Module 1. Conduction heat transfer
Lesson 1
CONDUCTION
1.1 Basic Law of Conduction
The concept of heat conduction embraces the process of heat propagation through direct contact between the particles of the substance. In gases, heat conduction occurs by molecular and atomic interaction. In fluids and solid dielectrics the process occurs by elastic molecular collision. In metals the flow of energy is mainly due to the diffusion of free electron. The process of heat propagation in general and that of heat conduction in particular is inseparably linked with the distribution of temperature, therefore it is necessary to define what is temperature field and temperature gradient.
1) Temperature field
Temperature is a property of substance characterizing the degree to which it is heated or the degree of its heat. Generally, temperature is a function of co-ordinates say X, Y, Z and time τ
t = f (x, y, z, τ)
The temperature field is the whole complex of temperatures of all space points at a given moment. The above equation is a mathematical expression of such a field. The temperature field can be transient or non-stationary if it depends on time.
If the temperature does not vary with time it is termed as steady state and stationary
t = f (x, y, z)
Two dimensional steady state temperature field is:
t = f (x, y)
Two dimensional unsteady state temperature field is:
t = f (x,y,τ)
One dimensional unsteady state temperature field is:
t = f (x, τ)
One dimensional steady state temperature field is:
t = f(x)
1.1.1 Isothermal surface
It is the locus of equal temperatures. Since one and the same space point cannot have two different temperatures, therefore, isothermal surfaces of different temperature do not intersect, so they either form a loop or end at the boundaries of a substance. The temperature in a substance varies only in the direction crossing the isothermal surfaces (Fig. 1.1). The most abrupt change in temperature occurs in direction normal to the isothermal surface. The rate of change in temperature ∆t in a substance with a normal distance ∆n in the isothermal surfaces is called temperature gradient. Mathematically, temperature gradient can be defined as
(Measured in °c/m)
The temperature gradient is a vector normal to the isothermal surface. This vector is positive in the direction of increasing temperature. The value of temperature gradient taken with negative sign is termed as temperature drop.
1.1.2 Heat flow
Thermal energy always flows from a region of higher temperature to one with lower temperature. The amount of transferred heat is termed as rate of heat flow (Q). This value usually refers to energy flow per unit time (hour). The rate of heat flow per unit area is called the specific rate of heat flow or sometimes thermal load of heating surface (q). Therefore, in technical systems of units “Q” has units kcal/hr, but in case of ‘q’ units are kcal/m2 hr. The value of Q as well as ‘q’ is a vector whose direction coincides with that in which heat propagates and runs in a direction opposite to that of temperature gradient.
1.1.3 Fourier’s law
Fourier discovered that the amount of transferred heat is proportional to the drop in temperature, time and area perpendicular to the direction in which heat flows. The relation of amount of heat transferred per unit area and per unit time can be written as
q = -λ grad.t (kcal/m2-hr)
This expression is the basic law of heat transfer i.e., Fourier’s law. The proportionality factor ‘λ’ in equation is termed as thermal conductivity, if is a physical property of a substance which characterizes the ability of substance to transfer heat therefore λ can be written as
Unit: kcal/m-hr-°C (F-area)
Therefore, the value of thermal conductivity determines the amount of heat passing per unit time, per unit area at a temperature drop of 1°C per unit length.
Thermal conductivity differs with each substance and in each case depends on structure, volume, weight, humidity, pressure and temperature. All these parameters hamper the correct choice of thermal conductivity.
Since temperature in the substance varies in the course of heat propagation, therefore, it is especially important to find out that how much thermal conductivity depends on temperature. So, experiments have shown that for most of the materials this dependence is linear i.e. λ = λ0 (1+b.t)
Where “λo” is value of thermal conductivity at 0°C and “b” is constant which is determined experimentally.
Thermal conductivity of gases ranges from 0.005 to 0.5 kcal/m-hr-°C. It increases with increase in temperatures and practically does not depend on pressure.
For gas mixtures the values of thermal conductivity may be determined only experimentally because the additivity law is inapplicable for thermal conductivity.
Thermal conductivity of liquids ranges from 0.08 to 0.6 kcal/m-hr-°C. For most liquids thermal conductivity decreases with rise in temperature, the only exceptions are water & glycerine.
Thermal conductivity of heat insulating materials ranges from 0.02 -2.5 kcal/m-hr-°C. Thermal conductivity increases with increase in temperature. As a rule, materials of high volume weight posses greater thermal conductivity. It also depends on the structure of the material, its porosity & humidity. Thermal conductivity of a damped material is considerably higher than thermal conductivity of the dry material and water taken separately. For instance, for dry brick λ = 0.3 and for water λ = 0.5 but damp brick soaked in water λ = 0.9. This phenomenon makes it necessary to proceed with special care while determining the thermal conductivity of a substance in heat transfer calculations.
Materials with low thermal conductivity i.e. less than 0.2 kcal/m-hr-°C are usually employed as heat insulating materials. Thermal conductivity of metals ranges from 2 to 360 kcal/m-hr-°C. The best conductor of heat is silver whose λ = 360 followed by red copper λ = 340 then gold (λ = 260), the aluminium (λ = 180). For most metals, thermal conductivity decreases with rise in temperature since the thermal & electric conductivity of the metal is because of diffusion of free electrons, for pure metal the two properties are proportional to each other. The thermal conductivity of a pure metal drops abruptly in presence of various elements & compounds. For example, “λ” for pure copper is 340 but if it contains traces of arsenic, the value is dropped to 122 kcal/m-hr-°C. The “λ” of steel containing 0.1% carbon is 45 kcal/m-hr-°C, 1% carbon is 34 kcal/m-hr-°C and 1.5% carbon is 31 kcal/m-hr-°C. The thermal conductivity of hardened carbon steel is 10-25% below that of soft steel.