Modeling of Convective Heat-transfer Coefficient for Leek Chips

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Annals of Warsaw University of Life Sciences – SGGWAgriculture No 54 (Agricultural and Forest Engineering) 2009: 57–63(Ann. Warsaw Univ. of Life Sci. – SGGW, Agricult. 54, 2009)

Modeling of convective heat-transfer coef ficient for leek chips

EWA GOLISZDepartment of Fundamental Engineering, Warsaw University of Life Sciences – SGGW

Abstract: Modeling of convective heat-transfer

coef ficient for leek chips. The values of convectiveheat-transfer coef ficient for leek were analyzed.The dependence of the product of this coef ficientand the surface of material being dried on dryingrate coef ficient and initial water content was de-termined. No unique dependence of convectiveheat-transfer coef ficient on drying air temperatureor water content could be found. Since determi-nation of the surface of material being dried wasdif ficult, it was assumed that it would be of ran-dom nature. Confirmation of random changes inthe surface of samples being dried was searchedfor by calculation and analysis of the dependenceof the product of convective heat-transfer coef-ficient α and the surface A of material on water

content. The changes in this productα

⋅

A wereapproximated with a quadratic polynomial, whereone of coef ficients was randomly selected fromthe generator of uniform distribution. The resultsare presented graphically on diagrams.

Key words: leek, convectional drying, convectiveheat-transfer coef ficient, randomness.

Notation:

A – surface of sample being dried, m2,Ms– dry matter mass, kg,

k – coef ficient of initial drying rate, 1/min,u – water content, kg H2O/kg d.m.,r – heat of evaporation, kJ/kg,t p – temperature of drying air, °C,tA – temperature of material surface, °C,tM – temperature of wet thermometer, °C,α – convective heat-transfer coef ficient,

W/(m2K),τ – drying time, min

INTRODUCTION

In the period, when the external exchange

conditions decide on the heat and masstransfer between a wet solid body anddrying air, it is assumed that the rate of water content changes can be describedwith the following equation [Pabis 1982]:

p A

s

du A(t t )

d M r

α ⋅= − −

τ(1)

This equation was used to calculate theconvective heat-transfer coef ficient at the beginning of drying process of wet agri-cultural products, e.g. garlic [Markowskiet al. 1994] or pumpkin [Sojak 2002].In this equation one can assume that thevalues r, Ms and t p do not change and tA == tM = const. The convective heat-trans-fer coef ficient for leek particle surfacewas determined by Kaleta et al. (1999),however, no methodic for determinationof surface A for sample being dried isgiven. The bulbous vegetables, including

leek ( Allium Porrum), behave differentlyduring drying than other vegetables, dueto their structure. The leeks being dried inthe form of slices fall into single rings asthey dry up, due to more intensive water exchange through the cross-section andthe heat stream though the side surfacesof slices. The phenomenon of irregular heating up of slices occur also, resulting



58 E. Golisz

from the heat transfer to particles of the leek chips being dried. Therefore,determination of the material surface isdif ficult.

DETERMINATIONOF CONVECTIVE HEAT-TRANSFER COEFFICIENT

The measured leek surface temperaturetA increases slightly at the beginning of drying, and drying rate du/dτ increasesalso, which is proved by the water con-tent measurements. However, a fairlyaccurate measurement of A surface is not possible. The equation (1) can be used tocalculate the product (α ⋅ A) of convec-tive heat-transfer coef ficient α and thesurface of material being dried A:

s 0

p A

rM k A

t(t )α ⋅ =

−(2)

The changes in water content in the leek slices were measured at drying air tem-

peratures: 40, 50, 60, 70, 80°C and theslice thickness 4, 8 and 12 mm [Golisz2003]. The drying rate was determined by linear regression method in inter-vals (ui, ui+1) basing on the measurementresults and the equation u(τ) = k iτ + ui.

The initial course of drying, amount-ed to 150–400 minutes, depending ondrying air temperature, was divided into periods of 25 or 50 minutes, and k coef-ficient was determined for each of them.Then, for the same time intervals thematerial surface temperature was read

off from the diagram. Depending on thenumber of thermocouples used (from 2to 4) there were obtained 2–4 tempera-ture measurements and the same number of results for temperature t p – tA and the product α ⋅ A.

The dependence of the product of con-vective heat-transfer coef ficient α andsurface A on the drying air coef ficient k was analyzed and presented on Figures1–3.

FIGURE 1. Changes in the product of convective heat-transfer coef ficient α and surface A of material being dried against coef ficient of drying rate k for slice thickness 4 mm and five drying temperatures

0.01 0.02 0.03 0.04 0.05

coefficient of drying rate k

0.20

0.40

0.60

0.80

1.00

1.20

1.40

p r o d u c t o f c o e f f i c i e n t

a n d s u r f a c e A

tickness of slice 4 mm

t = 40oC

t = 50oC

t = 60oC

t = 70oC

t = 80oC

α

t = 40°C

t = 50°C

t = 60°C

t = 70°C

t = 80°C




Modeling of convective heat-transfer coef ficient for leek chips 59

0.00 0.01 0.02 0.03 0.04


0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

p

r o d u c t o f c o e f f i c i e n t

a n d

s u r f a c e

A


t = 40 oC

t = 50oC

t = 60oC

t = 70oC

t = 80oC

α

t = 40°C

t = 50°C

t = 60°C

t = 70°C

t = 80°C


FIGURE 2. Changes in the product of convective heat-transfer coef ficient α and surface A of material being dried against coef ficient of drying rate k for slice thickness 8 mm and five drying temperatures

FIGURE 3. Changes in the product of convective heat-transfer coef ficient α and surface A of material

being dried against coef ficient of drying rate k for slice thickness 12 mm and four drying temperatures

0.00 0.01 0.02 0.03 0.04 0.05


0.10

0.20

0.30

0.40

0.50

0.60

p r o d u c t o f c o e f f i c i e n t

a n d

s u r f a c e

A


t = 50oC

t = 60oC

t = 70oC

t = 80oC

α

t = 50°C

t = 60°C

t = 70°C

t = 80°C



60 E. Golisz

Analyzing the diagrams one can findno unique effect of temperature on α

coef fi

cient, although for the same thick-ness of slice the k coef ficient has bigger value at higher temperature, especially atthe slice thickness 4 and 8 mm; however,it can not be found for 12 mm.

Then, the dependence between the product α ⋅ A and water content was pre-sented. To compare the obtained depen-dences, the variable were standardized bydividing the product of convective heat--transfer coef ficient and the surface by

their initial values, as well as water con-tent by its initial value. Therefore, point(1, 1) is an initial point of the diagram.

The values of product α ⋅ A, calculatedfor particular measurement series, wereapproximated with quadratic polynomialof a general form: y = ax2 – 2ax + (a + 1);after substitution of y = α ⋅ A/(α ⋅ A)o andx = u/uo it can be written as:

2

0 0 0

A u ua 2 a (a 1)

( A) u u

α ⋅ § ·= ⋅ − ⋅ ⋅ + +¨ ¸

α ⋅ © ¹(3)

It is evident from calculations andanalysis of diagrams that the values of obtained products α ⋅ A differ very muchfrom each other (although they were basedon the measurements carried out in thesame conditions); therefore, it can suggestthe randomness of investigated product.

Taking the above into consideration, thecoef ficient a of the polynomial (3) is ran-domly selected from uniform distribution.

At first, the minimal and maximal valuesof a coef ficient were selected for poly-nomials limiting the range of changes inconvective heat-transfer coef ficient α andsurface A, then the numbers for polyno-mials included in this range were selected.

The exemplary sets of twenty curvesas the diagrams of random functions are presented in Figures 4–6; one can assume

0.2 0.4 0.6 0.8 1.0

u/u

0.8

0.9

1.0

1.1

1.2

1.3

A / ( A ) o

t = 60oC, g= 4 mm

computational points

approximative polynomials

α

α

o

t = 60°C, g = 4 mm

FIGURE 4. Dependences of relative changes in convective heat-transfer coef ficient α and surface A of material being dried on relative changes in water content; diagrams of formulae approximating thesedependences for leek chips of slice thickness 4 mm and drying temperature 60°C




0.2 0.4 0.6 0.8 1.0

u/u

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

A / (

A ) o


t = 40oC

t = 50 oC

t = 60oC

t = 70oC

t = 80oC

α

α

o

t = 40°C

t = 50°C

t = 60°C

t = 70°C

t = 80°C


0.2 0.4 0.6 0.8 1.0

u/u

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

A / (

A ) o

temperature 60 C




α

α

o

temperature 60°C

FIGURE 5. Dependences of relative changes in convective heat-transfer coef ficient α and surface A of material being dried on relative changes in water content; diagrams of formulae approximating thesedependences for leek chips of slice thickness 4 mm at five drying temperatures

FIGURE 6. Dependences of relative changes in convective heat-transfer coef ficient α and surface A of material being dried on relative changes in water content; diagrams of formulae approximating thesedependences for leek chips of slice thickness 4 mm at drying temperature 160°C and three slicethickness values



62 E. Golisz

that changes in convective heat-transfer coef ficient are random also. Graphically,it is an area determined by many dia-grams of the same function (3). Figure4 presents the dependence α ⋅ A/(α ⋅ A)o for exemplary temperature 0°C and theslice thickness 4 mm. Figure 5 presentsthese dependences for one thickness of leek slice at five drying temperatures,while Figure 6 for exemplary tempera-ture and three slice thickness values.

The limit values of a coef ficient for polynomial (3) are presented for various

temperatures (Table 1) and leek slicethickness (Table 2).

TABLE 1. Values of a coef ficient for limit polyno-mials (3) at various drying temperatures

Drying temperature [°C] amin amax

50 –1.7 2.560 –1.2 170 –1.05 1.980 –1.2 3.3

TABLE 2. Values of a coef ficient for limit polyno-mials (3) at various leek slice thickness

Slice thickness [mm] amin amax

50 –2 3.560 –0.6 370 –1.3 0.8

SUMMARY

Basing on carried out analysis, no uniquedependence of convective heat-transfer coef ficient or the product α ⋅ A on dryingair temperature can be found, since thesequantities have similar values at differenttemperatures. Since the leek drying pro-cess is of random nature [Golisz 2003] itwas also assumed, that convective heat--transfer coef ficient can be of random

nature, too. Confirmation of randomchanges in the surface of samples beingdried was searched for by calculationsand analysis of the dependence of the product of convective heat-transfer coef-ficient α and material surface A and water content. Changes in product α ⋅ A wereapproximated with quadratic polynomialin the form of equation (3), for which thea coef ficient was randomly selected froma number generator of uniform distribu-tion. No unique effect of water contenton convective heat-transfer coef ficient α

can be found.

REFERENCES

GOLISZ E. 2003: Dedukcyjno-probabilistycznymodel konwekcyjnego suszenia warzyw cebulo-wych. Rozprawa doktorska, SGGW, Warszawa.

JAROS M. 1999: Kinetyka suszenia warzyw,Rozprawa habilitacyjna. Wydawnictwo Aka-demii Rolniczej w Lublinie.

KALETA A., GÓRNICKI K., STAKUN E. 1999:Współczynnik wnikania ciepła do plasterków

porów suszonych w warunkach konwekcjinaturalnej. Problemy Inżynierii Rolniczej, 2:47–53.

MARKOWSKI M., JAROS M., KALETA A.1994: Wyznaczanie współczynnika wnikaniaciepła do plasterków czosnku w warunkachkonwekcji swobodnej. Materiały konferencyjne,t II. VIII Sympozjum Suszarnictwa, s. 112–118.

PABIS S. 1982: Teoria konwekcyjnego suszenia produktów rolniczych. PWRiL, Warszawa.

SOJAK M. 2002: Wyznaczanie współczynnikawnikania ciepła w procesie konwekcyjnegosuszenia dyni spożywczej. Inżynieria Rolnicza,

nr 7a (40), s. 143–148.

Streszczenie: Modelowanie współ czynnika wni-

kania ciepł a krajanki pora. Przeprowadzono ana-lizę wartości współczynnika wnikania ciepła pora.Wyznaczono zależność iloczynu tego współczyn-nika i powierzchni suszonego materiału od współ-czynnika szybkości suszenia oraz od począ tkowejzawartości wody. Nie można wskazać jedno-znacznej zależności współczynnika wnikania cie-




pła od temperatury powietrza suszą cego i od za-wartości wody. Trudnym zadaniem jest określenie

powierzchni suszonego materiału i założono, że

ma ona charakter losowy. Potwierdzenia losowejzmiany powierzchni suszonych próbek poszuki-wano przez obliczenie i analizę zależności iloczy-nu współczynnika wnikania ciepła α i powierzch-ni A materiału, od zawartości wody. Zmiany tegoiloczynu α ⋅ A aproksymowano wielomianemdrugiego stopnia, w którym jeden ze współczyn-ników dobierano losowo z generatora rozk ładu

równomiernego. Wyniki przedstawiono graficz-nie na wykresach.

MS. received January 2009

Author’s address:Katedra Podstaw Inżynierii SGGWul. Nowoursynowska 164, 02-787 WarszawaPolande-mail: [email protected]

Modeling of Convective Heat-transfer Coefficient for Leek Chips

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