Heat Transfer Questions & Answers  


This is a good point. The problem is in the second equation (maybe you made a mistake when writing down what I wrote, or maybe I made a mistake on the blackboard — this happens sometimes): $$ \int_V\frac{\partial}{\partial t}(\rho cT)dV =\int_z\int_y\partial q_x^{"}dydz \int_z\int_x\partial q_y^{"}dxdz \int_y\int_x\partial q_z^{"}dxdy+\int_V SdV $$ There is an integral missing. This equation should be: $$ \int_V\frac{\partial}{\partial t}(\rho cT)dV =  \int_z\int_y\int_{x=0}^{x=L}\partial q_x^{"}dydz  \int_z\int_x\int_{y=0}^{y=H} \partial q_y^{"}dxdz  \int_y\int_x\int_{z=0}^{z=D} \partial q_z^{"}dxdy + \int_V SdV $$ Notice the missing integrals in front of the heat fluxes, and note that $L$, $H$, $D$ are the length, height, and depth of the domain we are integrating. The integrals of the heat fluxes can be easily done and this will yield: $$ \int_{x=0}^{x=L}\partial q_x^{"} = q_{x=L}^{"}q_{x=0}^{"} $$ and similarly for the other dimensions. I'll give you a 2 point bonus boost for your question.




Please typeset your question correctly using LATEX, and I will answer it afterwards: T1 should be $T_1$, q[radiation in] should be $q_{\rm rad,in}$, etc. This will make your question easier to read for me and for the class, and I can then answer it using the same notation as you used. Besides, LATEX skills can be an asset on a resume (LATEX is not only used on my website: it's commonly used to typeset mathematics within google docs, wikipedia, scholarpedia, scholarly publications and books, etc).




The net energy transfer must be from a hot body to a cold body. But this doesn't mean a cold body can not give energy to a hot body. If a cold body gives some energy to a hotter body, then the hotter body must give in return more energy to the cold body so that the net energy transfer is from hot to cold. I'll give you 1 point bonus boost for your question. Your question was good and I would have given more if you had typeset your mathematics correctly and if the spelling, grammar, and formulation of the sentences would be better.




Yes, you are right, the chart is not for a fin with an insulated tip (because such fins are not commonly used). Thus, if the tip is insulated, you need to remember that $L_{\rm c}=L$ instead of $L_{\rm c}=L+\frac{1}{2}t$. I think I mentioned something about this in class... But I feel generous so I'll give you 1.5 point bonus boost for this question.




Hmm, there is no “geometrical meaning” I can think of.. As for the normal fins, the circumferential fins were derived from the heat equation. Thus, the equations listed in the tables were simply derived making the standard fin assumptions. The question is not so clear, so I'll give you just 1 point bonus boost for it.




This is a good question. When they derived these shape factors, they had to make some assumptions and those are indicated in the rightmost column. When they indicate $L \gg D$ it means that $L$ should be infinitely greater than $D$ for the given exact solution to be valid. I'll give you 2 points bonus boost for your question.




Of course, you need to list your assumptions (as for any other problem), but I went quickly to give you a hint on how to solve the problem. I'll give you 0.5 point bonus boost for this question.




What is given is not the heat generation per surface area but the power in W. Thus, the heat generation per unit volume $S$ is the power divided by the volume. I'll give you 0.5 point bonus boost for your question because it's not really a question but a misunderstanding of the problem statement.




You need to fix your question before I can answer it. Please typeset your mathematics better by making the parentheses as large as the terms within them are. Also, phrase your question better. I can not understand what you don't understand. Explain in more detail what you don't understand with additional information and example(s) if necessary.




If you're seeking the center temperature in a cylinder, sphere or plane wall, then there's no need to use the “temperature distributions” charts. Just use the chart “centerline temperature” or “midplane temperature”. I think I mentioned this in class, so I'll give only 1 point bonus boost for this question.




You should always compute $\alpha$ using more fundamental properties $k$, $c$, etc, and never use $\alpha$ directly from the tables. This is because in the tables some solids have a range of $c$, $k$, etc, and not specific values. The value you choose for $k$ and $c$ at one point in your solution must be the same as those that are used to calculate $\alpha$, or your solution will be inconsistent. I think I mentioned this in class already, so I'll give you just 0.5 point bonus boost.




You can still use the Heisler charts if $\textrm{Fo}<0.2$. But keep in mind there will be a more substantial error on the term determined. In this case, the error is on the parameter $(T_0T_\infty)/(T_iT_\infty)$. Thus, if your Fourier number is much less than 0.2, and you have no choice but to use the Heisler chart to solve the problem, you need to make a statement in your solution that there may be significant error on a certain term (don't just say there is error, specify on which term the error is high). I liked your question but it was sloppily typed with no uppercases at the beginning of sentences and some wrong uppercases in the middle of sentences. I'll give you 1 point bonus boost for it.





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