Predicting segmental and overall ventilation of ensembles using an integrated bioheat and clothed cylinder ventilation models

Abstract

The purpose of this work is to implement an integrated ventilation and heat transport model of a clothed human subject to external wind. Each segment of the human body is modeled as a vertical cylinder with an air annulus separating the ventilated clothing from the skin with open or closed end to the environment. The developed ventilation model takes into consideration natural convection induced by the temperature difference between the skin and the annulus air. The steady-state mass and energy balance equations of the annulus microclimate air and clothing were solved numerically. The model is integrated with a segmental bioheat model to determine the local skin temperature, which is used as boundary condition. The clothing ventilation model was validated by conducting experiments on an isothermally heated vertical clothed cylinder with an open lower end aperture and subject to a uniform cross wind. Good agreement was found between the model predictions and experimental measurements of temperature at different angular and vertical locations in the air layer. The combined clothed cylinder and segmental bioheat model was validated with published experimental data on ensemble total ventilation for different permeabilities and wind speeds. It was found that clothed segments opened from the bottom increased ventilation by 40% when compared to clothed segments opened from the top. An increase of wind speed by 1.0 m/s leads to an increase in ventilation rate of about 36%. Natural convection was also found to enhance the ventilation of highly permeable clothing at low wind speeds compared to less permeable clothing.

Keywords

mixed convection clothing ventilation porous media bioheat modeling

Most comfort indices for the indoor environment are based on static ensemble clothing insulation and an assumed sedentary activity level at low air velocities.^1,2 To reduce building energy consumption, research has recently focused on increasing air velocity to bring thermal comfort at elevated air temperatures as well as localizing a higher air flow to upper body segments, enhancing associated ventilation.^3–9 Zhang and Zhao⁸ reported that upper body cooling could improve the thermal acceptability of room temperature to values ranging from 26℃ to 30.5℃. Similar findings were also reported by Ghaddar et al.^6,9 where air movement targeting upper body segments enhanced comfort. Different segments of the clothed body may have different ventilation rates and local thermal comfort becomes important in assessing whole-body comfort.^7–9

Research in local clothing ventilation took two tracks. One is empirical, which deals with the whole ensemble ventilation. Lotens¹⁰ used the tracer gas method, which is an effective experimental technique, and derived mathematical correlations of ventilation rate to obtain the effective wind velocity and air permeability of the outer fabric. However, these correlations were empirical and limited to the conditions at which they were deduced.¹⁰ The second research track for estimating local ventilation has been by mathematical modeling for predicting local and overall clothing ensemble ventilation rates. The modeling approach has a wider applicability since it can be combined with segmental bioheat models to predict local and overall comfort at different thermal conditions. This was implemented by Ghaddar et al.^11,12 to predict the local ventilation of the clothed human trunk. It is of interest to expand the ventilation model to clothed limbs which normally have the clothing aperture open at the bottom. Ghaddar et al.⁹ integrated measured local ventilation rates and corrected clothing dynamic resistance accordingly^13,14 to predict body heat loss and local and overall thermal comfort. The estimation of the clothing dynamic resistance, R_dynamic, is based on the dry fabric corrected by the ventilation flow rate. During ventilation, the insulation value of the clothing will decrease as a result of the air motion across the fabric. The decrease of the resistance is related to the radial mass air flow rate, as was suggested by Havenith et al.,^13,14 who elaborated this in their work.

There are several challenges to modeling local ventilation in the clothing which require separate consideration of each clothed body segment depending on its geometry, clothing aperture location, and the microclimate air-layer size. Clothing ventilation is essentially related to the flow characteristics in the air layer between skin and clothes,^15,16,21 the clothing permeability,¹⁰ and on the open clothing apertures.^11,14 The flow characteristics in the trapped air layer are the result of two phenomena: natural convection associated with warm body skin and forced flow induced by both wind that penetrates permeable clothing and by flow through the opening.^17–19 Chaves et al.²⁰ showed how the natural convection inside a heated open top cavity improved the forced convection penetrating through the vertical porous boundary. The relative importance of these mechanisms on ventilation and clothing insulation for different body segments is not the same (ventilation of clothed limbs vs. clothed trunk).

The upward flow induced by buoyancy is shown by Ghaddar et al.¹¹ to enhance dry heat loss from the inner isothermal cylinder. In their work, they present a dry model of coupled ventilation and mixed convection in a vertical air annulus of a clothed heated cylinder in uniform wind. They reported that natural convection enhanced ventilation of the annulus air for wind speeds less than 3 m/s and enhanced heat loss by 12% in highly permeable clothing at a uniform wind speed of 1 m/s. In a subsequent study, Ghaddar et al.¹² considered ventilation from a vertical clothed wet and heated cylinder with an open top subject to a uniform cross wind. They reported that the effect of adding a wet surface on the axial ventilation mass flow rate in a vertical annulus does not exceed 3% in comparison with a dry cylinder mixed convection at the same heat flux.¹² These results are applicable to the trunk segment when the top has an open aperture at the neck. However, the natural convection was not modeled for clothed limbs with sleeves and trousers, when ventilation from the bottom aperture could be significant since air is withdrawn at the ambient temperature. In addition, to be able to build the human thermal heat loss from the loss of various segments (trunk and limbs), the ventilation rate associated with clothed limbs and with the bottom opening should be accurately estimated. The study of the heat losses and ventilation rates of the whole body used a resistance model integrated with the bioheat model by considering a constant heat flux and constant metabolic rate.⁹ The literature does not report the effect of mixed convection when a clothed cylinder air gap aperture is open at the lower end (bottom), which represents the majority of the clothed segments of a human body.

This work aims to develop an integrated local clothing ventilation model with a segmental bioheat model to predict the clothed human thermal response when a human is subject to an external wind. Local ventilation rates for clothed limbs will be estimated based on modeling of mixed convection in a microclimate air layer of a clothed heated cylinder with a bottom open aperture. The coupling of the ventilation model to the bioheat model is incorporated through: (1) the segment skin temperature, which is used as boundary condition for the heated cylinder, and (2) through adjusting the clothing dynamic resistance due to ventilation. The clothed heated cylinder model with a bottom open aperture will be validated by experiments in a wind tunnel, while the predictions of total ensemble ventilation rates from an integrated clothed human will be validated by comparisons to ventilation values published in the literature for the whole ensemble. The novelty of this work arises from estimating the mixed convection for clothed limbs by modeling and the integration of this model to a bioheat model to estimate the ensemble total ventilation rate. Clothed limbs modeling is indispensable for the estimation of local ventilation rates, since the limbs are important in accelerating body cooling.²²

Methodology

The bioheat model used in this work is adapted from the work of Ghaddar et al.⁹ and divides the body into 17 segments: head, chest, back, pelvis, buttocks, lower arms, upper arms, hands, thighs, calves and feet. The model equations simulate the heat and moisture transport from the human skin through the clothing system using a modified Gagge's model.²³ Each body segment body is represented by Gagge's two-node model (1986), where two energy-balance equations are developed for the core node and the skin node. The sensible heat exchange from the skin to the air layer is expressed in terms of the convection heat loss at the skin surface and the radiation exchange between the skin and the fabric. The skin temperature is an input boundary condition obtained from the thermoregulatory model of the human body model, while the convective coefficients are obtained from empirical correlations.²⁴ For the latent heat transfer, the boundary condition is more complex. If liquid is present on the skin surface, then the skin boundary condition is the saturation pressure at the skin (sk) temperature P_sk = P^(T_sk). The vapor pressure at the skin surface is determined by a balance between the diffusion of vapor through the skin, the sweat secreted, and the transport of moisture away from the skin, as reported by Jones.²⁵ In the analysis of the present work, we assume that no liquid exists at the skin surface.^11,12

In this work, the bioheat model will be coupled through the skin temperature to the proposed mixed convection model for the clothed limb segments with open apertures to estimate local ventilation rates. The local ventilation rate is an important parameter because of its effect on clothing resistance. Therefore, the bioheat model incorporates as input a dynamic resistance that takes into account the correction for the dry resistance due to local ventilation. The most comprehensive data on dynamic clothing dry resistance originates from the work of Havenith et al.^13,14 describing the changes in clothing insulation values due to motion of the wearer and wind. The clothed trunk mixed convection model with an open top aperture uses the clothed cylinder model of Ghaddar et al.¹² to couple to the bioheat model. The clothed human body segments that are typically covered are the arms, the legs, and the trunk. Each segment consists of two concentric vertical cylinders (limb and its clothing) for a standing or walking person. The outer cylinder, representing the fabric, covers the inner cylinder, representing the human skin. Since our focus is on the mathematical modeling of clothed limbs, the air annulus between the two cylinders will be open at the bottom to study and model microclimate air-layer ventilation of a clothed heated cylinder subject to external wind. Based on recent knowledge from a bioheat model,⁹ the bioheat model and its clothed segments, including limbs, are coupled to the clothed cylinder ventilation model in order to determine the associated boundary condition of skin temperatures to be used as the inner cylinder thermal condition mentioned earlier.

Experiments are conducted in a wind tunnel in which a clothed vertical heated cylinder with an open-end aperture at the bottom is tested to validate the model by comparing it with experimentally obtained skin and air-layer temperatures. The integrated clothed cylinder model with a bioheat model will then be validated with published experimental data on ensembles at different external winds and clothing types. The total heat losses and ventilation rates will be predicted for different clothing types for a person subjected to a uniform cross wind.

Mathematical formulation

Clothed cylinder model

The physical configuration of the present study shown in Figure 1(a) consists of two concentric cylinders of radius R_s and R_f, and height L. A microclimate air annulus of thickness Y = R_f − R_s is trapped between the inner solid cylinder maintained at temperature T_skin and the outer porous cylinder represented by an isotropic fabric layer of permeability α and thickness e_f. The top end of the annulus is closed and adiabatic, while the bottom end is open to the environment at temperature T_amb. The configuration is placed perpendicular to an air flow at V_∞. Some air penetrates through the porous fabric into the air layer and is mixed with the incoming air from the bottom aperture at the environmental temperature and is then driven upward by the presence of the pressure gradient and the natural convection. The annulus-trapped air thickness Y is small compared to the length L and the inner radius cylinder R_s which permits the following assumptions:
plane Poiseuille flow in axial and angular directions;

negligible variations of pressure and temperature in the radial direction;

fully developed mixed buoyant upward flow;

transformation from a three dimensional (3D) to 2D problem.

Figure 1.
Schematic of (a) the physical configuration of the clothed vertical cylinders with an open end and (b) the cylindrical representation of a three-node fabric layer.
The annulus-trapped air thickness Y is considered constant. The ventilation across the clothing ensemble is for a standing posture and therefore a uniform gap width for a specific body segment is justifiable, although different segments can have a different average gap size. The outer fabric cylinder will be modeled using the three-node-fabric model of Ghali et al.,¹⁵ which uses a lumped layer of two fabric nodes (outer and inner) and an air void node to represent a thin fabric medium, as shown in Figure 1(b). The fabric outer node, which represents the exposed surface of the yarns, is in direct contact with the penetrating air in the air void node between yarns. However, the inner fabric node represents the inner portion of the solid yarn surrounded by the fabric outer node.

Governing equations

The mass and energy balance equations will be derived for the fabric and the air layer nodes. As we know, all the mass flow rates in this study are related to the difference in pressures (because of the use of the Poiseuille flow equation) except the mass flow rate induced from the natural convection, which is a function of the difference in temperature. Furthermore, the energy equation is dependent on ventilation and axial mass flow rates. Thus, the mass conservation equation and the energy balance equation are coupled and we cannot solve each equation alone. The mathematical formulation starts with the air annulus mass conservation equation given by:
$\frac{\partial (Y \overset{\cdot}{m} a z)}{\partial z} + \frac{\partial (Y \overset{\cdot}{m} a θ)}{R f \partial θ} - \overset{\cdot}{m} a Y = 0$
(1)
where $\overset{\cdot}{m} a z$ is the mass flux in the vertical z-direction in kg/m²·s, $\overset{\cdot}{m} a θ$ is the mass flux in the angular θ-direction, and $\overset{\cdot}{m} a Y$ is the radial infiltrating air flow rate through the fabric given by:
$\overset{\cdot}{m} a Y = \frac{α ρ a (P s - P a)}{Δ P m}$
(2)

From standard tests on fabrics' air permeability, the reference pressure difference is $Δ P m = 0.1245$ kPa.²²

Assuming Poiseuille flow in the θ-direction, the angular air mass flow rate per unit area is expressed in terms of the annular pressure as:
$\overset{\cdot}{m} a θ = - \frac{Y 2}{12 R f ν} \frac{d P a}{d θ}$
(3)
The upward mass flow rate is expressed in terms of pressure and driving temperature gradient of the air and temperature difference between the inner wall and the infiltrating air at fabric void temperature T_void. The upward mass flow rate per unit area due to forced flow is given by:
$\overset{\cdot}{m} a z = - \frac{Y 2}{12 ν} \frac{d P a }{d z}$
(4)

$P_{a}^{} = P a + ρ a gz$
(5)
while the induced buoyant mass flow rate term is:
$\overset{\cdot}{m} a n = ρ \frac{g β Y 2}{12 ν} (T a - T void)$
(6)
The radial, angular, and upward mass flow rates of equations (2), (3), (4), and (6) are substituted in equation (1) to give the mass conservation equation in pressure and temperature form as:
$\frac{α ρ a (P a - P s)}{Δ P m} + \frac{\partial}{R_{f}^{2} \partial θ} (\frac{Y 3}{12 ν} \frac{\partial P a}{\partial θ}) + \frac{\partial}{\partial z} (\frac{Y 3}{12 ν} \frac{\partial P a }{\partial z}) + \frac{ρ g β Y 3}{12 ν} \frac{d (T a - T void)}{d z} = 0$
(7)

The steady-state dry energy balance on the air spacing annulus is a balance of the dry convective heat transfer from the surface of the inner cylinder, the heat flow to the air associated with mass fluxes from the radial, angular, and upward directions, the heat diffusion from the void air of the thin fabric to the air layer, and the angular conduction of heat in the air layer. The dry energy balance of the air layer is then given by:
$h c (skin-air) (T skin - T a) + h c o, (T o - T a) + max (\overset{\cdot}{m} a y, 0) c p T void - max (- \overset{\cdot}{m} a y, 0) c p T a = \frac{\partial (y . q \overset{\cdot}{m} a z)}{\partial z} + \frac{\partial (y . q \overset{\cdot}{m} a θ)}{R f \partial θ} - \frac{k a}{R_{f}^{2}} \frac{\partial}{\partial θ} (Y \frac{\partial T a}{\partial θ}) - k a \frac{\partial}{\partial z} (Y \frac{\partial T a}{\partial z})$
(8)
where
$q \overset{\cdot}{m} a z = \overset{\cdot}{m} a z \cdot c p \cdot T \binom{a}{} q \overset{\cdot}{m} a θ = \overset{\cdot}{m} a θ \cdot c p \cdot T a$
(9)
where h_c(skin-air) is the convection coefficient from the skin to the air, h_c(o-air) is the convection coefficient from the fabric to the air, T_o is the fabric outer node temperature, k_a is the thermal conductivity of air in the trapped air annulus, and c_p is the air specific heat.

The dry energy balance of the fabric void node¹⁵ is given by:
$q \overset{\cdot}{m} a y + H co \times (T o - T void) = 0$
(10)

$q \overset{\cdot}{m} a y = \overset{\cdot}{m} a y \times c p fabric \times (T void - T a) if P s < P a q \overset{\cdot}{m} a y = \overset{\cdot}{m} a y \times c p fabric \times (T amb - T void) if P s > P a$
(11)
where H_co is the effective void space heat transfer coefficient between air passing through the void and the fabric outer node (yarns). The dry energy balance of the fabric outer node is then given by
$h c (o - air) \times (T amb - T o) + H co (T void - T o) + \frac{h r}{2} (T skin - T o) + \frac{h r}{2} (T amb - T o) = 0$
(12)

Convection and radiation coefficients in equations (10) and (12) are provided in Table 1.
Table 1.
Summary of convection heat transfer coefficients used in the model

Convection coefficient Value or equation (W/m²·K) Reference

h _{c_skin-air} 10.5 ²⁴

h _co 9.6 ²⁴

H _co if $\overset{\cdot}{m} a y > 0.00777$ then $H co = 495.72 * \overset{\cdot}{m} a y - 1.857$ else $H co = 2$ ²¹

h _c(o-air) $h c (o - air) = a . V \infty b where a = 4.8 * (2 . R f) - 0.33 and b = 0.5$ ³⁸

h _r 3.7 ³⁹

Boundary conditions

Modeling mixed convection in a small-thickness vertical annulus with an open bottom lower boundary, connecting to the environment, has relatively more complex boundary condition to tackle at the opening than mixed convection with an open top boundary. The complexity arises from the change in the direction of the flow (upward or downward) depending on angular position in the open annulus, which requires special treatment of the boundary conditions at the opening.²⁶ When the space between the two concentric cylinders is small enough, the velocity of the inlet or exit natural convection flow is only axial.^27–29 In this situation; the boundary conditions are influenced only by the pressure and temperature conditions.

Researchers identified two different types of pressure boundary condition that can be used for bottom openings. The first type is when the static pressure is equal to zero gauge pressure in both the inlet and the exit flow.^30,31 The second type is when the static pressure at the exit and the stagnation pressure at the inlet are equal to zero gauge pressure.^{27–29,32,33} Anil and Reji²⁷compared the two types and found that the second type of boundary condition is more reasonable and adequate to simulate a developing natural convection flow between parallel surfaces.

For thermal boundary conditions at the inlet and outlet of an annulus, researches have assumed that the inlet fluid temperature is uniform and ambient.^27–29 However, the gradient of the temperature of the fluid leaving the annulus is set equal to zero.^11,27–29 These conditions are applicable to the natural convection inflow and outflow only through one end of the annulus, top or bottom.

In the current model, the second type of pressure boundary condition is used for the open aperture at the lower end of the cylinder, while Bernoulli' equation is applied between P_∞ in the far environment to the opening at z = 0, using C_D, the loss coefficient, at the aperture of the domain dependent on area ratio of the aperture to the air layer thickness Y. To find the mass flow rate, we assume that the inlet stagnation pressure is zero. The associated boundary conditions for the air flow and pressure are summarized as follows:

At the lower opening:
$\overset{\cdot}{m} az (z = 0, θ) = C D [2 ρ a (P a - P \infty)] 12; P a (z = 0, θ) = P \infty - \frac{[\overset{\cdot}{m} a z (z = 0, θ)] 2}{2 ρ a}$
(13a)

At the closed end:
$\overset{\cdot}{m} a z (z = L, θ) = 0 \frac{d P a ((z = L, θ)}{d z} = 0$
(13b)

At the angular symmetry line:
$\overset{\cdot}{m} a θ (z, θ = 0 or π) = 0$
(13c)

Considering the inner cylinder as isothermal at T_skin and the environment temperature at T_∞, the associated thermal boundary conditions suggest a zero temperature gradient when the air is leaving or when the end is closed and ambient temperature when the flow is entering as follows:
$T a (z = 0, θ) = T \infty and \frac{d T a}{d z} (z = L, θ) = 0$
(14a)

$\frac{d T a}{d θ} (z, θ = 0 θ = π) = 0$
(14b)

Definitions of microclimate ventilation rate and sensible heat loss

Of interest is to calculate the total ventilation rate of the fabric, based on the renewal rate induced by external wind penetrating the porous fabric and enhanced by buoyancy. The total ventilation rate is calculated as the positive flow of air into the air annulus integrated per unit area of the clothed surface as follows:
$\overset{\cdot}{m} a (kg / s \cdot m 2) = \frac{1}{π L} \int_{0}^{L} \int_{0}^{π} max (0, \overset{\cdot}{m} a y) d θ d z$
(15a)

The segmental sensible heat loss Q_s from the skin to the microclimate air layer used in the clothed cylinder model and bioheat model is given by:
$Q s (W) = \int \int (h cskin-air R s (T skin (θ, z) - T a (θ, z)) + h r R s (T skin (θ, z) - T o (θ, z)) d θ d z$
(15b)

Definitions of clothing dynamic resistance

The dynamic resistance of the clothing defined by Ghali et al.³⁴ is given by:
$R dynamic = \frac{1}{(\frac{1}{R d} + \overset{\cdot}{m} a Y * Cp a)}$
(16)
where R_dynamic and R_d are, respectively, the dynamic resistance for ventilation through the fabric and the fabric dry resistance. The dynamic resistance is a parameter calculated from the clothed cylinder model to be used as an input to the bioheat model to correct for the ventilation effect on clothing resistance. As mentioned earlier, the ventilation rate is determined based on a sensible analysis since the skin vapor pressure effect is negligible on enhancing mixed convection within the microclimate air of the clothed body segments.¹² Once ventilation is determined, the bioheat model predicts the sensible and latent heat losses.

Numerical solution

The coupled pressure and energy equations of the clothed cylinder with associated boundary conditions (equations (1) to (14)) are discretized using a finite volume methodology, where the air layer zone is divided into N_Θ × N_z grids of size Δz and R_f.Δθ with thickness Y. Since the numerical methodology is similar to the methodology used by Ghaddar et al.,¹¹ the discrete formulation of the coupled pressure and heat balances will not be repeated here. Central differencing is used for second order terms in the air layer pressure and energy equations. For an assumed skin temperature, an initial assumption for the air temperature and pressure values for all nodes is that they are at ambient conditions and iterated until convergence is achieved when both coupled pressure and temperature equations converge such that the overall mass and energy are conserved in each grid and values of pressure and temperature are at a minimum relative error of 10⁻⁵. Once the pressure and temperature distribution in the air layer, the fabric void, and the fabric outer node are obtained, the mass flow rates of all directions are calculated to estimate the ventilation rates and to predict the sensible heat loss from the inner cylinder surface. The numerical solution is repeated for grid sizes to ensure that a grid-independent solution is obtained. In the presented simulations, the number of grid points were 400 with a maximum Δθ of 1 ° and minimum Δz of 0.001 m close to the opening.

In order to find the skin temperature associated with the bioheat model, the clothed cylinder ventilation is integrated to a segmental bioheat model that segments the body into cylindrical elements.⁹ Each body segment has known height and radius and will be associated if clothed with its respective air layer size and the type of aperture. The bioheat model accuracy in predicting comfort was validated by Ghaddar et al.⁹ with the results of a survey administered to subjects wearing typical clothing at different activity levels to record their overall and local thermal sensation and comfort in a transitional space. The validation of the bioheat model in predicting skin temperature was performed in previous studies in the literature reporting experiments performed in a controlled environment at different wind and ambient conditions.³⁵ We are using a validated bioheat model within the same range of environmental conditions reported in the literature and its results would then be valid in this work.

The predicted segmental ventilation from the cylinder model at the initial guess temperature of the skin is then used to accurately estimate the dynamic resistance (see equation (16)) to use as input to in the bioheat model. With the integration of the dynamic resistance, the bioheat model is then able to accurately predict human physiological responses, such as the body's skin temperature when the segment is exposed to external wind. To integrate with the bioheat model, the predicted local ventilation rate at the assumed skin temperature is used to estimate the clothing dynamic resistance. These data were then used as input to the corresponding clothed body segments and the bioheat model predicts a new segmental skin temperature which is used as the surface temperature of the clothed heated cylinder to recalculate the mixed ventilation rate. The alternating solution between the two models is repeated until a converging skin temperature is obtained between the bioheat model and the cylinder model. These alternating steps for coupling the two models are illustrated in Figure 2.
Figure 2.
Flow chart of the followed methodology.

The total ventilation rate of the clothing human body is constructed by considering the sum of the segmental ventilations. It is applicable to situations where clothing in not loose (gap between clothing and skin is small) and when interface ventilation between different segments is minimal (e.g. sleeved arm and clothed trunk). Significant attention must be given choosing in the type of clothing in the bioheat model. The integrated segmental bioheat and clothing model for estimation of ensemble total ventilation is then validated with the published work of Havenith et al. for several clothing ensembles at different wind speeds.¹⁴ The ensemble is formed by considering each segment and attributing to it several layers of fabric according to the ensemble description.

Experimental methodology

The aim of this experiment is to validate the predictions of the developed dry models for air gap apertures open at the bottom end that represent clothed arms and legs by measuring the air layer temperatures at different locations in the air layer of a clothed heated cylinder subject to external wind in a wind tunnel. The dry experiments were performed while subjecting the inner cylinder to a constant heat flux.

An open loop indoor wind tunnel was used to provide the steady flow at a room temperature of 25℃. The tunnel used a blow type variable speed axial fan with 0.7 m diameter and peak power of 1.6 kW. The wind tunnel has a square test cross-section of dimensions 0.8 m × 0.8 m and provides a uniform wind speed ranging from 0.5 to 4 m/s. Figure 3(a) shows a schematic of the clothed heated vertical cylinder experimental setup in the wind tunnel, measurement systems, and components. The clothed cylinder setup consisted of: (1) a copper hollow inner cylinder with a diameter of 25 mm and a length of 40 cm; (2) an outer fabric cylinder 40 cm long and 35 mm diameter resting on a thin metallic screen of 2 cm open squares where the cotton fabric is wrapped around and tightly fitted; (3) a support platform from the top; and (4) Styrofoam insulative material to insulate the top of the vertical cylinder and air annulus. The concentric cylinders are fixed at the upper end to the supporting insulated frame. The choice of a 5 mm gap is considered acceptable for use in testing the effect of natural convection due to the length of the cylinder compared to the gap size. Spencer-Smith³⁶ reported that natural convection is negligible for an air gap between clothing and skin below 8 mm when using a cylinder of length 20 cm. In our case, the height is 40 cm, increasing Grashof number by eightfold and resulting in a significant natural convection effect.
Figure 3.
Schematic of (a) the clothed cylinder in the wind tunnel representing the clothed arm and (b) the thermocouples mounted axially on the fabric cylinder to measure the temperature.

The selected fabric is cotton of permeability of 0.05 m³/m²·s. The cotton was obtained from Test fabrics Inc. (Middlesex, NJ 08846), and is made of unmercerized cotton duck, style #466 of thickness of 1 mm. The gap between the fabric and the cylinder represented the microclimate air zone at a thickness of 5 mm. The inner cylinder has a heating rod inserted to produce a constant heat flux condition at the cylinder surface given the small diameter and the cylinder conductive surface. The total power input to the heaters is regulated using a dimmer circuit to maintain power at 3.75 ± 0.03 W for all the experiments, resulting in a flux value of 53.4 W/m² to provide a uniform inner cylinder of temperature of 33 ± 0.3℃. Since the inner heated cylinder surface is highly conductive the non-uniformity in measured skin temperature was less than ± 0.3℃ at each experiment. The uniform cross wind experiments over the clothed cylinder were conducted at low-speeds and at steady-state conditions.

Seven thermocouples at different positions in the axial direction were mounted on the mesh cylinder to protrude half way into the air gap to measure the temperature of the air layer, as shown in Figure 3(b). The fabric and mesh cylinder are rotated at each time the experiment is repeated to measure the temperature variation in angular direction since we have a steady flow and thermal state. Data logger was used in order to measure all the temperature values. The accuracy of the temperature readings was ±0.1℃.

Results and discussion

This section focuses on the validation of the studied model with either experiment or a published paper. First, the ventilation model is validated with experiment. Second, the integrated bioheat model with the ventilation model is validated with published experimental data of clothed humans.¹⁴

Model validation by experiments

This section presents the computational and experimental results of ventilation rates and heat loss from a cotton-clothed vertical cylinder opened at the bottom end.

Figure 4 shows the predicted and experimentally measured values of (a) angular-averaged air annulus temperature as a function of height, and (b) axial-averaged air annulus temperature as a function of angle at a wind speed of 1 m/s. Good agreement is found between the experimental measurements of the air temperature and model predictions of the air layer temperature with a small maximum error of less than ±0.2℃. This is expected since the model of the air layer thickness is small and the lumped model assumption is valid.
Figure 4.
Plot of the predicted and experimentally measured values of (a) the angular-averaged air annulus temperature as a function of height, and (b) the axial-averaged air annulus temperature as a function of angle at a wind speed of 1 m/s.

Clothed heated cylinder mixed convection ventilation and thermal results

To understand further the effect of natural and mixed convection due to opening for clothed limbs, we analyze the cylinder model simulation results for a fixed cylinder surface temperature of ventilation rates and heat loss from a clothed vertical cylinder in the three cases of air gaps at an open bottom aperture, open top aperture, and closed end clothed, and heated vertical cylinders. We will compare the heat losses and ventilation rates between the three different clothed apertures and examine the effect of external wind for a clothed segment. The three simulated cases are:
(1) Annulus opened to the atmosphere from the top end and closed from the bottom end (representing a loose-fitting clothed trunk).¹¹

(2) Annulus opened to the atmosphere from the bottom end and closed from the top end (representing the clothed limb).

(3) Annulus closed from the two ends (representing a tight-fitting clothed limb or trunk).
Simulations are performed at an ambient temperature of 25℃ at dry conditions. The skin is taken to be at constant temperature of 34℃. In the simulations, the dimensions are the same in the three cases and are taken to be those of our current experiment clothed cylinder (L = 40 cm and R_f = 2.5 cm). However, when integrating the bioheat model, the dimensions will be changed to represent the actual dimensions of the human segment under consideration.

Figure 5 shows the angular-averaged microclimate air temperature along the axial distance. Regarding the first case where the opening is at the top, the flow is only entered through fabric and is driven upward and leaves the cylinder at its end.²⁷ For the open top end, the temperature decreases because of the high pressure at the bottom and the low pressure at the top that also draws in a higher air flow in the radial direction, contributing to the decreasing top temperature. At the exit, the temperature profile becomes stable with a zero temperature gradient. Regarding the air gap open bottom aperture case, it is obvious that the temperature increases with the axial distance. The air enters from the lowest open end, goes through fabric, and leaves from the top. The inlet flow is at ambient temperature; thus, the slope of the increasing temperature is large. The temperature becomes stable at the end. The importance of this comparison is to recognize the difference between the microclimate and the skin temperatures, which affects heat losses. Clearly, the heat losses are highest in the case of a clothing aperture open at the lower end of the limb at a total heat loss of 56.41 W/m², followed by the case of closed apertures at the top and bottom of the clothed cylinder at a total heat loss of 48 W/m², and then lastly the case of an open aperture at the top of the clothed cylinder at a total heat loss of 44.02 W/m². In fact, the entering air at ambient temperature causes the high heat loss since the ambient temperature is low compared to the skin temperature. However, in the case of open top annulus, air reaching the microclimate layer penetrates through the fabric only. When air penetrates through fabric, it does not reach the microclimate layer at the ambient conditions, because fabric nodes heat the air and allow it to enter at a high temperature relative to the ambient conditions. Thus, heat loss related to the difference of temperature between the skin and the microclimate air layer decreases. The top opening in this case does not enhance heat loss because the ambient air cannot enter the microclimate layer from this opening. A sensitivity analysis has been done in the limit that ambient temperature is kept lower than skin temperature by more than 4℃. An 8–16% increase in the ambient temperature leads to a decrease of 7–15% in ventilation rate and of 8–14% in total heat losses in both cases of open top and open bottom apertures for the air annulus. Another sensitivity analysis has been done on the air gap size. It is found that a 10% increase in air gap size leads to a 0.4% increase in the microclimate ventilation rate.
Figure 5.
Comparison between temperature profiles for open top, open bottom, and closed aperture for a clothed cylinder. For the open bottom aperture case, the air layer temperature has an ambient value at the entrance (z = 0), while for the open top (closed bottom) aperture case the temperature is highest since this would be its lowest ventilation rate.

A top open aperture leads to a decrease in the pressure difference between the microclimate layer and the outer fabric node. The penetrating mass flow rate through the fabric is caused by the difference in pressure between the microclimate and the outer fabric surface external pressure around the cylinder (equation (2)). Thus, this mass flow rate is the highest in the case of the open bottom aperture of the annulus followed by the closed ends followed by the open top annulus. This is obvious in Figure 6, representing the axial-average radial mass flow rate variation in the angular direction. Therefore, the microclimate ventilation related to the positive penetrating mass flow rate is also largest in the open bottom annulus case. Table 2 summarizes the comparison between these three different configurations in order to emphasize the importance of the opening location in increasing the total heat loss and the ventilation rate. This importance is also revealed by Ke et al.^37,38 when standing in wind conditions. Indeed, they prove experimentally that closing the bottom end decreases the ventilation rate intensively compared to other closing conditions.
Figure 6.
A plot of the axial-average radial mass flow rate variation in the angular direction for open top, open bottom, and closed ends aperture types of a clothed cylinder. The average microclimate ventilation is largest for the open bottom annulus case.

Table 2.
Comparison of total sensible heat loss and ventilation rate of a cylinder (R_f = 2.5 cm, L = 40 cm) subject to $V \infty$ = 1.0 m/s at T_amb = 25℃ for three clothed cylinder air annulus aperture types

Open top aperture of clothed cylinder air annulus Open bottom aperture of clothed cylinder air annulus Closed top and bottom apertures of clothed cylinder air annulus

Q (W/m²) 44.03 56.41 48.01

$\overset{\cdot}{m} a$ × 10⁻² (l/min) 13.83 20.82 14.73

If the natural convection term in the pressure equation (see equation (7)) is removed and the ventilation of a clothed human is solved based on forced convection, then we can compare with this mixed with the convection solution to estimate the wind speed below which natural convection is significant. At 0.1 m/s, the ventilation rate without natural convection decreased by 80% from the value estimated by the current model with natural convection, while at wind speeds of 1.0 m/s and 2.0 m/s, the percent difference between the two solutions is close to 4% and 1%, respectively. For wind velocities higher than 2 m/s, the natural convection effect can be neglected when estimating clothing ventilation.

Validation of the integrated bioheat and clothing ventilation model

Havenith et al. published experimental data on total clothing ventilation, vapor resistance, and permeability index for different human postures and wind velocities.¹⁴ The aim of this section is to estimate the ventilation rate of the body for different permeability ensembles subject to an external wind with open and closed apertures and compare the results with Havenith et al.'s experimental results. Three clothing ensembles were selected from their study for use in the validation of our segmentally constructed ventilation and bioheat model. The selected ensemble were
(1) Ensemble A (permeable case): workpants (cotton, Φ = 0.66, α = 2.121 m/s), poly-shirt (50% polyester, 50% cotton, Φ = 0.85, α = 5.465 m/s), and sweater (cotton, acrylic, Φ = 0.82, α = 4.13 m/s).

(2) Ensemble B (semi-permeable): workpants (cotton), poly-shirt (50% polyester, 50% cotton), sweater (cotton, acrylic), and coverall (cotton, Φ = 0.5, α = 2.161 m/s).

(3) Ensemble C (impermeable): workpants (cotton), poly-shirt (50% polyester, 50% cotton), sweater (cotton, acrylic), and rain coverall (nylon, Φ = 0.1, α = 2.3 × 10⁻³ m/s).

In all these ensembles, the human trunk is assumed to have an inner thin cotton liner layer with a very small dry resistance such that the skin temperature and inner layer temperature differ by less than 0.05℃. Therefore, the inner layer is disregarded. The clothed human body is divided into segments. In the model simulation, the research work of Song et al.⁴⁰ was adopted in selecting the different body segmental air gap sizes for an average-sized clothing ensemble The fabric used in each segment is set in the bioheat model, where the thermal properties of each fabric type play an important role in the heat transfer between the environment and the skin. After setting each ensemble appropriately in the bioheat model, the skin temperature calculated is integrated into the mathematical model. The segmental ventilation rate calculated by the mathematical model in each ensemble is obtained and entered into the bioheat model to predict the skin temperature. This iterative process is repeated until convergence is reached. The number of iterations depends on the initial temperature guess of the skin. In this work, we started with 34℃ for all segments. For this condition, it took 300–400 iterations to reach an error of 10⁻⁶ between the last skin temperature iterations. Once convergence is achieved, the total ventilation rate through each ensemble is then evaluated.

Figure 7 presents the comparison between model predictions and experimental results of ventilation. We should note that all the values predicted fall in the range of the standard deviation of the experimental study. Thus, good agreement was found between the parametric study predictions and experimental measurements of ventilation using the tracer gas. In high wind conditions, the predicted ventilation is under-estimated. This is related to the fact that blockage confronts the air when hitting arms and legs. In fact, when air strikes the hand which is not clothed, a high pressure region is created, increasing the air mass flow rate entering from the opening. This blockage phenomenon increases the ventilation. However, because it is not taken into account in the mathematical model, it is logical to see this difference in ventilation between experiment and modeling.
Figure 7.
Comparison between the parametric study predictions and published experimental results of Havenith et al.¹⁴ for three ensembles subject to several values of $V \infty$ .

The model has shown that when the wind speed increases, the opening in the bottom does not show any advantage in the suction of the ambient air. This is due to the diminishing effect of the natural convection for wind speeds above 2.0 m/s. The case of the open aperture at the lower end of the vertical annulus can be considered as a closed ends annulus and behaves in the same manner. The effect of permeability is clearly shown in enhancing the ventilation. A 36% increase in ventilation is obtained by moving from a semi-permeable to a permeable ensemble when only natural convection is present (no wind). However when $V \infty$ is larger than 2.0 m/s (forced convection), the increase in ventilation by moving from a semi-permeable to a permeable ensemble is 24%. Finally, in the mixed convection (0.0 < $V \infty$ < 2.0 m/s), the corresponding increase in ventilation by moving from a semi-permeable to a permeable ensemble is about 20%. In the case of the impermeable ensemble, the model has a limitation when wind speed increases above 1.0 m/s. This limitation is connected to the fabric model used in this work, where the effective convection coefficients of fabric nodes are based on permeable fabric. Moreover, the increase in ventilation rate when natural convection is included ranged from 9% to 13%, depending on the permeability. For a higher permeability ensemble, the natural convection enhancement effect on ventilation rate is more considerable than that encountered for a low permeability ensemble. Thus, the increase in permeability is indispensable to improve the ventilation in the no-wind condition because the natural convection enhancement effect on the flow within the air layer induces flow from the ambient air due to the pressure difference created between the ambient air and the air within the air gap. Total heat loss caused by the mixed convection is also a function of the fabric permeability and the wind speed. This is shown in Figure 8 where the heat losses follow approximately the same profiles as those of ventilation rates. Note that heat losses are computed by summing the segmental heat losses of clothed and nude body parts. Head, hands, and feet are nude and represent about 18% of the total body heat loss.
Figure 8.
A plot showing the predicted total body heat losses of the three different ensembles with respect to wind speed.

Table 3 shows the percentages ventilation for different segments compared to the total ventilation rate. As mentioned earlier, the clothed trunk is an open top clothing aperture for the air annulus while clothed limbs are normally assumed to have bottom open clothing apertures for the air annulus separating clothing from the skin. It is clear that at higher wind speeds, the clothed trunk ventilation is highest compared to that at the clothed limbs. However, at close to zero wind, the open clothing aperture at limbs at the lower end contribute most to the total ensemble ventilation since ventilation is enhanced considerably by natural convection. Clothed legs present a higher percent of total ventilation than arms because the natural convection effect is stronger for a bigger height enhancing the ventilation further.
Table 3.
Comparison of percentage of local to total ensemble ventilation rate

Percent of total ventilation (%)

Clothed limbs

Ensemble Posture Wind (m/s) Clothed trunk (R_s = 20 cm, H = 40 cm, Y = 8 mm) Arms (R_s = 4 cm, H = 60 cm, Y = 5 mm) Legs (R_s=20 cm, H = 110 cm, Y = 10 mm)

A Standing 0.0 14.75 21.15 64.09

0.7 30.29 18.11 51.60

4.1 68.11 8.73 23.15

B Standing 0.0 17.58 20.35 62.07

0.7 32.89 17.89 49.21

4.1 69.57 8.333 22.09

C Standing 0.0 19.31 19.68 61.01

0.7 34.89 16.69 48.41

Conclusions

A mathematical model derived from first principles is developed and is validated by experiments to simulate the coupled ventilation and heat convection in a vertical open bottom end air annulus between a concentric fabric cylinder and a heated solid cylinder placed perpendicular to airflow, representing the clothed limb. When subjected to a uniform cross wind, an open clothing aperture at the lower end of a clothed cylinder presents a significant ventilation rate when compared to that of the open top and closed ends cylinder. The reason is that the opening at the bottom end allows the ambient air to be sucked and then to ventilate the heated cylinder representing the skin.

The clothed cylinder model is integrated with a detailed segmental bioheat model to accurately determine the skin temperature associated with the clothed cylinder ventilation. The combination of the mathematical model and the segmental bioheat model is validated with published experimental data on ensemble total ventilation for different types of clothing permeability and wind speeds. The construction of total ventilation from segmental ventilation values of the trunk and limbs has led to an accurate prediction of total ventilation when compared with published data, particularly for wind speeds less than 2 m/s.

The combined bioheat and clothed cylinder ventilation model presents a valuable predictive tool for the prediction of clothed human ventilation, instead of depending on experimentation. It is applicable to situations where clothing is not loose (the gap between clothing and skin is small) and when interface ventilation between different segments is minimal (e.g. sleeved arm and clothed trunk).

Convection coefficient	Value or equation (W/m²·K)	Reference
h _{c_skin-air}	10.5	²⁴
h _co	9.6	²⁴
H _co	if $\overset{\cdot}{m} a y > 0.00777$ then $H co = 495.72 * \overset{\cdot}{m} a y - 1.857$ else $H co = 2$	²¹
h _c(o-air)	$h c (o - air) = a . V \infty b where a = 4.8 * (2 . R f) - 0.33 and b = 0.5$	³⁸
h _r	3.7	³⁹

	Open top aperture of clothed cylinder air annulus	Open bottom aperture of clothed cylinder air annulus	Closed top and bottom apertures of clothed cylinder air annulus
Q (W/m²)	44.03	56.41	48.01
$\overset{\cdot}{m} a$ × 10⁻² (l/min)	13.83	20.82	14.73

			Percent of total ventilation (%)
A	Standing	0.0	14.75	21.15	64.09
0.7	30.29	18.11	51.60
4.1	68.11	8.73	23.15
B	Standing	0.0	17.58	20.35	62.07
0.7	32.89	17.89	49.21
4.1	69.57	8.333	22.09
C	Standing	0.0	19.31	19.68	61.01
0.7	34.89	16.69	48.41

Footnotes

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			Percent of total ventilation (%)
				Clothed limbs
Ensemble	Posture	Wind (m/s)	Clothed trunk (R_s = 20 cm, H = 40 cm, Y = 8 mm)	Arms (R_s = 4 cm, H = 60 cm, Y = 5 mm)	Legs (R_s=20 cm, H = 110 cm, Y = 10 mm)
A	Standing	0.0	14.75	21.15	64.09
		0.7	30.29	18.11	51.60
		4.1	68.11	8.73	23.15
B	Standing	0.0	17.58	20.35	62.07
		0.7	32.89	17.89	49.21
		4.1	69.57	8.333	22.09
C	Standing	0.0	19.31	19.68	61.01
C	Standing	0.7	34.89	16.69	48.41