Recently, a conformable fractional derivative has been proposed to calculate the derivative of non-integer order of time functions. It has been shown that this new fractional derivative definition obeys many advantage...Recently, a conformable fractional derivative has been proposed to calculate the derivative of non-integer order of time functions. It has been shown that this new fractional derivative definition obeys many advantages over the preceding definitions. For mathematical models in applied sciences and to preserve the dimensionality of the physical quantities, an auxiliary parameter （~r） which has the dimension of seconds should be implemented in the fractional derivative definition. We obtain analytic solutions for the resulting conformable fractional differential equations describing the vertical velocity and the height of the falling body. It is shown that the dimensions of velocity and height are always correct without any restrictions on the auxiliary parameter cr which contradicts with the results in the literature when applying the Caputo definition to the same problem. This may open the door for many future works either to describe the role of such an auxiliary parameter or to derive a more suitable definition for the fractional derivative.展开更多
In recent years, there has been an increase of interest in the flow of gases at relatively high pressures and high temperatures. Hydrodynamic calculation of the energy losses in the flow of gases in conduits, as well ...In recent years, there has been an increase of interest in the flow of gases at relatively high pressures and high temperatures. Hydrodynamic calculation of the energy losses in the flow of gases in conduits, as well as through the porous media constituting natural petroleum reservoirs, requires knowledge of the viscosity of the fluid at the pressure and temperature involved. Although there are numerous publications concerning the viscosity of methane at atmospheric pressure, there appears to be little information available relating to the effect of pressure and temperature upon the viscosity. A survey of the literature reveals that the disagreements between published data on the viscosity of methane are common and that most investigations have been conducted over restricted temperature and pressure ranges. Experimental viscosity data for methane are presented for temperatures from 320 to 400 K and pressures from 3000 to 140000 kPa by using falling body viscometer. A summary is given to evaluate the available data for methane, and a comparison is presented for that data common to the experimental range reported in this paper. A new and reliable correlation for methane gas viscosity is presented. Predicted values are given for temperatures up to 400 K and pressures up to 140000 kPa with Average Absolute Percent Relative Error （EABS） of 0.794.展开更多
文摘Recently, a conformable fractional derivative has been proposed to calculate the derivative of non-integer order of time functions. It has been shown that this new fractional derivative definition obeys many advantages over the preceding definitions. For mathematical models in applied sciences and to preserve the dimensionality of the physical quantities, an auxiliary parameter （~r） which has the dimension of seconds should be implemented in the fractional derivative definition. We obtain analytic solutions for the resulting conformable fractional differential equations describing the vertical velocity and the height of the falling body. It is shown that the dimensions of velocity and height are always correct without any restrictions on the auxiliary parameter cr which contradicts with the results in the literature when applying the Caputo definition to the same problem. This may open the door for many future works either to describe the role of such an auxiliary parameter or to derive a more suitable definition for the fractional derivative.
基金supported by the Research Institute of Petroleum Industry-Kermanshah Campus.
文摘In recent years, there has been an increase of interest in the flow of gases at relatively high pressures and high temperatures. Hydrodynamic calculation of the energy losses in the flow of gases in conduits, as well as through the porous media constituting natural petroleum reservoirs, requires knowledge of the viscosity of the fluid at the pressure and temperature involved. Although there are numerous publications concerning the viscosity of methane at atmospheric pressure, there appears to be little information available relating to the effect of pressure and temperature upon the viscosity. A survey of the literature reveals that the disagreements between published data on the viscosity of methane are common and that most investigations have been conducted over restricted temperature and pressure ranges. Experimental viscosity data for methane are presented for temperatures from 320 to 400 K and pressures from 3000 to 140000 kPa by using falling body viscometer. A summary is given to evaluate the available data for methane, and a comparison is presented for that data common to the experimental range reported in this paper. A new and reliable correlation for methane gas viscosity is presented. Predicted values are given for temperatures up to 400 K and pressures up to 140000 kPa with Average Absolute Percent Relative Error （EABS） of 0.794.
