Why do we model? Partly because it is cheaper and quicker than making a product and measuring its performance, and partly because modelling allows the designer to estimate the effect of proposed changes and potential relayouts, and make informed decisions about aspects of a design that will affect its thermal performance without the need to “cut metal”.
As shown in Figure 1, this Unit is one of a linked set that describe how that modelling process works, both at the “on a back of a fag packet” level and at high levels of sophistication.
Whether we are carrying out a hand calculation, or making use of a computer-based numerical methods, we need to create a model of the system that we want to examine. In case this concept is unfamiliar, in this first section we are looking at the generic task of modelling. If you have already used a modelling tool such as FemLab in another module, then you will be able to read this section quite quickly; if not, we strongly recommend that you read the four linked documents, as these explain much more about how models work and what their limitations are.
A model in the context of this module is an interpretation of a physical problem that can be translated into one that can be solved mathematically. For example, we may wish to know how to best place components on a PCB to minimise component heating or to reduce electromagnetic interference. Now, we could set up a test circuit, and take measurements in a laboratory. We would then tweak the design, re-take the measurements and compare them with the first set, repeating this process until we found the best design solution. The problem lies in the number of iterations needed to optimise the design, which makes this approach not viable in terms of time and expense.
A better alternative is to use a model that allows us to represent a physical system numerically, so that we can test a design within a computer environment without actually having to build it. This provides a high level of flexibility, because we can change a design and recalculate the model to see what effect this has on the design behaviour. Furthermore, this can be done without significant cost and time, especially if the model is computer-based.
Not only can we see the effects of design changes, but we can also examine the “sensitivity” of the physical system both to expected changes during life, such as the clogging of filters, and to production variations in the properties of system components. A model also allows us to examine the effect of step changes in conditions such as occur when a system is powered up.
Modellers distinguish between ‘parametric’ and ‘physical’ models. In a parametric model, parameters within the model relate the outputs to the inputs. A good example of a parametric model is one that uses transfer matrices, such as those that might be used in a circuit analysis package like SPICE. In SPICE, electronic components are described by parameters such as resistance, inductance, capacitance, and gain. The overall response of a circuit containing these components is determined by the application of standard laws, such as Ohm’s Law, and is formed using standard matrix methods. The final answer is expressed in terms of a transfer matrix that gives the desired output parameter (such as voltage or current) for a particular component in terms of the system’s input parameter (usually voltage or current).
By entering a full description of the real system into the modelling software, a physical model attempts to replicate the real device as closely as possible, particularly in terms of physical layout and construction. This type of model is often used where a problem is difficult to solve analytically, or perhaps cannot be solved even partially by other means. This may be because there are too many parameters, or because their interaction is complex.
Physical models are particularly useful where non-linear parameters are involved. Take the case of a multilayer board with a number of layers, each layer having its own thermal properties. If these properties are linear, we can simplify the model by performing some kind of numerical averaging technique, such as a volume fraction, to obtain a representation of the multilayer board as a single layer. This will run more quickly, and might provide us with results that are adequate. However, if the parameters are non-linear and depend upon temperature, we cannot carry out such an averaging process, because we do not know in advance what the board temperature will be – which is why we are running the model in the first place! So we have no option but to describe the model as fully as possible, define the non-linearity of the parameters, and let the modelling application produce the results.
One of the concepts used in the modelling world is that of ‘black box’, ‘grey box’ and ‘white box’ models. If this means nothing to you, do read our short brief Choosing a model. The parametric model is a ‘black box’ in the sense that it creates a simple transfer matrix, although the parameters are based on physical principles. This level of modelling is achievable by hand calculation methods. The physical model will be either grey or white box, depending on how much detail we include. Typical thermal modelling software is technically grey box, because the only elements explicitly modelled are thermal aspects such as conduction and convection; a ‘multi-physics’ simulation, as with FemLab, would enable us to take into account mechanical aspects such as thermal expansion and vibration.
The great advantage of not treating the model as a black box is that we are able to allow for non-linearity in parameters. Our paper Linear and non-linear models contains one example of when we might want to use a more exact model for thermal analysis.
Whatever the model we build, we need to evaluate its results, which means ‘solving’ the model’s equations by finding values for the parameters that simultaneously satisfy all the equations. At the black box level, we may be fortunate enough to create a model with relatively simple equations that are amenable to ‘analytical’ methods. Otherwise referred to as ‘exact’ methods, these yield ‘closed-form’ solutions, with unique and precise answers.
Unfortunately, the more detailed the model, the more likely it is that the resulting equations will be complex, involving large systems of equations, non-linearities, differential operators, matrices, and similar, so that analytical solutions are not possible. In this case we are forced to use ‘numerical’ methods, which are techniques for expressing problems in such a way that they can be solved simply by repeating arithmetic operations.
Numerical methods are all about reaching approximate practical solutions whilst keeping errors within reasonable bounds, rather than looking for exact answers that in practice cannot be obtained. An early example of the approach was that employed by the Babylonians to calculate a value for the square root of 2 (needed for surveying purposes). This used a successive approximation method, the formula used each time (the ‘algorithm’) being:
. . . . . . . Eqn. 1
We start with an arbitrary guess F_{0}, calculate the value of F_{1} using the formula, and then substitute the value of F_{1} into the formula, so as to calculate F_{2}, and so on.
In this example we find that:
It is tempting, but highly dangerous, to derive these as general points, because the methods employed in numerical analysis vary widely:
Also remember that the computational routines used may create artefacts. For example, if you embody the algorithm of Equation 1 into an Excel spreadsheet, you will probably find that apparent convergence is achieved remarkably quickly, the reason being that only a limited number of significant figures can be handled by the software.
Natural convection provides a good example of where numerical methods come into their own. Compared with the case of forced convection, the contribution from free convection is more complex and lengthy to calculate analytically since it requires a trial-and-error solution. In most cases, the designer knows the ambient temperature of the fluid, and the temperature of the junction, but not the temperature of the extended surface. As the heat transfer coefficient depends on this surface temperature, the only way of proceeding is to make an estimate of the surface temperature and then solve the system of equations based on the estimate. If the estimate is different from the solved value, another estimate has to be made, and the process repeated until estimate and solved value converge.
For any modelling, we need to describe reality in terms of time and space, so that the ‘number crunching’ can take place. Traditionally a great deal of simplification was needed in order to allow the successive approximations involved in numerical methods to reach convergence within an acceptable time. Time can still be an issue – you will notice that most of the simulations you run will not give immediate results, and some may even need to run overnight. Fortunately, improvements in computing power, in parallel with more efficient computational methods, mean that we don’t necessarily have to use coarse meshes, simplify the situation to one or two dimensions, use symmetry to cut the computational work, or shy away from transient analysis. Nevertheless, we urge you to read our brief Representing time and space in a numerical model, because strategies like those could be important if you ever need to run more complex simulations, particularly in a multi-physics environment such as FemLab. And you might also need to think about making some degree of simplification if you find that the models in your assignments are running unacceptably slowly.
For most thermal simulation, a most important aspect is how one should represent space, because software models allow a degree of freedom in fitting the mesh. Many of the earlier programs were restricted to a rectilinear mesh, but this is not necessarily a good fit to round surfaces. Remember as you study Unit 11 to find out what you can about available programs in terms of the range of mesh types they will support, and in particular their ability to handle varying mesh sizes. There is a danger that looking in sufficiently fine detail to pick up small components on the surface, to see whether they contribute to turbulent flow, will lead to the simulation running out of memory when (unnecessarily) applying the same small size of mesh to a large and fairly invariant volume.
Sometimes it is only necessary to model small sections of a system. For example, it may not be necessary to model the entire board in order to model the thermal experience of a component, where a small section would suffice. Such decisions can only be made with experience, but model size is an important consideration and can save a lot of simulation time. Thermal analysis can be performed on different levels:
Typically we will combine modelling at the three levels, using library data as far as possible but modelling individual components where necessary, then assembling our board from these prefabricated elements, and finally building a system. However, whilst it may be possible to ‘drill down’ to the fine detail, the actual system-level analysis tool will probably use simplified models of components and boards based on the information that has been built up.
An important aspect to consider when designing a model is what is actually required of the model in terms of its precision. Were there no limiting factors, we would always design a model that precisely matched the physical system, but this is rarely possible, because it results in a very complex model that even the fastest computer may take weeks or months to run. We must therefore make sensible approximations in our model, in such a way that the precision of the model is not overly compromised and the model execution time is not too great.
Always be careful to understand the model that has been used, and be aware both of the limitations of the techniques and of the possibilities of error. And never, ever assume that a thermal simulation is correct without validating your predictions by measurement, even though, as you will have seen in Unit 7, this is not necessarily easy to do. Assuming, rather than demonstrating, is equivalent to designing a circuit with an embedded processor, and then relying solely on the fact that the ASIC worked correctly in the computer model. Would you ship the final product without checking a prototype?!
Read our brief Pitfalls of modelling to see some of the sources of error and the benefits of validation.
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Fortunately, complex models are not always needed. In fact, it is often more helpful to start with a simplistic view of a thermal situation, as a quick calculation can tell you how deep is the trouble you are in, without the need to think through all the modelling issues, build a computer model, and then wait for an extended computation. Also, the results of simple models remain useful, because they provide the basis for a sanity check on the results of complex simulations – does the temperature predicted by simulation lie within the range of the earlier estimate? – or has some error crept in? So let’s look at an example of a simple model, in this case for estimating the temperatures within an enclosure.
When in Unit 1 you discovered that each square metre of the Earth’s surface is irradiated by over 1kW of energy from the sun, you were probably surprised at the high figure. In fact, you might even have wondered why your surroundings on a hot summer’s afternoon are not closer to conditions on the planet Mercury! Hopefully, you followed up that thought with some reflection both on the ways in which heat is lost from the surface, and also on the wide variety of temperature outcomes over the face of our planet, based on aspects of the environment such as the prevailing wind, the composition of the surface, and any haze, smoke or cloud in the vicinity.
Yet the situation with the Earth’s surface is exactly the same as with any other “system”: once an allowance has been made for any heat generated within the system, and assuming that only thermal energy is involved, the total heat flow in must be balanced by the total heat flow out, in accordance with the First Law of Thermodynamics. [Note that the flows are totals for all routes, convection, conduction and radiation]
The temperature that our system reaches in steady state will depend on the mechanisms of heat transfer in and out, and on how these depend on temperature differences. With a system where we can take a simplified view of these mechanisms, we can even get useful information from a very broad brush approach.
To give an example of this:
The volume of air required is linked to the dissipation and the allowable temperature rise by the equation:
. . . . . . . Eqn. 2
where J is the amount of heat transferred in unit time, C_{p} is the specific heat of the fluid, ΔT is the temperature rise within the cabinet, and W is the mass flow of the fluid, defined as the volume in unit time multiplied by the density, ρ.
The equation is more useful in the form
. . . . . . . Eqn. 3
as this makes it possible to estimate the airflow needed to dissipate a given amount of heat for an allowable increase in temperature.
Using the parameters for dry air at 20°C and standard atmospheric temperature (101.3kPa), a combination that is the default for many fan manufacturers
specific heat ≈ 1.003J/(g·K) = 0.5572kJ/(kg·°F)
density = 1.204·10^{−3}g/cm^{3} = 0.03409kg/ft^{3}
the equation becomes:
. . . . . . . Eqn. 4
A final note of caution is that focusing on the allowable temperature rise tends to give the impression that, for a specified inlet temperature, the temperature at all other points in the cabinet will be no more than the combined figures of inlet temperature and temperature rise. This is very misleading:
Nevertheless, following good practice guidelines for the mean temperature rise (typically no greater than 10°C) will give some comfort about the likely maximum temperatures, and will give a first-pass estimate of the AHS specification that can form the basis for more detailed estimation or simulation.
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Many of the calculations we carry out will be in steady-state and consider the temperature gradients in just one dimension. Although simple, such models may be used to represent accurately many of the problems that we face.
At the most basic level, for conduction along a body, where we are seeking to establish the temperature at an intermediate point, we can devise a hydraulic analogy. Figure 1 shows this analogy – if you like, a model of a thermal system – in which header tank A, whose level is maintained automatically by a float valve mechanism, feeds intermediate tank B through a narrow tube, while tank B is at a lower level, and itself feeds into runaway C of infinite capacity.
Intuitively, the level of water in tank B can be seen to depend on the relative resistance to flow of the inlet and outlet pipes. A balance will be achieved at which the difference in levels h1, which is the “motive power” that drives fluid through pipe A–B, creates a flow into the intermediate tank that is balanced by the flow out through pipe B–C, itself generated by the “head” h2. If pipe B–C is of larger diameter than pipe A–B, the level in tank B will be relatively low; conversely, if pipe B–C is smaller, the level in tank B will be higher and the tank may even overflow. In this case, with a Newtonian fluid such as water, the flow rate will depend linearly both on the pressure drop and on the area of the pipe, from which calculating the height achieved in tank B is a trivial task.
Whilst this calculation depends on some simplifications, we can use the same balancing method to assess the condition in tank B for any fluid, provided that we can develop a relationship between pressure, pipe diameter, and flow rate for the particular fluid. For example, we could allow for liquids that need more than a certain pressure in order for flow to start, we could allow for the “viscous drag” near the inner surfaces of the tube, and we could take account of the changes in viscosity produced by changes in temperature. What we can be sure of is that flow will only take place when there is a pressure difference, which is an exact analogy to the Second Law of Thermodynamics, and that the rate of flow will be some function of the pressure difference, though this may be quite complicated.
We can also be sure that, at steady state, the total flow into any one point will be zero, with the flow in equalling the flow out. There is a slight complication that, in order to ensure this zero sum, we may have to extend the analogy to allow for fluid leaking or evaporating from the pipe and tanks, which are the equivalent of heat outflows through convection and radiation.
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However helpful the hydraulic analogy may be in reminding us that both heat and air flow, and that their distribution within a system will depend on the relative resistance to flow, it is probably less convenient than its electrical equivalent. Here we use the concept of “thermal resistance” and thermal circuits to model heat flow, in the same way that electrical circuits are used to model current flow. This is a topic on which we will be expanding in Unit 9, and one that is especially useful when modelling conduction, though it can also be applied to convection and radiation.
Table 1 lists the electrical properties where you will frequently find thermal analogues used, and gives both the analogue and its most common symbol:
electrical property | thermal analogue | alt. |
---|---|---|
voltage drop V | temperature drop ΔT | |
current flow I | rate of heat flow J | q'' |
electrical resistance R | thermal resistance R_{θ} | θ |
electrical conductivity σ | thermal conductivity k | |
capacitance C | thermal capacitance C_{θ} | Θ |
Consider the block of material illustrated in Figure 4, where heat flows between each face and the opposite face.
If we think of the opposition to heat flow as being a resistor connecting opposite faces, of heat flow itself as current, and of temperature as voltage, then the value of thermal resistance R_{θ} is given by
. . . . . . . Eqn. 5
where Δl represents the length of each side of the cube, and k is the thermal conductivity, the subscript indicating that this parameter is a function of temperature.
If we now consider the time taken for the block of material to change temperature then we have a capacitor whose capacitance is given by
. . . . . . . Eqn. 6
where S is the specific heat capacity and ρ the density of the material of the block. Using these equations, we can create an electrical analogy for heat conduction that takes the form of an RC network.
Convection can also be described by a heat transfer coefficient h and the following heat transfer equation:
. . . . . . . Eqn. 7
where q is the heat flow (units W), h the convection heat transfer coefficient (units W·m^{−2}K^{−1}), A the area across which heat transfer takes place (units m^{2}), and T_{solid} and T_{fluid} the surface temperature of the solid and the interfacial temperature of the liquid respectively. If we again take temperature to be analogous to potential difference, and heat flow as analogous to electrical current, then we can calculate a representative resistance that is given by:
. . . . . . . Eqn. 8
By devising a model that reflects the physical reality of the system as a network of interconnected thermal resistances, we can calculate the thermal characteristics of the system. In such a model, the resistance values are determined:
The greatest uncertainty lies in these estimates, which have to be based on information about the air flow and assumptions made about the nature of the local flow. For this reason, it is common to run the calculations using a range of values for h, so that the sensitivity of the model to this variable can be explored.
Although this simple approach has some limitations, considering thermal properties as being analogous to electrical properties is a useful tool for conceptualising and quantifying heat transfer problems that we will refer to throughout the module and see some examples of in Units 9 and 12.
An example of steady state heat transfer calculations using the electrical analogy.
A PCB has an average thermal conductivity of: 0.4J·s^{−1}m^{−1}k^{−1}
Compare your answers with the ones given here
At this link we have a more formal derivation of the analogy for conduction, convection and radiation which continues the theme in our supplementary information in Unit 5 on the heat diffusion equation.
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The path by which heat flows from a device to the ultimate heat sink is usually complicated. Some heat will be conducted directly to the package surface, from where it is removed by convection and radiation to the surroundings. Other heat is conducted through the leads and any mounting material to the PCB, arriving at the system enclosure by a mixture of conduction and convection processes. If we have a thermal analysis package available, then we can model all the mechanisms, though keeping the view simple helps create a simulation for which a convergent solution can be found without needing a supercomputer!
If a simulation package is not available, or we wish to simplify our view to make use of a package, then we need to understand more about the convection processes that take place. A greater understanding will also help us as we seek to make the best use of airflow, for example by optimising the board spacing so as to minimise the system impedance, and by enabling us to select the right AHS, having regard to grilles, filter, and the presence of packages in the air stream, all of which lead to a decrease in the flow rate and an increase in the pressure gradient.
In a basic model that it is possible to evaluate by hand calculation or using a spreadsheet, the thermal path from a junction to the environment has three components:
Of these three elements of thermal resistance, the third is the most difficult to estimate, as this depends both on the local conditions of airflow and on the temperature. However, given that a heat sink is made of highly conductive material, so tends towards having an even and well-defined temperature, it is possible to get useful results from simple formulae that involve a convection heat transfer coefficient. As we have seen earlier, the rate of heat transfer from the surface to the ambient (q) is related to the surface area A, the surface temperature T_{s}, and the ambient temperature T_{f} by:
. . . . . . . Eqn. 9
and the thermal resistance due to convection may be derived as:
. . . . . . . Eqn. 10
The area may be the actual area, in the case of a flat plate, or the effective area of a heat sink, based on its efficiency: more about this in Unit 14. From the thermal resistance(s) assigned to the convection path(s), combined with the thermal resistances for the conduction paths, we can then calculate the performance of the whole assembly.
But the real challenge when making simple models is to choose a realistic estimate for h, the convection heat transfer coefficient. As you will see in the following activity, this can be derived from estimates of flow and heat transfer. You will also find typical values in the literature, and these are valuable at least as starting points for modelling.
Read in Azar and Moffat Evaluation of different heat transfer coefficient definitions about how one can attempt to convert estimates of the convection flow rate and the amount of heat transferred into an estimate of the convective heat coefficient.
Then carry out a web search to establish the likely range of heat transfer coefficients, as measured and calculated by a number of authors. You will be able to use this information in your study of Unit 9 and again for Assignment 1.
Note that there are significant differences in the values quoted, depending on whether convection is forced or natural, on whether or not the flow is turbulent, and on the thickness of the boundary layer between fluid and surface.
As well as modelling the heat flow, a related challenge when modelling a system is to model the mass transport of the cooling fluid (gas or liquid), in order to quantify the flow rates throughout the system. We know intuitively that, even with forced convection, there will be variations in flow rate, depending on the physical layout of the assembly. Given the ways in which fluids flow, this is a difficult modelling exercise, but fortunately the concept of impedance to air flow can be brought into play. It is even possible to create a simplified view of airflow within a complex environment that gives a first-pass approximation to the real world without involving full and complex simulation – there will be more about this in Unit 9.
Velocities and pressure differences in natural convection are usually much smaller than with forced convection. Another key difference that affects analysis is that with forced convection the flow rate is generally known, whereas the natural convection flow is linked with and dependent on the local temperature field. This means that the flow cannot be known at the outset, so has to be determined by detailed consideration of the heat and mass transfer processes that are coupled with fluid flow. So, for practical purposes, computer simulation is the only realistic way by which the presence of turbulence can be assessed, other than by experiment. If performing hand calculations, it is important to use a heat transfer coefficient that is representative of the physical situation, and which has been measured or derived for a situation similar to the one you are interested in.
For many purposes engineers simply ignore heat transfer by radiation. But this can be misleading. In fact, Cathy Biber shows in her article^{1} A radiative “heat transfer coefficient” that, in the case of a closed box, the contributions of radiation and natural convection are actually similar.
We saw in Unit 5 that we can express radiation heat exchange in the form of Equation 11, where the radiation heat transfer coefficient is given by Equation 12:
. . . . . . Eqn. 11
. . . . . . Eqn. 12
Here we have modelled the radiation in a manner similar to convection, and linearised the radiation rate equation. However, the radiation heat transfer coefficient used is strongly dependent on temperature. Dr Biber suggests a useful simplification, which is to approximate the heat transfer coefficient by:
. . . . . . . Eqn. 13
where σ is the Stefan-Boltzmann constant and ε is the grey-body emissivity of the object whose absolute temperature is T_{m}, the arithmetic mean of the temperature of the object and the temperature of the surrounding walls.
It is also possible to combine convection and radiation into a single heat transfer coefficient as Bruce Guenin does in Convection and radiation. However, this may not be particularly helpful in the kind of review typically required early in the study, in which the role of radiation heat transfer is uncertain.
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Each of these lists is in the order in which the material is referenced in the Unit text. However, note that links to SAQ answers are not included!
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