Chapter 3 Thermal-electrical analogy: thermal network 3.1 Expressions for resistances Recall from circuit theory that resistance �"#"$ across an element is defined as the ratio of electric potential difference Δ� across that element, to electric current I traveling through that element, according to Ohm’s law, � � = "#"$ � (3.1) Within the context of heat transfer, the respective analogues of electric potential and current are temperature difference Δ� and heat rate q, respectively. Thus we can establish “thermal circuits” if we similarly establish thermal resistances R according to Δ� � = � (3.2) Medium: λ, cross-sectional area A Temperature T Temperature T Distance r Medium: λ, length L L Distance x (into the page) Figure 3.1: System geometries: planar wall (left), cylindrical wall (right) Planar wall conductive resistance: Referring to Figure 3.1 left, we see that thermal resistance may be obtained according to T1,3 − T1,5 � �, $-./ = = (3.3) �, �� The resistance increases with length L (it is harder for heat to flow), decreases with area A (there is more area for the heat to flow through) and decreases with conductivity �. Materials like styrofoam have high resistance to heat flow (they make good thermal insulators) while metals tend to have high low resistance to heat flow (they make poor insulators, but transmit heat well). Cylindrical wall conductive resistance: Referring to Figure 3.1 right, we have conductive resistance through a cylindrical wall according to T1,3 − T1,5 ln �5 �3 �9 $-./ = = (3.4) �9 2��� Convective resistance: The form of Newton’s law of cooling lends itself to a direct form of convective resistance, valid for either geometry. T1 − T@ � �, $-.? = = (3.5) �, ℎ� 3.2 Series and parallel thermal networks With expressions for calculating thermal resistances in hand, we move on to the important task of choosing appropriate models for thermal networks. The utility of thermal resistances exists in the ease with which otherwise complicated thermal systems are modeled. This section considers series and parallel thermal networks, drawing analogies to circuit theory’s rules for equivalent resistance. For simplicity, only Cartesian geometries are considered. Series networks: Recall from circuit theory that resistors in series produce an equivalent resistance between input-output terminals that is the sum of individual resistances, owing to the fact that each individual resistor has the same current flowing through it. �1"9B"1 "C = �B B (3.6) Likewise, systems in which multiple elements are intercepted by a single heat flowline are modeled serially. Figure 3.2: Layered planar wall. Figure 3.2 illustrates an example series model and circuit schematic for a double-exposed, layered window (note the presence of both conduction and convection transfer modes). In this example, the equivalent thermal resistance would be the sum of each resistance shown, and would relate overall temperature difference across the network according to T@,3 − T@,D 1 1 �G �H �I 1 �"C = = + + + + (3.7) �, � ℎ3 �G �H �I ℎ5 Parallel networks: Recall that electrical resistors in parallel produce equivalent resistance L3 L3 �JK9K##"# "C = �B B (3.8) Such a model exists when multiple current paths exist between two nodes. A similar situation occurs in thermal networks when two paths (through different media) exist between two points of the same temperature. An example system is shown in Figure 3.3, with two equivalent schematics. Figure 3.3: Parallel conduction network. The equivalent resistance between the left and right faces of material F (or G) is thus L3 L3 L3 �M �N �"C = + (3.9) �M � 2 �N � 2 Note that this equivalent resistance will be less than either individual component. Note that this circuit model is valid only when heat flow is assumed approximately one-dimensional (if significant heat flow occurred vertically between materials F and G, the resistors-in-parallel model would be invalidated). 3.3 Elements of thermal network There are two fundamental physical elements that make up thermal networks, thermal resistances and thermal capacitance. There are also three sources of heat, a power source, a temperature source, and fluid flow. A note on temperature In practice temperature when we discuss temperature we will use degrees Celsius (°C), while SI unit for temperature is to use Kelvins (0°K = -273.15°C). However, we will generally be interested in temperature differences, not absolute temperatures (much as electrical circuits deal with voltage differences). Therefore, we will generally take a reference temperature (which we will label T1), and measure all temperatures relative to this reference. We will also assume that the reference temperature is constant. Thus, if T1 is =25°C, and the temperature of interest is Ti=32°C, we will say that Ti=7° above reference. Note: this is consistent with electrical systems in which we assign one voltage to be ground (and assume that it is constant) and assign it the value of zero volts. We then measure all voltages relative to ground. Thermal resistance Consider the situation in which there is a wall, one side of which is at a temperature T1, with the other side at temperature T2. The wall has a thermal resistance of R12. This is depicted in Figure 3.4 in three different, but equivalent, diagrams. Figure 3.4: Thermal-electrical analogy of conduction heat transfer. Thermal capacitance In addition to thermal resistance, objects can also have thermal capacitance (also called thermal mass). The thermal capacitance of an object is a measure of how much heat it can store. If an object has thermal capacitance its temperature will rise as heat flows into the object, and the temperature will lower as heat flows out. To understand this, envision a rock in the sun. During the day heat goes in to the rock from the sunlight, and the temperature of the rock increases as energy is stored in the rock as an increased temperature. At night energy is released, and the rock cools down. We represent a thermal capacitance in isolation in diagrams (and equations) as shown in Figure 3.5 (in the drawing at the left the coil represents a power source and the stippled object is the thermal capacitance). In the thermal analogy, one end of the capacitor is always connected to the constant ambient temperature. Figure 3.5: Thermal-electrical analogy of thermal capacitance. The electrical model will always have one side of the capacitance connected to ground, or / PQLPR reference. Also, we could write the equation as � = � but since T1 is constant, it can be /S removed from the derivative. The thermal capacitance of an object is determined by its mass and specific heat � = ��J (3.10) where C is the thermal capacitance, m is the mass in kilograms, and cp is the specific heat in J/(kg- °K). It is always assumed that the capacitor is at a single uniform temperature, though this is obviously a simplification in many cases. Example The specific heat of water is 4.2 kJ/kg-°C. - What is the thermal capacitance of 5 liters of water? - If the water starts at θc=25°C, how hot will it be if it is heated with a 1 kW heater for 1 minute? Solution. 3 a) C = m·cp, and 5 liters of water has a mass of 5 kg. So C = 5·(4.2·10 ) = 21 kJ/°C. b) First, calculate the rate of increase of temperature �� ��$ � 1�� 1 � 1 = = = = °� �� � �� �� 21 � 21 °� 21 °� then find the total increase: �� 1 ∆� = $ ∆� = °� ∙ 60� = 2.9°� $ �� 21 � so the final temperature is 25 + 2.9 = 27.9°� Power source (or heat source) A common part of a thermal model is a controlled power source that generates a predetermined amount of power, or heat, in a system. This power can either be constant or a function of time. In the electrical analogy, the power source is represented by a current source. An example of a power source is the quantity q in the diagrams for the thermal capacitance, above. In practice a power source is often an electrical heating element comprised of a coil of wire that is heated by a current flowing through it. Therefore, we use a diagram of a coil of wire to represent the power source. An ideal power source generates power that is independent of temperature. Temperature source Another common source used in thermal systems is a controlled temperature source that maintains a constant temperature. An ideal temperature source maintains a given temperature independent of the amount of power required. A refrigerator is an example of such a source. Another such source is the ambient surroundings. We will assume that the temperature of the ambient surroundings is constant regardless of the heat flow in or out (we will also take ambient temperature to be our reference temperature). Mass transfer (fluid flow) If fluid with specific heat cp J/(kg-°K) flows into a system with a flow rate of G kg/sec and a temperature of Tin °C above reference, and flows out at a temperature of Tout °C below reference then the rate of heat flow into the system is given by �� � (3.11) � = � ∙ � ∙ � − � °� = �� � − � � B. ��� J �� � B. -iS J B. -iS We can cancel the K and °C since a temperature difference (Tin-Tout) is the same in Kelvin or Celsius. If you carefully observe this equation, it makes sense intuitively. Heat into a system goes up with mass flow rate into the system (increased mass flow, yields increased heat flow). Heat into a system also goes up with the specific heat of the mass (higher specific heat indicates increased capacity to store heat). Finally, heat into a system increases with an increased inflow temperature, or a decreased outflow temperature (if the temperature difference between inflow and outflow increases, more heat is being taken from the fluid).