3.3.2. Initial and Boundary Conditions. The pipe is subject to atmospheric pressure Po at time zero this represented initial condition . Boundary condition represented as at the pipe inlet profiles, uniform pressure Pinlet and its value change with concentration of nanofluid [φ%] and inlet temperature T0 = 60 ºC can be measured for all experimental . On the pipe wall, the pressure was atmospheric pressure Po . Moreover, flow and thermal fields are assumed symmetrical with respect to the axial plane .
Initial condition :
P(r ,0 ,0) = P0 , P(0 ,z ,0) = P0 , P(r,z,0) = P0
Boundary condition : P(r ,0 ,t) = Pinlet , P(0 ,0 ,t) = Pinlet
∂P/∂z=0 when z=0
3.3.3. Numerical Analysis …show more content…
In the explicit formation, there is only one point at layer of (n+1) ,and the pressure at that point can be calculated from the values of the previous time layer .The finite difference method is used to solve parabolic PDEs.Where λ is the convergence factor , and assume Δr = Δz ,and Using λ=1/7 to minimize truncation error ,therefore to this study Δr = Δz =0.008 mm, the number of nodes in r-coordinate is 5 nodes and number of nodes in z- coordinate is 750 node, where λ=(α_e ∆t)/(∆r)^2 ,and assuming Δr = …show more content…
At r=0 , 〖 P〗_(i+1,j)^n=〖 P〗_(i-1,j)^n , also at z=0 when ∂P/∂z=0 then 〖 P〗_(i,j+1)^n=〖 P〗_(i,j-1)^n substituting in equation (3.14 ) becomes:
〖〖〖P_(i,j)^(n+1)=4λ P〗_(i+1,j)^n+(1-6λ) P〗_(i,j)^n+2λ P〗_(i,j+1)^n ….(3.16)
Figure (3-3) pressure distribution at point (a) A numerical scheme is adopted to solve the problem step by step upon the initial and boundary conditions .The grid increment will influence the calculated results . See figure (3-2),equations which calculated the pressure distribution against r and z – coordinates are described below .A computational grid is shown in figure (3-2).
2- The pressure distribution through the r- coordinate was employed upon the boundary condition was assumed as shown in figure (3-4) ,where i=2 to m-1 and j=1 to 1 ,this mean r>0, z=0 ,when ∂P/∂z=0 .Then the pressure distribution through r- coordinate is presented
Initial condition :
P(r ,0 ,0) = P0 , P(0 ,z ,0) = P0 , P(r,z,0) = P0
Boundary condition : P(r ,0 ,t) = Pinlet , P(0 ,0 ,t) = Pinlet
∂P/∂z=0 when z=0
3.3.3. Numerical Analysis …show more content…
In the explicit formation, there is only one point at layer of (n+1) ,and the pressure at that point can be calculated from the values of the previous time layer .The finite difference method is used to solve parabolic PDEs.Where λ is the convergence factor , and assume Δr = Δz ,and Using λ=1/7 to minimize truncation error ,therefore to this study Δr = Δz =0.008 mm, the number of nodes in r-coordinate is 5 nodes and number of nodes in z- coordinate is 750 node, where λ=(α_e ∆t)/(∆r)^2 ,and assuming Δr = …show more content…
At r=0 , 〖 P〗_(i+1,j)^n=〖 P〗_(i-1,j)^n , also at z=0 when ∂P/∂z=0 then 〖 P〗_(i,j+1)^n=〖 P〗_(i,j-1)^n substituting in equation (3.14 ) becomes:
〖〖〖P_(i,j)^(n+1)=4λ P〗_(i+1,j)^n+(1-6λ) P〗_(i,j)^n+2λ P〗_(i,j+1)^n ….(3.16)
Figure (3-3) pressure distribution at point (a) A numerical scheme is adopted to solve the problem step by step upon the initial and boundary conditions .The grid increment will influence the calculated results . See figure (3-2),equations which calculated the pressure distribution against r and z – coordinates are described below .A computational grid is shown in figure (3-2).
2- The pressure distribution through the r- coordinate was employed upon the boundary condition was assumed as shown in figure (3-4) ,where i=2 to m-1 and j=1 to 1 ,this mean r>0, z=0 ,when ∂P/∂z=0 .Then the pressure distribution through r- coordinate is presented