Question

How is the Reynolds analogy expressed? What is the value of it? What are its limitations?

Short answer

Answer: The limitations of the Reynolds analogy include: (a) assuming Prandtl number is equal to 1, (b) only valid for turbulent flow regime, (c) assuming constant temperature and fluid properties, and (d) not considering factors such as variable viscosity, surface roughness, and non-uniform temperature distributions. More accurate methods like correlations or numerical simulations should be used when these limitations significantly impact the accuracy, or when the Prandtl number differs significantly from 1.

Step-by-step solution

1. Expression of Reynolds Analogy

The Reynolds analogy is expressed as a relationship between the friction factor, heat transfer coefficient, and the Reynolds number for a turbulent flow. The analogy can be written as: Rex * Pr = (St * Dex * k) / μ Here, - Rex is the Reynolds number, which is a dimensionless quantity representing the ratio of inertial forces to viscous forces in the fluid flow - Pr is the Prandtl number, another dimensionless quantity representing the ratio of momentum diffusivity to thermal diffusivity - St is the Stanton number, a dimensionless quantity representing the ratio of heat transferred by convection to the heat transferred by conduction across a boundary layer - Dex is the hydraulic diameter of the channel where the flow occurs - k is the thermal conductivity of the fluid - μ is the dynamic viscosity of the fluid

2. Value of the Reynolds Analogy

The value of the Reynolds analogy relates the Stanton number (St) to the friction factor (f) and the Prandtl number (Pr), as shown below: St = Cf / (2 * Pr) Where Cf is the friction factor or the Fanning friction factor. This relationship makes it possible to estimate the heat transfer coefficient from flow friction data or vice versa.

3. Limitations of the Reynolds Analogy

Although the Reynolds analogy is a useful tool for estimating heat transfer coefficients in turbulent flow, it has several limitations: a. The Reynolds analogy assumes that the Prandtl number is equal to 1, which is not always the case for all fluids. The accuracy of the analogy decreases when the Prandtl number differs significantly from 1. b. The analogy only holds for the turbulent flow regime and is not generally valid for laminar or transitional flows. c. It assumes a constant temperature and constant properties of the fluid, which may not always hold true in real-world applications. d. The Reynolds analogy doesn't consider factors that may affect heat transfer, such as variable viscosity, roughness of the surface and wall, and non-uniform temperature distributions. Despite these limitations, the Reynolds analogy can still provide an acceptable approximation for heat transfer in turbulent flows when the Prandtl number is near 1 and under certain simplifying conditions. Nevertheless, more accurate correlations or numerical simulations should be used for cases where the limitations of the analogy significantly impact its accuracy.