(b) Temperature Transfer Coefficients, Aik The following is a simplified example of the estimation of temperature transfer coefficients for points downstream of a river discharge. It is assumed that the heat is completely mixed with the stream flow at the point of discharge, and that the downstream temperature distribution is largely determined by the combined effects of longitudinal advection and surface cooling (i.e., longitudinal mixing is neglected). T* is first defined as the excess temperature in the stream, i.e., the difference between the temperature of the stream below the discharge point and the undisturbed ambient temperature. The stream is assumed to be of constant width, so that the change in T✶ with respect to the distance travelled from the point of discharge is given by the classical differential equation yielding exponential die-away: where T* is the initial mixed excess temperature (=R;). Since both T* and T* are in degrees, it is evident that the -kx temperature transfer coefficient, e must be a dimensionless quantity involving the appropriate surface cooling parameters. A typical surface cooling coefficient, K, is 150 Btu per square foot of surface area per day per degree above equilibrium temperature (see Reference (1)). The factors required to convert this into the die-away coefficient, k, above, are the stream flow, Q, the surface width, w (ft), the density of water, p, and the specific heat of water, thus: If the stream flow is expressed in terms of its dependence on the depth, h, width, w, and mean velocity, U (ft/sec), this equation becomes: The subroutine reads the coordinates of the possible plant sites and of the sampling stations at which temperature standards must be met, and computes the distance from each plant site to each stream sampling station for insertion in the equation: Table VI shows the output obtained from the zero-one integer model using the basic data for the sample problem (see Table II, Run 4). The input for this run was presented in Table V in Section B3 of this appendix. The first part of the output simply contains the input information in tabular form. First, the load centers are listed with their power demands and distribution cost rates. Next, the transmission cost rates involved in serving each load center from each site are printed. Following this, the plant alternatives are detailed with their location, intercept cost for production, load-dependent production cost rate, maximum generating capacity and mixed temperature rise, i.e., the number of degrees the mixed stream temperature will be raised by each MW of power production at that site. Then the total unit cost rate (per MW) of serving each load center from each alternative is determined by summing the load-dependent production cost rate of the appropriate plant alternative, the associated transmission cost rate and the distribution cost rate of the appropriate load center. The power demand of each load center is then repeated and is multiplied by the total unit cost to determine the total load-dependent cost per week for serving each load center from each plant alternative. After the cost information, the downstream distances of each site and each stream point are listed (in thousands of feet from Site 1), followed by the temperature transfer coefficients from each site to each stream point and the maximum permissible excess temperature at each stream point. The optimal solution to the problem is printed after the input data and includes the total weekly cost of the solution; the plant alternatives to be built, along with their design generating capacity and intercept cost for production; and finally a listing of the load centers to be served by each plant alternative and the corresponding total load-dependent costs. (Note: Error messages of "underflow" may be encountered in the output of the program. These may be ignored, however, since the extremely small numbers resulting in underflow messages are generated as a consequence of the algorithm used to solve the problem.) -B 82 |