The thermo-physical properties of the fluid used are presented in Table 1. This is comprised of density, viscosity, specific heat capacity and thermal conductivity.
Fluid Density (kg/m3) Viscosity (Pa•s) Specific heat capacity (J/(kg•K)) Thermal conductivity (W/(m•K)) Chyme 1 000 1.000 4 180 0.600 Pap (Ogi) 1 024 1.095 1 840 0.536 Soymilk (Soya) 920 0.95 3 970 0.501 Hibiscus sabdariffa roselle (Sobo) 800 0.316 2 470 0.491 Source: Berk, 2013
Table 1. Thermo-physical properties of fluids considered in this study.
The research focuses on incompressible and viscous fluid flow through the mouth to the lower part of the gastrointestinal tract (GIT) and small intestine. AUTODESK INVENTOR 2017 commercial version was used for the computational modelling of the human intestine as well as the steady flow field generated inside the modelled intestine. The movement of the contraction waves within the alimentary canal was simulated using a commercial version of CFD solver, ANSYS FLUENT 16.0. Navier-stokes equations of fluid flow with deformed boundary walls were used for the model. Fig. 2 shows the mesh generated domain and the modelled intestine.
The model is governed by the continuity equation and Navier-Stokes equations with the following assumptions: (ⅰ) The flow is a laminar natural convection; (ⅱ) It is assumed to be a steady-state hydrodynamics incompressible fluid flow; (ⅲ) The inlet temperature is 303 K; (ⅳ) The velocity profile is fully developed in the model; (ⅴ) The velocity along the radial direction is assumed to be zero, i.e., velocity along the radial direction is insignificant when compared with axial velocity; (ⅵ) The effect of gravity is neglected; (ⅶ) No slip boundary condition at the walls is employed; (ⅷ) The thermal boundary condition of constant heat flux is applied.
The continuity equation that governs the numerical model is presented in Equation (1):
where Ux is non-zero velocity component, r is non-dimensional radial coordinate, ϕ is non-dimensional azimuthal coordinate, and x is non-dimensional axial coordinate, thus, Ur = Uϕ = 0. There, Equation (1) is reduced to ∂Ux/∂x = 0. It confirms that Ux does not depend on the radial component of the tube which implies a thinned wall tube and also Ux = Ux(r) = 0 is the same for all values of x, thus Ux = Ux(r).
where ρ is fluid desity, t is time, μ is dynamic fluid viscosity, and g is acceleration due to gravity.
Substituting Ur = Uϕ = 0 and ∂Ux/∂x = 0 into equations (2) and (3), gives ∂ρ/∂r = 0 and ∂ρ/∂ϕ = 0. ∂ρ/∂r = 0 and ∂ρ/∂ϕ = 0 show that pressure depends only on axial direction and imposing steady state condition yields ∂Ux/∂t = 0.
Neglecting gravitational effect, Equation (6) is the momentum transport equation for the model.
The energy transport equation for a steady for flow is:
where k is thermal conductivity, T is fluid temperature, Cp is specific heat capacity, ϕ(i) is viscous dissipation term.
Eliminating the viscous dissipation term ϕ(i):
Energy Transport Equation for flow within the tube is presented in Equatoin (9):
where α = (k/ρCp), constant.
Since the intestine is a porous medium and the flow is a laminar natural convection, which is also assumed to be a steady-state hydrodynamics incompressible fluid flow. Therefore, the Darcy's law is applied and is presented in Equation (10):
where P is pressure, L is length of flow, q is fluid flow rate.
The relationship between the shear stress and shear rate for a power law non-newtonian fluid is presented in Equation (11):
where σ is shear stress, γ is shear rate, H is consistency of the power-law fluid, and n is power-law index.
Nusselt number represents the rate of heat transfer across the wall. The tendency of the model and the pulsating part in providing heat transfer augmentation is critically examined and governed by Equation (12):
where Tw is wall temperature, Tb is bulk fluid temperature, q is fluid flow rate, and dh is hydraulic diameter of intestinal model.