# Chapter 11

# One-Dimensional Flows with Friction and Heat Transfer

## 11.1. Friction force and heat transfer on a wall

We study a one-dimensional flow in a pipe with the section *A*(*x*), where a friction force is exerted on the wall at the same time as heat is exchanged with the fluid. A supply can be made by burning kerosene or any other mode of exchange: wall heating, radiation, electromagnetic effect, and mixing of hot gases. If this exchange is negative, the wall extracts heat from the fluid and cools it down (cooling problems). The case of the friction effect without heat supply corresponds to flows in pipes transporting fluids, such as gas or oil pipelines.

To address these problems, one-dimensional models have been proposed where the properties of the flow are assumed to be constant in any section *A*(*x*). They combine a great simplicity of implementation with the advantage of providing sufficient accuracy for many industrial applications. In reality, the flow is two-dimensional, or three-dimensional, a boundary layer with the thickness *δ* increasing with distance *x* forming on the pipe wall. The skin friction *τ*_{w} is a consequence of the viscosity of the fluid creating a force, or contact action exerted by the fluid on the wall (see Chapter 4). The effect of the wall on the fluid is, of course, equal but with an opposite sign. The skin friction *τ*_{w} is expressed within the approximation framework of the boundary layer:

where *μ*_{w} is the value of the molecular viscosity at the wall, *y* the ...