Unruled Notebook

Entries from February 2007

Notes on Scale Analysis

February 28, 2007 · 4 Comments

Scale analysis or order of magnitude analysis is a back of the envelope method for deriving the most information with a reasonable understanding of the physics of a phenomenon. In other words, it is a method to yield first cut estimates about the relationships between several parameters involved in a particular problem. Let us take an example [1] to see how this works.

Suppose we have a thin plate of transverse width D (see figure) is immersed in a hotter fluid at time t = 0. The question we have is at what time the center of the plate will “feel the heat” so that the temperature of that location starts rising from then on, from its initial temperature. (Quenching, a process of engineering interest, is the reverse of this process, where the plate is “cooled” by the fluid.)

 

scale_example_1.png

Figure 1

To answer this question let us simplify things a bit and assume that the height and thickness of the plate are such that one can consider an unsteady (changing in time) conduction heat transfer process to govern the temperature increase in the plate. The one dimensional heat equation governing the transient conduction heat transfer process in this thin plate is then given by

rho c_Pfrac{partial T}{partial t}=kfrac{partial ^2T}{partial x^2} cdots (1)

where T is the temperature, t the time, x the spatial position, cP is the specific heat capacity at constant pressure, k the thermal conductivity and rho the density of the material that undergoes the transient conduction heat transfer.

Owing to the symmetry of the situation (see figure) we can concentrate on one half of the plate (D/2) for making an order of magnitude estimate of each term in Eq. (1) above. The LHS of Eq. (1) for instance can be “scaled” as

rho c_Pfrac{partial T}{partial t} sim rho c_Pfrac{Delta T}{t} cdots (2)

Observe the Tilda instead of the equality symbol in Eq. (2). Also, the Delta T in Eq. (2) is the temperature difference possible for the system (Fig. 1) in the time t (unknown) and is unknown as such and we can make a guess about it once we write the scale for the RHS of Eq. (1) as well. Proceeding to do so, the RHS of Eq. (1) can be “scaled” as

k frac{partial ^2T}{partial x^2} = k frac{partial}{partial x}Big( frac{partial T}{partial x}Big) sim k frac{1}{D/2)}frac{Delta T}{(D/2)} = kfrac{Delta T}{(D/2)^2} cdots (3)

Observe in Eq. (3) as well there appears a Delta T whose exact value is unknown. Using Eqs. (2) and (3) in (1), we can find a time scale as

t sim frac{(D/2)^2}{alpha} cdots (4)

where alpha is the thermal diffusivity of the material of the plate.

If Eq. (4) is the answer to our problem of when the center of the plate will start feeling the heat or will begin to rise its temperature, then the two unknown Delta T values in Eq. (2) and (3) must be equal. In fact, it is the only instance when the Delta T across the spatial distance of D/2 (see figure) will match exactly with the Delta T for the time duration t (measured from the initial time of t = 0). Hence our assumption about Delta T while finding the answer in Eq. (4) is correct. The result in Eq. (4) for determining the heat penetration time compares well with the results from the exact analysis to the problem.

And the solution method is that simple.

Now for some notes and observations on the basic rules [1] for using this technique on other situations/equations.

It is necessary to define limits for the spatial region in which we perform the analysis. For instance, in the above example the x-axis is extended between 0 and D/2 (and not beyond that).

The philosophy behind this technique is every equation, in a physical sense, is a balance between two dominant scales resulting in the effect or phenomenon that equation is trying to describe. If this is not so, that is, if there are more than a few terms in an equation, one must find out the dominant terms for a particular situation so that the equation could be interpreted (reduced) as a balance between two terms.

For identifying the dominant scales, in general, these rules could be used

For an equation of the form c = a + b, if the order of magnitude of one term is greater than the other, i.e. O(a) > O(b), then the order of magnitude of the sum c = a + b is determined by the dominant term, i. e. c = a + b can be interpreted in a scaling sense as O(c) = O(a). Similar conclusions hold for c = a – b.

If on the other hand, in an equation c = a + b, O(a) = O(b), then one term cannot be thrown out in comparison with the other. Both of them have to be treated with respect and the only scaling argument one can derive is O(c) ~ O(a) ~ O(b).

In a product c = ab, the small is as weighty as the big and so the only scaling rule is of the form O(c) = O(a)O(b). Similar arguments hold for fractions (c = a/b) as well.

While interpreting the result from a scale analysis, one should be careful about certain things. For instance, the result of such an analysis will be accurate only to an order of magnitude, i.e. if the result predicts a scale of order 1, the actual answer could be anywhere within the next order on both sides (0.1 to 10). This may sometimes lead to erroneous and confusing interpretations. For instance if the result from a scale analysis suggests certain parameter group is of order one, say, C ~ 1 and if the actual answer is found to be C2 = 0.1, it doesn’t mean the scaling analysis is necessarily wrong. The correct interpretation lies in the fact that since the scale analysis yields a result of order one for C, it allows C to be even less than 1, which when squared (i.e. C2) could lead to an answer (the correct one that matches with the exact answer, in this case) that is even smaller.

For instance, such an occasion appears when one compares the results from scaling arguments with the more exact results for the prediction of the thermal entrance length in a duct flow. The scaling could still be correct; only the interpretation isn’t. More on this perhaps in a separate post.

On the other hand, scale analysis as a solution technique finds more use and relevance with differential equations rather with integral equations, a reason for its lack of use in such fields as radiation heat transfer, wherein the phenomenon is modeled more with integral and/or integro-differential equations.

To end, scale analysis or order of magnitude analysis is practiced both as a research and pedagogical tool for more than a century. The earliest example in the field of fluid flow (one of my interests) could be the order of magnitude analysis performed by Ludwig Prandtl to reduce the formidable Navier-Stokes equations into a slightly less formidable boundary layer equations, valid inside a “boundary layer” – the region of fluid flow near a solid wall, where viscous effects dominate in the fluid.

Many researchers have used this method with remarkable success in delineating good physical insights about complex problems of scientific relevance. Some applications of this method to a wide spectrum of problems are discussed in [2].

[1] the example and the basic rules are discussed in Convection Heat Transfer by A. Bejan (the same example is also discussed in [2] below). Some observations are mine.

[2] Qualitative Methods in Physical Kinetics and Hydrodynamics by Vladimir Krainov.

Categories: Lecture Notes · Research Notes · Science Notes · Thermal Sciences
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Nano-aluminium and Rocket Science

February 17, 2007 · 22 Comments

Let us talk rocket science. And explore in a long essay, the curious case of the conjecture of using nanometer sized aluminium particles to save rockets from exploding…

A particle of metallic aluminium can be of any desirable size. In the beginning of this post let us have it in microns. These aluminium particles are used in rocket propellants (fuel) for two reasons.

The first one, which is apparent, is that since they are metals, while they burn along with the propellants at over 4,100 K and releases a large amount of energy that significantly expands the gases within the combustion chamber. The expanding combustion gases within a fixed volume results in pressure increase leading to higher exit velocity for the exhaust gases that escape through the nozzle. This results in increased specific impulse of the motor.

The second one is less apparent and a bit more involved.

While burning, these aluminum particles turn into alumina droplets inside the combustion chamber of the rockets. These particles are carried along with the burnt gases that provide the thrust for the rocket – and come out from the rocket as smoke that we see.

Meanwhile, the burnt propellants in gaseous from that are ejecting out of the combustion chamber cause acoustic oscillations, which affect the entire rocket in many ways. For instance, the rocket motor by default has a natural frequency of vibration and if these acoustic oscillations match their frequency, the resulting resonance could cause irrevocable damage. Solid rockets suffer from unsteady acoustical waves that may resonate within the combustion chamber and cause pressure spikes [1]. Further, these acoustic oscillations may lead to the propellant getting burnt very strongly and quickly (faster rate of combustion) ending up at times in the explosion of the rocket nozzle (flanges will fall apart due to excess internal thrust). These acoustic oscillations also could mess with the electronic controls of the rocket leading to their malfunction and disrupt the guidance and control of the rocket.

The second reason for using aluminium particles with rocket propellants is that the resulting alumina particles mix in the gaseous propellant exhaust. They interfere with the traveling acoustic waves mentioned earlier and dampen them, resulting in a steadier, uniform, burning of the propellant.

Figure 1: Cutaway drawing of a solid rocket motor (courtesy [1])

But there are enough caveats in this. For instance, these alumina particles render the exhaust as two phase flow (gaseous flow with solid alumina particles). This two phase flow is locally “dragged” by the solid alumina particles. The resulting locally lagging flows (from the main flow of the exhaust) lead to reduced thrust (momentum loss) for the rocket motor.

Further, when the aluminium particles burn, they don’t do it in tandem with the propellants. Once the propellant begin to burn, the aluminium particles begin to melt a while later upon coming into contact with the flame released by the propellants and they form a sort of a big flame ball fusing many such aluminium particles together. The resulting oxidized aluminum bulk forms alumina droplets of a certain size, which while traveling with the exhaust, can reduce the acoustic oscillations.

But not always.

Just like a flautist uses a shorter flute to play a Jagadhanandhakaraka in Naattai raga at the start of the concert and a longer flute at the end, while playing, say, Jagadhodharana in Kaapi raga, larger the rocket motor, shorter their natural frequency. Hence the size of alumina particles to dampen the acoustic oscillations should be suitably tailored for a particular size rocket motor.

So the question one is left with is what should be the size of the original aluminium particles that are to be added to the propellants, so that after burning they would result in alumina droplets of suitable size that are able to dampen the acoustic oscillations of a particular rocket motor.

To answer this question, one should first answer the question of how pure the initial micron-sized aluminium particle should be.

These aluminum particles are very reactive with atmospheric oxygen, and a thin passivation layer of alumina (Al2 O3) quickly forms on any exposed aluminium surface. This layer envelops the micron size aluminium particle as a hard spherical shell of nanometer thickness (~ 5nm) and protects the metal from further oxidation. This alumina in the annular shell can exist in three forms, the a, b and g types – the former two in crystalline form while the latter g in amorphous form. Alumina in all of these forms is harder than aluminium and has a melting point a few times larger (~ 2300 C) than the aluminium metal (660 C) that resides inside the core.

Figure 2: Aluminum oxide particles photographed using a scanning electron microscope (enlarged 8000x) (courtesy [1])

If such an “impure” aluminium particle (of micrometer size) coated with nanometer sized outer shell of alumina is used in the propellant, while combustion, the aluminium will melt first, become a liquid with about 6 percent volume increase and try to come out by cracking the hard alumina shell. The successful liquid aluminium oozing out of the shell at random places will grow on the surface of the shell by further oxidation and form a bridge with a similarly behaving adjacent aluminium particle. This leads to a porous aluminium/alumina bed, which then burns in the flame (sintering) resulting from the combustion and forms the fireball that was mentioned earlier, resulting in the alumina droplets.

In the above scenario, the efficiency and size of the alumina droplet formation is determined by how much of the aluminum actually comes out of the original “shell” and was able to burn to form the alumina particles of a particular homogeneous size. For instance, if not enough aluminium coalesce in the first place to form the porous bed, the end result of alumina will be of one size – which may not be the desired one to control acoustic oscillations.

As we saw earlier, the alumina can exist in three forms of variable hardness. So the oozing of the liquid aluminium from the shell of alumina could be a random event, as where exactly on the surface the shell will crack depends on the composition of the alumina.

We can appreciate now the importance of the purity of the aluminium particles that are to be mixed with the propellants. As far as possible aluminum particles with an outer shell of alumina are to be avoided (although this is impossible on Earth – see Satya’s comments below) so that one doesn’t have to address the issue of random cracking and insufficient aluminum during combustion.

But, for instance, this was not known to rocket engineers when they started practicing these technologies and by 1967 NASA US (military? – see Satya’s comment below) lost a rocket because of insufficient damping of acoustic oscillations – a result of mismatch of alumina droplet size and rocket motor size, which in turn is a result of impure aluminium mixed in the propellant in the first place. A simple case of giving the contract for preparing the aluminium to two companies, neither of them being aware of the implications of the issue and preparing aluminium particles by solidification of liquid aluminum with two different cooling rates, resulting in particles with a shell of alumina of markedly different compositions (think of amorphous versus crystalline). The 1967 mishap was explained only by 1994 after understating the entire process of how aluminium and alumina particles dampen acoustic oscillations.

Now, where does nanometer sized aluminium come into the picture?

In the original micrometer sized particle, there will be around a million aluminum atoms. In the nanometer sized particles this number is reduced to about a few hundreds. Such an agglomeration results in asymmetric atomic level forces, preventing strong oxidation of the aluminium particle. In other words, in a nanometer sized aluminium particle, the alumina shell, a result of the inevitable oxidation, is absent rendered non-uniform and may be even absent in a few places on its surface (because of the non-uniform atomic force fields, becuase of only the agglomerate having only a few hundred atoms – see Satya’s comment below for further clarifications). This results in a chain of advantages.

For instance, there is direct immediate ignition of aluminum particles in the combustion chamber as there is no non-uniform protective alumina shell around it (so at “weak spots” on the surface of the aluminium particle where the alumina shell is minimal or absent, ignition chance is enhanced – see the comments section below for a discussion). So there is no sintering effect and no aluminium fireball formed as before. The aluminum particles burn and directly result in alumina droplets. Of course, the alumina size is still a variable but the factors that influence the formation of alumina are now minimized because of the use of nanometer sized aluminium particles. Further, the nanometer sized aluminum particles result in superior combustion purposes because of reduced melting point and direct ignition and complete combustion. The resulting alumina droplets even if are not in proper size – only micron sized alumina is useful for acoustic oscillation damping; not a nano-sized alumina droplet – it is still an advantage. One can inject separately micro-sized alumina particles directly into the combustion chamber to control acoustic oscillation.

We shall stop here exploring the case of the nano-aluminium saving rocket explosions. All of this is very much under the research stage. Much of it involved simple hard work and years of patient experiments, unlike a good detective story.

After all, it is just rocket science.

[1] http://www.aerospaceweb.org/question/propulsion/q0246.shtml

[I thank Dr. Satya Chakravarthy, my colleague and collaborator from the Aerospace Department of IIT Madras for sharing this exciting story - one of his research interests - with me.]

Categories: Research Notes · Science Notes · Thermal Sciences
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Predicting Flow Transition in Porous Media

February 11, 2007 · Comments Off

How to predict the Flow Transition in Porous Media? Transition shifts the dependence of pressure-drop from a linear in velocity drag term to a quadratic in velocity drag term, resulting in a shift in pressure-drop its prediction is useful in engineering.

Use of a characteristic Reynolds number as a parameter for the transition criterion indicating the departure from linear Darcy flow, has been proposed in several studies. Early examples include Wyckoff et al. (1934), Ward (1964), Ahmed and Sunada (1969) etc. while more recent ones include Fand et al. (1987) and Firdauss et al. (1997). See Lage (1998) and Lage and Antohe (2000) for an elaborate reference list.

One such Reynolds number, for example, is written as Re_p = UD_e/\nu where U (m/s) is the global seepage speed; D_e , (m) is the average particle diameter of the solid particles constituent of the porous medium; \nu , m^2/s ) is the kinematic viscosity of the flowing fluid. Observe here, that the procedure of exchanging U with the local, pore velocity u is valid only for an isothermal flow through the porous media. Only then, we can view the flow to maintain a “slug flow” with u everywhere identically equal to U, the global seepage speed.

The use of particle diameter as the representative scale in the Re_p is consistent with the use of packed non-consolidated spherical particles as the solid matrix of a porous medium. However, from Scheidegger (1960), we learn that, the value of the Re_p above which transition results, for different porous media, ranges between 0.1 and 75 – an uncertainty factor of 750 for a transition predicting parameter!

This is partly due to the differences in the pore structure of the porous media tested. The other important reason is the characteristic length scale used by Re_p is a microscopic (fluid-continuum level) information, while the transition (from Darcy equation) that it tries to predict is a phenomenon identified at the global level, i.e. above the macroscopic porous-continuum level. (see Representative Elemental Volume concept and Flow Through Porous Media Summary )

This suggests the use of a global representative length scale for defining a Reynolds number. For example, using K^{1/2} (permeability raised to half) as the length scale, we may define a Re for the porous medium flows as

Re_K = UK^{1/2}/\nu \cdots (1)

The viscosity in Eq. (1) is evaluated at a representative temperature, usually the inlet temperature of the flow configuration. Transition happens when this Re is of the order of 10 or higher. This was verified experimentally, among others, by Ward (1964).

Although this Re_K is more restrictive in its numerical values, to predict transition, it still is incomplete as a parameter for establishing a transition criterion because it is devoid of the form coefficient, a representative parameter of the structure of any porous medium and which is very much present in the ~Hazen-Dupuit-Darcy (HDD) model of momentum conservation, as given below.

\frac{\Delta P_0}{L} = \frac{\mu_0}{K_0}U + \rho C_0U^2 = D_{\mu _0} + D_{C_0} \cdots (2)

Clearly, any parameter that predicts the transition from viscous to form dominated flow regime should include parameters that are representative of both of the flow regimes.

Based on scaling arguments for Eq. (2), an effective alternate, as proposed in Lage (1998), is \lambda , the ratio of global form-drag and global viscous-drag forces along a porous channel with uniform cross-section. It is given by

\frac{\Delta P |_0}{L} \sim D_{\mu_{0}} \rightarrow \frac{\Delta P |_0}{L} \sim D_{C_0}

\lambda = \frac{\text{form-drag}}{\text{viscous-drag}} = \frac {D_{C_0}}{D_{\mu_{0}}} = \left(\frac{\rho C_0 K_0}{\mu_0}\right)U \cdots (3)

where K_0 and C_0 are the permeability and form coefficient of the porous medium obtained from isothermal experiments and U is the cross-section averaged Darcy (or seepage) fluid speed. When \lambda > 1 , the flow is said to have departed from Darcy flow, into the quadratic flow regime.

Observe, \lambda as a transition predicting parameter, considers the global HDD model in its entirety, i.e., Eq. (2), unlike any of the transition predicting options discussed earlier. To predict the transition from the dominance of one drag to the other, it is necessary to correctly compare the relative strength of either of the drags that oppose the flow.

Naturally, as observed from Eq. (3), l includes all of the parameters that should affect the transition. In general, for a given porous configuration (i.e., for a chosen set of \rho, \mu_0, K_0 \text{and} C_0 ), \lambda is a function only of U. One can expect with certainty, transition from Darcy flow to happen beyond the flow velocity where the strength of the drags are equal ( \lambda \sim 1 ). This procedure also reduces the scatter and uncertainty inherent in the use of the characteristic Reynolds numbers, discussed earlier.

Ahmed and Sunada (1969) and Geertsma (1974) have previously used an alternate Reynolds number, similar to the RHS of (3). It was used in their studies as an effective non-dimensional parameter, along with a friction-factor fK, to plot pressure-drop versus flow velocity results. Citing Ahmed and Sunada (1969), later authors, like Civian and Tiab (1989), Civian and Evans (1996), have also used this Re in their work. However the interpretation of the Re as a ratio of the drags, as introduced in Eq. (3), is not clearly seen in any of these works.

We close this discussion with one final observation. Whether it is a characteristic Reynolds number, like the one defined in Eq. (1) or l, Eq. (3), the tacit assumption behind the use of any of these parameters for establishing the transition criterion is that the global HDD model, Eq. (2), is fundamentally valid for the flow configuration considered.

Related Notes

Flow Transition in Porous Media | Flow Through Porous Media Summary

References

*Ahmed, N. and Sunada, D. K., 1969 Nonlinear flow in porous media. ASCE Journal of Hydraulics Division, 95, HY 6, 1847-1857.
*Civian, F. and Tiab, D., 1989 Second Law analysis of petroleum reservoirs for optimized performance. SPE 18855, Proceedings of the SPE Production Operations Symposium, Oklahoma.
*Civian, F. and Evans, R. D., 1996 Determination of non-Darcy flow parameters using a differential formulation of the Forchheimer equation. SPE 35621, Proceedings of the SPE Gas Technology Symposium, Calgary, Alberta, Canada.
*Geertsma, J., 1974 Estimating the coefficient of inertial resistance in fluid flow through porous media, Soc. Pet. Eng. Journal, 14, 445-450.
*Fand, R. M., Kim, B. Y. K., Lam, A. C. C. and Phan, R. T., 1987 Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. ASME J. of Fluids Engineering, 109, 268-274.
*Lage, J. L., 1998 The Fundamental Theory of Flow through Permeable Media: from Darcy to Turbulence. Transport Phenomena in Porous Media, D. B. Ingham and I. Pop, eds., Pergamon, New York, 1-30.
*Lage, J. L. and Antohe, B. V., 2000 Darcy’s experiments and the deviation to nonlinear flow regime. ASME J. of Fluids Engineering, 122, 619-625.
*Scheidegger, A. E., 1960 The Physics of Flow through Porous Media (1st edition). University of Toronto Press, Toronto.
*Scheidegger, A. E., 1974 The Physics of Flow through Porous Media (3rd edition). University of Toronto Press, Toronto.
*Ward, J. C., 1964 Turbulent flow in porous media. ASCE Journal of Hydraulics Division, 90, HY 5, 1-12.
*Wyckoff, R. D., Botset, H. G., Muskat, M. and Reed, D. W., 1934 Measurement of permeability of porous media. Bulletin of the American Association of Petroleum Geologists, 18, 161-190.

Categories: Lecture Notes · Porous Medium · Research Notes
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Flow Transition in Porous Media

February 10, 2007 · Comments Off

For low speeds, Flow Through Porous Media is modeled using the Darcy Law, a linear relation between the global pressure-drop ( \Delta P / L, N/m^2 ) and global fluid velocity U (m/s). As the global velocity U increases, departure from this simple flow modeling law results. The departure results eventually in the global pressure-drop across a porous medium to be governed by the form drag, which depends on the fluid density and quadratic global velocity ( U^2 ).

Knowledge of the departure from Darcy flow (linear velocity, viscous drag dominated), in terms of the global velocity U, is very important in porous medium flows. For example, this knowledge prescribes the pressure-drop for a typical engineering porous medium configuration, which eventually determines the required pump power.

bulk-temperature-1.jpg

Figure 1′: Schematic of a porous medium flow configuration

In Fig. 1, a porous medium channel that is uniformly heated with constant heat flux is shown. But for our discussion here, an isothermal (non-heated) configuration is sufficient. For an isothermal flow through the porous medium channel of Fig. 1, with D_{C_0} = \rho C_0 U^2 representing the global form-drag and D_{\mu _0} = \mu _0 U/K_0 , the global viscous-drag (with viscosity evaluated at the inlet fluid temperature, i.e., \mu _0 = \mu (T_0 = T_{in}) ) acting within the porous medium, the global momentum conservation statement is the Hazen Dupuit Darcy (HDD) model that reads

\frac{\Delta P|_0}{L} = \frac{\mu _0}{K_0}U + \rho C_0 U^2 = D_{\mu _0} + D_{C_0} \cdots (1)

Here \Delta P / L|_0 refers to the pressure-drop across the channel for isothermal flows.

It is widely understood that the flow through porous media is characterized by two distinct regimes (see Dullien (1992) and Nield and Bejan (2006) for further details). For speeds much smaller than unity, as the U^2 term is much smaller than the U term, the pressure gradient on the LHS of Eq. (1) is likely to be balanced more by the linear in velocity term (i.e. \Delta P / L |_0 \sim D_{\mu _0} ). Hence Eq. (1) become identical to the [[Darcy Law]]. As the velocity increases, the balance is shifted to the quadratic velocity term, i.e., \Delta P|_0 /L \sim D_{C_0} hence the //departure// from Darcy flow.

\frac{\Delta P|_0}{L} \sim D_{\mu _0} \rightarrow \frac{\Delta P|_0}{L} \sim D_{C_0}

In this context it is instructive to keep in mind that both of these drag terms are always present, immaterial of the magnitude of the velocity. The flow situation is always best governed by Eq. (1). Only the strength of the individual contribution of the drag terms in the RHS changes from viscous drag to form drag, as the flow velocity increases.

Further, since we began with Eq. (1), the departure from Darcy flow meant the flow is governed more by the form drag term and hence, form drag dominant. However, if one were to include the Brinkman term (see Flow Through Porous Media), then the departure from Darcy could also result in other flow situations. In general, the above explanation of transition is valid wherever Eq. (1) is valid for predicting the porous medium flow – i.e., low permeability porous medium flows.

Related Note

Predicting Flow Transition in Porous Media

References

*Dullien, F. A. L., 1992 Porous Media: Fluid Transport and Pore Strucuture (2nd edition). Academic Press, San Deigo.

*Nield, D. A. and Bejan, A., 2006 Convection in Porous Media. 3rd edition, Springer Verlag, New York.

Categories: Lecture Notes · Porous Medium · Research Notes
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Variable Viscosity Effects Explained

February 9, 2007 · 1 Comment

With the notes discussed in earlier (set of one, two and three) posts we can now think of what does variable viscosity or more precisely, temperature-dependent viscosity do in a forced convection situation.

For simplicity, we will ask this question in the context of a liquid flow through a parallel-plate channel as shown in Fig. 1.

bulk-temperature-1.jpg

Figure 1: Schematic of a parallel plate channel sustaining non-isothermal flow

Heating (or cooling) the liquid flow inside the channel makes the local temperature vary everywhere inside the channel. The temperature distribution of this process is got, in principle, by solving the general energy transport equation.

To solve the energy equation (that results in the temperature distribution knowledge inside the channel), a priori knowledge about the local velocity distribution is essential, which is got usually from the general momentum transport equation. While solving the momentum equation (for the convection process), we must in principle take into account the local viscosity variation, as it is a function of the local temperature. In essence, we are to solve a system of coupled partial differential equations.

Notice here, the word coupled represents a more intricate coupling than the already existing one due to the appearance of velocity in the energy equation. Local viscosity variation creates a two-way coupling because of which we cannot solve separately, either of the conservation statements while assuming it a constant (in the former case) allows us to solve the momentum equation separately. Figure 2 explains the situation with a collage.

variableviscosity-2.jpg

Figure 2: a) Drawing Hands drawn in 1948 by Maurice Escher (Courtesy: Cordon Art B. V.-Baarn-the Netherlands) b) effect of temperature dependency of viscosity

To illustrate the relevance of the problem to engineering, we take a look at the specific case of the channel in Fig. 1. Treating the viscosity a constant everywhere inside the channel, the Navier-Stokes equation for maintaining a hydro-dynamically fully developed laminar flow reduces to the algebraic form of the well known, Hagen-Poiseuille equation. Using this equation, for a given average velocity U, we predict the global longitudinal pressure-drop for the channel in Fig. 1 as

\frac{\Delta P}{L} = \left(\frac{12\mu (T)}{(2H \text(or) D)^2}\right)U \cdots (1)

This equation represents the balance between the longitudinal pressure force that has to be imposed to overcome the net friction force across the channel length (exerted by the channel wall on the flowing fluid) to maintain a flow with average velocity U. This pressure-drop is very important because that value determines how much pump power one would require for pumping the fluid across the channel length. This translates to the expenditure in electricity (usually) to drive the pump. Less the pressure-drop, less is the cost incurred.

For intance, for liquids, since the viscosity decreases with increase in temperature, the hotter the channel is, the “easier” the resulting flow; in other words, for a mass flow rate to cross the length of the channel, the pressure-drop (hence the pumping power) required would be lesser than the corresponding case of constant and uniform viscosity (isothermal channel flow).

When the channel in Fig. 1 is isothermal (not heated or cooled), the viscosity in Eq. (1) is evaluated at the inlet (reference) fluid temperature. For non-isothermal flows (channel is heated or cooled), since the local viscosity is spatially varying, one of the following averaging options is usually adopted to find a suitable viscosity.

But how to predict this pressure-drop for non-isothermal channel flows? The first option is to evaluate the viscosity of Eq. (1) at the simple arithmetic average of the bulk temperature Tb(x), defined as

T_b(x) = \frac{1}{2HU}\int ^{2H}_0 (uT)_x dy \cdots (2)

evaluated at the entry (x = 0) and exit (x = L) of the channel of Fig. 1.

The second option is to evaluate the viscosity at the log mean temperature difference given by

T_{LMD} = \frac{T_b(L) - T_b(0)}{ln\left[\frac{T_b(L)}{T_b(0)}\right]} \cdots (3)

and the third is to calculate an average viscosity along the entire channel, using the known temperature dependency \mu (T) in terms of the bulk temperature, i.e., \mu (T_b) , hence

\bar{\mu} = \frac{1}{L}\int ^L_0 \mu (T_b(x))dx \cdots (4)

and write Eq. (1) as

\frac{\Delta P}{L} = \left(\frac{12\bar{\mu}}{(2H \text(or) D)^2}\right) U \cdots (5)

Observe all of these options rely on bulk temperature. The bulk temperature, Eq. (2), in turn requires the prior knowledge of the velocity profile, which is influenced by the local viscosity!

This circularity is ironed-off by using the viscosity found from the averaging options, for both the differential (N-S equation) and algebraic (Eq. (1), in our case) momentum conservation statements. This makes the viscosity uniform and constant everywhere inside the channel and in principle, assumes the problem away.

However, all of these procedures for evaluating the viscosity are valid only when the outcome compares well with experimental results. Even after using the “averaged” viscosity, if the global pressure-drop predicted by Eq. (1) deviates largely from the experimental results, we could deduce that the averaging options have failed to capture entirely, the effects of local viscosity variation in the channel.

Obviously, for this situation, Eq. (1) in its present form, is rendered useless for the practicing engineer and a suitable modification of it is sought.

A similar line of argument based for instance on temperature dependent thermal conductivity, can question the use of the heat transfer solutions based on constant property assumption. Extensive investigations on the hydrodynamic and thermal (heat transfer) implications of variable properties in clear (of porous medium) fluid convection have been conducted for the past several decades.

Categories: Fluid Sciences · Lecture Notes · Thermal Sciences
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