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How to Solve the Heat Flux Equation

Key Takeaways

  • The importance of thermal fluid flow for aerodynamic systems.

  • Understanding the basic heat flux equation.

  • Analyzing thermal conditions for high-speed aerospace vehicles. 

Example of heat fluid flow increase when reentering earth’s atmosphere

Extreme heat fluctuation when reentering the earth's atmosphere

It is critical that we understand how temperature changes as aerospace vehicles travel in, out, and through the atmosphere. For fluid flow systems, this is typically much more complex than simply noting that heat tends to go from a higher temperature to a lower one. In fact, heat variation is important for virtually all fluid dynamics equation implementations, as temperature can impact fluid density, velocity, pressure, viscosity, and other important parameters. 

When analyzing thermal properties, the heat flux equation is a tool that is often employed. Therefore, it is critical that this equation be understood, including how to apply and obtain solutions for different systems. For aerodynamic systems, this is best appreciated by knowing the impact of fluid heat flow.

The Importance of Thermal Fluid Flow Conditions for Aerodynamic Systems

Most of us are aware that adverse weather conditions--like storms or extreme cold --can affect aircraft. However, the same is true for high heat, which may be due to climate or occur during flight. Aircraft rely on the ability to get and maintain lift to fly. Lift is created when the air pressure above a surface is lower than the pressure of the flow beneath it. Variation in temperature can affect these pressures. The degree to which these changes occur and the implications vary depending upon aircraft speed and, consequently, the velocity of the airflow. At higher temperatures, these implications can be quite significant and create problems for aerodynamic system performance.  

Aerodynamic System Problems from High-Temperature Fluid Flow

  • Difficulty acquiring and/or maintaining lift
    On extremely hot or high humidity days, it is not uncommon for some flights to be grounded. This is due to the lower density of air, which makes it harder for aircraft to degenerate enough speed to produce the necessary lift to get or remain airborne. This phenomenon is due to the fact that gas molecules are spaced farther apart--lower density--and sufficient lift force is difficult to generate or maintain. This is primarily an issue for heavier, slower jets used for passenger travel. 

  • Structural deformation and/or breakdown
    At the other end of the spectrum, faster planes, such as supersonic or hypersonic jets, are also subject to problems due to high thermal flow. In these cases, high-temperature fluid flow interacts with the object through convection and/or radiation and can cause structural issues that may range up to material degradation.

As these potential problems indicate, knowing how your system will respond to heat flow is imperative. This begins with an understanding of the heat flux equation.  

Understanding the Basic Heat Flux Equation

In order to understand the heat flux equation, it is helpful to first define a few terms.


Heat density is the amount of heat transferred per unit area.

Heat flux or heat flux rate is the amount of thermal energy that is transferred per unit area with respect to time.

Thermal conductivity is the rate--per unit of time--at which thermal energy passes through a specific material--per unit of area. 

As these definitions indicate, all of the above terms describe the transfer of thermal energy through some specified material. However, only thermal conductivity is actually considered a material property. The heat flux equation relates the thermal conductivity to a specific temperature change or gradient, as shown below.

    q = -k∇T   

The equation above is based on work done by Joseph Fourier, which led to the Fourier series of infinite integrals. This equation is readily solvable for simple one-dimensional evaluations. However, for complex heat flux flow evaluations that are necessary to study high-speed aerodynamic systems, higher-order analysis and more advanced techniques are necessary.

Analyzing Thermal Conditions for High-Speed Aerospace Vehicles

Due to the high temperatures generated for aircraft flying at supersonic--Mach 1--and above speeds, many parameters that can be neglected for slower aircraft must be considered. For example, heat flow must be looked at from several perspectives. These include thermal energy flow both inside the object’s body and external to it, as there will be both convective and radiative concerns. Moreover, there are multiple areas--airfoils, nose, engines, body, etc.--for which analysis is required. This level of complexity necessitates the use of CFD tools such as those utilized in the example shown below.

Example of heat differences for high-speed aircraft vehicle flight

High-speed aircraft fluid flow in Omnis

The criticality of accurately solving the heat flow equation for high-speed aircraft requires that advanced CFD tools such as Cadence’s Omnis (shown above) be employed when designing these high-performance vehicles. 

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