Introduction & Context

The unsteady state heating of an infinite slab is a fundamental problem in heat transfer, representing scenarios where a solid material is subjected to a sudden change in ambient temperature. In process engineering, this model is critical for analyzing thermal processing of materials, such as the curing of polymer sheets, the heat treatment of metal plates, or the sterilization of food products in flat containers. By assuming the slab is infinite in two dimensions, we simplify the heat conduction equation to a one-dimensional problem, allowing engineers to predict the temperature evolution at the center of the slab over time.

Methodology & Formulas

The analysis utilizes the one-term approximation of the Heisler charts, which is valid for Fourier numbers greater than 0.2. The process begins by calculating the dimensionless Biot number (Bi) and Fourier number (Fo) to characterize the heat transfer regime.

The Biot number represents the ratio of internal conductive resistance to external convective resistance:

\[ Bi = \frac{h L}{k} \]

The Fourier number represents the dimensionless time, characterizing the rate of heat conduction relative to the rate of thermal storage:

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

The dimensionless temperature at the center of the slab is determined using the first eigenvalue and constant derived from the transcendental equation for the specific geometry:

\[ \theta_0 = \frac{T_0 - T_{\infty}}{T_i - T_{\infty}} = C_1 \exp(-\zeta_1^2 Fo) \]

The actual center temperature is then recovered by rearranging the dimensionless temperature definition:

\[ T_0 = \theta_0 (T_i - T_{\infty}) + T_{\infty} \]

Parameter	Condition	Engineering Implication
Fourier Number (Fo)	Fo < 0.2	One-term approximation is inaccurate; use infinite series solution.
Fourier Number (Fo)	Fo ≥ 0.2	One-term approximation is valid.
Biot Number (Bi)	Bi < 0.1	Lumped system analysis is recommended (negligible internal resistance).
Biot Number (Bi)	Bi > 0.1	One-term approximation is applicable (internal resistance is significant).

The infinite slab model is a conservative approximation used when the dimensions of the object in two directions are significantly larger than the third. You can apply this model when:

The heat transfer occurs primarily through one face of the material.
The length and width of the slab are at least ten times greater than the thickness.
Edge effects or heat losses from the sides are negligible compared to the primary surface flux.

The Biot number (Bi) dictates the internal temperature distribution by comparing internal conductive resistance to external convective resistance.

If Bi is less than 0.1, the internal resistance is negligible, and you can use the lumped capacitance method.
If Bi is greater than 0.1, internal temperature gradients are significant, and you must use the Heisler charts or the analytical series solution for unsteady state conduction.

The one-term approximation is a simplified mathematical approach valid only for specific conditions. You should be aware of these constraints:

It is only accurate for Fourier numbers (Fo) greater than 0.2.
It assumes constant thermal properties, which may not hold true if the material undergoes a phase change or significant temperature-dependent property shifts.
It assumes a uniform initial temperature throughout the slab.

Worked Example: Transient Thermal Processing of a Steel Plate

A process engineer is evaluating the thermal treatment of a large, thin steel plate used in a manufacturing assembly. The plate is initially at a uniform temperature and is suddenly exposed to a high-temperature convective environment. We must determine the temperature at the mid-plane of the slab after 300 seconds of exposure.

Knowns:

Thermal conductivity (k): 50.0 W/m·K
Thermal diffusivity (alpha): 0.000015 m²/s
Half-thickness (L): 0.05 m
Convective heat transfer coefficient (h): 100.0 W/m²·K
Initial temperature (Ti): 20.0 °C
Fluid temperature (Tinf): 300.0 °C
Time (t): 300.0 s
Biot number (Bi): 0.1
Fourier number (Fo): 1.8
First coefficient (C1): 1.016
First eigenvalue (zeta1): 0.311 rad

Step-by-Step Calculation:

Calculate the dimensionless temperature at the mid-plane (x = 0) using the one-term approximation for the infinite slab: \[ \theta_0 = \frac{T_0 - T_{\infty}}{T_i - T_{\infty}} = C_1 \exp(-\zeta_1^2 \cdot Fo) \]
Substitute the known values into the equation: \[ \theta_0 = 1.016 \cdot \exp(-(0.311)^2 \cdot 1.8) \]
Compute the exponent: \[ -(0.311)^2 \cdot 1.8 = -0.0967 \cdot 1.8 = -0.174 \]
Calculate the dimensionless temperature: \[ \theta_0 = 1.016 \cdot \exp(-0.174) = 1.016 \cdot 0.840 = 0.854 \]
Solve for the mid-plane temperature (T₀): \[ T_0 = T_{\infty} + \theta_0(T_i - T_{\infty}) \] \[ T_0 = 300.0 + 0.854(20.0 - 300.0) \] \[ T_0 = 300.0 + 0.854(-280.0) \] \[ T_0 = 300.0 - 239.12 = 60.880 \text{ °C} \]

Final Answer:

The temperature at the mid-plane of the slab after 300 seconds is 60.880 °C.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle