In equation 6 of their recent (October 2022) paper [1], Kleparnik, Zemcik, Treeby and Jaros define the volume rate of heat deposition, Q, as
Q = -div(I_avg) = - d(Ix_avg)/dx - d(Iy_avg)/dy - d(Iz_avg)/dz (1),
(the second equality has been introduced by me for explicitness). This is in line with early expressions such as e.g. the one given by Nyborg in [2].
To my knowledge, eq. (1) is not used in any of the k-wave examples. When I use it to compare with the temperature output that other expressions, such as
Q = p^2/(density*sound_speed) (2) -- plane wave approximation
or
Q = alpha_np * abs(I_avg) (3),
lead to in example_diff_focused_ultrasound_heating, I get very different results -- the ones given by (1) seem particularly odd. More precisely, what I am doing is running this example for the different Qs shown below:
============================
% Heating By A Focused Ultrasound Transducer
[...]
% set the sensor mask to cover the entire grid
sensor.mask = ones(Nx, Ny);
sensor.record = {'p', 'p_max_all', 'I_avg'}; %Include I_avg in recorded data
[...]
% =========================================================================
% CALCULATE HEATING
% =========================================================================
% convert the absorption coefficient to nepers/m
alpha_np = db2neper(medium.alpha_coeff, medium.alpha_power) * ...
(2 * pi * freq).^medium.alpha_power;
% extract the pressure amplitude at each position
p = extractAmpPhase(sensor_data.p, 1/kgrid.dt, freq);
% reshape the data, and calculate the volume rate of heat deposition
p = reshape(p, Nx, Ny);
%Q = alpha_np .* p.^2 ./ (medium.density .* medium.sound_speed); % Plane wave approximation
I_avg_x = reshape(sensor_data.Ix_avg, Nx,Ny);
I_avg_y = reshape(sensor_data.Iy_avg, Nx,Ny);
%Q = alpha_np .* sqrt(I_avg_x.^2 + I_avg_y.^2);
Q = -divergence(I_avg_x, I_avg_y);
[...]====================================
MY QUESTION IS: how should eq. (1) be used in a thermal simulation in k-wave? Any idea why seemingly inconsistent answers are obtained when using it in the aforementioned example?
This post mentions the gradient (?) of the intensity, but no further details are provided: http://www.k-wave.org/forum/topic/doubts-converting-pressure-into-heat-deposition
Many thanks in advance for any clarification you may provide.
Manuel
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[1] Kleparnik, P., Zemcik, P., Treeby, B. E., & Jaros, J. (2022). On-the-Fly Calculation of Time-Averaged Acoustic Intensity in Time-Domain Ultrasound Simulations Using a k-Space Pseudospectral Method. IEEE transactions on ultrasonics, ferroelectrics, and frequency control, 69(10), 2917–2929. https://doi.org/10.1109/TUFFC.2022.3199173
[2] Nyborg (1981). Heat generation by ultrasound in a relaxing medium. The Journal of the Acoustical Society of America 70, 310 (1981); https://doi.org/10.1121/1.386778