  # EQUATRAN Tutorial 4: Non-linear Simultaneous Equations

In this tutorial we will use the previous example of a heat exchanger calculation and add more detail. This introduces non-linearity into the system of equations.

The heat transfer rate can be related to the overall heat transfer coefficient U and the logarithmic mean temperature difference ΔT by the equation:

 Q = UAΔTlm

For a simple counter-current exchanger the log mean temperature difference is calculated from the temperatures of the streams entering and leaving the heat exchanger:

 ΔTlm = ((Thin-Tcout)-(Thout-Tcin))/ln((Thin-Tcout)/(Thout-Tcin))

We can now calculate the value of the heat transfer coefficient U, given that the heat transfer area A is known. The revised source text is:

 // Heat Exchanger Calculation Q = Fh*Cph*(Thin-Thout) Q = Fc*Cpc*(Tcout-Tcin) Q = U*A*DTlm DTlm = ((Thin-Tcout)-(Thout-Tcin))/LOGE((Thin-Tcout)/(Thout-Tcin)) Cph = 0.62;  Cpc = 1.0 // Input Fh = 1000              // Flow rate Thin = 60;  Thout = 30 // Temperatures Tcin = 18;  Tcout = 29 A = 1.5

Results:

 // Heat Exchanger Calculation <<   results   >> Q               =  18600          Fc              =  1690.909       U               =  619.3999

That was still a simple calculation, but now we will use the same set of equations with different variables as the unknowns. We will specify the flows and intlet temperatures of both streams, and the values of U and A. EQUATRAN must now calculate the heat duty and the two outlet temperatures, but this is a non-linear problem and cannot be solved directly by rearranging the equations.

Modify the input and output sections of the source text, but leave the equations unchanged.

 // Heat Exchanger Calculation Q = Fh*Cph*(Thin-Thout) Q = Fc*Cpc*(Tcout-Tcin) Q = U*A*DTlm DTlm = ((Thin-Tcout)-(Thout-Tcin))/LOGE((Thin-Tcout)/(Thout-Tcin)) Cph = 0.62;  Cpc = 1.0 // Input Fh = 1000;  Fc = 1800  // Flow rate Thin = 60;  Tcin = 18; // Temperatures A = 0.5 U = 500 OUTPUT Q, Thout, Tcout

Now when the problem is run, EQUATRAN must estimate a value for Q and solve the problem iteratively. The user is prompted to enter an initial estimate for Q. Enter the initial value as 1, then click ‘Go’.  The new results are:

 // Heat Exchanger Calculation <<   results   >> Q               =  8223.211       Thout           =  46.73676       Tcout           =  22.56845