How can I model temperature advection due to percolating rain water?

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Hi, I have a vertical column of a rock layer with the following vertical temperature distribution in Kelvin:
Td = [305.557543064521
283.841779658518
280.511694532208
279.032758050297
277.585218159758
276.266803864024
275.215826036983
274.441444325940
273.888287309701
273.482750462903
273.150000000000]
I want to model how a raindrop advects heat if it enters on top of the layer and travels down via: QR = cW*pW*P*dT (See eq. 44 here: http://escholarship.org/uc/item/8r38754s) with from left to right the specific heat capacity of water cW (4184 J kg-1 K-1), the density of water pW (1000 kg m-3), the rainfall rate P (m s-1) and the temperature difference dT between the raindrop and the layer. I have been trying some stuff, but it seems to result in extremely low temperature increases at the bottom layer (only 0.0056 degrees). Can anybody help me on checking if this is correct or I am doing something wrong? Thanks!
rhow = 1000; % Density of water (kg m^-3)
cW = 4184; % Water specific heat capacity (J kg-1 K-1)
P = 0.0113; % Precipitation rate (m 3h-1)
sechr = 10800; % Seconds in 3 hour
vol_heat_cap_deb = 2784795.435; % Volumetric heat capacity (J m-3 K-1)
% Calculate energy fux
for j = 2:11
Q_rain_deb(j,1) = ((max(0,rhow.*cW.*(P./(sechr)).*(Td(j-1,1)-Td(j,1)))));
end
% Calculate temperature differencee
for j = 1:11
Q_rain_temp(j,1) = (sechr).*((Q_rain_deb(j,1)) ./ (vol_heat_cap_deb));
end

Answers (1)

Torsten
Torsten on 11 Apr 2022
Edited: Torsten on 11 Apr 2022
I think the equation to calculate the increase in temperature T of the falling raindrop is
-u*rho*cp*dT/dx = 3/R * alpha * (T-T_L), T(x=0) = T0
where T_L is the layer temperature.
So the radius of the raindrop, the heat transfer coefficient and the height of the layers must somehow enter your equations.
The equation from above can be solved analytically.
  4 Comments
Torsten
Torsten on 12 Apr 2022
Edited: Torsten on 12 Apr 2022
Maybe it's better to consider the complete massflow of rain through the ground as to consider a single droplet:
mdot*cp*dT/dx = alpha*(T-T_L) * A/V
where "mdot" is the mass flow of water, "V" is the volume of water and "A" is the surface area of water exposed to heat transfer.
You will find this formula in every textbook on convective heat transfer.

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