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Next: Radial Model Up: THE THROHPUT CODE Previous: Summary Description of THROHPUT

Axial Model Equation Set

Currently, THROHPUT uses the area-averaged Navier-Stokes equations to model the behavior of the working fluid (solid, liquid and gas) and the noncondensible gas. Submodels are used to calculate the effects of interphasic transfer processes (Radial Model), diffusive terms, friction loss terms, state equations, the axial capillary force term, and the capillary pressure relationship (Surface Model). All of these submodels are formulated in such a way that they can be inserted into the Axial Model equation set and solved simultaneously, in a fully implicit manner.

The Axial Model equation set models the overall thermohydraulic behavior of the heat pipe system using fifteen separate equations at each node. These equations are:

Continuity (Conservation of Mass) Equations

Mixture of Gases (Working Fluid and Noncondensible) Continuity:

$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_m\rho_m }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$$\displaystyle \left.\vphantom{ \alpha_m\rho_m }\right)$ + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m\rho_m V_m }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$Vm$\displaystyle \left.\vphantom{ \alpha_m\rho_m V_m }\right)$ = $\displaystyle \sum_{{x=l,s}}^{}$$\displaystyle \Gamma_{{xg}}^{}$ (1)
Noncondensible Continuity:

$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_m\rho_m X_n }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$Xn$\displaystyle \left.\vphantom{ \alpha_m\rho_m X_n }\right)$ + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m\rho_m X_n V_m }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$XnVm$\displaystyle \left.\vphantom{ \alpha_m\rho_m X_n V_m }\right)$ = $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m D_n^X \frac{\partial X_n}{\partial z} }\right.$$\displaystyle \alpha_{m}^{}$DnX$\displaystyle {\frac{{\partial X_n}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_m D_n^X \frac{\partial X_n}{\partial z} }\right)$ + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m D_n^\rho \frac{\partial \rho_m}{\partial z} }\right.$$\displaystyle \alpha_{m}^{}$Dn$\scriptstyle \rho$$\displaystyle {\frac{{\partial \rho_m}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_m D_n^\rho \frac{\partial \rho_m}{\partial z} }\right)$ (2)
Liquid Continuity:

$\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_l\rho_l }\right.$$\displaystyle \alpha_{l}^{}$$\displaystyle \rho_{l}^{}$$\displaystyle \left.\vphantom{ \alpha_l\rho_l }\right)$ + $\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_l\rho_l V_l }\right.$$\displaystyle \alpha_{l}^{}$$\displaystyle \rho_{l}^{}$Vl$\displaystyle \left.\vphantom{ \alpha_l\rho_l V_l }\right)$ = $\displaystyle \sum_{{x=g,s}}^{}$$\displaystyle \Gamma_{{xl}}^{}$ (3)
Solid Continuity:

$\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_s\rho_s }\right.$$\displaystyle \alpha_{s}^{}$$\displaystyle \rho_{s}^{}$$\displaystyle \left.\vphantom{ \alpha_s\rho_s }\right)$ = $\displaystyle \sum_{{x=g,l}}^{}$$\displaystyle \Gamma_{{xs}}^{}$ (4)

Conservation of Internal Energy Equations

Mixture of Gases (Working Fluid and Noncondensible) Internal Energy:

    $\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_m \rho_m U_m }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$Um$\displaystyle \left.\vphantom{ \alpha_m \rho_m U_m }\right)$ + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m \rho_m U_m V_m }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$UmVm$\displaystyle \left.\vphantom{ \alpha_m \rho_m U_m V_m }\right)$ = - Pm$\displaystyle \left(\vphantom{ \frac{\partial}{\partial z} \left( \alpha_m V_m \right) + \frac{\partial \alpha_m}{\partial t} }\right.$$\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m V_m }\right.$$\displaystyle \alpha_{m}^{}$Vm$\displaystyle \left.\vphantom{ \alpha_m V_m }\right)$ + $\displaystyle {\frac{{\partial \alpha_m}}{{\partial t}}}$$\displaystyle \left.\vphantom{ \frac{\partial}{\partial z} \left( \alpha_m V_m \right) + \frac{\partial \alpha_m}{\partial t} }\right)$ (5)
    + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m D_n^X \left( h_n-h_g \right) \frac{\partial X_n}{\partial z} }\right.$$\displaystyle \alpha_{m}^{}$DnX$\displaystyle \left(\vphantom{ h_n-h_g }\right.$hn - hg$\displaystyle \left.\vphantom{ h_n-h_g }\right)$$\displaystyle {\frac{{\partial X_n}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_m D_n^X \left( h_n-h_g \right) \frac{\partial X_n}{\partial z} }\right)$ + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m D_n^\rho \left( h_n-h_g \right) \frac{\partial \rho_m}{\partial z} }\right.$$\displaystyle \alpha_{m}^{}$Dn$\scriptstyle \rho$$\displaystyle \left(\vphantom{ h_n-h_g }\right.$hn - hg$\displaystyle \left.\vphantom{ h_n-h_g }\right)$$\displaystyle {\frac{{\partial \rho_m}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_m D_n^\rho \left( h_n-h_g \right) \frac{\partial \rho_m}{\partial z} }\right)$  
    + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m k_m \frac{\partial T_m}{\partial z} }\right.$$\displaystyle \alpha_{m}^{}$km$\displaystyle {\frac{{\partial T_m}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_m k_m \frac{\partial T_m}{\partial z} }\right)$ + $\displaystyle \sum_{{x=l,s}}^{}$$\displaystyle \left(\vphantom{ Q_{xm} + Q^\Gamma_{xg} }\right.$Qxm + Q$\scriptstyle \Gamma$xg$\displaystyle \left.\vphantom{ Q_{xm} + Q^\Gamma_{xg} }\right)$  

Liquid Internal Energy:
    $\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_l\rho_l U_l }\right.$$\displaystyle \alpha_{l}^{}$$\displaystyle \rho_{l}^{}$Ul$\displaystyle \left.\vphantom{ \alpha_l\rho_l U_l }\right)$ + $\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_l\rho_l U_l V_l }\right.$$\displaystyle \alpha_{l}^{}$$\displaystyle \rho_{l}^{}$UlVl$\displaystyle \left.\vphantom{ \alpha_l\rho_l U_l V_l }\right)$ = - $\displaystyle \epsilon_{v}^{}$Pl$\displaystyle \left(\vphantom{ \frac{\partial \alpha_l V_l}{\partial z} + \frac{\partial \alpha_l}{\partial t} }\right.$$\displaystyle {\frac{{\partial \alpha_l V_l}}{{\partial z}}}$ + $\displaystyle {\frac{{\partial \alpha_l}}{{\partial t}}}$$\displaystyle \left.\vphantom{ \frac{\partial \alpha_l V_l}{\partial z} + \frac{\partial \alpha_l}{\partial t} }\right)$ (6)
    + $\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_l k_l \frac{\partial T_l}{\partial z} }\right.$$\displaystyle \alpha_{l}^{}$kl$\displaystyle {\frac{{\partial T_l}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_l k_l \frac{\partial T_l}{\partial z} }\right)$ + $\displaystyle \sum_{{x=m,s,w}}^{}$Qxl + $\displaystyle \sum_{{x=g,s}}^{}$Qxl$\scriptstyle \Gamma$  

Solid Internal Energy:

$\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_s\rho_s U_s }\right.$$\displaystyle \alpha_{s}^{}$$\displaystyle \rho_{s}^{}$Us$\displaystyle \left.\vphantom{ \alpha_s\rho_s U_s }\right)$ = $\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_s k_s \frac{\partial T_s}{\partial z} }\right.$$\displaystyle \alpha_{s}^{}$ks$\displaystyle {\frac{{\partial T_s}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_s k_s \frac{\partial T_s}{\partial z} }\right)$ + $\displaystyle \sum_{{x=m,l,w}}^{}$Qxs + $\displaystyle \sum_{{x=g,l}}^{}$Qxs$\scriptstyle \Gamma$ (7)
Wall Internal Energy:

$\displaystyle \rho_{w}^{}$cpw$\displaystyle {\frac{{\partial T_w}}{{\partial t}}}$ = $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_w k_w \frac{\partial T_w}{\partial z} }\right.$$\displaystyle \alpha_{w}^{}$kw$\displaystyle {\frac{{\partial T_w}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \alpha_w k_w \frac{\partial T_w}{\partial z} }\right)$ + Qin + $\displaystyle \sum_{{x=l,s}}^{}$Qxw (8)

Conservation of Momentum Equations

Mixture of Gases (Working Fluid and Noncondensible) Momentum:

$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_m\rho_m V_m }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$Vm$\displaystyle \left.\vphantom{ \alpha_m\rho_m V_m }\right)$ + $\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_m \rho_m V_m^2 }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$Vm2$\displaystyle \left.\vphantom{ \alpha_m \rho_m V_m^2 }\right)$ = - $\displaystyle \alpha_{m}^{}$$\displaystyle {\frac{{\partial P_m}}{{\partial z}}}$ - $\displaystyle \mathcal {F}$mVm + $\displaystyle \alpha_{m}^{}$$\displaystyle \rho_{m}^{}$gz (9)
Liquid Momentum:

$\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \left(\vphantom{ \alpha_l\rho_l V_l }\right.$$\displaystyle \alpha_{l}^{}$$\displaystyle \rho_{l}^{}$Vl$\displaystyle \left.\vphantom{ \alpha_l\rho_l V_l }\right)$ + $\displaystyle \epsilon_{v}^{}$$\displaystyle {\frac{{\partial}}{{\partial z}}}$$\displaystyle \left(\vphantom{ \alpha_l \rho_l V_l^2 }\right.$$\displaystyle \alpha_{l}^{}$$\displaystyle \rho_{l}^{}$Vl2$\displaystyle \left.\vphantom{ \alpha_l \rho_l V_l^2 }\right)$ = - $\displaystyle \epsilon_{v}^{}$$\displaystyle \alpha_{l}^{}$$\displaystyle {\frac{{\partial P_l}}{{\partial z}}}$ - $\displaystyle \mathcal {F}$lVl - $\displaystyle \Delta$Pcap$\scriptstyle \nu$$\displaystyle {\frac{{\partial \alpha_l}}{{\partial z}}}$ + $\displaystyle \epsilon_{v}^{}$$\displaystyle \alpha_{l}^{}$$\displaystyle \rho_{l}^{}$gz (10)

Thermodynamic State Equations

Mixture of Gases (Working Fluid and Noncondensible) State:

Pm = $\displaystyle \rho_{m}^{}$Tm$\displaystyle \left(\vphantom{ X_n R_n+\left( 1-X_n \right) R_g }\right.$XnRn + $\displaystyle \left(\vphantom{ 1-X_n }\right.$1 - Xn$\displaystyle \left.\vphantom{ 1-X_n }\right)$Rg$\displaystyle \left.\vphantom{ X_n R_n+\left( 1-X_n \right) R_g }\right)$ (11)
Liquid State:

$\displaystyle \rho_{l}^{}$ = $\displaystyle \rho_{l}^{}$$\displaystyle \left(\vphantom{ P_l,T_l }\right.$Pl, Tl$\displaystyle \left.\vphantom{ P_l,T_l }\right)$ (12)
Solid State:

$\displaystyle \rho_{s}^{}$ = $\displaystyle \rho_{s}^{}$$\displaystyle \left(\vphantom{ T_s }\right.$Ts$\displaystyle \left.\vphantom{ T_s }\right)$ (13)

Closure Equations

Volume Fraction Sum:

$\displaystyle \sum_{{x=m,l,s}}^{}$$\displaystyle \alpha_{x}^{}$ = 1 (14)
Capillary Pressure Relation:

Pm - Pl = $\displaystyle \mathcal {L}$$\displaystyle \left(\vphantom{ \Delta P_{cap} \left( \alpha_m \right) }\right.$$\displaystyle \Delta$Pcap$\displaystyle \left(\vphantom{ \alpha_m }\right.$$\displaystyle \alpha_{m}^{}$$\displaystyle \left.\vphantom{ \alpha_m }\right)$$\displaystyle \left.\vphantom{ \Delta P_{cap} \left( \alpha_m \right) }\right)$ (15)

The system variables in these equations are: $ \rho_{m}^{}$, $ \rho_{l}^{}$, $ \rho_{s}^{}$, Xn, $ \alpha_{m}^{}$, $ \alpha_{l}^{}$, $ \alpha_{s}^{}$, Pm, Pl, Vm, Vl, Tm, Tl, Ts and Tw. The interphase transfer terms ( $ \Gamma_{{xy}}^{}$, Qxy, Qxy$\scriptstyle \Gamma$) are all functions of $ \rho_{m}^{}$, Xn, Tm, Tl, Ts, and Tw and are defined in the Radial Model. A volume-based porosity has been included in the appropriate terms of the liquid and solid equations to account for the volume taken up by the wick structure. The internal energies of the phases are expressed in terms of the temperatures using other state equations in the final model. Boundary conditions for the Axial Model are simply the specification of zero velocity and zero heat flux on both ends of the heat pipe.

The axial model equation set is spatially discretized using a staggered mesh, with cell-boundary values for cell-centered properties being donored according to the appropriate velocity. The temporal discretization in THROHPUT is done in a fully implicit manner, resulting in a nonlinear equation which is solved via a Newton iteration. The Jacobian system matrix is block tridiagonal, which allows many terms to be differenced implicitly without significantly adding to the complexity of the solution scheme. Multiple Newton iterations are taken through the Jacobian system to solve the nonlinear system.

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Next: Radial Model Up: THE THROHPUT CODE Previous: Summary Description of THROHPUT
Michael L. Hall