# Solving the Convection-Diffusion Equation for this Pipe with a Heat Sink

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• HumanistEngineer
In summary: All we are interested in is the length-integrated heat transfer coefficient, ##\rho C_p\left(\frac{\pi d_i^2}{4}\Delta x\right)##.
HumanistEngineer
TL;DR Summary
Temperature propagation through an insulated pipe in time
The solution of convection-diffusion equation with a heat sink (heat loss from pipe to the ground)
Hi Again,

I try to solve the transient temperature propagation through a buried insulated pipe by means of solving the convection-diffusion equation with a heat sink that is the heat loss from the water mass to the ground. Below you can see the details of my calculation steps in my numerical analysis.
The problem is that my results are oscillating and/or abnormal. Would you please check my finite difference approximations, if they are correct or not?

Details:

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It looks OK. So??

Here is the result (for different mesh numbers). The maximum (steady-state) temperature should be at around 70 °C.

That last term in the differential equation should involve a heat transfer coefficient. Please provide the derivation of that last term in the equation. What is the inlet condition and the initial condition? What is the temperature of the ground?

I get $$c=\frac{h}{\rho C}\frac{4}{D}$$where h is the heat transfer coefficient

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Thank you Chestermiller for your time. You indicated c as c = h/(rho cp) (4/D) . Why is there this 4/D?

I use the thermal resistance, R [K/W], in the last term c, which means that h=1/R. Here how R was derived:

The initial condition for the pipe is that the water temperature is initially at 50 °C and in my numerical example the inlet water temperature is increased to 70 °C. Then the aim is to see how the temperature propagates through the pipe length in time. The ground temperature is constant at 10 °C.

If U is the overall heat transfer coefficient based on the inside diameter of the pipe, then the balance equation over a length ##\Delta x## of pipe, in finite difference form, goes like:
$$\rho C_p\left(\frac{\pi d_i^2}{4}\Delta x\right)\frac{\partial T}{\partial t}+\ ... \ =\ ...\ +\pi d_i\Delta xU(T-T_{ground})$$
If we divide this by ##\rho C_p\left(\frac{\pi d_i^2}{4}\Delta x\right)## and take the limit as ##\Delta x## approaches zero, we obtain: $$\frac{\partial T}{\partial t}+\ ...\ =\ ...\ +\frac{U}{\rho C_p}\frac{4}{d_i}(T-T_{ground})$$
Here, $$\frac{1}{Ud_i}=\frac{1}{h_{flow}d_i}+\frac{1}{2\lambda}\ln{(d_{o}/d_{i})}+...$$

Note that ##\Delta x## is not present here.

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## 1. What is the Convection-Diffusion Equation?

The Convection-Diffusion Equation is a mathematical model used to describe the transport of heat, mass, or momentum in a fluid. It takes into account both convection (the transfer of heat due to fluid motion) and diffusion (the transfer of heat due to temperature differences) in a system.

## 2. How is the Convection-Diffusion Equation applied to a pipe with a heat sink?

In the context of a pipe with a heat sink, the Convection-Diffusion Equation is used to model the transfer of heat from the fluid inside the pipe to the surrounding environment. The heat sink acts as a boundary condition, absorbing or dissipating heat from the fluid as it flows through the pipe.

## 3. What factors affect the solution of the Convection-Diffusion Equation for this pipe with a heat sink?

The solution of the Convection-Diffusion Equation for a pipe with a heat sink is affected by various factors such as the fluid flow rate, the temperature difference between the fluid and the environment, the size and material of the pipe and heat sink, and the properties of the fluid (e.g. viscosity, density, thermal conductivity).

## 4. How is the Convection-Diffusion Equation solved for this problem?

The Convection-Diffusion Equation can be solved using various numerical methods, such as finite difference, finite element, or finite volume methods. These methods discretize the domain (i.e. the pipe and heat sink) into smaller elements and solve the equation for each element, taking into account the boundary conditions and initial conditions.

## 5. What are the practical applications of solving the Convection-Diffusion Equation for a pipe with a heat sink?

The solution of the Convection-Diffusion Equation for a pipe with a heat sink has various practical applications, such as designing heat exchangers, optimizing cooling systems, and predicting the temperature distribution in industrial processes. It is also used in the fields of fluid dynamics, thermodynamics, and heat transfer for understanding and analyzing heat and mass transfer phenomena.

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