I'm trying to remember how simple heat transfer differentials work and keep hurting myself in my confusion.I took everything except multivariable calc back in uni, so every resource's insistence on using partial diffs is throwing me for a loop.Basically, I want to model heat transfer from an oven to food. But with an ideal / simplified model to save my sanity.Let's say I have the following:1. A uniform sphere with radius r.2. The sphere's thermal conductivity, k, which we will pretend is a constant?3. A steady external heat source at temperature u_e applying uniformly to the surface area of the sphere.4. The sphere's uniform starting temperature at time (t) = 0, u_0.Then I want to solve the following things:1. The time it takes for the center point of the sphere to reach target temperature (u0_f), t_f.2. The temperature of the sphere's surface area at time t_f, s_f. (s = u at radius r.)3. The temperature of a point halfway into the sphere at time t_f, h_f. (h = u at 0.5r.)What is the generic equation supposed to look like? Is it not a simple single variable equation that I merely have to integrate to get distinct solutions?I could have sworn heat transfer was used to introduce basic differential equations to us after we started to get comfortable with integration. Am I crazy?As a topic of discussion beyond the above... has anybody else played around with (re)heating their food by using a high starting temperature and then dropping the temperature every few minutes? I can't be the only one who does this, right?
>>16485914just use a FEM solver lmaohttps://reference.wolfram.com/language/tutorial/NDSolveOverview.html
>>16485938>just use a FEM solver lmaoI want to understand how I'm getting an answer so I can intuit how changing variables will affect the solutions, and so I can slowly make the model more complex and varied, not just get a quick answer.Besides, I'm having a problem with the fundamental formulas. There's no point trying to use a solver if I can't even write a formula to solve.The minimum thing I need to figure out is if my example is even possible to model without partial differentials. If it isn't then I must be misremembering what we actually reviewed in my calc 2 class so many years ago, and I can go dig out my notebooks from storage to set myself straight.I'd like to avoid teaching myself multivariable calculus if it's not necessary for me to test a few proof of concepts.
Have you tried learning partial differentials?Unironically just watch some math video on youtube by someone who speaks in English and not some useless Latin alphabet on blackboard.
>>16486453I touched upon them very briefly (probably calc 4), but never had to apply them to anything and forgot everything about them.I'm not against learning partial differentials. But we don't learn by trying to understand the most complex system and then simplifying from there. We learn by understanding simplified systems and then add complexity. There's a reason we don't teach acceleration and potential / kinetic energy transfer to high school kids by bombarding them with air resistance and linear algebra matrices.
