熱時定数（Thermal Time Constant）

Exactly the analogy. The thermal time constant $\tau$ is the time it takes for an object to eliminate 63.2% of the temperature difference with its surrounding environment. The defining equation is:

Here $\rho$ is density, $V$ is volume, $c_p$ is specific heat, $h$ is the heat transfer coefficient, and $A$ is the heat dissipation area.

That's correct. The numerator $\rho V c_p$ is the thermal capacitance $C_\mathrm{th}$, representing how much thermal energy the object can store. The denominator $hA$ is the thermal conductance, whose reciprocal is the thermal resistance $R_\mathrm{th} = 1/(hA)$. In other words:

If you connect "a huge tank (large $C_\mathrm{th}$) to a thin pipe (large $R_\mathrm{th}$)", it takes time for the water level in the tank to change. Similarly, objects with large thermal capacity and poor heat dissipation have a larger $\tau$.

They are mathematically completely equivalent. This is called the "Electro-Thermal Analogy," a correspondence that dates back to Clapeyron's research in 1834:

Because of this correspondence, a method widely used in electronic device thermal design is to incorporate thermal circuits into SPICE simulators for transient analysis. The "thermal equivalent circuit" found in power semiconductor datasheets is exactly this.

Good question. The criterion is the Biot number:

$L_c = V/A$ (characteristic length), $k$ is the thermal conductivity of the object. This value represents the ratio of "surface convective resistance" to "internal conductive resistance."

• $\mathrm{Bi} < 0.1$: Internal heat conduction is fast, object is nearly uniform temperature. → Lumped Capacitance Method can be used, describable by a single $\tau$
• $\mathrm{Bi} \geq 0.1$: Temperature gradient develops inside. → A single $\tau$ is insufficient. Distributed parameter analysis (e.g., FEM) is needed

Let's think about a smartphone. An SoC chip (5mm square, silicon $k \approx 150$ W/(m·K)) has $\mathrm{Bi} \approx 0.001$, completely a lumped system. Its $\tau \approx 0.3$ to $0.5$ seconds, very short. On the other hand, the entire aluminum chassis ($k \approx 200$ W/(m·K) but large volume) has $\tau \approx 200$ to $400$ seconds.

So, the SoC temperature rises instantly, but it takes minutes for the chassis to warm up. That time lag you feel when you start using a smartphone—"the screen is hot but the back is still cool"—is precisely the difference in thermal time constants.

The starting point for the lumped capacitance method is the overall energy balance of the object. Defining the excess temperature $\theta(t) = T(t) - T_\infty$ ($T_\infty$ is ambient temperature):

This is a first-order linear ordinary differential equation, and the solution is exponential decay:

That is:

Correct. What's important in practice is the achieved percentage after multiple $\tau$:

In thermal shock testing, setting "hold time to $5\tau$" is to ensure reaching 99.3% thermal equilibrium. This concept is also used in JIS C 60068 and IEC 60068.

In that case, the temperature response becomes a superposition of multiple time constants (multi-mode). The transient heat conduction equation for a distributed system:

Solved by separation of variables, the solution takes the form of an infinite series:

Here $\phi_n$ are spatial eigenfunctions, and $\tau_n = 1/\lambda_n$ is the time constant of the $n$-th mode. In the order $\tau_1 > \tau_2 > \tau_3 > \cdots$, as time passes, higher-order modes (smaller $\tau$) decay quickly, leaving the longest $\tau_1$ (dominant time constant).

Excellent analogy. Exactly right. For example, for a one-dimensional flat plate (thickness $2L$) cooled by convection, the eigenvalues $\zeta_n$ are the roots of the transcendental equation $\zeta_n \tan \zeta_n = \mathrm{Bi}$, and the time constant for each mode is:

$\alpha = k/(\rho c_p)$ is the thermal diffusivity. $\zeta_1$ is the smallest, so $\tau_1$ is the largest. When the Fourier number $\mathrm{Fo} = \alpha t / L^2 > 0.2$, approximation with only the first term yields error within 1%—this is the condition for the Heisler chart to be valid.

In FEM, the discretized equation for unsteady heat conduction is:

$[C]$ is the thermal capacitance matrix, $[K]$ is the thermal conductivity matrix (including convection boundaries), $\{F\}$ is the external heat load vector. For $\{F\} = 0$, the homogeneous equation, substituting $\{T\} = \{\phi\}e^{-\lambda t}$ gives:

This is a generalized eigenvalue problem. Each eigenvalue $\lambda_i$ gives the time constant $\tau_i = 1/\lambda_i$. The smallest eigenvalue $\lambda_1$ corresponds to the longest dominant time constant.

Exactly the same structure. In structures, the mass matrix $[M]$ plays the role of "inertia"; in heat, the thermal capacitance matrix $[C]$ does. It can be directly obtained in Ansys Mechanical with *THERMAL_EIGENVALUE or in Abaqus with *HEAT TRANSFER, STEADY STATE=NO, EIGENVECTOR. However, many solvers don't have a direct "thermal eigenvalue" command, so fitting from transient analysis results is often more practical.

There are three most commonly used methods in practice:

1. 63.2% Method: Output the temperature history of the point of interest from transient analysis and read the time when it reaches 63.2% of the initial temperature difference. Simplest, but in multi-mode systems, influence from modes other than the first causes error.
2. Logarithmic Plot Method: Plot $\ln(\theta/\theta_i)$ against time. For pure exponential decay, it becomes a straight line, and the slope $-1/\tau$ gives the time constant. The non-linear part of the curve is the influence of higher-order modes.
3. Curve Fitting: Fit the temperature history data to $\theta = \sum_{n=1}^{N} A_n e^{-t/\tau_n}$ to identify multiple time constants and amplitudes. Called the Prony series method, covering most practical cases with 3-5 modes.

Exactly. Viscoelastic relaxation time, electrical RC time constants, thermal time constants—all are superpositions of exponential decay, and the Prony series is a common analysis tool. In Python, you can easily implement it with SciPy's curve_fit, and MATLAB also has the fit function.

This is a very important point in practice. The guidelines are:

• Implicit Methods (Backward Euler, Crank-Nicolson): Unconditionally stable, but if $\Delta t > \tau$, it "skips over" the response and fails to capture the physics. Guideline is $\Delta t \leq \tau_\mathrm{min}/5$.
• Explicit Methods (Forward Euler): Stability condition $\Delta t < \tau_\mathrm{min}/2$ is mandatory. Constrained by the element-level minimum time constant $\tau_e = \rho c_p (\Delta x)^2 / (2k)$.
• Automatic Time Stepping: Many commercial solvers adaptively adjust $\Delta t$ based on the rate of temperature change. In Ansys, DELTIM; in Abaqus, CETOL (allowable temperature change) in *HEAT TRANSFER are control parameters.

For example, in a model mixing an SoC chip ($\tau \approx 0.5$ s) and a chassis ($\tau \approx 300$ s), initially use $\Delta t \approx 0.05$ s to capture the chip's transient, then gradually increase $\Delta t$ to efficiently compute the chassis's slow change. It's like "zooming a telescope."

There are many use cases. Some typical ones are:

1. Electronic Cooling Design: A power MOSFET's junction temperature rises instantaneously ($\tau_\mathrm{junction} \approx 0.01$ to $1$ s), but the heat sink follows slowly ($\tau_\mathrm{HS} \approx 100$ to $1000$ s). For pulsed loads, the short $\tau$ is important; for steady loads...