Conductive Heat Transfer 3

[mikelfew]

We need to calculate how long it takes for an aluminium plate to assume the same temperature as the heated air in an oven. To a first approximation it may be assumed that the amount of heat available for the heated air in the oven is effectively infinite compared to the mass of the aluminium plate. Assuming the plate is 12"x12"x1" and loses no heat through conduction to the outer wall of the oven how long will it take before the aluminium is soaked through to the same temperature as the heated air inside the oven? Typically the oven and the plate start at a temperature of 22°C and then the aluminium plate is heated using air which is heated in the oven at the rate of 1°C/minute controlled by an electronic controller.

Thanks very much

 

[IRstuff]

Essentially it's an inverse convection cooling problem, so a first order estimate heating rate might be close to the effective time constant of the system, and multiplying by 3 might be close to the total heating time.

TTFN

FAQ731-376

 

[mikelfew]

Thanks for your input - that's pretty much what we are doing now. Effectively an estimate. However, I was hoping for a more definitive theoretical approach to the problem.

Thanks again though - I appreciate your input.

 

[ione]

The high thermal conductivity of aluminium, coupled with a relatively small thickness, implies a low Biot number and so I would approach the problem through a lumped capacitance model.

If the air temperature of the oven were constant the plate temperature would asymptotically tend to Tamb:

T(t)=Tamb - (Tamb - Ti)*exp[-A*h*t/(m*cp)]

Being :

Tamb = ambient/oven temperature
Ti = starting temperature
A = plate exposed area
h = heat transfer coefficient (convection + radiation)
m = plate mass
cp = aluminium specific heat.
t = time

Now the problem could be a bit more tricky as your Tamb is not constant but varies at the rate of 1°C/minute.

 

[mikelfew]

Thanks Ione, that's really helpful. I can assume that the oven will ramp up to ambient + 50°C in 30 minutes and if I assume that no heat transfer takes place to the plate during that period I will be on the safe side and this will simplify the math. Can I assume the units of the various elements are: °K, metres, kg, seconds, cp = J/g°K, h = W/sq metre*K? Lastly, have you any idea on a number for the heat transfer coefficient for air/aluminium.

I really appreciate your help on this one.

 

[ione]

My miss to not having reported units in my previous post, anyway you are right.

If both the down side and the upper side of the plate are exposed to still air (natural convection) I would consider approx 6 W/(m^2*K). This value takes into account also the radiative contribute.

 

[mikelfew]

Thanks very much - I'll go away and do the sums!

 

[IRstuff]

Just bear in mind that a commercial convection conveyor oven can fully cook a pizza in under 7 minutes and there are high-end ovens that can do it in less than 4 minutes.

TTFN

FAQ731-376

 

[Dave41A]

For the "ramp" part of the heating cycle (where the oven is warming up at a steady rate), it may be helpful to note that the response of a first order system (i.e. lumped capacitance thermal system) is itself a ramp function that simply lags behind the ambient temperature by a time period equal to the time constant (the first few seconds are not a perfect ramp, but close enough).

Then, when the oven reaches steady-state, the response of the system is that is a first-order system to a steady input...i.e. the temperature of the plate will asymptotically approach that of the oven.

You should be able to model the temperature of the plate to a high degree of accuracy in this piece-wise manner.

Good luck,

Dave

 

[zekeman]

You have to solve the ODE with the ramp (part 1).Using some of Ione's notation

MCpdT/dt=hA(To+at-T)

where Tamb= T0+at
t= time
a= ramp rate 60 deg C/hr
MCp/hA= time constant of system

The solution is:

T=(T0+aMCp/hA)*exp-(t(hA/MCp)+at-aMCp/hA
After heating the oven to its final temperature Tf, at time s, the temperature of the aluminum is Tal<Tf so the ODE becomes for t>s (part2)

MCpdT/dt=hA(Tf-T)
and the solution to this part is
T=Tal+(Tf-Tal)exp-[(hA/MCp)(t-s)]

 

[zekeman]

correction
The last equation should read

T=Tf-(Tf-Tal)exp-[(hA/MCp)(t-s)] or

T=Tal +(Tf-Tal){(1-exp-[(hA/MCp)(t-s)]

which shows that the aluminum reaches Tal after the ramp rise of oven temperature and then goes asymtotically toward Tf

 

[ione]

Long time since I’ve taken my math course, so “handle with extreme care”, anyhow the solution of the first differential equation (the one concerning the temperature profile during the ramp time) should be:

T(t) = (aMCp)/(hA)*[exp(-hAt/(MCp)) -1 ] + T0 + at

 

[zekeman]

Ione,

Thanks for the correction.

Have to pay more attention to the particular solution.