Problems

Problem 8.1 (TINKER)

Draw a linear graph of the fluid system with schematic below.

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Problem 8.2 (TAILOR)

Draw a linear graph of the fluid system with schematic below.

A linear graph representation is shown below.

Problem 8.3 (SOLDIER)

(a) Draw a linear graph of the fluid system with schematic below. (b) Draw a normal tree and identify the state variables and system order.

A linear graph representation is shown below with a normal tree in green.

The state variables are $P_{C_1}$, $P_{C_2}$, and $Q_I$ and therefore the system order is $n = 3$.

Problem 8.4 (TPAIN)

Consider an apparatus with two chambers filled with gas at potentially different temperatures illustrated in figure 8.9. Temperature sensors are embedded in the two “sensor blocks,” made of copper for low thermal resistance and made large enough to provide enough thermal capacitance to smooth out temperature fluctuations.¹ The “structural conduit” is made of steel, less thermally conductive, but conductive nonetheless. The conduit provides structure to the apparatus and is hollow to allow the sensor wires to run through.

Figure 8.9: A diagram of the two-chamber apparatus.

A concern with this apparatus is that the temperature in one chamber will affect the temperature in the other, most conspicuously by heat conducting through the structural conduit.

We will begin an analysis of the thermal isolation of the two chambers and temperature measurements. Develop a thermal lumped-parameter model as follows.

Describe the lumped-parameter elements you will use to model the system.
Draw a linear graph of the lumped-parameter model.
Superimpose a normal tree on the graph, identify the system order, and choose the state variables.

Chambers, sensor blocks, and heat sources

The temperature (average?) of a chamber is likely distinct from that of its sensor block in transient response, so a node for each chamber is reasonable, as is a node for each sensor block.²

We are not told how heat flows into the chambers. It would be reasonable to model the chambers as input temperature sources. However, it is likely that the origins of the heating will be well-understood, so it is perhaps better to model each chamber, which will hold some heat energy, as a thermal capacitor ($C_1$, $C_4$) with a heat source applied ($q_{S_1}$, $q_{S_2}$).

Ambient temperature

There’s the matter of the ambient air temperature as well. Heat may flow through the chamber walls or via the conduit return to escape the system. We choose to assume the only possible path to the outside temperature is via the conduit (and not via outer insulation, but it is reasonable to include both). The conduit has a “T” in it and each sensor block is influenced by the external air through that "T," the center of which we choose to model as a node.

Together with a reference/ground temperature node, that brings us to seven nodes.

Temperature source

The outside air temperature can be modeled as a temperature sources $T_S$ input provided from outside the dynamic system.

Thermal capacitance of the sensor blocks

There is a question of which nodes have thermal capacitance. We have already said that the chambers should have capacitance. In addition, the sensor blocks, which are made of a material that has significant heat capacity, should have capacitance. All thermal capacitors connect to ground in order to account for the internal thermal energy of the element, so it’s a matter of determining which should have capacitors.

Thermal resistance

The flow of heat among the seven nodes identified above is intirely through elements that provide thermal resistance.

Certainly, a chamber and its corresponding sensor block transfer heat via convection, which we can model with a resistance in each case.

The aim of our analysis is to determine the impact of heat flow through the structural conduit, so clearly there must be a heat path through it, between the sensor blocks. Here we have another consideration. Above, we decided to model the effect of ambient air temperature via a node that connects to the conduit. The conduit path is a “T,” and heat flows through the common center node or “junction” among the sensor block nodes and the ambient temperature node. Each of these connections has a resistance of its own, with the junction-sensor block paths appearing to have equal thermal resistance (it appears symmetric).

Insulation assumption

We could include heat path (resistor) that connects the two chambers directly and a heat path from each chamber to the ambient temperature; however, as we mentioned above, we will assume the insulation is sufficient to minimize those effects to negligibility.

Linear graph etc

See the linear graph and normal tree of . This is effectively a translation of the above into linear graph form and therefore needs little explanation at this point.

Figure : A linear graph of the thermal model with normal tree superimposed.

From the linear graph, we identify the four independent energy storage elements to be $C_1$, $C_2$, $C_3$, and $C_4$, and therefore the system order is $n = 4$ and the state variables are \[\begin{aligned} T_{C_1}, T_{C_2}, T_{C_3}, T_{C_4}. \end{aligned}\]

As mentioned above, it is possible to get a different order of model under different assumptions, but the model presented is a balance of analytic minimalism and descriptivity.

Problem 8.5 (DRAMP)

Consider a device with four amplifiers in an array on a printed circuit board (PCB), as illustrated in figure 8.10. The amplifiers generate significant heat (as a byproduct), and they must be cooled. For this reason, each amplifier has mounted on top a heat sink device with fins. A fan forces airflow over the fins to dissipate the heat via convection.

Figure 8.10: Top view of four amplifiers in a chassis.

As the designer of the chassis housing the amplifiers, you would like to develop a lumped-parameter thermal model of the system to ensure that, under different heat generation loads, the amplifiers remain within their acceptable temperature range.

Describe the lumped-parameter elements you will use to model the system, including inputs.
Draw a linear graph of the lumped-parameter model.
Superimpose a normal tree on the graph, identify the system order, and choose the state variables.

The amplifiers, which we can assume to be integrated circuits (ICs), can be modeled as heat sources $Q_{S_i}$. The ICs themselves cannot hold much heat, but they are usually connected to their heat sinks through thermal paste, which have low thermal resistance, so we can model the heat sources $Q_{S_i}$ as flowing directly into thermal capacitances $C_i$ that model, in part, the function of the heat sinks. The other function of the heat sinks is to dissipate heat through convection, in concert with the fan. This convection depends on the geometry of the heat sinks, the (average) temperature of the air flowing over the heat sinks, and the speed of airflow. We model the heat sink convection as thermal resistances $R_{S_i}$.

The PCB absorbs some heat through conduction (from each amplifier IC), which can be modeled as thermal resistances $R_{P_i}$. It would be reasonable to ignore this conduction and assume all the heat is dissipated through the heat sinks, but we will not ignore it in this solution. The PCB can store some heat, i.e., it has thermal capacitance. We could model multiple regions of the board as thermal capacitances with resistances among them, but we choose to model the board as a single thermal capacitance $C_\text{PCB}$. Where does the heat from the PCB go? Like the heat sinks, the PCB exchanges heat with the air through convection ($R_\text{PCB}$), but it usually has much less surface area. There may be some conduction from the PCB to the chassis, through structural elements (like bolts), but we will ignore these effects (ignoring these effects make our model more conservative).

The temperature of the air in the chamber certainly varies, throughout. However, we will assume that the air flow from the fan is sufficient to model it as a single temperature. With enough air flow, this temperature should be fairly close to the temperature of the air outside the chassis, which can be modeled as a temperature source $T_S$.
See the linear graph and normal tree of . This is effectively a translation of the above into linear graph form and therefore needs little explanation at this point.

{#fig:dramp-01 fig width=“1\linewidth”}

From the linear graph, we identify the five independent energy storage elements to be $C_1$, $C_2$, $C_3$, $C_4$, and $C_\text{PCB}$, and therefore the system order is $n = 5$ and the state variables are $$
\[\begin{aligned} T_{C_1}, T_{C_2}, T_{C_3}, T_{C_4}, T_{C_\text{PCB}}. \end{aligned}\]
$$

It is possible to get a different order of model under different assumptions, but the model presented is a balance of analytic minimalism and descriptivity.

Problem 8.6 (UP)

Consider the diagram of the first stages of a drinking water treatment plant shown in figure 8.11. The water to be treated comes from a reservoir and is pumped by Pump 1 into the coagulation tank. The suspended particles are too small to settle via gravity, and their generally negative charges repel each other, keeping them from clumping. Here a small amount of a chemical coagulant with positive charge is rapidly mixed in with paddles. Coagulation is the resultant clumping of the particles.

The water with coagulated particles flows through a long, thin pipe and enters a series of 3 flocculation tanks. In the flocculation tanks, which mix at decreasing rates as the fluid progresses, the coagulated particles join into increasingly larger pieces called flocs. Note that the placement of the inlet and outlet of each tank has a sorting effect.

After the third tank, the water flows through a short, wide pipe into the sedimentation tank. Now the flocs are large enough to settle via gravity, and Pump 2 on the outlet pumps the water sans flocs to filtration and disinfection stages of the purification process.

The quality of the process is highly dependent on the volumetric flow rates and pressures in the tanks. Develop a lumped-parameter linear graph model with the following steps:

Describe the lumped-parameter elements you will use to model the system, including inputs.
Draw a linear graph of the lumped-parameter model.
Superimpose a normal tree on the graph, identify the system order, and choose the state variables.

(a) Each of the tanks will be modeled as a fluid capacitor. The coagulation tank is modeled as a fluid capacitor. The long pipe connecting it to the first flocculation tank is modeled as a resistor and inertance in series.

The first and second flocculation tanks have different outlet and inlet heights, with an orifice modeled as a resistor, between. Each tank could be modeled as a constant pressure source to bridge the height differential, then a fluid capacitor for the remaining fluid level. There are two disadvantages to this: we have extra (constant) inputs and most of the energy stored in the tanks isn’t counted. Fortunately, the fluid resistance between the tanks depends only on the height difference, so we can simplify things by modeling the orifice resistor connecting between the two tanks at the full-height pressures.

In the third flocculation tank, our model accounts for the height difference explictly via a constant pressure input corresponding to the pressure head. We do lose most of the energy with this choice, but it has the advantage of allowing us to ignore the depths of the sedimentation tank, which may have considerable sedimentation.

The short pipe connecting the third flocculation tank to the sedimentation tank is modeled as a fluid resistor.

The two pumps, on the left and the right, are modeled as volumetric flowrate sources. In most cases, the flowrates of the two pumps will be regulated (and approximately equal to each other).

(b and c) See for the linear graph and normal tree for the lumped-parameter model. The state variables are the across-variables on A-type branches and through-variables on T-type links; that is, \[\begin{aligned} P_{C_1}, P_{C_2}, P_{C_3}, P_{C_4}, P_{C_5}, Q_{I_1}. \end{aligned}\] With $6$ state variables, the system order is $n = 6$.

Figure : Water treatment plant linear graph model.

It would be reasonable to subdivide each chamber into several small “finite elements,” but then there’s the question of how to model the gas motion (which takes us beyond a thermal model). Another option is to assume the temperature of the chamber and the block are identical. This is reasonable, but probably misses some of the important dynamics going on.↩︎
It would be reasonable to subdivide each chamber into several small “finite elements,” but then there’s the question of how to model the gas motion (which takes us beyond a thermal model). Another option is to assume the temperature of the chamber and the block are identical. This is reasonable, but probably misses some of the important dynamics going on.↩︎

Online Resources for Section 8.6

No online resources.