Heat and Mass Transfer

, Volume 43, Issue 3, pp 255–264

Numerical simulation of forced convection in a duct subjected to microwave heating

Authors

  • J. Zhu
    • Department of Mechanical and Aerospace EngineeringNorth Carolina State University
    • Department of Mechanical and Aerospace EngineeringNorth Carolina State University
  • K. P. Sandeep
    • Department of Food ScienceNorth Carolina State University
Original

DOI: 10.1007/s00231-006-0105-y

Cite this article as:
Zhu, J., Kuznetsov, A.V. & Sandeep, K.P. Heat Mass Transfer (2007) 43: 255. doi:10.1007/s00231-006-0105-y
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Abstract

In this paper, forced convection in a rectangular duct subjected to microwave heating is investigated. Three types of non-Newtonian liquids flowing through the duct are considered, specifically, apple sauce, skim milk, and tomato sauce. A finite difference time domain method is used to solve Maxwell’s equations simulating the electromagnetic field. The three-dimensional temperature field is determined by solving the coupled momentum, energy, and Maxwell’s equations. Numerical results show that the heating pattern strongly depends on the dielectric properties of the fluid in the duct and the geometry of the microwave heating system.

List of symbols

A

area (m2)

Cp

specific heat capacity (J/(kg K))

c

phase velocity of the electromagnetic propagation wave (m/s)

E

electric field intensity (V/m)

f

frequency of the incident wave (Hz)

h

effective heat transfer coefficient (W/(m2 K))

H

magnetic field intensity (A/m)

L

standard deviation of temperature (°C)

k

thermal conductivity (W/(m K))

m

fluid consistency coefficient, (Pa sn)

n

flow behavior index

Nt

number of time steps

p

pressure (Pa)

q

electromagnetic heat generation intensity (W/m3)

Q

volume flow rate (m3/s)

T

temperature (°C)

t

time (s)

tan δ

loss tangent

w

velocity component in the z direction (m/s)

W

width of the cavity (m)

ZTE

wave impedance (Ω)

Greek symbols

η

apparent viscosity (Pa s)

ε

electric permittivity (F/m)

ɛ′

dielectric constant

ɛ′′

effective loss factor

λg

wave length in the cavity (m)

μ

magnetic permeability (H/m)

ρ

density (kg/m3)

σ

electric conductivity (S/m)

Superscripts

τ

instantaneous value

Subscripts

ambient condition

0

free space, air

inc

incident plane

in

inlet

x, y, z

coordinate system of the applicator

X, Y, Z

coordinate system of the microwave cavity

1 Introduction

Microwave heating has been utilized in the food industry for decades. It has been used predominantly as a batch processing and sporadically as a continuous process. Microwave heating has well-known advantages over traditional heating methods, such as fast heating and high energy efficiency as well as heating without direct contact with high-temperature surfaces. However, microwave heating has also been known to heat products non-uniformly [13]. Several factors affect the magnitude and uniformity of absorption of electromagnetic energy. These include dielectric properties, ionic concentration, volume, and shape of a product [4]. The power and temperature distributions inside the product can be predicted by solving the coupled momentum, energy, and Maxwell’s equations. Due to a large number of factors that affect heating and the complexity of the equations involved, numerical modeling is the only viable approach for conducting realistic process simulations [5].

A number of studies have been reported that dealt with numerical modeling of the microwave heating problem in a cavity containing a lossy material. Solutions of Maxwell’s equations for a number of simplified cases are presented in de Pourcq [2], Webb et al. [6], and Ayappa et al. [7]. Electromagnetic field and microwave power distributions for three-dimensional cavities are obtained in Liu et al. [8], Zhao and Turner [9], and Zhang et al. [10]. As for the simulation of heat transfer induced by microwave treatment, most previous works focused on the solid lossy material inside the microwave cavity. Zhang and Datta [11] coupled two separate finite element softwares to predict the temperature distribution inside solid foods due to heating in a domestic microwave oven. Liu et al. [8] applied a finite difference time domain (FDTD) algorithm to a three-dimensional problem and investigated heating inside a partially loaded cavity, demonstrating the significant effect of relative location of the dielectric material within the cavity. There are also recent studies of a multi-dimensional heating process of a liquid by a microwave field. Zhang et al. [5] developed a three-dimensional model by coupling the momentum, energy, and Maxwell’s equations to investigate natural convection of a contained liquid induced by microwave heating. In their model, the local microwave power dissipation in a liquid is predicted by employing an FDTD method and the transient temperature and flow patterns in the liquid are simulated using the SIMPLER algorithm [12]; the two modules are coupled by temperature dependent dielectric properties of the liquid. A similar algorithm is adopted in the present study.

This study reports a numerical prediction of forced convection heat transfer occurring in a rectangular applicator within a three-dimensional microwave cavity. Three types of liquid foods are considered in this study to investigate the effect of dielectric properties on heating by simulating the steady-state temperature distributions in various liquids.

2 Numerical model

The microwave heating system, as illustrated in Fig. 1, consists of a single mode microwave resonant cavity and a vertical applicator tube. The liquid flows through the applicator tube vertically upward, absorbing microwave energy during the process which heats the liquid.
https://static-content.springer.com/image/art%3A10.1007%2Fs00231-006-0105-y/MediaObjects/231_2006_105_Fig1_HTML.gif
Fig. 1

Schematic diagram of the microwave cavity and the applicator

2.1 Geometry of the system

As shown in Fig. 1, the cavity dimensions (Cx × Cy × Cz) are 406×305×124 mm. The applicator tube dimensions (Ax × Ay × Az) are 46×46×124 mm. The applicator is located in the center of the cavity so that the centerline of the applicator is located at Dx = Cx/2 = 203 mm and Dy = Cy/2 = 152.5 mm. The microwave cavity is excited in TE10 mode [13] operating at a frequency of 915 MHz by imposing a plane polarized source at the incident plane (X=21 mm). The reflected microwave energy is absorbed at the absorbing plane (X=0 mm).

2.2 Mathematical model formulation

2.2.1 Electromagnetic field

The equations governing the electromagnetic field are based on the Maxwell curl relation. The three-dimensional unsteady Maxwell’s equations in Cartesian coordinates are:
$$\frac{{\partial H_{X}}}{{\partial t}} = \frac{1}{\mu}{\left({\frac{{\partial E_{Y}}}{{\partial Z}} - \frac{{\partial E_{Z}}}{{\partial Y}}} \right)},$$
(1)
$$\frac{{\partial H_{Y}}}{{\partial t}} = \frac{1}{\mu}{\left({\frac{{\partial E_{Z}}}{{\partial X}} - \frac{{\partial E_{X}}}{{\partial Z}}} \right)}, $$
(2)
$$\frac{{\partial H_{Z}}}{{\partial t}} = \frac{1}{\mu}{\left({\frac{{\partial E_{X}}}{{\partial Y}} - \frac{{\partial E_{Y}}}{{\partial X}}} \right)},$$
(3)
$$\frac{{\partial E_{X}}}{{\partial t}} = \frac{1}{\varepsilon}{\left({\frac{{\partial H_{Z}}}{{\partial Y}} - \frac{{\partial H_{Y}}}{{\partial Z}} - \sigma E_{X}} \right)},$$
(4)
$$\frac{{\partial E_{Y}}}{{\partial t}} = \frac{1}{\varepsilon}{\left({\frac{{\partial H_{X}}}{{\partial Z}} - \frac{{\partial H_{Z}}}{{\partial X}} - \sigma E_{Y}} \right)},$$
(5)
$$\frac{{\partial E_{Z}}}{{\partial t}} = \frac{1}{\varepsilon}{\left({\frac{{\partial H_{Y}}}{{\partial X}} - \frac{{\partial H_{X}}}{{\partial Y}} - \sigma E_{Z}} \right)},$$
(6)
where E and H are the electric and magnetic field intensities, σ is the electric conductivity, μ is the magnetic permeability, and ε is the electric permittivity.
The boundary conditions for the electromagnetic fields are:
  • At the surface of the wall of the cavity, a perfect conducting condition is utilized. Therefore, normal components of magnetic fields and tangential components of electric fields are assumed to vanish:
    $$H_{\rm n} = 0,\quad E_{\rm t} = 0$$
    (7)
  • At the absorbing plane, Mur’s [14] first order absorbing condition is utilized:
    $${\left({\frac{\partial}{{\partial Z}} - \frac{1}{c}\frac{\partial}{{\partial t}}} \right)}E_{Z} \left| {_{{X = 0}} = 0}, \right.$$
    (8)
    where c is the phase velocity of the propagation wave.
  • At the incident plane, the input microwave source is simulated by the following equations:
    $$E_{{Z,{\rm inc}}} = - E_{{{\rm in}}} \sin {\left({\frac{{\pi Y}}{{W}}} \right)}\cos {\left[ {2\pi {\left({ft - \frac{{X_{{\rm in}}}}{{\lambda _{g}}}} \right)}} \right]}$$
    (9)
    $$H_{{Y,{\rm inc}}} = \frac{{E_{{{\rm in}}}}}{{Z_{{\rm TE}}}}\sin {\left({\frac{{\pi Y}}{{W}}} \right)}\cos {\left[ {2\pi {\left({ft - \frac{{X_{{{\rm in}}}}}{{\lambda _{g}}}} \right)}} \right]},$$
    (10)
    where Ein is the input value of the electric field intensity, W is the width of the cavity, ZTE is the wave impedance, and λ g is the wave length of a microwave in the cavity.

2.2.2 Heat and mass transport equations

It is assumed that the applicator, where the liquid is subjected to microwave heating, is located past the hydrodynamic entry section of the pipe; therefore, the flow in the applicator is hydrodynamically fully developed; only the streamwise velocity component is non-zero. Experiments conducted at the Food Rheology Laboratory of North Carolina State University indicate that for the non-Newtonian liquids considered in this research (apple sauce, skim milk, and tomato sauce) the consistency and flow behavior indices (their experimentally determined values are summarized in Table 1) are well approximated as temperature-independent. The momentum equation is then presented as:
$$\frac{\partial}{{\partial x}}{\left({\eta \frac{{\partial w}}{{\partial x}}} \right)} + \frac{\partial}{{\partial y}}{\left({\eta \frac{{\partial w}}{{\partial y}}} \right)} - \frac{{{\rm d}p}}{{{\rm d}z}} = 0$$
(11)
where η is the apparent viscosity for the non-Newtonian fluid, which in this study is assumed to obey the power-law. The apparent viscosity for the power-law fluid is given by:
$$\eta = m{\left[ {{\left({\frac{{\partial w}}{{\partial x}}} \right)}^{2} + {\left({\frac{{\partial w}}{{\partial y}}} \right)}^{2}} \right]}^ \frac{{(n - 1)}}{{2}}, $$
(12)
where m and n are the fluid consistency coefficient and the flow behavior index, respectively.
Table 1

Parameter values utilized in computations

 

Apple sauce

Skim milk

Tomato sauce

f (MHz)

915

915

915

E0 (V/m)

9,000

9,000

9,000

μ (H/m)

4π×10−7

4π×10−7

4π×10−7

ɛ0 (F/m)

8.854×10−12

8.854×10−12

8.854×10−12

k (W/(m K))

0.5350

0.5678

0.5774

cp (J/(kg K))

3703.3

3943.7

4000.0

h (W/(m2K))

30

30

30

ρ (kg/m3)

1104.9

1047.7

1036.9

Q (m3/s)

6.0×10−6

6.0×10−6

6.0×10−6

m (Pa sn)

32.734

0.0059

3.9124

n

0.197

0.98

0.097

The temperature distribution in the liquid is obtained by solving the following energy equation wherein the microwave power absorption is accounted for by an electromagnetic heat source term:
$$\rho C_{p} {\left({\frac{{\partial T}}{{\partial t}} + w\frac{{\partial T}}{{\partial z}}} \right)} = k{\left({\frac{{\partial ^{2} T}}{{\partial x^{2}}} + \frac{{\partial ^{2} T}}{{\partial y^{2}}}} \right)} + q,$$
(13)
where q stands for the local electromagnetic heat generation intensity term, which is a function of dielectric properties of the liquid and the electric field intensity:
$$q = 2\pi f\varepsilon _{0} {\varepsilon}' (\tan \delta)E^{2} $$
(14)
In (14), ɛ 0 is the permittivity of the air, ɛ′ is the dielectric constant of the liquid, and tan δ is the loss tangent, a dimensionless parameter defined as:
$$\tan \delta = \frac{{{\varepsilon}''}}{{{\varepsilon}'}},$$
(15)
where ɛ′′ stands for the effective loss factor. The dielectric constant, ɛ′, characterizes the penetration of the microwave energy into the product, while the effective loss factor, ɛ′′, indicates the ability of the product to convert the microwave energy into heat. Both ɛ′ and ɛ′′ are dependent on the microwave frequency and the temperature of the product. tan δ indicates the ability of the product to absorb microwave energy.
The following boundary conditions are utilized. At the inner surface of the applicator tube, a hydrodynamic no-slip boundary condition is used. At the inlet to the applicator, a uniform, fully developed velocity profile is imposed; it is specified by the inlet volume flow rate, Q. Heat transfer at the applicator wall is modeled as follows. The wall is assumed to lose heat by natural convection, which is modeled by the following equations:
$$\hbox{at the walls normal to the } x\hbox{-direction: }\quad - k\frac{{\partial T}}{{\partial x}} = h{\left({T - T_{\infty}} \right)}$$
(16)
$$\hbox{at the walls normal to the } y\hbox{-direction: }\quad - k\frac{{\partial T}}{{\partial y}} = h{\left({T - T_{\infty}} \right)},$$
(17)
where k is the thermal conductivity of the liquid and h is the effective heat transfer coefficient defined as:
$$h = \frac{1}{{1/h_{{\rm air}} + L_{{\rm wall}} /k_{{\rm wall}} + 1/h_{{\rm liquid}}}}$$
(18)
where hair stands for the heat transfer coefficient from the applicator wall to the air in the cavity and hliquid stands for the heat transfer coefficient from the liquid inside the applicator to the wall. Lwall is the thickness of the applicator wall and kwall is the thermal conductivity of the wall. hliquid is (18) depends on the calculated temperature field; however, since hairhliquid, the last term in the denominator of (18) is neglected compared to the first term and h is set to a constant value given in Table 1.
The inlet liquid temperature is set uniform and equal to the temperature of the free space outside the applicator, T. The initial temperature distribution in the applicator is assumed uniform and also equal to T, as follows:
$$T = T_{\infty} \; \hbox{at}\; t = 0.$$
(19)
As time is increased, the temperature distribution evolves until it reaches steady-state, it is this steady-state temperature distribution that is of most engineering significance; therefore, figures display the steady-state temperature field.

2.2.3 Numerical solution

Two different time steps are utilized to update the electromagnetic and thermal-flow fields. An FDTD method [15] is used to solve Maxwell’s equations (1, 2, 3, 4, 5, 6). The obtained electromagnetic fields are used to calculate the electromagnetic heat source, given by (14), which represents the heating effect of the microwave field on the liquid. Since in (14) the dielectric constant, ɛ′, and the loss tangent, tan  δ, are temperature dependent, an iterative scheme is required to resolve the coupling of the energy and Maxwell’s equations. The time scale for electromagnetic transients (a picosecond scale) is much smaller than that for the flow and thermal transport (a second scale). A time step in the block of the code that solves Maxwell’s equations must satisfy the stability requirement of the FDTD scheme [16] written as:
$$\Delta t \leq \frac{1}{{c{\sqrt {\frac{1}{{\Delta X^{2}}} + \frac{1}{{\Delta Y^{2}}} + \frac{1}{{\Delta Z^{2}}}}}}}.$$
(20)
The energy equation is solved by utilizing a fully implicit scheme; a time step of 1 s is used in these computations. The electromagnetic heat source, q, defined by (14), is computed in terms of the time-average field, \(\bar{E},\) which is treated as a constant over one time step for the thermal-flow computation (to average out high-frequency oscillations of the electric field from the electromagnetic heat source, q), and defined as:
$$\bar{E} = \frac{1}{{N_{\rm t}}}{\sum\limits_{\tau = 1}^{N_{\rm t}} {E^{\tau}}},$$
(21)
where Nt is the number of time steps in each period of the microwave and Eτ is the instantaneous E field. The details of the numerical scheme used in this study are given in ref. [17].

3 Results and discussion

Table 1 shows electromagnetic and thermo-physical properties used in computations. The temperature-dependent data for the dielectric constant and loss tangent are plotted versus temperature in Fig. 2.
https://static-content.springer.com/image/art%3A10.1007%2Fs00231-006-0105-y/MediaObjects/231_2006_105_Fig2_HTML.gif
Fig. 2

Temperature dependence of the dielectric properties: a dielectric constant, ɛ′; b loss tangent, tan δ

3.1 Heating patterns for liquids with different dielectric properties

Figure 3 displays steady-state temperature distributions at the outlet of the applicator for the three different liquids as well as the corresponding electromagnetic heat generation intensity distributions. The heat generation intensity is also shown at the applicator outlet; however, its dependence on z (different from that of the temperature) is insignificant. This figure illuminates the interaction of the electromagnetic field and forced convection in the liquid. It illustrates that the temperature and electromagnetic heat generation intensity are nonuniformly distributed at the applicator outlet for all liquids. For the apple sauce, from Fig. 3-a(1), the electromagnetic heat generation intensity distribution at the applicator outlet exhibits two well-defined peaks near the central area. Since the electromagnetic heat generation intensity determines the temperature distribution, Fig. 3-a(2) depicts two hot spots around the central area and four hot spots at the corners of the applicator tube. Similar behavior of electromagnetic heat generation intensity and temperature can be observed in Fig. 3-b(1–2) for the skim milk, while the intensities of the hot spots near the center of the tube are smaller. For the tomato sauce, from Fig. 3-c(1–2), the peaks of the electromagnetic heat generation intensity disappear and there are no hot spots of the temperature in the central area of the applicator tube, only four hot spots at the corners. From this analysis it can be concluded that although the difference of dielectric properties of these three liquids is not great (see Fig.  2), it causes a significant difference in their heating as they pass through the microwave cavity. In order to evaluate the uniformity of the temperature distribution quantitatively, a standard deviation of temperature is introduced as:
$$L = {\sqrt {\frac{1}{A}\iint {{\left({T - T_{\rm m}} \right)}}^{2} {\rm d}A}},$$
(22)
where A is the area of a cross-section perpendicular to the streamwise direction, and Tm is the mean temperature. Figure 4 displays the standard deviation of temperature versus the streamwise locations for the three liquids considered in this study. A larger standard deviation of temperature corresponds to a more nonuniform temperature distribution. Figure 4 shows that at the outlet of the applicator, the apple sauce has the most uniform temperature distribution and the tomato sauce has the most nonuniform temperature distribution. This can be attributed to the difference of their dielectric properties. From Fig. 2, one can see that the dielectric constants ɛ′ of the three liquids are almost the same; however, the loss tangent, tan δ, is significantly different for these three products. Consider the heat generation intensity equation, (14), if the frequency of the microwave and the dielectric constant are held constant. In this case the electromagnetic heat generation is proportional to the loss tangent. Thus, being a high loss liquid (with a large loss tangent), the tomato sauce absorbs more microwave energy than the skim milk (which is characterized by a medium loss tangent) and the apple sauce (which is characterized by the smallest loss tangent of the three products), which results in the peak value of the electromagnetic heat generation intensity; also, the temperature range (TmaxTmin) for the tomato sauce is larger than that for the skim milk and the apple sauce. The result is that the tomato sauce has the largest standard deviation of temperature at the outlet (the most nonuniform temperature distribution at the outlet) and the apple sauce has the most uniform temperature distribution.
https://static-content.springer.com/image/art%3A10.1007%2Fs00231-006-0105-y/MediaObjects/231_2006_105_Fig3_HTML.gif
Fig. 3

Electromagnetic heat generation intensity and temperature distributions at the outlet of the applicator: [a(1)–c(1)] electromagnetic heat generation intensity distributions (W/m3) for the apple sauce, skim milk, and tomato sauce, respectively; [a(2)–c(2)] temperature distributions (°C) for the apple sauce, skim milk, and tomato sauce, respectively

https://static-content.springer.com/image/art%3A10.1007%2Fs00231-006-0105-y/MediaObjects/231_2006_105_Fig4_HTML.gif
Fig. 4

Standard deviation of the temperature distribution for the apple sauce, skim milk, and tomato sauce

3.2 Effect of different locations of the applicator on the heating process

This section discusses the effect of positioning the applicator at different locations in the microwave cavity. As previously mentioned, Fig. 3 shows the heat generation intensity and temperature distributions at the outlet of the applicator for the three liquids when the applicator is located at the center of the microwave cavity (which is considered as its base position). Figure 5 shows similar distributions with the applicator displaced by 141 mm forward in the X-direction from the base position (see Fig. 1). A comparison between Figs. 3 and 5 suggests that there is a great difference between the distributions and magnitudes of the electromagnetic heat generation intensity and temperature for the two cases. In particular, in Fig. 3-a(2), which shows the temperature distribution at the outlet of the applicator for the apple sauce, there are two hot spots near the center of the applicator while in Fig. 5-a(2) there is only one hot spot positioned almost exactly in the center of the applicator. Also, the peak value of the temperature in Fig. 5-a(2) is about 1.7 times greater than that in Fig. 3-a(2). Comparing Figs. 3-a(1) and 5-a(1), one can find similar differences in the distributions of the electromagnetic heat generation intensity. This proves the significant effect of positioning the applicator tube in the microwave cavity on heating the product.
https://static-content.springer.com/image/art%3A10.1007%2Fs00231-006-0105-y/MediaObjects/231_2006_105_Fig5_HTML.gif
Fig. 5

Effect of the location of the applicator on heating the product: [a(1)–c(1)] electromagnetic heat generation intensity (W/m3) distributions at the outlet for the apple sauce (a), skim milk (b), and tomato sauce (c), respectively; [a(2)–c(2)] temperature (°C) distributions at the outlet for the apple sauce (a), skim milk (b), and tomato sauce (c), respectively, for the applicator having 141 mm off the original location in the X-direction

3.3 Effect of the size of the applicator

The effect of the size of the applicator tube on heating patterns for the apple sauce is discussed in this paragraph. Figure 6 shows the electromagnetic heat generation intensity and temperature distributions of the three liquids at the outlet of the applicator with dimensions of 60×60×124 mm. This applicator is larger than the applicator of the base size (46×46×124 mm) although the enlarged applicator is positioned similarly in the center of the cavity, at the same location as the applicator of the base size. Comparing Figs. 3 and 6, the distributions of the electromagnetic heat generation intensity and temperature are greatly affected by enlarging the applicator. For example, in the skim milk the peak value of the electromagnetic heat generation intensity and temperature in the applicator of a larger size are twice and three times larger than those in the applicator of the base size, respectively. This can be attributed to the fact that the larger applicator has a larger cross-sectional area allowing for more absorption of the microwave energy; also, for the same inlet volume flow rate, the flow in a larger applicator has a lower flow rate thus allowing fluid particles have larger residence time, which makes it possible for them to absorb more microwave energy.
https://static-content.springer.com/image/art%3A10.1007%2Fs00231-006-0105-y/MediaObjects/231_2006_105_Fig6_HTML.gif
Fig. 6

Effect of the size of the applicator on heating the product: [a(1)–c(1)] electromagnetic heat generation intensity (W/m3) distributions at the outlet for the apple sauce (a), skim milk (b), and tomato sauce (c), respectively; [a(2)–c(2)] temperature (°C) distributions at the outlet for the apple sauce (a), skim milk (b), and tomato sauce (c), respectively, for the applicator size of 60×60×124 mm

4 Conclusions

A numerical model is developed to simulate forced convection in a rectangular duct subjected to microwave heating. The results reveal a complicated interaction between electromagnetic field and convection. Dielectric properties of a liquid flowing in the applicator tube play an important role in the heating process. Even a small difference in dielectric properties can result in a completely different heating pattern. It is also found that the electromagnetic heat generation intensity and the temperature distributions in the liquid are sensitive to the size and the location of the applicator. Both the magnitude and the distribution of the electromagnetic heat generation intensity and the temperature depend strongly on the geometry of the microwave heating system. This illustrates the importance of numerical modeling to design an optimal microwave heating device.

Acknowledgements

The authors acknowledge with gratitude a USDA grant that provided support for this work and the assistance of the Food Rheology Laboratory at North Carolina State University. The calibrations of the fluid consistency coefficients and the flow behavior indexes for the non-Newtonian liquids considered in this study by Ms. S. Ramsey are greatly appreciated.

Copyright information

© Springer-Verlag 2006