Update Content - 2025-07-18

content/zettels/heat_transfer.md

@@ -8,52 +8,107 @@ Tags
 :
 
 
+## Electrical Analogy - Lumped Mass Modeling {#electrical-analogy-lumped-mass-modeling}
+
+The difference in temperature \\(\Delta T\\) is driving potential energy flow \\(Q\\) (in watts):
+
+\begin{equation}
+  \Delta T = R\_{th} \cdot Q
+\end{equation}
+
+\\(R\_{th}\\) is the analogy of a "thermal resistance", and is expressed in K/W.
+
+
 ## Conduction (diffusion) {#conduction--diffusion}
 
-The _conduction_ corresponds to the heat transfer \\(P\\) (in watt) through molecular agitation within a material and is specified with:
-\\[ P = \frac{\lambda \cdot A \cdot \Delta T}{L} \quad [W] \\]
+The _conduction_ corresponds to the heat transfer \\(Q\\) (in watt) through molecular agitation within a material.
+
+\begin{equation}
+R\_{th} = \frac{d}{\lambda A}
+\end{equation}
+
 with:
 
 -   \\(\lambda\\) the thermal conductivity in \\([W/m \cdot K]\\)
 -   \\(A\\) the surface area in \\([m^2]\\)
--   \\(\Delta T\\) the temperature difference in \\([K]\\)
--   \\(L\\) the length of the barrier in \\([m]\\)
+-   \\(d\\) the length of the barrier in \\([m]\\)
 
 
 ## Convection {#convection}
 
-The convection corresponds to the heat transfer through flow of a fluid.
+The convection corresponds to the heat transfer \\(Q\\) through flow of a fluid.
 It can be either _natural_ or _forced_.
 
-The _forced convection_ \\(P\\) (in watt) can be described with:
-\\[ P = h A (T\_0 - T\_f) \quad [W] \\]
+\begin{equation}
+R\_{th} = \frac{1}{h A}
+\end{equation}
+
 with:
 
 -   \\(h\\) the convection heat transfer coefficient in \\([W/m^2 \cdot K]\\).
     \\(h \approx 10.5 - v + 10\sqrt{v}\\) with \\(v\\) the velocity of the object through the fluid in \\([m/s]\\)
+    Typically:
+    -   \\(h = 5 - 10\ W/m^2/K\\) for free convection with air
+    -   \\(h = 500 - 5000\ W/m^2/K\\) for forced water cooling in a tube of 5mm diameter
 -   \\(A\\) the surface area in \\([m^2]\\)
--   \\(T\_0\\) the temperature of the object in \\([K]\\)
--   \\(T\_f\\) the temperature of the convecting fluid in \\([K]\\)
 
-Note that clean-room air flow should be considered as forced convection.
+Note that clean-room air flow should be considered as forced convection, and \\(h \approx 10 W/m^2/K\\).
 
 
 ## Radiation {#radiation}
 
-_Radiation_ corresponds to the heat transfer \\(P\\) (in watt) through the emission of electromagnetic waves from the emitter to its surroundings is:
-\\[ P = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) \\]
+_Radiation_ corresponds to the heat transfer \\(Q\\) (in watt) through the emission of electromagnetic waves from the emitter to its surroundings.
+
+In the general case, we have:
+\\[ Q = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) \\]
 with:
 
 -   \\(\epsilon\\) the emissivity which corresponds to the ability of a surface to emit energy through radiation relative to a black body surface at equal temperature.
     It is between 0 (no emissivity) and 1 (maximum emissivity)
 -   \\(\sigma\\) the Stefan-Boltzmann constant: \\(\sigma = 5.67 \cdot 10^{-8} \\, \frac{W}{m^2 K^4}\\)
--   \\(A\\) the surface in \\([m^2]\\)
 -   \\(T\_r\\) the temperature of the emitter in \\([K]\\)
 -   \\(T\_s\\) the temperature of the surrounding in \\([K]\\)
 
+In order to use the lumped mass approximation, the equations can be linearized to obtain:
+
+\begin{equation}
+R\_{th} = \frac{1}{h\_{rad} A}
+\end{equation}
+
+with:
+
+-   \\(h\_{rad}\\) the effective heat transfer coefficient for radiation in \\(W/m^2 \cdot K\\)
+-   \\(A\\) the surface in \\([m^2]\\)
+
+
+### Practical Cases {#practical-cases}
+
+Two parallel plates:
+
+\begin{equation}
+h\_{rad} = \frac{\sigma}{1/\epsilon\_1 + 1/\epsilon\_2 - 1} (T\_1^2 + T\_2^2)(T\_1 + T\_2)
+\end{equation}
+
+Two concentric cylinders:
+
+\begin{equation}
+h\_{rad} = \frac{\sigma}{1/\epsilon\_1 + r\_1/r\_2 (1/\epsilon\_2 - 1)} (T\_1^2 + T\_2^2)(T\_1 + T\_2)
+\end{equation}
+
+A small object enclosed in a large volume:
+
+\begin{equation}
+h\_{rad} = \epsilon\_1 \sigma (T\_1^2 + T\_2^2)(T\_1 + T\_2)
+\end{equation}
+
+
+### Emissivity {#emissivity}
+
 The emissivity of materials highly depend on the surface finish (the more polished, the lower the emissivity).
 Some examples are given in <tab:emissivity_examples>.
 
+Gold coating gives also a very low emissivity and is typically used in cryogenic applications.
+
 <a id="table--tab:emissivity-examples"></a>
 <div class="table-caption">
   <span class="table-number"><a href="#table--tab:emissivity-examples">Table 1</a>:</span>
@@ -114,6 +169,35 @@ The required heat is then:
 | Water at 25 degrees | 4182                            |
 
 
+## Heat Transport (i.e. Water cooling) {#heat-transport--i-dot-e-dot-water-cooling}
+
+<a id="figure--fig:heat-transfer-fluid"></a>
+
+{{< figure src="/ox-hugo/heat_transfer_fluid.png" caption="<span class=\"figure-number\">Figure 1: </span>Heat transfered to the fluid" >}}
+
+\begin{equation}
+Q\_{in} = h \cdot A \cdot (T\_{wall} - T\_{mean})
+\end{equation}
+
+<a id="figure--fig:heat-transport"></a>
+
+{{< figure src="/ox-hugo/heat_transport.png" caption="<span class=\"figure-number\">Figure 2: </span>Heat Transport in the fluid" >}}
+
+\begin{equation}
+Q\_{out} = \phi \rho c\_p (T\_{mean,in} - T\_{mean,out})
+\end{equation}
+
+with:
+
+-   \\(Q\_{out}\\) the transported heat in W
+-   \\(\phi\\) the flow in \\(m^3/s\\)
+-   \\(\rho\\) the fluid density in \\(kg/m^3\\)
+-   \\(c\_p\\) the specific heat capacity of the fluid in \\(J/(kg \cdot K)\\)
+-   \\(T\_{mean}\\) the mean incoming and outgoing fluid temperature
+
+Because of energy balance, we have in the stationary condition: \\(Q\_{in} = Q\_{out}\\)
+
+
 ## Heat flow {#heat-flow}
 
 The heat flow \\(P\\) (in watt) is the derivative of the heat: