Find an expression of thermal current
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Mᴀᴛʜᴇᴍᴀᴛɪᴄs ᴏғ Hᴇᴀᴛ Cᴜʀʀᴇɴᴛ
Bᴇᴄᴀᴜsᴇ ʜᴇᴀᴛ ᴄᴜʀʀᴇɴᴛ ʀᴇᴘʀᴇsᴇɴᴛs ᴛʜᴇ ғʟᴏᴡ ᴏғ ʜᴇᴀᴛ ᴇɴᴇʀɢʏ ᴏᴠᴇʀ ᴛɪᴍᴇ, ʏᴏᴜ ᴄᴀɴ ᴛʜɪɴᴋ ᴀʙᴏᴜᴛ ɪᴛ ᴀs ʀᴇᴘʀᴇsᴇɴᴛɪɴɢ ᴀ ᴛɪɴʏ ᴀᴍᴏᴜɴᴛ ᴏғ ʜᴇᴀᴛ ᴇɴᴇʀɢʏ, ᴅQ (Q ɪs ᴛʜᴇ ᴠᴀʀɪᴀʙʟᴇ ᴄᴏᴍᴍᴏɴʟʏ ᴜsᴇᴅ ᴛᴏ ʀᴇᴘʀᴇsᴇɴᴛ ʜᴇᴀᴛ ᴇɴᴇʀɢʏ), ᴛʀᴀɴsᴍɪᴛᴛᴇᴅ ᴏᴠᴇʀ ᴀ ᴛɪɴʏ ᴀᴍᴏᴜɴᴛ ᴏғ ᴛɪᴍᴇ, ᴅᴛ. Usɪɴɢ ᴛʜᴇ ᴠᴀʀɪᴀʙʟᴇ H ᴛᴏ ʀᴇᴘʀᴇsᴇɴᴛ ʜᴇᴀᴛ ᴄᴜʀʀᴇɴᴛ, ᴛʜɪs ɢɪᴠᴇs ʏᴏᴜ ᴛʜᴇ ᴇϙᴜᴀᴛɪᴏɴ:
➡H = ᴅQ / ᴅᴛ
Iғ ʏᴏᴜ'ᴠᴇ ᴛᴀᴋᴇɴ ᴘʀᴇ-ᴄᴀʟᴄᴜʟᴜs ᴏʀ ᴄᴀʟᴄᴜʟᴜs, ʏᴏᴜ ᴍɪɢʜᴛ ʀᴇᴀʟɪᴢᴇ ᴛʜᴀᴛ ᴀ ʀᴀᴛᴇ ᴏғ ᴄʜᴀɴɢᴇ ʟɪᴋᴇ ᴛʜɪs ɪs ᴀ ᴘʀɪᴍᴇ ᴇxᴀᴍᴘʟᴇ ᴏғ ᴡʜᴇɴ ʏᴏᴜ ᴡᴏᴜʟᴅ ᴡᴀɴᴛ ᴛᴏ ᴛᴀᴋᴇ ᴀ ʟɪᴍɪᴛ ᴀs ᴛʜᴇ ᴛɪᴍᴇ ᴀᴘᴘʀᴏᴀᴄʜᴇs ᴢᴇʀᴏ. Exᴘᴇʀɪᴍᴇɴᴛᴀʟʟʏ, ʏᴏᴜ ᴄᴀɴ ᴅᴏ ᴛʜᴀᴛ ʙʏ ᴍᴇᴀsᴜʀɪɴɢ ᴛʜᴇ ʜᴇᴀᴛ ᴄʜᴀɴɢᴇ ᴀᴛ sᴍᴀʟʟᴇʀ ᴀɴᴅ sᴍᴀʟʟᴇʀ ᴛɪᴍᴇ ɪɴᴛᴇʀᴠᴀʟs.
Exᴘᴇʀɪᴍᴇɴᴛs ᴄᴏɴᴅᴜᴄᴛᴇᴅ ᴛᴏ ᴅᴇᴛᴇʀᴍɪɴᴇ ᴛʜᴇ ʜᴇᴀᴛ ᴄᴜʀʀᴇɴᴛ ʜᴀᴠᴇ ɪᴅᴇɴᴛɪғɪᴇᴅ ᴛʜᴇ ғᴏʟʟᴏᴡɪɴɢ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄᴀʟ ʀᴇʟᴀᴛɪᴏɴsʜɪᴘ:
➡H = ᴅQ / ᴅᴛ = ᴋA (T(H) - T(C)) / L
Tʜᴀᴛ ᴍᴀʏ sᴇᴇᴍ ʟɪᴋᴇ ᴀɴ ɪɴᴛɪᴍɪᴅᴀᴛɪɴɢ ᴀʀʀᴀʏ ᴏғ ᴠᴀʀɪᴀʙʟᴇs, sᴏ ʟᴇᴛ's ʙʀᴇᴀᴋ ᴛʜᴏsᴇ ᴅᴏᴡɴ (sᴏᴍᴇ ᴏғ ᴡʜɪᴄʜ ʜᴀᴠᴇ ᴀʟʀᴇᴀᴅʏ ʙᴇᴇɴ ᴇxᴘʟᴀɪɴᴇᴅ):
- H: ʜᴇᴀᴛ ᴄᴜʀʀᴇɴᴛ
- ᴅQ: sᴍᴀʟʟ ᴀᴍᴏᴜɴᴛ ᴏғ ʜᴇᴀᴛ ᴛʀᴀɴsғᴇʀʀᴇᴅ ᴏᴠᴇʀ ᴀ ᴛɪᴍᴇ ᴅᴛ
- ᴅᴛ: sᴍᴀʟʟ ᴀᴍᴏᴜɴᴛ ᴏғ ᴛɪᴍᴇ ᴏᴠᴇʀ ᴡʜɪᴄʜ ᴅQ ᴡᴀs ᴛʀᴀɴsғᴇʀʀᴇᴅ
- ᴋ: ᴛʜᴇʀᴍᴀʟ ᴄᴏɴᴅᴜᴄᴛɪᴠɪᴛʏ ᴏғ ᴛʜᴇ ᴍᴀᴛᴇʀɪᴀʟ
- A: ᴄʀᴏss-sᴇᴄᴛɪᴏɴᴀʟ ᴀʀᴇᴀ ᴏғ ᴛʜᴇ ᴏʙᴊᴇᴄᴛ
- TH - TC: ᴛʜᴇ ᴛᴇᴍᴘᴇʀᴀᴛᴜʀᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ ᴡᴀʀᴍᴇsᴛ ᴀɴᴅ ᴄᴏᴏʟᴇsᴛ ᴛᴇᴍᴘᴇʀᴀᴛᴜʀᴇs ɪɴ ᴛʜᴇ ᴍᴀᴛᴇʀɪᴀʟ
- L: ᴛʜᴇ ʟᴇɴɢᴛʜ ᴀᴄʀᴏss ᴡʜɪᴄʜ ᴛʜᴇ ʜᴇᴀᴛ ɪs ʙᴇɪɴɢ ᴛʀᴀɴsғᴇʀʀᴇᴅ
- Tʜᴇʀᴇ's ᴏɴᴇ ᴇʟᴇᴍᴇɴᴛ ᴏғ ᴛʜᴇ ᴇϙᴜᴀᴛɪᴏɴ ᴛʜᴀᴛ sʜᴏᴜʟᴅ ʙᴇ ᴄᴏɴsɪᴅᴇʀᴇᴅ ɪɴᴅᴇᴘᴇɴᴅᴇɴᴛʟʏ:
➡(TH - TC) / L
Tʜɪs ɪs ᴛʜᴇ ᴛᴇᴍᴘᴇʀᴀᴛᴜʀᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ᴘᴇʀ ᴜɴɪᴛ ʟᴇɴɢᴛʜ, ᴋɴᴏᴡɴ ᴀs ᴛʜᴇ ᴛᴇᴍᴘᴇʀᴀᴛᴜʀᴇ ɢʀᴀᴅɪᴇɴᴛ.
Hope it helps..... ❤❤