Preface ix
Chapter 1. Building a Model for a Coupled Problem 1
1.1. Basic equations of the models (Appendix 1) 2
1.2. Boundary layers 3
1.2.1. Forced convection 4
1.2.2. Natural convection 6
1.3. Heat balance for a "system" and boundary conditions 8
1.4. On the problem of cooling of a cup of tea 11
1.4.1. Balance equations 12
1.4.2. Research of transfer correlations13
1.4.3. Surface temperature as a function of average temperature of the liquid 15
1.4.4. Liquid temperature as a function of time 16
1.5. Bather on a beach 19
Chapter 2. Approximate Determination of Transfer Coefficients 25
2.1. Natural convection around an isolated sphere 25
2.1.1. Equations of boundary layers depending on velocity and temperature 26
2.1.2. Integration over the boundary layer thickness 28
2.1.3. Dimensionless formulation 32
2.1.4. Numerical solution 33
2.2. Coupled exchanges around the head of a baby lying down 37
2.2.1. System of equations 38
2.2.2. Boundary layers for the horizontal disk 40
2.2.3. Boundary layers on curved surfaces 41
2.3. Forced convection around a cylinder 43
2.3.1. System of equations 44
2.3.2. Integration of the equations of the dynamic boundary layer 46
2.3.3. Dimensionless integral equation 48
2.3.4. Resolution of the upwind dynamic boundary layer 50
2.3.5. Resolution of the downwind dynamic boundary layer 55
2.3.6. Resolution of the thermal boundary layer 56
Chapter 3. Human Thermal Models 61
3.1. The Fanger model: from climatic chamber to standard 61
3.1.1. Environment and human body physical parameters 62
3.1.2. Equilibrium balance equation in the Fanger model 69
3.1.3. Examples of ambient environment qualifications 72
3.2. Gagge model 76
3.2.1. A simple, unsteady and regulated geometrical model 76
3.2.2. Response of "human system" to a sudden change in metabolism 78
3.3. Stolwijk 25 node model 80
3.4. Thermal model of a baby lying down 82
3.4.1. Geometrical division 82
3.4.2. Metabolism and respiration 83
3.4.3. Exchanges of the uncovered part of the head 84
3.4.4. Conduction between body layers 85
3.4.5. Sensible heat exchanges of the trunk 87
3.4.6. Trunk evaporation 88
3.4.7. Blood convection 89
3.4.8. System of equations 90
3.4.9. Simulation results 91
Chapter 4. Heat and Humidity Transfer in Clothing 97
4.1. From heterogeneous porous to continuous model media 98
4.2. Heat diffusion and convection 100
4.3. Vapor diffusion 101
4.4. The effect of bound water 105
4.5. Liquid water diffusion 111
4.6. Mass and energy balances 119
4.7. Limit conditions 121
4.8. Processing for a numerical resolution 123
4.9. First example: condensation in a multilayer 124
4.10. Convection and diffusion 128
4.11. Taking account of radiation 130
4.12. Second example: firefighters' clothing 135
4.13. Traditional warm weather clothing 137
Appendices 143
Appendix 1 145
Appendix 2 151
Appendix 3 155
Bibliography 157
Index 161
1
Building a Model for a Coupled Problem
There are numerous and varied heat and humidity exchange coupled problems in the environment, and more specifically in man's surrounding environment (comfort, habitat, clothing, etc.) and a common methodology to approach these can be established. First of all, we need to position ourselves in relation to the digital/IT tools currently offered in "the market" and which allow for a resolution of numerous physics problems. The readers may be under the impression that the difficulty resides rather in making a choice among all these tools/software. It is common for specialized research departments to use software adapted to their fields (habitat, aviation, automobile, etc.) though a layman perceives them as some sort of magic "black box". When results come out, the reliability interval is often uncertain, as the given problem was never treated for a neighboring configuration. It should be noted that solving a mathematical model numerically with elaborated software presupposes the formulation of a number of simplifying hypotheses that may be valid for a given configuration, but risky for another. To take an extreme example, outside of our field of study, media report on the progress of IPCC works concerning climate heating predictions while they highlight the uncertainty of 20-year predictions. At planetary scale, ocean/atmosphere models are particularly complex.
Let us therefore consider a "system" whose thermal and hydric behavior in particular conditions is to be determined: an individual in a room, a manned vehicle, an incubator, a piece of sportswear, etc. It is always possible to set the proper orders of magnitude for the behavior of a system under thermal constraints by scale analysis of the equations of an adapted model and by using the theoretical and experimental data in the literature. This first model can be preliminary to the use of software that is more complex but more difficult to interpret under the relative influence of input parameters. Through several simple examples, we will examine the implementation of such models.
1.1. Basic equations of the models (Appendix 1)
A fluid medium (humid air, liquid water, etc.) put in motion by a machine (forced ventilation, pump, etc.), wind, temperature gradients (natural convection), can be described by a number of variables depending on space and time: pressure p, temperature T, velocity , density ?, enthalpy h, etc.
The (quite) general conservation equations given here are written in condensed notation, using a pseudo vector (nabla), or gradient, which in Cartesian coordinates x, y, z can be written: .
Mass conservation can be written as:
[1.1a] Or if we use the differential operator in the direction of movement , we then have:
[1.1b] For vapor contained in incompressible air, mass conservation is written as:
[1.2] This equation is based on Fick's law of diffusion, which gives the mass diffusive flux (kg/m2s) of the vapor species (?v) in the air (?): where Dv is the diffusion coefficient. This law is valid for humid air with . For gas mixtures where this order of magnitude is no longer valid, a more precise law is applied.
The momentum conservation for the "volume forces" limited to gravity, for "Newtonian" fluids with constant viscosity coefficient µ is written as:
[1.3] The first member is the inertia term and the second member contains pressure, gravity and viscosity terms.
The energy conservation or the first law of thermodynamics is written as:
[1.4] The first member expresses enthalpy conservation, and the second one contains the "heat" corresponding to compression or expansion, an important quantity for certain machines, the conductive/convective transfer and the viscous dissipation. In certain cases, a radiation energy term can be added. Viscous dissipation, always positive, is expressed as a function of the derivatives of velocity components. At low velocities it is negligible. Therefore, the most common form of [1.4] applicable at low velocities is:
[1.5] We should note that equations [1.2] and [1.5] are similar in form, and this will have very practical analogical consequences.
We add to these the so-called equations of "state" of the fluid. For a perfect gas, which is the case of air at human environment temperature and atmospheric pressure, we have:
[1.6] For a liquid at moderate pressures and temperatures we have:
[1.7] 1.2. Boundary layers
A fluid in movement is limited by a solid wall surrounded by "boundary layers", regions that despite their low thickness play a key role in the exchanges (of momentum, heat, etc.), and where important gradients (of velocity, temperature, etc.) develop. This concept originated in the development of aviation in the 1920s. Airplanes with airfoils that generated the best lift for a wide incidence range had to be designed.
1.2.1. Forced convection [SCH 60]
The typical example treated by fluid mechanics texts is that of a heated cylinder, with diameter D, placed perpendicular to a uniform fluid flow of velocity V. The development of boundary layers (in terms of velocity and temperature) starting from the stagnation point, then a detachment that generates unsteady wake behind the cylinder can be observed (Figure 1.1). Another velocity layer accompanies the wake's vortices. The thermal boundary layer, for heat conduction "across" the stream filaments, is continuous and adapts to velocity heterogeneities. The Reynolds number Re = VD/?, where ? is the kinematic viscosity, describes the flow regime as subcritical when the dynamic boundary layer remains laminar before detachment, or supercritical when the dynamic boundary layer transits from laminar to turbulent regime before detachment. The point of detachment moves downstream when turbulence emerges in the boundary layer. The critical Reynolds number is close to 4.5.105.
Figure 1.1. Boundary layers around a cylinder in forced convection
Let us suppose that this cylinder has a surface that is being heated at constant temperature and also kept humid (saturated) by an internal device. A third boundary in humidity (?v) is thus established around the cylinder. The thermal boundary layers, which are key to heat exchange, are "globalized" by a heat transfer coefficient hcv (not to be confused with the enthalpy h) whose definition by heat flux density is the following:
[1.8] Tp and To are, respectively, the temperature of the wall and of the fluid stream away from the wall, and the temperature gradient is defined at the wall along the exterior wall surface normal. For the cylinder example, hcv is a function of the curvilinear abscissa. This transfer coefficient is rendered dimensionless in the form of a Nusselt number Nu = hcvL/?, where L is a length characteristic to the problem (D for a cylinder). Integrating over the surface we get an average transfer coefficient and an average Nusselt number often denoted by In the case of forced convection, experimental or theoretical data take the form Nu = f (Re, Pr) which is often expressed as Nu ~ Ren Prp, where Pr is Prandtl number, a characteristic of the fluid written as
Similarly, in order to characterize the "vapor" boundary layer, a mass transfer coefficient denoted by k is defined by the mass density flux:
[1.9] The dimensionless form of this coefficient is the Sherwood number Sh = kL/Dv, a function of Reynolds number characterizing the flow, and of Schmidt number Sc = ?/Dv, which connects the physical properties of fluids (vapor and air in the mentioned example). In general, the data refer to surface averaged values. For a wall kept humid in such a manner that it can be considered a liquid surface, we have ?vp = ?vsat(Tp), which is the condition for air to be saturated with water vapor.
In the same context of forced convection the analogy of equations [1.2], [1.5] allows us to affirm that for similar boundary conditions (constant temperature and humidity at the wall, for example) the relations Nu = f(Re, Pr) and Sh = f(Re, Sc) will be similar in form.
1.2.2. Natural convection [BEJ 84, BEJ 93]
Let us now consider the previous example heated cylinder in still air. Natural convection occurring around the cylinder under the influence of thermal body forces (warm air is lighter than cold air) generates a thermal boundary layer and a "plume"-like wake above it (Figure 1.2).
Figure 1.2. Natural convection around a heated cylinder
The heating intensity (wall temperature Tp) determines the nature of the flow (laminar, turbulent) and is expressed by a number, obtained by rendering the equations dimensionless, namely the Rayleigh number , where g is gravitational acceleration, ß = 1/T for a perfect gas (T Kelvin), L is a length characteristic to the problem (here, D). An alternative to Ra is the Grashof number Gr = Ra/Pr. The laminar/turbulent transition takes place around Gr = 109. The experimental correlations...
