Infrared Drying

Finally, consideration must b e given to the most easily produced radiation in the electromagnetic spectrum—infrared radiation. Its wave­

length is shorter than those of the other sources considered, lying in the range 7 X 1 0 ~5 cm. to about 1 X 1 ( ) -2 cm.

Its chief virtue is that it provides a convenient and efficient method of transferring heat to irregular surfaces and for this reason it has been used frequently to dry both porous and nonporous materials. Although it can be used to dry solid foodstuffs in air, infrared heating offers no great advantage over convective heat transfer because the surface cannot be heated beyond the temperature at which damage occurs and because transfer of water vapor must take place by convection in an air stream in any case.

The application of infrared radiation in freeze drying would provide a solution to the problem of conveying heat to and through the outer dried layer if, over a certain range of wavelength, the dried layer was transparent to the radiation but the remaining frozen material was opaque or absorbent. A search has been m a d e by Preston (1958) for a

"window" in freeze-dried cod muscle in the range of wavelength from 7 χ 1 0 ~⁵ cm. to 2.5 χ 1 0^{- 3} cm., but without success. Owing to consistently high absorption, infrared radiation of any wavelength in this range can do no more than heat the surface. Its usefulness is limited, therefore, to providing heat at the surface of an irregular-shaped material or where contact between a heated plate and the material is not possible.

D . PHYSICS OF Am D R Y I N G

The following treatment is an elaboration of the considerations given in Section II, A above and is concerned in showing that the drying behavior of fish is not arbitrary, as has often been supposed, but conforms largely to well-known laws of physics.

Evaporative cooling of the surface of a fish is an important factor in certain drying processes, as it enables high air temperatures to b e maintained, while at the same time preventing the surface from becoming overheated, as, for instance, in the cold smoking process. Equation ( 7 a )

shows that the magnitude of the cooling effect is linearly related to the pressure difference across the boundary layer because L, kh, and kw

are practically constant. Thus Equation ( 6 ) suggests that —dW/dt oc (θα — θ8) throughout both periods of drying, except at the lowest water contents. Figure 4 is an example showing this to b e true.

0 1.0 2 . 0 3 . 0 4.0 5.0 6.0 BQ - 0, CC.)

F I G . 4. Relationship b e t w e e n surface cooling a n d rate of d r y i n g for c o d fillets e x p o s e d to a n air stream of velocity 3 0 c m . / s e c . D r y - b u l b t e m p e r a t u r e 3 0 ° C ; wet-bulb temperature 1 8 ° C .

In the absence of radiation or conduction, 6S is equal to the wet-bulb temperature 6W during the constant-rate period, since the same con­

siderations apply equally to the surface of a fish and a wet-bulb.

Equation ( 6 ) together with the equation of Apjohn (1835) Pw — pa = ΒΡ(θα — θ„)

(which relates the difference pw — pa between the saturation vapor pressure at the temperature of the wet-bulb and that in the air, and the wet-bulb depression, where Β is a constant and Ρ is atmospheric pres­

sure), indicate that at a given air velocity the rate of evaporation per unit area

—(dW/dt )/Α = ε = constant X (θα — 0W) ( 9 ) This shows that the rate of drying is proportional to the wet-bulb

depression and is independent of the air temperature. This is seen in Fig. 5 to hold fairly well over a range of air temperature from 20°

to 100°C.

Drying rate is shown in Fig. 6 to be related to air velocity u by a power law of the form

ε = const. X {e^a — O^w)uⁿ (10)

This is of the same form as the heat transfer equation recommended (Marshall and Friedman, 1950) for the constant-rate in drying from plane water-saturated surfaces and the product of the constant term and uⁿ corresponds to the coefficient of heat transfer k^h. The values of these constants are compared in the following tabulation.

η const. (= kh/un)

S u r f a c e of fish m u s c l e 0.77 1.65 X 1 0 - 8 W a t e r - s a t u r a t e d s u r f a c e 0.80 1.67 X 1 0 - 8 «

a In the original reference ( J a s o n , 1 9 5 9 ) , a n error w a s m a d e in the conversion from f o o t - p o u n d - h o u r units to c.g.s. units w h i c h i n d i c a t e d a less satisfactory agreement.

Equation ( 9 ) is a useful empirical equation for calculating drying rates and is sufficiently accurate for most purposes. More detailed considerations (Powell, 1940), involving aerodynamic behavior in the vicinity of finite surfaces have been shown (Jason, 1959) to apply equally well to fish fillet pieces. Such considerations lead to an equation

εΖ / ( ρ^β_ ρ^β) = 2.12 χ Ι Ο "⁷ l°'⁷⁷(l + 0.121 w⁰-^{8 5}) (11) which represents the drying behavior of a plane surface of length I under conditions of streamlined flow.

The way in which the concentration of water at the surface C^s of a slab varies with time can be derived (Carslaw and Jaeger, 1947, p. 104) from Equation ( 2 ) and is expressed in the equation

sc (Dt 1 2 ^ 1 /n2n2Dt\ )

C0 being the initial concentration, assumed to be uniform throughout the slab of thickness 2c.

Equation (12) breaks down when the rate of diffusion within the slab is insufficient to support the saturated condition at the surface and the concentration therefore rapidly falls to an equilibrium value C6.

300 0a-0w(°C.)

Ο 0o=2O°C. Χ 0a=25°C. • 0a=3O°C. + 0α = 40°C. Δ 0a=5O°C. V 0a=6O°C. • 0G = 7O°C. • 0a = 8O°C. • 0a= 90 °C. • 0a=IOO°C. FIG. 5. Effect of wet-bulb depression on rate of drying per unit area of cod fillets exposed to air stream of velocity 366 cm./sec. at various dry-bulb temperatures. (Vertical bar through point and number indicate limits of standard devia­ tion and number of determinations respectively where the latter exceed 4 determinations.)

u (cm./sec.) FIG. 6. Relationship between logarithm of rate of evaporation per unit wet-bulb depression and logarithm of air velocity for cod fillets. (Vertical bars through point and number indicate limits of standard deviation and num­ ber of determinations respectively where the latter exceed 4 determinations.)

This occurs at a time tc, which may be called the critical time. Inserting these values into Equation (12) and rearranging gives

The left hand side of Equation (13) forms a dimensionless group E C / D ( C0 — Ce) which is a function of another dimensionless group Dt^c/c². This function has been plotted in Fig. 7 and may be used to derive the critical time for any given initial rate of evaporation, provided that the diffusion constant is known at the temperature Os. Figure 8 shows that there is good agreement between experiment and theory over a range of two orders of magnitude in ε.

The mean water content in a slab of a material at the critical time t^c is known as the critical moisture content (Perry, 1950) and is given by the expression

Wc-Wn W0—(Wn + Atc)

= (14)

wn wn

where Wn is the weight of nonaqueous material contained in a total initial weight W0, and where Wc is the total weight at t = tc. It is clear that the value of the critical moisture content is not constant, as is frequently assumed, but varies with each of the parameters involved in Equation ( 1 3 ) .

The drying behavior during the falling-rate period can be described by solving the Fourier equation and putting in the boundary conditions.

In a system of rectangular coordinates x, y, z, Equation ( 2 ) becomes dC d2C d2C d2C

— zzz Dx \-Dy \-Dz (15a)

dt dx2 y dy2 ^ dz2 K 1

where Ό^χ, D^y, and Ό^ζ respectively represent the diffusion coefficients along these axes.

But since it has been found that diffusion is practically isotropic in fish muscle (Jason, 1959) D^w = D^y = D^z = D so that Equation ( 1 5 a ) may be written

dC /d2C d2C d2C\

= D ( Η 1 ) (15b)

dt \ dx2 dy2 dz2 J

For the region

—a

^

^

—b

^

i.e., for a slab of dimensions 2a, 2b, 2c (neglecting, for the moment, the existence of a constant-rate period) the boundary conditions are

( i ) C = C⁰ when t = 0

(ii) C = C⁶ at χ — —a, a; y == —b, b; ζ — —c, c when t > 0

In other words, it is assumed that the water content at the surface attains its equilibrium value immediately drying commences.

0.10 1.00 10.0 40.0

€ c / D ( C0- Ce)

F I G . 7. e c/ D ( C0 — Ce) as a function of Dtc/c2.

F I G . 8 . Effect of rate of evaporation on duration of constant-rate period at θα = 3 0 ° C . for c o d fillet 0 . 5 c m . thick, showing theoretical curves for θ8= 1 5 ° , 2 0 ° , 2 5 ° C .

The solution of Equation (15b) has been given by Carslaw and Jaeger (1947, p. 163) as

OA oo oo oo / I \l + m + n

c - C e = ™ ( c 0 - c e) γ γ γ — Li>

Ätf (^{2 l} + ! ) ( 2 m + l ) ( 2 n + 1) ( 2 Ζ + 1 ) π χ ( 2 m + 1)jw/ (2n + 1)πζ

X cos cos — cos exp (—ahm,nt)

2a 2b 2c ( 1 6 ) Όπ*Γ(21+1)* ( 2 m + 1) 2 ( 2 n + l ) * - |

w h e r e c w = _ - ^ _ + + _ _ J (17)