Extending the Validity of Basic Equations for One-dimensional Flow in Tubes with Distributed Mass Sources and Varying Cross Sections

Flow problems are solved using so-called fundamental equations and the corresponding initial and boundary conditions. The fundamental equations are the motion equation, the continuity equation, the energy conservation equation, and the state equations. In our paper, we extend the validity of the equation of motion used to describe one-dimensional, steady-state tubular flow to a case in which the mass flow of the medium changes along the tubular axis during the flow. Such flows occur in perforated and/or porous pipes and air ducts. The research in this direction was motivated by the fact that the extension and formulation of the equation of motion in this direction has not been carried out with completely general validity. In the equation of motion used to solve the problems, the isochoric and isotherm nature were assumed. In our paper, we present fundamental equations that formulate differential equations to describe polytrophic and expanding flows.


Introduction
Space theory for one-dimensional flows within tubes has only been analyzed for the constant mass flow case. In this paper, we extend the basic equations to mass source spaces. This expansion is necessary to solve problems that occur during the design of ventilation and air duct systems, as well as to complete the theoretical study of such problems. The description should be applied for tubes with uniform porosity and perforation.
The basic equations consist of the equation of motion, the energy equation, the continuity equation and the state equation of the medium. We regard the flow in the tube as one-dimensional, and disregard the radial and tangential velocity components. We regard the flow to be uniform across the entire cross section of the tube. We assume that the flow is steady. We describe the motion equation as a balance of momentum flow rates. The reaction forces at the boundary surface are cancelled out by the compression forces. The enthalpy, translational kinetic energy and external heat convection appear in the energy equation. The flow is horizontal.
The momentum loss or momentum gain is accounted for in the axial direction velocity component of the mass outflow. Mass outflow usually happens in the direction normal to the boundary surface. The translational kinetic energy is calculated from the absolute value of the mass outflow velocity. The geo-metric models appear in Fig. 1.
Related literature is available by Wang [1], Wang et al. [2], Garbai [3] and Czétány [4][5][6][7]. The application of the developed model can be utilized e.g. in the further development Fig. 1 Reaction force on the lateral surface of the transpired solar collector's mathematical modeling, described by Fawaier et al. [8] Distributed mass flow source has also been described by Sánta [9] during the energy analysis of heat pump systems with different refrigerants.
2 Fundamental equations characterizing flow in tubes with mass and momentum sources 2.1 Motion equation for momentum flow rate and momentum source The motion equation is a modified balance of momentums. The flow momentum balance is the sum of the work of normal forces, the change in momentum flow rate, the mass inflow or outflow momentum through the lateral surface of the tube, and the energy dissipation due to tube friction.
Therefore, the balance of momentum is as follows: further: In a different form: Dividing by Aρ: Rearranging the equation: Finally: Special cases can be written as: In this case the motion equation is Eq. (8): that's the typical form of the equation.
Then the motion equation is the same as in case of '1.' in the start of this listing: Then we again get case of '1.' in the start of this listing: Then the motion equation is: If ρ = constant: After the differentiating:˙ After simplification: Finally: If dm/dx = Ap × w py /L then we obtain Wang's formula. However, we can define a function for dm/dx, such as dm/dx = u = constant for uniform loss. If: then: If the momentum flow rate source = 0, because w px ≈ 0, and ρ = constant, then: In engineering practice, the above equation is the basic equation for sizing airducts for perforated tubes. If we want isobar flow then we can determine the change in the cross section along the tube by using this equation.

Deriving the energy equation
The energy equation for infinitesimal space in steady state is the balance of enthalpy and translational kinetic energy. The equation accounts for energy inflows and outflows at the front and rear of the space segment, as well as through the lateral surface. When the enthalpy of the inside flow and outflowing medium is the same h, the balance of the infinitesimal energy according to Fig. 1. is the following: After multiplication: Further: After substituting this into the energy equation: Multiplying each element by (k-1): After expansion: Multiplying both sides by v k−1 : Rearranging the equation: Finally: Alternatively:

Polytrophic flow in porous tube with friction, other forms based on the previous equations
The following equations can be derived from Eq. (46) which are useful in engineering practice.
For uniform loss:

Special cases
If the flow m = constant, and there is no source, then the state equation of the flow is Eq. (51): Substituting into Eq. (57): After simplification: (60) For an air duct with a circular cross section: The differential equation can be solved by direct integration.

Proceeding from the energy equation
The state equation can be derived from the energy equation with the following assumptions: dp dx dV dx = = 0 0 , .