How to calculate time to change the temperature of metal

[JoeSalerno]

So I'm looking to calculate the time required to raise the temperature of a piece of lead. What I know so far: Mass of metal: 4 grams
Initial temperature: 250 degrees Celsius
Final temperature: 350 degrees Celsius
Specific heat of lead: 0.13 kj/kg
I'm very new to thermodynamics and don't know if this will be a simple equation or a complex one, but my current math understanding rests in pre-calculus. If there are any missing variables let me know, but I'd like to figure out how long it would take to heat up the mass of lead (I know it will go through a phase change) at a given temperature. Any help is greatly appreciated.

 

[Baluncore]

How much energy is needed to heat the metal to the mp, melt the lead at 327.5 °C, then heat the liquid lead to 350 °C ?
How will you deliver the heat to the lead? How fast can you get heat into the lead?

 

[JoeSalerno]

Is it correct that to find the energy needed to heat the lead through the process can be found by Qf=Q1+Q2+Q3 when Q1=mass*change in temperature from 250 to 327.5(melting point)*Cp of lead, Q2= mass* latent heat of lead, and Q3=mass* change in temperature from 327.5 to 350(Cp of lead? If not, how do you find out the required energy. I am trying to determine the method of heating between radiation, Joule heating, or another form of non conductive heating. I do not know of a way to determine how fast heat can be "absorbed" (for lack of a better term) by the lead.

 

[Baluncore]

Yes, Qf=Q1+Q2+Q3 is the right approach.
The time will be a problem unless you can find the thermal resistance or you can deliver the energy as a fixed power beam.
Can you do it in a way that will reduce heat losses?

 

[JoeSalerno]

This is purely a hypothetical problem, not an experiment, so I can theoretically use any method of heating, whatever would be most efficient. I'm not aware of what the most efficient method of heating is, but since it's theoretical, anything is possible to use. I was thinking of using Joule heating by incorporating it into a circuit, or possibly just using an oven style heating up the ambient air in the system. If you know a more efficient way, please inform me as I'm not all that experienced in this field. Is there a way to calculate the thermal resistance or is it a constant dependent on the material like specific heat and latent heat?

 

[JoeSalerno]

So for the Qf=Q1+Q2+Q3 this is what came out:
Qf=(4g*77.5[delta T]*0.031cal/gC)+(4g*22400J/kg)+(4g*22.5[delta T]*0.031cal/gC)
Qf=(9.61)+(89600)+(2.79)
Qf=89,612.4 Joules
Does this sound right? I know the units are a bit of a cluster, but did the end result come out properly?

 

[Baluncore]

Is the thermal capacity of 0.031 cal/g.C for solid lead the same as liquid lead ?
Can you plot a graph of energy against temperature from 250°C to 350°C ?
You should expect the phase change to dominate the energy graph.

 

[JoeSalerno]

To find the Cp of the lead, would I use Cp=Q/mdeltaT? I'm not sure of how to plot a graph of the energy against temperature, whether I have to fit a graph with the only known point being (0,250) or is it something that can just be Googled.

 

[Baluncore]

I'm not sure of how to plot a graph of the energy against temperature, whether I have to fit a graph with the only known point being (0,250)

The graph starts at 250°C, the slope is gentle, then there is the phase change step at the mp, followed by another gentle slope to 350°C. The slope is dQ / dTemp.

 

[JoeSalerno]

Wouldn't that make it linear as y=(q/t)x+250? Although temperature would be the variable and not x. I feel as if I have this messed up.

 

[Baluncore]

Plot with temp as the x axis, energy as the y axis.