# Liquid Nanofilms' Condensation Inside a Heat Exchanger by Mixed Convection
## 2 Numerical Modelling
### The Dimensionless Governing Equations
#### 2.2.3 Boundary Conditions  
* For the inlet conditions (at \(\xi=0\)): \[\begin{array}{c}T(0,\eta)=T_{0};\ c(0,\eta)=c_{0};\ u(0,\eta)=u_{0};\ p(0,\eta)=p_{0}\end{array}\] (26) \[\begin{array}{c}T_{L}(0,\eta_{L})=T_{0L};\ \int\limits_{0}^{1}\rho_{0L}\ \delta_{0}\ u_{L}(0,\eta_{L})\ d \eta_{L}=m_{0L};\ \Phi=\Phi_{0}\end{array}\] (27)
* For the dry plate (at \(\eta\) = 1): \[\begin{array}{c}u(\xi,1)=v(\xi,1)=0;\ \frac{\partial c}{\partial\eta} \Bigg{)}_{\eta=1}=0;\ T(\xi,1)=T_{w}\end{array}\] (28)
* For the wet plate (at \(\eta_{L}\) = 0): \[\begin{array}{c}u_{nf}(\xi,0)=v_{nf}(\xi,0)=0;q_{w}=-\lambda_{L}\frac{1}{ \delta}\frac{\partial T_{L}}{\partial\eta_{L}}\Bigg{)}_{\eta_{L}=0};D_{B} \frac{\partial\Phi}{\partial\eta_{L}}=-\frac{D_{T}}{T_{nf}}\frac{\partial T_{nf }}{\partial\eta_{L}}\end{array}\] (29)
* For the gas-liquid interface (at \(\eta\) = 0 and \(\eta_{L}\) = 1): \[\begin{array}{c}u_{nf}(\xi,\eta_{L}=1)=v_{nf}(\xi,\eta=0);\ T_{nf}(\xi,\eta_ {L}=1)=T_{nf}(\xi,\eta=0)\end{array}\] (30)  
\[\begin{array}{c}v(\xi,\eta=0)=-\frac{D}{1-c(\xi,\eta=0)}\frac{\partial c}{ \partial\eta}\Bigg{)}_{\eta=0},\ \mbox{where}\ c(\xi,\eta=0)=\ c_{sat}(T(\xi,\eta=0)),\end{array} \tag{31}\]  
where \(\begin{array}{c}c_{sat}(T(\xi,\eta=0))=\frac{M_{0}}{M_{S}}\frac{P_{\frac{1}{ \eta}}}{P_{S}}\end{array}\).  
The continuities of the shear stress give:  
\[\begin{array}{c}\frac{1}{\delta}\mu_{L}\frac{\partial u_{L}}{\partial\eta_{ L}}\Bigg{)}_{\eta_{L}=1}=\frac{1}{d-\delta}\mu\frac{\partial u}{\partial\eta} \Bigg{)}_{\eta=0}\end{array} \tag{32}\]The energy balance at the liquid-gas interface is:  
\[-\frac{1}{\delta}\lambda_{nf}\frac{\partial T_{nf}}{\partial\eta_{L}}\bigg{)}_{ \eta_{L}=1}=-\frac{1}{d-\delta}\lambda\frac{\partial T}{\partial\eta}\bigg{)}_{ \eta=0}-\frac{\frac{1}{d-\delta}\rho L_{v}D\frac{\partial\zeta}{\partial\eta} \bigg{)}_{\eta=0}}{1-c(\xi,\eta=0)} \tag{33}\]  
We introduce the following quantities:  
* The latent heat flux is given by: \[q_{L}=-\frac{\rho\ L_{v}D\ \frac{\partial\zeta}{\partial\eta}\bigg{)}_{\eta=0}}{(d- \delta)(1-c(\xi,\eta=0))}\] (34)
* The sensible heat flux is given by: \[q_{s}=-\lambda\frac{1}{d-\delta}\frac{\partial T}{\partial\eta}\bigg{)}_{\eta=0}\] (35)
* The local condensation rate is: \[\overset{\bullet}{m}(\xi)=-\frac{\rho\ D\ \frac{\partial\zeta}{\partial\eta} \bigg{)}_{\eta=0}}{(d-\delta)(1-c(\xi,\eta=0))}\] (36)
* The cumulated condensation rate is: \[Mr(\xi)=\int\limits_{0}^{x}\overset{\bullet}{m}(\xi)d\xi\] (37)  
The density \(\rho_{nf}\) and heat capacity \((\rho c_{P})_{nf}\) of the nanofluids are obtained from the following equations:  
\[\rho_{nf}=\Phi\rho_{p}+(1-\Phi)\rho_{L} \tag{38}\]  
\[(\rho c_{P})_{nf}=\Phi\rho_{p}c_{PP}+(1-\Phi)\rho_{L}c_{PL} \tag{39}\]  
The Maxwell model gives the thermal conductivity of nanofluid by:  
\[\frac{\lambda_{nf}}{\lambda_{L}}=\frac{\lambda_{P+}2\lambda_{L}-2\Phi(\lambda _{L}-\lambda_{P})}{\lambda_{P+}2\lambda_{L}+\Phi(\lambda_{L}-\lambda_{P})} \tag{40}\]  
The effective dynamic viscosity of the nanofluid is calculated according to the Brinkman model:  
\[\mu_{nf}=\frac{\mu_{L}}{(1-\Phi)^{2.5}} \tag{41}\]