Leaf Boundary Layer Conductance
Table of contents
Code
Main program
sp_10_01.m | View on GitHub
% Supplemental program 10.1
% -----------------------------------------
% Calculate leaf boundary layer conductance
% -----------------------------------------
% Physical constants
tfrz = 273.15; % Freezing point of water (K)
g = 9.80616; % Gravitational acceleration (m/s2)
rgas = 8.31447; % Universal gas constant (J/K/mol)
visc0 = 13.3e-06; % Kinematic viscosity at 0C and 1013.25 hPa (m2/s)
Dh0 = 18.9e-06; % Molecular diffusivity (heat) at 0C and 1013.25 hPa (m2/s)
Dv0 = 21.8e-06; % Molecular diffusivity (H2O) at 0C and 1013.25 hPa (m2/s)
Dc0 = 13.8e-06; % Molecular diffusivity (CO2) at 0C and 1013.25 hPa (m2/s)
% Calculations are for 100 data points that vary in wind speed and/or leaf temperature
num = 100;
% Input variables
for p = 1:num
tair(p) = tfrz + 15; % Air temperature (K)
pref(p) = 101325; % Pressure (Pa)
rhomol(p) = pref(p) / (rgas * tair(p)); % Molar density (mol/m3)
wind(p) = p / 10; % Wind speed (m/s): 0.1 -> 10 m/s by 0.1
dleaf(p) = 0.05; % Leaf dimension (m)
tleaf(p) = tair(p) + p / 10; % tleaf - tair: 0.1 -> 10 K by 0.1 (for free convection)
end
for p = 1:num
% Adjust diffusivity for temperature and pressure
fac = 101325 / pref(p) * (tair(p) / tfrz)^1.81;
visc = visc0 * fac; % Kinematic viscosity (m2/s)
Dh = Dh0 * fac; % Molecular diffusivity, heat (m2/s)
Dv = Dv0 * fac; % Molecular diffusivity, H2O (m2/s)
Dc = Dc0 * fac; % Molecular diffusivity, CO2 (m2/s)
% Dimensionless numbers
Re(p) = wind(p) * dleaf(p) / visc; % Reynolds number
Pr = visc / Dh; % Prandtl number
Scv = visc / Dv; % Schmidt number for H2O
Scc = visc / Dc; % Schmidt number for CO2
Gr = g * dleaf(p)^3 * max(tleaf(p) - tair(p), 0) / (tair(p) * visc * visc); % Grashof number
% Forced convection - laminar flow
b1 = 1.5; % Empirical correction factor for Nu and Sh
Nu = b1 * 0.66 * Pr^0.33 * Re(p)^0.5; % Nusselt number
Shv = b1 * 0.66 * Scv^0.33 * Re(p)^0.5; % Sherwood number, H2O
Shc = b1 * 0.66 * Scc^0.33 * Re(p)^0.5; % Sherwood number, CO2
gbh1(p) = Dh * Nu / dleaf(p) * rhomol(p); % Boundary layer conductance, heat (mol/m2/s)
gbw1(p) = Dv * Shv / dleaf(p) * rhomol(p); % Boundary layer conductance, H2O (mol/m2/s)
gbc1(p) = Dc * Shc / dleaf(p) * rhomol(p); % Boundary layer conductance, CO2 (mol/m2/s)
% Forced convection - turbulent flow
Nu = b1 * 0.036 * Pr^0.33 * Re(p)^0.8; % Nusselt number
Shv = b1 * 0.036 * Scv^0.33 * Re(p)^0.8; % Sherwood number, H2O
Shc = b1 * 0.036 * Scc^0.33 * Re(p)^0.8; % Sherwood number, CO2
gbh2(p) = Dh * Nu / dleaf(p) * rhomol(p); % Boundary layer conductance, heat (mol/m2/s)
gbw2(p) = Dv * Shv / dleaf(p) * rhomol(p); % Boundary layer conductance, H2O (mol/m2/s)
gbc2(p) = Dc * Shc / dleaf(p) * rhomol(p); % Boundary layer conductance, CO2 (mol/m2/s)
% Free convection
Nu = 0.54 * Pr^0.25 * Gr^0.25; % Nusselt number
Shv = 0.54 * Scv^0.25 * Gr^0.25; % Sherwood number, H2O
Shc = 0.54 * Scc^0.25 * Gr^0.25; % Sherwood number, CO2
gbh3(p) = Dh * Nu / dleaf(p) * rhomol(p); % Boundary layer conductance, heat (mol/m2/s)
gbw3(p) = Dv * Shv / dleaf(p) * rhomol(p); % Boundary layer conductance, H2O (mol/m2/s)
gbc3(p) = Dc * Shc / dleaf(p) * rhomol(p); % Boundary layer conductance, CO2 (mol/m2/s)
end
% Make graph
plot(wind,gbh1,'b-',wind,gbh2,'r-')
title('Heat')
xlabel('Wind speed (m s^{-1})')
ylabel('Boundary layer conductance (mol m^{-2} s^{-1})')
legend('Laminar','Turbulent','Location','best')
% Write output to files
A = [wind; Re; gbh1];
fileID = fopen('forced-laminar.dat','w');
fprintf(fileID,'%10s %10s %10s\n','wind','Re','gbh');
fprintf(fileID,'%10.1f %10.2f %10.4f\n', A);
fclose(fileID);
B = [wind; Re; gbh2];
fileID = fopen('forced-turbulent.dat','w');
fprintf(fileID,'%10s %10s %10s\n','wind','Re','gbh');
fprintf(fileID,'%10.1f %10.2f %10.4f\n', B);
fclose(fileID);
C = [tleaf-tair; gbh3];
fileID = fopen('free-convection.dat','w');
fprintf(fileID,'%10s %10s\n','Tl-Ta','gbh');
fprintf(fileID,'%10.1f %10.4f\n', C);
fclose(fileID);
Output
Figures
Figure 1
Text
forced-laminar.dat | View on GitHub | View raw
wind Re gbh
0.1 341.27 0.2868
0.2 682.53 0.4056
0.3 1023.80 0.4968
0.4 1365.07 0.5736
0.5 1706.34 0.6413
0.6 2047.60 0.7025
0.7 2388.87 0.7588
0.8 2730.14 0.8112
0.9 3071.41 0.8604
1.0 3412.67 0.9070
1.1 3753.94 0.9513
1.2 4095.21 0.9936
1.3 4436.48 1.0341
1.4 4777.74 1.0732
1.5 5119.01 1.1108
1.6 5460.28 1.1473
1.7 5801.55 1.1826
1.8 6142.81 1.2168
1.9 6484.08 1.2502
2.0 6825.35 1.2827
2.1 7166.62 1.3143
2.2 7507.88 1.3453
2.3 7849.15 1.3755
2.4 8190.42 1.4051
2.5 8531.69 1.4341
2.6 8872.95 1.4625
2.7 9214.22 1.4903
2.8 9555.49 1.5177
2.9 9896.76 1.5445
3.0 10238.02 1.5709
3.1 10579.29 1.5969
3.2 10920.56 1.6225
3.3 11261.83 1.6476
3.4 11603.09 1.6724
3.5 11944.36 1.6968
3.6 12285.63 1.7209
3.7 12626.89 1.7446
3.8 12968.16 1.7680
3.9 13309.43 1.7912
4.0 13650.70 1.8140
4.1 13991.96 1.8365
4.2 14333.23 1.8588
4.3 14674.50 1.8808
4.4 15015.77 1.9025
4.5 15357.03 1.9240
4.6 15698.30 1.9453
4.7 16039.57 1.9663
4.8 16380.84 1.9871
4.9 16722.10 2.0077
5.0 17063.37 2.0281
5.1 17404.64 2.0483
5.2 17745.91 2.0682
5.3 18087.17 2.0880
5.4 18428.44 2.1076
5.5 18769.71 2.1271
5.6 19110.98 2.1463
5.7 19452.24 2.1654
5.8 19793.51 2.1843
5.9 20134.78 2.2031
6.0 20476.05 2.2216
6.1 20817.31 2.2401
6.2 21158.58 2.2584
6.3 21499.85 2.2765
6.4 21841.12 2.2945
6.5 22182.38 2.3124
6.6 22523.65 2.3301
6.7 22864.92 2.3477
6.8 23206.18 2.3651
6.9 23547.45 2.3825
7.0 23888.72 2.3997
7.1 24229.99 2.4167
7.2 24571.25 2.4337
7.3 24912.52 2.4505
7.4 25253.79 2.4673
7.5 25595.06 2.4839
7.6 25936.32 2.5004
7.7 26277.59 2.5168
7.8 26618.86 2.5331
7.9 26960.13 2.5493
8.0 27301.39 2.5653
8.1 27642.66 2.5813
8.2 27983.93 2.5972
8.3 28325.20 2.6130
8.4 28666.46 2.6287
8.5 29007.73 2.6443
8.6 29349.00 2.6598
8.7 29690.27 2.6752
8.8 30031.53 2.6906
8.9 30372.80 2.7058
9.0 30714.07 2.7210
9.1 31055.34 2.7360
9.2 31396.60 2.7510
9.3 31737.87 2.7659
9.4 32079.14 2.7808
9.5 32420.41 2.7955
9.6 32761.67 2.8102
9.7 33102.94 2.8248
9.8 33444.21 2.8393
9.9 33785.48 2.8538
10.0 34126.74 2.8681
forced-turbulent.dat | View on GitHub | View raw
wind Re gbh
0.1 341.27 0.0900
0.2 682.53 0.1567
0.3 1023.80 0.2168
0.4 1365.07 0.2729
0.5 1706.34 0.3262
0.6 2047.60 0.3774
0.7 2388.87 0.4269
0.8 2730.14 0.4751
0.9 3071.41 0.5220
1.0 3412.67 0.5679
1.1 3753.94 0.6129
1.2 4095.21 0.6571
1.3 4436.48 0.7006
1.4 4777.74 0.7433
1.5 5119.01 0.7855
1.6 5460.28 0.8271
1.7 5801.55 0.8683
1.8 6142.81 0.9089
1.9 6484.08 0.9491
2.0 6825.35 0.9888
2.1 7166.62 1.0282
2.2 7507.88 1.0671
2.3 7849.15 1.1058
2.4 8190.42 1.1441
2.5 8531.69 1.1821
2.6 8872.95 1.2197
2.7 9214.22 1.2571
2.8 9555.49 1.2942
2.9 9896.76 1.3311
3.0 10238.02 1.3677
3.1 10579.29 1.4040
3.2 10920.56 1.4401
3.3 11261.83 1.4760
3.4 11603.09 1.5117
3.5 11944.36 1.5472
3.6 12285.63 1.5824
3.7 12626.89 1.6175
3.8 12968.16 1.6524
3.9 13309.43 1.6871
4.0 13650.70 1.7216
4.1 13991.96 1.7560
4.2 14333.23 1.7901
4.3 14674.50 1.8242
4.4 15015.77 1.8580
4.5 15357.03 1.8917
4.6 15698.30 1.9253
4.7 16039.57 1.9587
4.8 16380.84 1.9920
4.9 16722.10 2.0251
5.0 17063.37 2.0581
5.1 17404.64 2.0909
5.2 17745.91 2.1237
5.3 18087.17 2.1563
5.4 18428.44 2.1888
5.5 18769.71 2.2211
5.6 19110.98 2.2534
5.7 19452.24 2.2855
5.8 19793.51 2.3176
5.9 20134.78 2.3495
6.0 20476.05 2.3813
6.1 20817.31 2.4130
6.2 21158.58 2.4446
6.3 21499.85 2.4761
6.4 21841.12 2.5074
6.5 22182.38 2.5387
6.6 22523.65 2.5699
6.7 22864.92 2.6010
6.8 23206.18 2.6321
6.9 23547.45 2.6630
7.0 23888.72 2.6938
7.1 24229.99 2.7245
7.2 24571.25 2.7552
7.3 24912.52 2.7858
7.4 25253.79 2.8163
7.5 25595.06 2.8467
7.6 25936.32 2.8770
7.7 26277.59 2.9072
7.8 26618.86 2.9374
7.9 26960.13 2.9675
8.0 27301.39 2.9975
8.1 27642.66 3.0274
8.2 27983.93 3.0573
8.3 28325.20 3.0871
8.4 28666.46 3.1168
8.5 29007.73 3.1465
8.6 29349.00 3.1760
8.7 29690.27 3.2055
8.8 30031.53 3.2350
8.9 30372.80 3.2644
9.0 30714.07 3.2937
9.1 31055.34 3.3229
9.2 31396.60 3.3521
9.3 31737.87 3.3812
9.4 32079.14 3.4103
9.5 32420.41 3.4393
9.6 32761.67 3.4682
9.7 33102.94 3.4971
9.8 33444.21 3.5259
9.9 33785.48 3.5546
10.0 34126.74 3.5833
free-convection.dat | View on GitHub | View raw
Tl-Ta gbh
0.1 0.0581
0.2 0.0691
0.3 0.0765
0.4 0.0822
0.5 0.0869
0.6 0.0910
0.7 0.0945
0.8 0.0977
0.9 0.1007
1.0 0.1033
1.1 0.1058
1.2 0.1082
1.3 0.1103
1.4 0.1124
1.5 0.1144
1.6 0.1162
1.7 0.1180
1.8 0.1197
1.9 0.1213
2.0 0.1229
2.1 0.1244
2.2 0.1259
2.3 0.1273
2.4 0.1286
2.5 0.1299
2.6 0.1312
2.7 0.1325
2.8 0.1337
2.9 0.1349
3.0 0.1360
3.1 0.1371
3.2 0.1382
3.3 0.1393
3.4 0.1403
3.5 0.1414
3.6 0.1423
3.7 0.1433
3.8 0.1443
3.9 0.1452
4.0 0.1461
4.1 0.1471
4.2 0.1479
4.3 0.1488
4.4 0.1497
4.5 0.1505
4.6 0.1513
4.7 0.1522
4.8 0.1530
4.9 0.1538
5.0 0.1545
5.1 0.1553
5.2 0.1561
5.3 0.1568
5.4 0.1575
5.5 0.1583
5.6 0.1590
5.7 0.1597
5.8 0.1604
5.9 0.1611
6.0 0.1617
6.1 0.1624
6.2 0.1631
6.3 0.1637
6.4 0.1644
6.5 0.1650
6.6 0.1656
6.7 0.1663
6.8 0.1669
6.9 0.1675
7.0 0.1681
7.1 0.1687
7.2 0.1693
7.3 0.1699
7.4 0.1704
7.5 0.1710
7.6 0.1716
7.7 0.1721
7.8 0.1727
7.9 0.1733
8.0 0.1738
8.1 0.1743
8.2 0.1749
8.3 0.1754
8.4 0.1759
8.5 0.1765
8.6 0.1770
8.7 0.1775
8.8 0.1780
8.9 0.1785
9.0 0.1790
9.1 0.1795
9.2 0.1800
9.3 0.1805
9.4 0.1810
9.5 0.1814
9.6 0.1819
9.7 0.1824
9.8 0.1828
9.9 0.1833
10.0 0.1838