非結(jié)構(gòu)化網(wǎng)格中圓管流動(dòng)的傳熱計(jì)算

1、 Duct Flow Heat Transfer Calculations Using UnstructuredGridsZhang Junbo 1，Zhang Min 1，John C. Chai2，Xu Bin 1，Liu Shaopei 12School of Power Eng.，Nanjing University of Science & Technology，Nanjing (210094) School of Mechanical and Aero spacing Eng.，Nanyang Tech. University，Singapore (639798)Abstr

2、act 1An interesting and important class of the applications of the heat conduction equation is the fully developed laminar flow and heat transfer in ducts. In this paper, the velocity and energy governing equations that are similar with the heat conduction equation were solved using unstructured gri

3、ds in duct geometries and several thermal boundary conditions. Meanwhile, the solutions were analyzed and compared with the results using structured grids so that the procedures are evidenced exactly and actually.Keywords: Structured/unstructured grids/meshes，heat conduction equation，fully developed

4、 laminar flow1. IntroductionThe problem of fully developed laminar flow, forced convection heat transfer in circular ducts is primary importance to the practical design of a wide variety of heat exchangers1-7。In this paper, the velocity and energy governing equations that are similar with the heat c

5、onduction equation were solved using unstructured grids in duct geometries and several thermal boundary conditions. Meanwhile, the solutions were analyzed and compared with the results using structured grids so that the procedures are evidenced exactly and actually.2. Governing EquationIn Cartesian

6、coordinates, the steady state general forms of the conservation equations are in tensor notation as below,(u)(v)(w)+=+S +xyzxxyyzz(2.1)where, is the dependent (scalar) variable, it may be velocity u（X direction），v（Y direction）, w（Zis density of fluid, S is source term in single volume, is the direct

7、ion）, and temperature. corresponding diffusion coefficient for .Velocity Field:In a duct flow, the main (axial) flow direction z can be divided into developing and fully developed regions as shown in Figure 1. In the developing region, the velocity distribution changes with z. The three velocity com

8、ponents are w = w(x, y, z), u = u(x, y, z), and v = v(x, y, z). In the fully developed region, the velocity distribution is invariant with z. Therefore, w = w(x, y), u = u(x, y), and v = v(x, y). In this paper, a special class of fully developed flows is considered. In this class of simple fully dev

9、eloped flows, no cross-stream velocity exists. Therefore, w = w(x, y), u = 0, and v = 0. The z-momentum equation for simple fully developed flows is2w2wdpµ=0 +x2y2dz(2.2)where µ and p are the fluid viscosity and pressure, respectively. The velocity on the pipe surface is set to zero. The f

10、riction factor f is calculated from- 1 - dp2Dh f=2dzwave2dp2Dh fRedzwµave(2.3a) (2.3b)where , Dh, and wave are the fluid density, the pipe water diameter, and the average velocity, respectively. The Reynolds number Re is defined asRewaveDhµ (2.4)The product f Re is a constant for a laminar

11、 fully developed flow. Its value is 64 for the circular pipe and 96 for the parallel-plate channel. Most duct flows have values between 64 and 96.Figure 1. Developing and fully developed regionsTemperature Field:With a fully developed velocity field, the temperature (T) becomes independent of z only

12、 under certain boundary conditions, such as different temperatures of two walls (see Figure 2). Then, the heat flow enters the duct at one wall and an equal amount leaves at the other. There is no net heat transfer to the flowing fluid. The computational problem is that of pure heat conduction with

13、a zero source term. Such fully developed heat transfer problems are “uninteresting” and have only limited practical utility.Figure 2. Developing and fully developed temperature fieldsWhen there is net heat transfer to the fluid, its temperature T does not become independent of z; but a thermal fully

14、 developed region can be defined for which some dimensionless temperature becomes- 2 - invariant with z and the heat transfer coefficient becomes constant. The energy equation in the thermally developed region isTkxxT+kyyTc=0 pz(2.5)The dimensionless heat transfer coefficient is called the Nusselt n

15、umber. Similar to the friction factor, it can be defined in many ways. One definition isNuhDh k(2.6)where h and k are the heat transfer coefficient and the thermal conductivity, respectively. The heat transfer coefficient is defined ashq TwTb(2.7)where q, Tw, and Tb are the wall heat flux, the wall

16、temperature, and the bulk temperature, respectively. The bulk temperature is defined ascpwTdA TbcpwdA(2.8)For simple fully developed flows, the Nusselt number is constant and independent of the Reynolds number and the Prandtl number in the thermally developed region. The energy equations for two typ

17、es of thermal boundary conditions are discussed next.3. Laminar Flow In A Circular PipeFully developed laminar flow in a circular duct is the most basic problem. The friction factor f depends on the Reynold number Re. The Nusselt number Nu depends on the thermal boundary condition. For laminar flow

18、in round duct, the product f Re is constant and equals to 64. Two types of thermal boundary conditions, namely uniform heat flux qw and uniform temperature Tw are examined (see Figure 3). The Nu for the uniform heat flux qw and the uniform temperature Tw conditions are 4.364 and 3.657, respectively.

19、 The following values are used in this problem.µ=1,dp=1,=1,D=2 (3.1) dzTwTbTTwTbzz(a) (b)Figure 3. Wall and bulk temperatures (a) with constant heat flux and (b) with uniform wall temperature condition- 3 - . For this type of problem, all temperatures rise linearly with z at the same rate. Ther

20、eforeTdTwdTb=constant zdzdzFrom energy balance (3.2) dTbqP =dzcpwaveA(3.3)where A and P are the cross-sectional area and the perimeter, respectively. From Equation (3.2), the difference between the wall and the bulk temperatures is a constant in the fully developed region. Therefore, any convenient

21、temperature can be assigned as the wall temperature. The following values are used in this part of the problem.k=1,q=5,cp=1 (3.4) The dimensionless temperature used in the presentation of the results is=k(TwT)k(TwT)= 4qDcpwaveD2bdz(3.5) The Nusselt number is obtained fromNuhDhDhDhqQw= kkTwTbk(TwTb)P

22、(3.6) same as up one. For this type of boundary condition, the dimensionless temperature defined in Equation (3.7) is independent of z in the thermally fully developed region.=TwT TwTb(3.7) The Nusselt number is same as relation (3.6). Whereq=From Equation (3.7) cpwaveAdTbPdz (3.8) dTT=b zdzThe foll

23、owing values are used in this calculation. (3.9) k=1,dTb=1,cp=1,Tw=0 dz(3.10)A total of 430 and 1660 triangular control volumes (Fig. 4) are used to model this problem. The dimensionless velocity w/wave and the dimensionless temperature distributions are shown in Fig. 5. It can be seen that both the

24、 course and the fine grids produce reasonable solutions.Table 1 shows quantitative comparisons of f Re, Nu, and w/wave with the exact solutions. The errors (the quantities inside the parenthesis) are calculated fromE=YnumYexactYexact×100% (3.11)where Y stands for the variable under consideratio

25、n. It can be seen that all the numerical solutions are accurate to within 1% of the exact solutions.- 4 -430 control volumes 1600 control volumesFigure 4. Computational grids for a pipeFigure 5. Distributions of (a) the dimensionless velocity w/wave, (b) the dimensionless temperature for the constan

26、twall heat flux condition and (c) the dimensionless temperature for the constant wall temperature situationTable 1. f Re Nu、w/waveExact f Re w/wave Nuq* Nut* 64 2 4.364 3.657 430 64.293(0.46%) 1.992(0.4%) 4.395(0.71%) 3.659(0.055%) 1660 64.288(0.45%) 2.0004(0.02%) 4.374(0.23%) 3.658(0.027%)4. Duct F

27、low With Tempearture-Dependent ViscosityFor many fluids, the viscosity depends on the temperature. If the variation of the viscosity is substantial, it will have a significant effect on the velocity distribution in a duct. A problem that a lead to a fully developed flow is examined (Fig. 6). If any

28、of the thermal boundary conditions used in the previous examples are adopted, the velocity field with a temperature-dependent viscosity will not become fully developed. The reason is that these boundary conditions lead to a continuous increase or decrease of the fluid temperature in the z direction.

29、 As a result, the viscosity field will vary in the z direction. Therefore, it is not possible to get a velocity distribution that does not change in the z direction under such thermal boundary conditions.What is needed is a temperature variation in the duct cross section, but no temperature change i

30、n the z direction. This is the “uninteresting” type of fully developed heat transfer. Although such temperature fields are uninteresting for convective heat transfer, they are suitable candidates for studying the effect of the temperature-dependent viscosity.- 5 -Figure 6. Schematic of problemThe th

31、ermal boundary condition is given by a constant temperature T1 on the inner boundary and the sidewalls of the duct, and a constant temperature T2 on the outer boundary as shown in Fig. 6. The fluid viscosity µ is a linear function of the temperature T. Thusµ=1.0+0.04Tk=1,µ=1,cp=1,=/3,

32、Rin=0.2,Rout=1T1=100,T2=0,dp=1,dzT=0 z(3.12) The governing equations for the problem are same up one. The following values are used in this demonstration, (3.13) (3.14)Figure 7 shows the four computational grids used in this study. Figures 8 and 9 show the effect of gird refinement on the velocity f

33、ield and the temperature field, respectively. It can be seen from both figures that the temperature field is not affected by the spatial grids. The velocity field however is quite sensitive to the spatial grids. This is because the viscosity depends on the temperature.(a) 20×20 30×30- 6 -(

34、b) 400 control volumes 900 control volumesFigure7. Computational grids: (a) structured grids and (b) unstructured grids20×20 30×30Figure8. Structured grids: effect of grid refinement on (a) velocity and (b) temperature fieldsFigure9. Unstructured grids: effect of grid refinement on (a) velocity and (b) temperature fields5. CloseTwo test problems for the fully developed flow and heat transfer in ducts were examined. The results show that the proposed procedure was capable of producing accurate solutions.- 7 - R