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Forced Convection Heat Transfer Characteristics of Developed Laminar Flow in the Octagonal Channels of Octo-Square Asymmetric Particulate Filters

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Abstract

The forced convection heat transfer characteristics of fully developed laminar flow in an octagonal channel with alternating sides of different lengths have been investigated. Such channels occur in octo-square asymmetric diesel and gasoline particulate filters (PFs). The cross section of these channels resembles a square with the corners “cut off”. As the size of the cutoff is increased, the channel cross section varies from square (zero cutoff) to a regular octagon. The temperature field has been solved, and Nu was determined for the constant wall heat flux (H2) and constant wall temperature (T) boundary conditions (BCs) for the full range of possible channel geometries from a square to a regular octagon. This work compliments earlier work by the author where the temperature field for the H1 BC, as well as the velocity field, was solved. Together, these two papers provide all the parameters required in the momentum and energy balance equations of a 1-dimensional model for an octo-square PF (Nu, friction factor, momentum flux correction factor, etc.). Heat transfer is more effective for a square channel than a regular octagonal one with the H1 and T BCs (where the wall temperature is constant), but the converse is true for the H2 BC. For typical commercially available octo-square PFs, the error induced in a 1-dimensional model by incorrectly assuming that the octagonal channels are square is relatively small for the constant wall temperature BCs (H1 & T) but is potentially significant for the constant wall heat flux (H2) BC.

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Fig. 1

© 2019 T.C. Watling

Fig. 2

© 2019 T.C. Watling

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Data Availability

Calculated results are tabulated in the Supplementary Material (Section S6, Table S5). There is no experimental data.

Code Availability

The code used is given in the Supplementary Material (Section S7).

Notes

  1. A corrigendum to the previous work is given in Section S8 of the Supplementary Material.

  2. To be clear, with the H1 BC, the heat flux at the wall can vary around the wall, just not axially.

  3. For mass transfer, the equivalent to the H2 and T BCs are constant wall mass flux and constant wall concentration, respectively.

  4. While the effect of through-wall flow on Nu has typically been neglected in 1-dimensional PF models, this does not mean that the effect of wall flow on heat transfer has been neglected in all models. In general (before Ref. [29, 30]), there have been two forms of the energy balance equations used in 1-dimensional models, differing in whether gas leaving the inlet channel and entering the PF wall was assumed to be at the wall temperature [15, 18, 31] or at the mixing cup temperature of the gas in the inlet channel [15, 27]. The latter results in convective heat transfer from the gas in the inlet channel to the PF wall, which (of course) increases with the through-wall flow.

  5. This is a recap from previous work [20].

  6. Actually, incompressible flow is not a requirement for V to appear outside the derivative as this equation contains the product ρV (equal to the mass flux along the channel), which is independent of pressure and temperature.

  7. If the temperature solution had terms containing z with r and/or θ then the flux at the wall (\(-\mathbf{\nabla} T\cdot \mathbf{n}\)) would vary with z as \(\partial T/\partial r\) and/or \(\partial T/\partial \theta\) would be functions of z.

  8. Since \({T}^{*}\) has units (m4), it is referred to as “normalised” rather than “dimensionless”.

  9. For the H1 BC, \({T}^{*}\) was defined as \({T}^{*}=\lambda {d}_{H}^{3}\left(T-{T}_{\Gamma }\right)/8fRe h\left({T}_{m}-{T}_{\Gamma }\right)\) [20], i.e. with \(h({T}_{m}-{T}_{\Gamma })\) instead of \({q}_{\Gamma }\) and the addition of \({T}_{\Gamma }\) in the numerator compared to Eq. 9. Different definitions of \({T}^{*}\) are used for the H1 and H2 BCs for future convenience in the respective derivations. A key reason for these differences is that \({T}_{\Gamma }\) is a constant around the channel wall with the H1 BC but not with H2.

  10. This equation also applies equally to the H1 BC [20], but has been written differently compared to the previous work [20] to separate out the c0 term, which, unlike the other constants, is not determined (Sect. 2.3.2).

  11. It is worth noting that numeric evaluation of Eq. 32 using the trapezium rule with a ½° interval (as was done when evaluating Eq. 31) resulted in small oscillations in the value of Nu calculated with γ. The use of an analytic solution (Eq. 35), on the other hand, results in a completely smooth trend (Figs. 7 and 8) and therefore is the preferred method.

  12. This definition applies to the H1 and T BCs, where the wall temperature is constant. For the H2 BC, replace \({T}_{\Gamma }\) with \({T}_{\Gamma m}\).

  13. In this case, integral2 transforms the domain for integration into a rectangle and then applies a vectorised, adaptive quadrature algorithm using the Gauss–Kronrod (3,7) pair.

  14. These values are approximate as they had to be read off a graph as tabulated values were not given.

  15. Or, for the H2 BC, \({T}^{\mathrm{\prime}}\) is larger in the centre of the channel than the mean value of \({T}^{\mathrm{\prime}}\) around the channel wall.

  16. Values of the maximum deviation of the heat flux at the wall from its intended value as a percentage of the intended heat flux for the H2 BC can come out quite large compared to the maximum deviation of the dimensionless temperature at the wall from its intended value for the H1 BC, which was used as a measure of accuracy in previous work [20]. This is at least partly due to the calculation of these values which involves dividing by the intended heat flux at the wall for the H2 BC and dividing by the mixing cup temperature of the fluid in the channel for the H1 BC; the latter calculation is likely to involve dividing by a larger number and hence to give a smaller value.

  17. The error is given by the following: \(\%Error=100\left({Nu}_{p,Sq}-{Nu}_{p,Oct}\right)/{Nu}_{p,Oct}\), where \({Nu}_{p,Sq}\) and \({Nu}_{p,Oct}\) are values of the p-based Nusselt number for the assumed-square and correct (octagonal) channel geometry.

Abbreviations

BC:

Boundary condition

H1:

Constant wall temperature and axially constant heat flux at the wall

H2:

Constant wall heat flux

PF:

Particulate filter

S&L:

Shah and London (book)

T:

Constant wall temperature

\({\varvec{A}}\) :

Column matrix whose elements are \({a}_{j}\); defined by Eq. 20

\({A}_{C}\) :

Cross-sectional area of the channel, m2

\({a}_{j}\) :

Constants in Eq. 4 and Table 2, m2−jN

\(b\) :

A constant with any real value, -

\({\varvec{C}}\) :

Column matrix whose elements are \({c}_{j}\); defined by Eq. 21

\({c}_{j}\) :

Constant in Eq. 11, m4−jN

\({C}_{p}\) :

Specific heat capacity of gas at constant pressure, J kg−1 K−1

\(d\) :

Width of channel, i.e. distance between opposing straight sides of the channel, m

\(D{a}_{II}\) :

Second Damköhler number, i.e. the ratio of the rate of reaction to the rate of diffusive mass transport, -

\({d}_{H}\) :

Hydraulic diameter of channel, \(4{A}_{C}/p\), m

\({d}_{Reg}\) :

Length of the side of a regular octagon of width \(d\), m

\(ds\) :

An element of length on the channel wall, m

\({d}_{SL}\) :

Length of a “slanted side” of the octagonal channel, m

\({d}_{ST}\) :

Length of a “straight side” of the octagonal channel, m

\({\varvec{E}}\) :

Matrix defined by Eq. 22

\({\mathbf{e}}_{\mathbf{r}}\), \({\mathbf{e}}_{{\mathbf{\uptheta}}}\) :

Plane polar unit vectors in the radial and angular directions, -

\(f\) :

(Fanning) friction factor, -

\(g\) :

Dimensionless local axial gas velocity, \(V/{V}_{m}\), -

\(\mathrm{g}\left(z\right)\) :

The z-dependant part of the temperature solution, K

\(h\) :

Heat transfer coefficient for heat transfer between gas in the channel and the channel wall, W m−2 K−1

\(i\) :

Any integer; positive, negative or zero, -

\(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) :

Cartesian unit vectors in the x-, y- and z-direction, -

\(j\), \(k\) :

Indices in summations; non-negative integers, -

\(m\) :

Number of points on the channel wall at which the boundary condition is evaluated, -

\(N\) :

Order of rotational symmetry (= 4), -

\(\mathbf{n}\) :

Outward unit vector normal to the channel wall, -

\(n\) :

Upper limit on \(j\) and \(k\) in the summations, -

\({n}_{r}\), \({n}_{\theta }\) :

Radial and angular components of \(\mathbf{n}\), defined by Eq. 18, -

\({Nu}_{dH}\) :

Hydraulic-diameter-based Nusselt number, \(h {d}_{H}/\lambda\), -

\({Nu}_{p}\) :

Channel-perimeter-based Nusselt number, \(hp/\lambda\), -

\(P\) :

Pressure of gas in the channel, Pa

\(p\) :

Channel perimeter, m

\({q}_{\Gamma }\) :

Outward heat flux normal to the channel wall at the channel wall, W m−2

\(r\) :

Radial coordinate in polar coordinates relative to the channel centre, m

\(Re\) :

Reynolds number, \(\rho {V}_{m} {d}_{H}/\mu\), -

\({r}_{\Gamma }\) :

Radial coordinate of the channel wall, m

\(T\) :

Local temperature of the gas in the channel, K

\({T}^{*}\) :

Normalised local temperature, defined by Eq. 9, m4

\({T}^{\mathrm{\prime}}\)  :

Dimensionless temperature, defined by Eq. 36, -

\({T}_{int}^{\mathrm{\prime}}\) :

Solution for the dimensionless temperature field initially obtained by solution of Eq. 41, -

\({T}_{m}\) :

Mixing cup temperature of the gas in the channel, K

\({T}_{m}^{*}\) :

Mixing cup version of temperature \({T}^{*}\), defined by Eq. 30, m4

\({T}_{\Gamma }\) :

Temperature of the channel wall, K

\({T}_{\Gamma m}\) :

Mean channel wall temperature, i.e. the channel wall temperature averaged around the channel perimeter at a given axial location, K

\({T}_{\Gamma m}^{*}\) :

Normalised mean channel wall temperature, i.e. normalised version of \({T}_{\Gamma m}\), defined by Eq. 32, m4

\({\varvec{U}}\) :

Matrix defined by Eq. 23

\(V\) :

Local axial gas velocity, m s−1

\({V}_{m}\) :

Mean axial velocity of gas in the channel, \(\langle V\rangle\), m s−1

\({\varvec{W}}\) :

Matrix defined by Eq. 24

\(x\), \(y\) :

Cartesian coordinates across the channel, m

\(z\) :

Axial distance or coordinate, m

\(\gamma\) :

Channel shape factor, defined by Eq. 2, -

\(\theta\) :

Angular coordinate in polar coordinates, rad

\(\Lambda\) :

Separation constant (Eq. 38), m−1

\(\lambda\) :

Thermal conductivity of the gas in the channel, W m−1 K−1

\(\mu\) :

Gas viscosity, Pa s

\(\rho\) :

Gas density, kg m−3

\(\Psi \left(r,\theta \right)\) :

The \(r\)- and \(\theta\)-dependant part of the temperature solution, K

\(\langle \ \rangle\) :

Average of the bracketed quantity over a cross section

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Appendix. Evaluation of the integral giving \({{\varvec{T}}}_{{\varvec{\Gamma}}{\varvec{m}}}^{\boldsymbol{*}}\)

Appendix. Evaluation of the integral giving \({{\varvec{T}}}_{{\varvec{\Gamma}}{\varvec{m}}}^{\boldsymbol{*}}\)

This appendix describes the evaluation of the integral in Eq. 32 to give \({T}_{\Gamma m}^{*}\) (Eq. 35). Substitution of Eqs. 3, 11 and 34 into Eq. 32 and evaluating the integral gives:

$$\begin{array}{c}{T}_{\Gamma m}^{*}=\frac{2N}{p}\left(\left[\frac{{d}^{5}\left(6\mathrm{cos}2\theta +\mathrm{cos}4\theta +8\right)\mathrm{sin}\theta }{30720{\mathrm{ cos}}^{5}\theta }\right.\right.\\ -\sum\limits_{j=0}^{n}\frac{{a}_{j}{d}^{jN+3}\left(\left[jN+3\right]\mathrm{sin}\left(\left[jN+1\right]\theta \right)+\mathrm{sin}\left(\left[jN+3\right]\theta \right)\right)}{{2}^{jN+5}\left(jN+1\right)\left(jN+2\right)\left(jN+3\right){\mathrm{cos}}^{\mathrm{jN}+3}\theta }-\sum\limits_{j=1}^{n}{\left.\frac{{c}_{j}{d}^{jN+1}\mathrm{sin}\left(\left[jN+1\right]\theta \right)}{{2}^{jN+1}\left(jN+1\right){\mathrm{cos}}^{\mathrm{jN}+1}\theta }\right]}_{0}^{A\widehat{O}C}\\ \begin{array}{c}-\left[\frac{{\left({d}^{2}+{d}_{ST}^{2}-{d}_{SL}^{2}\right)}^{5/2}\left(6\mathrm{cos}\left(2\left[\pi /N-\theta \right]\right)+\mathrm{cos}\left(4\left[\pi /N-\theta \right]\right)\left.+8\right)\mathrm{sin}\left[\pi /N-\theta \right]\right.}{30720 {\mathrm{cos}}^{5} \left[\pi /N-\theta \right]}\right.\\ -\sum\limits_{j=0}^{n}\frac{{a}_{j}{\left({d}^{2}+{d}_{ST}^{2}-{d}_{SL}^{2}\right)}^{\left[jN+3\right]/2}{\left(-1\right)}^{j}\left(\left[jN+3\right]\mathrm{sin}\left(\left[jN+1\right]\left[\pi /N-\theta \right]\right)+\mathrm{sin}\left(\left[jN+3\right]\left[\pi /N-\theta \right]\right)\right)}{{2}^{jN+5}\left(jN+1\right)\left(jN+2\right)\left(jN+3\right){\mathrm{cos}}^{\mathrm{jN}+3}\left[\pi /N-\theta \right]}\\ -\left.\sum\limits_{j=1}^{n}{\left.{\frac{{c}_{j}{\left({d}^{2}+{d}_{ST}^{2}-{d}_{SL}^{2}\right)}^{\left[jN+1\right]/2}{\left(-1\right)}^{j}\mathrm{sin}\left(\left[jN+1\right]\left[\pi /N-\theta \right]\right)}{{2}^{jN+1}\left(jN+1\right){\mathrm{cos}}^{\mathrm{jN}+1}\left[\pi /N-\theta \right]}}\right]}_{A\widehat{O}C}^{\pi /N}\right)-{c}_{0}\end{array}\end{array}$$
(43)

Evaluating the limits in this gives Eq. 35. Note that the \(\theta =0\) and \(\theta =\pi /N\) limits evaluate to zero. Further information on the integrals used is given in the Supplementary Material (Section S2).

When evaluating Eq. 32 over \(A\widehat{O}C\le \theta \le \pi /N\), it is useful to note the following identities:

$${\int }_{A\widehat{O}C}^{\pi /N}\frac{\mathrm{cos}jN\theta }{{\mathrm{cos}}^{\mathrm{jN}+2}\left[\pi /N-\theta \right]}d\theta =-{\int }_{\pi /N-A\widehat{O}C}^{0}\frac{{\left(-1\right)}^{j}\mathrm{cos}jN\phi }{{\mathrm{cos}}^{\mathrm{jN}+2}\phi }d\phi =-\frac{{\left(-1\right)}^{j}}{jN+1}{\left[\frac{\mathrm{sin}\left(\left[jN+1\right]\left[\pi /N-\theta \right]\right)}{{\mathrm{cos}}^{\mathrm{jN}+1}\left[\pi /N-\theta \right]}\right]}_{A\widehat{O}C}^{\pi /N}$$
(44)
$$\begin{array}{c}{\int }_{A\widehat{O}C}^{\pi /N}\frac{\mathrm{cos}jN\theta }{{\mathrm{cos}}^{\mathrm{jN}+4}\left[\pi /N-\theta \right]}d\theta =\\ -{\int }_{\pi /N-A\widehat{O}C}^{0}\frac{{\left(-1\right)}^{j}\mathrm{cos}jN\phi }{{\mathrm{cos}}^{\mathrm{jN}+4}\phi }d\phi =\\ -\frac{{\left(-1\right)}^{j}}{\left(jN+2\right)\left(jN+3\right)}{\left[\frac{\left[jN+3\right]\mathrm{sin}\left(\left[jN+1\right]\left[\pi /N-\theta \right]\right)+\mathrm{sin}\left(\left[jN+3\right]\left[\pi /N-\theta \right]\right)}{{\mathrm{cos}}^{\mathrm{jN}+3}\left[\pi /N-\theta \right]}\right]}_{A\widehat{O}C}^{\pi /N}\end{array}$$
(45)

In both cases, the second term is obtained by substituting \(\phi =\pi /N-\theta\) into the first term and noting that \(\mathrm{cos} \left(jN\left[\pi /N-\phi \right] \right)={\left(-1\right)}^{j}\mathrm{cos}jN\phi\). The third term is obtained from the second by evaluating the integral and substituting in \(\phi =\pi /N-\theta\).

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Watling, T.C. Forced Convection Heat Transfer Characteristics of Developed Laminar Flow in the Octagonal Channels of Octo-Square Asymmetric Particulate Filters. Emiss. Control Sci. Technol. (2024). https://doi.org/10.1007/s40825-023-00233-0

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