Stodola’s law
In its most common definition, Stodola’s law, or the cone rule, expresses, for a steam turbine operating at constant speed, a nearly quadratic relationship between mass flow rate and upstream and downstream pressures, which results in an almost linear dependence between flow rate and inlet pressure for a fixed downstream pressure. It is often used in a simplified form that relates the reduced flow to the square root of the difference of the squares of the pressure roots, and serves as the natural behavior law of condensing turbines in off-design operation.
As we will see, this definition is only valid when it can be assumed that the working fluid can be treated as an ideal gas.
In this case, and if no specific assumptions are made about the relationship between upstream and downstream pressures, the expression of Stodola’s law is as follows:
$$ \frac{{T_{\text{in}}} \dot{m}^2}{K^2} = P_{\text{in}}^2 - P_{\text{out}}^2 \tag{9.4.11}$$In this expression, the values with the ‘in’ subscript relate to the turbine’s inlet, and those with the ‘out’ subscript to its outlet. K is called the Stodola constant.
If we assume that the downstream pressure is negligible compared to the upstream pressure, it simplifies and becomes:
$$ \frac{\dot{m} \sqrt{T_{\text{in}}}}{P_{\text{in}}} = \text{Const} \tag{9.4.12} $$It can be shown that, in the general case, it can be expressed as follows:
$$ \dot{m} \sqrt{\frac{(k+1)v_{\text{in}}}{k P_{\text{in}}} } = K_0 \sqrt{1 - \left[\frac{P_{\text{out}}}{P_{\text{in}}}\right]^{(k+1)/k}} \tag{9.4.14} $$The main differences are that if k is the coefficient of the law passing through the upstream and downstream points (which has no reason to be a polytropic):
the temperature disappears from the square root on the left-hand side of the equation
the exponent of the pressure ratio that was 2 is replaced by the term (k+1)/k, which depends on the properties of the fluid and therefore varies according to the turbine’s inlet and outlet conditions
the square root of the inverse of this term multiplies Stodola’s ‘constant’, which therefore also varies depending on the operating conditions of the turbine.
This expression is much more complex than the simplified Stodola law usually used, which corresponds to equation (9.4.12).