In the operation of a power plant superheater, exacting demands are made on the steam temperature maintenance at the outlet. For temperature control at the outlet of a superheater, the relevant system state is the temperature pattern along the superheater tube. This is described by a distributed-parameter system, which involves an infinite number of state variables. To derive a simplified model for control purposes, the superheater is divided into segments, and a lumped model is derived, which represents a finite number of intermediate temperatures.

Assuming that the pressure inside the tube is constant, the enthalpy of the steam satisfies the relation $dH = C_pdT(kcal/kg)$ , where $C_p(kcal/kg^{\circ}C)$ is the constant-pressure specific heat. Hence, we conclude that the heat supplied to the following fluid(steam) only increases its enthalpy, $dH = dQ$ , where $Q$ denotes the heat. In the above equations, it is assumed that convection is the exclusive heat transfer mode for the superheater. Hence the heat transfer from to metal $Q_{ms}(kcal/s)$ and from gas to metal $Q_{gm}(kcal/s)$ are expressed in terms of the heat transfer rates from gas to metal $\alpha_{gm}(kcal/m^2s^{\circ}C)$ and from metal to steam $\alpha_{ms}(kcal/m^2s^{\circ}C)$ and heating surface $S(m^2)$ :

$$\begin{align*}
\alpha_{ms}S_1(T(l,t)-T(l,t)) &=Q_{ms} \\
\alpha_{gm}S_2(T_m(l,t)-T(l,t)) &=Q_{gm}.
\end{align*}$$

It is also assumed that the heat transfer rates $\alpha_{gm}$ and $\alpha_{ms}$ are constants.

Now, to simulate the profile of superheated steam precisely, it is necessary to divide the superheater into $n$ segments as shown in the attached image.

In the first segment, the desuperheater is included and system is modified as follows:

$$\begin{align*}
V_s\rho C_p\frac{{\mathrm d} x_1}{{\mathrm d}t} &={C_{p}}{T_{i}}{w_{i}}-{C_{p}}({w_{i}}+{w_{d}}){x_{1}} +{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}})+{C_{pd}}{T_{d}}{w_{d}}\\
M_mC_m \frac{{\mathrm d}z_1}{{\mathrm d}t} &={\alpha_{gm}}{S_{2}}(T{g_{1}}-{z_{1}})-{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}}),
\end{align*}$$

where $x=[x_1,x_2,\ldots,x_n]^{\mathrm T}=[T_1,T_2,\ldots,T_n]^{\mathrm T}$, $z=[z_1,z_2,\ldots,z_n]^{\mathrm T}=[T_{m1},T_{m2},\ldots,t_{mn}]^{\mathrm T}$, and $T_{mi}(^{\circ}C)$ are metal temperature, $T_i(^{\circ}C)$ are steam temperature, $i=1,\ldots,n$.