Heat exchanger wall¶
The wall separates the hot fluid circulating in the tube from the cold water out of the tube. Heat is exchanged between both fluids through the wall.
The heat exchanger wall models the heat flow through a cylindric pipe wall under following assumptions:
The energy accumulation is considered in each mesh cell.
The heat flow through the wall is supposed to be positive when it is going from the outside to the inside of the pipe.
The phenomenon of longitudinal heat conduction in the wall is neglected.
The thermal conductivity of the wall is constant in space and time. The two latter hypotheses justify the 1D-modeling of the cylindric wall.
Modelica component model¶
The equations mentioned below are implemented in the component HeatExchangerWall, located in the Thermal.HeatTransfer sub-library. This component has 2 connectors:
WT1: thermal port on internal side,
WT2: thermal port on external side.
Nomenclature¶
| Symbol | Description | Unit | Definition | Modelica name |
|---|---|---|---|---|
| \(c_{p, w}\) | Specific heat capacity of the wall | \(\mathrm{J} / \mathrm{kg} / \mathrm{K}\) | cpw | |
| \(D\) | Internal diameter of the pipes | \(\mathrm{m}\) | D | |
| \(e\) | Wall thickness | \(\mathrm{m}\) | e | |
| \(L\) | Pipe length | \(\mathrm{m}\) | L | |
| \(N_{\mathrm{S}}\) | Number of sections inside the wall | \(-\) | Ns | |
| \(N_{\mathrm{t}}\) | Number of pipes in parallel | \(-\) | ntubes | |
| \(T_{\mathrm{m}}\) | Melting temperature of the tubes metal | \(\mathrm{K}\) | - | |
| \(T_{\mathrm{w}, i}\) | Average wall temperature in section \(i\) | \(\mathrm{K}\) | Tp[i] | |
| \(T_{\mathrm{w} 1, i}\) | Wall temperature in section \(i\) of side 1 (internal wall side) | \(\mathrm{K}\) | Tp1[i] | |
| \(T_{\mathrm{w} 2, i}\) | Wall temperature in section \(i\) of side 2 (external wall side) | \(\mathrm{K}\) | Tp1[i] | |
| \(\Delta M_{\mathrm{w}}\) | Mass of a wall section | \(\mathrm{kg}\) | dM | |
| \(\Delta W_{l, i}\) | Thermal power transferred by conduction from the center of the wall to the wall internal surface, for each section \(i\) | \(\mathrm{W}\) | dW1[i] | |
| \(\Delta W_{2, i}\) | Thermal power transferred by conduction from the wall external surface to the center of the wall, for each section \(i\) | \(\mathrm{W}\) | dW2[i] | |
| \(\Delta x\) | Wall section length | \(\mathrm{m}\) | \(L / N_{\mathrm{S}}\) | dx |
| \(\lambda_{\mathrm{w}}\) | Wall thermal conductivity | \(\mathrm{W} / \mathrm{m} / \mathrm{K}\) | cpw | |
| \(\rho_{\mathrm{w}}\) | Wall density | \(\mathrm{kg} / \mathrm{m}^{3}\) | rhow |
Governing equations¶
The heat flux in the wall is computed using the formulation of Fourier’s equation expressed in cylindrical coordinates.
Dynamic energy balance equation for the wall¶
Validity domain:
\(\forall T_{\mathrm{w}, i}\)
Mathematical formulation:
\[\Delta M_{\mathrm{w}} \cdot c_{p, \mathrm{w}} \cdot \frac{\mathrm{d} T_{\mathrm{w}, i}}{\mathrm{d} t}=\Delta W_{2, i}\Delta W_{l, i}\]
Comments:
\(\Delta M_{\mathrm{w}}\) is given by \(\Delta M_{\mathrm{W}}=N_{\mathrm{t}} \cdot \rho_{\mathrm{W}} \cdot \pi \cdot \frac{(D+2 \cdot e)^{2}D^{2}}{4} \cdot \Delta x\)
Fourier’s equation in cylindrical coordinates (conduction through the internal side of the wall)¶
Validity domain:
\(\forall T_{\mathrm{w}, i}\) and \(\forall T_{\mathrm{w} 1, i}\)
Mathematical formulation:
\[\Delta W_{l, i}=N_{\mathrm{t}} \cdot \lambda_{\mathrm{w}} \cdot \frac{2 \cdot \pi \cdot \Delta x}{\ln ((e+D) / D)} \cdot\left(T_{\mathrm{w}, i}T_{\mathrm{w} 1, i}\right)\]
Comments:
Fourier’s equation in cylindrical coordinates (conduction through the external side of the wall)¶
Validity domain:
\(\forall T_{\mathrm{w}, i}\) and \(\forall T_{\mathrm{w} 2, i}\)
Mathematical formulation:
\[\Delta W_{2, i}=N_{\mathrm{t}} \cdot \lambda_{\mathrm{w}} \cdot \frac{2 \cdot \pi \cdot \Delta x}{\ln ((2 \cdot e+D) /(e+D))} \cdot\left(T_{\mathrm{w} 2, i}T_{\mathrm{w}, i}\right)\]
Comments:
References¶
El Hefni, Baligh and Bouskela, Daniel (2019). Modeling and Simulation of Thermal Power Plants with ThermoSysPro, sect. 9.4.4. Springer Nature Switzerland AG.