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Theoretical Models

This section focuses on the general equations and laws that SWHSNoPCM is based on.

RefnameTM:consThermE
LabelConservation of thermal energy
Equation-∇\cdot{}\boldsymbol{q}+g=ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}
Description
  • is the gradient (Unitless)
  • \boldsymbol{q} is the thermal flux vector (\frac{\text{W}}{\text{m}^{2}})
  • g is the volumetric heat generation per unit volume (\frac{\text{W}}{\text{m}^{3}})
  • ρ is the density (\frac{\text{kg}}{\text{m}^{3}})
  • C is the specific heat capacity (\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}})
  • t is the time ({\text{s}})
  • T is the temperature ({{}^{\circ}\text{C}})
Notes
  • The above equation gives the law of conservation of energy for transient heat transfer in a given material.
  • For this equation to apply, other forms of energy, such as mechanical energy, are assumed to be negligible in the system (A:Thermal-Energy-Only).
SourceFourier Law of Heat Conduction and Heat Equation
RefByGD:rocTempSimp
RefnameTM:sensHtE
LabelSensible heat energy (no state change)
EquationE={C^{\text{L}}}\,m\,ΔT
Description
  • E is the sensible heat ({\text{J}})
  • {C^{\text{L}}} is the specific heat capacity of a liquid (\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}})
  • m is the mass ({\text{kg}})
  • ΔT is the change in temperature ({{}^{\circ}\text{C}})
Notes
  • E occurs as long as the material does not reach a temperature where a phase change occurs, as assumed in A:Water-Always-Liquid.
SourceDefinition of Sensible Heat
RefByIM:heatEInWtr
RefnameTM:nwtnCooling
LabelNewton’s law of cooling
Equationq\left(t\right)=h\,ΔT\left(t\right)
Description
  • q is the heat flux (\frac{\text{W}}{\text{m}^{2}})
  • t is the time ({\text{s}})
  • h is the convective heat transfer coefficient (\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}})
  • ΔT is the change in temperature ({{}^{\circ}\text{C}})
Notes
  • Newton’s law of cooling describes convective cooling from a surface. The law is stated as: the rate of heat loss from a body is proportional to the difference in temperatures between the body and its surroundings.
  • h is assumed to be independent of T (from A:Heat-Transfer-Coeffs-Constant).
  • ΔT\left(t\right)=T\left(t\right)-{T_{\text{env}}}\left(t\right) is the time-dependant thermal gradient between the environment and the object.
SourceincroperaEtAl2007 (pg. 8)
RefByGD:htFluxWaterFromCoil