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16 Nov 2019
In the case of a dry-adiabatic process, it is possible to relate any pair of the three state variables, p, rho, and T, to each other; the relationships are referred to as Poisson's relations. Figure 6.2 of Ahrens and Henson, 11th ed., p. 146, illustrates dry-adiabatic ascent and descent of an air parcel, referring to expansion and cooling of the ascending parcel and to compression and warming of the descending parcel. The Poisson's relations applicable to Fig. 6-2 relate pressure to temperature, p2 = p1 (T_2/T_1)^c_p/R, and specific volume to temperature, alpha_2 = alpha_1 (T_1/T_2)^c_V/R In the equations given above, the subscripts "1" and "2" refer to initial and final states, respectively, for a process, c_P/R = 7/2 is the ratio of the specific heat capacity at constant pressure for dry air to the gas constant for dry air, c_V/R = 5/2 is the ratio of the specific heat capacity at constant volume for dry air to the gas constant for dry air, and alpha, the specific volume, is defined as the inverse of density (i.e., alpha rho^-1). a) Assume that an altitude of 0 m in Fig. 6.2 corresponds to a pressure of 1000 hPa. Use the ideal gas law to calculate the specific volume in units of m^3 kg^-1 corresponding to the temperature indicated for this level. b) Use the applicable Poisson relation given in the heading of this problem to calculate the pressure in units of hPa for the parcel in Fig. 6.2 when it reaches an altitude of 2000 m. c) Use the applicable Poisson relation given in the heading of this problem to calculate the specific volume in units of m^3 kg^-1 for the parcel in Fig. 6.2 when it reaches an altitude of 2000 m. d) The text in Fig. 6.2 states that a rising parcel expands and cools. Is the result of the calculation performed in part (c) consistent with the stated expansion of the parcel? Explain your reasoning.
In the case of a dry-adiabatic process, it is possible to relate any pair of the three state variables, p, rho, and T, to each other; the relationships are referred to as Poisson's relations. Figure 6.2 of Ahrens and Henson, 11th ed., p. 146, illustrates dry-adiabatic ascent and descent of an air parcel, referring to expansion and cooling of the ascending parcel and to compression and warming of the descending parcel. The Poisson's relations applicable to Fig. 6-2 relate pressure to temperature, p2 = p1 (T_2/T_1)^c_p/R, and specific volume to temperature, alpha_2 = alpha_1 (T_1/T_2)^c_V/R In the equations given above, the subscripts "1" and "2" refer to initial and final states, respectively, for a process, c_P/R = 7/2 is the ratio of the specific heat capacity at constant pressure for dry air to the gas constant for dry air, c_V/R = 5/2 is the ratio of the specific heat capacity at constant volume for dry air to the gas constant for dry air, and alpha, the specific volume, is defined as the inverse of density (i.e., alpha rho^-1). a) Assume that an altitude of 0 m in Fig. 6.2 corresponds to a pressure of 1000 hPa. Use the ideal gas law to calculate the specific volume in units of m^3 kg^-1 corresponding to the temperature indicated for this level. b) Use the applicable Poisson relation given in the heading of this problem to calculate the pressure in units of hPa for the parcel in Fig. 6.2 when it reaches an altitude of 2000 m. c) Use the applicable Poisson relation given in the heading of this problem to calculate the specific volume in units of m^3 kg^-1 for the parcel in Fig. 6.2 when it reaches an altitude of 2000 m. d) The text in Fig. 6.2 states that a rising parcel expands and cools. Is the result of the calculation performed in part (c) consistent with the stated expansion of the parcel? Explain your reasoning.