This section focuses on the general equations and laws that SWHSNoPCM is based on.
Refname TM:consThermE
Label Conservation 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 ).
Source Fourier Law of Heat Conduction and Heat Equation RefBy GD:rocTempSimp
Refname TM:sensHtE
Label Sensible heat energy (no state change) Equation \[E={C^{\text{L}}}\,m\,ΔT\] 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 .
Source Definition of Sensible Heat RefBy IM:heatEInWtr
Refname TM:nwtnCooling
Label Newton’s law of cooling Equation \[q\left(t\right)=h\,ΔT\left(t\right)\] 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.
Source incroperaEtAl2007 (pg. 8)RefBy GD:htFluxWaterFromCoil