# Pipe flow with precipitation at boundary

• Sizhe
In summary, the problem at hand involves a plug flow with uniform velocity in a circular pipe, where a supersaturated solution with concentration c is present. At the pipe wall, solute precipitation occurs at a net reaction rate per unit area of R'=k(c-c^{sat}), where k and c^{sat} are independent of time. The goal is to determine the solute concentration as a function of axial and radial position, as well as the thickness of the precipitation layer at a given axial position. This problem can be solved using the analogy of a heat transfer problem, with the rate constant k being equivalent to the heat transfer coefficient. However, the condition that the diffusion layer is thin compared to the diameter may require a boundary layer approximation,
Sizhe

## Homework Statement

Consider a plug flow (slippery walls) with a uniform velocity ##U## in a circular pipe of diameter,##d## and length ##L##. The fluid is a supersaturated solution with a concentration ##c##. At the pipe wall, precipitation of solute happens at a net reaction rate per unit area of ##R'=k(c-c^{sat})##. rate constant ##k## and ##c^{sat}## are independent of time.

## Homework Equations

(1) Determine an expression for the solute concentration as a function of axial and radial position. The precipitated solute does not influence the slipperiness of the wall and bulk flow. And the diffusion layer is thin relative to diameter.
(2) determine an expression for the thickness of precipitation layer as a function of axial position ##x##, if the solid density is ##\rho##.
Hint: the governing equations of this problem bear similarities with a heat transfer problem for which analytical solutions are readily available. And all the transfer coefficients are known.

## The Attempt at a Solution

In a cylindrical coordinate system, the mass conservation of solute in bulk at equilibrium state is (neglecting axial diffusion):
$$U\frac{\partial c}{\partial x} = \frac{D_{salt}}{r}\frac{\partial}{\partial r}(r\frac{\partial c}{\partial r})$$
with the B.C:
$$-D_{salt}\frac{\partial c}{\partial r}=k(c-c^{sat}), @ r=d/2$$
and ##\frac{\partial c}{\partial r}=0 @ r=0##. With this governing equation, I think the analytical solution for the forced convection problem can be used because this problem is similar to the convection heat transfer with constant surface heat flux. However, I suspect that the constant surface flux condition might not hold as the solute flux at the surface depends on the precipitation. Any idea of using the similarity between the heat and mass transfer in this case?

This is indeed analogous to a heat transfer problem. If you let t = x/U, then the equations reduce to those for transient cooling of a cylinder, with a convective heat transfer coefficient to the surroundings at its surface. The rate constant k is basically the heat transfer coefficient. The analytic solution to this problem is presented in Carslaw and Jaeger, Conduction of Heat in Solids, and in Heat Transmission by McAdams.

The only puzzling part of the problem statement relates to the condition that the diffusion layer is thin compared to the diameter. This suggests that you are supposed to solve for the boundary layer approximation at short times (allowing you to neglect the curvature). However, I don't know of a boundary layer solution to this problem with the given boundary condition (although there are boundary layer solutions both to the constant wall temperature case and the constant wall heat flux case). So I don't see how they can ask you for this thin diffusion layer approximate solution.

## 1. What is "pipe flow with precipitation at boundary"?

"Pipe flow with precipitation at boundary" refers to the movement of fluids (such as water or air) through a pipe or conduit, while also taking into account the presence of precipitation (such as rain or snow) at the boundary of the pipe. This type of flow is important to consider in engineering and environmental applications, as it can affect the performance and efficiency of pipes and drainage systems.

## 2. How does precipitation affect pipe flow?

Precipitation at the boundary of a pipe can significantly impact the flow of fluids through the pipe. For example, if the precipitation is heavy enough, it can create a blockage or decrease the diameter of the pipe, leading to reduced flow rates and potential clogging. Precipitation can also change the flow regime within the pipe, resulting in different flow patterns and velocities.

## 3. What factors influence pipe flow with precipitation at boundary?

There are several factors that can influence pipe flow with precipitation at boundary, including the type and intensity of precipitation, the size and material of the pipe, the slope of the pipe, and the properties of the fluid (such as viscosity and density). Other factors such as pipe roughness, pipe fittings, and any obstructions or blockages in the pipe can also impact the flow.

## 4. How is pipe flow with precipitation at boundary studied?

Scientists and engineers use various methods to study pipe flow with precipitation at boundary, including experimental testing, mathematical modeling, and computer simulations. These methods allow for the analysis and prediction of flow patterns, velocities, and other variables under different conditions, helping to optimize the design and operation of pipe systems.

## 5. What are the practical applications of studying pipe flow with precipitation at boundary?

The study of pipe flow with precipitation at boundary has important practical applications in various fields, such as civil engineering (for designing drainage systems and predicting flood risks), environmental science (for understanding the impact of precipitation on water quality and ecosystems), and urban planning (for managing stormwater runoff and reducing the risk of infrastructure damage). It can also help improve the efficiency and sustainability of water and wastewater systems.

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