Transient Conduction: One-Dimensional Solutions

When a hot metal slab is quenched in oil or a food product is rapidly frozen, the temperature inside the object changes with time and position. Predicting these internal temperature profiles is the essence of transient conduction. This topic is critical for engineers designing heat treatment processes, food storage systems, electronic cooling, and any application where controlling thermal stress or process timing is paramount. Mastering the analytical solutions for simple geometries provides the foundational skills to tackle more complex, real-world scenarios.

The Breakdown of Lumped Capacitance: The Role of the Biot Number

The simplest model for transient heat transfer is the lumped capacitance method. It assumes the temperature within an object is uniform at any instant because its internal conduction resistance is negligible compared to its surface convection resistance. This assumption is valid only when the Biot number is small. The Biot number ($Bi$) is a dimensionless parameter defined as the ratio of internal conduction resistance to external convection resistance: $$Bi=\frac{h{L}_c}{k}$$ where $h$ is the convection heat transfer coefficient, $k$ is the thermal conductivity of the solid, and $L_c$ is the characteristic length (Volume/Surface Area).

A Biot number less than 0.1 typically justifies the lumped capacitance assumption. However, when $Bi>0.1$, internal temperature gradients become significant. You can no longer treat the object as having a single, uniform temperature. This is the regime where the more sophisticated one-dimensional solutions discussed here are required. For a plane wall of thickness $2L$ cooled symmetrically on both sides, $L_c=L$. A high $Bi$ indicates that heat cannot conduct to the interior fast enough to keep up with heat loss at the surface, creating steep internal temperature gradients.

Exact Analytical Solutions: Eigenfunction Expansions

For one-dimensional transient conduction in simple shapes (plane walls, long cylinders, spheres) with uniform initial temperature and constant boundary conditions, the governing heat equation can be solved analytically using separation of variables. The solutions take the form of infinite series called eigenfunction expansions.

For a plane wall with an initial temperature $T_i$ suddenly exposed to a convective environment at $T_{\infty }$, the solution is: $$\theta (x^{\ast },t^{\ast })=\sum _{n=1}^{\infty }{C}_n\exp (-{\zeta }_n^2{t}^{\ast })\cos ({\zeta }_n{x}^{\ast })$$ where:

• $\theta =(T-T_{\infty })/(T_i-T_{\infty })$ is the dimensionless temperature.
• $x^{\ast }=x/L$ is the dimensionless spatial coordinate.
• $t^{\ast }=\alpha t/L^2$ is the dimensionless time, known as the Fourier number ($Fo$). Here, $\alpha$ is the thermal diffusivity.
• ${\zeta }_n$ are the eigenvalues (roots of the transcendental equation $\zeta \tan \zeta =Bi$).
• $C_n$ are constants dependent on $Bi$.

Similar series solutions exist for cylinders and spheres, involving Bessel functions and radial coordinates. These exact solutions are mathematically precise but computationally intensive due to the infinite series. They reveal that the temperature at any point is a function of three dimensionless groups: the dimensionless position ($x^{\ast }$ or $r^{\ast }$), the Fourier number ($Fo$), and the Biot number ($Bi$).

Engineering Tools: The Heisler Charts

To circumvent the tedious calculations of the infinite series, the results have been conveniently tabulated and plotted graphically. These are known as Heisler charts. They plot the dimensionless centerline temperature (for a plane wall, cylinder, or sphere) against the Fourier number ($Fo$), with the Biot number ($Bi$) as a parameter on a set of curves.

For example, to find the center temperature of a large steel plate ($T_i=500°C$, $T_{\infty }=100°C$) after 5 minutes of cooling, you would:

1. Calculate $Bi$ and $Fo$.
2. Locate the appropriate chart for a plane wall.
3. Find the curve corresponding to your calculated $Bi$ (interpolating if necessary).
4. Read the value of the center temperature ratio ${\theta }_{center}$ at your calculated $Fo$.
5. Solve for the actual center temperature: $T_{center}=T_{\infty }+{\theta }_{center}(T_i-T_{\infty })$.

A second set of charts (position correction charts) then allows you to find the temperature at any other location within the object using the already-determined center temperature. These charts are an invaluable graphical tool for quick engineering estimates where high computational precision is not required.

The One-Term Approximation: A Powerful Simplification

For many practical engineering problems, particularly those involving longer times, the infinite series solution converges very rapidly. It is observed that for a Fourier number greater than 0.2, the first term of the infinite series (the $n=1$ term) dominates the solution. This leads to the highly useful one-term approximation.

For a plane wall, the solution simplifies to: $$\theta (x^{\ast },t^{\ast })\approx C_1\exp (-{\zeta }_1^2{t}^{\ast })\cos ({\zeta }_1{x}^{\ast })$$ where $C_1$ and ${\zeta }_1$ are now simple functions of $Bi$, readily available in standard tables. The analogous forms for cylinders and spheres use $J_0({\zeta }_1{r}^{\ast })$ and $[\sin ({\zeta }_1{r}^{\ast })/({\zeta }_1{r}^{\ast })]$, respectively.

This approximation is remarkably accurate for $Fo>0.2$ and is the standard method used in most engineering applications. It reduces the problem to a straightforward algebraic calculation once you look up two tabulated coefficients ($C_1$ and ${\zeta }_1$). For instance, calculating the time required for the center of a spherical meatball to reach a safe temperature in a freezer becomes a direct computation instead of a graphical estimation or an infinite series summation.

Applying the Solutions to Different Geometries

The principles are identical for the three canonical shapes, but the characteristic length and functional forms differ.

• Plane Wall (Slab) of thickness 2L: Symmetric cooling/heating on both faces. Characteristic length for $Bi$ and $Fo$ is $L$, the half-thickness.
• Long Cylinder of radius $r_o$: Radial conduction only. Characteristic length is the radius $r_o$.
• Sphere of radius $r_o$: Radial conduction only. Characteristic length is the radius $r_o$.

For objects that can be modeled as the intersection of two or more one-dimensional geometries (e.g., a short cylinder is the intersection of an infinite cylinder and an infinite plane wall), the solution is found using the product rule: ${\theta }_{shortcylinder}(r,x,t)={\theta }_{infcylinder}(r,t)\times {\theta }_{infslab}(x,t)$. This multiplicative property vastly extends the applicability of these one-dimensional solutions to more complex, multi-dimensional shapes.

Common Pitfalls

1. Misapplying the Lumped Capacitance Model: The most frequent error is using the simple $T(t)=T_{\infty }+(T_i-T_{\infty })\exp (-hAt/\rho cV)$ formula when $Bi>0.1$. This will significantly overestimate the object's cooling or heating rate and completely miss the internal temperature gradients. Always calculate $Bi$ first to select the appropriate solution method.

1. Misreading Heisler Charts or Coefficient Tables: These tools require careful interpolation, especially for intermediate $Bi$ values. Confusing the chart for a plane wall with that for a cylinder is another easy mistake. Double-check the geometry and ensure you are using the correct characteristic length ($L$ for a wall, $r_o$ for a cylinder/sphere) in your $Bi$ and $Fo$ calculations.

1. Using the One-Term Approximation Outside Its Validity: The one-term approximation is not accurate for very small times ($Fo<0.2$). At the very beginning of a transient process, many terms in the infinite series are needed to capture the sharp temperature change at the surface. Using the one-term approximation here will give incorrect results. Recognize that $Fo>0.2$ is the general rule for its safe application.

1. Neglecting the Assumptions: These solutions assume constant properties ($k$, $\alpha$), a uniform initial temperature, and a constant convective boundary condition ($T_{\infty }$, $h$). In real processes, properties vary with temperature, the initial state may not be uniform, and $h$ can change. Engineers must judge when these idealizations are acceptable or when a numerical solution is necessary.

Summary

• The Biot number ($Bi$) is the key criterion: $Bi>0.1$ signifies significant internal temperature gradients, necessitating the solutions covered here.
• Exact solutions are eigenfunction expansions (infinite series) that express temperature as a function of dimensionless position ($x^{\ast }$), Fourier number ($Fo$), and $Bi$.
• Heisler charts provide a graphical tool to quickly find centerline and off-center temperatures without solving the series.
• For most engineering applications with $Fo>0.2$, the one-term approximation of the infinite series offers an accurate and computationally simple algebraic method.
• These solutions apply directly to plane walls, long cylinders, and spheres, and can be combined via the product solution method for multi-dimensional shapes like short cylinders.