, two-dimensional case for the preceding physical problem. Derive

the governing dimensionless parameters using the nondimensionaliza￾tion of the equations and the boundary conditions.

3.6. During the heat treatment of steel bolts, the bolts are placed on a con￾veyor belt that passes through a long furnace at speed U as shown in

Figure P3.6. In the first section, the bolts are heated at a constant heat

flux q. In the second and third sections, they lose energy by convection

D

x q h, Ta

T U

FIGURE P3.4