Mathematical modeling of air duct heater using the finite difference method

Mathematical modeling of air duct heater using the finite difference method In this research, mathematical modeling of a duct heater has been performed using energy conservation law, Stefan-Boltzman law in thermal radiation, Fourier's law in conduction heat transfer, and Newton's law of cooling in convection heat transfer. The duct was divided to some elements with equal length. Each element has been studied separately and air physical properties in each element have been used based on its temperature. The derived equations have been solved using the finite difference method and consequently air temperature, internal and external temperatures of the wall, internal and external convection heat transfer coefficients, and the quantity of heat transferred have been calculated in each element and effects of the variation of heat transfer parameters have been surveyed. The results of modelling presented in this paper can be used for the design and optimization of heat exchangers.


INTRODUCTION
The importance of optimization and design of heat exchangers in industrial fi elds is known to engineers. Today's industrial designers are making a wide effort to achieve new economic, precise, optimized and high yield methods. The procedures of designing the heat exchangers are different depending on the heat exchangers applications. The fi nite difference is one of the confi dent, easy and useful methods for designing and solving heat transfer problems 1 . In addition, using the numerical method, including the variable physical properties for the design of heat exchangers enhances the quality and precision of the calculation steps. In simple and ordinary methods, the physical properties are estimated in T f (fi lm temperature which is defi ned as an arithmetic average of input and output temperatures of an element) or caloric temperature for more convenience 2 . It is important to consider this fact that physical properties are changing through the length of heat exchanger and it is not reasonable to estimate the physical properties in average or constant caloric temperature 3-7 . In this study, the numerical method is applied to analyse the heat transfer performance of a duct heater. Duct heaters are mainly used for transferring the heated or even cooled air, particularly in air condition systems and central heater of buildings and industries. Subsequently, the length of the heat exchanger has been divided into some intervals and furthermore, the physical properties of each segment have been estimated based on the fi lm temperature independently. This numerical method (fi nite difference) has made new perspectives for accurate design of heat exchangers which leads to increase the operation yield and improves the economics factor simultaneously 8 .

MODELING
For mathematical modeling, an air duct heater has been proposed for heating the environment. For simplicity, the shape of the duct heater is considered as cylinder and convection and radiation heat transfers to the environment have been considered from each element. The schematic of the considered geometry and a segment of channel with the length of x and cylinder side perimeter P are shown in Fig. 1. By writing the total energy balance equation for the considered segment of the channel, Eq. (1) is obtained: (1) Where is air mass fl ow rate and hi is defi ned as an internal heat transfer coeffi cient which is related to Reynolds number according to many empirical correlations existing in this fi eld. In this study, hi has been estimated using Eqs. (2) and (3) for the laminar and turbulent fl ow, respectively 9 : (2) (3) Physical properties required in the above mentioned equations have been estimated at T m,a indicating the air bulk temperature. Energy balance on the wall of duct channel is defi ned as Eq. (4): (4) By substitution of the Newton's cooling law and Stefan-Boltzmann radiation law into Eq. (4), the following equation is obtained: where T m,w,o and T m,w,i are outside and inside wall  (6) The outside heat transfer coeffi cient for the outside duct wall that is exposed to stagnant air with environment temperature value is obtained by Eq. (7) 2 : By substitution of Eq. (7) into Eq. (5), the following equation has been obtained: (8) Note that Eq. (9) must be used for calculating and estimating the outside temperature of each segment, which leads to Eq. (9): (9) Briefl y speaking, the following algorithm is used in a computer based programming for the calculation of heat transfer rate: 1) An element length (x) is selected. It should be mentioned that the accuracy of this method strongly depends on the number of segments or the selected length of each segment. Smaller length of elements will increase the accuracy of the results.
2) Calculation will start with the air inlet position which is kept as (x = 0) and all the physical properties for start point must have been estimated in T bulk-inlet .
3) Calculate the Reynolds number; select the proper Eqs.  10) and (11) are subsequently substituted into Eq. (7). If it is not consistent, return to step 4 for swapping the assumption. 6) Solve Eq. (9) for T m+1,a . This value will be considered as the air inlet temperature in the next segment. 7) Calculation steps (1-6) are subsequently repeated for the next segments until the end of the duct heater. However, heat fl ux in each segment can be calculated using Eq. (13): (13) In Table 1, the geometry of the considered channel has been exhibited. For accuracy and simplifying in calculation steps, MATLAB computer based programming has been used.

Effect of various parameters
The effects of some important parameters such as fl ow rate of inlet air fl ow, internal diameter of the channel, and channel construction material on the heat transfer ability of the duct heater have been studied in separate following parts of this paper.

Flow rate of inlet air
The fl ow rate of the inlet air was changed in order to make two different fl ow regimes (namely laminar and turbulent fl ows). It has been particularly done for investigating on the values of heat transfer coeffi cients in two different regimes of air fl ow. Figs. 2-7 show the results obtained due to the air inlet variations.
As shown in Fig. 2, increasing the fl ow rate of inlet air, raises the temperature of the air through the effective length of channel. However, the temperature of the air is reduced with increasing the length of the duct. Table 2 typically represents the status of T m,w,i besides the T m,w,o with variation of air fl ow rate through the length of duct. Increasing the mass fl ow rate of the air raises the internal heat transfer coeffi cients of the air in the channel length. These data are given in Fig. 3.
Furthermore, the infl uence of air mass fl ow rate on the external heat transfer coeffi cient has been given in Fig.  4, which implies that increasing the mass fl ow rate of water, the external heat transfer coeffi cient is increased in contrast to this value is decreased through the length of channel as well as heat fl ux will be increased with raising the mass fl ow rate of air and will be decreased with length of duct which have been truly shown in Fig. 5.

Effect of internal diameter of the channel
To investigate the effects of internal diameter on other parameters, different values of internal diameter have been employed at air fl ow rate 2.3kg.s -1 . For better understanding the results are given in the next coming fi gures. Air output temperature is decreased with increasing of the internal diameter and is decreased through the length of the channel, too. Figs. 6-9 present the infl uence of variations of diameter on: air temperature, internal heat transfer coeffi cient, external heat transfer coeffi cient and heat fl ux respectively.

Channel material of construction
In this section, two channels with two different materials of construction are simulated in order to evaluate the effect of channel material on its heat transfer performance. The only properties which have infl uences on the heat transfer from this duct heater are emissivity and conductivity of the metals. Table 3 summarized the properties of the two channels which were used for the simulation in this paper.
As shown in Fig. 10, temperature drop in each element of the oxidized copper channel is less than that in the oxidized iron channel. The main reason for this Table 3. Emissivity and conduction coeffi cient of selected materials  Moreover, because of that, it is concluded that the internal and external wall temperature of copper channel is less than iron specimen due to the high conduction heat transfer coeffi cient of copper that enhances the rate of heat transfer from the internal wall towards outside. Additionally, higher emissivity of oxidized copper in compared to oxidized iron leads to more radiative heat transfer between the outside wall and ambient, subsequently the outside wall temperature is smaller in comparison with the oxidized iron channel.
As shown in Figs. 11-12, the internal and external convection coeffi cient of oxidized copper in each element of the channel is higher than that of the oxidized iron channel. In fact, the outside wall temperature of the oxidized copper channel is smaller than for oxidized iron channel and according to Eq. (7) if the outside temperature of the channel decreases, the outside convection coeffi cient of the channel decreases, too. Through the channel, according to Eq. (2), the Nusselt number is proportional to the multiplication of the Reynolds number and the Prandtl number. Furthermore, reducing the temperature through the length of channel results in increasing the air density, subsequently leads to decreasing its viscosity, too. Accordingly, the Reynolds number will increase through the length of the channel. Meanwhile, the Prandtl number is decreased with decreasing the temperature. Therefore, the heat transfer coeffi cient through the channel is decreased. In addition, this infl uence is explicitly sensible for the oxidized copper channel in comparison to the oxidized iron channel and cause the reduction of heat transfer coeffi cient of the oxidized copper channel.
Also, heat fl ux in each element of the oxidized copper channel is higher in comparison to the oxidized iron channel. The oxidized copper emissivity and conduction coeffi cients are higher than for oxidized iron. The internal and external heat transfer coeffi cients of the oxidized iron are higher than the oxidized copper. At the outlet of these two channels, according to Figs. 11-12, heat transfer coeffi cients are closer to each other and the infl uence of emissivity and conduction coeffi cient lead to large values of transferred heat. But at the outlet of the channel, decrease of heat transfer coeffi cient in the oxidized copper is more than that of iron channel. It is the major reason of the identifi cation of heat fl uxes for two different materials of channels. The results of the obtained heat fl uxes are given in Fig. 13. Table 4. Comparison between the results obtained using LMTD and fi nite difference methods  Table 4 gives a rough comparison between LMTD (Log Mean Temperature Difference method) and the fi nite difference mathematical method in estimating the heat transfer rate. As shown, there is maladjustment between the LMTD obtained results and the fi nite difference method. One of the major reasons of this difference refers to the estimating condition of variable physical properties. There is an important point that the physical properties are not constant throughout the heat exchanger which was mentioned before and this point must be considered. Since in our study, the variable physical properties are considered, the values of Reynolds are not constant. Therefore the range of Re and Pr are reported in Table 5.

CONCLUSIONS
In this research, the effect of some parameters such as mass fl ow rate, the internal diameter of the channel and the material quality of the channel have been directly or indirectly investigated on heat transfer values and parameters. The results show that, with increasing the fl ow rate values, air temperature will increase so do all the parameters, such as the inside wall temperature and the outside wall temperature and all the heat transfer coeffi cients. Also, reducing the channel diagonal leads to an increase of all the properties, such as heat transfer coeffi cient and air stream temperatures. Likewise the quality and the material of the skin channel composer, has an undeniable infl uence on heat transfer values, furthermore the heat fl ux will increase. The effective length of the duct channel has no effect on the internal heat transfer coeffi cient, too. Also, the materials which were selected to be used for designing of the air duct channel were tested separately for measuring the heat transfer coeffi cient.