3.4. Steady-State Thermal Hydraulics

The steady-state thermal hydraulics calculations for a channel using the single pin per subassembly option consist of direct solutions of the relevant steady-state equations, rather than running the transient calculations until they converge to a steady-state solution.

For the steady-sate calculations, the user specifies the coolant flow rate for each channel, the coolant inlet temperature and exit pressure, and the power in each node of each channel. The code then calculates the remaining coolant temperatures and pressures, as well as the temperatures in the fuel, cladding, structure, and reflectors. First, the coolant temperatures in a channel are calculated, starting at the inlet and working upward. The steady-state coolant temperature calculation requires only the coolant flow rate, the total power in each axial node, and the coolant heat capacity; so coolant temperatures can be calculated before the fuel and cladding temperatures are known. The second step is to calculate the coolant pressures, starting at the subassembly outlet and working down. Inlet orifice coefficients are adjusted so that all channels have the same total pressure drop. The pressure calculations are described in Section 3.9. The third step is to set the structure and reflector temperatures equal to the coolant temperatures everywhere except in the core and axial blankets. The gas plenum temperatures are also set equal to the coolant temperature in this region, and the cladding temperature in the gas plenum region is also set equal to the coolant temperature. Fourth, the fuel-pin temperatures are calculated for each axial node in the core and axial blankets, starting at the cladding outer surface and working inward. Last, the structure temperatures in the core and axial blankets are calculated.

3.4.1. Basic Equations

The basic heat-transfer equations used in the steady-state calculations are the same as those used for the transient solution, except that all of the time derivatives are dropped in the steady-state solution. These equations include Eq. 3.3-1 and Eqs. 3.4-4 to Eqs. 3.3-10. Also, the spatial finite differencing used in the steady-state is the same as that used in the transient.

For the steady-state calculations, eq. Eqs. 3.3-5 becomes

(3.4‑1)

\[\frac{\text{d}}{\text{dz}} \left( wc_{\text{c}}T \right) = Q_{\text{ct}}A_{\text{c}}\]

where the total heat source per unit volume, \(Q_{\text{ct}}\), at node \(jc\) is

(3.4‑2)

\[Q_{\text{ct}} \left( jc \right) = Q_{\text{c}} \left( jc \right) + Q_{\text{ec}} \left( jc \right) + Q_{\text{sc}} \left( jc \right) = \frac{ \overline{P} \left( jc \right)}{A_{\text{c}}} \Delta z \left( jc \right)\]

and \(\overline{P}\left( jc \right)\) is the total steady-state power (watts) in the node. For this equation, it is assumed that all heat generated in the fuel, cladding, and structure ends up in the coolant. Note that outside the core and axial blankets \(\overline{P}\left( jc \right)\) and \(Q_{\text{ct}}\left( jc \right)\) are zero.

For the steady-state fuel and cladding calculations, Eq. 3.3-1 is multiplied by \(2 \pi r\) and integrated from the fuel inner surface, \(r_{\text{if}}\), to give

(3.4‑3)

\[2\pi kr \frac{\text{dT}}{\text{dr}} = - 2 \pi \int_{r_{\text{if}}}^{r}{r' Q \left( r' \right)} dr'\]

where the adiabatic boundary condition at \(r_{\text{if}}\) has been used.

3.4.2. Coolant Temperatures

The finite difference form for Eq. 3.4-1 may be written as

(3.4‑4)

\[w{\overline{c}}_{\text{c}}\left( \text{j} \right) \frac{\left\lbrack T_{c}\left( jc + 1 \right) - T_{c}\left( jc \right) \right\rbrack}{\Delta z \left( jc \right)} = \frac{\overline{P}\left( jc \right)}{ \Delta z \left( jc \right)}\]

or

(3.4‑5)

\[T_{\text{c}}\left( jc + 1 \right) = T_{\text{c}}\left( jc \right) + \frac{\overline{P}\left( jc \right)}{w{\overline{c}}_{\text{c}}\left( jc \right)}\]

where \({\overline{c}}_{\text{c}}\left( jc \right)\) is the specific heat evaluated at the average temperature, \({\overline{T}}_{\text{c}}\left( jc \right)\), given by

(3.4‑6)

\[{\overline{T}}_{\text{c}}\left( jc \right) = \frac{T_{\text{c}}\left( jc \right) + T_{\text{c}}\left( jc + 1 \right)}{2}\]

Also, \(T_{\text{c}}\left( 1 \right)\) is equal to the inlet temperature:

(3.4‑7)

\[T_{\text{c}}\left( 1 \right) = T_{\text{in}}\]

Starting from jc = 1, Eq. 3.4-5 is used to match up the channel. An iteration is used to obtain consistency between \({\overline{c}}_{\text{c}}\left( jc \right)\) and \({\overline{T}}_{\text{c}}\left( jc \right)\).

3.4.3. Fuel and Cladding Temperatures in the Core and Axial Blankets

At each axial node, the radial node powers, \(Q(i)\), are calculated using Eqs. 3.3-22 to 3.3-25. Note that the \(Q\) in Eq. 3.4-3 is a power per unit volume, whereas \(Q(i)\) is an integral value for a node:

(3.4‑8)

\[Q \left( i \right) = \int_{z \left( j \right)}^{z \left( j + 1 \right)}\int_{r_{\text{i}}}^{r_{i + 1}}{\ \ 2\pi r\ \ Qdr\ \ \text{dz}}\]

The sums, \(Q_{\text{sm}} (i)\), are calculated as

(3.4‑9)

\[Q_{\text{sm}}\left( i \right) = \sum_{ii = 1}^{i}{\text{Q}\left( ii \right)}\]

Equation 3.4-3 becomes

(3.4‑10)

\[2\pi\ {\overline{k}}_{\text{i,i} + 1}\ \frac{r \left( i + 1 \right) \left \lbrack T \left( i + 1 \right) - T\left( i \right) \right\rbrack}{\Delta r_{\text{i,i} + 1}}\ = \ \frac{Q_{\text{sm}}\left( i \right)}{ \Delta z }\]

or

(3.4‑11)

\[T\left( i \right) = T\left( i + 1 \right) + \frac{\Delta r_{\text{i,i} + 1} Q_{\text{sm}}\left( i \right)}{2\pi k_{\text{i,i} + 1} r \left( i + 1 \right) \Delta z }\]

where \(\Delta r_{i,i + 1}\) and \({\overline{k}}_{i,i + 1}\) are given by Eqs. 3.3-21 and 3.3-26.

The calculations for an axial node start with the coolant temperature that has already been calculated, as in the section above:

(3.4‑12)

\[T \left( \text{NC} ,j \right) = \overline{T} \left( jc \right)\]

Then the cladding surface temperature is given by

(3.4‑13)

\[T\left( \text{NE}' \right) = T\left( \text{NC} \right) + \frac{Q_{\text{sm}}\left( \text{NE}' \right)}{2\pi r\left( \text{NE}' \right) \Delta z~ h_{\text{c}}}\]

Cladding temperatures at nodes \(\text{NE}\) and \(\text{NE}''\) are calculated using Eq. 3.4-11. Since \({\overline{k}}_{\text{ii} + 1}\) can be a function of \(T_{\text{i}}\), a simple iteration between Eq. 3.4-11 and Eq. 3.3-26 is used.

The equation used for the fuel surface temperature is

(3.4‑14)

\[2\pi r\left( \text{NR} \right) \left\{ h_{\text{b}}\left\lbrack T \left( \text{NT} \right) - T\left( \text{NE}'' \right)\ \right\rbrack + \varepsilon\sigma\left\lbrack T\left( \text{NT} \right)^{4} - T\left( \text{NE}'' \right)^{4} \right\rbrack \right\} = \frac{Q_{\text{sm}}\left( \text{NT} \right)}{\Delta z}\]

or

(3.4‑15)

\[T\left( \text{NT} \right) = d_{1} - d_{2} T \left( \text{NT} \right)^{4}\]

where

(3.4‑16)

\[d_{1} = T\left( \text{NE}'' \right) + \frac{Q_{\text{sm}}\left( \text{NT} \right)}{2\pi r\left( \text{NR} \right)\Delta z~ h_{\text{b}}} + \frac{\epsilon\sigma T\left( \text{NE}'' \right)^{4}}{h_{\text{b}}}\]

and

(3.4‑17)

\[d_{2} = \frac{\epsilon \sigma}{h_{\text{b}}}\]

Equation 3.4-15 is solved by iteration.

After the fuel surface temperature has been calculated, the inner fuel node temperatures are calculated one at a time, starting at the outside and working inward, by iterating between Eqs. 3.4-11 and 3.3-26. In this procedure, T(i) is to be found after T(i+1) is known. First, T(i) is set equal to \(T(i + 1)\). Second, k(i) is to be found after T(i+1) is known. First, T(i) is set equal to \(T(i + 1)\). Second, k(i) is calculated as a function of the temperature, T(i). Third, \({\overline{k}}_{\text{i,i} + 1}\)is calculated using Eq. 3.3-26. Fourth, a new value for T(i) is calculated, using Eq. 3.4-11. Fifth, the new T(i) from the fourth step is compared with old value used in the second step. If the two values differ by less than a user-specified convergence criterion, then the iteration is finished, and the code goes on to the next node. Otherwise, the code goes back to the second step, using the new value of T(i), and repeats the process.

3.4.4. Structure Temperatures in the Core Axial Blankets

The inner structure node temperature is calculated using

(3.4‑18)

\[\Delta z~S_{\text{pr}}\ H_{\text{sic}}\ \left\lbrack T \left( \text{NSI} \right) - T \left( \text{NC} \right) \right\rbrack = \gamma_{\text{s}}\overline{P}\left( j \right)\]

or

(3.4‑19)

\[T\left( \text{NSI} \right) = T\left( \text{NC} \right) + \frac{\gamma_{\text{s}}\overline{P}\left( j \right)}{ \Delta z ~S_{\text{pr}}\ H_{\text{sic}}}\]

The outer structure node is then calculated using

(3.4‑20)

\[T\left( \text{NSO} \right) = T\left( \text{NSI} \right) + \frac{\gamma_{\text{s}}\overline{P}\left( j \right)\ d_{\text{sto}}}{ \Delta z ~S_{\text{pr}}\ H_{\text{stio}}\ \left( d_{\text{sti}} + d_{\text{sto}} \right)}\]

3.4.5. Reflector, Structure, Cladding, and Gas Plenum Temperature Outside the Core and Axial Blankets

Outside the core and axial blankets no power sources are considered, so the reflector and structure temperatures at an axial node are the same as the coolant temperature for the steady-state. The coolant temperatures are the same at all axial nodes in the gas plenum region, and the cladding and gas temperatures in this region are equal to the coolant temperatures.