Temperature Profiles

HeatQuiz was first designed to teach the fundamentals of conductive heat transport. The main task is to develop temperature profiles within solid bodies. The entire problem is governed by the heat equation (Fourier’s law).

 \dot{Q}(x)=- \lambda \cdot A \cdot \frac{\partial T}{\partial x}

In the simplest case, the heat flux  \dot{Q} , the area  A , and the thermal conductivity  \lambda are constant throughout the body. In this simple case, the temperature gradient  {\partial T}/{\partial x} is constant, leading to a straight temperature profile in the body as shown below.

Conduction in a wall

Note that the temperature rises from left to right. The heat flux is in opposite direction, which is accounted for in the heat equation by the minus sign.

In principle, every term in this equation can vary and can be dependent on the spatial coordinate. For instance, a variation of the thermal conductivity  \lambda can arise in a multi-body system with sharp changes across the different materials. If the area and the heat flux remain constant, the temperature gradient must change contrary to the change in thermal conductivity. A typical example is given in the figure below. Note that you also need to specify the direction of the temperature kink.

Two-body Problem

In a similar way, the area  A can vary, leading also to a changing temperature gradient for a fixed heat flux. A variable area is typical for heat transfer in pipes or spheres. In those cases, the temperature gradient decreases with increasing radial direction due to the increasing surface area.

Conduction in pipe

Heat sources and heat sinks lead to an increasing or decreasing heat flux. Recalling Fourier’s law, a rising heat flux increases the thermal gradient and vice versa. The figure below shows a wall comprising of three different layers. The center layer is heated. At the boundaries of the outer two layers, an equal ambient temperature exists. In this case, the highest temperature is in the heated layer with a maximum shifted to the right side. This shift results from the lower thermal resistance on the right side compared to the thermal resistance on the left side.

Conduction with internal heat source

In all the described cases above, the upper and lower side wall are adiabatic. Heat transfer is only in the horizontal direction. A very common type of problems in heat transfer is fins, commonly known from CPU cooler in the computer. In a classical fin problem, heat is conducted through the material and heat is released to the ambience by convection.

In HeatQuiz, boundaries with convective heat transfer are marked with a blue curve, a given temperature, and a given heat transfer coefficient  \alpha . The figure below shows a fin problem. From the left side, a constant heat flux is imposed. This heat flux is released to the ambient such that the temperature decreases continuously from the left to the right side. Heat is released to the ambience. Thus, the heat flux decreases from the left to the right side, consequently decreasing the slope.

Fin diabatic head

Fin problems differ in the boundary condition of fins head (right side in the figure above). Here, the fin head is characterized by convective boundary condition. Thus, there is heat transfer from the fin head to the ambience and a non-zero temperature gradient. The other type of boundary condition is an adiabatic fin head as shown in the figure below. Adiabatic walls are characterized by crosses. Due to the adiabatic boundary condition at the fin head, the temperature gradient at the boundary is zero.

Fin adiabatic head

Basically, the physical explanations given above are sufficient to solve all heat transfer problems in HeatQuiz. The combination of the different effects however yields to more complex problems.