# Heat generation in general heat conduction equation

In the derivation of a general heat conduction equation for a differential element we apply the principle of conservation of energy. According to which

(Net heat conducted into element)+(Internal heat generated)=(Increase in internal energy)

Isn't the internal heat generation happening at the expense of internal energy so how can it increase the internal energy. For example if we consider an exothermic chemical reaction happening inside a control space then wouldn't the heat generation happen at the expense of internal energy of the system assuming the chemical substance is part of the system.

• Please explain what you mean by "at the expense of internal energy," which you use twice. If resistive heating or an exothermic chemical reaction is generating thermal energy at the rate of 100 W, for example, what is the complication with simply using this rate for the "internal heat generated" term? – Chemomechanics Apr 28 '18 at 16:57
• @Chemomechanics If the chemical substance or the resistive wire is part of the system then wouldn't their chemical or electrical energy be part of that system's internal energy ? The heat would be generated at the expense of this energy. My question is how can then the internal energy of system increase due to this heat generation – Siddharth Prakash Apr 28 '18 at 18:40

## 3 Answers

There are two equivalent ways of doing the analysis. In one method, the internal energy includes both changes due to sensible heat and due to reaction, in which case the heat generation term would be zero. The other method lumps the heat generation from reaction into the "generation term" and only includes sensible heat changes in the "internal energy" term. If you work it out, you will find that the two approaches are equivalent.

The general form of a conservation law is

$\mathrm{(What\ we\ started\ with)} + \mathrm{(What\ we\ added)} = \mathrm{(What\ we\ have\ now)}$

That’s often rearranged to

$\mathrm{(What\ we\ have\ now)} - \mathrm{(What\ we\ started\ with)} = \mathrm{(What\ we\ added)}$

If there are multiple forms involved, each of those terms can be a sum.

$(E_\mathrm{thermal,now} + E_\mathrm{mechanical,now}) - (E_\mathrm{thermal,before} + E_\mathrm{mechanical,before}) = \mathrm{Added\ energy\ from\ chemical\ reactions}$

That is often written as the individual changes in each:

$\Delta E_\mathrm{thermal} + \Delta E_\mathrm{mechanical} = \mathrm{Added\ energy\ from\ chemical\ reactions}$

So you, the person considering conservation of energy, get to decide where each type of energy appears: as a quantity that changes, or as something added from outside. Each type can appear either way, but only once.

• Could you please explain with an example? – Siddharth Prakash Apr 28 '18 at 18:40
• @SiddharthPrakash edited it to make it a bit more specific – Bob Jacobsen Apr 28 '18 at 22:25

For understanding Internal heat generation , you can take the classic example of passage of electricity through an electric wire. The energy which is getting dissipated as heat is not the internal energy stored within the system(electric wire).

• then why is it called internal heat generated? – Siddharth Prakash Apr 28 '18 at 18:36
• Since heat generation (conversion of electrical energy to heat)occurs within the system, it is called internal heat generation. – Jithin Apr 29 '18 at 3:34
{"ft_lang_label":"__label__en","ft_lang_prob":0.8929283,"math_prob":0.9817349,"size":600,"snap":"2021-04-2021-17","text_gpt3_token_len":107,"char_repetition_ratio":0.17617449,"word_repetition_ratio":0.045454547,"special_character_ratio":0.17333333,"punctuation_ratio":0.030927835,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9931261,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-19T06:35:19Z\",\"WARC-Record-ID\":\"<urn:uuid:3f5ffd94-376b-4aab-ab98-22e40c6a9136>\",\"Content-Length\":\"183923\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1fe1380d-3ec5-4008-9e01-102718fb0be1>\",\"WARC-Concurrent-To\":\"<urn:uuid:e1d68465-9deb-4cd0-bc47-6c286692bf2f>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/402685/heat-generation-in-general-heat-conduction-equation/402717\",\"WARC-Payload-Digest\":\"sha1:73DCTS5T7K46355B7WPGXA3QGJMVYDS6\",\"WARC-Block-Digest\":\"sha1:2VGRM7KFBJP7CSRT777HDOYN6IE46WTJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038878326.67_warc_CC-MAIN-20210419045820-20210419075820-00049.warc.gz\"}"}