Last updated: 2022-03-07

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1 Preparing data

1.1 Estimating dry heat loss at 30C and solar influx

1.1.1 Solar influx

Here we calculate solar influx and the temperature corrected heat loss for each experiment.

Solar influx (difference between heat loss with radiation on and off) is calculated as:

\[Solar\ influx \ (W/m^2) = heatflux_{(radiation)}- heatflux_{(no \ radiation)}\]

Here we plot the solar influx.

Version	Author	Date
05389ae	Tina Lasisi	2022-01-19
bb99f1d	Tina Lasisi	2022-01-08

A plot of solar influx shows an outlier in the wet experiments where the heat loss with solar radiation was somehow less than the heat loss without solar radiation. We will exclude this outlier.

1.1.2 Temperature correction (to 30 C)

For the dry heat loss experiments (measuring dry heat resistance), we had to run some experiments at different temperatures to create a large enough gradient between skin/surface temperature and ambient temperature. The manikin is only able to produce results by measuring heat loss, and at 0.3 m/s wind speed, we found that the straight hair and nude (no hair) conditions led to overheating of the manikin. Subsequently, we had to adjust the temperatures to create a temperature gradient.

We thus need to bring all the measurements of heat loss to the same temperature. As heat resistance is temperature independent, we can use this to calculate the expected heat loss at various temperatures. We estimate heat loss at 30C with the following calculation.

\[Heat\ loss (W/m^2)_{(30C\ no\ radiation)} = 5/heat\ resistance_{(no \ radiation)} \]

To calculate the heat loss at 30C with radiation, we subtract the solar influx from the temperature corrected heat loss:

\[Heat\ loss (W/m^2)_{(30C\ with\ radiation)} = Heat\ loss (W/m^2)_{(30C\ no\ radiation)}- solar\ influx(W/m^2) \]

2 Exploratory plots

Version	Author	Date
c0ce5d2	Tina Lasisi	2022-03-06

2.1 Sweating

We use the dry and wet data to infer the amount of sweat that a scalp could evaporate under conditions of solar radiation at 30C (maximum sweat capacity) and how much evaporative cooling from sweat would be needed to cancel out any heat gain (zero heat gain sweat).

Here, we plot the sweat rate potential (left) and the sweat rate required to cancel out heat gain at \(T_{ambient} = 30^\circ C\).

The quantity of sweat that can be maximally evaporated (left) and that is required for zero heat gain (right) with various head coverings under three wind speeds

Version	Author	Date
c0ce5d2	Tina Lasisi	2022-03-06

What emerges is that while heat loss potential is higher without hair as a barrier (i.e. the “nude” condition), the potential sweat far exceeds the physiologically possible sweat rate for humans. The plot for zero heat gain shoes that a nude scalp requires the most sweat and this requirement is inversely correlated with hair curvature.

3 Regression models

3.1 Dry heat loss

3.1.1 Radiation off

Here, we model the effect of the wig variable on the off (heat loss without radiation) variable while correcting for wind.

Without radiation, having hair will reduce the heat loss.


Call:
lm(formula = off ~ wind + wig, data = df_dry_off)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.0046 -4.0646  0.1855  2.7241 14.8000 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   46.192      2.279   20.27  < 2e-16 ***
wind          11.317      1.022   11.07 2.65e-12 ***
wigLowCurv   -40.449      2.653  -15.25 5.91e-16 ***
wigMidCurv   -40.838      2.653  -15.39 4.54e-16 ***
wigHighCurv  -38.445      2.653  -14.49 2.37e-15 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.627 on 31 degrees of freedom
Multiple R-squared:  0.9373,    Adjusted R-squared:  0.9292 
F-statistic: 115.8 on 4 and 31 DF,  p-value: < 2.2e-16

3.1.2 Radiation on

With radiation, there is a net increase in heat (i.e. heat gain) without any hair. Additonally, we observe that heat gain decreases with increasingly curled hair.


Call:
lm(formula = on ~ wind + wig, data = df_dry_on)

Residuals:
     Min       1Q   Median       3Q      Max 
-13.6579  -4.8720  -0.0048   3.3664  18.9682 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -129.287      2.854  -45.29  < 2e-16 ***
wind          17.450      1.280   13.63 1.23e-14 ***
wigLowCurv    69.729      3.322   20.99  < 2e-16 ***
wigMidCurv    91.439      3.322   27.52  < 2e-16 ***
wigHighCurv  113.588      3.322   34.19  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.048 on 31 degrees of freedom
Multiple R-squared:  0.9798,    Adjusted R-squared:  0.9771 
F-statistic:   375 on 4 and 31 DF,  p-value: < 2.2e-16

3.1.3 Solar influx

Here, we model the effect of the wig variable on influx while correcting for wind.

In the dry heat loss experiments, we see that all hair (regardless of curliness) decreases the solar influx. Additionally, the curlier the hair, the lower the solar influx.


Call:
lm(formula = influx ~ wind + wig, data = df_dry)

Residuals:
   Min     1Q Median     3Q    Max 
-8.106 -3.844  1.099  2.763  9.053 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  175.4796     2.0111  87.254  < 2e-16 ***
wind          -6.1330     0.9018  -6.801 1.29e-07 ***
wigLowCurv  -110.1778     2.3409 -47.067  < 2e-16 ***
wigMidCurv  -132.2778     2.3409 -56.508  < 2e-16 ***
wigHighCurv -152.0333     2.3409 -64.948  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.966 on 31 degrees of freedom
Multiple R-squared:  0.994, Adjusted R-squared:  0.9932 
F-statistic:  1275 on 4 and 31 DF,  p-value: < 2.2e-16

3.1.4 Summary of dry heat loss regression models

3.1.4.1 All separate

	Radiation Off	Radiation On	Solar Influx
(Intercept)	46.19 ***	-129.29 ***	175.48 ***
	[41.54, 50.84]	[-135.11, -123.47]	[171.38, 179.58]
wind	11.32 ***	17.45 ***	-6.13 ***
	[9.23, 13.40]	[14.84, 20.06]	[-7.97, -4.29]
wigLowCurv	-40.45 ***	69.73 ***	-110.18 ***
	[-45.86, -35.04]	[62.95, 76.50]	[-114.95, -105.40]
wigMidCurv	-40.84 ***	91.44 ***	-132.28 ***
	[-46.25, -35.43]	[84.66, 98.22]	[-137.05, -127.50]
wigHighCurv	-38.45 ***	113.59 ***	-152.03 ***
	[-43.86, -33.04]	[106.81, 120.36]	[-156.81, -147.26]
N	36	36	36
R2	0.94	0.98	0.99
* p < 0.001; p < 0.01; * p < 0.05.

Regression coefficients across regression models.

3.1.4.2 Radiation conditions combined


Call:
lm(formula = heatloss ~ wind + wig + var + var * wig, data = df_dry_radcombo)

Residuals:
     Min       1Q   Median       3Q      Max 
-14.4757  -4.5094  -0.4139   4.7580  22.7503 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)         42.3079     2.5983   16.28   <2e-16 ***
wind                14.3839     0.8996   15.99   <2e-16 ***
wigLowCurv         -40.4490     3.3023  -12.25   <2e-16 ***
wigMidCurv         -40.8384     3.3023  -12.37   <2e-16 ***
wigHighCurv        -38.4455     3.3023  -11.64   <2e-16 ***
varon             -167.7111     3.3023  -50.79   <2e-16 ***
wigLowCurv:varon   110.1778     4.6702   23.59   <2e-16 ***
wigMidCurv:varon   132.2778     4.6702   28.32   <2e-16 ***
wigHighCurv:varon  152.0333     4.6702   32.55   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.005 on 63 degrees of freedom
Multiple R-squared:  0.9826,    Adjusted R-squared:  0.9804 
F-statistic: 444.5 on 8 and 63 DF,  p-value: < 2.2e-16

	Model 1
(Intercept)	42.31 ***
	[37.12, 47.50]
wind	14.38 ***
	[12.59, 16.18]
wigLowCurv	-40.45 ***
	[-47.05, -33.85]
wigMidCurv	-40.84 ***
	[-47.44, -34.24]
wigHighCurv	-38.45 ***
	[-45.04, -31.85]
varon	-167.71 ***
	[-174.31, -161.11]
wigLowCurv:varon	110.18 ***
	[100.85, 119.51]
wigMidCurv:varon	132.28 ***
	[122.95, 141.61]
wigHighCurv:varon	152.03 ***
	[142.70, 161.37]
N	72
R2	0.98
* p < 0.001; p < 0.01; * p < 0.05.

Regression coefficients for dry heatloss

3.2 Evaporative resistance (wet experiments)

Here, we repeat the same modelling process for the evaporative resistance data from the wet experiments.

3.2.1 Radiation off

Here, we model the effect of the wig variable on the off (heat loss without radiation) variable while correcting for wind.

Without solar radiation, all hair (regardless of texture) decreases evaporative resistance.


Call:
lm(formula = off ~ wind + wig, data = df_wet_off)

Residuals:
    Min      1Q  Median      3Q     Max 
-32.423  -5.988   2.672   5.875  40.867 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  171.047      9.142   18.71 8.88e-13 ***
wind          42.586      3.933   10.83 4.77e-09 ***
wigLowCurv  -116.039     10.191  -11.39 2.24e-09 ***
wigMidCurv  -129.155     10.191  -12.67 4.35e-10 ***
wigHighCurv -134.404     10.707  -12.55 5.04e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 16.83 on 17 degrees of freedom
Multiple R-squared:  0.955, Adjusted R-squared:  0.9444 
F-statistic: 90.24 on 4 and 17 DF,  p-value: 3.238e-11

3.2.2 Radiation on

With radiation, hair decreases evaporative resistance.


Call:
lm(formula = on ~ wind + wig, data = df_wet_on)

Residuals:
    Min      1Q  Median      3Q     Max 
-47.469 -11.283   4.403   6.848  54.322 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  117.536     12.729   9.234 4.91e-08 ***
wind          55.442      5.476  10.124 1.29e-08 ***
wigLowCurv  -106.646     14.189  -7.516 8.45e-07 ***
wigMidCurv  -113.896     14.189  -8.027 3.49e-07 ***
wigHighCurv -123.867     14.908  -8.309 2.17e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 23.43 on 17 degrees of freedom
Multiple R-squared:  0.9255,    Adjusted R-squared:  0.9079 
F-statistic: 52.77 on 4 and 17 DF,  p-value: 2.307e-09

3.2.3 Solar influx

Combining the above data to calculate solar influx, we see that there is not a considerable effect of radiation on evaporative resistance.


Call:
lm(formula = influx ~ wind + wig, data = df_wet)

Residuals:
     Min       1Q   Median       3Q      Max 
-13.4541  -4.2173  -0.8525   3.9887  15.0463 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   53.511      4.577  11.690 1.50e-09 ***
wind         -12.857      1.969  -6.529 5.15e-06 ***
wigLowCurv    -9.392      5.102  -1.841  0.08318 .  
wigMidCurv   -15.259      5.102  -2.991  0.00822 ** 
wigHighCurv  -10.537      5.361  -1.966  0.06591 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 8.425 on 17 degrees of freedom
Multiple R-squared:  0.7498,    Adjusted R-squared:  0.6909 
F-statistic: 12.74 on 4 and 17 DF,  p-value: 5.664e-05

3.2.4 Summary of evaporative heat loss regression models

3.2.4.1 All separate

	Radiation Off	Radiation On	Solar Influx
(Intercept)	171.05 ***	117.54 ***	53.51 ***
	[151.76, 190.34]	[90.68, 144.39]	[43.85, 63.17]
wind	42.59 ***	55.44 ***	-12.86 ***
	[34.29, 50.88]	[43.89, 67.00]	[-17.01, -8.70]
wigLowCurv	-116.04 ***	-106.65 ***	-9.39
	[-137.54, -94.54]	[-136.58, -76.71]	[-20.16, 1.37]
wigMidCurv	-129.16 ***	-113.90 ***	-15.26 **
	[-150.66, -107.65]	[-143.83, -83.96]	[-26.02, -4.49]
wigHighCurv	-134.40 ***	-123.87 ***	-10.54
	[-157.00, -111.81]	[-155.32, -92.41]	[-21.85, 0.77]
N	22	22	22
R2	0.96	0.93	0.75
* p < 0.001; p < 0.01; * p < 0.05.