FIGURE 1.6  Pictorial depiction of the plane wall problem.

Example 1.1 Consider a vertical plate 1 m in height, and 1 m in the direction perpendicular to the plane of the paper and 5 mm thick. The plate is at 100 °C and the surroundings at 30 °C. Free convection from both sides of the plate gives a h value of 6 W/m2 K. The emissivity of the plate is 0.9. Calculate (1) convection heat transfer from both the sides, (2) radiation heat transfer from both the sides, and (3) total heat transfer and the fraction of heat transferred by radiation.

Solution:

A pictorial depiction of the given problem is shown in Fig. 1.6

1. The convection heat transfer is given by

=1115.3 W

3.The total heat transfer from the surface is given by

The fraction of heat transferred by radiation is

= 11151955..33 = 57%

12 CHAPTER 1  Introduction

1.7  Combined modes of heat transfer

There are many situations where more than one mode of heat transfer is involved. The ubiquitous problem of condensation of the working medium in a refrigerator or an air conditioner involves combined surface radiation and convection. Exercise problem 1.1, which concerns heat transfer in a filament bulb, is a classic example of multimode heat transfer. The challenge in these problems is that the resultant heat transfer is not additive. The individual modes of heat transfer often interact, making the resulting problem quite formidable to solve.

1.8  Phase-change heat transfer

Heat transfer processes can also occur isothermally in principle when phase change takes place. This can be solid-liquid or liquid-solid or liquid-vapor or vapor-liquid change of phase. Boiling and condensation are two-phase-change processes widely seen in nature, as well as in several engineering applications. The hydrological processes in the Earth’s atmosphere involve evaporation of water and condensation back into rainfall. Water is converted into steam in a boiler of a thermal power plant, and this is converted to useful mechanical work in a turbine. The expanded steam is then condensed in a condenser to complete the cycle.

Phase change heat transfer can also occur in conduction problems involving melting and solidification. A classic example is a heat sink with phase change material used for the cooling of electronics (Baby and Balaji, 2019). The phase change material in the heat sink absorbs the heat isothermally at its melting temperature and undergoes a phase change (solid to liquid). As a consequence, the electronic device is maintained at a particular temperature. Another widely used device involving a phase change heat transfer in convection, for application of electronic cooling, is a heat pipe. In general, a heat pipe is a hollow tube that contains an evaporator section, an adiabatic section, and a condenser section and is filled with the working fluid at saturation temperature and corresponding saturation pressure. The working fluid at the evaporator section takes the heat, is converted into vapor and travels to the condenser section. At the condenser section, the vapor releases heat and is condensed into a liquid and travels back to the evaporator through gravity or a wick (in the case of a wicked heat pipe).

1.9  Concept of continuum

Much of the heat transfer theory that we know and are going to study through this book critically hinges on a key assumption known as the “continuum hypothesis.” According to this, we model the behavior of materials or substances or media based on the assumption that they are a continuous mass rather than discrete particles. This assumption is fine as long as the particle density is high enough so that even though particles enter or leave a system boundary, the number of particles inside the system may be deemed to be constant. In convection, the continuum hypothesis breaks down

in highly rarefied flows. In radiation, for example, the continuum breaks down in the atmosphere beyond 50 km from the mean sea level.

Problems

1.1.Consider the heat transfer from an evacuated filament bulb. List all the pertinent heat transfer processes associated with the bulb.

1.2.A widely used method to determine the thermal conductivity of solid material is to make a thin sheet out of it, thereby reducing the heat transfer across the material to be one-dimensional. A smart way of measuring the thermal conductivity would be to use a foil heater and sandwich it between two identical samples of the material, as shown in Fig. 1.7. The four sides (left, right, front, and back) are insulated to reduce heat losses.

FIGURE 1.7  Foil heater and sample arrangement.

Consider an experiment where two 6 mm plates of the material whose thermal conductivity is to be estimated are used. The sides of the plates are 20 cm by 20 cm. The heater is energized, and temperatures T1 ,T2 ,T3 , and T4 are recorded. After some time, steady-state is reached and the following temperatures are recorded (see Table 1.3).

Assuming that the thermal conductivity of the material is invariant with respect to temperature, determine the thermal conductivity of the material when the electric power used is 40 W.

1.3.Consider a typical freezer compartment that is 1.65 m high, 0.75 m wide, and 0.75 m deep. The freezer is insulated with the help of polyurethane foam with a thermal conductivity of 0.028 W/m/K. Consider insulation with this material with a thickness of 200 mm all around the freezer except on the bottom side. If the outside and inside of the insulation are at 40 and −12 °C, what is the heat leak into the freezer?

Table 1.3  Recorded temperature data for problem 1.2.

14CHAPTER 1  Introduction

1.4.Consider a plane wall whose one side is maintained at a temperature of 150 °C. The other side is exposed to free convection with a heat transfer coefficient of

6 W/m2.K, and T∞ is equal to 30 °C. The thermal conductivity of the wall is 1.7 W/m.K, and the thickness of the wall is 50 cm. Determine the heat transfer through the wall in W/m2. Also determine the temperature at the side of the wall exposed to the convection environment. When will this temperature approach the free stream temperature? When will it approach 150 °C?

1.5.In a coal-fired steam power plant, hot steam at 540 °C, 160 bar is transported from a boiler to the turbine in a pipe. The internal diameter of the pipe is 400 mm, and the outer diameter of the pipe is 410 mm and is made of stainless steel (SS-304) with k = 15 W/m.K.

a. Determine the heat loss per meter length of the pipe if natural convection

exists on the outside with h = 6 W/m2.K and T∞ = 40 °C if the pipe is assumed to be at 540 °C.

b. If the conduction inside the pipe is considered, what is the approach to determine the temperature at the surface of the pipe. What would be an engineering solution to say bring down the outer surface temperature to 55 °C or less?

1.6.The solar flux in W/m2 falling on the Earth has a value of 1368 W/m2. Assuming that the Earth absorbs 70% of this and is in radiative equilibrium. Determine the equivalent black body temperature of the earth if the radius of the earth is 6370 km.

1.7.A 1-m-long mild steel plate has a thickness of 5 mm and is 250 mm wide. It is suspended vertically in still air and is energized by a foil heater with a uniform power of Q Watts. The emissivity of the plate is 0.85, and the heat transfer coefficient associated with natural convection is 6 W/m2.K. The ambient temperature is 30 °C, and this can also be considered to be the temperature of the surroundings for radiative heat transfer. The plate loses heat by convection and radiation from both the sides.

a. Write down the equation governing for the variation of temperature with time of the plate, assuming the whole plate to be spatially isothermal.

b. Determine the value of Q, for which the plate reaches a steady-state temperature of 80 °C.

c. If the thermal conductivity of the plate is 45 W/m.K, and its density and specific heat capacity are 7850 kg/m3 and 500 J/kg.K respectively, what is the cooling rate of the plate when the power is switched off after the steadystate is reached?

d. If you neglect radiation, what will be the error in your estimate of the cooling rate?

References

Baby, R., Balaji, C., 2019. Thermal Management of Electronics, Volume I: Phase Change Material-Based Composite Heat Sinks - An Experimental Approach, vol.1 Momentum Press, pp. 1–165.

Incropera, F.P., Lavine, A.S., Bergman, T.L., DeWitt, D.P., 2013. Principles of Heat and Mass Transfer, seventh ed. Wiley, pp. 1–1076.