4-5 Flow of Vapor through Holes

130 Chapter 4 Source Models

The-cross-sectional area of the pipe is

The mass flow rate is given by

This represents a significant flow rate. Assuming a 15-min emergency response period to stop the release, a total of 26,000 kg of hazardous waste will be spilled. In addition to the material released by the flow, the liquid contained within the pipe between the valve and the rupture will also spill. An alternative system must be designed to limit the release. This could include a reduction in the emer- gency response period, replacement of the pipe by one with a smaller diameter, or modification of the piping system to include additional control valves to stop the flow.

4-5 Flow of Vapor through Holes 131

G a s P r e s s u r i z e d W i t h i n

P r o c e s s U n i t E x t e r n a l S u r r o u n d i n g s

A t T h r o a t :

Figure 4-9 A free expansion gas leak. The gas expands isentropically through the hole. The gas properties (P, T) and velocity change during the expansion.

A discharge coefficient C,, is defined in a similiar fashion to the coefficient defined in sec- tion 4-2:

Equation 4-42 is combined with Equation 4-41 and integrated between any two conven- ient points. An initial point (denoted by subscript "0") is selected where the velocity is zero and the pressure is Po. The integration is carried to any arbitrary final point (denoted without a subscript). The result is

For any ideal gas undergoing an isentropic expansion,

pvy = - P = constant, (4-44)

py

where y is the ratio of the heat capacities, y = C,IC,. Substituting Equation 4-44 into Equa- tion 4-43, defining a new discharge coefficient C, identical to that in Equation 4-5, and inte-

132 Chapter 4 Source Models

grating results in an equation representing the velocity of the fluid at any point during the isen- tropic expansion:

The second form incorporates the ideal gas law for the initial density p,. R, is the ideal gas con- stant, and To is the temperature of the source. Using the continuity equation

and the ideal gas law for isentropic expansions in the form

results in an expression for the mass flow rate:

Equation 4-48 describes the mass flow rate at any point during the isentropic expansion.

For many safety studies the maximum flow rate of vapor through the hole is required.

This is determined by differentiating Equation 4-48 with respect to PIP, and setting the deriv- ative equal to zero. The result is solved for the pressure ratio resulting in the maximum flow:

The choked pressure Pchoked is the maximum downstream pressure resulting in maximum flow through the hole or pipe. For downstream pressures less than PC,,,,, the following statements are valid: (1) The velocity of the fluid at the throat of the leak is the velocity of sound at the prevailing conditions, and (2) the velocity and mass flow rate cannot be increased further by reducing the downstream pressure; they are independent of the downstream conditions. This type of flow is called choked, critical, or sonicjlow and is illustrated in Figure 4-10.

An interesting feature of Equation 4-49 is that for ideal gases the choked pressure is a function only of the heat capacity ratio y. Thus:

4-5 Flow of Vapor through Holes 133

G a s P r e s s u r i z e d P r o c e s s U n i t

W i t h i n

E x t e r n a l S u r r o u n d i

<

^'choked

At T h r o a t :

'choked -

U r S o n i c V e l o c i t y

Figure 4-10 Choked flow of gas through a hole. The gas velocity is sonic at the throat. The mass flow rate is independent of the downstream pressure.

Gas Y Pchoked

Monotonic -1.67 0.487P0

Diatomic and air ~ 1 . 4 0 0.528P0

Triatomic -1.32 0.542P0

For an air leak to atmospheric conditions (PChoked = 14.7 psia), if the upstream pressure is greater than 14.710.528 = 27.8 psia, or 13.1 psig, the flow will be choked and maximized through the leak. Conditions leading to choked flow are common in the process industries.

The maximum flow is determined by substituting Equation 4-49 into Equation 4-48:

where

M is the molecular weight of the escapilig vapor or gas, To is the temperature of the source, and

R, is the ideal gas constant.

134 Chapter 4 Source Models

Table 4-3 Heat Capacity Ratios y for Selected Gases1

Gas

Heat Chemical Approximate capacity

formula molecular ratio or symbol weight (M) y = C,,IC, Acetylene

Air Ammonia Argon Butane Carbon dioxide Carbon monoxide Chlorine Ethane Ethylene Helium

Hydrogen chloride Hydrogen Hydrogen sulfide Methane Methyl chloride Natural gas Nitric oxide Nitrogen Nitrous oxide Oxygen Propane

Propene (propylene) Sulfur dioxide

- - - - - - -

lCrane Co., Flow of Flutds Through Valves, Fzttzngs, and Pipes, Technical Paper 410 (New York: Crane Co., 1986).

For sharp-edged orifices with Reynolds numbers greater than 30,000 (and not choked), a constant discharge coefficient C, of 0.61 is indicated. However, for choked flows the discharge coefficient increases as the downstream pressure decrease^.^ For these flows and for situations where C, is uncertain, a conservative value of 1.0 is recommended.

Values for the heat capacity ratio y for a variety of gases are provided in Table 4-3.

9Robert H. Perry and Cecil H. Chilton, Chemical Engineers Handbook, 7th ed. (New York: McGraw-Hill, 1997), pp. 10-16.

4-5 Flow of Vapor through Holes 135

Example 4-4

A 0.1-in hole forms in a tank containing nitrogen at 200 psig and 80°F. Determine the mass flow rate through this leak.

Solution

From Table 4-3, for nitrogen y = 1.41. Then from Equation 4-49

Thus

PCh,,,, = 0.527(200

+

14.7) psia = 113.1 psia

An external pressure less than 113.1 psia will result in choked flow through the leak. Because the external pressure is atmospheric in this case, choked flow is expected and Equation 4-50 applies.

The area of the hole is

The discharge coefficient C, is assumed to be 1.0. Also, Po = 200

+

^{14.7 =} 214.7 psia, T o = 80

+

^{460 = 540°R,}

Then, using Equation 4-50,

136 Chapter 4 Source Models