_{1}

This paper describes a new method of calculation of one-dimensional steady compressible gas flows in channels with possible heat and mass exchange through perforated sidewalls. The channel is divided into small elements of a finite size for which mass, energy and momentum conservation laws are written in the integral form, assuming linear distribution of the parameters along the length. As a result, the calculation is reduced to finding the roots of a quadratic algebraic equation, thus providing an alternative to numerical methods based on differential equations. The advantage of this method is its high tolerance to coarse discretization of the calculation area as well as its good applicability for transonic flow calculations.

Let us consider a compressible gas flow along a channel with a variable cross- section area and perforated sidewalls through which heat and mass exchange is possible.

Let us divide this channel into many small elements of finite length

The mass, energy and momentum conservation laws for this element can be written in the following one-dimensional integral form:

Here:

Indices 1, 2, and “per” designate cross-sections 1 and 2 and perforation.

As the element’s length

As such, the third equation of the system (1) can be re-written as follows:

Here:

The walls’ friction force

where

Let us transform the Equation (2) by dividing both its parts by

After grouping the terms we shall have:

Now we should note that:

and, besides:

where:

After some transformations we shall obtain a quadratic algebraic equation for the characteristic Mach number at the element’s exit:

where:

The total temperature at the element’s exit

1) The additional gas is injected into the element through the perforation holes

Here:

2) The gas is partly sucked away from the element through the perforation holes

3) There is no mass exchange

4) There is no mass- and heat exchange

Thus, the characteristic Mach number at the element’s exit

where the coefficients

In case of a constant cross-section area and the absence of external influences (i.e. when

The second solution describes a normal shock wave and therefore may physically exist just for

Then all other gas parameters at the cross-section 2 can be easily found:

In such a manner, by dividing the channel into small discrete elements and sequentially calculating the flow parameters in each element with the help of relationships (10)-(13) we may calculate the gas flow along the whole channel. To complete this system of equations three additional conditions have to be taken into account:

・ The total pressure losses in the channel should be equal to the external pressure difference between the channel’s inlet and outlet. For subsonic flows this requirement can be met by iterative adjustment of the flow velocity at the channel’s entry. For supersonic flows this requirement can be met by iterative adjustment of the shock wave’s position inside the channel (i.e. finding the element in which the sign before the root in the Equation (10) should be changed from “+” to “−”).

・ The mass flow through the perforation holes

・ The heat flow

along the length (

One more advantage of this method is its good applicability for transonic flow calculations. Unlike numerical methods based on the known differential equation:

this method makes it possible to calculate flows at

Due to this feature, this method can be used for calculations of transonic gas flows in channels.

As can be seen from

1) An algebraic method of calculation of one-dimensional steady compressible gas flows in channels is presented, which is an alternative to finite-difference methods.

2) The advantage of this method is its high tolerance to coarse discretization of the calculation area as well as its good applicability for transonic flow calculations.

Tolmachev, A. (2017) Algebraic Calculation Method of One-Dimensional Steady Compressible Gas Flow. Open Journal of Fluid Dynamics, 7, 83-88. https://doi.org/10.4236/ojfd.2017.71006