Problem Statement       

Consider an array of heated tubes submerged in a vessel with fluid flowing past them. Neglecting end effects, the flowfield can be assumed 2-D in planes with normals parallel to the tube axes. Further, for modest fluid onset velocity, a steady state solution can be sought. Finally, because of resultant planes of symmetry, the flowfield solution domain W can be the small region between the hatched lines shown in Figure 1.

 

 

 

 

 

 

 

Figure 1. Schematic of the DE simplified analysis problem.

      This problem statement requires that DM, DP and DE all be addressed. In FEMLAB nomenclature, these conservation laws are expressed as

            DM:                                                                                                           (1)

            DP:                                                                 (2)

            DE:                                                                   (3)

 where r is density,  is dynamic viscosity, F is a body force, cp is specific heat at constant pressure, k is thermal conductivity, and Q is a heat source.

      Your classical undergraduate coursework characterizes heat transfer modes via non-dimensional groups. For the typical case of onset flow at small Mach number (Ma < 0.3), the fluid density can be assumed constant (ro) everywhere except in the body force term F in (2). Then one invokes the Boussinesq model for thermal buoyancy effects, whereby 

                                                                                                                           (4)

 Assuming all other thermal data are constant, and that the flowfield is laminar, the non-dimensional forms of DM, DP, DE equivalent to (1)-(3) are 

            DM:                                                                                                           (5)

            DP:                                                  (6)

            DE:                                                                          (7)

 The definitions for the non-dimensional groups Re (Reynolds number), Gr (Grashoff number), and Pr (Prandtl number) in terms of FEMLAB variables are 

                                                                                                            (8)

           

 wherin the reference scales are: U Þ velocity, D Þ tube diameter, b Þ thermal coefficient of expansion (Tabs)-1,  and g Þ gravity constant, and the non-dimensional (potential) temperature definition is 

                                                                                                                  (9)

       Finally, the convection heat transfer coefficient h, in the BC for DE, becomes replaced by Nusselt number Nu, and natural convection sometimes employs the Rayleigh number Ra. These definitions are  

(10)

 
     

           

  which completes the non-dimensional group definitions for the problem statement.

      The heat transfer modes as defined in your textbooks are

forced convection for 

            natural convection for                                                                             (11)

            mixed convection for 

The resultant convective heat transfer is correlated by Nusselt number, Nu. Available literature correlations for natural and forced convection, NuN and NuF, are: 

            ; RaD £ 1012                                              (12)

            ; RePr > 0.2              (13)

        For mixed convection, NunM º NunF  ± NunN, and n is 4 for a cylinder. The plus sign is used when the tube temperature exceeds the fluid temperature and vice versa.

      The conservation law system (5)-(13) describes the analysis problem at hand. Note that any consistent sets of units are admissible, and the input to FEMLAB is dimensional. Hence Re, Pr, Gr, and Nu will be computed in the code from input data. 

Setting up the problem in FEMLAB

The base case data specification is: 

To (inlet temperature of the fluid)

298K

T1 (outside temperature of the tube)

303K

D (outside diameter of the tube)

0.005m

ro (density of the fluid)

1e3kg/m3

*  (dynamic viscosity of the fluid)

1e-3kg/ms

cp (heat capacity of fluid)

4.2e3J/kg×K

kc (thermal conductivity of fluid)

0.6W/m×K

b (volume expansion coefficient)

0.18e-3 K-1

g (acceleration due to gravity)

9.8m/s2

 In the subdomain settings define the buoyancy body force Fy as

            Fy = rogb (T-To)

      The steady incompressible Navier-Stokes state variable solution {Q} is established via GWSh + qTS implementation using the FE k = 1 natural coordinate basis on a mesh of triangles.