An analytical dynamic model of heat transfer from the heating body to the heated room

On the base of mathematical description of thermal balance the dynamic model of the hot-water heating body (radiator) was designed. The radiator is mathematically described as a heat transfer system between heating water and warmed-up air layer. Similarly, the dynamic model of heat transfer through the wall from the heated space to the outdoor environment was design. Both models were interconnected into dynamic model of heat transfer from the heating body to the heated room and they will be implemented into simulation model of the heating system in Matlab/Simulink environment. 1 Hot-water heating body An analytic identification method based on mathematical description of a heat exchanger was chosen for design of the hot-water heating body dynamic model [1]. A hot-water heating body can be described as a heat transfer system between heating water and warmed-up air layer [2]. Heating water circulates inside radiator and delivers a heat through surface layer. External side of radiator is surrounded by air layer, which is warmed-up and heated air naturally flows up due to difference of specific weight [3]. It was considered ideal mixing of the heating water in internal space of the radiator and ideal air mixing in boundary layer of the radiator.


Hot-water heating body
An analytic identification method based on mathematical description of a heat exchanger was chosen for design of the hot-water heating body dynamic model [1].
A hot-water heating body can be described as a heat transfer system between heating water and warmed-up air layer [2].Heating water circulates inside radiator and delivers a heat through surface layer.External side of radiator is surrounded by air layer, which is warmed-up and heated air naturally flows up due to difference of specific weight [3].It was considered ideal mixing of the heating water in internal space of the radiator and ideal air mixing in boundary layer of the radiator.The heating water with mass flow q b and temperature T b1 inputs into radiator (Fig. 1).At every time point there is the volume of heating water V b with density U b in internal space of radiator.This water delivers a heat through surface layer of the radiator with plane A r which is surrounded by air layer.Temperature of the return heating water from radiator is T b2 .The air flow rate is q a , volume of air in boundary layer of the radiator is V a , its input temperature to the radiator surface is T a1 and its output temperature after heating is T a2 .Exchanged heat flow from heating water to the radiator wall is ) b and from the radiator wall to room space is ) a [4,5].

Mathematical description of heat transfer in the heating body
Generally, for dynamic balance describing of heat energy increasing or decreasing in system it is valid, that difference of input and output heat flows is equal to the time variation of the accumulated energy in a system [6].Then the heat accumulation equation is valid for air heating in the boundary layer of the radiator: where ) a is heat flow from the radiator wall to the surrounding [W], q a is air flow in the boundary layer of the radiator [kgs -1 ], c a is specific heat capacity (specific T a2 is output air temperature [K], U a is air density [kgm -3 ] and V a is air volume in the boundary layer of the radiator [m 3 ].For heat exchanging from the heating water to the radiator wall this heat accumulation equation is valid: where ) b is heat flow from the heating water to the radiator wall [W], q b is mass flow rate of the heating water [kgs -1 ], c b is specific heat capacity (specific heat) of water [Jkg -1 K -1 ], T b1 is input water temperature into the radiator [K], T b2 is temperature of the return heating water from the radiator [K], U b is water density [kgm -3 ]   and V b is volume of water in the radiator [m 3 ].It is necessary for design of automatic control to know the dynamic dependence of the control deviation according to changes of variables which have effect on the deviation [7][8][9][10].For that reason, we express dependent variables in equations ( 1) and ( 2) by their values in initial steady-state and their increments.Next, to simplify computation, we express each of dependences in non-dimensional form as relative changes of variables and we get system of differential equations describing the radiator:

Dynamic model of the hot-water heating body
After Laplace transform of differential equations ( 3) and ( 4) and other mathematical operations it is valid for relative change of air output temperature: To make mathematical description of dynamic characteristics of the radiator complete, it is necessary to supplement equation ( 5) and ( 6) with relations for relative changes of heat flows x )a and x )b .If heat transfer coefficients h a and h b are constant in unsteadystates, relative variations of heat flows are dependent only on temperature changes of the heating water and air.Generally, we can express it as: Transfer functions in relations ( 7) and ( 8) are [2]: and time constants W b , W r : where we get by reciprocity of indexes "a" and "b" in terms (9).
Based on the terms (5) till ( 9) for non-dimensional variables the block diagram of dynamic model of the hot-water heating body was designed (Fig. 2).

X Tb1
Heat transfer through the radiator wall Input variables into model are non-dimensional variables or constants: input temperature of air X Ta1 , air flow rate X qa , temperature of the input heating water into radiator X Tb1 and mass flow rate of the heating water X qb .Output variables from model are non-dimensional variables: output temperature of air after heating by the radiator surface X Ta2 and temperature of the return heating water from the radiator X Tb2 .

Heat transfer through the wall
For design of the dynamic model of heat transfer through the wall we have considered a plane wall, where the wall has been considered as continuum with continuously distributed thermal resistance and capacity [13,14].We have chosen elementary layer with following parameters (Fig. 3 ) d) Fig. 3. Heat transfer through the plane wall.

Mathematical description of heat transfer through the wall
The heat energy does not originate either does not lose in considered elementary layer of the wall.Then difference of the input heat and output heat in the layer has to be equal to the time variation of the energy in the layer [15].
Let's c is specific heat capacity (specific heat) and U is volume weight of the wall material, then: Considering that heat flow d) is: If specific heat capacity c and volume weight U of the used wall material are constant, then [16]: According to Fourier's law the heat flow is directly proportional to the temperature gradient: where O is heat conductivity coefficient of the used wall material.
Partial differential equations ( 16) and ( 17) with relevant initial and border conditions completely describe non-stationary one-dimensional heat flow [17].
As we mentioned above we have expressed dependent variables by their values and increments and by substitutions and subtractions we have got partial differential equations system of heat transfer dynamics through the wall: To simplify computation, we have expressed each of dependences in non-dimensional form and then we can express partial differential equations ( 18) and (19) 16), (17) or ( 20), ( 21) still need to be supplemented by equations of heat transfer on both sides of the wall surfaces.In dimensionless form for indoor surface area it is valid: where * x 1 ) includes also external effects on the heat flow transfer into the wall (temperature changes or heat transfer coefficient changes, etc.), O N Similarly, for outdoor surface area it is valid: where * x 2 ) includes external conditions of the heat flow and h 2 is heat transfer coefficient between wall surface and air [Wm -2 K -1 ].

Dynamic model of heat transfer through the wall
By the Laplace transform of partial differential equations (20) and (21) and by the other mathematical operations it has been possible to get a system of equations, which describes dependence of non-dimensional variables for heat flows ) 1 , ) 2 and temperatures T w1 , T w2 .
Laplace transform image of partial differential equations system is: Using inverse Laplace transform we have got following equations: .And simultaneously because that the unknowns variables are the heat flow ) 2 taken away from the refrigerated wall surface and temperature T w1 of the heated wall surface, we express  Based on the terms (28) till (35) for non-dimensional variables the block diagram of dynamic model of heat transfer through the wall was designed (Fig. 4).
Input variables into model are non-dimensional variables: temperature of the heated air in the room X Ta2 and outdoor temperature X To Output variables from model are non-dimensional variables of the wall temperatures X Tw1 and X Tw2 [18][19][20].

Conclusion
Dynamic model of heat transfer from the heating body to the heated room consists of two relative separated models: model of the heating body and model of heat transfer through the wall to the outdoor environment.Interconnection of these two models is presented in Fig. 5.
The designed dynamic model will be the basis for creating the simulation model of the heating system in Matlab/Simulink environment [21][22][23][24][25].

Fig. 2 .
Fig. 2. Block diagram of dynamic model of the hot-water heating body.
): thickness of the plane wall dw [m], layer thickness in the plane wall dy [m], distance of layer from the heated surface y [m].The heat flow ) 1 [W] is supplied into heated wall surface which temperature is T w1 [K], the heat flow ) 2 [W] is taken away from the refrigerated wall surface which temperature is ) 2 [W].The heat flow ) inputs into the unit surface of layer dy, and the heat flow ) + d) outputs from it.Temperature of the elementary layer is T w [K].y dy heat transfer coefficient between air and wall surface [Wm -2 K -1 ].

2 Fig. 4 .
Fig. 4. Block diagram of dynamic model of heat transfer through the wall.

Fig. 5 .
Fig. 5. Block diagram of dynamic model of heat transfer from the heating body to the heated room.
other variables not mentioned above are: A r is plane of the radiator surface [m 2 ], h a is heat transfer coefficient between radiator surface and air [Wm -2 K -1 ], h b is heat transfer coefficient between heating water and radiator body [Wm -2 K -1 ], d r is thickness of the radiator wall [m], O r is heat conductivity coefficient of the radiator body material [Wm -1 K -1 ], U [11,12]ecific weight of the radiator body material [kgm-3] and c r is specific heat capacity (specific capacity) of the radiator body material [Jkg -1 K -1 ][11,12].