Conduction is the transfer of thermal energy within a body due to the random motion of molecules. The average energy of the molecules is proportional to the temperature. Consider a bar of length d and cross-sectional area A, with one end at a fixed temperature T, and the other at a fixed temperature T2, where T₁ > T2. The more energetic molecules at the hot end transfer kinetic energy to the less energetic molecules at the cold end. In the steady state, the rate of flow of heat Q is constant along the length of the bar and is given by Fourier's law of heat conduction: Q=KA (T₁-T₂) d (2.4) where k is called the thermal conductivity. It should be noted that eqn (2.4) applies only in the steady state. In practice, it takes time for a solid body to establish a steady-state temperature distribution. For unsteady heat conduc- tion, the timescale to establish a steady state is determined by the characteristic time t for an isotherm to diffuse a distance x, and for a material with thermal diffusivity, K= k/pc(m² s¯¹), where k is the thermal conductivity, c the specific heat, and p the density, is given by: (2.5) The algebraic form of eqn (2.5) is easily derived by dimensional analysis (see Exercise 2.1), and how heat diffuses along a bar is explored in Exercise 2.2. 2.1 Derive the form of eqn (2.5) using dimensional analysis. Estimate the characteristic timescale for heat to conduct through a heat shield of thickness 1 cm (p= 5 x 10³ kg m³, k 10¹ Wm¹ °C, c 10³ J kg-¹ °C-¹.)
Conduction is the transfer of thermal energy within a body due to the random motion of molecules. The average energy of the molecules is proportional to the temperature. Consider a bar of length d and cross-sectional area A, with one end at a fixed temperature T, and the other at a fixed temperature T2, where T₁ > T2. The more energetic molecules at the hot end transfer kinetic energy to the less energetic molecules at the cold end. In the steady state, the rate of flow of heat Q is constant along the length of the bar and is given by Fourier's law of heat conduction: Q=KA (T₁-T₂) d (2.4) where k is called the thermal conductivity. It should be noted that eqn (2.4) applies only in the steady state. In practice, it takes time for a solid body to establish a steady-state temperature distribution. For unsteady heat conduc- tion, the timescale to establish a steady state is determined by the characteristic time t for an isotherm to diffuse a distance x, and for a material with thermal diffusivity, K= k/pc(m² s¯¹), where k is the thermal conductivity, c the specific heat, and p the density, is given by: (2.5) The algebraic form of eqn (2.5) is easily derived by dimensional analysis (see Exercise 2.1), and how heat diffuses along a bar is explored in Exercise 2.2. 2.1 Derive the form of eqn (2.5) using dimensional analysis. Estimate the characteristic timescale for heat to conduct through a heat shield of thickness 1 cm (p= 5 x 10³ kg m³, k 10¹ Wm¹ °C, c 10³ J kg-¹ °C-¹.)
Chapter2: The Kinetic Theory Of Gases
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Transcribed Image Text:2.2.1 Conduction
Conduction is the transfer of thermal energy within a body due to the random motion of
molecules. The average energy of the molecules is proportional to the temperature. Consider
a bar of length d and cross-sectional area A, with one end at a fixed temperature T₁ and the
other at a fixed temperature T₂, where T₁ > T₂. The more energetic molecules at the hot end
transfer kinetic energy to the less energetic molecules at the cold end. In the steady state, the
rate of flow of heat Q is constant along the length of the bar and is given by Fourier's law of
heat conduction:
Q=KA (T₁-T₂)
d
(2.4)
where k is called the thermal conductivity.
It should be noted that eqn (2.4) applies only in the steady state. In practice, it takes time for
a solid body to establish a steady-state temperature distribution. For unsteady heat conduc-
tion, the timescale to establish a steady state is determined by the characteristic time t for an
isotherm to diffuse a distance x, and for a material with thermal diffusivity, K = k/pc(m² s¯¹),
where k is the thermal conductivity, c the specific heat, and p the density, is given by:
(2.5)
The algebraic form of eqn (2.5) is easily derived by dimensional analysis (see Exercise 2.1),
and how heat diffuses along a bar is explored in Exercise 2.2.
K
2.1 Derive the form of eqn (2.5) using dimensional analysis. Estimate the characteristic
timescale for heat to conduct through a heat shield of thickness 1 cm (p = 5 x 10³ kg
m³,k 10¹ Wm¹°C,c 10³ J kg-¹ °C-¹.)
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