Investigation of the heat transfer coefficient for a red clay brick

There is a need for the development of materials for thermal isolation. Heat energy is used in different ways, for example, for house warming or production of electricity from solar power plants. However, to use the heat energy efficiently isolating materials are needed. There are different materials available to isolate houses and heat storage containers. However, those materials are expensive and some are over engineered. The best available material for thermal isolation would be air that has the thermal coefficient λair=0.02446WmK$\lambda _{air}=0.02446\frac{W}{mK}$ . Thus, materials that trap a large amount of air are good isolators. On the other hand, some isolating materials are harmful for the environment, for example, glass wool, asbestos. An alternative are red clay bricks because red clay is a natural product that is present everywhere on the planet. But the thermal coefficient of red clay bricks is high λclay=0.6WmK$\lambda _{clay}=0.6\frac{W}{mK}$ . To reduce the thermal coefficient cavities can be added in the brick. The cavities trap air and reduce the thermal coefficient of the brick.


F I G U R E 1
The geometry of the brick (left), where ,  and ℎ are the width, length and height of the brick.The width of the cavities (holes) is   , while it is assumed that the cavities have all the same length and width.The heat flux through a cross section of hollow brick with the cavities is sketched on the right.The cavities are white and solid walls are gray colored.

INTRODUCTION
Red clay bricks are under investigation to improve the construction (e.g., [1]) and heat transfer properties.Because there is an abundance of clay on the planet and red clay bricks are cheap to produce, we can find many up to date research regarding the heat transfer of red clay brick, for example, the combination of solid (red clay) and liquid (air) are used to improve the heat isolation properties of the bricks [2, 3] or alternative materials for example agricultural wastes are investigated to reduce heat transfer [4].Chen and Liu [5] presented an optimized construction of a heat saving hollow brick.Hollow bricks are employed to increase the thermal isolation because air has a smaller thermal conductivity coefficient than clay.
Costa [6] investigated the heat transfer of a two dimensional hollow brick considering radiation heat transfer.It was shown that the radiation heat transfer has to be accounted for in numerical simulations.Jamal et al. [7] investigated heated two dimensional hollow red clay brick walls, while Ouakarrouch et al. [8] investigated the heat transfer of a three dimensional hollow brick.
Here we simulate the heat transfer through a hollow red clay brick.Thus we have two different domains, where the red clay walls of the brick are modelled as solid and the air that is enclosed in the brick holes is modelled as fluid.The holes are considered to be cavities.To solve the fluid flow in the cavities we employed the Boundary-Domain Integral method.This method is based on Green's second identity.A more detailed explanation on the Boundary-Domain Integral method is presented in [9].The drawback of the method is the computational complexity which scales quadratic ( 2 ), where  is the number of unknowns in the domain.In order to reduce the computational cost fast methods are employed, for example, the Fast Multipole method [10], -matrices in combination with the Adaptive Cross Approximation [11,12] or the  2 -methodology presented by Börm in [13].For this research we employed  2 -matrices to accelerate the BDIM, where the integral kernel is approximated with the Lagrange interpolation.In [14], the impact of the  2 -approximation on the accuracy of the proposed method was investigated.It was shown that the approximation reduces the computational complexity to almost linear complexity ().

PROBLEM SETTING
The geometry is a square hollow brick.Two opposite side walls of the brick are heated at different temperatures   and   , respectively.The brick is divided into solid parts, where heat is transferred with conduction and parts with holes (considered as cavities), where heat is transferred with conduction, convection and radiation.The geometry of the hollow brick is presented on the left panel in Figure 1.The width was varied from 5.0  to 20.0 , the height was 50  and length 38 .On the right in Figure 1 we present the heat flux through the cross section of the hollow brick.
For the solid part the Fourier constitutive model can be employed.To determine the heat transfer coefficient of the cross section it is considered that the heat flux is not constant  * ≠  * * in the cavities and the energy equations are formulated to solve the unknown heat transfer through the cross section for  cavities where   the heat transfer of solid brick and    ,    are the hot and cold wall heat transfer coefficients.The thickness of the solid   and width of the cavities   is the same   =   .The fluid temperature   is the average temperature between the hot and cold walls of the cavities   =   + +1

2
. From this we can form a linear system of equations for  cavities (2) The unknowns are the temperatures at the solid walls   .However, the heat transfer coefficients    ,    are also unknown.To obtain them the fluid flow in the cavity is solved.The non-dimensional velocity-vorticity formulation of the Navier-Stokes equations and the energy equation are is the Prandtl number.The first equation in (3) is the modified Helmholtz kinematic equation that is employed to solve the velocity ⃗  for a guessed vorticity ⃗ , while the time is not actual physical time but false time to increase the stability of the computation.A detailed explanation on the false time is presented in [14].The second is the transport vorticity equation and the last is the energy equation.A more detailed explanation of the procedure to obtain this form of the equations can be found in [9].The equations in (3) were cast into integral equations to solve the equations with the Boundary-Domain Integral method.For that we consider a domain Ω with the boundary Γ and the normal vector ⃗ .Firstly we present the integral form of the modified Helmholtz kinematic equation (i.e., first equation in (3)) where  is an under-relaxation parameter and  * ( ⃗ , ⃗ ) is the modified Helmholtz fundamental solution, where ⃗  is the source and ⃗  the collocation point.To obtain the boundary vorticity additionally the tangential form of the kinematic modified Helmholtz equation where  = 1, 2, 3 and    is the flux component of the vorticity.The dominating differential operator in the transport vorticity equation is the Laplacian which allows to employ Green's second identity with the Laplace fundamental solution  *  ( ⃗ , ⃗ ) to obtain the integral form (6). Lastly we present the energy equation (i.e., the third equation in (3)) where   is the dimensionless heat flux.To solve the unknown velocity, vorticity and temperature the vorticity is guessed in the volume.With this the values of the boundary vorticity are computed with (5).Next, the obtained vorticity is employed to solve the velocity in (4).With the known velocity the remaining unknown vorticity is obtained from (6).Lastly, the temperature in the volume is solved with the energy equation ( 7).The process is repeated until a prescribed residuum is reached.A detailed description on the iterative process can be found in [9].When the fluid flow is solved the heat flux coefficients    ,    are obtained from the Nusselt number  = , where   is the fluid thermal conductivity.Thus the unknown temperatures can be solved using (2).With the known temperatures the Rayleigh numbers for the next iteration are computed.This allows then to solve the fluid flow in the new iteration.The computation is repeated until a prescribed residuum is reached.

Heat Radiation
The heat transfer on the side walls of the cavities is not adiabatic.It is considered that the heat is transferred with radiation to the fluid [15] .All the surfaces of the cavity are assumed to be diffuse-gray and opaque, with a uniform emissivity .The air that is enclosed in the cavity is fully translucent and does not influence the radiation.Heat flux leaving the surface is evaluated as the difference between the net energy and outgoing radiation where Δ  =   −  +1 is the temperature difference between the hot and cold wall of the cavity.The outgoing radiation  is defined as the sum of the incoming radiation   from other surfaces and the net energy of a gray body while the incoming radiation is given by 4  Γ , where ⃗  = ⃗  − ⃗ , and ⃗ ( ⃗ ), ⃗ ( ⃗ ) are the normal vectors on the surfaces.Inserting the above integral into (9) and rearranging the equation gives With the outgoing radiation  the heat flux   is determined using (8).This is employed for the heat flux boundary condition on the side walls of the cavity.

Discretization
The domain Ω and the boundary Γ is discretised by domain cells and boundary elements, respectively.On the mesh usual polynomial shape functions ( ⃗ ) and Φ( ⃗ ) are defined to approximate the unknowns in the above given set of integral equations.The discrete form of ( 4) is where [], [ ⃗   ], [ ⃗ ] and [] are the matrices obtained from the discretization.To obtain the boundary vorticity the above equation is multiplied with the normal vector ⃗  and the vector ⃗  is split into the boundary and inner domain part }  and rearranging the equation gives where { ⃗ }  = ⃗  ⋅ ⃗ , { ⃗ }  = ⃗  ⋅ ⃗  are the normal vorticitiy, velocity and [ ⃗ ] is a matrix with the normal vector components on the diagonal.To discretise the remaining Equations ( 6) and ( 7) the subdomain technique is applied.A full description on the numerical procedure is presented in [14].

Approximation
The matrices in (4) are fully populated and, hence, the  2 -methodology will be applied to reduce the storage and computing time.This is essential for handling the domain integral [14].First, a -structure is build.For that the geometry is divided into clusters that are then combined into cluster tress.The formed cluster trees are combined into block clusters that are tested for admissibility.A detailed description on the construction of the block clusters and the admissibility conditions is presented in [14].The integral kernel in the admissible block clusters is approximated with Lagrange interpolation.To reach linear computational complexity, a nested cluster basis is employed to the approximation of the integral kernel [14].

Boundary conditions
The presented methodology was employed to simulate the fluid flow and heat transfer in the hollow brick.It was considered that the holes are cavities where fluid flow is present because of natural convection.The geometry of the cavities is a unit cube.Air is enclosed in the cavities at temperature 0  .The fluid flow in the cavities is solved with equations in (3).To solve the boundary vorticity in Equation ( 5) and domain velocity in (4) no-slip boundary condition were employed on the walls ⃗  = 0, while the vorticity ⃗ , vorticity flux ⃗   and temperature , heat flux   boundary conditions in ( 7) are shown on Figure 2. In order to solve the fluid flow in the cavities parallel computing was employed, where each processor solved the fluid flow in one cavity.We determined the average Nusselt number on the hot and cold walls of the cavities  = 1 Γ ∫   Γ to obtain the heat transfer coefficients    ,    , while from the heat transfer coefficients the unknown temperatures in (2) could be obtained.

NUMERICAL RESULTS
We present the obtained results of the numerical simulations with the presented method and boundary conditions.We observed the influence of the brick width and number of cavities  on the heat transfer coefficient.The number of cavities was changed from 1 to 20.In each of the following figures, several lines are plotted which correspond to the solutions in the respective cavity.As a consequence the lines have different length, that is, for only one cavity only one result exist and for ten cavities ten lines.In figure 3, we show the Nusselt number at the cold wall on the last cavity, for different widths of the brick, where the brick width  is presented in centimeters.It is observed that the heat flux drops when the number of cavities increases and the width of the brick decreases.From the definition of the Rayleigh number below Equation (3) it is clear that the width of the cavity   has a bigger influence on the Rayleigh number than the temperature difference Δ  , because the   has a power of three.However, a bigger number of cavities does not necessarily give a smaller heat flux.It can be seen that after ten cavities the heat flux decreases slower than for a smaller number of cavities.In Figure 4, we present the change of the Rayleigh number depending on the width of the cavities.The width of the cavities depends on the width of the brick .Increasing the number of cavities will decrease their size.Thus the Rayleigh number is smaller and the characteristic velocity of the fluid flow drops.This is also shown with the Nusselt number from Figure 3 where the number drops towards one when we increase the number of cavities.Hence at  = 1, only diffusion of the heat is present.Lastly we show the average temperature in the cavities at different heating temperatures Δ =   −   and widths.

I G U R E 4
The Rayleigh number for three different brick widths [] at Δ = 100[] and  number of cavities in the brick cross section.

F I G U R E 5
The average cavity temperatures for differently heated bricks Δ,  = 1.0  (top) and differently width bricks [] (bottom) and  number of cavities in the brick cross section.Each line stands for a number of cavities in the hollow brick.
In the first row in Figure 5 the temperature difference on hot and cold wall of the brick Δ is changed and the width of the brick is constant.It is clear that the temperate is not dropping linearly.Thus convective heat transfer is present.While in the second row the Δ is constant and width changes.The temperature profile changes from non-linear to linear.One can expect that the isolation properties of the red clay brick increases when holes are added to the brick design.However, the number of holes determine the overall heat isolating properties of the hollow brick.

CONCLUSIONS
The velocity-vorticity formulation of the Navier-Stokes equations for incompressible flows has been realized as boundary domain integral equation. 2 -approximation has been used to save storage, especially, for the domain integrals.The proposed method was employed to simulate the heat transfer through the hollow brick.It was shown that for a limited number of holes in the brick the heat transfer decreases.However, increasing the amount of holes in the brick further, would not decrease the heat transfer significantly.Also the temperature difference between the hot and cold wall and the width of the brick have an influence on the heat flux.Certainly, it is meaningless to increase the number of cavities to infinity.Thus, the material properties of the brick would have to be changed in order to further decrease the heat transfer.
From the obtained heat transfer coefficients one can determine the effective heat transfer coefficient of the hollow brick.

A C K N O W L E D G M E N T S
The authors would like to acknowledge the use of HPC resources provided by the ZID of Graz University of Technology.

the
where   is current times step and  −1 previous time step,  is the dimensionless temperature, Rayleigh number, where the Boussinesq approximation is used and  =  0  0

I G U R E 2
The vorticity ⃗  and vorticity flux ⃗   boundary conditions.The arrows show the surface normal vector.F I G U R E 3The Nusselt number on the cold wall of the cavities for three different brick widths [] at Δ = 100[] and  number of cavities in the brick cross section.Each line stands for a number of cavities in the hollow brick.
is used.The integral form of the transport vorticity equation (i.e., the second equation in (3)) is