## 5 Thermal Analysis of Raised-Field Agricultural Systems
To examine the heat storage effect related to heat conduction transfer times in moist berm soil mixtures depending on soil thermal diffusivity \(\kappa\), the well-known transient heat conduction equation is delineated:  
\[\varpropto{\left(\partial{\rm T}/\partial{\rm t}\right)}=\kappa\left(\partial^{2 }{\rm T}/\partial{\rm x}^{2}\right) \tag{1}\]  
where \(\varpropto{\rm=density}\left({\rm lb}/{\rm ft}^{3}\right)\)  
T = temperature (\({}^{\circ}\)F),  
c = specific heat (BTU/hr lb \({}^{\circ}\)F),  
t = time (hr),  
k = thermal conductivity (BTU/hr ft \({}^{\circ}\)F),  
and \(\kappa\) = thermal diffusivity (ft\({}^{2}\)/hr). Air = 0.187,  
Rock = 0.0188,  
Soil (Av) = 0.0046,  
Soil (sandy, dry) = 0.0028,  
Soil (sandy, moist) = 0.0033,  
Soil (soil mix, moist) = 0.0040,  
Water = 0.0014. A solution to Equation (1) is  
\[{\rm x}/2(\kappa\cdot{\rm t})^{1/2}\]  
\[{\rm T}\left({\rm x},{\rm t}\right)=(2\;{\rm T}_{0}\;\pi^{-1/2})\int\exp\;(-{ \rm s}^{2})\;{\rm ds}={\rm T}_{0}\;{\rm erf}\;[{\rm x}/2\;(\kappa\cdot{\rm t}) ^{1/}0 \tag{2}\]  
For a berm soil block at T = 0 at t = 0 for \(0<\kappa<\infty\) depth range with a given \(\kappa\) value, where the \({\rm x}=0\) top boundary is maintained at T\({}_{0}>\) T, the heat conduction solution gives the ensuing T (x, t) in the berm soil block for t \(\geq\) 0, \(0<\kappa<\infty\). From Equation (2), for a location \({\rm x}_{1}\) where T = T\({}_{0}/2\), the time for this heat transfer to produce this effect is given by t = \(\kappa^{-1}\) [\({\rm x}_{1}/2\;(0.477)\)]\({}^{2}\), using the standard error function reference ([PERSON] and [PERSON] 1964). The smaller the \(\kappa\) value, the longer the time for heat transfer to progress. The mechanism for temperature progression is the continuous heat flow from the berm block from the constant temperature T\({}_{0}\) boundary at x = 0. From this solution, the time for a temperature reduction to T\({}_{0}/2\) at x\({}_{1}\) depends inversely on thermal diffusivity \(\kappa\). From the above table of \(\kappa\) values for different media, it is noted that when the medium of heat conduction is air, then cooling at a depth \({\rm x}_{1}\) to T\({}_{0}/2\) occurs rapidly owing to \(\kappa\) being large for air, so that the time to cool to T\({}_{0}/2\) is a small value. If the berm soil medium for heat conduction is wet or substantially moist, then the time to cool at x\({}_{1}\) is considerably larger. If the medium is water, then the time to cool to T\({}_{0}/2\) is approximately 133 times more than if the medium is air. Thus, the lower the \(\kappa\) value, the greater the heat storage effect. Based upon these observations, given the soil medium characteristic of raised fields, if heat can be absorbed internally by means of daily solar flux input, then the possibility for heat retention during cold altiplano nights is high, as the time required for heated moist berm soil to cool is low due to its small \(\kappa\) value. The heat retention capability then acts as a further protective enhancement to protect root crops against altiplano temperatures, which would ordinarily destroy crops through freezing. The presence of water in the raised-field systems' swales further enhances the heat storage capacity of the system, as swale water's dark base of decomposed plant debris provides a radiation trap, producing elevated water temperatures. As water's \(\kappa\) value is low, it will retain heat for extended periods, and this serves as a supplemental heat source to the berm interior during long cold nights. Through these means, an extension of the growing season may be achieved by possibly a few weeks, enabling double cropping of certain types of products or at least survival of existing crops to a further level of maturity, as was demonstrated empirically by contemporary farmers on rehabilitated raised fields [24].