LITERATURE DATA OF HEAT TRANSFER COEFFICIENTS IN FOOD PROCESSING

ABSTRACT

Published data on the surface heat transfer coefficients in food processing operations have been selected from the literature, organized into a database, and presented. Useful information concerning the proposed equation, the system geometry, the experimental conditions, the range of application, and the relevant references, are tabulated.

NOMENCLATURE

Latin Symbols
A	=	surface area (m2)
a	=	parameter
Bi	=	Biot number (hL/ks)
cp	=	specific heat (J/kg°C)
C	=	factor
D	=	diameter (m) or (mm)
d	=	particle diameter (m) or (mm)
Fo	=	Fourier number (kt/ρcpL2)
g	=	acceleration of gravity (=9.81 m/s2)
G	=	mass velocity (kg/m2 s)
Gr	=	Grashof number (L3ρ2gβΔT/μ 2)
h	=	surface heat transfer coefficient (W/m2K)
hvol	=	volumetric heat transfer coefficient (W/m3K)
Jc	=	Colburn J-factor ((Nu Pr)2/3/(Re Pr))
JH	=	heat transfer factor (St Pr2/3)
K	=	constant
l	=	half diameter (m) or (mm)
L	=	characteristic dimension (m) or (mm)
L*	=	characteristic dimension in Reynolds number (L*=dp ϵ/(1−ϵ)) (m) or (mm)
M	=	mass flow rate (kg/s)
m	=	parameter
n	=	parameter
N	=	speed of rotation (RPM)
Nu	=	Nusselt number (hL/k)
Pr	=	Prandtl number (cp μ/k)
Q	=	volumetric flow rate (m3/s)
R	=	radius (m) or (mm)
Re	=	Reynolds number (ρLV/μ)
RH	=	relative humidity of air (%)
St	=	Stanton number (h/Vρcp)
T	=	temperature (°C or K)
V	=	velocity (m/s)
Vd	=	bulk volume of dryer (m3)
Vs	=	sleep velocity (m/s)
V	=	total volume of material in drying (m3)
W	=	moisture content of moist solids (kg/kg wb) Greek Symbols
β	=	thermal expansion coefficient (−)
ΔT	=	temperature difference (°C)
ϵ	=	porosity (−)
λ	=	thermal conductivity (W/m°C)
η	=	dynamic viscocity (kg/m s)
ρ	=	density (kg/m3)
Ψ	=	particle sphericity
Ω	=	shape factor Subscripts
a	=	air
b	=	bean
f	=	fluid
m	=	mash room
p	=	product (or particles)
s	=	solid (or steam)
sh	=	shrimp
t	=	tube
w	=	water (or wall)

INTRODUCTION

The interface heat and mass transfer coefficients are important in the design of food processes and processing equipment, and in the control of food packaging and storage. Heat transfer coefficients are essential in thermal processing, in cooling or freezing and in storage operations. Mass transfer coefficients are important in drying and storage of foods, and in separation processes. One basic feature of both coefficients is that they are affected strongly by the characteristics of the processing equipment and the geometry of the system.

The surface heat transfer coefficient (h) is defined in the Newton's law of heat transfer:

Where, Q is the heat flow rate transferred from solid walls to the surrounding fluid (W), A is the heat transfer area (m²), ΔT is the temperature gradient between the solid wall the fluid bulk (°C or K), and h is the surface heat transfer coefficient (W/m²K). The latter depends on the local rheological and thermophysical properties of the fluid film in contact with the solid surface, as well as, on the geometry of the system.

In some processes of solids, th the volumetric heat transfer coefficient (h_vol), is preferred in many cases. It is defined as:

Where, α is the specific surface (m⁻¹), defined as:

Where, A is the effective surface area (m²) and V is the total volume of the material (m³). The values of heat transfer coefficient can be determined experimentally, or calculated from empirical equations involving the dimensionless groups listed in Table 1.

Table 1. Dimensionless Groups of Physical Properties

Dimensionless Group	Definition
Biot number	Bi=hL/λ s
Fourier number	Fo=λt/ρcpL^2
Grashof number	Gr=L^3ρ^2gβΔT/η^2
Heat transfer factor	jH=St Pr^2/3
Nusselt number	Nu=hL/λ
Prandtl number	Pr=cpη/λ
Reynolds number	Re=ρLu/η
Stanton number	St=h/uρcp

h, heat transfer coefficient (W/m² K); L, characteristic dimension of system geometry (i.e., for horizontal or vertical pipes and solid particles, L equals the diameter D) (m); λ _s, thermal conductivity of solid (W/m K); λ, thermal conductivity of fluid (W/m K); t, time (s); ρ, density of the fluid (kg/m³); C_p, the specific heat of the fluid (J/kg K); β, thermal expansion coefficient; ΔT, temperature difference between solid walls and fluid bulk (K); η, dynamic viscosity of fluid (kg/m s); u, fluid velocity (m/s).

The scope of this paper is to select all the available data in the literature and to present them analytically for each process and each food material.

DATA

Empirical equations, involving the above dimensionless groups, are available in classical Chemical Engineering literature, most of which are summarized by Rahman[1] in a comprehensive review, in which food engineering literature data are also included. The present review is further updated by a detailed literature search in international food engineering and food science journals during the recent years, as follows:

	Drying Technology, 1987–1999
	Journal of Food Science, 1985–1999
	International Journal of Food Science and Technology, 1989–1999
	Journal of Food Engineering, 1985–1999
	Transactions of the ASAE, 1978–1999
	International Journal of Food Properties, 1998–2000

A total number of 60 papers were retrieved from the above journals. The data refer to 9 different processes, which count 91 empirical equations (Table 2) and include about 50 food materials (Table 3). A brief description of the relevant food processes is also included (Table 2).

Table 2. Surface Heat Transfer Coefficient Equation Count in Food Processes

Process (description)	Type	No. of Eqns.
Baking
A heat processing of foods involving temperature-dependent complex chemical reactions, like browning, dextrinisation, starch gelatinization, volatilisation, oxidation, change in flavor carried out at 100–250°C. The heat is transferred by convection, conduction and radiation, directly from the exhaust gases of the combustion chamber, or indirectly, using electricity and hot air as the heating medium.	Forced Convection	3
	Natural Convection	2
		Sub total: 05
Blanching
A heat process during which the food products (usually fruits or vegetables) are exposed to hot water or steam for a predetermined time period. The aim of this process is to inactivate enzymes, so as to inhibit undesirable changes in odors, flavors and colors. The heat is transferred from the heating medium mainly by natural convection.	Natural Convection	2
		Sub total: 02
Cooling
During this process the reduction of the product temperature is desirable, mainly for preservation or further processing reasons. For preservation, the final product temperature is a few degrees above zero (e.g., 0–7°C), while for further processing the final product temperature is usually close to ambient temperatures. Cooling medium can be air, water, brine and water/alcohol mixtures. Air at forced convection is the most commonly used, but in special cases, immersion into the cooling medium, at forced or natural convection, is also used.	Forced Convection	12
	Natural Convection	12
		Sub total: 14
Drying
It is the most widely old used method of food preservation. During this process simultaneous heat and mass transfer is taking place, resulting in moisture migration from the food material to the surroundings. The migration of water through the solid follows various mechanisms leading to its vaporization and transfer to the surrounding air. The moisture content of the food material is reduced to a minimum, increasing thus its preservation life. The most common type of drying is the “convective” one, where heat is transferred by forced convection from a hot air stream to the product. In “conductive” drying the heat is transferred to the product by conduction, usually from a hot plate. In “fluidised bed” the food material particles are fluidised with the air stream improving the drying performance. The same result is obtained with “rotary” drying, where air and food product are mixed into a rotating drum.	Conductive	1
	Convective	17
	Fluidized bed	2
	Rotary	6
		Sub total: 26
Freezing
It is a process similar to cooling with the main difference in the final product temperature, which is lowered to negative values, usually less than −18°C, resulting to a long preservation period. Except the above-mentioned cooling media, cryogenic liquids, such as N2 or CO2, can also be used. In this case freezing is carried out either by immersion or by spraying of the cryogenic liquid directly on to the product.	Forced Convection	11
	Natural Convection	1
		Sub total: 12
Sterilization Processes
Aseptic Processing
A thermal sterilization process applied to liquid/particle food systems. The fluid containing the suspension of the particles flows into a tube, e.g., the inner tube of a concentric pipe heat exchanger, while the heat is transferred rapidly from the tube walls to the fluid and from the fluid to the food particles.	Forced Convection	13
	Mixed Flow	2
	Natural Convection	1
		Sub total: 16
PASTEURIZATION	Forced Convection	3
Pasteurization is a heat processes that is carried out with the aim to reduce the concentration of harmful microorganisms and enzymes, to prevent food deterioration and extend storage life. The usual equipment used is steam jacketed vessels and heat exchangers (i.e., scraped surface or plate). An important stage is the cooling period, which follows the heating one. The lethal effect on microorganisms during this period is considered significant. Heat is transferred from the heating (or cooling) medium to the walls (of the heat exchanger or the food package) and from the walls to the liquid and/or particles of the food product.		Sub total: 03
Retorting (Canning)
Retorting or Canning is a sterilization process similar to pasteurization, the main difference being the extent of the thermal destruction of microorganisms. In contrast to pasteurization, sterilization is carried out at higher temperatures and time resulting in longer storage life and lower organoleptic quality of the product. The usual equipment used is rotating or agitating steam autoclaves, known as retorts.	Forced Convection	3
	Mixed Flow	3
	Natural Convection	1
		Sub total: 07
STORAGE
Used for raw materials as well as final products. It can be defined as the establishment of suitable environmental conditions, so that to preserve the food materials with the less possible deterioration. Examples of storage are the aerated silos used for grains and cereals, the packaging of foods in controlled atmosphere, the cold storage of fruits and vegetables in air-conditioned cooling rooms, or the storage of ice cream and frozen food in industrial freezers. In this process, the heat is transferred from the gas mixture of the environment to the food package and from the packaging material to the food product, or vice versa.	Forced Convection	4
	Natural Convection	2
		Sub total: 06
		Grand Total: 91

Table 3. Number of Heat Transfer Coefficient Equations Retrieved for Each Food Material

	MATERIAL	No. of Eqns.
1	Apples	1
2	Apricots	1
3	Barley	2
4	Beef	1
5	Cakes	1
6	Calcium Alginate Gel	1
7	Canola Seeds	1
8	Carrot	1
9	Corn	2
10	Corn Starch	1
11	Figs	1
12	Fish	1
13	Grapes	3
14	Green Beans	1
15	Hamburger	2
16	Maize	1
17	Malt	1
18	Meat Carcass	1
19	Model Food	4
20	Newtonian Liquids	1
21	Non-Food Material	3
22	Particulate Liquid Foods	3
23	Peaches	1
24	Placebo	1
25	Potatoes	2
26	Poultry Excreta	1
27	Raspberries	1
28	Rice	1
29	Soya	2
30	Soybean	1
31	Strawberries	1
32	Sugar	1
33	Wheat	3
34	Spherical Particles	1
35	Corn Cream	1
36	Rapeseed	1
37	Meatball	1
38	Tomatoes	1
Total count of equations		54

DATABASE DESCRIPTION

The selected data were organized into a database. Every record in the database includes the following fields:

	Process
	Type: If more than one type of process is available
	Material
	Geometry
	Experimental Conditions
	Empirical Equations
	Range of Application
	Citation: The paper from which the data are retrieved

All the available records of the database are presented in Table 4. It should be noted that the empirical equations contained in this table have been tested under the particular experimental and process conditions and should be used with care. Another important factor that should be treated with care is the geometry of the system and the resulting characteristic dimension (L) that is involved in many dimensionless groups in Table 1.

Table 4. Empirical Equations for Surface Heat Transfer Coefficient in Food Processing

BAKING

Forced Convection

Cakes. Baking in industrial tunnel oven. Cakes were moulded in aluminium cylindrical plates. Ta=186∼225°C, V=0.02∼0.44 m/s.

Nu=0.664 Re^0.5 Pr^0.33 Re<5×10^5 (laminar flow)

Nu=0.7 Re^0.61 Re>40 (equivalent diam. Method)

h=0.0887 Ta + 61.4 V(Empirical Method)

Baik, O.D.; Grabowski, S.; Trigui, M.; Marcotte, M.; Castaigne, F. [2]

Natural Convection

Cakes. Baking in industrial tunnel oven. Cakes were moulded in aluminium cylindrical plates. Ta=186∼225°C, V=0.02∼0.44 m/s.

Nu=0.82 (Pr Gr)^0.2 Pr^0.034

Baik, O.D.; Grabowski, S.; Trigui, M.; Marcotte, M.; Castaigne, F. [2]

Roti meal. Traditional Indian meal in a tandoor oven. Ta=470°C, Tp=130–175°C.

h=0.28 (ΔT/D)^0.25

ΔT=refractory/product temp. diff., D=diam. of roti=446 mm

Saxena, D.C.; Haridas, P.; Raghava, K. [3]

BLANCHING

Natural Convection

Green beans. Preheating with air at natural convection. Ta=50–80°C, RH=80%,Tb=25–30°C.

Nu=0.59 Gr^0.25

Ta=air temp., Tb=bean center temp., D=bean diamm.=10 mm

Zhang, Q.; Cavalieri, R. [4]

Green beans. Heating with steam. Ts=80–105°C, Tb=30–100°C.

h=0.06[(k^3ρ^2 g λ)/(Ts–Tb) µ; D)]^0.25

Ts= steam temp., Tb=bean center temp., D=bean diamm.=10 mm

Zhang, Q.; Cavalieri, R. [4]

COOLING

Forced Convection

Apples. Air cooling of apples in packed bed. Ta=−6∼0°C, Tp=2∼30°C, V=1∼10 m/s.

Nu=0.027 Re^0.714

Apples diam.=58–63 mm

Fikiin, A.G.; Fikiin, K.A.; Triphonov, S.D. [5]

Apricots. Air cooling of apricots in packed bed. Ta=−6∼0°C, Tp=2∼30°C, V=1∼10 m/s.

Nu=0.101 Re^0.56

Apricots diam.=30–33 mm

Fikiin, A.G.; Fikiin, K.A.; Triphonov, S.D. [5]

Figs. Air cooling of spherical products. Ta=4°C, V=1.0–2.5 m/s.

h=(kw/D) (2.0 + 0.552 Re^0.53 Pr^0.33)

1<Re<48,000, kw=thermal conductivity of water, D=fig diameter=47 mm

Dincer, I. [6]

Grapes. Air cooling of grapes in packed bed. Ta=−6∼0°C, Tp=2∼30°C, V=1∼10 m/s.

Nu=0.419 Re^0.484

Grapes diam.=20–22 mm

Fikiin, A.G.; Fikiin, K.A.; Triphonov, S.D. [5]

Meat carcass. Various chilling schemes. Ta=0°C, V=0.5 m/s.

Nu=0.228 Re^0.731 Pr^0.33

Mallikarjunan, P.; Mittal, G.S. [8]

Model food. Air cooling of a bed of spheres (simulated agricultural product). Ta=60∼20°C, V=0.6∼2.3 m/s.

Nu=2.0+3.78 Re^0.44 Pr^0.33 Tv^0.33

Diam.=75 mm, Tv=0.01 ((v−vm)^2/vm)^1/2, v=local velocity, vm=mean air velocity, 4000<Re<15,000, 0.2<Tv<0.49

Alvarez, G.; Flick, D. [9]

Peaches. Air cooling of peaches in packed bed. Ta=−6∼0°C, Tp=2∼30°C, V=1∼10 m/s.

Nu=0.165 Re^0.50

Peaches diam.=54–56 mm

Fikiin, A.G.; Fikiin, K.A.; Triphonov, S.D. [5]

Raspberries. Air cooling of raspberries in packed bed. Ta=−6∼0°C, Tp=2∼30°C, V=1∼10 m/s.

Nu=0.026 Re^0.68

Raspberries diam.=19–20 mm

Fikiin, A. G.; Fikiin, K. A.; Triphonov, S.D. [5]

Shrimp. Heating in agitated water. Tw=0°C, Tsh=90–20°C.

Nu=0.48 Gr^0.26

Tsh=shrimp temp., Tw=water temp., D=shrimp diamm.=12.7–17.5 mm

Chau, K.; Snyder, G. [10]

Strawberries. Air cooling of strawberries in packed bed. Ta=−6∼0°C, Tp=2∼30°C, V=1∼10 m/s.

Nu=0.121 Re^0.56

Strawb. diam.=28–31 mm

Fikiin, A.G.; Fikiin, K.A.; Triphonov, S.D. [5]

Tomatoes, pears and cucumbers. Air cooling of spherical or cylindrical products. Ta=4°C, Tp=21.5°C.

Nu= 0.267 Re^0.45 Pr^0.33

Rtom=35 mm, Rpear=30 mm, Rcucumb=19 mm, Lcucumb=160 mm, 1000<Re<24,000

Dincer, I. [7]

Tomatoes, pears and cucumbers. Water cooling of spherical or cylindrical products. Tw=1°C, V=0.05 m/s, Tp=21.5°C.

Nu =1.264 Re^0.45 Pr^0.33

Rtom=35 mm, Rpear=30 mm, Rcucumb=19 mm, Lcucumb=160 mm, 1000<Re<70,000

Dincer, I.[^62]

Natural Convection

Mushrooms. Immersed in water. Tw=80–40°C, ΔT=20°C.

Nu=0.08 Gr^0.27

Nu=1×10^2 Fo^−0.23

log Gr=17.8 – 0.75 Fo

Tw=water temp., ΔT=max temp. difference., D= 28 mm

Alhamdan, A.; Sastry, S.; Blaisdell, J. [11]

Tomato Paste. Hot filled cylindrical glass jar left to cool in ambient. Ta=19–26°C.

h=C (Tp−Ta)^n/(2l)^1–3n

C=0.19, n=0.33, 10^9<Gr<10^12, Tp=40–95°C, Vol=200 or 500 cm3, 2l=D

Sandoval, A.J.; Barreiro, J.A.; Mendoza, S. [18]

DRYING

Conductive

Starch. Gelatinized rice starch on a hot plate. T=80–120°C

Nu=0.82 Gr^1/5 Pr^0.034

Daud, W.; Armstrong, W.D. [19]

Convective

Barley. Thin Layer of Grains. Ta=40–175°C, V=0.6 m/s.

JH=St Pr^2/3=3.26 Re^−0.65

20<Re<1000

Sokhansanj, S. [20]

Canola seeds. Fixed bed. Ta=30–65°C, V=0.28–0.42 m/s

Nu=0.143Re^0.67

T (K), P(Pa), D=3.6 mm

Lang, W.; Sokhansanj, S.; Rohani, S. [21]

Carrot. Cubes in basket. Ta=30–70°C, Ga=7000 kg/m2h.

JH=St Pr^2/3=0.692 Re^− 0.486

500<Re<5000

Mulet A.; Berna A. Rosselo C. [22]

Corn. Crossflow moving bed.

JH=St Pr^2/3=2.06 Re^− 0.65/ϵ

ϵ=bed porosity

Torrez, N.; Gustafsson, M.; Schreil, A.; Martinez, J. [23]

Corn. Thin layer of corn kernels. Ta=27–150°C, V=1.4–2.0 m/s.

Nu=2.0+0.6 Re^0.5 Pr^0.33

L=D=corn kernel equivalent diameter=8.3×10^− 3 (m)

Fortes, M.; Okos, M. [26]

Grapes. Air drying in trays with mixed air flow. Ta=−50∼70°C, Tp=26∼70°C, V=0.03 m/s (estimated).

Nu=0.664 Re^0.5 Pr^0.33

Ghiaus, A.G.; Margaris, D.P.; Papanikas, D.G. [27]

Grapes. Seedless Sultana grapes in rectangular metal pan. Ta=50–76°C, V=2.0–2.5 m/s.

Nu=0.74 Re^0.57 Pr^0.33

Vagenas, G.; Marinos-Kouris, D.; Saravacos, G. [28]

Malt. Deep layer. Ta=30–90°C, V=0.25 m/s.

hvol=49.32×10^3 Ga ^0.6

Ga (kg/m^2 s)

Lopez, A.; Virseda, P.; Martinez, G.; Llorka M. [29]

Placebo. Spouted bed. Ta=55–65°C.

Nu=2.0+K(cond) Re^0.5

K(cond)=empirical constant depended on the experimental conditions=1.8

Taranto, O.; Rocha, S.; Raghavan, G. [25]

Potatoes. Slabs, parallel flow. Ta=40–70°C, V=4 m/s, slab dimensions: 10×20×45 mm^3.

Nu=0.23 Re^0.8 Pr^0.4

Wang, N.; Brennan, J.G. [30]

Poultry excreta. Rectangular samples of various thicknesses. Ta=21–32°C, V=0.25–1.47 m/s.

h=0.377 G^0.4/D^0.6

G=mass velocity=0.28–1.7 kg/m^2 s, D=characteristic dimension =0.1064–0.1192 m

Dixon, J.; Wells, G.; Esmay, M. [31]

Rice. Crossflow moving bed.

JH=St Pr^2/3=2.06 Re^−0.65/ϵ

ϵ=bed porosity

Torrez, N.; Gustafsson, M.; Schreil, A.; Martinez, J. [23]

Rice. Packed bed/cross flow.

hvol=8.69×10^4 Ga^1.3

50<T<60, 0.25<Ga<0.5

Patt, D.; Rumsey, T.; Palazoglu, A. [24]

Soybean. Spouted bed. T=55–65°C.

Nu=2.0+K(cond) Re^0.5

K(cond)=empirical constant depended on the experimental conditions=1.8

Taranto, O.; Rocha, S.; Raghavan, G. [25]

Wheet. Fixed bed. T=30–65°C, V=0.28–0.42 m/s.

h=9.73×10^5 (Ga T/Patm)^0.6

T (K), P(Pa), D=3.6 mm

Lang, W.; Sokhansanj, S.; Rohani, S. [21]

Wheet. Thin Layer of Grains. Ta=40–175°C, V=0.6 m/s.

JH=St Pr^2/3=3.26 Re ^−0.65

20<Re<1000

Sokhansanj, S. [20]

Windsor bean. Thin layer of beans in combined convection dielectric drying. T=−3–15°C, V=0.2–0.8 m/s.

JH=St Pr^2/3=3.26 Re ^−0.65

20<Re<1000

Ptasznik, W.; Zygmunt, S.; Kudra, T. [32]

Fluidized Bed

Corn starch. Rotary-stream fluidized bed drying (combination of fluidized bed and pneumatic drying). Ta=100°C.

Nu=−0.937 + 0.113 Ret ^0.622 − 0.42 Ret ^−1.285 Bs ^−0.790

Nu & Ret are based on particle diam., Bs=mass flow ratio of solid particles to air=0.17∼0.863, 12<Ret<189, dryer diam.=150 mm

Shu-De, Q.; Fang-Zhen, G. [33]

Maize. Fluidized bed of maize-sand mixture. Ta=60–120°C.

Nu=6.45 Re^0.1(dm/ds)^0.464

dm=maize mean diam., ds=sand mean diam., specific experimental value of (dm/ds)=2.85

Mourad, M.; Hemati, M.; Steinmetz, D.; Laguerie, C. [34]

Rotary

Fish meal. Ta=250–450°C, V=0.7–1.5 m/s.

hvol=2.37×10^2 Ga ^0.67/D

D=dryer diameter (m)

Shene, C.; Cubillos, F.; Perez, R.; Alvarez, P. [35]

Soya meal. Ta=250–450°C, V=0.7–1.5 m/s.

hvol=2.69×10^2 Ga ^0.686/D

D=dryer diameter (m)

Alvarez, P.; Shene, C. [36]

Soya meal. Ta=250–450°C, V=0.7–1.5 m/s.

Nu=0.005 Re^1.3

Alvarez, P.; Shene, C. [36]

Soya meal. Ta=250–450°C, V=0.7–1.5 m/s.

hvol=2.37×10^2 Ga ^0.67/D

D=dryer diameter (m)

Shene, C.; Cubillos, F.; Perez, R.; Alvarez, P. [35]

Sugar. Granular in rotary dryer. Ta=12.5–30°C, Ma=8.1–29.8 kg/s.

hvol=0.51×(Vs100/Vd)^0.45Ga 0.16

Vs=bulk volume of sugar (m^3), Vd= volume of dryer (m^3), Ga=air mass velocity (kg/m^2 h)

Douglas, P.; Kwade, A.; Lee, P.; Mallick, S. [37]

Sugar. Granular in rotary dryer. Ta=12.5–30°C, Ma=8.1–29.8 kg/s.

Nu=2.0+0.6 Re^0.5 Pr^0.33

Wang, F.; Cameron, I.; Lister, J.; Douglas, P. [38]

FREEZING

Forced Convection

Beef. Air freezing in flat plate freezer. Ta=−28.4∼−17.8°C, Va=0∼14 m/s.

Nu=0.579 Re^0.582

Plate thickness=0.945∼1.89 cm, 1×10^4<Re<5×10^5

Heldman, D.R. [39]

Calcium alginate gel. Balls suspended by strings in a deep freezer. Ta=−22.8°C, Va=10 cm/s, D=4−7 cm, Tp=−22.8–27°C.

Bi=2 + 0.6 Re^0.5 Pr^0.33

Bi=h D/kp, kp=thermal conductivity of product

Sheng, H.L. [40]

Hamburger. Air freezing of disk shaped product on a conveyor belt. Air flow downward to belt. Ta=−60°C, Tp=50°C, Va=1∼15 m/s.

Nu=0.326 Re^0.64

Disk diam.= characteristic length=10 cm, thikness=1 cm, 20×10^−3<Re<70×10^−3

Flores, E.S.; Mascheroni, R.H. [41]

Hamburger. Air freezing of disk shaped product on a conveyor belt. Air flow parallel to belt, packed hamburger. Ta=−60°C, Tp=50°C, Va=1∼15 m/s.

Nu=4.293 Re^0.395

Disk diam.= characteristic length=10 cm, thikness=1 cm, 20×10^−3<Re <70×10^−3

Flores, E.S.; Mascheroni, R.H. [41]

Hamburger. Air freezing of disk shaped product on a conveyor belt. Air flow parallel to belt, unpacked hamburger. Ta=−60°C, Tp=50°C, Va=1∼15 m/s.

Nu=7.891 Re^0.328

Disk diam.= characteristic length=10 cm, thikness=1 cm, 20×10^−3<Re<70×10^−3

Meat. Beef hamburgers refrigerated on a conveyor belt tunnel, cross air flow (downwards). Ta=−35°C∼−25°C, Tp(initial)=−60°C, V=1∼7.5 m/s.

Nu=0.363 Re^0.64 Pr^0.33

10,000<Re<70,000, 120<Nu<410, L×D=1×10 cm^2

Tocci, A.M.; Mascheroni, R.H. [42]

Meat. Beef hamburgers refrigerated on a conveyor belt tunnel, cross air flow (upwards). Ta=−35°C∼−25°C, Tp(initial)=−60°C, V=1∼7.5 m/s.

Nu=4.666 Re^0.355 Pr^0.33

10,000<Re<70,000, 110<Nu<220, L×D=1×10 cm^2

Tocci, A.M.; Mascheroni, R.H. [42]

Meat. Beef hamburgers refrigerated on a conveyor belt tunnel, parallel air flow. Ta=−35°C∼−25°C, Tp(initial)=−60°C, V=1∼7.5 m/s.

Nu=8.77 Re^0.328 Pr^0.33

10,000<Re<70,000, 160<Nu<300, L×D=1×10 cm^2

Tocci, A.M.; Mascheroni, R.H. [42]

Meat. Beef meatballs refrigerated on a conveyor belt tunnel, cross air flow (downwards). Ta=−35°C∼−25°C, Tp(initial)=−60°C, V=1∼7.5 m/s.

Nu=0.448 Re^0.552 Pr^0.33

2500<Re<25,000, 30<Nu<110, D=3.8 cm

Tocci, A.M.; Mascheroni, R.H. [42]

Meat. Beef meatballs refrigerated on a conveyor belt tunnel, cross air flow (upwards). Ta=−35°C∼−25°C, Tp(initial)=−60°C, V=1∼7.5 m/s.

Nu=0.535 Re^0.515 Pr^0.33

2500<Re<25,000, 25<Nu<90, D=3.8 cm

Tocci, A.M.; Mascheroni, R.H. [42]

Meat. Beef meatballs refrigerated on a conveyor belt tunnel, parallel air flow. Ta=−35°C∼−25°C, Tp(initial)=−60°C, V=1∼7.5 m/s.

Nu=0.175 Re^0.642 Pr^0.33

2500<Re<25,000, 25<Nu<105, D=3.8 cm

Tocci, A.M.; Mascheroni, R.H. [42]

Natural Convection

Dried Figs. Dried fig slabs (40×40×10 mm) in deep freezer. Ta=−22°C.

h=(kFL)/(α-0.375FL^2)

k=(0.148+0.493W), α=(0.088×10 ^−6+(aw−0.088×10 ^−6)W), aw=0.148×10^−6m^2/s, W=moisture content (∼0.18)

Dincer, I. [7]

STERILISATION PROCESSES

Aseptic Processing

Forced Convection

Model food. Brass and aluminium particles into water flowing in tube. Volumetric flow rate=1.33∼7.98×10 ^−4 m^3/s.

Nu=26.81 + 0.00455 Re (Dp/DT)

Re & Nu are based on particle diam.=1.33∼2.39 cm, Re=3600∼27,300, (Dp/DT)=0.2618∼0.6273

Sastry, S.K.; Lima, M.; Brim, J.; Brunn, T.; Heskitt, B.F. [43]

Model food. Heat transfer from fluid (sodium carboxymethylcellulose solution) to particles (brass or polysterene cubes) flowing into a tube. Tf=115.5°C, Tp=100∼135°C, volumetric flow rate=2.52∼5.05×10 ^−4 m^3/s.

Nu=2.0 + 0.6 Re^0.5 Pr^0.33 (Ranz Marshall corellation)

tube diam.=50.8 mm, Length=3 m, dp=particle size=11.7∼19.9 mm

Balasubramaniam, V.M.; Sastry, S.K. [44]

Model food. Heat transfer from fluid (sodium carboxymethylcellulose solution) to particles (brass or polysterene cubes) flowing into a tube. Tf=115.5°C, Tp=100∼135°C, volumetric flow rate=2.52∼5.05×10 ^−4 m^3/s.

Nu=0.332 Re^0.5 Pr^0.33 (relative velocity method)

tube diam.=50.8 mm, Length=3 m, dp=particle size=11.7∼19.9 mm

Balasubramaniam, V.M.; Sastry, S.K. [44]

Model food. Heat transfer from silicon cubes to water. Tf=115.6°C, Tsurface=129.4°C, V=0.43∼1.21 m/s.

Nu=2.0 + 0.0333 Re^1.08

Laminar flow, 287.3<Re<880.76

Chandarana, D.I.; Gavin, A.; Wheaton, F.W. [45]

Model food. Heat transfer from silicon cubes to water/starch solution (2%). Tf=115.6°C, Tsurface=129.4°C, V=0.43∼1.21 m/s.

Nu=2.0 + 0.0282 Re^1.6 Pr^0.89

Laminar flow, 1.23<Re<27.38, 9.47<Pr<376.18

Chandarana, D.I.; Gavin, A.; Wheaton, F.W. [45]

Model food. Mashroom shaped particle (aluminium cast). Fluid (sodium carboxymethylcellulose, “CMC” 1.3∼2.2% wt) is flowing around particle in rest. Tf=71°C, m=0.076∼0.287 kg/s.

Nu=2 + 28.37 Re^0.233 Pr^0.143 (Dm/Dt)^1.787

Dm=diameter of mashroom cap projected area=2.065∼2.817 cm, Dt=tube diam.=5 cm, Lt= tube length=1.5 m, (Dm/Dt)=0.413∼0.5634, particle volume=5×10 ^−6∼10×10 ^−6 m^3, particle surface area=13.37×10 ^−4∼ 22.88×10 ^−4 m^2

Zuritz, C.A.; McCoy, S.C.; Sastry, S.K. [46]

Non-food material. Flow past immersed bodies (fluid to particle heat transfer).

Nu=2.0 +1.3 Pr^0.15 + 0.66 Pr^0.31 Res ^0.5

Res=Reynolds number based on sleep velocity of particles=Vp−Vf

Kramers, H. [47]

Non-food material. Flow past immersed bodies (fluid to particle heat transfer).

Nu=2.0 +0.6 Pr^0.33 Res ^0.5

Res=Reynolds number based on sleep velocity of particles=Vp−Vf

Ranz, W.E.; Marshall, W.R. [48]

Non-food material. Flow past immersed bodies (fluid to particle heat transfer).

Nu=2.0 + (0.4 Res ^0.5 + 0.06 Res ^2/3) Pr^0.4 (μ/μ w)^0.25

Res=Reynolds number based on sleep velocity of particles=up−uf, μ w= viscosity at wall

Whitaker, S. [49]

Particulate liquid foods. Alginate Particles (0--30%) in 4% starch solution, flowing in tubular heat exchanger. T=25–45°C, m=1100–2000 kg/h.

Nu=0.05 Re^0.8 Pr^0.33

D=tube diameter=35.6 mm, Dp=particles diam.=5.6 mm

Sannervik, J.; Bolmstedt, U.; Tragardh, C. [50]

Particulate liquid foods. Packed bed of food simulated particles of copper in water or glycerol solution. Tf=13–18°C, Tp=23–35°C, Q=0.6–5 m^3/h, Vf=0.021–0.17 m/s.

Nu=0.5 Relocal ^0.6 Pr^0.28

Dt=tube diameter=100 mm, dp=particles diam.=15 mm, Relocal=ρf dp Vf/(μ ϵ), ϵ=solids voidage %

Mankad, S.; Nixon, K.M.; Fryer, P.J. [51]

Particulate liquid foods. Variable voidage bed of food simulated particles (copper) in water or glycerol solution. Tf=13–18°C, Tp=23–35°C, Q=0.6–5 m^3/h, Vf=0.021–0.17 m/s.

Nu=2.7+1.1 ϵ ^−1.1−Relocal ^0.46 Pr^0.23

D=tube diameter=100 mm, Dp=particles diam.=15 mm, Relocal=ρf dp Vf/(μ ϵ), ϵ=solids voidage %

Mankad, S.; Nixon, K.M.; Fryer, P.J. [51]

Various types spherical particles. Moving spherical particles in rest, stirred or rotating fluid. Tf=80°C, Vp=0.31 m/s.

Nu=8 + 2.53 Re^0.54

Particles move at 50 mm distance from the rotation axis

Astrom, A.; Bark, G. [52]

Mixed flow^2

Carrots. Heating of carot cylinders in (0∼1%) carboxy methyl cellulose solution (CMC), flowing into a tube. Tf=50∼80°C, V=0.2∼0.7×10^−3 m/s.

Nu=2.45 (Gr Pr)^0.108

L=20∼40 mm, D=16∼23 mm, 3×10^3<(Gr Pr)<6×10^4

Awuah, G.B.; Ramaswamy, H.S.; Simson, B.K. [53] ^1

Potatoes. Heating of potato cylinders in (0∼1%) carboxy methyl cellulose solution (CMC), flowing into a tube. Tf=50∼80°C, V=0.2∼0.7×10^−3 m/s.

Nu=2.02 (Gr Pr)^0.113

L=20∼40 mm, D=16∼23 mm, 3×103<(Gr Pr)<6×10^4

Awuah, G.B.; Ramaswamy, H.S.; Simson, B.K. [53]

Natural Convection

Mushrooms. Immersed in water. Tw=40–80°C, ΔT=20°C.

Nu=5.53 Gr^0.21 Nu=6.3×10^2 Fo^−0.18 log Gr=17.5 – 1.15 Fo

Tw=water temp., ΔT=max temp. difference., D=28 mm

Alhamdan, A.; Sastry, S.; Blaisdell, J. [11]

Pasteurization

Forced convection

Carrots. Cubes in packed bed sterilized by hot water circulation. Tw=135∼136.7°C, V=0∼1.58 cm/s.

Nu=Jh Re Pr^0.33

for l=1 cm, Jh=3.26∼4.67 10^−2, for l=2 cm, Jh=1.57∼2.52 10^−2

Chang, S.Y.; Toledo, R.T. [12]

Orange juice. Plate heat exchanger. Tp=80°C∼90°C.

Nu=1.12×10^−5 Re^1.39 Pr^1.63

156<Re<567, 41<Pr<98

Kim, H.B.; Tadini, C.C.; Singh, R.K. [13]

Orange juice. Plate heat exchanger. Tp=80°C∼90°C.

h=−1309+58358.5 μ+7810 V

0.0062<μ<0.0148 (Pa s), 0.32<V<0.69 (m/s)

Kim, H.B.; Tadini, C.C.; Singh, R.K. [13]

Retorting (Canning)

Forced convection

Corn (cream-style). Retorting of rotating cans. Heating with steam and cooling with water. Ts=132.2°C, Tw=15.6°C, N=8 RPM, Tp(initial)= 82.2°C.

Nu=0.492 Re^0.667 Pr^0.37 (H/L)^0.848

4<Re<268, 414<Pr<12,339, 0.06<H/L<0.21, H=head space of can (m), L=can's height (m), can type: 300×407

Zaman, S.; Rotstein, E.; Valentas, K.J. [14]

Model food. Heat transfer from water/sugar solutions (30 and 60%) to aluminium bean-shaped particles, packed bed in 303×406 cans rotated in a retort at 2, 4, 6 and 8 RPM. Ts=115.6°C, Tp=15°C.

Jc=26,900 Re^−0.706Ω^6.98

Jc=Colburn J-factor=(Nu Pr)2/3/(RePr), Ω=shape factor=(surface area of a sphere with volume equal to that of particle)/(surface area of particle), Characteristic length=particle diameter

Fernandez, C.L.; Rao, M.A.; Rajavasireddi, S.P.; Sastry, S.K. [15]

Particulate liquid foods. Heat transfer from fluid to particles in a can rotated (end-over-end) in a retort. T=110–130°C, N=10–20 RPM, R=radius of rotation=0.09–0.27 m.

Nu=0.167 (Re Pr)^0.61 (kp/kl)^0.48 (d/D)^−0.7 (e/(100−e))^0.067Ψ^0.23

k=thermal conductivity, p=particle, l=liquid, d=particle diam., D=can diam., e=% volume concentration of particles, Ψ=particle sphericity=equivalent diam. particle surface/real particle surface, 28<Re<1550, 1180<(Re Pr)<7710, 0.56<(kp/kl)<2.24, 0.22<d/D, 0.29, 20%<ϵ<40%, 0.806<Ψ<1

Sablani, S.S.; Ramaswamy, H.S.; Mujumdar, A.S. [16]

Mixed flow

Newtonian liquids. Water, glycerin and sucrose solutions (30, 40, 50, 60% w/v), were processed in cans with steam retorting. Inside heat transfer (can wall to liquid). Ts=110∼115.6°C, Tf=30∼70°C, N=2, 4, 6& 8 RPM.

Nu=0.135 (Gr Pr)^0.323 3.91×10^−3(Re Pr D/L)^1.369

D=can diam.=characteristic length, can types=#303 & #10, L=length of can, L/D=1.13∼1.37, Nu=24∼272, Re=0.4∼458, Pr=2.8∼476, Gr=2×104∼2.9×10^9

Rao, M.A.; Cooley, H.J.; Anantheswaran, R.C.; Ennis, R.W. [17]

Newtonian liquids. Water, glycerin and sucrose solutions (30, 40, 50, 60% wt), were processed in cans with steam end over end rotation. Inside heat transfer (can wall to liquid). Ts=100°C, Tf=25∼100°C, N=0∼38.6 RPM.

Nu=2.9 Re^0.436 Pr^0.287

Nu & Re based on Dr=Rotation diam.=0∼7.5 cm, can type=303×406 L/D=0.73∼1.37, Re=83∼2.1×105, Pr=2.8∼49

Anantheswaran, R.C.; Rao, M.A. [55]

Non-Newtonian liquids. Aqueous guar gum solutions (0.30, 0.40, 0.50, 0.75% w/v), were processed in cans with steam retorting. Inside heat transfer (can wall to liquid). Ts=110∼115.6°C, Tf=30∼70°C, N=2, 4, 6 & 8 RPM.

Nu=2.6 (Gr Pr)^0.205 7.15×10^−7(Re Pr D/L)^1.837

D=can diam.=characteristic length, can types=#303 & #10, L=length of can, L/D=1.13∼1.37, Nu=24∼160, Re=0.4∼96, Pr=205∼2100, Gr=100∼5.2×10^5

Rao, M.A.; Cooley, H.J.; Anantheswaran, R.C.; Ennis, R.W. [17]

Natural Convection

Model food. Heat transfer from cooling water to plastic cans or blocks during cooling period of sterilization. Tw=14.9∼16°C, Tp=121∼16.4°C.

Nu=0.47 (Gr Pr)^0.25

can size (L×D)=62×57 mm^2, block dimensions=26×91×141 mm^3, ΔT=1∼105°C

Tucker, G.S.; Clark, P. [56]

Storage

Forced Convection

Barley. Bed of grains in cylindrical storage bins exposed to weather conditions (Convection from cylindrical wall). Ta=− 10–25°C.

Nu=0.227 Re^0.634

L=D=bin diameter=5.56 (m)

Alagusundaram, K.; Jayas, D.; White, N.; Muir, W. [57]

Potatoes. Cooling and storage of a cylindrical bed of potatoes in aerated silos. Ta=6.7°C, Tp=6.7∼15.5°C, V=0.011 m/s.

Nu=(0.5 Re^1/2 + 0.2 Re^2/3) Pr^1/3

Bed, height=2.4 m, diam.=0.7 m, Dimensions of potatoe tube (L×D)=95×51 mm^2, dp=equivalent potatoe diam.=65 mm, ϵ=porosity=0.61, L*=characteristic length in Reynolds number=dp ϵ/(1−ϵ), 10<Re<10^4

Xu, Y.; Burfoot, D. [60]

Rapeseed (canola). Bed of grains in cylindrical storage bins exposed to weather conditions (Convection from cylindrical wall). Ta=− 10–25°C.

Nu=0.227 Re^0.633

L=D=bin diameter=5.56 (m)

Alagusundaram, K.; Jayas, D.; White, N.; Muir, W. [57]

Wheat. Bed of grains in aerated cylindrical storage bins. Ta=− 10–40°C, V=0.004–0.018 m/s.

Nu=0.012 Re^0.805

L=D=bin diameter=6.6 (m)

Chang, C.; Converse, H.; Steele, J. [58]

Natural Convection

Onions. Horizontal cylindrical samples stored in controlled atmospheres of air, O2 and CO2. Ta=0°C, Tp=0–10°C.

h=(1.32) [(Ta−Tp)/D]^0.25

Dimensions (L×D)=0.5×7 cm^2 for red onions, 2.2×1.05 cm^2 for green onions

Kang, J. S.; Lee, D.S. [59]

Onions. Sperical samples stored in controlled atmospheres of air, O2 and CO2. Ta=0°C, Tp=0–10°C.

h=ka (2 + 0.39 Gr^0.25)/D

D=9 cm for apples (estimated), k=thermal conductivity of air

Kang, J.S.; Lee, D.S. [59]

1 Data have been collected from, Rahman, M.S.: 1995.[1]

2 Definition of “Mixed flow” is given by Rahman, M.S.: 1995.[1]

3 Data have been collected from, Maesmans, G.; Hendrickx, M.; DeCordt, S.; Fransis, A.; Tobback, P.[61]

Table 5. Sample Calculation of the Information Use

d (m)	T (°C)	u (m/s)	ñ (kg/m^3)	n (m^2/s)	Cp (J/kg°C)	k (W/m°C)	Re	Pr	Nu	h (W/m^2 °C)	St	JH
0.005	50	2.00	1.079	1.93E−05	1008.7	0.0282	559	0.691	24.1	135.9	0.062	0.049
0.005	50	2.25	1.079	1.93E−05	1008.7	0.0282	628	0.691	25.8	145.4	0.059	0.046
0.005	50	2.50	1.079	1.93E−05	1008.7	0.0282	698	0.691	27.4	154.4	0.057	0.044
0.005	63	2.00	1.036	1.99E−05	1009.1	0.0292	520	0.688	23.1	135.0	0.065	0.050
0.005	63	2.25	1.036	1.99E−05	1009.1	0.0292	585	0.688	24.7	144.4	0.061	0.048
0.005	63	2.50	1.036	1.99E−05	1009.1	0.0292	650	0.688	26.3	153.3	0.059	0.046
0.005	76	2.00	0.998	2.05E−05	1009.6	0.0302	487	0.686	22.2	134.2	0.067	0.052
0.005	76	2.25	0.998	2.05E−05	1009.6	0.0302	547	0.686	23.8	143.5	0.063	0.049
0.005	76	2.50	0.998	2.05E−05	1009.6	0.0302	608	0.686	25.2	152.4	0.061	0.047
0.01	50	2.00	1.079	1.93E−05	1008.7	0.0282	1117	0.691	35.8	100.9	0.046	0.036
0.01	50	2.25	1.079	1.93E−05	1008.7	0.0282	1257	0.691	38.3	107.9	0.044	0.034
0.01	50	2.50	1.079	1.93E−05	1008.7	0.0282	1396	0.691	40.6	114.6	0.042	0.033
0.01	63	2.00	1.036	1.99E−05	1009.1	0.0292	1041	0.688	34.3	100.2	0.048	0.037
0.01	63	2.25	1.036	1.99E−05	1009.1	0.0292	1171	0.688	36.7	107.2	0.046	0.036
0.01	63	2.50	1.036	1.99E−05	1009.1	0.0292	1301	0.688	39.0	113.8	0.044	0.034
0.01	76	2.00	0.998	2.05E−05	1009.6	0.0302	973	0.686	33.0	99.6	0.049	0.038
0.01	76	2.25	0.998	2.05E−05	1009.6	0.0302	1095	0.686	35.3	106.5	0.047	0.037
0.01	76	2.50	0.998	2.05E−05	1009.6	0.0302	1216	0.686	37.5	113.1	0.045	0.035
0.015	50	2.00	1.079	1.93E−05	1008.7	0.0282	1676	0.691	45.1	84.7	0.039	0.030
0.015	50	2.25	1.079	1.93E−05	1008.7	0.0282	1885	0.691	48.2	90.6	0.037	0.029
0.015	50	2.50	1.079	1.93E−05	1008.7	0.0282	2094	0.691	51.2	96.2	0.035	0.028
0.015	63	2.00	1.036	1.99E−05	1009.1	0.0292	1561	0.688	43.2	84.2	0.040	0.031
0.015	63	2.25	1.036	1.99E−05	1009.1	0.0292	1756	0.688	46.2	90.0	0.038	0.030
0.015	63	2.50	1.036	1.99E−05	1009.1	0.0292	1951	0.688	49.1	95.6	0.037	0.029
0.015	76	2.00	0.998	2.05E−05	1009.6	0.0302	1460	0.686	41.6	83.7	0.042	0.032
0.015	76	2.25	0.998	2.05E−05	1009.6	0.0302	1642	0.686	44.5	89.5	0.039	0.031
0.015	76	2.50	0.998	2.05E−05	1009.6	0.0302	1825	0.686	47.2	95.0	0.038	0.029

Examining the equations of Table 4 for “forced convection”, it can be seen that the majority of them may by grouped into a function of Nu, Re and Pr numbers by the general form:

Where, a, n, m are suitable parameters. Something similar can also be observed for “natural convection” or for “mixed flow”, where the dimensionless numbers can be grouped by the preceding form, substituting the Reynolds by the Grashof number, or by a more simple expression of the type:

Where, a, n are parameters

RESULTS AND DISCUSSION

The presented equations can be used for a rough estimation of the heat transfer coefficient for various foods and various food processes. Unfortunately the accuracy of this estimation was not presented in the literature. In most of the papers this information was not included. However, in some cases more than one literature equations are available. In these cases the estimation of the accuracy could be based on the different values obtained from the different sources. The following example shows how these equations could be used. Suppose that we are drying grapes in trays using air at 607deg;C with velocity 2 m/s. The corresponding heat transfer coefficient can be calculated from two sources:

a.	Ghiaus et al.[1] [27] ⇒ Nu=0.664 Re0.5 Pr0.33 ⇒ h=114.6 W/m2K
b.	Vagenas et al.[28]⇒ Nu=0.74 Re0.57 Pr0.33 ⇒ h=100.9 W/m2K

As it can be seen these values appear a deviation of 14%. It is proposed to gather more than two values using similar products or similar processes in order to estimate the accuracy of the heat transfer coefficient. For safe design the smallest value must be selected.

CONCLUSION

Literature data of the surface heat transfer coefficient of foods were organized into a data base and presented. They can be used for rough estimation of the heat transfer coefficient for various processes and various food materials. The accuracy of estimation could be obtained only in the cases that more than one literature data are available for the same material and same process.

Acknowledgments