Unsteady Conduction: Multidimensional Product Solutions

Analyzing how heat spreads through a solid object over time is crucial for engineering applications ranging from quenching metal parts to designing thermal protection systems. While one-dimensional solutions are well-established, real components like short cylinders or rectangular blocks require a two or three-dimensional analysis. The product solution technique provides an elegant and powerful method to solve these complex transient problems by combining simpler, one-dimensional solutions, saving significant analytical and computational effort.

From One-Dimensional Limitations to Multidimensional Reality

In transient or unsteady conduction, temperature within a solid changes with both time and position. For simple geometries like a large plane wall, an infinitely long cylinder, or a sphere, one-dimensional solutions are available, often presented graphically via Heisler charts. These charts plot dimensionless temperature against the Fourier number (a dimensionless time, $Fo=\alpha t/L_c^2$) with the Biot number ($Bi=h{L}_c/k$) as a parameter.

However, a finite cylinder—think of a short metal rod—is not truly one-dimensional. Heat transfers radially from the curved surface and axially from the flat ends. A three-dimensional rectangular block exchanges heat from all six faces. Solving the governing differential equation directly for these geometries is mathematically intensive. The product solution method circumvents this difficulty by recognizing that under specific conditions, the temperature distribution in a multidimensional geometry can be constructed as the product of the distributions from its constituent one-dimensional shapes.

The Core Principle and Its Critical Assumptions

The product solution states that the dimensionless temperature in a finite geometry is equal to the product of the dimensionless temperatures from the one-dimensional geometries whose intersection forms it. For a finite cylinder of radius $r_o$ and half-length $L$, this is the intersection of an infinite cylinder and a plane wall. The solution is expressed as:

$${\left(\frac{\theta }{{\theta }_i}\right)}_{\text{finite cylinder}}={\left(\frac{\theta }{{\theta }_i}\right)}_{\text{infinite cylinder}}\times {\left(\frac{\theta }{{\theta }_i}\right)}_{\text{plane wall}}$$

Where $\theta =T-T_{\infty }$ and ${\theta }_i=T_i-T_{\infty }$. Similarly, for a three-dimensional rectangular block with half-lengths $L_x$, $L_y$, and $L_z$:

$${\left(\frac{\theta }{{\theta }_i}\right)}_{\text{rectangular block}}={\left(\frac{\theta }{{\theta }_i}\right)}_{\text{wall, x}}\times {\left(\frac{\theta }{{\theta }_i}\right)}_{\text{wall, y}}\times {\left(\frac{\theta }{{\theta }_i}\right)}_{\text{wall, z}}$$

This method rests on several key assumptions:

1. The initial temperature is uniform.
2. The thermal properties are constant.
3. The convective heat transfer coefficient $h$ is uniform and constant over all exposed surfaces.
4. There is no heat generation within the solid.

Crucially, the one-dimensional solutions used in the product are for slabs of thickness 2L (where L is the half-thickness from the center to the surface) and infinite cylinders of radius $r_o$.

Applying the Method: A Step-by-Step Workflow

Let's walk through the process of finding the centerline temperature in a short, cylindrical stainless steel billet suddenly cooled by air.

Step 1: Decompose the Geometry. Identify the one-dimensional components. A finite cylinder is the intersection of an infinite cylinder (radius $r_o$) and a plane wall (half-thickness $L$).

Step 2: Calculate Dimensionless Parameters for Each Direction. You must compute the correct Biot number ($Bi$) and Fourier number ($Fo$) for each one-dimensional component.

• For the infinite cylinder component: Use the characteristic length $L_c=r_o$. So, $B{i}_{\text{cyl}}=h{r}_o/k$ and $F{o}_{\text{cyl}}=\alpha t/r_o^2$.
• For the plane wall component: Use the characteristic length $L_c=L$ (the half-thickness). So, $B{i}_{\text{wall}}=hL/k$ and $F{o}_{\text{wall}}=\alpha t/L^2$.

Step 3: Find One-Dimensional Solutions. Using the calculated $Bi$ and $Fo$ for each component, find the corresponding centerline dimensionless temperature (${\theta }_0/{\theta }_i$). This is typically done using the appropriate Heisler chart (Chart 1 for cylinders, Chart 1 for plane walls) or the corresponding analytical approximation.

Step 4: Construct the Product Solution. Multiply the one-dimensional solutions together to get the multidimensional result. For the center temperature ($r=0,x=0$) of the finite cylinder: $${\left(\frac{{\theta }_0}{{\theta }_i}\right)}_{\text{finite cyl}}={\left(\frac{{\theta }_0}{{\theta }_i}\right)}_{\text{infinite cyl}}\times {\left(\frac{{\theta }_0}{{\theta }_i}\right)}_{\text{plane wall}}$$

Step 5: Calculate Temperature. Finally, solve for the actual center temperature: $T_{\text{center}}=T_{\infty }+{\theta }_i\cdot ({\theta }_0/{\theta }_i{)}_{\text{finite cyl}}$.

Extending Heisler Charts to Two and Three Dimensions

The product solution technique directly extends the utility of standard Heisler charts. Engineers are not limited to charts specifically for finite cylinders or blocks; they can use the classic charts for infinite cylinders and plane walls to solve for these more complex shapes. This is possible because the charts are graphical solutions to the one-dimensional governing equations, and the product method is mathematically valid under the stated assumptions.

For example, to find the temperature at an off-center location in a rectangular block, say at $(x_1,y_1,z_1)$, you would first find the one-dimensional dimensionless temperatures at those relative positions ($x_1/L_x$, etc.) for each wall using Chart 2 (the position-correcting chart). You would then multiply these three position-corrected ratios by the three centerline temperature ratios from Chart 1. The full solution becomes: $${\left(\frac{\theta }{{\theta }_i}\right)}_{\text{block}}={\left(\frac{{\theta }_0}{{\theta }_i}\cdot \frac{\theta }{{\theta }_0}\right)}_{\text{wall, x}}\times {\left(\frac{{\theta }_0}{{\theta }_i}\cdot \frac{\theta }{{\theta }_0}\right)}_{\text{wall, y}}\times {\left(\frac{{\theta }_0}{{\theta }_i}\cdot \frac{\theta }{{\theta }_0}\right)}_{\text{wall, z}}$$

Common Pitfalls

Using the Wrong Characteristic Length for Bi and Fo. This is the most frequent error. Remember that for the plane wall component of a finite cylinder, the relevant length is the axial half-length ($L$), not the radius. For each one-dimensional solution, $L_c$ is the length from the mid-plane (or centerline) to the surface in that specific direction. Confusing these will yield incorrect dimensionless numbers and an invalid product.

Applying the Method Outside Its Assumptions. The product solution fails if the initial temperature is not uniform, if heat generation is present, or if the boundary conditions are not convective (or equivalent) on all surfaces. For instance, an insulated face changes the problem fundamentally, as it is no longer the intersection of standard one-dimensional geometries.

Misinterpreting the One-Dimensional Solutions. The "plane wall" solution is for a wall of thickness 2L with convection on both sides. Ensure you are using the correct chart or equation for this configuration. Using a solution for a semi-infinite solid, for example, would invalidate the entire process.

Forgetting that the Product is for Dimensionless Temperature. You multiply the dimensionless temperature ratios ($\theta /{\theta }_i$), not the actual temperatures. Always work in the dimensionless form until the final step of converting back to $T$.

Summary

• The product solution technique allows for the analysis of transient heat conduction in finite multidimensional geometries (e.g., short cylinders, rectangular blocks) by combining the solutions for their constituent one-dimensional shapes.
• The dimensionless temperature at a point in the finite object is found by multiplying the dimensionless temperatures from the corresponding one-dimensional solutions for each coordinate direction.
• Each one-dimensional solution requires calculating the appropriate Biot number and Fourier number using the characteristic length from the center to the surface in that specific direction.
• This method effectively extends Heisler chart results to two and three dimensions, leveraging existing charts for infinite cylinders and plane walls.
• Successful application is contingent on strict adherence to the underlying assumptions: uniform initial temperature, constant properties and convection coefficient, and no internal heat generation.