Sphere heating by convection - boundary condition(numerical)

[headrein]

Howdy,

I've been fighting with coding a numerical solution to the unsteady temperature distribution of a cool solid sphere being heated by forced convection of hot gas surrounding it. The problem for me lies in the surface boundary condition.

I must specify heat flux by equating the heat transfered away from the surface to the heat transferred in by convection:

-k(dT/dr) = h(Tsurface - Tgas)

Now, in finite difference form,

dT/dr = (Ts1 - Ts2)/dr

dr = spacial step size now, Ts1 = surface temp, Ts2 is the temperature one step into the sphere from the surface.
Solving for surface temp:

Ts = ((k/dr/h)*Ts2 + Tgas)/(k/dr/h + 1)

This just doesn't work somehow, and I can't figure out why.. I'm dealing with small spheres, and the above equation gives me a surface temp about equal to that of the next layer in, and the damn sphere heats up very very slowly.. I've also compared it to the analytical solution which confirms it's not right.

Does anybody know of a numerical method for handling this boundary which works?

many thanks,
eric

(sorry for the lengthy post)

 

[sailoday28]

I believe that this problem is being over simplified and might recieve more responses if placed on the Heat Transfer and Thermdynamics portion of the forum.

If you consider quasi-steady flow:
Your heat transfer coefficient, h, will vary over the surface. Consider where the flow hits the sphere dead on and where it leaves,180 degrees later. The velocity distribution and therefore its effect on h can be significant.
Further, I believe you are implying with the boundary condition dT/dr = (Ts1 - Ts2)/dr, that the outer surface of the sphere is only a function of time and not coordinate. If that is true, then you are using 1D equation for temp distribution in the sphere which could lead to error.
I would look for quasi-steady heat transfer coef as a function of geometry and model with more than one dimension.

 

[IRstuff]

Your premise appears incorrect.

The basis of the standard heat flow equation starts with a source of heat that travels through a series of layers and ends up in a heat sink, which is the air.

The convection coefficient of the air layer presumes an infinite sink at t[sub]ambient[/sub].

T[sub]surface[/sub] is a function of the heatflow across the interface and the interior temperature of the sphere, which is not constant. Your equations need to account for that.

TTFN

 

[onebug]

1) the problem is 2D-axisymmetric.
2) do you have sufficient mesh resolution to solve for the boundary layer?
3) is the flow laminar or turbulent?
4) the heat transfer coeff is not a constant, but a function of space for steady state.
5) at the surface, there's only balance of conduction heat transfer: heat flux of conduction of gas to surface equals to that of solid. another condition is temperature continuity. depend on your gas temperature, you can neglect radiation.

 

[mfontenot29]

One way to do this is to include the solid in the computation domain. In the solid of course, all velocities are set to zero, and the specific heat and other properties must be set accordingly. This is called a conjugate heat transfer problem and is not uncommon.

Hope this has helped.

 

[corus]

The problem isn't 2D axisymmetric and your boundary conditions are correct if you're assuming h to be constant everywhere. For small objects you can assume that the temperature is constant throughout and simplify the heat flow equation to (4/3)p.C.r^3.dT/dt=-h.4.r^2,(T-Ta) for a sphere, or dT/dt = -3h.(T-Ta)/(p.C.r), which can easily be solve.

corus

 

[sailoday28]

corus (Mechanical) STATES
The problem isn't 2D
Clearly, if you assume the sphere temp constant, the convection coef and surrounding ambient constant, then the problem is one dimensional.

Corus--What is your basis for the above assumptions?

 

[corus]

The basis is that the questioner refers to small spheres and hence the sphere is approximately at a uniform temperature, as you said. With that the heat diffusion equation reduces to a form of dT/dt = f(T,r) and can be solved easily. p is density and C is specific heat by the way.

You could also consider it as 1 dimensional with spherical co-ordinates, or in 2 dimensions when, as modelled as an arc, it would be axisymmetric. There'd be no need for 2D though with a single heat transfer coefficient and either 1D spherical co-ordinates or my approximate analytical method would do.

corus

 

[sailoday28]

Corus-Your approach is that of a lumped system. Generally that approximation is true, only if the convection coef is small compared to thermal conductivity of the sphere.

I suggest this approach if you compare closed form solutions of a sphere with the approx film coef, same diam and thermal diffusivity. If temp at center of sphere compares with lumped analysis, then lumped is satisfactory way to go.

 

[corus]

sailoday28,
Sinilar to what you describe, I have seen a formula for the 'biot modulus' based upon the epxression hL/k for a cylinder of length L. What it is for a sphere I have no idea. Howver, the test of whether the approximation leads to negigible error is if this biot modulus is small. What small is, is another matter.

corus

 

[Kess]

For the closed form solution - see page 287 and 288 of Conduction Heat Transfer by Vedat Arpact (Addision Wesley).