Intermediate Flow of Gases - LAb

1 Intermediate Flow of Gases As we have seen earlier, the conductance of a tube in viscous regime (high pressures) is directly proportional to pressures as given by the formula C = Q/ (P1-P2) = [ D4/128 L] P In which D = Diameter L = Length of Tube = Poisson s number (in poises) P = Pressure C = conductance As per this formula, the conductance should fall to zero when P = 0. However, this does not take into account the slip which occurs at the walls of the container, given by a slip co efficient , which is = P [2M/ ( Ro T)] [f / (2-f)] Where f = fraction of molecules that are adsorbed and reemitted and 1-f is the fraction that are specularly reflected. Since the velocity of the gas is not zero at the wall, Q = [ D4/ (128 L)] P (P1-P2) {1 + [8h/ ( D)]} Thus it results that Q = (c1PD4 + c2D3) (P1-P2)/ L Where c1 = /128 c2 = ( /16) [( /2) (Ro T/M) ] [(2-f)/f] Thus the Conduction is given by C = (c1PD4 + c2D3) / L Knudsen defined this equation as: C = [ D4/ (128 L)] P + 1/6 [2M/ ( Ro T)] D3/L x [1 + [2M/ ( Ro T)] DP/ ] / [1 + [2M/ ( Ro T)] DP/ ] c1 = /128 c2 = x 103 (T/M) x [1 + x 10-4 (M/T) DP/ ] / [1 + x 10-4 (M/T) DP/ ] At small values of average Pressure, the second term can be neglected and so c2 = x 103 (T/M).

2 On comparing with earlier equation we find that f = This means that at low pressures, 74% of the molecules are adsorbed and reemitted and 26% are reflected. At high pressures, we get c2 = x 10 3 (T/M) Comparing with equation again, we get f = Knudsen s results simply imply that the fraction of the molecules adsorbed and reemitted changes slowly in the Intermediate region. Knudsen s equation gives the conductance in any regime, assuming that over the whole length of the tube L, the regime is the same (Molecular, Intermediate , And Viscous) Minimum conductance Condition: Knudsen s Equation can be rewritten as: y = ax + b (1 + cx)/ (1 + dx) Where y = CL/D3 x = PD/ a = x 10 -2 b = (T/M) c = (M/T) d = (M/T) Differentiating this with respect to x and setting the resultant equation equal to zero, the value of x at which y has a minimum value is determined, this is Xmin = [b (d-c) /a -1]/d Thus (PD/ ) min = (T/M) Thus we have = P (M/T) Thus we have D/ min = Thus we see that according to the equation of minimum conductance , it occurs when mean free path of the molecules is times the diameter of the tube.

3 For larger than this value, the conductance increases asymptotically toward that given for molecular flow and for values less than this value, the conductance follows molecular flow. Transition Pressure: The transition pressure is defined as the value of pressure for which the viscous term c1PD4 is equal to the non viscous term c2D3. This means ax = b (1 + cx) / (1 + dx) From which x = (1/2ad) {(bc-a) +/- [(bc-a) 2 + 4abd] } Taking positive sign, (PD/ ) t = (T/M) Therefore, D/ t = Therefore the transition flow represents a region of viscous and non viscous flow, both of which are significant. The Limits of Intermediate Range: The limits of Intermediate range can be considered as being those where the contribution of one of the flow conditions predominates, where the contribution of one of them is an order of magnitude more important that that of the other.

4 Therefore the upper limit of the Intermediate range that above which the flow can be considered viscous is given by ax = 10 b (1 + cx) / (1 + dx) Thus x = (1/2ad) {(10bc-a) +/- [(10bc-a) 2 + 4abd] } Using the positive sign again, we get (PD/ ) u = 942 (T/M) And D/ u = 111 The lower limit of the Intermediate range, the range below which the flow can be considered molecular is given by ax = .1 b (1 + cx) / (1 + dx) Thus x = (1/2ad) {(.1bc-a) +/- [(.1bc-a) 2 + 4abd] } Using the positive sign again, we get (PD/ ) l = 10 (T/M) And D/ u = This implies that Pu = 10 Pt and Pl = .1 Pt Therefore the Intermediate range of pressure extends over two magnitudes of pressure. General Equation of Flow: This is given by C = Cm J Where Cm is the conductance of the molecular flow Cm = (1/6) [2 RoT/M] 1/2 (D3/L) And J = Cv /Cm + [1 + (M/ RoT) DP/ ]/ [(1 + (M/ RoT) DP/ ] Where CV is the conductance for viscous flow.)

5 CV = [ D4/128 L] P Molecular Flow through L-Shaped Regions The molecules (in a molecular flow) through an L shaped tube can be divided in two categories: 1) Molecules which collide with the wall in the region of the elbow, and 2) Molecules which pass across the tube Molecules having the path 1 will see the opening of the tube as an impedance, thus the conductance of the elbow will be given for path 1 by C = (T/M) D3 / [L1 + L2 + ] Molecules having path 2 will pass the elbow without feeling its influence. Thus, the conductance of the molecules is C = (T/M) D3 / [L1 + L2] According to the above equations, an L-shaped tube can be represented as a tube with the diameter D, having an equivalent length Le, which will be situated between Lax< Lc < (Lax + D) Where Lax = L1 + L2 is the length as measure along the axis.

6 For a more precise evaluation, it can be considered that all the molecules will travel along according to path 1, when the shape of the elbow is that of a hairpin, thus the bend at = 1800. Considering that the number of molecules having path 1 is proportional to the angle t of the bend, it results that the equivalent length is: Le = Lax + D/180