1
Modeling of Heat Transfer

This chapter presents a reminder of courses on heat transfer limited to what is necessary to understand and master the methods of measuring the thermal properties of materials which will be described in the rest of this book.

1.1. The different modes of heat transfer

1.1.1. Introduction and definitions

We will first define the main quantities involved in solving a heat transfer problem.

1.1.1.1. Temperature field

Energy transfers are determined from the evolution of the temperature in space and time: T = f (x, y, z, t). The instantaneous value of the temperature at any point of space is a scalar quantity called a temperature field. We will distinguish two cases:

• - time-independent temperature field: the regime is called steady state or stationary;
• - evolution of the temperature field over time: the regime is called variable, unsteady or transient.

1.1.1.2. Temperature gradient

If all the points of space which have the same temperature are combined, an isothermal surface is obtained. The temperature variation per unit length is maximal in the direction normal to the isothermal surface. This variation is characterized by the temperature gradient:

[1.1]

where: is the normal unit vector;

is the derivative of the temperature along the normal direction.

Figure 1.1. Isothermal surface and thermal gradient

1.1.1.3. Heat flux

Heat flows under the influence of a temperature gradient from high to low temperatures. The quantity of heat transmitted per unit time and per unit area of the isothermal surface is called the heat flux ? (W m-2):

[1.2]

where S is the surface area (m2).

f (W) is called the heat flow rate and is the quantity of heat transmitted to the surface S per unit time:

[1.3]

1.1.1.4. Energy balance

The determination of the temperature field involves the writing of one or more energy balances. First, a system (S) must be defined by its limits in space and the different heat flow rates that influence the state of the system must be established and they can be:

Figure 1.2. System and energy balance

The first principle of thermodynamics is then applied to establish the energy balance of the system (S):

[1.4]

After having replaced each of the terms by its expression as a function of the temperature, we obtain a differential equation whose resolution, taking into account the boundary conditions of the system, makes it possible to establish the temperature field. We will first give the possible expressions of the heat flow rates that can enter or exit a system by conduction, convection or radiation before giving an expression of the flux stored by sensible heat.

1.1.2. Conduction

Conduction is the transfer of heat within an opaque medium, without displacement of matter, under the influence of a temperature difference. The transfer of heat via conduction within a body takes place according to two distinct mechanisms: transmission via atomic or molecular vibrations and transmission via free electrons.

Figure 1.3. Conductive heat transfer scheme

The theory of conduction is based on the Fourier hypothesis: the heat flux is proportional to the temperature gradient:

[1.5]

The heat flow rate transmitted by conduction in the direction Ox can therefore be written in algebraic form:

[1.6] where: ? is the conductive heat flux (W m-2); f is the conductive heat flow rate (W); ? is the thermal conductivity of the medium (W m-1 K-1); x is the space variable in the heat flow's direction (m); S is the surface area of the passage of the heat flux (m2).

The values of the thermal conductivity ? of some of the most common materials are given in Table 1.1. A more complete table is given in Appendices A.1 and A.2.

Table 1.1. Thermal conductivity of certain materials at room temperature

Material ? (W m-1 K-1) Silver 419 Copper 386 Aluminum 204 Mild steel 45 Stainless steel 15 Ice 1.9 Concrete 1.4 Clay brick 1.1 Glass 1.0 Water 0.60 Plaster 0.48 Asbestos 0.16 Wood (hard, soft wood) 0.12-0.23 Cork 0.044-0.049 Stone wool 0.038-0.041 Glass wool 0.035-0.051 Expanded polystyrene 0.036-0.047 Polyurethane (foam) 0.030-0.045 Extruded polystyrene 0.028 Air 0.026

1.1.3. Convection

Here, we will only consider the heat transfer between a solid and a fluid, the energy being transmitted by the fluid's displacement. A good representation of this transfer mechanism is given by Newton's law:

Figure 1.4. Convective heat transfer scheme

[1.7] where: f is the heat flow rate transmitted by convection (W); hc is the convective heat transfer coefficient (W m-2 K-1); Tp is the solid's surface temperature (K); T8 is the temperature of fluid away from solid surface (K); S is the area of solid/fluid contact surface (m2).

The value of the convective heat transfer coefficient hc is a function of the fluid's nature, temperature, velocity or the temperature difference and the geometrical characteristics of the solid/fluid contact surface. The correlations in the most common cases of natural convection are given in Appendix 3, i.e. when the fluid's movement is due to temperature differences (no pump or fan).

Thermal characterization aims to measure the conductive and diffusing properties of a material. Convection most often occurs as a mode of "parasitic" transfer on the boundaries of the system by influencing the internal temperature field. We therefore have to take this into account. The correlations presented in Appendix 3 show that the coefficient of heat transfer by natural convection depends on the temperature difference between the surface and the surrounding fluid. Most often this difference is not perfectly uniform on surfaces and varies over time. It is therefore not possible to calculate it precisely and it will most often have to be estimated.

In natural convection, its value is generally between 2 and 5 W m-2 K-1. The radiation heat transfer coefficient that will be defined below is of the same order of magnitude. It will therefore be noted that placing a device under vacuum makes it possible to reduce losses by decreasing convective transfers but not canceling them, because radiation transfer is not affected by pressure.

1.1.4. Radiation

Radiation is a transfer of energy by electromagnetic waves (it does not need material support and even exists in a vacuum). We will only focus here on the transfer between two surfaces. In conduction problems, we take into account the radiation between a solid (whose surface is assumed to be gray and diffusing) and the surrounding environment (of large dimensions). In this case, we have the equation:

[1.8] where: f is the radiation heat flow rate (W); s is Stefan's constant (5,67.10-8 W m-2 K-4); ep is the surface emission factor; Tp is the surface temperature (K); T8 is the temperature of the medium surrounding the surface (K); S is the area of surface (m2).

Figure 1.5. Radiation heat transfer...