Solution of High Speed Gas Problems

We will illustrate the procedure for solving the three types of pipe flow problems for high-speed gas flows: unknown driving force, unknown flow rate, and unknown diameter.

1. Unknown Driving Force

The unknown driving force could be either the upstream pressure, Pj, or the downstream pressure, P2. However, one of these must be known, and the other can be determined as follows.

j. Calculate NRe = DG/pj and use this to findfj from the Moody diagram or the Churchill equation.

2. Calculate NMaj = (G/PJ)(RTj/kM)j/2. Use this with Eqs. (9-62), (9-64), and (9-65) or Fig. 9-3 or Appendix I to find (4fL*/D)J, Pj/P*, and Tj/T*. From these values and the given quantities, calculate L*, P*, and T*.

3. Calculate L* = L* — L, and use this to calculate (4fi L*/D)2. Use this with Fig. 9-3 or Appendix I or Eqs. (9-62), (9-64), and (9-65) to get NMa2, P2/P*, and T2/T*. [Note that Eq. (9-62) is implicit for NMa2]. From these values, determine P2 and T2.

4. Revise p by evaluating it at an average temperature, (Tj + T2)/2, and pressure, (Pj + P2)/2. Use this to revise NRe and thus f, and repeat steps 3 and 4 until no change occurs.

Figure 9-3 Fanno line functions for k = 1.4. ([a] From Hall, 1951 and [b] from Shapiro, 1953.)

2. Unknown Flow Rate

The mass velocity (G) is the unknown, which is equivalent to the mass flow rate because the pipe diameter is known. This requires a trial and error procedure, because neither the Reynolds nor Mach numbers can be calculated a priori.

1. Assume a value for NMa1. Use Eqs. (9-62), (9-64), and (9-65) or Fig. 9-3 or Appendix I with this value to find P1 /P*, T1/T*, and (4/L*/D)j. From these and known quantities, determine P* and T*.

2. Calculate G1 = NMa1P1(kM/RT1)1/2 and NRe1 = DG/p. From the latter, find f1 from the Moody diagram or Churchill equation.

3. Calculate (4fL*2/D)2 = (4fL*/D)1 - 4f1L/D. Use this with Eq. (9-62) (implicit) an2d Eqs. (9-614) and (9-65) or Fig. 9-3 or Appendix I to find NMa2, P2/P*, and T2/T* at point 2.

4. Calculate P2 = (P2/P*)P*, T2 = (T2/T*)T*, G2 = NMMkM/ RT2)1/2, and NRe = DG2/p. Use the latter to determine a revised value off = f2.

5. Using f = (f + f2)/2 for the revised friction factor, repeat steps 3 and 4 until there is no change.

6. Compare the given value of P2 with the calculated value from step 4. If they agree, the answer is the calculated value of G2 from step 4. If they do not agree, return to step 1 with a new guess for NMa1, and repeat the procedure until agreement is achieved.

3. Unknown Diameter

The procedure for an unknown diameter involves a trial-and-error procedure similar to the one for the unknown flow rate.

1. Assume a value for NMa1 and use Eqs. (9-62), (9-64), and (9-65) or Fig. 9-3, or Appendix I to find P1/P*, T1/T* and (4fL*/D)1. Also, calculate G = NMa1P1(kM/RT1)1/2, D = (4m/^G)1/2, and NRe1 = DG/p. Use NRe1 to find f1 from the Moody diagram or Churchill equation.

2. Calculate P2/P* = (P1/P*)(P2/P1), and use this with Fig. 9-3 or Appendix I or Eqs. (9-64) (implicitly), (9-62), and (9-65) to find NMa2, (4fL*/D)2, and T2/T*. Calculate T2 = (T2/T*)(T*/T1)T1, and use P2 and T2 to determine p2. Then use p2 to determine NRe2 = DG/p2, which determines f2 from the Moody diagram or Churchill equation.

3. Using f = (f1 + f2)/2, calculate L = L* - L* = \(4fL\/D\ -(4fL*/D)2 ](D/4f).

4. Compare the value of L calculated in step 3 with the given value. If they agree, the value of D determined in step 1 is correct. If they do not agree, return to step 1, revise the assumed value of NMaj, and repeat the entire procedure until agreement is achieved.

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