# Example of Drying Process Calculation

The problems experienced in drying process calculations can be divided into two categories: the boundary layer factors outside the material and hu­midity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differ­ences inside the material during the drying process. The mathematical dis­cussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its depen­dency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here.

We will cover a simple drying model to examine the radiation drier of coated paper. We assume there are no major temperature or humidity varia­tions in the direction of the paper web thickness, and that temperature T and humidity u are constant in the direction of thickness. This assumption requires that the capillary action be ignored, and the pressure gradient of water is zero on the assumption du/dx = dT/dx = 0. How is it possible that the humidity distribution remains uniform?

The only approach is to ignore the capillary flow and to assume water vaporization takes place evenly in the thickness of the paper web. With a

Radiation drier, this approach is reasonable if the radiation energy is ab­sorbed evenly inside the web. Assuming the boundary layers on both sides of the web are similar, the vapor flow is distributed symmetrically to the center of the web. The model of vaporization and water drift is shown in Fig. 4.38.

To derive formula (4.318) for vapor flow in a porous material, we ap­proximate the pressure gradient in Eq. (4.318) with

DpA = Pa(s)~Pa(T, u) , 9

Bx Ax ’ ‘ ;

Where Ax = s/4 and s is the thickness of the paper web (Fig. 4.38). The vapor flow through the boundary layer can be represented as

N v = k’ |rr P ~Pa(j) ia 3201

PA^Ax kc RT[np-pA(Sy (4.32U)

Where k’c = (a/pcp)Le1_” (Eq. (4.300)).

Equation (4.314) follows from Eqs. (4.250), (4.253), (4.256a), and (4.259):

1a = kc(cA(s) — cA(y)) (Eq. (4.256a))

= k’crЈ-(cA(s)-cA(y)) (Eq. (4.259))

Pbm

= K — f-(pA{s) — pA(y)) (Eq. (4.250))

Pbm

Cl n&&L>

— ^ — PA(y)) (Eq. (4.253))

C pBiy)-pB(s)

= k’ (p = pA + Pb)

P-Pa(s)

 Gp
 T
 PA (y) PA (s)

‘pA(y)

 I L ___ — Pa(s) I i X,, I ■- pAiT, u) M ________ Pa(T, u) — Pa(s)

‘PMV)

 ( Ј

 PA (a, T) PA (s)

 Gr

PAly)

(b)

FIGURE 4.38 (a) The uniform vaporization of water in paper, (b) Resistance web analogy for steam flow; I /Gp = resistance due to the paper, I /Gr = resistance due to the boundary layer.

With reference to Eq. (4.251),

V — Mi = b’ ^A-PlnP~

PAvAx Maja kc RT Inp_pA(sy

Or Eq. (4.320).

 {atzlAy} = in" P-Pa(s) The logarithmic function in Eq. (4.320) can be written as

1 , PA(s)-pA(y)] P-Pa(s) y

When (pA(s) — pA(p>))/(p — pA(s)) < 1, a logarithmic function to a series (ln(l + x) = x-x /s+ •••) can be developed, and using the first term gives the approximation

Empim. (4.32D

P-PaM p-m7»

In this approximation pA(s) in the denominator is replaced by pA(T, u).

We use the following notations:

G,= K f? Tp-pl{T, u) (4.322)

(4323)

Using approximation (4.319) in Eq. (4.318) and Eq. (4.317) in Eq. (4.320) leads to the steam flow formula

PaVax = Gp{pA(T, u)-pA(s)) = Gr(pA(s)-pA(y)) .

Eliminating pA(s) gives

Pavax = G(Pa(T, u) — pA(y)), (4.324)

The total conductance is determined from

H = i+<rp- <4-325>

The total vaporization on both sides of the web is determined by Eq. (4.324) multiplied by 2. If the radiation power absorbed in the web is Q"(W rrf"), the energy balance can be written as

Q" = C"^ + 2G(pA(T, u)-pA(y))l(T, u) , (4.326)

Where l(T, u) is the vaporization heat of water and C" is the heat capacity of the web ( J ). This can be calculated from

C" = m"(cpl + ucp2), (4.327)

Where m" is the square mass of the dry substance of the paper web (kg m is the specific heat of the dry substance, and cp2 is the specific heat of water.

The humidity change due to vaporization is

-m"Xt = 2G^T’ u)-Pa(v)) ■ (4.328)

The negative sign in Eq. (4.328) is due to the fact that (du/dt) < 0 repre­sents the net vaporization to the surroundings.

Calculating the temperature rise at each time t with Eq. (4.326) and Eq. (4.328) gives the corresponding change in humidity. As the function G and the pressure p{T, u) are complex, numerical solutions are used.

Example 6

Calculate the humidity change and the temperature rise in a paper web at the time when the web humidity « = 0.20 and temperature 0 = 70 °C. As­sume the heat transfer factor on the web surface is a = 40 W m-2 K 1 and the humidity of the surrounding air is x = 0.05 kg H20/kg dry air. The radiation power density absorbed by the web surface is 250 kW m-2.

The vaporization heat of water, which depends on the humidity, is accu­rately determined by

L(T, u) = l0(T) + r(T, u), (4.329)

Where

/0(T) is the vaporization heat of free water

R(T, u) is the required auxiliary heat (sorption heat)

The sorption heat for the newsprint is calculated as

 Ax-l
 T — T 1 cr 1
 R = A6uAzexp(A4u)T‘ (4.330)

Where A2 =-1.3820, A3 = 7.557, A4 =-3.372, Ab = 8.633 x 10~3 kj kg"1 K2, A7 = 696.0 K, and Tcr = 647.3 K. Substituting u = 0.20 and T = 343 K in this equation gives r = 21.1 kjkg-1. Table 4.7 gives l0(6 = 70 °C) = 2333.3 kj kg“1; hence, l(T, u) = 2354 x 103 Jkg-1.

The partial pressure of the surrounding air pA(y) is calculated using the humidity x = 0.05 kg H20/kg dry air:

PA(y) = 0 622 + xP = 0.622+50.051C)5 = 7440 Pa

The steam pressure inside the paper web pA{T, u) calculated in the previ­ous example was pA(T, u) 11.7 x 103 Pa and the diffusion resistance fac­tor e = 0.50. If the thickness of the paper web s = 0.09 mm, the mean diffusion distance inside the paper is Ax = s/4 = 0.0225 mm.

Substituting the numerical values in Eq. (4.323) to determine the conductance,

 / c > 105 { 18 x 10-3 ‘ ^105 — 27.7 x 103y 1^8.314×343
 Gp = 0.5

 X 36.3 x 3 x 10-6-

To calculate the conductance of the boundary layer we first calculate the mass transfer factor using Eq. (4.300);

K’r =— Le1“«.

‘ Pct>

Assume that n = 0.4. Table 4.7 gives (6 = 70 °C):

PA = 0.1981 kgirf3, pB = 0.715 kgirf5

Dab = 36.3 x 10‘6m2 s"1, A = 0.02418 W m"1 K_1 pcp = pAcpA + pBcpB = 0.1.981 x 1850 + 0.715 x 1006 = 1086 J itT3 K “1

Le = QmEp = 1.63 A

K’c = x 1.631 "°-4 = 4.93 x 10“2 m s-1

Boundary layer conductance is, from Eq. (4.322),

Gr = 4.93 x 10~21^ X 103 x 105 1

R 8.314×343 1 o5 — 27.7 x 105

= 0.432 x 10~6 kg rcf2 s“1 Pa-1.

Total conductance is, from Eq. (4.325),

 G 0.432 xlO"6 7.04 x 10“й = 0.407 x 10 6 kg m 2 s 1 Pa’

Vaporization at time t = 2 x 0.407 x 10~6(27.7 x 103 -7.44 x 105) = 16.49 x 10_3kgm_2s. The humidity change rate is calculated from Eq.(4.328) when the square mass of the dry substance of the paper web is m" = 40.5 x 10-3 kg m-2.

Du _ 16.49 x 10 _ a-i — i

Dt ~ 40.5 xl O’3 ‘

Heat capacity C" = 40.5 x 10“3(1400 + 0.2 x 4186) = 90.6 J nr2 K~‘.

The temperature rise, Eq. (4.326), is

^ = ^-g(250 x 103 — 16.49 xl0’3 x 2354 x 103) = 2330 K s’1.

The velocity of a coated paper web is 17 m s-1 and the width of the IR drier is 0.4 m. Thus the delay time in one drier is 0.4/17 = 0.0235 s. This yields an indication of the processes inside the drier, using the above calculated values:

|Am| = 0.41 x 0.0235 = 0.01 AT = 2330 x 0.0235 = 55 K

An accurate indication is achieved by carrying out the calculations in small time steps, such as At = 0.004 s, and then by calculating the vaporiza­tion, humidity change, and corresponding temperature rise at each time step. This is the numerical solution of differential equations (4.326) and (4.328). The results of a calculation of this type are shown in Table 4.12.

TABLE 4.12 Calculations for Infrared Drier

 Infra power = 250 kW/m1 Time Temperature Humidity (kg water/ « (°C) Kg dry air) 0.0040 47.58 0.2317 0.0080 57.32 0.2310 0.0120 66.05 0.2294 0.0160 73.57 0.2265 0.0200 79.59 0.2220 0.0240 83.83 0.2157 0.0280 77.01 0.2091 0.0320 72.21 0.2046 0.0360 68.43 0.2010 0.0400 65.31 0.1981 0.0440 62.64 0.1957 0.0480 60.34 0.1936 0.0520 58.32 0.1918 0.0560 56.53 0.1902 0.0600 54.93 0.1887 0.0640 53.50 0.1875 0.0680 52.22 0.1863 0.0720 51.05 0.1853 0.0760 49.99 0.1844 0.0800 49.03 0.1835 0.0840 48.16 0.1827 0.0880 47.36 0.1821 0.0920 46.63 0.1814 0.0960 45.96 0.1808 0.1000 45.34 0.1803 0.1040 44.78 0.1798 0.1080 44.26 0.1794 0.1120 43.78 0.1789 0.1160 43.33 0.1786 0.1200 42.93 0.1782

 Note: Initial web temperature is 37 °C and the initial humidity is 0.23. After the infrared drier (t ‘ 0.024 s), there is a free draw, where the water vaporizes from the web to the surroundings (x — 0.03 kg H20/kg dry air), resulting in the cooling down of the web. The heat transfer factor in the drier and after the drier is a =40 W/m2 K.