Thermal system elements
Systems in which heat flow is of interest are called thermal systems. For instance, heat generated by an engine or a server farm flows through several bodies via the three modes of heat transfer: conduction, convection, and radiation. This is, of course, a dynamic process.
A detailed model would require a spatial continuum. However, we are often concerned with, say, the maximum temperature an engine will reach for different speeds or the maximum density of a server farm while avoiding overheating. Or, more precisely, how a given heat generation affects the temperature response of system components.
As with electrical, mechanical, and fluid systems, we can describe thermal systems as consisting of discrete lumped-parameter elements. The dynamic models that can be developed from considering these elements are often precisely the right granularity for system-level design.
We now introduce a few lumped-parameter elements for modeling thermal systems. Let a heat flow rate \(q\) (SI units W) and temperature \(T\) (SI units K or C) be input to a port in a thermal element. There are three structural differences between thermal systems and the other types we’ve considered. We are confronted with the first, here, when we consider that heat power is typically not considered to be the product of two variables; rather, the heat flow rate \(q\) is already power: \[\begin{aligned} \mathcal{P}(t) &= q(t). \end{aligned}\] A thermal element has two distinct locations between which its temperature drop is defined. We call a reference temperature ground.
The heat energy \(H\) of a system with initial heat \(H(0)\) is \[\begin{aligned} H(t) = \int_0^t \mathcal{P}(\tau) d\tau + H(0). \end{aligned}\]
We now consider an element that can store energy, called an energy storage element; an element that resists power flow; and two elements that can supply power from outside a system, called source elements. The second difference is that there is only one type of energy storage element in the thermal domain.
Thermal capacitors
When heat is stored in an object, it stores potential energy via its temperature \(T\). This is analogous to how an electronic capacitor stores its energy via its voltage. For this reason, we call such thermal elements thermal capacitors.
A linear thermal capacitor with thermal capacitance \(C\) (SI units J/K), temperature \(T\), and heat \(H\) has the constitutive equation \[\begin{aligned} H = C T. \end{aligned}\] Once again, time-differentiating the constitutive equation gives us the elemental equation: \[\begin{aligned} \frac{d T} {d t} = \frac{1} {C} q,\end{aligned}\]
The thermal capacitance \(C\) is an extensive property—that is, it depends on the amount of its substance. This is opposed to the specific heat capacity \(c\) (units J/K/kg), an intensive property: one that is independent of the amount of its substance. These quantities are related for an object of mass \(m\) by the equation \[\begin{aligned} \label{eq:lumped_thermal_capacitance} C = m c. \end{aligned}\]
Thermal resistors
Thermal resistors are defined as elements for which the heat flowrate \(q\) through the element is a monotonic function of the temperature drop \(T\) across it. Linear thermal resistors have constitutive equation (and, it turns out, elemental equation) \[\begin{aligned} q = \frac{1} {R} T \end{aligned}\] where \(R\) is called the thermal resistance.
Thermal resistors do not dissipate energy from the system, which is the third difference between thermal and other energy domains we’ve considered. After all, the other “resistive” elements all dissipate energy to heat. Rather than dissipate energy, they simply impede its flow.
All three modes of heat transfer are modeled by thermal resistors, but only two of them are well-approximated as linear for a significant range of temperature.
- conduction
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Heat conduction is the transfer of heat through an object’s microscopic particle interaction.1 It is characterized by a thermal resistance \[\begin{aligned} \label{eq:lumped_thermal_resistance} R = \frac{L} {\rho A}, \end{aligned}\] where \(L\) is the length of the object in the direction of heat transfer, \(A\) is the transverse cross-sectional area, and \(\rho\) is the material’s thermal conductivity (SI units W/K/m).2
- convection
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Heat convection is the transfer of heat via fluid advection: the bulk motion of a fluid. It is characterized by a thermal resistance \[\begin{aligned} R = \frac{1} {h A}, \end{aligned}\] where \(h\) is the convection heat transfer coefficient (SI units W/m\(^2\)/K) and \(A\) is the area of fluid-object contact (SI units m\(^2\)). The convection heat transfer coefficient \(h\) is highly and nonlinearly dependent on the velocity of the fluid. Furthermore, the geometry of the objects and the fluid composition affect \(h\).
- radiation
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Radiative heat transfer is electromagnetic radiation emitted from one body and absorbed by another. For \(T_1\) the absolute temperature of a “hot” body, \(T_2\) the absolute temperature of a “cold” body, \(\varepsilon\) the effective emissivity/absorptivity,3 and \(A\) the area of the exposed surfaces, the heat transfer is characterized by \[\begin{aligned} q = \varepsilon \sigma A (T_1^4 - T_2^4), \end{aligned}\] where \(\sigma\) is the Stefan-Boltzmann constant \[\begin{aligned} \sigma = 5.67 \cdot 10^{-8}\ \frac{\textrm{W}}{\textrm{m}^2 K^4}. \end{aligned}\] Clearly, this heat transfer is highly nonlinear. Linearization of this heat transfer is problematic because the temperature difference \(T\) between the bodies does not appear in the expression. For many system models, radiative heat transfer is assumed negligible. We must be cautious with this assumption, however, especially when high operating temperatures are anticipated.
Heat flow rate and temperature sources
Thermal sources include many physical processes—almost everything generates heat!
An ideal heat flow rate source is an element that provides arbitrary heat flow rate \(Q_s\) to a system, independent of the temperature across it, which depends on the system.
An ideal temperature source is an element that provides arbitrary temperature \(T_s\) to a system, independent of the heat flow rate through it, which depends on the system.
Generalized element and variable types
In keeping with the definitions of , temperature \(T\) is an across-variable and heat flow rate \(q\) is a through-variable.
Consequently, the thermal capacitor is considered an A-type energy storage element. A thermal resistor is considered to be a D-type energy dissipative element, although it does not actually dissipate energy. It does, however, resist its flow and relates its across- and through-variables algebraically, both signature characteristics of D-type elements.
Temperature sources are, then, across-variable sources and heat flow rate sources are through-variable sources.
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We use the term “object” loosely, here, to mean a grouping of continuous matter in any phase.↩︎
Thermal resistance can also be defined as an intensive property \(\rho^{-1}\), the reciprocal of the thermal conductivity. Due to our lumped-parameter perspective, we choose the extensive definition.↩︎
The parameter \(\varepsilon\) is taken to be the combined “gray body” emissivity/absorptivity. Consult a heat transfer text for details.↩︎
Online Resources for Section 8.2
No online resources.