Pressure difference across an aerated seed bulk for some common duct and store cross-sections

J. agric. Engng Res. (1983) 28, 437-450

Pressure Difference across an Aerated Seed Bulk for some Common Duct and Store Cross-sections A. J. HUNTER*

Bulk seed in store is often aerated for preservation purposes. The design of the associated fan and duct system is dependent upon estimation of the pressure difference across the bulk for a given airflow and duct and store geometry. Conformal mapping techniques are used to predict the component of pressure difference arising from the linear term in Ergun’s equation, and the components arising from the square-law term are estimated by assuming radial flow near the duct and parallel flow remote from the duct. The formulae derived in this paper are applicable to most common duct store situations. The required resistance coefficients are given for airflow through 27 different seeds.

1.

Introduction

Bulk seed in a store is commonly aerated to reduce and even out its temperature and so reduce the rate of deterioration through insect and mould attack. The fan pressure required to force a certain flow rate of air through the bulk is a function of the seed type, the seed depth and the geometry of the aeration ducting. The modelling of the variation of pressure gradient with air velocity is frequently based on the logarithmic expression of Hukill and Ives.’ This expression fits the data of Shedd* no better than the quadratic expression of Ergur? as is apparent from Table 1. Ergun’s3 equation is far simpler to apply to non-uniform airflow distributions as will be demonstrated. TABLE 1 Comparison between Hukill and Ives’’ expression for pressure gradient, and Ergun’s’ equation with the data of Shedd’ for wheat

Shedd,’ Palm

18.5 35.9 74.8

0.0056 0.0112 0.0228 0.0342 0.0508 0.0666 0.1016 0.1524 0.2033

116.8 189.5 259.0 443.6 735.3 1051.5

Hukill and Ives,’ Pa/m Percentage deviation 17.2 34.9 74.7 117.6 185.4 256.1 433.8 738.3 1094.7

-7.1 -3.0 -0.1 0.6 -2.2 -1.1 -2.2 0.4 4.1

Ergun, Palm Percenfage deviation 18.0 36.4 76.9 119.9 186.9 256.1 429.3 727.2 1080.7

-2.7 1.1 2.8 26 -1.4 -1.1 -3.2 -1.1 2.8

The pressure difference can be predicted from potential flow models,4 by finite difference and finite element method? or from small scale experiments. 6-8 The purpose of this article is to present some analytical solutions which are easy to use and yet cover most common duct and store cross-sections. Spencer9 applied conformal mapping to non-uniform airflow distributions *Agricultural Engineering Group, Division of Chemical and Wood Technology, Organization. Highett. Victoria, Australia Received 10 January 1983; accepted in rewed form 14 April 1983

Commonwealth

Scientific

and Industrial

Research

437 0021-8634/83/050437

+ 14 $03.00/O

@ 1983 The British Society for Research in Agricultural

Engineering

438

PRESSURE

DIFFERENCE

ACROSS

SEED

BULK

exactly as is done here in section 3.1, but for the symmetrical case only. However, he used the Hukill and Ives,’ logarithmic expression for resistance to airflow which made it necessary for him to resort to a graphical presentation of his results, a separate graph set being required for each seed type. The present work gives formulae for several airflow distribution patterns, in terms of seed resistance parameters. The formulae are therefore applicable to all seed types, and indeed to packed beds in general. A further advantage of solutions in this form is that a single mathematical expression covers all examples of a given geometrical type.

NOTATION length parameter, section 3.6, m displacement of duct from centreline of seed cross-section, sections 3.1, 3.2 and 3.3, : Fig. 3 (right), m b real value of w at which sources are placed, section 3.4, m length parameter, section 3.6, m b length parameter, section 3.2, m duct diameter, section 3.2, m f; d half-width of perforated floor duct, section 3.3, m imaginary unit, 4-l airflow resistance parameter, Eqn (3), m* Pa-l s-~ f, 1 length of store and duct, m potential function, air pressure, real part of F, Pa P total air pressure difference across seed bulk, Pa AP linear term pressure difference, Pa API Ap,,Ap, square-law term pressure difference caused by duct constriction and parallel flow, respectively, Pa stream function, imaginary part of F 4 duct radius, sections 3.1 and 3.5, m r r polar co-ordinate, section 3.2, Fig. 3 (right), m 1 complex variable, section 3.4, Fig. 4 face air velocity, m/s V W complex variable, section 3.4, Fig. 4 length co-ordinates, Fig. 3, m X,Y z complex variable, = x+iJl constant, Eqn (13) c F function of the complex variable z; F = p+iq H height of the seed bulk above the floor at the section through the duct, m perforated perimeter of duct section, m PP total airflow, m3/s Q R first Ergun coefficient, Pa s m-2 s second Ergun coefficient, Pa s2 m-3 W width of seed store cross-section, m constants, section 2.1, Pa s2 m-3, s/m a# 6 scaling parameter, section 3.4 seed bulk porosity parameter, Eqn (29) t e polar co-ordinate, section 3.2, Fig. 3 (right), rad viscosity of air, Pa s t* density of air, kg/m3 parameter, Eqn (20) ?

A.

J. HUNTER

439

It is assumed that the flow pattern is determined by the linear term in Ergun’? equation (potential flow), then the effect of non-linear terms is included by superposition. Each solution consists of a linear term and two square-law terms which account for the variation of pressure gradient with air velocity. Sutherland and Ghaly’O have previously presented formulae equivalent to Eqns (49) and (50) which are included for completeness of presentation. 2. 2.1.

Theory

Variation of pressure gradient with velocity

When air flows through a cylindrical parcel of bulk seed in a direction parallel to the generator of the cylinder, the pressure gradient along the flow is proportional to the velocity at very low velocities (laminar flow), and proportional to the square of the velocity at high velocities (turbulent flow). Over the full range of velocities the pressure gradient may be estimated from Ergun’? equation which may be written . ..(l) p,i = Rv~+Svi~, where vi is the face velocity in the ith direction, and R and S are constants depending on the seed type. The subscript ,i indicates the operation a/ax,. A least squares fit of Eqn (1) to Shedd’s* data yielded the values of R and S in Table 2 where sufficient data were available. For some seeds which are indicated by an asterisk, Shedd’s* data were over an insufficient range for reasonable values of both R and S to be obtained. Therefore TABLE 2 Values of R and S in Eqn (1) for several seed types

Seed, moisture Alfalfa, 7% Barley, 12% Clover, alsike, dry Clover, crimson 8% Clover, red, dry *Corn, clean ear, lot 2, 16% *Corn. ear. lot 1. as harvested. 20% Corn, shelied, 12.4% ‘_ Fescue, I 1y0 Flax, 11% Grass seed, brome, 10.5% Grass seed, rescue, 13y0 Kobe Lespedeza, 15.5% Lupin seed, blue, 7.5% Oats, 13% *Pea beans, 15% *Peanuts in shell, 4.4% Popcorn, shelled, yellow pearl type, 12% Popcorn, white rice type, 14% Rice, rough, 13% Sericea Lesnedeza. 13% Sorghum, grain, 13% _ Soybeans, 10% Wheat, 11% Linseed, glenelg, 7.9% Rapeseed, tower, 5.7% Safflower, gila, 5.9% Sunflower, commercial crushing, 79%

R, Pa

s

me2

16 318 1676 27 263 10 455 17 626 6.19 128 719 4722 10 421 1535 709 3167 512 1816 435 29.0 1046 1766 1952 16 318 2664 646 3131 14 907 7097 1207 1593

S, Pa s2 mm3

__

21 841 8348 26 142 21 050 27 231 396 1802 6855 9817 37 056 5855 3152 7211 4313 9619 3323 856 6727 9377 10 419 21 841 6908 3760 10 756 32 594 13 899 6701 8744

*Insufficient data were available to enable accurate values of-R to be established in these cases (see section 2.1)

directly

440

PRESSURE

DIFFERENCE

ACROSS

SEED

BULK

the following procedure was adopted. Porosity values for those seeds were estimated from the literature”*‘* and 0.4 was found to be a suitable value in each case. Ergun’? equation may be written with the substitution R = kS2, . ..(2) where k = 49 @ (3) 2’ P

The first and second Ergun coefficients are therefore related by the fluid properties and seed bulk porosity. Using standard atmospheric properties and a porosity of 0.4, Eqn (3) gives a k value of 39.4x 10-6. Eliminating R between Eqns (1) and (2) and substituting this value for k, S can be found by fitting Shedd’s* data at a single point. Eqn (2) then gives R. Values calculated in this way are given in Table 2. Values for the last four seeds were calculated from Sutherland and Ghaly’O using their values for air properties of 18.1 x 1O-6 Pa s for viscosity and 1.19 kg/m3 for density and their table of values of u = R/p and b = S/p. The Hukill and Ives’ expression for p,c is avi2/ln( 1+pvi) and values of R and S can be estimated directly from a and 8. Equating expressions for the low velocity asymptote gives R = a//3. However, since data from which a and /I are derived are usually from the transition region, the author suggests equating the two expressions at say Rv = Sv* and Rv = 2Sv2. This gives R = I.12 aI/3and S = O-346 a. Care should be taken with units. Values of R and S for some seeds were calculated in this way from Hukill and Ives’ and checked against values in Table 2. Good agreement was observed. Eliminating k between Eqns (2) and (3) $

-

P2R=-, 0.191

49,us2 which may be used to estimate porosity. were evaluated.

p2

ah

. .(4)

p and p are the air properties at which R and S or a and /J

2.2. Division of pressure drop into linear and square-law components The square-law term in Eqn (1) influences the pressure gradient but does not significantly affect the airflow field. The effect of nonlinearity has been examined by Brooker13 for the case of flow from a centrally located, open sided rectangular duct, and the airflow paths are indicated by lines drawn perpendicular to the isobars. These paths show very little difference between the true non-linear pressure gradient case and the potential flow solution. There are three cases where the square-law term will not affect the airflow field at all. These are radial flow in two or three dimensions, and parallel flow. It follows that the flow pattern is determined by geometry alone for radial flow and parallel flow. In other cases, since the linear pressure gradient term is dominant, this term can be used to determine the flow field and the linear term pressure difference Ap,. The effect of non-linear terms in the pressure gradient may be obtained by superposition of effects for the regions where the flow field may be approximated by radial flow (Ap,), and parallel flow (Ap,). We have then JSv2ds = Ap,+Ap,,

. ..(5)

where ds is an increment of distance along a streamline. Expressions for Ap, and Ap, are derived in section 4. The total pressure difference will then be estimated as the sum AP = ~,+AP,+~J,.

. ..(6)

2.3. Governing equation for flow field Neglecting the square-law term in Eqn (l), p,i = Rv~.

. . .(7)

A.

J.

HUNTER

Neglecting

441 air density changes,

the conservation

equation

is . ..(8)

ri.i = 0, the summation convention being understood. Differentiating Eqn (7) and using Eqn (8) to eliminate P,ii so

=

I’Zwe find . ..(9)

0

that the pressure field satisfies Laplace’s equation.

2.4. Use of the complex potential function F If F = p+iq is an analytic function of the complex variable z = x+iy, where x and y are space co-ordinates, then bothp and q satisfy Laplace’s equation and all that remains is to find a suitable F(z) to solve a specific problem. p is known as the potential function and q the stream function. Since in the F plane, lines of constant p and constant q are perpendicular and an analytic function defines a conformal transformation, then in the z plane, lines of constant p and q are again perpendicular. It follows that lines of constant q are lines of maximum p.i and coincide with airflow lines in the z plane.

3. Flow field solutions and linear term pressure differences for different duct and store cross-sections . The length of the duct is denoted 1 and the width of the seed store cross-section served by the duct is denoted W. The height of the seed bulk above the floor measured at the section through the duct is denoted Hand the volume flow of air from the duct is Q. The displacement of the duct from the centreline of the store is denoted b in sections 3.1, 3.2 and 3.3. The flow fields presented in this section are for flow in two dimensions only and are derived by Most seed bulk aeration systems consist of straight sections of conformal mapping techniques. duct arranged on the floor of the store. If the layout consists of duct sections at right angles, or straight sections in a circular silo, the plan area served by each duct can be estimated and equated to the product IW. An approximation to the pressure difference can then be obtained. The flow field in the first four examples presented is derived for a seed bulk of infinite height. The finite height solution is obtained by approximating the top surface of the seed bulk by a surface of constant pressure within the flow field. The flow fields for the systems analysed are shown in Figs I and 2. For clarity, the duct sizes shown are larger in proportion to the store than they would be in practice. The flow streamlines shown divide the flow into 10 equal parts so that relative air velocities can be estimated. The co-ordinate systems used in the following sections are shown in Fig. 3. 3.1. Half-round duct [Fig. I (top, left)] By placing a single point source at x = b, y = 0 and using the sine transformation the sides of the store we find the complex potential function sing

F(z) = -Fin

.

-sing

. 410)

I Pressure

is given by the real part of Fand p(y) = - 2

to fold up

for x = b we have

In [sh2-$-2

sin2 g(ch

If the radius of the duct is r, the linear term pressure difference AP, = p(r) -P(H).

s-l)]

.

., .(I])

is given by . ..(12)

Fig. I. Flow fields approximating aeration of a seed store with a flat floor and vertical sides using: (top, leji) a halfround duct; (top, right) a round duct; (bottom, left) a perforatedfloor duct and (bottom, right) a symmetrical pair of louvre ducts

Fig. 2. Flow fields approximating

crossflow aeration of a circular cylindrical silo using: (left) half-round ducts and (right) round ducts

Fig. 3. (Left) Plan view of circular cylindrical silo showing co-ordinate system and (right) cross-section of shed store showing co-ordinate system

A.

J. HUNTER

443

Pressure contours which may be used to approximate a top surface of the seed bulk are found We find by equating the pressure function to a constant. ch$=sinzsinW+

rb W

7ib f , w I

C+cos2~cos2w

. .(13)

which was used to plot the top surfaces of the seed bulks in Fig. 1. If H/W is assumed large, r/W small and b zero, Eqn (12) reduces to

.

.., (14)

This equation is accurate within 10% if W > 6P, and H > 0.3 W, and within I y. if W > 6P, and H > 0.6 W when compared with Eqn (12) for the half-round duct. Pp is the perforated area per unit length of duct, and for the half-round duct is equal to w. Eqn (14) gives an estimate of dp, and may be used for any horizontal duct and store system. to a round duct [Fig. 1 (top, right)] 3.2. Approximation Placing a point source at x = b, y = c and another at x = b, J’= -c and again using the sine transformation, the resulting complex potential function is F(Z)= Pressure

-$(ln

[sin;-sins((b+ic)]+ln[sin$-sini(b-ir)]).

. ..(15)

is given by the real part of F and for x = b,

..,416) The contour of potential around two sources distant sources, is given in polar co-ordinates by

2c of like sign which crosses between

the

. ..(17)

r2 = 2c2 cos 26.

The height of the upper surface of the contour is therefore cJ~ If this height is taken to be the diameter of the duct d, then a slightly higher pressure will be estimated than the actual because the true duct encloses the contour. The effect of this approximation was estimated by replacing the contour by a circle of the same area (c2), concentric with the duct cross-section. The pressure drop across the annular portion between circle and duct was then estimated using Eqn (49) of section 4.3. The error in dp, was found to be 0.036 RQ/l; which comes to 28.4 Pa in the first numerical example of section 6. This is 59/o of the calculated value of dp, and is a measure of the amount by which the estimate of dp, may exceed the true value. The linear term pressure difference is found from Eqn (16) as . ..(18)

AP, = p(d) -P(H).

Eqn (13) may be used to approximate the constant pressure surface since the flow field remote from the duct will approximate that of the previous example for the same off-centre distance b. 3.3. Perforatedfloor duct [Fig. I (bottom, left)] The cash transformation for potential flow through a slit is well known with the sine transformation gives the complex potential function F(z) = &f where 2d denotes

arcosh

sin (~z/ W) - sin (nb/ W) cos (ad/W) cos (zb/ W) sin (nd/ W) i

the width of the slit.

and combining

1 ’

this

. ..(19)

PRESSURE

444

DIFFERENCE

ACROSS

SEED

BULK

Pressure is given by the real part of F and for x = b we have p(v) = $7 arsh

[++ d+2+4 sin2 (nd/ W) cos* (nb/ W) sh2 (ny/ IV)]*

. ..(20)

>

43 cos (nb/ W) sin (nd/ W)

(

where C$= cos2 (rrb/ W) [sh2 (TV/ W) -sin2 (?rd/ W)]+sin2 (ab/ W) [ch (ry/ W) -cos (nd/ W)]“. Since p(O) = 0, the linear term pressure difference is given by dp, = p(H). The velocity distribution at the surface of the duct is found from v = -(l/R) given by v(x) =

(Q//W

(dp/dy) and is

~0s (4W

. ..(21)

[sin (n/W) (b+d) -sin (TX/ W)]*[sin (nx/ IV) -sin (n/ W)(b -d)]*

Again, Eqn (13) may be used to approximate the constant pressure surface as in the previous example. 3.4. Approximation The transformation

to a louvre duct system [Fig. 1 (bottom,

right)]

w+b

t = w+ln a+i7

. . .(22)

was obtained from Kober” and is shown in Fig. 4. The sine transformation is used to fold up the sides of the store, and causes the slits in the t plane to approximate louvres. Thus t =$

..(23)

sin;,

with 6 to be determined.

u

c” A”

B”

G”

E” A’ C’

B’

G’

F’

E’

F” E’

C’ D’

D”

, A

B

C

f-plane

z-plane

Fig. 4. Transformations

I D

, E

, F

I G

w-plane

used in approximating the louvre duct system

The air supply is simulated by placing point sources at w = &b. function cannot be expressed explicitly. We have F(z) = - 2

rr

In [w”(z) -b2] ,

The complex potential

. ..(24)

with b to be determined. Pressure is given by the real part of F and for x = 0,

P(Y)= -

$7In

Sh*(‘$W)+b2 .

I

.(25)

For the surface of the seed bulk in the duct positioned as shown in Fig. I (bottom, right), the supply pressure is given by p = - 8

x

In 2b.

. . .(26)

445

A. J. HUNTER

The linear term pressure

difference

for a seed bulk of centreline

height H is therefore ..(27)

It remains to find b and 6 for a given louvre geometry. NOW w = 2b+b2 maps into the edge of the louvre, so equating the imaginary parts in Eqns (22) and (23), gives the function describing the louvre as cos(nx/W) sh(ry/W) = rrS. . ..(28) Substitution of the coordinates Eqns (22) and (23) gives 5=

x0, y, of the louvre

edge determines

6. Equating

= ~2b+b2+kf;;;~;:;;1

ntt;;;;O;oW()

real parts

,

in

.(29)

from which b can be calculated. The right-hand

side of Eqn (29) has asymptotes

21/26and

b and it was found that the expression

5 = [(2 J~)‘~~+b’~~]~~r~~ is a very good approximation

(maximum

deviation

b = [J10-556+[1.8 3.5. Flow across a circular A source at z = W/2 and a sink circular flow paths. Flow from one is therefore simulated by the complex

.(30)


We have, therefore,

-3a250]‘“‘9.

. ..(31)

cylindrical silo between half-round ducts [Fig. 2 (left)] at z = -W/2 gives rise to circular pressure contours half-round duct to another across a circular cylindrical potential function

and silo

432) Pressure

is given by the real part of F and so p(x

For flow between

two half-round

,

y)

_

QR

2d

ln

(x+ WV2+y2

. .(33)

(X - w/2)2+y” I

ducts of radius r we find

3.6. Flow across a circular cylindrical silo between two round ducts [Fig. 2 (right)] Two sources at z = -b&a and two sinks at z = b&a give rise to a flow field which includes a circular flow line at x2+y2 = b2-a2. .(35) If this circle is taken to represent F(z)

the silo wall then the complex _

potential

function

QR ln @S-b -4 (z+b+a) 2771 (z-b-a)(z-b+a)

may be used to simulate flow across a circular cylindrical diameter of the silo is given by W = 2 Jb2-a2.

silo between

.(36) two round

ducts.

The

.(37) If the ducts are assumedto be in contact with the silo wall then the duct pressures are those corresponding to x = 5 tib2 -a2, y = 0. Substituting in Eqn (36) and taking the real part it is found that

Ap, =

2QR arth bJb2--ai .

rrl

.(38)

446

PRESSURE

DIFFERENCE

ACROSS

SEED

BULK

The values of x other than &1/b2--a2 at which the pressure contour which forms the duct surface, cuts the x axis, relate the size of the ducts to a and 6. We find x=f

b2&a J2b2 -a2 Jb2 I .

. (39)

x=f

b2-a J2b2--a2 JF_QZ I *

. ..(40)

We are interested in

since the other roots are outside the silo. As in section 3.2, above, we take our duct to enclose the relevant contour so that the approximation leads to a slightly higher pressure difference. We have then . . X41)

Solving for a and b between Eqns (37) and (41), . ..(42) and b = Ja2+( W/2)2 .

. ..(43)

Substituting from Eqns (42) and (43) into Eqn (38) we find . ..(44)

4.

Square term pressure difference and radial flow

The square term pressure difference is calculated in two parts as discussed in section 2.2. One part is associated with the constriction of airflow in the vicinity of the duct, and the other results from the parallel flow through the remainder of the bulk. 4.1.

Constriction square term pressure diference Ap,

The most suitable airhow pattern on which an estimate might be based is that of radial flow from a half-round duct on a flat floor. From continuity . ..(45) v = Q/r&. Using Eqn (1) and integrating from the duct radius to infinity AP,

=

SQ” .rr12v ’

. .(46)

where P, is the perforated perimeter. 4.2. Parallel flow square term pressure diference Ap, The square term resistance to airflow may be of significance remote from the duct particularly for high aeration rates or for tall silos. Using Eqn (1) the parallel flow square term pressure difference is given by AP, =

SQ2H 12

.

..

(47)

A.

J.

HUNTER

447

4.3. Radial flow between two concentric circular cylindrical shells Many seed dryers make use of radial flow between two concentric circular cylinders.15 This design combines compactness with a minimum of depth in the direction of airflow. If the cylinders are of radii rl and r2 the velocity at radius r is . ..(48)

v = Q/Znrl. Using Eqn (1) and integrating

from r1 to r2, dp, = 2

. ..(49)

In z

and

4.4. Radialflow between two concentric hemispherical shells Radial flow between two concentric hemispherical shells is relevant to a mound flat floor aerated from a central source. The velocity at radius r is v = Q/2m2,

of seed on a . ..(51)

from which by Eqn (1) .(52) and

5.

Comparison of results with those of others

5.1. Half-round duct Spencer9 gave sufficient airflow and geometrical information for some comparisons to be made between his predictions of pressure drop, the experiments of Barrowman and Boyce,16 and the present work. Referring to Spencer’s9 section 3.1, the geometry concerned is one of a duct with vertical sides and a semi-circular top, centrally placed in a store having vertical sides. The values of a and p given by Spencer9 are 19 144 and 6.49, respectively. Using the conversions in section 2.1 we find R = 3302 and S = 6624. The relevant equations of the present work are Eqns (1 I), (12), (46) and (47) which combined give the total pressure difference figure in Table 3. TABLE

3

Comparison of pressure drop predictions (in Pa) with experiments of Barrowman and BoyceI and theory of Spencer’ Case (i) Barrowman Spencer* Hunter

and Boyceq6

1196 1091 1141

I

Case(ii) 934 852 902

Referring to Spencer’s9 sections 2.2 and 3.2, the CCand /3 values used are 27 213 and 13.38, respectively. From section 2.1 of the present work, the corresponding values of R and S are 2278 and 9416. The geometry concerned is first one of a centrally placed half-round duct in a

PRESSURE

448

DIFFERENCE

ACROSS

SEED

BULK

store having vertical sides. Secondly, by way of comparison, two ducts of the same size are used and placed in the same store so as to divide the store into two symmetrical halves. The relevant equations of the present work are again Eqns (1 l), (12), (46) and (47). Spencer’s9 pressure drop values of 623 and 369 Pa are to be compared with values calculated by the author of 690 and 406 Pa, respectively. 5.2. RadialJIow Hukill and Ives’ have given a comparison of theory and experiment for radial flow of air through.shelled corn. R and S for shelled corn based on Shedd’? data are given in Table 2, and Eqns (49) and (50) enabled a comparison of predictions and experiment to be made (Table 4). TABLE 4

Comparison of pressure drop predictions (in Pa) with experiments and predictions of Hukill and Ives’ Increment, m Radial depth, m Average observed values Hukill and Ives’ Hunter

0.15 0.30 513 431 501

0.30 0.46 167 179 192

0.46 0.61 102

6.

0.91 1.22

107

0.61 0.91 125 117

I .22 1.52

1.52 1.83

1.83 2.13

2.13 244

77

52

39

29

21

65

52

36

29

105

117

19

67

45

34

26

22

Numerical examples

Three sets of numerical examples are given in Table 5. Each may be used as a computational check for users of the formulae, and provides an indication of the relative importance of the linear term and the two square-law term resistances in real systems. In all the examples the seed used has Ergun coefficients R = 3161 and S = 10 268. Also the width of the stores (diameter in the case of crossflow) was 10 m. TABLE 5

Numerical results for wheat system

___~

Half-round duct Round duct Perforated floor duct Louvre ducts Crossflow, half-round ducts Crossflow, round ducts Half-round duct Round duct Perforated floor duct Louvre ducts Approximate formula, Eqn (14) ___Crossflow, half-round ducts Crossflow, round ducts

H, m

b, m

5 5

P,, m

2 1 2

3.14 4.44 2.00 5.66 3.14 6.28

10 10 10 10

0 0 0

1 1 1

10

0

7

7 -

-

-

1

-

1 ____-

Ap,, Pa

APZ, Pa

AP,, Pa

AP, Pa

616 567 1027 925 1251 1014

521 491 883 847 1126 888 ___478 512 503 726

65 46 102 36 2x65 2x33

30 30 42 42 60 60

33 33 33 33

10 10 10 10

521 555 546 169

478

33

10

521

~-

1

687

2x33

10

763

1

757

2x33

10

833

The first set of numerical examples correspond exactly to the geometrical arrangements shown in Figs 1 and 2. The airflow Q was 5 and the duct length 20 m. Values of b are given in Table 5. The second set of numerical examples is presented to facilitate a comparison between resistances to airflow of duct and store systems of different duct types. For this purpose the seed bulk

A.

449

J. HUNTER

height H is set at 10 m and the perforated perimeter of each duct is set at 1 m. The approximate values for dp, obtained using Eqn (14) are also included. Other parameter values used were: Q = 1; 1 = 10; W = 10; b = 0. From the results we see that the approximate formula Eqn (14) should not be used for louvre ducts. This is not surprising considering the difference in airflow patterns [Fig. I (top, !$) and (bottom, right)]. The louvre duct system clearly has a disadvantage relative to the other duct types with regard to pressure drop since the air passes through a full n radians where the airflow path is narrowest. Against this, however, is the very significant advantage that the louvre duct is self cleaning when the store is emptied, and also it is readily inspected for insect pests. The third set of numerical results gives a comparison between round ducts and half-round ducts of equivalent perforated perimeter, in the crossflow geometry. Parameter values used were: Q = I, I = 10, W = 10 (1 here is the height of the circular cylinder and W is the diameter of the cylinder). To understand why the pressure difference in the half-round case is less than that for the fully round case it is necessary to realize that for the same perforated perimeter, the round ducts would be contained by the half-round ducts, and so there is clearly additional seed for the air to pass through in the case of round ducts. Round ducts will often be used in crossflow aeration systems however because the resistance to airflow along the duct will be the governing design factor and the round duct has a greater hydraulic diameter.

7.

Discussion

The formulae derived herein cover commonly-used aeration duct arrangements. The approximate formula Eqn (14) will be found quite useful because the conditions for its use apply to many installations and it has a very simple form. To use the formulae derived, the seed parameters required are the Ergun coefficients R and S. As many of these as possible are presented, together with expressions for calculating them from the commonly-used Hukill and Ives’ coefficients. An expression for the porosity is given in terms of R and S and also a and is. If the seed bulk porosity is known these expressions will provide a check on the values of the coefficients. If the top surface of the seed bulk were horizontal instead of that shown in Fig. I, but had the same average height, the linear term pressure difference will in most cases be less than that given by the formulae. The crossflow formulae will be approximate because of the three dimensional effect near the grain surface. As the aspect ratio of the silo becomes greater, the error will become less. The square-law term pressure differences dp, and dp, are modelled rather crudely but in most cases they will be very much the minor part of the total pressure difference. As the airflow increases, the predictions will remain within reasonable bounds. The comparison of predictions with those of Spencer’ described in section 5.1 is in effect a confirmation of the procedure whereby linear term and square-law term pressure differences are calculated separately. The agreement with the experiments of Barrowman and Boyce16 are better than those of Spencer9 and agreement with Spencer’s’ own example is within about loo/,. The comparison between the present work and the theory and experiments of Hukill and Ives’ provides significant support for the approach used in the present work. The integration of the Hukill and Ives’ pressure gradient function leads to a complicated series of logarithms which are replaced by Eqns (49) and (50) in the present work. The agreement between experiment and the present theory appears to be at least as good as the Hukill and Ives’ expression. The numerical examples serve to demonstrate the use of the formulae and show the relative merits of the different duct types. The uncertainty in LIP, as estimated in section 3.2 for the first numerical example is of the order of dp, and dp,. It is important to include ilp, and dp, in the estimate of dp, however, because they contribute to the upper bound value, whereas the uncertainty in dp, does not.

450

PRESSURE

8.

DIFFERENCE

ACROSS

SEED

BULK

Conclusions

The prediction of pressure drop across a seed bulk to be aerated is often made on the assumption that the sizes of ducts conform with conventional practice. ” The formulae presented here enable predictions to be made for arbitrary sizes of duct and so facilitate evaluation of alternative design options. Acknowledgements The work described in this paper was supported by the Australian

Wheat Industry Research Council.

REFERENCES

Hukill, W. V.; Ives, N. C. Radial airflow resistance ofgruin. Agric. Engng, 1955 36 (5) 332-335 z Shedd, C. K. Resistance of grains and seeds to air-ow. Agric. Engng, 1953 34 (9) 616-619 3 Ergun, S. Fluidflow through packed columns. Chem. Engng Progr., 1952 48 89-94 ’ Hohner, G. A.; Brooker, D. B. A hydrodynamic unulogue study of grain aeration cooling. Res. Bull. 851, University of Missouri, College of Agriculture, Agricultural Experiment Station, February 1964 ’ Smith, E. A. 3-Dimensional analysis of air velocity andpressure in beds of grain and hay. J. agric. Engng Res., 198227 101-117 6 Boyce, D. S.; Davies, J. K. Air distribution from a lateral duct with different escape areas in barley. J. agric. Engng Res., 1965 10 230-234 ’ Williamson, W. F. Pressure losses and drying rates in grain ventilated with various on floor duct systems. J. agric. Engng Res., 1965 10 (4) 271-276 8 Brooker, D. B. Lateral duct uirjlow patterns in grain drying bins. Agric. Engng, 1958 39 (6) 348-351 9 Spencer, H. B. Pressure drop in on-floor duct drying systems. J. agric. Engng Res., 1969 14 (2) 165-l 72 ‘OSutherland, J. W.; Ghaly, T. F. Heated air drying of oilseeds. J. Stored Prod. Res., 1982 18 (2) 43-54 ‘I Jones, J. D. Intergt&nular spaces in some storedfoods. Food, 1943 12 325-328 ta Zink, F. J. Specific gravity and air space of grains and seeds. Agric. Engng, 1935 16 (11) 439-440 ” Brooker, D. B. Computing air pressure and velocity distribution when airflows through a porous medium and nonlinear velocity-pressure relationships exist. Trans. Am. Sot. agric. Engrs, 1969 12(l) 118-120 ” Kober, H. Dictionary of Conformal Representutions. New York: Dover, 1957 151-152 ‘s Sutherland, J. W. Butch grain drier design and performance prediction. J. agric. Engng Res., 1975 20 423-432 ‘( Barrowman, R.; Boyce, D. S. Air distribution from lateral ducts in burley. J. agric. Engng Res., 1966 11(4) 243-247 I7 Holman, L. E. Aeration of grain in commercial stores. U.S. Dept. Agric. Mktg Res. Rep. No. 178 l