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

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

Answer 1

[Torsten]

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.

[Yoni Verhaegen -WE-1718-]

I see, but I want to model how the rain drop changes the temperature of the environment as it travells down taking heat from the surface, and not how the environment changes the temperature of the rain drop.

[Torsten]

It doesn't matter. The heat transfered to the droplet(s) is equal to the heat taken from the environment.

[Yoni Verhaegen -WE-1718-]

Good, thank you. Do you maybe have a reference for that formula?

[Torsten]

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.