The governing equations for the steady state condition are obtained by dropping the time dependent terms. Also, by definition w (the non-dimensional flow rate) is unity at steady state. Therefore, the steady state momentum and energy equations can be written as:

The steady state solutions for the temperature of the various segments of the loop can be obtained from Eq. (13) (see Vijayan (1999)). Using these steady state solutions, the integral in

equation (12) can be calculated as (jd dZ = 1,for the loop shown in Fig. 1. Hence, the steady

state flow rate can be expressed as

Where C=(2/p)r and r = 1/(3-b). Thus, knowing the value of p and b the constants C and r in equation (14) can be estimated. For laminar flow (where p=64, and b=1) equation (14) can be rewritten as

Similarly, assuming Blassius friction factor correlation (p=0.316 and b=0.25) to be valid for turbulent flow we can obtain the following equation

In the transition region, one can expect a continuous change in the exponent of equation (14) from 0.5 to 0.364 as well as for the constant from 0.1768 to 1.96. It may be noted that both equations (15) and (16) are the exact solutions of equation (12) assuming the same friction factor correlation to be applicable to the entire loop. For closed loops, often the same friction factor correlation may not be applicable for the entire loop even if the loop is fully laminar or fully turbulent. An example is a fully laminar loop with part of the loop having rectangular cross section and remaining part having circular cross section. Hence, one has to keep in mind the assumptions made in deriving the relationship (14) while applying it.

The relationship expressed by Eq. (14) is derived for a rectangular loop with both the heater and cooler having the horizontal orientation. For other orientations also, it can be easily shown that the same relationship holds good if the loop height in the definition of the Grm is replaced with the centre-line elevation difference between the cooler and the heater, Dz (see

Vijayan et al. (2000)). The same is true for other loop geometries like the figure-of-eight loop used in Pressurised Heavy Water Reactors (PHWRs). For the figure-of-eight loop, the heater power used in the Grm is the total power of both heaters. The equations are also applicable for identical parallel-channel or parallel-loops systems if the parallel channels/loops are replaced by an equivalent path having the same hydraulic diameter and total flow area.