Tube with Hot Laminar Flow and Outer Heat Transfer

Let us now consider an incompressible laminar flow of hot fluid through an externally cooled circular tube.

The fluid flow has fully developed velocity and temperature profiles at the tube inlet, whereas the heat transfer conditions specified at the tube outer surface surrounded by a cooling medium sustain the self-consistent fluid temperature profile throughout the tube. The geometry model is shown in Figure 1.

Figure 1. Laminar flow in a tube cooled externally: (1) liquid, (2) polystyrene.

In accordance with Ref. 12, a laminar tube flow with a fully developed velocity profile has a self-consistent fully developed temperature profile if the following two conditions are satisfied: the fluid's properties are temperature-independent and the heat flux from the tube inner surface to the fluid (or vise versa) is constant along the tube. These conditions provide the following fully developed tube flow temperature profile:

where

T is the fluid temperature,

r is a radial coordinate (r = 0 corresponds to the tube axis, r = Ri corresponds to the tube inner surface, i.e., Ri is the tube inner radius),

z is an axial coordinate (z = zin corresponds to the tube inlet),

qw is a constant heat flux from the fluid to the tube inner surface,

k is the fluid thermal conductivity,

ρ is the fluid density,

Cp is the fluid specific heat under constant pressure,

umax is the maximum fluid velocity of the fully developed velocity profile:

Since the tube under consideration has no heat sinks and is cooled by surrounding fluid medium, let us assume that the fluid medium surrounding the tube has certain fixed temperature Te, and the heat transfer between this medium and the tube outer surface is determined by a specified constant heat transfer coefficient αe.

By assuming a constant thermal conductivity of the tube material, ks, specifying an arbitrary αe, and omitting intermediate expressions, we can obtain the following expression for Te:

where Ro is the tube outer radius.

In the validation example under consideration (Figure 2) the following tube and fluid characteristics have been specified: Ri = 0.05 m, Ro = 0.07 m, z - zin = 0.1 m, the tube material is polystyrene with thermal conductivity ks = 0.082 W/(m∙K), umax = 0.002 m/s, T(r = 0, z = zin) = 363 K, qw = 147.56 W/m2, k = 0.3 W/(m∙K), Cp = 1000 J/(kg∙K), fluid dynamic viscosity μ = 0.001 Pa∙s, ρ = 1000 kg/m3 (these fluid properties provide a laminar flow condition since the tube flow Reynolds number based on the tube diameter is equal to Red = 100). The T(r, zin) and u(r) profiles at the tube inlet, the Te(z) distribution along the tube, αe = 5 W/(m2∙K), and tube outlet static pressure Pout = 1 atm have been specified as the boundary conditions.

The inlet flow velocity and temperature profiles have been specified as the initial conditions along the tube.

To reduce the computational domain, the calculations have been performed with the Y = 0 and X = 0 flow symmetry planes. The calculations have been performed at result resolution level 7.

The fluid and solid temperature profiles predicted at z = 0 are shown in Figure 3 together with the theoretical curve. It is seen that the prediction practically coincides with the theoretical curve (the prediction error does not exceed 0.4%).

Figure 2. The model used for calculating the 3D flow and the conjugate heat transfer in the tube with Flow Simulation: (1) sketch line for temperature profile determination, (2) computational domain.

Figure 3. Fluid and solid temperature profiles across the tube, predicted with Flow Simulation and compared with the theoretical curve.