Help needed transport problems involving bessel eqn

In summary, the conversation discusses a problem involving the burning of a wooden dowel in a stream of hot air. The goal is to estimate the time it takes for the surface temperature of the dowel to reach 750°F, assuming constant wood properties and neglecting thermal radiation. The initial assumptions include radial transfer, negligible transfer in the z direction, and constant wood properties. The conversation also mentions the use of a lumped capacitance approach and suggests referencing Incropera and Dewitt's Fundamentals of Heat and Mass Transfer for further information.
  • #1
aelisha1079
2
0
Hi everybody... i would like to seek help for the problem below. the assumptions I've considered is that transfer is radial only since it is a very long cylinder (infinitely long) that transfer in z direction is negligible, thermal radiation is zero, and wood properties are constant. Starting from the general energy equation I've come with a form similar to a modified bessel function of zero order but I'm stuck after that. i hope you may be able to help me. thanks a lot in advance.


Problem : Burning of a wooden dowel


In a fire research experiment, a long wooden dowel ( around rod), diameter 1.00 inch, temperature 74°F, is placed in a stream of 1,500°F air. The heat transfer coefficient between gas and surface is 5 Btu/h-ft2-F. Estimate the exposure time up to ignition of the rod on the assumption that ignition takes place when the surface temperature reaches 750°F.
Wood Properties at 74°F
Ρ= 50 lb/ft3
k = 0.10 Btu/h-ft-F
cp = 0.6 Btu/lb-F

assume that the wood properties are constant up to 750°F. Neglect the possible role of thermal radiation.
 
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  • #2
Hi aelisha1079, welcome to PF. I would check first if conduction within the rod can be ignored (i.e., calculate the Biot number). If it can, then a lumped capacitance approach can be used. If not, then the standard approach is to start with the time-dependent solution for the heat equation in cylindrical geometry; there's no reason to try to re-derive it.
 
  • #3
hi mapes! thanks for the reply. I've tried to derive the equation because i can't find a reference with the same case as unsteady heat conduction in an infinitely long cylinder. I would appreciate if you can provide me some referance for the problem. thanks so much
 
  • #4
Check Incropera and Dewitt's Fundamentals of Heat and Mass Transfer.
 

1. What is the Bessel equation?

The Bessel equation is a second-order linear differential equation that is widely used in physics and engineering to describe wave-like phenomena, such as heat transfer, sound waves, and electromagnetic waves. It is named after the mathematician Friedrich Bessel who first studied it in the early 19th century.

2. How is the Bessel equation used in transport problems?

The Bessel equation is used in transport problems to model the behavior of waves or vibrations in a given system. It is particularly useful in problems involving cylindrical or spherical geometry, where the solution to the Bessel equation can represent the radial component of the wave or vibration.

3. What are the typical applications of the Bessel equation in transport problems?

The Bessel equation is commonly used in a variety of transport problems, such as heat transfer in a cylindrical or spherical object, sound propagation in a cylindrical pipe, and electromagnetic waves in a cylindrical waveguide. It is also used in problems involving spherical symmetry, such as the gravitational potential around a spherical object.

4. What are the key properties of solutions to the Bessel equation?

One of the most important properties of solutions to the Bessel equation is that they exhibit oscillatory behavior, with an infinite number of oscillations as the radius or time approaches infinity. Additionally, the solutions can be expressed as a linear combination of two types of Bessel functions: the Jn and Yn functions, which are known as the Bessel functions of the first and second kind, respectively.

5. What are some common techniques for solving transport problems involving the Bessel equation?

There are several techniques for solving transport problems involving the Bessel equation, depending on the specific application and boundary conditions. These include separation of variables, integral transforms, and the use of recurrence relations. In some cases, numerical methods may also be used to approximate the solution.

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