1. Chapter 8: TRANSIENT ANALYSIS METHODS FOR BUILDING ELEMENTS
Agami Reddy (July 2016)
Why dynamic models
Terminology
Importance of storage effects in buildings
Types of transient solutions
Finite difference numerical methods
6. Linear models and concept of superposition
7. Transfer function models for building elements
8. Conduction time series model
9. Thermal network models
HCB 3- Chap 8: Transient Analysis

2. What are dynamic models and when are they used?
In the previous chapter, heat transfer has been assumed to be steady-state
For the case of a simple wall, at any given time:
Heat in = Heat out
In certain cases this is a very poor assumption since the thermal heat capacity of the element cannot be neglected
For a Trombe wall element, the more appropriate expression to use is:
Heat stored = Heat in – Heat out
+ Heat generated inside the element
HCB 3- Chap 8: Transient Analysis

3. Other instances where dynamic models are necessary
Embedded piping in ceiling walls or floors
HCB 3- Chap 8: Transient Analysis

4. Building Elements Requiring Transient Analysis
thermal mass of the building envelope
subject to changes in outdoor ambient
air conditions and in solar radiation
over the day,
(b) transmitted solar radiation which is
first absorbed in the internal mass of
the various elements within a
room/space (such as slabs, internal
walls and furniture) and released later in time,
(c) internal mass interacting with operational schedules causing diurnal fluctuations in internal load (lights, equip. and occupants),
(d) internal mass interaction with control strategies of the indoor environment (diurnal thermostat set-up or set-back or variations in supply air volume).
HCB 3- Chap 8: Transient Analysis

5. HCB 3- Chap 8: Transient Analysis
Terminology
Static models
- Steady-state solution (or response)
- Quasi-steady state solution
Dynamic models
- Transient solution
- Steady-periodic transient solution
HCB 3- Chap 8: Transient Analysis

6. HCB 3- Chap 8: Transient Analysis

7. Conduction Transfer Coefficients- Roofs and Walls
Decrement factor
Time lag
Table 8.3
contd…
HCB 3- Chap 8: Transient Analysis

8. HCB 3- Chap 8: Transient Analysis
contd…
HCB 3- Chap 8: Transient Analysis

9. HCB 3- Chap 8: Transient Analysis
Temperature Profiles in a Wall Under Transient Conditions
Forward and
Backward trains
Santamouris, M. and D. Asimakopoulos (1996)
HCB 3- Chap 8: Transient Analysis

10. HCB 3- Chap 8: Transient Analysis
System model
f(x)
A system model is a description of the system. Empirical and mechanistic models are made up of three components: (i) input variables (also referred to as regressor, forcing, exciting, exogenous or independent variables which act on the system, and : - can be controlled by the experimenter, - are uncontrollable or extraneous such as climatic variables; (ii) system structure and parameters/properties provide the necessary physical description of the systems in terms of physical and material constants ; for example, thermal mass, overall heat transfer coefficients, mechanical properties of the elements; and (iii) output variables (also called response, state, endogenous or dependent variables) which describe system response to the input variables.
Models must be sufficiently detailed to describe phenomenon yet be simple enough for practical use
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11. HCB 3- Chap 8: Transient Analysis
If we add effect of wooden support beams in walls, floor and roof, total = 11,000
Distinguish between dynamic and static heat capacity
HCB 3- Chap 8: Transient Analysis

12. HCB 3- Chap 8: Transient Analysis

13. HCB 3- Chap 8: Transient Analysis

14. Different Transient Solutions to Dynamic Analysis of Buildings
a) Distributed models
- Harmonic/admittance methods- closed form solutions
b) Numerical methods (finite differences)- most flexible
c) Thermal network models: lumped and 3C models
d) Transfer function models for both conduction and rooms
(used by eQuest, DOE-2, TRACE, HAP…)
- CLTD/CLF method is a simplified older version (now obsolete)
e) Transfer function models for conduction and then heat balance method for room response (EnergyPlus)
- Simplified approach: Radiant Time Series method
Equipment sizing: like choose cloth
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15. HCB 3- Chap 8: Transient Analysis
Coverage
Building elements- Chap 8
Walls, roof, slabs
subject to outdoor ambient air and solar radiation changes
Entire rooms/spaces- Chap 9
- treatment of transmitted solar
- treatment of internal loads
- combining various individual heat loads
- peak load calculation methods
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16. HCB 3- Chap 8: Transient Analysis
Finite Difference Method
The finite difference method relies on discretizing a function on a grid.
A very flexible and widely used numerical method for solving differential equations with any type of boundary condition
It involves discretizing the derivatives of the temperature profile both spatially and temporally into discrete quantity
The solid, or say the building wall assumed to be a homogeneous element with constant conductivity, is divided into a number of equally spaced finite layers of thickness
HCB 3- Chap 8: Transient Analysis

17. HCB 3- Chap 8: Transient Analysis
Each of the layers is assumed to be at a uniform temperature i.e., all the mass of the wall element is located at the node. An energy balance on node m yields:
Heat flow in from node (m-1)- heat flow out to node (m+1) = change in heat capacity of node m, or
with thermal diffusivity
HCB 3- Chap 8: Transient Analysis

18. HCB 3- Chap 8: Transient Analysis
Equation from previous slide:
Explicit forward difference formulation
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19. HCB 3- Chap 8: Transient Analysis
The smaller the time step, the more accurate the solution. However, computing time increases greatly and so it is advantageous to use as large a time step as possible.
Though the explicit finite difference method is easy to use, there is a restriction in the maximum time step which can be selected because of instability issues. The maximum time step is given by the stability criterion which states that for all interior nodes be selected such that
For example, for a brick wall,
If a mesh size of 0.01 m is selected, then the simulation time step is 111 s = 1.85 min.
HCB 3- Chap 8: Transient Analysis

20. HCB 3- Chap 8: Transient Analysis
Transient response of a wall subjected to a step change forcing function
Consider a concrete wall 20 cm thick of a conditioned space which is subject to an abrupt change in the outdoor air temperature.
Initially, the wall is at 20o C throughout its thickness. The outdoor air temperature is abruptly increased to 35o C.
Calculate the time evolution of the temperature profiles inside the wall and determine the heat flow inwards.
Explicit finite difference assuming the wall to be discretized into 5 nodes (nodal spacing is 5 cm) - nodes are numbered 0, with 0 being the node representing the exterior wall surface and node 5 representing the interior wall surface.
Node 1 2 3 4
hrs
HCB 3- Chap 8: Transient Analysis

21. Concept of Superposition for Linear Systems
(Used in most Building Energy Simulation Programs)
HCB 3- Chap 8: Transient Analysis

22. Transfer Function Model Formulation
HCB 3- Chap 8: Transient Analysis

23. Conductive Heat Gain
Step 1: Instantaneous heat gain using conduction transfer functions
HCB 3- Chap 8: Transient Analysis

24. HCB 3- Chap 8: Transient Analysis
The transfer function coefficients are subject to physical constraints
under steady-state limit:
HCB 3- Chap 8: Transient Analysis

25. Conduction Transfer Coefficients- Roofs and Walls
HCB 3- Chap 8: Transient Analysis

26. HCB 3- Chap 8: Transient Analysis

27. HCB 3- Chap 8: Transient Analysis
Table 8.5:
Calculation
results of
Example 8.2
Conduction
transfer model
HCB 3- Chap 8: Transient Analysis

28. HCB 3- Chap 8: Transient Analysis
Sol-air temp.
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29. HCB 3- Chap 8: Transient Analysis
Comparison of Transient Responses of Lightweight and Heavy Walls
Sol-air temp.
HCB 3- Chap 8: Transient Analysis

30. Conduction Time Series (CTS) Model Formulation
Used to calculate hourly 1-D conductive heat flows through opaque building elements such as roof and wall surfaces is part of the Radiant Transfer Series (RTS) method (described in Chap 9)
CTS model is akin to the conduction transfer function model in the sense that both of them use a time series model involving lagged values, and also assume that the indoor space temperature is held constant.
The conduction transfer function model uses an iterative process to calculate the conductive heat flow across the opaque elements of the building envelope
The novelty of the CTS model is that it replaces this iteration with a simple summation which does not involve lagged values of the conductive heat gains
HCB 3- Chap 8: Transient Analysis

31. HCB 3- Chap 8: Transient Analysis
CTS Model
HCB 3- Chap 8: Transient Analysis

32. HCB 3- Chap 8: Transient Analysis
The CTS factors, comprising of 24 coefficients, for various types of walls and roofs can be found from lookup tables similar to those created for the CTF coefficients (Table 8.6). These factors can be viewed as hourly adjustments to the steady-state heat transfer expression which correct for heat absorbed in the wall or roof during previous hours and released during this hour.
Figure 8.11 Conduction time series factors for light and heavy walls
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34. HCB 3- Chap 8: Transient Analysis
TABLE 8.6 Calculation results for Example 8.3 meant to illustrate the use of the conduction time series model. The wall is a 100 mm concrete light weight wall with board insulation and gypsum board.
- Conductive gain reaches its maximum at 19:00, which is 3 h after the peak of the sol-air temperature.
- As a check, verify that the solution satisfies the steady-state limit.
Multiplying the average U.(Tos−Ti )= (37.95−25)o C x = 8.71 W/m2
(same as shown in table)
HCB 3- Chap 8: Transient Analysis

35. HCB 3- Chap 8: Transient Analysis.
Thermal Network
Models
models for a wall
provide less
accurate solutions
and are used with
larger time steps
- 15 min or 1 h
Lumped model
2R1C lumped model
n th order lumped model
Thermal networks to model heat flow through a homogeneous plane wall of surface area A and wall thickness
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36. HCB 3- Chap 8: Transient Analysis
Example of
3-node
discretization
1/2 of the face brick thermal mass is assigned to the central node
and 1/4th to each of the two outside nodes
RC networks are not unique
Discretion of user involved
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37. HCB 3- Chap 8: Transient Analysis
1R1C Networks for Simple One-zone Buildings
The above eqn can be re- expressed as: T(t+1)+a1 T(t) =0
which is the time series formulation of the transfer function approach
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38. HCB 3- Chap 8: Transient Analysis
Time Constant
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39. HCB 3- Chap 8: Transient Analysis
1R1C Networks can be used for rough estimation;
for example, cool-down and set-up times in residences
Fig. 8.16
HCB 3- Chap 8: Transient Analysis

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41. HCB 3- Chap 8: Transient Analysis
Outcomes
Understand basic difference between steady state and dynamic models
Understand instances where dynamic models are necessary
Familiarity with different transient solution methods
Knowledge of the finite difference method
Understand concept of superposition of linear systems
Be able to apply the transfer function model to analyze conduction through simple building envelop elements
Be able to apply the conduction time series model to analyze conduction through simple building envelop elements
Familiarity with how heat transfer through composite envelop elements can be modeled as RC networks
Be able to apply the 1R1C network to analyze heat-up and cool-down of one-zone buildings
HCB 3- Chap 8: Transient Analysis