# Simulation of Coupled Heat-Mass Transfer in Sea Cucumbers with Heat Pump Drying
## 2 Materials and Methods
### Governing Equations  
The sea cucumber is a heterogeneous porous medium [20]. Its volume component is defined as  
\[V=V_{s}+V_{\rm w}+V_{g} \tag{4}\]  
The porosity \(\phi\) is described as  
\[\phi=\frac{V_{\rm w}+V_{g}}{V} \tag{5}\]  
where \(V_{s}\) is the solid phase component, \(V_{\rm w}\) is the liquid phase component, \(V_{g}\) is the gas phase component, and \(\phi\) is the porosity. The water saturation \(S_{\rm w}\) and the air saturation \(S_{\rm g}\) are defined as the volume fractions of water and gas in the pore volume, respectively. \[S_{\rm w}=\frac{\Delta V_{\rm w}}{\Delta V_{\rm w}+\Delta V_{\rm g}}=\frac{ \Delta V_{\rm w}}{\phi\Delta V} \tag{6}\]  
\[S_{\rm g}=\frac{\Delta V_{\rm g}}{\Delta V_{\rm w}+\Delta V_{\rm g}}=\frac{ \Delta V_{\rm g}}{\phi\Delta V} \tag{7}\]  
The water concentration of the porous medium is defined as [21]  
\[c_{w}=S_{w}\times\phi\times\rho_{w}/M_{w} \tag{8}\]  
where \(\phi\) is the porosity, \(\rho_{w}\) is the density of the fluid, \(S_{w}\) is the water saturation, and \(M_{w}\) is the molecular weight. The vapor concentration at ambient temperature is defined as [21]  
\[c_{v}=610.7*10^{\circ}(7.5*(T-273.1)/(T-35.8))/R/T \tag{9}\]  
where \(T\) is the air temperature, \(R\) is the gas constant, and \(c_{v}\) is the vapor concentration. The evaporation source phase \(R_{evap}\) is defined as [21]  
\[R_{evap}=K_{evap}\frac{M_{w}}{RT}(c_{v,sat}-c_{v})S_{g}\varnothing \tag{10}\]  
where \(K_{evap}\) is the evaporation rate constant, \(M_{w}\) is the molecular weight, \(c_{v}\) is the vapor concentration, and \(c_{v,sat}\) is the saturated vapor concentration. The dry basis moisture content of sea cucumber is defined as [21]  
\[M_{db}=\frac{c_{\rm w}+c_{v}}{(1-\phi)\rho_{\rm s}} \tag{11}\]  
where \(M_{db}\) is the moisture content on dry basis, \(c_{w}\) is the water concentration, \(c_{v}\) is the vapor concentration, \(\varnothing\) is the porosity, and \(\rho_{\rm s}\) is the density of the solid skeleton. The momentum equation in the drying oven is  
\[\rho_{g}\frac{\partial u}{\partial t}+\rho_{g}(u\cdot\
abla)u=-\
abla p+ \
abla\cdot\left[\left(\mu_{\rm g}+\mu_{\rm T}\right)\left(\
abla u+\left( \
abla u\right)^{\rm T}\right)\right.-\frac{2}{3}(\mu_{\rm g}+\mu_{\rm T}) \
abla\cdot u] \tag{12}\]  
The momentum equation in the porous medium region is  
\[ \begin{array}{rl}\frac{\rho_{g}}{S_{g}\phi}\frac{\partial u}{\partial t}&+ \frac{\rho_{g}}{S_{g}\phi}(u\cdot\
abla)u\frac{1}{\phi S_{\rm g}}=-\
abla p+ \
abla\cdot\left[\frac{\mu_{g}}{\phi S_{\rm g}}\left(\
abla u+\left(\
abla u \right)^{\rm T}\right)\right.\\ &\left.-\frac{2}{3}\frac{\mu_{g}}{\phi S_{\rm g}}(\
abla\cdot u)\right]-\left( \mu_{g}{k_{\rm gr}}^{-1}+\frac{1}{2}\rho_{g}F|u|\right)u\end{array}  \tag{13}\]  
where \(\rho_{g}\) is the density \(\mu_{g}\) is the dynamic viscosity of air, \(\mu_{T}\) is the turbulent viscosity, \(t\) is the time, \(F\) is the relaxation function, and \(\varnothing\) is the porosity. The mass transfer of vapor occurs in the drying oven and porous media, whereas that of water occurs only in the porous media. The mass equations for the drying oven and porous media are described as follows:  
Mass equations for vapor and water in the porous media:  
\[\frac{\partial c_{y}}{\partial t}+\
abla\cdot n_{\rm v}=R_{\rm evap} \tag{14}\]  
\[\frac{\partial c_{w}}{\partial t}+\
abla\cdot n_{w}=-R_{\rm evap} \tag{15}\]  
Mass equation for vapor in the drying oven:  
\[\frac{\partial c_{y}}{\partial t}+\
abla\cdot n_{\rm v}=0 \tag{16}\]  
where \(n_{\rm v}\) and \(n_{w}\) are the mass fluxes of vapor and water, respectively. \(c_{w}\) is the water concentration, and \(c_{v}\) is the vapor concentration.