There are various types of gasket constitutive equations, and their mechanical behaviors differ significantly. Through extensive testing under both ambient and high-temperature conditions, it has been observed that the loading and unloading curve of a gasket—commonly referred to as the compression-rebound curve—is nonlinear and non-conservative. During preloading and operational phases at a constant temperature, the displacement of the gasket can be expressed as:
Dk = (Sk / Ac)^(1/Nc) (1)
Dg = (Sg / As)^(1/Ns) + Dp (2)
Where Sk and Sg represent the pre-tightening stress and residual stress during operation, respectively; Ac, As, Nc, and Ns are regression coefficients; Dp is the plastic deformation after full unloading, which depends on the gasket's material, type, initial pre-tightening pressure, and operating temperature.
Under preloading and operational conditions, the bolt lengths are given by:
lb1 = l0 + qb * W1 (4)
lb2 = l0 + qtb * W2 + 2 + bc * l0 + tb(T2 - T1)l0 (5)
Here, l0 is the initial length of the bolt; qb and qtb are the elastic coefficients of the bolt at pre-tightening temperature T1 and working temperature T2, respectively; tb is the linear expansion coefficient of the bolt material at T2; bc represents the creep strain of the bolt at T2; W1 and W2 are the bolt loads during preloading and operation, respectively:
W1 = Ag * Sk
W2 = Ag * Sg + (Dm² * p) / 4
Where Ag is the full area of the gasket in mm², Dm is the average diameter of the gasket in mm, and p is the medium operating pressure in MPa.
Based on the deformation compatibility condition of the flange connection system, the following equation holds:
Dk - (Dg + Dgc) = (lb2 - lb1) + 2(Df2 - Df1) - 2tf (9)
Substituting equations (1) through (8) into equation (9), we obtain the deformation compatibility equation for the high-temperature flange connection system:
(Sk / Ac)^(1/Nc) [1 - (a + bT2) ln t] - (Sg / As)^(1/Ns) - Dp + 2tftf(T2 - T1) - [qtbW2 - qbw1] - 2(qtfM2 + qpp - qfM1) = 0 (10)
3. Bolt Cyclic Stress Calculation
Assuming that the temperature difference between the bolt and flange is zero when the medium pressure and temperature fluctuate, only the difference in linear expansion coefficients between the two materials is considered. Additionally, changes in bolt and gasket creep within the temperature fluctuation range are ignored. Let the asterisk (*) denote physical parameters under an unstable condition. Then, the deformation compatibility equation under unstable conditions becomes:
(Sk / Ac)^(1/Nc) [1 - (a + bT*) ln t] - (Sg* / As)^(1/Ns) - Dp + 2tftf(T* - T1) - [qtbW*] - 2(qtfM* + qpp* - qfM1) = 0 (11)
Subtracting equation (11) from (10) gives:
(Sg* / As)^(1/Ns) - (Sg / As)^(1/Ns) + qtb(W* - W2) + Tbl0(T* - T2) + 2qtf(M* - M2) + 2qp(p* - p) - 2tftf(T* - T2) = 0 (12)
Which simplifies to:
(Sg* / As)^(1/Ns) - (Sg / As)^(1/Ns) + qtbW + tbl0T + 2qtfM + 2qpp - 2tftfT = 0 (13)
Where W = W* - W2 = (Dm² * p) / 4 + Ag(Sg* - Sg) (14)
The change in flange bending moment due to temperature and pressure fluctuations is given by:
M = M* - M2 = Di²pl1/4 + (Dm² - Di²)pl2/4 + Ag(Sg* - Sg)l3
Due to the nonlinearity of the gasket's constitutive relationship, equations (13) and (14) alone are insufficient to solve for Sg, Sg*, and W. To determine these three unknowns, the residual gasket pressure Sg under stable conditions must first be obtained using equation (10) based on initial conditions, such as the known bolt preload. Subsequently, Sg* and W can be calculated using equations (13) and (14).
Radiator Valve,Radiator Thermostat,Lock Valve Radiator,Traditional Thermostatic Radiator Valves
Ningbo Safewell Plumbing Co., Ltd. , https://www.safewellbrass.com