Unsteady State Heating of Finite Solids (Newman's Law)

Reference ID: MET-65C7 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

This reference sheet describes the calculation of the center temperature of a finite solid object (e.g., a canned food item modeled as a short cylinder) during unsteady-state heating or cooling. The method is based on Newman’s Law, which combines one-dimensional transient conduction solutions for each principal direction of a multidimensional geometry. It is essential in process engineering for designing thermal processes such as retorting, pasteurization, and cooling of packaged foods, where accurate prediction of internal temperature histories ensures product safety and quality.

Methodology & Formulas

1. Dimensionless Temperature

\[ \theta \;=\; \frac{T - T_{\infty}}{T_{i} - T_{\infty}} \]

2. Dimensionless Groups

\[ \tau \;=\; \frac{\alpha \, t}{L_{c}^{2}} \qquad\text{(Fourier number)} \] \[ Bi \;=\; \frac{h \, L_{c}}{k} \qquad\text{(Biot number)} \]

3. One-Term Approximation for an Infinite Shape

\[ \theta_{\text{shape}} \;=\; A_{1}\,\exp\!\bigl(-\lambda_{1}^{2}\,\tau\bigr)\,C_{1} \]

For a plane wall (half-thickness \(L\)): \(C_{1}= \cos\!\bigl(\lambda_{1}\,x/L\bigr)\). At the center \(x=0\) so \(C_{1}=1\).
For an infinite cylinder (radius \(r_{o}\)): \(C_{1}= J_{0}\!\bigl(\lambda_{1}\,r/r_{o}\bigr)\). At the axis \(r=0\) so \(C_{1}=1\).

4. Newman’s Law (Product Solution)

\[ \theta_{\text{finite}} \;=\; \theta_{\text{shape1}}\;\times\;\theta_{\text{shape2}}\;\times\;\theta_{\text{shape3}} \]

For a short cylinder the product reduces to two dimensions (radial cylinder and axial plane wall):

\[ \theta_{\text{center}} \;=\; \theta_{\text{cyl}} \;\times\; \theta_{\text{wall}} \]

5. Step-by-Step Calculation

Determine characteristic lengths:
- Radial direction: \(L_{c,\text{cyl}} = r_{o}\)
- Axial direction: \(L_{c,\text{wall}} = L\) (half-height)
Compute Biot numbers: \[ Bi_{\text{cyl}} \;=\; \frac{h\,r_{o}}{k} \] \[ Bi_{\text{wall}} \;=\; \frac{h\,L}{k} \]
Compute Fourier numbers: \[ \tau_{\text{cyl}} \;=\; \frac{\alpha\,t}{r_{o}^{2}} \] \[ \tau_{\text{wall}} \;=\; \frac{\alpha\,t}{L^{2}} \]
Obtain coefficients \(A_{1}\) and eigenvalues \(\lambda_{1}\) from standard tables as functions of the respective Biot numbers. For high Biot numbers (\(Bi \gtrapprox 40\)), the common approximations are: \[ \lambda_{1,\text{cyl}} \approx 2.4048 \quad\text{(first root of }J_{0}\text{)} \] \[ \lambda_{1,\text{wall}} \approx \frac{\pi}{2} = 1.5708 \] \[ A_{1,\text{cyl}} \approx 1.000,\qquad A_{1,\text{wall}} \approx 1.000 \]
Calculate dimensionless center temperatures: \[ \theta_{\text{cyl}} \;=\; A_{1,\text{cyl}} \,\exp\!\bigl(-\lambda_{1,\text{cyl}}^{2}\,\tau_{\text{cyl}}\bigr) \] \[ \theta_{\text{wall}} \;=\; A_{1,\text{wall}} \,\exp\!\bigl(-\lambda_{1,\text{wall}}^{2}\,\tau_{\text{wall}}\bigr) \]
Apply Newman’s Law: \[ \theta_{\text{center}} \;=\; \theta_{\text{cyl}} \,\theta_{\text{wall}} \]
Convert back to actual temperature: \[ T_{\text{center}} \;=\; T_{\infty} \;+\; \theta_{\text{center}}\,(T_{i} - T_{\infty}) \]

Validity Checks & Regime Limits

Dimensionless Group	Valid Range for One-Term Approximation
Biot number \(Bi\)	\(0.01 \;\le\; Bi \;\le\; 100\) (values at the bounds require careful interpretation)
Fourier number \(\tau\)	\(\tau \;>\; 0.2\) (accuracy improves significantly for \(\tau > 0.5\))

When the calculated \(Bi\) and \(\tau\) for each principal direction satisfy the limits above, the one-term solution and Newman’s product formulation provide an accurate estimate of the transient temperature field. For \(Bi\) values at the limits or \(\tau\) values near 0.2, the approximation error increases.

Newman's Law (or the product solution method) allows process engineers to determine the temperature in a finite solid by calculating the product of the dimensionless temperature solutions of intersecting infinite (one-dimensional) geometries. This approach is valid under the following conditions:

The solid must be homogeneous and isotropic with constant thermal properties.
The initial temperature must be uniform throughout the solid.
The surface heat transfer coefficient \(h\) and the ambient fluid temperature \(T_{\infty}\) must be constant and uniform over each surface.
The temperature field in each intersecting infinite shape must be accurately represented by its own one-dimensional transient conduction solution (e.g., using the one-term approximation when \(\tau > 0.2\)).

The method transforms a complex multi-dimensional problem into simpler, solvable one-dimensional problems.

The one-term approximation (using only the first term of the infinite series solution) is generally considered sufficiently accurate for engineering design when the Fourier number \(\tau\) is greater than approximately 0.2. At this threshold, the contributions of the higher-order terms in the series become negligible. The approximation's accuracy improves as \(\tau\) increases. The Biot number should also be within a practical range (typically \(0.01 \le Bi \le 100\)) for the tabulated coefficients \(A_1\) and \(\lambda_1\) to be readily available and accurate.

While powerful, Newman's Law has specific constraints that process engineers must consider:

It assumes constant thermal properties (\(k, \alpha, \rho, c_p\)), which may not hold over large temperature ranges.
It is applicable only to geometries that can be constructed by the intersection of standard infinite shapes (e.g., finite cylinders from an infinite cylinder and an infinite plate, rectangular bricks from three infinite plates). It cannot handle arbitrary, complex geometries.
It does not account for internal heat generation within the solid.
It assumes a constant ambient fluid temperature \(T_{\infty}\). In batch processes where the heating or cooling medium temperature changes significantly, this assumption may lead to error.
It requires uniform boundary conditions (constant \(h\) on all surfaces), which may not be realistic in all processing equipment.

For a finite cylinder, you must calculate two independent Biot numbers, one for each principal direction:

Radial Direction (Infinite Cylinder): The characteristic length is the cylinder radius \(r_o\). The Biot number is \(Bi_{\text{cyl}} = \dfrac{h \, r_o}{k}\).
Axial Direction (Infinite Plane Wall): The characteristic length is the half-height \(L\) of the cylinder. The Biot number is \(Bi_{\text{wall}} = \dfrac{h \, L}{k}\).

These Biot numbers are then used to obtain the appropriate coefficients (\(\lambda_1\), \(A_1\)) from standard tables for the infinite cylinder and infinite plane wall solutions, respectively. The product of the resulting dimensionless temperatures gives the solution for the finite cylinder.

Worked Example: Unsteady State Heating of a Canned Food Item

A canned food product, modeled as a short cylinder, is heated in a retort to ensure sterilization. The center temperature must be determined after a specified heating time to verify process safety.

Known Parameters:

Geometry: Radius \( r_o = 0.040 \, \text{m} \), half-height \( L = 0.050 \, \text{m} \)
Material properties: Thermal conductivity \( k = 0.600 \, \text{W/m·K} \), thermal diffusivity \( \alpha = 1.500 \times 10^{-7} \, \text{m}^2/\text{s} \)
Boundary conditions: Heat transfer coefficient \( h = 1200 \, \text{W/m}^2\cdot\text{K} \), surrounding fluid temperature \( T_\infty = 121.000 \, ^\circ\text{C} \)
Initial condition: Uniform initial temperature \( T_i = 25.000 \, ^\circ\text{C} \)
Process time: \( t = 3500.000 \, \text{s} \)

Step-by-Step Calculation:

Calculate the Biot and Fourier numbers for each dimensionless dimension.
- For the infinite cylinder (radial direction): \[ Bi_{\text{cyl}} = \frac{h \, r_o}{k} = \frac{1200 \cdot 0.040}{0.600} = 80.000 \] \[ \tau_{\text{cyl}} = \frac{\alpha \, t}{r_o^{2}} = \frac{1.500 \times 10^{-7} \cdot 3500}{(0.040)^{2}} = 0.3281 \]
- For the infinite plane wall (axial direction): \[ Bi_{\text{wall}} = \frac{h \, L}{k} = \frac{1200 \cdot 0.050}{0.600} = 100.000 \] \[ \tau_{\text{wall}} = \frac{\alpha \, t}{L^{2}} = \frac{1.500 \times 10^{-7} \cdot 3500}{(0.050)^{2}} = 0.2100 \]
Validity Check: Both Biot numbers are within the stated range of \(0.01 \le Bi \le 100\), though \(Bi_{\text{wall}} = 100\) is at the upper limit. Both Fourier numbers are greater than 0.2, satisfying the criterion for using the one-term approximation, although \(\tau_{\text{wall}} = 0.210\) is near the lower bound where approximation error is higher.
Determine the coefficients \( \lambda_1 \) and \( A_1 \) from standard one-term approximation tables for each Biot number. For these high Biot numbers, we use the asymptotic values.
- For \( Bi_{\text{cyl}} = 80.000 \): \( \lambda_{1,\text{cyl}} \approx 2.4048 \), \( A_{1,\text{cyl}} \approx 1.000 \)
- For \( Bi_{\text{wall}} = 100.000 \): \( \lambda_{1,\text{wall}} \approx \pi/2 = 1.5708 \), \( A_{1,\text{wall}} \approx 1.000 \)
Calculate the dimensionless center temperature for each infinite shape using the one-term approximation.
- For the infinite cylinder (center, \( r=0 \)): \[ \theta_{\text{cyl}} = A_{1,\text{cyl}} \exp(-\lambda_{1,\text{cyl}}^{2} \tau_{\text{cyl}}) = 1.000 \cdot \exp(-(2.4048^{2}) \cdot 0.3281) = \exp(-1.8977) = 0.1498 \]
- For the infinite plane wall (mid-plane, \( x=0 \)): \[ \theta_{\text{wall}} = A_{1,\text{wall}} \exp(-\lambda_{1,\text{wall}}^{2} \tau_{\text{wall}}) = 1.000 \cdot \exp(-(1.5708^{2}) \cdot 0.2100) = \exp(-0.5182) = 0.5956 \]
Apply Newman's Law (product solution) to find the dimensionless temperature for the finite short cylinder.
- \( \theta_{\text{center}} = \theta_{\text{cyl}} \cdot \theta_{\text{wall}} = 0.1498 \cdot 0.5956 = 0.08920 \)
Convert the dimensionless temperature back to the actual center temperature.
- \( T_{\text{center}} = T_\infty + \theta_{\text{center}} (T_i - T_\infty) = 121.000 + 0.08920 \cdot (25.000 - 121.000) \)
- \( T_{\text{center}} = 121.000 + 0.08920 \cdot (-96.000) = 121.000 - 8.5632 = 112.4368 \, ^\circ\text{C} \)

Final Answer: The center temperature of the canned food after 3500 seconds is approximately \( T_{\text{center}} = 112.44 \, ^\circ\text{C} \). (Note: Using values from standard tables for the exact \(Bi\) might yield a slightly different result, as the high-\(Bi\) approximations were used here.)

"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