Introduction & Context

Fourier's Second Law, also known as the heat diffusion equation, describes the time-dependent distribution of temperature in a medium. In process engineering, this calculation is critical for analyzing unsteady-state heat transfer, where temperatures change over time rather than reaching a steady state. This is particularly vital in applications such as the quenching of metal components, the sterilization of food products in autoclaves, and the thermal processing of polymers. By predicting the temperature profile within a solid, engineers can ensure material integrity, optimize cycle times, and prevent thermal stress-induced failures.

Methodology & Formulas

The calculation utilizes the one-term approximation for a plane wall subjected to a constant surface temperature (equivalent to an infinite Biot number). This approach simplifies the infinite series solution of the heat equation, which is valid for sufficiently large values of the Fourier number.

The dimensionless temperature ratio, θ, is defined as:

\[ \theta = \frac{T(z, t) - T_{\infty}}{T_i - T_{\infty}} \]

The Fourier number (Fo), which characterizes the ratio of the rate of heat conduction to the rate of thermal energy storage, is calculated as:

\[ Fo = \frac{\alpha t}{L_c^2} \]

The temperature at a specific position z and time t is determined by the following one-term approximation:

\[ \theta = C_1 \exp(-\zeta_1^2 Fo) \cos\left(\zeta_1 \frac{z}{L_c}\right) \]

Where the constants for a plane wall with constant surface temperature are defined as:

\[ \zeta_1 = \frac{\pi}{2} \] \[ C_1 = \frac{4}{\pi} \]

Parameter	Condition/Threshold	Engineering Significance
Fourier Number (Fo)	Fo < 0.2	Approximation is inaccurate; higher-order terms are required.
Fourier Number (Fo)	Fo ≥ 0.2	One-term approximation is valid for engineering estimates.
Biot Number (Bi)	Bi → ∞	Assumes negligible surface convection resistance (constant surface temperature).

Fourier's Second Law, also known as the heat diffusion equation, accounts for the change in temperature over time, whereas the steady-state equation assumes temperature remains constant at any given point. Key differences include:

The inclusion of a time-dependent term (partial derivative of temperature with respect to time).
The requirement for thermal diffusivity, which represents how quickly heat propagates through a material.
The necessity of defining initial conditions in addition to boundary conditions to solve the differential equation.

The lumped capacitance method is a simplified approach used when internal thermal resistance is negligible compared to external convective resistance. You should consider this method when:

The Biot number is less than 0.1.
The object has high thermal conductivity.
The object has a small characteristic length or a very low convective heat transfer coefficient.
You need a rapid estimation of temperature change without solving complex partial differential equations.

To accurately model unsteady-state heat transfer, you must define the following material properties:

Thermal conductivity (k), which dictates the rate of heat flow through the material.
Density (ρ), which influences the thermal mass of the system.
Specific heat capacity (C_p), which determines the energy required to change the temperature of the material.
Thermal diffusivity (α), which is the ratio of thermal conductivity to the product of density and specific heat capacity.

Boundary conditions define how the system interacts with its environment at the surfaces. Common types encountered in process engineering include:

Dirichlet conditions: Specifying a constant surface temperature.
Neumann conditions: Specifying a constant heat flux at the surface.
Robin conditions: Specifying convective heat transfer, where the surface heat flux is proportional to the temperature difference between the surface and the fluid.

Worked Example: Unsteady State Heat Conduction in a Ceramic Slab

A process engineer is evaluating the thermal treatment of a thin ceramic slab used in high-temperature electronics. The slab, initially at a uniform temperature, is suddenly exposed to a hot air stream to initiate a curing process. We must determine the temperature at the center of the slab after 300 seconds of exposure.

Knowns:

Thermal diffusivity (α): 1.2e-05 m²/s
Slab thickness (2L): 0.1 m (Characteristic length L_c = 0.05 m)
Initial temperature (T_i): 293.15 K
Ambient fluid temperature (T_∞): 473.15 K
Time elapsed (t): 300 s
Position of interest (z): 0.0 m (Centerline)

Step-by-Step Calculation:

Calculate the Fourier number (Fo) to determine the dimensionless time: \[ Fo = \frac{\alpha \cdot t}{L_c^2} = \frac{1.2 \times 10^{-5} \cdot 300}{0.05^2} = 1.440 \]
Determine the first eigenvalue (ζ₁) and the corresponding coefficient (C₁) for a plane wall with infinite Biot number (assuming perfect surface contact): \[ \zeta_1 = \frac{\pi}{2} \approx 1.571 \] \[ C_1 = \frac{4}{\pi} \approx 1.273 \]
Calculate the dimensionless temperature (θ) at the centerline using the one-term approximation: \[ \theta = C_1 \cdot \exp(-\zeta_1^2 \cdot Fo) = 1.273 \cdot \exp(-1.571^2 \cdot 1.440) = 0.036 \]
Convert the dimensionless temperature back to the absolute temperature (T_z): \[ T_z = T_{∞} - \theta \cdot (T_{∞} - T_i) = 473.15 - 0.036 \cdot (473.15 - 293.15) = 466.587 \text{ K} \]
Convert the result to Celsius: \[ T_{z,C} = 466.587 - 273.15 = 193.437 \text{ °C} \]

Final Answer: The temperature at the center of the ceramic slab after 300 seconds is 193.437 °C.

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