<p>I'd like to calculate the power lost through the filament being extruded (or in other words, at how many Watts I'd have to run an ideal heater that loses heat ONLY through filament so that it stays at constant temperature).</p>
<ul>
<li>Power is defined in Watts as <span class="math-container">$\text W =\frac{\text J}{\text s}$</span></li>
<li><a href="https://en.wikipedia.org/wiki/Specific_heat_capacity" rel="nofollow noreferrer">Specific heat capacity</a> of a material is defined as <span class="math-container">$C =\frac{\text J}{ \text K \times\text{kg}}$</span></li>
</ul>
<p>When extrusion happens, the filament of higher temperature leaves the hotend while the same weight of filament of lower temperature enters the hotend.</p>
<p>Let's say the specific heat capacity of the filament is <span class="math-container">$C$</span> and the extrusion rate <span class="math-container">$r$</span> is given with units kg/s. The temperatures are <span class="math-container">$T$</span>.</p>
<p>Is it correct then to say that the power consumption of filament extrusion is <span class="math-container">$W = (c\times T_\text{Nozzle} - c\times T_\text{Environment}) \times r$</span></p>
<p>meaning that if I would run a heater cartridge at exactly &quot;W&quot; watts and extrude filament with rate &quot;r&quot; and the block would not loose heat through any other means than through the filament extrusion, then the nozzle temperature would stay constant?</p>

[{'answer_id': 10174, 'author': '0scar', 'author_id': 5740, 'author_profile': 'https://3dprinting.stackexchange.com/users/5740', 'pm_score': 3, 'selected': False, 'text': '<p>This is very simply stated, in fact the specific heat is a function of temperature and state of the material (liquid or solid). Also you need to consider which type of specific heat you use, e.g. the one for constant volume <span class="math-container">$C_V$</span> or for constant pressure <span class="math-container">$C_P$</span>. Constant pressure is probably preferred considering the mechanics of the printer (pressing filament into the nozzle-heatbreak assembly).</p>\n\n<p>A very interesting source of information is the <a href="https://polymerdatabase.com/polymer%20physics/HeatCapacity.html" rel="noreferrer">PolymerDatabase.com</a>.</p>\n\n<p>This source confirms that:</p>\n\n<blockquote>\n  <p>In the case of polymers, we have to distinguish between the heat capacity of liquid, rubbery and glassy polymers. The heat capacity increases with increasing temperature, therefore, a liquid or rubbery polymer can hold more energy than a solid polymer. All materials show this increase in heat capacity with temperature.</p>\n</blockquote>\n\n<p>also:</p>\n\n<blockquote>\n  <p>Specific heat capacities as a function of temperature have been published for only a limited number of polymers. In many cases, the heat capacity (at constant pressure) as a function of temperature can be approximated by straight lines.</p>\n</blockquote>\n\n<p>In such cases you can use the value of the specific heat at a predefined temperature (in thermodynamics that is frequently 298 K) to get approximations for your thermoplastic material. These formulae can then be used to integrate over the temperature rise.</p>\n\n<p>Please remember that a cartridge is of a certain value of Watt; to have a lower power, the cartridge modulates voltage to keep the heating block within a predefined temperature range.</p>\n'}, {'answer_id': 10175, 'author': 'Trish', 'author_id': 8884, 'author_profile': 'https://3dprinting.stackexchange.com/users/8884', 'pm_score': 2, 'selected': False, 'text': '<h1>No.</h1>\n<p>Your formula is quite off, and it starts with the nomenclature:</p>\n<h2>Watt</h2>\n<p>Watt is the <strong>unit</strong> of <strong>energy transfer</strong> which equals <strong>power</strong>.</p>\n<p>The commonly used term &quot;wattage&quot; does not exist in science. It is a very despised shorthand <em>only</em> used in terms of <em>electric power</em> <span class="math-container">$P=UI$</span>.</p>\n<p>Both power <span class="math-container">$P$</span> (like work over time) and heat energy transfer <span class="math-container">$\\Delta Q$</span> (which is one variant of power) use the unit <span class="math-container">$\\text W=\\frac {\\text J} {\\text s}$</span>, which is confusing but a necessary distinction. Always remember that <span class="math-container">$P_\\text{total}=\\sum_{i=1}^nP_i$</span> - the total power in and out of an object is the sum of all partial powers!</p>\n<h2>Heat Energy transfer</h2>\n<p>The Heat energy transfer through an object is defined as the <em>change</em> of the heat energy <span class="math-container">$Q$</span> stored inside an object. <span class="math-container">$Q$</span> is given in <span class="math-container">$\\text J$</span>, so its change <span class="math-container">$\\Delta Q$</span> is given in <span class="math-container">$\\text J$</span> too. To get to the power, the energy change needs to be measured at several spots in time, so we make the derivate over time and get the power in <span class="math-container">$\\text W$</span>. We\'re looking at <span class="math-container">$\\dot Q=\\frac {\\delta Q} {\\delta t}$</span>.</p>\n<p>The absolute change of heat energy of an item is defined as <span class="math-container">$\\Delta Q(t)=m(t) c \\Delta T$</span>: Increasing the temperature <span class="math-container">$T$</span> of an object with the mass <span class="math-container">$m$</span> and specific heat capacity <span class="math-container">$c$</span> by <span class="math-container">$\\delta T$</span> (between times <span class="math-container">$t_0$</span> and <span class="math-container">$t$</span>) results in a change of the stored energy by <span class="math-container">$\\Delta Q(t-t_0)$</span>.</p>\n<p>So, we know <span class="math-container">$Q=c m \\Delta T$</span> and <span class="math-container">$P=\\dot Q=\\frac \\delta {\\delta t} c m \\Delta T$</span></p>\n<h2>Problem in question</h2>\n<p>We know that the drain (<em>loss</em>) of thermal energy from the system is via three ways:</p>\n<ul>\n<li>melting plastic (phase transition)</li>\n<li>extruding heated plastic</li>\n<li>convective heat loss to the air</li>\n<li>black body radiation of the heater block</li>\n</ul>\n<p>We know that the total balance in equilibrium should be <span class="math-container">$P_\\text{total}=P_\\text{heating}+P_\\text{melting}+P_\\text{extrusion}+P_\\text{convection}+P_\\text{bb}=0$</span>.</p>\n<h2>heat deposited into the system</h2>\n<p>Let\'s start at the simplest: we simply know the nominal heating power of the cartridge, it is usually written upon the cartridge itself, usually something in the area of 20 to 40\xa0W. In praxis, it is not exactly that, but the ballpark fits. Otherwise, we\'d plug in <span class="math-container">$P_\\text{heating}=\\epsilon \\frac {U^2} {R}$</span> for our specific resistor, where <span class="math-container">$\\epsilon$</span> is a coefficient between 0 and 1 telling us how good it is in converting electric to heat energy. Remember that since <span class="math-container">$U$</span> is technically a function of time (it is modulated to control heating behavior), our heating power also is, even though not explicit!</p>\n<h2>black body radiation loss <span class="math-container">$P_\\text{bb}$</span></h2>\n<p>Black body radiation: <span class="math-container">$P_\\text{bb}=A \\sigma T^4$</span> where <span class="math-container">$A$</span> is the surface area of the object, <span class="math-container">$\\sigma$</span> is a constant called <a href="https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_constant" rel="nofollow noreferrer">Stefan-Boltzmann Constant</a>. That much thermal energy is just lost due to radiation via photons, even if we don\'t see it glowing.</p>\n<h2>convection loss <span class="math-container">$P_\\text{convection}$</span></h2>\n<p>The change of heat energy via heat convection is roughly defined as <span class="math-container">$H=\\theta A (T-T_f)$</span> which brings us another coefficient <span class="math-container">$\\theta$</span> about how good the block heats the air and the temperature of the medium (air) around <span class="math-container">$T_f$</span> - which we can replace as <span class="math-container">$(T-T_f)=\\Delta T_a$</span>.</p>\n<p>And then we get to the biggest can of worms: the thermal heat transfer for melting the plastic and how much thermal energy is extruded from the system. For one of them, we can estimate some ballpark numbers, for the other, we will get into problems.</p>\n<h2>extrusion loss <span class="math-container">$P_\\text{extrusion}$</span></h2>\n<p>The heat energy removed from the system by extruding plastic we can estimate from what we already established about thermal energy back in the Heat Energy transfer paragraph: <span class="math-container">$Q=mc(T_0+\\Delta T)$</span> using the specific heat capacity <span class="math-container">$c [\\frac {\\text{J}}{\\text kg K}] $</span> of the molten plastic as it is extruded (more about that later). But that\'s not the loss per time, but the heat energy stored in it in Joules. What factor is changing? In this case, it is the mass <span class="math-container">$m=r\\times t$</span> where <span class="math-container">$r=\\frac {\\text kg} {\\text s}$</span> is the extrusion rate. So <span class="math-container">$Q_\\text{extrusion}=rtc\\ \\Delta T_\\text{extrusion}$</span> and subsequently <span class="math-container">$P_\\text{extrusion}=rc\\ \\Delta T_\\text{extrusion}$</span></p>\n<p>This leaves us with the big problem: as <a href="https://3dprinting.stackexchange.com/a/10174/8884">0scar</a> correctly pointed out by directing to the <a href="https://polymerdatabase.com/polymer%20physics/HeatCapacity.html" rel="nofollow noreferrer">PolymerDatabase</a> the specific heat capacity is <em>not</em> a constant and <em>not</em> linear but changes depending on the aggregate of the substance. We can make some estimate about it though from how we formulated the total power and adding a few absolutes for convenience:</p>\n<p><span class="math-container">$$P_\\text{total}=P_\\text{heating}+P_\\text{melting}+P_\\text{extrusion}+H_\\text{convection}+P_\\text{bb}=0$$</span></p>\n<p><span class="math-container">$$P_\\text{heating}-H_\\text{convection}-P_\\text{bb}-P_\\text{extrusion}=P_\\text{melting}$$</span></p>\n<p><span class="math-container">$$\\epsilon \\frac {U^2}{R}-\\theta A \\Delta T_a-A \\sigma T^4-rc\\ \\Delta T_\\text{extrusion}=P_\\text{melting}$$</span></p>\n<p>Remember, that <span class="math-container">$U$</span> is a time-dependent factor (because of the control board activating it or disabling it), <span class="math-container">$T_f$</span> is <em>also</em> not a steady thing and changes depending on the airflow (though we can just pin it for our thought experiment) and thus <span class="math-container">$T$</span> itself might change over time as a result. <span class="math-container">$T$</span> is not equal to <span class="math-container">$\\ \\Delta T_\\text{Extrusion}$</span> but is the temperature of the heater block system as a whole. <span class="math-container">$\\ \\Delta T_\\text{Extrusion}$</span>, on the contrary, is the temperature increase <em>of the filament</em> and not necessarily the same <span class="math-container">$\\Delta T_{air}$</span>, the differential between the heater block and the air. Why this differentiation is necessary becomes apparent if one realizes that the path of the filament might benefit from the heat that is <em>lost</em> from the heater block along that path, pre-heating the filament.</p>\n<h2>Phase Transition <span class="math-container">$P_\\text{melting}$</span></h2>\n<p><sub><span class="math-container">$\\propto$</span> is the proportionality sign and indicates that I might skip factors or constants.</sub></p>\n<p>What is that last part? That <span class="math-container">$P_\\text{melting}$</span>? It is the power of the <a href="https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/States_of_Matter/Phase_Transitions/Fundamentals_of_Phase_Transitions" rel="nofollow noreferrer">Phase Transition</a>. <a href="https://en.wikipedia.org/wiki/Enthalpy_of_fusion" rel="nofollow noreferrer">Matter does <em>not</em> shift between phases <em>freely</em></a>. There is energy stored in the state itself! So when transitioning from one phase to the other, that energy either has to be added (when going from solid to liquid or liquid to gas) or removed (when going the other way).</p>\n<p>The &quot;heat of fusion&quot; is a material constant. For this look, I\'ll call it <span class="math-container">$\\phi [\\frac {\\text{J}} {\\text {g}}]$</span>. We can make an estimation for the power that is put into melting the filament: there\'s an amount of filament that gets an amount of heat and undergoes the phase transition per time increment <span class="math-container">$$P_\\text{melting}\\propto \\frac \\delta {\\delta t}\\phi m_\\text{melting}=\\phi*\\dot m$$</span> Now, we have the product of specific &quot;latent heat&quot; and melting mass derived over time... We had earlier the flow rate of material <span class="math-container">$m=r\\times t$</span> and the &quot;latent heat&quot; is a constant. So, we pull out <span class="math-container">$\\dot m=r$</span> again. So in the end we get that the power that is needed to melt our filament is proportional to the flow of the material and the material constant. <span class="math-container">$$P_\\text{melting}\\propto \\phi r$$</span></p>\n<h2>Conclusion tl;dr</h2>\n<p>When eliminating the loss via convection and black body radiation and assuming them 0 or neglectable, we assume our heater is packed in perfect isolation - and call them losses. Assuming <span class="math-container">$\\epsilon=1$</span> for a perfect heater, we are left with this equilibrium situation:</p>\n<p><span class="math-container">$\\frac {U^2}{R}-P_\\text{losses}=P_\\text{melting}+rc\\ \\Delta T_\\text{extrusion}$</span></p>\n<p><strong>The sum of the power of the phase transition (melting of the filament) and the energy stored in the extruded filament per time <span class="math-container">$(\\frac{dQ}{dt})$</span> is equal to the energy deposited into the hotend over time [minus losses over time]</strong></p>\n'}]
['https://3dprinting.stackexchange.com/questions/10173', 'https://3dprinting.stackexchange.com', 'https://3dprinting.stackexchange.com/users/16467/']