# Heat equation: Convection

• tjackson3
In summary: It doesn't seem like it will converge very quickly, and it doesn't feel like something you can easily analyze the behavior of.In summary, the problem involves finding the transient and steady-state temperature distribution in a thin rod with one end heated to temperature T_0 and the other cooled by convection in a fluid of temperature T_f. Assuming unit thermal diffusivity and heat transfer coefficient, the heat equation u_t = u_xx is used to solve the problem. The boundary conditions and initial temperature distribution are discussed, including the use of Fourier's law of heat conduction. Separation of variables is used to obtain a solution, but the convergence and behavior of the solution may be difficult to analyze.
tjackson3

## Homework Statement

A thin rod of length ∏ is heated at one end to temperature $T_0$. It is insulated along its length and cooled at the other end by convection in a fluid of temperature $T_f$ . Find the transient and steady-state temperature distribution in the rod, assuming unit thermal diff usivity (K = 1) and unit heat transfer coefficent ( $\gamma = 1$). Also assume arbitrary initial temperature distribution.

## Homework Equations

The heat equation: $u_t = u_xx$

## The Attempt at a Solution

For one thing, I'm not sure I'm setting this up entirely correctly. The equation I'm attempting to solve is

$$u_t = u_xx$$

Subject to

$$\begin{eqnarray*} u(0,t) &= & T_0 \\ u(x,0) & = & f(x) \\ u_x(\pi,t) & = & u(\pi,t) - T_f \end{eqnarray*}$$

That last equation comes from Fourier's law of heat conduction and ensures that the heat flux is properly matched at the boundary. If this is the correct equation with the correct set of initial/boundary conditions, them I'm still lost. Even if I try separation of variables, clearly the eigenvalue problem has to be the problem in x. However, since you don't have a pair of homogeneous boundary conditions. Even if you were to transform the first one into a homogeneous boundary condition, I have no idea what you'd do with the Robin one.

One last note: We did a similar example in class. I don't think that it was stated that the rod was insulated along its length (in particular, this example was a cooling fin). The equation we had in that case was:

$$\rho c u_t = u_xx - (p/A)(T-T_f)$$

where $\rho,c,p,A$ were density, specific heat, perimeter, and area, respectively. However, we arrived at this equation by considering a small slice of surface area ΔS, generated by looking at a small slice of length Δx, and equating the rate of increase of stored energy in the corresponding ΔV with heat flow by conduction - heat flow by convection through ΔS. But if it's insulated along its length, wouldn't conduction be the only mechanic at work?

Thank you so much!

Alright, I tried making a change of variables, and here's what I got:

I defined $u^0(x,t) = \frac{T_f-T_0}{1+\pi}x + T_f$. This satisfies both boundary conditions. Then define $v(x,t) = u(x,t) - \frac{T_f-T_0}{1+\pi}x - T_f$. Then v(x,t) satisfies the same old heat equation, with boundary conditions $v(0,t) = 0, v_x(\pi,t)-v(\pi,t) = 0$.

Then using separation of variables, you get

$$v(x,t) = \sum_{n=1}^{\infty}\ c_n e^{-\lambda^2t}\sin\lambda x$$

where $\lambda$ is given implicitly as the roots of the equation $\lambda = \tan\lambda\pi$ (which looks familiar, like something out of a PDE course I took last year). The coefficients are given in the usual Fourier series way, but you can't really come up with an analytic form for them, since it involves integrating $\sin\lambda x$. Then the final answer is

$$u(x,t) = \frac{T_f-T_0}{1+\pi}x + T_f + \sum_{n=1}^{\infty}\ (\int_0^{\pi}\ \sin\lambda_nx (f(x)-\frac{T_f-T_0}{1+\pi}x - T_f)\ dx) e^{-\lambda^2t}\sin\lambda x$$

But that solution feels weird

Last edited:

## 1. What is the heat equation?

The heat equation is a mathematical equation that describes how heat is transferred through a material. It takes into account the temperature, thermal conductivity, and heat capacity of the material to calculate the rate of heat transfer.

## 2. What is convection?

Convection is a type of heat transfer that occurs when a fluid, such as air or water, moves and carries heat with it. This can happen through natural convection, where the fluid moves due to temperature differences, or forced convection, where an external force, such as a fan or pump, is used to move the fluid.

## 3. How does the heat equation relate to convection?

The heat equation can be used to model convection by incorporating the fluid's movement and heat transfer rate into the equation. This allows us to predict the temperature distribution and heat transfer within a system experiencing convection.

## 4. What are the boundary conditions for the heat equation in convection?

The boundary conditions for the heat equation in convection include the temperature at the surface of the material, the fluid velocity near the surface, and the heat flux at the surface. These conditions are necessary to accurately model the heat transfer through convection.

## 5. How is the heat equation in convection solved?

The heat equation in convection is typically solved using numerical methods, such as finite difference or finite element methods. These methods involve discretizing the material into smaller parts and solving the equation for each part to approximate the temperature distribution and heat transfer within the system.

• Calculus and Beyond Homework Help
Replies
5
Views
263
• Calculus and Beyond Homework Help
Replies
7
Views
829
• Calculus and Beyond Homework Help
Replies
0
Views
448
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Differential Equations
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
2K