One dimensional transient heat flow of a hoop

[crawfs3]

1. Problem: derive the energy balance from first principles of a hoop of inner radius ri, outer radius ro. the hoop material has a density of rho (p), heat capacity of c and thermal conductivity k. the center of the hoop has a temperature of T1 and the gas inside the hoop has a convection coefficient of h1. The temperature outside the hoop (not at the outer surface of the hoop) is T2 and the gas outside has a convection coefficient of h2.


2. Attempted solution: since its a hoop, cylindrical coordinates should be used. And since its a one dimensional problem the heat transfer in the y and z components is zero as well as the thermal generation term. I get the following equation: (1/r)d/dr(krdT/dr) = pcdT/dt. This is the equation for the conduction through the hoop and can be solved fully through integration and boundary conditions.

I can derive the temperature profile from the centre of the hoop to the edge of the inner radius using the equivalent thermal circuit but it wants the energy balance so I don't know what to for convection. Please help. Thanks.

 

[KataKoniK]

Hello, I understand your confusion about how to incorporate convection into the energy balance equation for this problem. Let me break it down for you step by step.

Firstly, we need to consider the energy balance for the hoop as a whole. This includes the energy transfer through conduction, as you have mentioned, but also the energy transfer through convection.

The energy balance equation for the hoop can be written as follows:

Energy in - Energy out = Energy stored

Energy in: This includes the heat transfer through convection from the gas inside the hoop to the inner surface of the hoop, as well as the heat transfer through conduction from the inner surface to the outer surface of the hoop.

Energy out: This includes the heat transfer through convection from the outer surface of the hoop to the gas outside, as well as any radiation heat transfer (if applicable).

Energy stored: This refers to the change in internal energy of the hoop material, which is represented by the heat capacity term in the equation.

Now, let's break down the energy transfer terms in more detail. The heat transfer through convection is given by the following equation:

Q = hA(T1-T2)

Where Q is the heat transfer, h is the convection coefficient, A is the surface area, and T1 and T2 are the temperatures at the inner and outer surfaces of the hoop respectively.

For the heat transfer through conduction, we can use Fourier's law:

Q = -kA(dT/dr)

Where Q is the heat transfer, k is the thermal conductivity, A is the cross-sectional area, and (dT/dr) is the temperature gradient in the radial direction.

Using these equations, we can write the energy balance equation for the hoop as:

h1A(T1-T2) - h2A(T2-T∞) - k(2πrL)(dT/dr) = pVc(dT/dt)

Where L is the length of the hoop, V is the volume, and c is the heat capacity of the hoop material.

Note that the first term on the left side represents the energy transfer from the gas inside the hoop to the inner surface, the second term represents the energy transfer from the outer surface to the gas outside, and the third term represents the energy transfer through conduction from the inner to the outer surface. The right side represents the change in internal energy of the hoop material.

Now, to solve this equation, we need to apply

 

[Mene]

3. Your approach using cylindrical coordinates and the heat transfer equation for conduction through the hoop is correct. However, to fully solve for the energy balance, you will also need to consider the effects of convection on both sides of the hoop. This can be done by applying the Newton's Law of Cooling, which states that the rate of heat transfer due to convection is proportional to the temperature difference between the surface and the surrounding fluid, and the heat transfer coefficient.

In this case, you will have two equations for convection, one for the inside surface of the hoop and one for the outside surface. These equations will include the convection coefficients and the temperature differences between the surface and the surrounding fluid.

Once you have these equations, you can combine them with the heat transfer equation for conduction and the given boundary conditions to solve for the temperature profile and the energy balance of the hoop. It may also be helpful to draw a thermal circuit diagram to visualize the different heat transfer mechanisms and how they are connected in the hoop.

I hope this helps in solving the problem. Good luck!