Calculating Soak Time in Oven to Reach Ambient Temperature 4

[KimWonGun]

Is the following logic correct to estimate the time required for a metal object in an oven (convection) to be heated up to ambient temperature?

Using the equation (in steady state) P = k* A * T^4, where k is the Stefan-Boltzmann constant, A is the surface area, and T is the ambient temperature, I solve for P to get power.

Using the equation W = (T2-T1)*C * m, where T2 is the oven temperature, T1 is room temperature, C is the object's specific heat, and m is the object's mass, I solve for W to get energy.

Dividing power P into energy W gives me the time required to raise the object's temperature (uniform) to the oven temperature.

I just need a reasonable estimate to ensure that the entire object is heated to the ambient temperature; overshooting is acceptable in this case.

 

[IRstuff]

Only if you're assuming that the only means of losing heat is by radiation, which, for terrestrial applications, is usually a bad assumption, particularly if your statement that it's a convection oven is correct. Additionally, I don't see an emissivity term in your radiation expression, and it's highly unlikely that a typical metal object would be a perfect blackbody. In general, radiative transfer is two-way, not one, so you need to have an expression for the emission of the object.

Since it's supposed to be in a convection oven, then you need to have a forced convection heat transfer expression to account for that. And, depending on the object's thermal conductivity, you may need to constrain the equations to ensure that you don't grossly underestimate the time.

TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[chicopee]

One of the parameter affecting the heating time is when the mass of the object to be heated is thick another word Boit modulus > .1 or 10%.

 

[ione]

You’ve to perform a transient analysis. Look for lumped capacitance approach. Dealing with a metal you should get a good estimate, anyway check Biot number (Bi<0.1 implies conductive heat transfer prevails over convective heat transfer mechanism).

 

[KimWonGun]

Thanks for your helpful recommendations. I attempted to incorporate them in the following analysis and ask for further feedback:

Assuming a heat transfer coefficient of 40 W/(m^2-K) for forced indoor air, a characteristic length of 0.075 m, and a thermal conductivity of 16 W/(m-K) for stainless steel, the Boit number is 0.19.

Because that value falls outside the 10% rule, is the lumped capacitance method not valid?

Is it reasonable to just consider the radiative heat transfer when (a) I wish to obtain a conservative time estimate and (b) the oven has highly reflective aluminum walls and ceiling?

Using the equation (in steady state) P = k * E * A * T^4, where k is the Stefan-Boltzmann constant, E is the emissivity of the object, A is the surface area of the object, and T is the ambient temperature, I solve for P to get power.

Using the equation W = (T2-T1)*C * m, where T2 is the oven temperature, T1 is room temperature, C is the object's specific heat, and m is the object's mass, I solve for W to get energy.

Dividing power P into energy W gives me the time required to raise the object's temperature (uniform) to the oven temperature.

 

[IRstuff]

Because that value falls outside the 10% rule, is the lumped capacitance method not valid?

Not necessarily, it may just mean that you have to consider a multilayer lumped model, rather than a single RC-type.

Is it reasonable to just consider the radiative heat transfer when (a) I wish to obtain a conservative time estimate and (b) the oven has highly reflective aluminum walls and ceiling?

Actually, the converse would be true. Highly reflective surfaces have lower emissivities, potentially making convection more dominant.

Otherwise, your basic concept is roughly correct.

Q.object = Q.convection + Q.radiation_in - Q.radiation_out


TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[KimWonGun]

IRstuff:

Thanks for your correction. Based on what you just shared, is my logic now correct for steady-state conditions?

Q.obj = Q.conv + Q.rad_in - Q.rad_out = k*A1*(T2-T1) + E1*s*A1*T2^4 - E2*s*A2*T2^4 where:

k = heat transfer coefficient of (forced) air
A1 = surface area of heating element
T2 = desired temperature
T1 = start temperature
E1 = emissivity of heating element material
s = Stefan-Boltzmann's constant
E2 = emissivity of object material
A2 = surface area of object

Then using the equation W = (T2-T1)*C * m, where T2 is the desired temperature, T1 is start temperature, C is the object's specific heat, and m is the object's mass, I solve for W to get energy.

Dividing power P into energy W gives me the time required to raise the object's temperature (uniform) to the oven temperature.

 

[KimWonGun]

Using the values below, I appear to be missing a critical element since the calculated result is negative. Can anyone help?

Q.object = Q.convect + Q.radiate_in - Q.radiate_out= kA1(T2-T1) + E1sA1T2^4-E2sA2T2^4

k= 40 W/m^2-K
A1= 0.1022 m^2
T2= 366.15 K
T1= 288.15 K
E1= 0.03 (nickel - plated)
s= 5.67E-08 W/m^2-K^4
E2= 0.85 (stainless steel)
A2= 2.3117 m^2

Q.convect= 318.71 W
Q.radiate_in= 30.57 W
Q.radiate_out= 2002.47 W
Q.object= -1653.18 W
-1.65 kW

 

[IRstuff]

The posititve radiation term should be the oven's temperature, while the negative term should something like the logmean of the start and end temperatures. A similar deal for the convection term. The exercise is not to find the steady state solution, since that should be in equilibrium. You don't want the starting condition, since that has the highest heat flux, and will underestimate the time.

TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[zekeman]

"Assuming a heat transfer coefficient of 40 W/(m^2-K) for forced indoor air, a characteristic length of 0.075 m, and a thermal conductivity of 16 W/(m-K) for stainless steel, the Boit number is 0.19. "

The length term in Biot is the thickness of the piece, not the length.

I suspect the thickness to be much less and bring the Biot number proportionately smaller and in the range where the steady state might be allowed.

On another note, depending on the spaciousness of the oven vs the size of the mass, a low wall emissivity due to multiple reflections can effectively be closer to 1 or a black body than the basic emissivity, so the correct equation for radiative exchange being
Am*sigma *e*(Tw^4-T^4)
Am=radiative area of mass
T surface temperature of mass
Tw wall temperature
where e will range from e of the mass the product of e and the effective wall emissivity. In most cases you can use e as the overall emissivity.
To be conservative here, I would use 1/2 the emissivity of the mass.

 

[zekeman]

I should also add that the h in the Biot number calculation has to include the effective h due to radiation, hr , so

h=hc+hr

hr=e*(Tw^4-T$^4)/(Tw-T)

 

[ione]

I usually adopt Bi=0.1 as threshold to establish whether the lumped capacitance model is applicable, anyway Bi=0.2 is accepted as well (

http://www.engr.mun.ca/~yuri/Courses/MechanicalSystems/ThermoFluids.pdf).

And how accurate have you been in computing the convective heat transfer coefficient?

 

[KimWonGun]

zekeman:

Thanks for your suggestions.

I am assuming that the "characteristic length" equals the part thickness; is this nomenclature not standard? Therefore, the 0.075 m value is the measured part thickness.

Given that the oven is almost 72X larger (by volume) than the object, your recommendation for the radiative exchange equation appears viable.

But based on comments from IRstuff, I may not be explaining the objective clearly or do not properly understand his point. So permit me to clarify my assignment with additional details:

I need to oven-cure a paint that coats the cavity side of a wheelhouse-shaped stainless steel object. Because the cure temperature at the substrate interface is critical, I want to estimate the time required for the part to reach the oven temperature ALREADY AT STEADY STATE. The transient time is not my focus since I need to do a precise ramp up to the set temperature anyway.

Because the part is quite thick and stainless steel is not particularly conductive, I am assuming the heating time ONCE THE OVEN REACHES THE SET TEMPERATURE is considerable. And because I seek a conservative estimate, I thought I could simplify the problem as one of radiative heat transfer only.

So given the additional data my questions are:

1) Is my proposed initial solution (2/24) viable (but including an emissivity value)?

2) Should I assume as the radiative area of a cavity-shaped object only one side (volume/thickness) or both sides (volume/thickness*2)?

 

[ione]

Transient is referred to the piece to be soaked and not to the oven.

 

[zekeman]

" Because the cure temperature at the substrate interface is critical,"

How critical? Do you mean that when the paint reaches the cure temperature you would pull the object out of the oven? If that is approximately correct, then post the cure temperature, the temperature of the oven, and the maximum oven wall temperature

You must remember that the temperature profile thru the thickness is never uniform but will always have a gradient. Also, only one side exchanges heat.

An analysis could give you an estimate of this and also the approximate time to do it but as was pointed out , a transient solution must be done.

I have graphical solutions that can help you. If this is a one-time problem, I could help you use the charts.

 

[zekeman]

Also, if you could give the dimensions of the open wheelbarrow surface sitting on the floor of the oven and the dimensions of the oven, the emissitity may be better estimated .

 

[KimWonGun]

ione:

Thanks for the correction.


zekeman:

The cure time for the paint starts when the metal surface reaches the oven temperature.

Thanks for the reminders of basic principles.

Thanks for your kind offer to share a graphical solution, but I need to develop a general algorithm for this scenario.

 

[ione]

It is understood that a gradient of temperature will establish between the external surface of the piece to be cured and its core. But it seems to me what matters here is the skin (external surface) temperature. It’s that temperature which “activates” the paint. So you can undoubtedly apply a lumped capacitance model as the characteristic length is not the overall thickness of the item to be treated, but a much thinner layer (I’m not able to quantify how thick, but a person skilled in paint processes should be).

 

[KimWonGun]

ione:

Thanks for your comment.

The curing time to bond the paint and metal is my concern, hence the focus on the metal temperature.

 

[KimWonGun]

Attempting to better estimate the convective heat transfer coefficient, I would appreciate any feedback to the following:

The heating element used in the oven is rated at 36 W/in^2 (55,800 W/m^2). I attached thermocouples at the inlet and outlet sides to measure the temperature difference. Running the fan at 2,000 ft^3/min (0.943 m^3/s) I see a steady-state temperature rise of 30 degrees F (272 K).

Using the relationship h = q / A*dT, I divide 272 K into 55,800 W/m^2 to get a forced convective heat transfer coefficient of 205 W/m^2-K.

Question 1: Does this seem logical?

Question 2: Acknowledging that the air velocity will be lower away from the fan in the oven, is the 40 W/m^2-K rule-of-thumb more reasonable?