The Temperature Propagation near the Peltier Cooling Elements

The regulation of the human body temperature becomes more relevant problem for climate conditions cardinal exchanges. When environmental temperature increases or decreases too much, human has no possibility to achieve his thermal comfort by himself, additional help is necessary for this reason. In our case we offer additional temperature regulation system of the human body integrated into wearing clothes [1]. We can summarize this system as a system human – clothing – environment. The cooling elements based on the thermoelectric effect are used for this system development, they called Peltier elements [2, 3]. The basic advantages of this elements are – reliability, safety for human, „green“ technology, precision, long exploitation time, absence of mechanical parts, possibility to use the same system for heating too. So suppose the value of the eliminated heat flow from the human body is QΣ. We can reach this value with one powerful Peltier element, or with some quantity of those elements with less power. Obviously, the human will feel enough comfortable only in this case, when whole his body simultaneously is cooled [4]. But that isn’t very rational solution the entire body to lay with Peltier elements. Let’s try to find how we need to distribute well–balanced Peltier elements PE for effectiveness of the human body cooling. We analyze the case when body is covered by clothes and don’t has direct contact with external surroundings. In this case thermal resistance of the surface layer of the human body is considerably less than resistance of the system air– clothes–ambience. So we can carry that the heat from the body is transferred only by Peltier cooling element.


Introduction
The regulation of the human body temperature becomes more relevant problem for climate conditions cardinal exchanges.When environmental temperature increases or decreases too much, human has no possibility to achieve his thermal comfort by himself, additional help is necessary for this reason.In our case we offer additional temperature regulation system of the human body integrated into wearing clothes [1].We can summarize this system as a system human -clothing -environment.The cooling elements based on the thermoelectric effect are used for this system development, they called Peltier elements [2,3].The basic advantages of this elements are -reliability, safety for human, "green" technology, precision, long exploitation time, absence of mechanical parts, possibility to use the same system for heating too.
So suppose the value of the eliminated heat flow from the human body is Q Σ .We can reach this value with one powerful Peltier element, or with some quantity of those elements with less power.Obviously, the human will feel enough comfortable only in this case, when whole his body simultaneously is cooled [4].But that isn't very rational solution the entire body to lay with Peltier elements.Let's try to find how we need to distribute well-balanced Peltier elements PE for effectiveness of the human body cooling.We analyze the case when body is covered by clothes and don't has direct contact with external surroundings.In this case thermal resistance of the surface layer of the human body is considerably less than resistance of the system airclothes-ambience.So we can carry that the heat from the body is transferred only by Peltier cooling element.

Temperature propagation near one Peltier element
We suppose that we cool the human back, which surface we can approximate as plane.Besides for this problem solution we accept the heat is flowing only in particular upper layer of the human body, and whole rest part of body which is under upper tissue is maintaining a constant temperature T p .
Suppose we have heat (cold) source, its form is disc and its outside radius is r 0 .Source temperature differences with temperature T p in value T 0 .Let's keep the layer of the body surface, in which heat flow is circulating, is solid in form of the parallelepiped rectangular.Its height is h, and it has no limit in the different directions.
Suppose the thermal resistivity of the skin per unit length is R q0 [(K•m)/W] and thermal conductivity of the skin per unit length is G q0 [W/(K•m)].Let's say both of the parameters does not depend on coordinates.It means the skin is homogeneous in longitudinal and transverse directions.Let us solve problem of the temperature distribution on a surface of the skin based on these assumptions by applying an analogy to long line theory with only difference that a heat from such disc type source in homogeneous environment propagates uniformly in all directions.It means that thermal resistivity of the unit volume of the skin in longitudinal direction and conductivity of the skin in transverse direction depends on a distance from a source.Cylindrical coordinate system will be used.Origin point of the model will be on the line which is also the axis of the Peltier element and in the point where the line intersects with a bottom layer of the surface body.Whole volume of the surface layer will be segmented to elementary volumes, each of these occupy the same angle α.Elementary volume of the surface layer is shown in Fig. 1. ( So, the longitudinal thermal resistivity dR q could be expressed Heat in a transverse direction travels through an area dS and distance h Thermal conductivity of the elementary volume dG q in a transverse direction is Equivalent electric circuit of the thermal elementary circuit under consideration is shown in Fig. 2.
The equation could be overwritten if the values of the second smallness order are ignored by formula The Kichhoff's voltage law for a loop shown in figure 2 After simplification it will be After the (8) equation differentiation, application of the (2) equality and after ignoring second smallness order values we get . 1 from equation ( 6) and Q q from equation (8) by taking in consideration ( 2) and ( 4).Angle variable α after some mathematical actions is canceled out.So we have just one variable r in the obtained equation.Further, we apply total but not partial derivatives and obtain equation Characteristic equation of this differential equation is The equation has two roots.One of the roots is positive and the other is negative.Positive root is related with the reflected wave which is not present for the case because the area under consideration is homogeneous until the infinity.Therefore the real process is related with just a negative root , 4 Solution of the (10)-th equation could be expressed by an exponential function where A is the constant of integration.Value of A could be established by taking in consideration the fact that temperature changes by the exponential law just beyond the limit of the Peltier element.The limit is defined by a circle rr 0 , where the temperature is .
where T q (r) and T 0 in this equation is the difference of the appropriate point temperature with an internal temperature of the body T p .dR q dG q dR q dG q dr T(r)

Temperature distribution in the case of two Peltier elements is present
The case is shown in Fig. 3.

Fig. 3. An interaction between two Peltier elements
In a case when the two thermal energy sources are present temperature distribution in the area around them could be calculated by applying superposition principle.However the fact that each of the elements interacts to another and one creates the uneven temperature of the other source must be taken into account.Therefore reflected waves appear in the case.Besides the local system of axes of each PE are not concentric so to obtain the distribution of the temperature by analytical expressions is a hard task.We are mostly considered about the temperature in the point P, which is in the middle of the line which connects the centers of PE1 and PE2, which is in a middle of the line which connects point O and point O' in a case temperatures of the both elements are the same and are equal to T 0 .Axis x of the global system of axes will be superposed with the line crossing point O and point O ' .Suppose the distance from a point O to a point O ' is d.Also d is a distance from center of the one PE to center of the other PE.We will get an expression of temperature distribution in the line OO' as a superposition of two waves which travels from PE1 to PE2 in opposite directions where A and B are the constants of integration.The constants define transmitted and reflected waves to one and to the opposite directions.Function  (x) in the equation ( 16) could be expressed where Constants of integrations are obtained from the boundary conditions: T=T 0 , when x=d/2.By applying the values and by taking in consideration (x) is twin function we will get: By applying some mathematical operations i.e. the second equation difference from the first we will get equality AB.After the summing of these equations we will get Temperature dependence on a coordinate x in the line OO' could be expressed The multiplier K represents T P value's difference from a case temperature travels in one coordinate of Cartesian coordinate system.

Temperature distribution in a case when several regularly placed Peltier elements is present
Under consideration is the case of Peltier elements placed in squares as shown in a Fig.  difference between D and PE is lower.Temperature difference in all the other points of any square area is lower then T D .We can make sure temperature in the point D is the highest by drawing the lines of the same temperature as it is shown in Fig. 4.

Assurance of the desirable distance with permissible temperature change
Placement the Peltier elements must assure desirable temperature in the problematic points.If the PE placement manner is chosen as shown in Fig. 4 the problematic point is point D. Let suppose the temperature difference is T D .Value of r could be increased by decreasing the value of M. We can make sure from the equation ( 12) that M  1 in the case resistivity of the body surface is uniform.Value of the M could be decreased by decreasing the value of longitudinal thermal resistivity R q0 .The goal could be reached by applying some textile materials adherent to a body.Total longitudinal thermal resistivity of the system body -fabric could be decreased to a desirable level by applying the materials with a sufficiently small resistivity R qa0 .Total thermal resistance could be obtained considering fabric and body surface layer in longitudinal direction resistances are connected in parallel.Let's designate body thermal resistivity in longitudinal direction R qk0 .Suppose thickness of the fabric is l a .Let's assume that h is total fabric and body surface layer thickness.Therefore part of the fabric in a total body surface layer -fabric volume will be d/h.Fabric thermal resistance in the length r and in a segment α will be

Conclusions
1.An efficiency of the human body cooling (heating) system directly depends on distribution of the cooling elements; 2. The propagation of the temperature near Peltier elements we can get by using analogy of the long lines; 3. The temperature around one Peltier element varies exponentially; 4. Temperature distribution in the line connecting two cooling elements is descriptive by circuit function; 5. We can identify the biggest temperature differences near Peltier elements using received expression.These Peltier elements are arranged by grid step.

Fig. 1 .
Fig. 1.The basic volume of the surface layer in the surroundings of the Peltier element PE Heat in the elementary volume length of which is dr travels through an area S in longitudinal direction

Fig. 2 .
Fig. 2. The part of the thermal circuit, situated in interval [r, r+dr] and inclusive angle α Applying Kichhoff's current law to the node I of the circuit shown in Fig.2 yields

Fig. 4 .
Fig. 4. Distribution of the Peltier elements on the cooling surface However some parts of the thermal flow travel to the other PE through the point D, too.We must evaluate the reflections at other PE when we use the circuit theory for temperature T D calculation.It can be made by modeling.The temperature T D which can be obtained by expression (21) is the estimation of temperature difference maximum between the point D and PE.The real temperature