Establishment of the Unsteady Heat Conduction Differential Equation
In the study of heat transfer, thermal processes are broadly categorized into two regimes: steady-state conduction and unsteady-state (transient) conduction. Steady-state conduction describes a system that has reached thermal equilibrium, where the temperature at any given point remains constant over time and is solely a function of spatial coordinates.
However, most real-world engineering phenomena—such as the rapid quenching of steel, the thermal management of high-performance microelectronics, or the heating of building envelopes—are inherently time-dependent. In these scenarios, the temperature field fluctuates as energy moves through the medium. To model these processes, we must establish the unsteady heat conduction differential equation, a partial differential equation (PDE) that describes how temperature $T$ evolves with respect to both position $\mathbf{r}$ and time $t$.
The derivation of the unsteady conduction equation relies on two fundamental principles of physics: the Law of Conservation of Energy and Fourier’s Law of Heat Conduction.
1. The Law of Conservation of Energy
According to the First Law of Thermodynamics, for any infinitesimal control volume ($dV$), the rate of change of internal energy must be balanced by the net rate of heat energy entering the volume plus the rate at which heat is generated within it. Mathematically, this is expressed as:
$$\text{Rate of Energy Accumulation} = \text{Net Heat Flux Inflow} + \text{Rate of Internal Heat Generation}$$
2. Fourier’s Law of Heat Conduction
Fourier’s Law provides the constitutive relationship between the heat flux and the temperature gradient. It states that the local heat flux density $\mathbf{q}$ is proportional to the negative gradient of the temperature:
$$\mathbf{q} = -k \nabla T$$
In this expression, $k$ represents the thermal conductivity of the material. The negative sign is physically significant: it dictates that heat naturally flows from regions of higher temperature to regions of lower temperature, following the direction of the steepest temperature decrease.
Mathematical Derivation of the Governing Equation
To derive the differential form of the heat equation, we consider a microscopic control volume $dV$ within a solid medium.
Step 1: Quantifying Energy Accumulation
Assuming the material has a density $\rho$ and a specific heat capacity at constant pressure $c_p$, the internal energy per unit volume is $\rho c_p T$. Therefore, the rate at which energy accumulates within the control volume $dV$ over time is:
$$\rho c_p \frac{\partial T}{\partial t} dV$$
Step 2: Calculating Net Heat Inflow
The net heat entering the control volume through its surface $A$ is the integral of the heat flux over that surface. To convert this surface integral into a volume integral, we apply Gauss’s Divergence Theorem:
$$\oint_A \mathbf{q} \cdot \mathbf{n} , dA = \int_V (\nabla \cdot \mathbf{q}) , dV$$
Since we are interested in the net inflow (the energy entering the volume), we take the negative of the divergence:
$$\text{Net Inflow} = -\int_V (\nabla \cdot \mathbf{q}) , dV$$
Step 3: Accounting for Internal Heat Sources
If the medium contains internal heat sources—such as those generated by electrical resistance (Joule heating), chemical reactions, or nuclear decay—we denote the volumetric heat generation rate as $\dot{q}$. The total heat generated within the volume is:
$$\int_V \dot{q} , dV$$
Step 4: Synthesis of the General Equation
By substituting these three components into the energy conservation balance, we obtain:
$$\int_V \rho c_p \frac{\partial T}{\partial t} , dV = -\int_V (\nabla \cdot \mathbf{q}) , dV + \int_V \dot{q} , dV$$
Since this equality must hold for any arbitrary control volume $dV$, the integrands themselves must be equal. This yields the general differential form:
$$\rho c_p \frac{\partial T}{\partial t} = -\nabla \cdot \mathbf{q} + \dot{q}$$
Finally, by substituting Fourier’s Law ($\mathbf{q} = -k \nabla T$) into the equation, we arrive at the fundamental unsteady heat conduction equation:
$$\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + \dot{q}$$
Simplifications and the Role of Thermal Diffusivity
In many engineering applications, the complexity of the general equation can be reduced through specific assumptions.
Isotropic Media and Constant Thermal Conductivity
If the material is isotropic (properties are identical in all directions) and the thermal conductivity $k$ is assumed to be independent of temperature, the term $\nabla \cdot (k \nabla T)$ simplifies to $k \nabla^2 T$, where $\nabla^2$ is the Laplacian operator. The equation becomes:
$$\frac{\partial T}{\partial t} = \frac{k}{\rho c_p} \nabla^2 T + \frac{\dot{q}}{\rho c_p}$$
At this stage, we introduce a critical thermophysical property known as thermal diffusivity ($\alpha$):
$$\alpha = \frac{k}{\rho c_p}$$
Thermal diffusivity measures the ability of a material to conduct thermal energy relative to its ability to store it. A high $\alpha$ implies that the material responds rapidly to thermal changes, whereas a low $\alpha$ indicates a slower, more "sluggish" thermal response. The simplified equation is then:
$$\frac{\partial T}{\partial t} = \alpha \nabla^2 T + \frac{\dot{q}}{\rho c_p}$$
One-Dimensional Approximation
For geometries such as thin plates or long wires where lateral heat loss is negligible, we can simplify the spatial component to a single dimension ($x$):
$$\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} + \frac{\dot{q}}{\rho c_p}$$
Practical Application: Transient Heating of a Uniform Rod
To illustrate how this mathematical framework is applied, consider the following scenario:
Problem Statement:
A uniform metal rod of length $L$ has constant properties ($\rho, c_p, k$). At $t=0$, the entire rod is at a uniform temperature $T_0$. The ends of the rod are suddenly brought into contact with two heat reservoirs maintained at constant temperatures $T_1$ and $T_2$.
Mathematical Modeling:
To solve for the temperature distribution $T(x, t)$, we must define three components:
- Governing Equation: Since there is no internal heat source ($\dot{q}=0$):
$$\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$$ - Initial Condition (IC): Defines the state at the start:
$$T(x, 0) = T_0 \quad \text{for } 0 \le x \le L$$ - Boundary Conditions (BC): Defines the interaction with the environment:
$$T(0, t) = T_1 \quad \text{and} \quad T(L, t) = T_2 \quad \text{for } t > 0$$
Solution Strategy:
This is a non-homogeneous boundary value problem. The standard analytical approach is to decompose the solution into a steady-state component $T_s(x)$, which satisfies the boundary conditions, and a transient component $T_t(x, t)$, which satisfies homogeneous boundary conditions. The transient part is typically solved using the separation of variables method.
Conclusion
The establishment of the unsteady heat conduction equation is a cornerstone of thermal science. By synthesizing the conservation of energy with Fourier's Law, we create a powerful mathematical tool capable of predicting temperature evolution in complex systems. Successful thermal modeling requires not only the correct governing equation but also a precise definition of:
- Material Properties: Specifically the thermal diffusivity ($\alpha$).
- Initial Conditions: The starting thermal state of the system.
- Boundary Conditions: The nature of the thermal exchange at the system's limits (e.g., Dirichlet, Neumann, or Robin conditions).
Mastering these elements is essential for anyone performing advanced thermal simulations or designing efficient heat transfer systems in modern engineering.