Loads that are applied near-instantaneously, and then remain constant for a reasonable duration of time, such as fluid hammer and reUef valve loads, effectively are applied with a Dynamic Load Factor (dynamic multiplier) between 0.0 and 2.0. This is evident by assuming the worst case — no damping and instantaneous application of a constant force — and performing a time history analysis of the dynamic equation:
M x(t) + K x(t) = F(t)
Equating energies (where the kinetic energy added to the mass is Fx(t), while the crumpling energy of the spring is [Kx{(t)^2}/2]:
Fx(t) = Kx(t)2/2, or Kx(t) = 2 F(t)
The term Kx(t) represents internally induced forces/moments within the system. The DLF is the ratio of the induced forces to the applied forces, or K x(t)max / F(t), which in this case has its maximum value of 2.0. It is often highly conservative to apply twice the calculated force as a static load, but this is still often done. As the load ramp-up time (such as the opening time of a relief valve) increases, or the load duration decreases (such as fluid hammer in a short piping leg), the DLF will decrease as well. In order to take advantage of the "true" (reduced) DLF, it is necessary to perform a dynamic analysis, such as a time history analysis or a response spectrum analysis. In lieu of a dynamic analysis, the user can only estimate a DLF, estimate the applied load, and apply a concentrated static force equal to the DLF times the applied load to the piping system.
Fluid Hammer: It is not always so easy to calculate the applied loads. One method of estimating fluid hammer loads is described in Crocker & King's Piping Handbook as:
F = p c dv A /144g
Where:
F = fluid hammer force (exclusive of DLF), lb p = density of fluid, lbm/ft^3
= 62.4 for water
= 0.0003 for saturated steam at atmospheric pressure
= 1.85 for superheated steam at 1000°F and 1500 psig
c = speed of sound in a fluid, ft/sec
= for liquid: 12 [g Ef / (1 + D Ef /1 Ep)]
= approximately 3000-4000 ft/sec for water in typical pipe sizes
= for gas: (kgRT)^1/2
= approximately 2000-2500 ft/sec for steam in typical pipe sizes
g = acceleration gravity, ft/sec^2 Ef = bulk modulus of fluid, psi
= approximately 300,000 psi for water and other fluids
D = inside diameter of pipe, in t = wall thickness of pipe, in Ep = modulus of elasticity of pipe material, psi k = ratio of specific heats for gas, dimensionless
= 1.3 for steam, 1.24 for ethylene, 1.27 for natural gas
R = gas constant, ft-lb/lbm-°R
= 85 for steam, 55.1 for ethylene, 79.1 for natural gas
T = temperature of gas, °R dv = change in fluid velocity causing fluid hammer, ft/sec A = internal area of pipe, in^2
Relief valves: Relief valves are used in piping to provide an outlet on those occasions when pressure builds up beyond that desired for safe operation. When the pressure setting is reached, the valve opens, allowing sufficient fluid to escape from the piping system to lower the pressure. This discharge initiates ajet force, which must be resisted by the piping system. Valve opening time and duration of the jet load affect the dynamic response of the system, thus affecting the developed loads.
Relief valve jet loads are normally provided by the valve manufacturer. If this is not the case, the loads can be estimated by a thorough thermodynamic analysis. This method is discussed in detail in Section 5 of these course notes.
In lieu of thermodynamic and dynamic analyses, the B31.1 code provides a means of estimating the discharge force (as an equivalent static force) of a relief valve venting steam to atmosphere. The force is estimated as such:
F = DLF (M V / g + P A)
Where:
F = static equivalent discharge force, lb DLF = dynamic load factor (as calculated below), dimensionless M = mass flow rate from valve x 1.11 (factor of safety), lbm/sec V = fluid exit velocity, ft/sec
= [(2gJ)(h0-a)/(2b-l)]^1/2
J = conversion constant, 778.16 ft-lb/Btu h0 = stagnation enthalpy of steam, Btu/lbm.
The stagnation enthalpy of the steam-water mixtures generally lie within the relatively narrow limits 350 to 600 Btu/lb
a,b = steam constants as per following table:
g = gravitational constant = 32.2 fit/sec^2 P = static pressure at discharge, psig
= {[ M (b -1)/Ab ] [ 2J(h0 - a)/g(2b -1)]^1/2} - Pa
A = internal area of discharge pipe, in^2 Pa = atmospheric pressure = 14.7 psi
The dynamic load factor (DLF) is used to account for the increased load caused by the sudden application of the discharge force. (Note that DLFs are discussed in great detail in Sections 4 and 5 of these seminar notes.) For the purposes of this estimate, the DLF varies between 1.1 and 2.0, depending upon the rigidity of the valve installation and the opening time of the valve. If the piping system is relatively rigidly restrained, the DLF can be calculated by finding the natural period of vibration of the valve installation, treating it as a single degree-of-freedom oscillator:
T = 0.1846 [WH^3/gEI]^1/2
Where:
T = natural period of vibration, sec
W = weight of relief valve installation, lb
H = distance, run pipe to center of outlet pipe (see Figure 2-36), in
g = gravitational constant = 386.4 in/sec^2
E = modulus of elasticity of pipe material, psi
I = moment of inertia of inlet pipe, in^4
Next, the ratio of the valve opening time, to, to the fundamental period of vibration of the valve installation, T, should be found. This ratio is then used to determine the DLF from the chart in Figure 2-37. (Note that in the event that the opening time is not known, a conservative value of 2.0 for the DLF should be used.)
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