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The Commercial Module implements commercial building models. Version 1.0 of this module only supports single-zone office buildings. Support for additional commercial buildings types is planned, including multi-zone office, schools, stores, and refrigerated warehouses.


Small Office Building

Figure 1 - The Equivalent Thermal Parameters (ETP) model

The Commercial Module uses a simple Equivalent Thermal Parameters (ETP) model for small single-zone office buildings (Taylor and Pratt 1988), shown in Figure 1, with first-order ordinary differential equations (ODEs):

[math] \begin{align} T_i' & = \frac{1}{C_a} \left [ T_m H_m - T_i \left ( U_a - H_m \right ) + \sum_{end\ uses}Q_x + T_o U_a \right ] \\ T_m' & = \frac{1}{C_m} \left [ H_m \left ( T_i - T_m \right ) + Q_m \right ] \end{align} [/math]

where

  • [math]T_i[/math] = the temperature of the air inside the building
  • [math]T_i'[/math] = dTi/dt
  • [math]T_m[/math] = the temperature of the mass inside the building (for example, furniture, inside walls)
  • [math]T_m'[/math] = dTm/dt
  • [math]T_o[/math] = the ambient temperature outside air
  • [math]U_a[/math] = the UA of the building envelope
  • [math]H_m[/math] = the UA of the mass of the furniture, inside walls, etc.
  • [math]C_m[/math] = the heat capacity of the mass of the furniture inside the walls, etc.
  • [math]C_a[/math] = the heat capacity of the air inside the building
  • [math]Q_i[/math] = the heat rate from internal heat gains of the building (for example, plugs, lights, people)
  • [math]Q_h[/math] = the heat rate from heating, ventilating, and air conditioning unit
  • [math]Q_s[/math] = the heat rate from the sun to air (solar heating through windows, etc.)
  • [math]Q_m[/math] = the heat rate direct to the mass (e.g, solar radiation direct to mass)

The general first order ODEs (C1-C5 defined by inspection above) is

[math] \begin{align} T_i' & = c_1 T_i + c_2 T_m + c_3 \\ T_m' & = c_4 T_i + c_5 T_m + c_6 \end{align} [/math]

with the constants [math]c_1[/math] through [math]c_6[/math] defined as

  • [math]c_1 = - (U_a+U_m)/C_a[/math]
  • [math]c_2 = U_m/C_a[/math]
  • [math]c_3 = (Q_a+U_a T_o)/C_a[/math]
  • [math]c_4 = U_m/C_m[/math]
  • [math]c_5 = - U_m/C_m[/math]
  • [math]c_6 = Q_m/C_m[/math]

The general form of the second-order ODE is [math]p_1 T_i'' + p_2 T_i' + p_3 T_i = p_4[/math]. The solutions to the second-order ODEs for indoor and mass temperatures are

[math] \begin{align} T_i(t) & = k_1 e^{r_1 t} + k_2 e^{r_2 t} + \frac{p_4}{p_3} \\ T_m(t) & = \frac{T_i'(t) - c_1 T_i(t) - c3}{c2} \end{align} [/math]

where:

  • [math]p_1 = 1/c_2[/math]
  • [math]p_2 = -(c_1+c_5)/c_2[/math]
  • [math]p_3 = c_1 c_5 / c_2 - c_4[/math]
  • [math]p_4 = -c_3 c_5 / c_2 + c_6[/math]
  • [math]r_1,r_2[/math] are the roots of the [math]p_1 r^2 + p_2 r + p_3 = 0[/math]
  • [math]k_1 = [ r_1 T_i(0) - r_2 p_4/p_3 - T_i'(0) ] / (r_2 - r1)[/math]
  • [math]k_2 = [T_i'(0) - r_1 k_1] / r_2[/math]
  • [math]t[/math] = the elapsed time
  • [math]T_i(t)[/math] = the temperature of the air inside the building at time [math]t[/math]
  • [math]T_i'(t)[/math] = the rate of temperature change of the air inside the building at time [math]t[/math]

so that

[math] T_i'(0) = c_2 T_m(0) + c_1 T_i(0) - \left ( c_1 + c_2) T_o + c_7 \right ) [/math]

and

[math] T_m(t) = k_1 \frac{r_1 - c_1}{c_2} e^{r_1 t} + k_2 \frac{r_2-c_1}{c_2} e^{r_2 t} + \frac{p_4}{p_3} + \frac{c_6}{c_2} [/math]

with

  • [math]c_7 = Q_a / C_a[/math]

Defaults

All end-use power factor default to 1.0.

The outdoor air defaults to 59 F, relative humidity to 75%, and solar exposures as follows

  • South : 0
  • South-east : 0
  • South-west : 0
  • East : 0
  • West : 0
  • North-east : 0
  • North-west : 0
  • North : 0
  • Horizontal : 0

The default interior air and mass temperatures are set to the default outdoor air temperature.

The control defaults are as follows:

  • Heating setpoint : 70F
  • Cooling setpoint : 75F
  • Auxiliary cut-in: 20F
  • Economizer cut-in: 60F
  • Setpoint deadband : 1F
  • Ventilation fraction : 1 /h
  • Lighting fraction : 0.5 pu

The default occupancy schedule is M-F 8-17h. When occupied, the default occupancy is 0.002 occupants/sf.

Initialization

The default heating capacity is computed by solving the heat flow equation for the peak heating condition, which gives:

[math]Q_{max_{heat}}=UA ( T_{set_{heat}}-T_{design_{heat}} )[/math]

The default cooling capacity is computed by solving the heat flow equation for the peak cooling condition, which gives:

[math] Q_{max_{cool}} = UA ( T_{design_{cool}}-T_{set_{cool}}) + \sum_{windows}{A_{window} Q_{solar} c_{glazing}} + \sum_{loads}{Q_{load}} +Q_{vent} [/math]

where

  • [math]Q_{vent}=0.2402 \times 0.0735(T_{design_{cool}}-T_{set_{cool}}) \times V_{air} \times ACH [/math]

The heating COP is given by [math]COP_{heat} = Dist_{triangle}(1,2)[/math] and the cooling COP is given by [math]COP_{cool} = Dist_{triangle}(3,5)[/math].

Controls

The HVAC system has 6 control modes:

OFF 
In the OFF mode, the HVAC system is completely off. No ventilation and

no heating or cooling of any kind of performed. This mode is engaged whenever [math]T_{off_{heat}} \lt T_{air} \lt T_{off_{cool}}[/math] and [math]occupancy = 0[/math].

VENT 
In the VENT mode, the HVAC system is ventilating the zone at the

minimum_ach rate. This mode is engaged whenever [math]T_{off_{heat}} \lt T_{air} \lt T_{off_{cool}}[/math] and [math]occupancy \gt 0[/math].

HEAT 
In the HEAT mode, the primary heating (COP>1) system is on and the building is

ventilating at the minimum_ach rate only if occupancy is non-zero. This mode is engaged whenever [math]T_{cutin_{aux}} \lt T_{air} \le T_{on_{heat}}[/math].

AUX 
In the AUX mode, the secondary heating (COP=1) system is on and the building is

ventilating at the minimum_ach rate only if occupancy is non-zero. This mode is engaged whenever [math]T_{air} \le T_{cutin_{aux}}[/math].

COOL 
In the COOL mode, the active cooling (COP>1) system is on and the building

ventilating at the minimum_ach rate only if occupancy is non-zero. This mode is engaged whenever [math]T_{air} \ge T_{on_{cool}} and T_{out} \gt T_{cutin_{econ}}[/math].

ECON 
In the ECON mode, the passive cooling (COP=&infty;) system is on and the building

is ventilating using the rate required to cool using outdoor air only. This mode is engaged whenever [math]T_{air} \ge T_{on_{cool}} and T_{out} \le T_{cutin_{econ}}[/math].

Power calculations

Except as noted below, when ventilation is required, [math]Pvent = floor_area (0.1 - 0.01\imath) [/math] VA/sf, and [math]Qvent=0.2402 \times 0.0735 (T_{out}-T_{air}) V_{air} \times ventilation_rate[/math].

HVAC

OFF 
COP = 0, Qactive = Qpassive = 0, Pvent = 0
VENT 
COP = 0, Qactive = 0, Qpassive = Qvent
HEAT 
COP = 1.0 + (COP_{heating}-1) (T_{out} - T_{aux}) / Trange, zone.hvac.heating.capacity + zone.hvac.heating.capacity_perF*(zone.hvac.heating.balance_temperature-Tout), Qpassive = Qvent
AUX 
COP = 1.0, Qactive = COP * heating_capacity, Qpassive = Qvent
COOL 
COP = -1.0 - (zone.hvac.cooling.cop+1)*(Tout-TmaxCool)/(TmaxCool-Tecon), zone.hvac.cooling.capacity - zone.hvac.cooling.capacity_perF*(Tout-zone.hvac.cooling.balance_temperature), Qpassive = Qvent
ECON 
COP = 0, Qactive = 0, Qpassive = Qvent

Lighting

lights.load = lights.fraction*(lights.capacity + (lights.capacity/lights.power_factor)*(sin(arccos(lights.power_factor)))J) (kW)

lights.heatgain = lights.load*lights.heatgain_fraction (kW)

Plugs

plugs.load = plugs.fraction*(plugs.capacity + (plugs.capacity/plugs.power_factor)*(sin(arccos(plugs.power_factor)))J) (kW)

plugs.heatgain = plugs.load * plugs.heatgain_fraction (kW)

Other loads

Large Office Buildings

The general formulation for a multizone state-space solution

For every node n, the heat transfer equation is

[math] C_n\frac{dT_n(t)}{dt} = Q_n(t) + \sum_{m=1}^M{ U_{nm} \left[ T_m(t)-T_n(t) \right] } [/math]

For a state space representation form a continuous, linear, time-invariant system is

[math]\begin{align} \dot x & = Ax + Bu \\ y & = Cx + Du \end{align}[/math]

And from [1], the solution is:

[math] y_t= \sum_{j=0}^n(S_j u_{t-j\delta})-\sum_{j=1}^n(e_j y_{t-j\delta}) [/math]

where

[math]\begin{align} S_0 & = CR_0 \Gamma_2+D \\ S_j & = C \left[ R_{j-1} ( \Gamma_1 -\Gamma_2 ) + R_j \Gamma_2 \right] + e_j D, for 1 \le j \le n-1 \\ S_n & = CR_{n-1} (\Gamma_1 -\Gamma_2 ) + e_n D \end{align}[/math]

Caveat

Prior to Hassayampa (Version 3.0) Multizone offices are created by connecting several single-zone objects with a multizone object linking them together with a user-defined UA. Unfortunately, this UA is not integrated into the thermal solution but used to add or remove heat from the zones based on the temperature difference. This method works fine as long as the temperature difference between the two zones remains relatively constant. The multizone object limits the time step to avoid problem with excessive changes in the heat transfer rate. Nevertheless, this method is only an approximation and can introduce some error when large temperature fluctuations occur.

At this time, the multizone offices only support unitary HVAC. There is no support for central/VAV HVAC systems.

Multizone ETP Linearization

 *** WORKING DRAFT *** 
Please review, edit and comment as necessary.


Tech:Commercial - Linearized solution of the Equivalent Thermal Parameters method for modeling multizone commercial buildings

Introduction

 TODO:  Describe purpose of multizone ETP solver and reason for linearization.

Nomenclature

[math]C_n[/math] 
The heat capacity of [math]n[/math]th node (Btu/degF).
[math]\Delta t[/math] 
The time-step taken for the next iteration (h).
[math]\Delta T_n[/math] 
The difference between the observed temperature and the setpoint temperature of the [math]n[/math]th node, if controlled.
[math]k[/math] 
The proportional control gain of the HVAC control system (unitless).
[math]N[/math] 
The number of nodes in a thermal network (integer).
[math]n[/math] 
A reference to the [math]n[/math]th of [math]N[/math] nodes (integer).
[math]m[/math] 
A reference to the [math]m[/math]th of [math]N[/math] nodes (integer).
[math]Mode_n[/math] 
The HVAC mode of the system interacting with the [math]n[/math]th zone (heat, cool, etc.).
[math]Q_n[/math] 
The HVAC heat flow into (positive for heating) or out of (negative for cooling) the [math]n[/math]th node, if controlled (Btu/h).
[math]Q_{nm}[/math] 
The heat flow from the [math]n[/math]th to [math]m[/math]th nodes (Btu/h).
[math]Q_{mode-cap,n}[/math] 
The heating (positive) or cooling (negative) capacity of the HVAC system interacting with the [math]n[/math]th node, if controlled (Btu/h).
[math]T_n[/math] 
The temperature of the [math]n[/math]th node (degF).
[math]T_{cool,n}[/math] 
The cooling setpoint temperature of the [math]n[/math]th node, if controlled (degF).
[math]T_{heat,n}[/math] 
The heating setpoint temperature of the [math]n[/math]th node, if controlled (degF).
[math]U_{nm}[/math] 
The thermal conductance of the path from the [math]n[/math]th and [math]m[/math]th nodes (Btu/degF/h).

Methodology

Figure 1. 6-Node Thermal Network

Known parameter values and boundary conditions are shown in red. Illustrated in Figure 1 is an arbitrary six-node thermal network. Note that the top horizontal branch is the original ETP circuit model, where [math]T_0[/math] is the outdoor air temperature (known boundary condition), [math]T_1[/math] is the indoor air temperature, and [math]T_2[/math] is the mass temperature. Node 3 might represent an unconditioned atrium space. Nodes 4 and 5 are present to illustrate series solutions of massless nodes, and might also represent an atrium with an additional buffer space to the outdoors.

Two additional massless paths are added to illustrate additional circumstances. A node n is characterized by its temperature Tn, its mass [math]C_n[/math] (which can be zero), and an exogenous heat gain [math]Q_n[/math] (from solar, internal, and/or HVAC, which is a boundary condition that can also be zero).

Nodes [math]m[/math] and [math]n[/math] are connected by conductance’s [math]U_{mn}[/math]. A conductance might represent the UA of a building, an air flow rate between two nodes, or the product of a convective surface heat transfer coefficient and the surface area, for example.

To solve this circuit explicitly, the heat flow from any node [math]m[/math] through the conductance to any other node [math]n[/math] at the time [math]t[/math]is denoted as [math]Q_{mn}(t)[/math]. Thus

[math]Q_{mn}(t)=U_{mn}\left[T_m(t)-T_n(t)\right][/math]
(1)

Note that this implies a sign convention for heat flow into a node as positive. Thus, by definition, [math]Q_{mn}=-Q_{nm}[/math].

Assume a "small" time step [math]\Delta t[/math] during which all temperatures change by a "small" amount relative to the temperature differences between the nodes, i.e. [math]Q_{mn}[/math] can be considered a constant. This is the linearizing assumption. The maximum time-step [math]\Delta t[/math] permissible that satisfies the linearizing assumption is limited by two considerations:

(1) How long until an HVAC [math]Mode_n[/math] changes or internal gains [math]Q_n[/math] change?
This is determined by the time to the next expected change in mode, which can be estimated for each node based on the type of control (bang-bang or proportional).
(2) How long until the change in the temperature difference between a two nodes [math]n[/math] and [math]m[/math] exceeds a preset limit?
This is determined by computing the rate of change of the temperature difference and estimating the time until that rate of change exceeds a preset limit.

Step 1: Compute temperatures of all massive nodes

Then the heat balance on a massive node [math]n[/math], i.e., [math]C_n \gt 0[/math], from a set of connected nodes [math]m=1[/math] to [math]M[/math], from time [math]t[/math] to time [math]t+\Delta t[/math] is:

[math]Q_n(t)+\sum_{m=1}^M Q_{mn}(t)=\frac{C_n\left[T_n(t+\Delta t)-T_n(t) \right]}{\Delta t}[/math]
(2)

where [math]Q_n(t)[/math] is assumed to be constant boundary condition over the interval.

Solving this for the temperature of the node at the next time step:

[math]\frac{C_n T_n(t+\Delta t)}{\Delta t} = Q_n(t)+\sum_{m=1}^M Q_{nm}(t)+\frac{C_n T_n(t)}{\Delta t}[/math]
(3)
[math]T_n(t+\Delta t) = \frac{\Delta t}{C_n} \left[Q_n(t)+\sum_{m=1}^M Q_{nm}(t)\right] + T_n(t)[/math]
(4)

Substituting the definition for [math]Q_{nm}[/math] from Equation (1):

[math]T_n(t+\Delta t) = \frac{\Delta t}{C_n} \left[Q_n(t)+\sum_{m=1}^M U_{mn}\left[T_m(t)-T_n(t)\right]\right] + T_n (t)[/math]
(5)

or

[math]T_n(t+\Delta t) = \frac{\Delta t}{C_n} Q_n(t) + \frac{\Delta t}{C_n} \sum_{m=1}^M U_{mn} T_m(t) + T_n(t) \left[1-\frac{\Delta t}{C_n}\sum_{m=1}^M U_{mn}\right][/math]
(6)

Step 2: Compute temperatures of all massless nodes

If node [math]n[/math] is massless, i.e., [math]C_n = 0[/math], then it must be in thermal equilibrium with all adjacent nodes at any time. For massless nodes, Equation (2) reduces to

[math]Q_n(t)+\sum_{m=1}^M Q_{nm}(t+\Delta t)=0[/math]
(7)

where over the time interval from [math]t[/math] to [math]t+\Delta t[/math] [math]Qn[/math] is a constant boundary condition.

Substituting the definition for [math]Q_{mn}[/math] from Equation (1):

[math]Q_n(t)+\sum_{m=1}^M U_{mn} \left[ T_m(t+\Delta t)-T_n(t+\Delta t) \right]=0[/math]
(8)

Solving for the temperature of the node at time t+Δt:

[math]T_n(t+\Delta t)=\frac{Q_n(t)+\sum_{m=1}^M U_{mn}T_m(t+\Delta t)}{\sum_{m=1}^M U_{mn}}[/math]
(9)
Figure 2. Reduced equivalent of 6-node thermal network

Computing the temperature of massless nodes in the thermal network assumes all temperatures of adjacent nodes are known at the end of a time-step, either as boundary conditions, or because they are massive and have had their temperatures computed in Step 1.

The simplest way to resolve this is to reduce the network to an equivalent network when massless nodes are in series, parallel, or "wye" configuration.

Series massless nodes configurations

Figure 3a - Series massless node reduction

A series configuration of two nodes, such as the thermal network in Figure 1, can be reduced as shown in Figure 2, where Node 5 is eliminated and the equivalent series conductance from Node 0 to Node 4 is

[math]U_{xy} = \frac{U_{xw} + U_{wy}}{U_{xw} U_{wy}}[/math]
(10a)

Parallel massless nodes configurations

Figure 3b - Parallel thermal path reduction

A parallel configuration of two nodes can be reduced to a single node thus:

[math]U_{xy} = U_{A} + U_{B} \![/math]
(10b)

Delta massless nodes configurations

A "delta" configuration of 3 nodes [math](x, y, z)[/math] can be transformed to a "wye" configuration of four nodes [math](x,y,z,w)[/math] as follows:

[math]\begin{align} U_{xw} &= U_{xy}U_{xz}\left[ \frac{1}{U_{xy}} + \frac{1}{U_{yz}} + \frac{1}{U_{xz}} \right] \\ U_{yw} &= U_{xy}U_{yz}\left[ \frac{1}{U_{xy}} + \frac{1}{U_{yz}} + \frac{1}{U_{xz}} \right] \\ U_{zw} &= U_{xz}U_{yz}\left[ \frac{1}{U_{xy}} + \frac{1}{U_{yz}} + \frac{1}{U_{xz}} \right] \end{align}[/math]
(10c)

Note that for the general "star-mesh" case, the transformation is:

[math] U_{ij} = U_iw U_jw \sum_{n=1}^N \frac{1}{U_n} \![/math]
(10d)

where [math]N=1[/math] is the dangling node case (which eliminates the node), [math]N=2[/math] is the series node case (see above) and [math]N=3[/math] is the "delta-wye" transformation case. The "star-mesh" transformation has no general inverse without additional constraints, so all mesh configurations must be simplified using a series of appropriate wye-delta transformations.

In such all cases, the temperature of the new node, [math]w[/math] is calculated as follows:

[math] T_w = \frac{\sum_{n=1}^N U_{nw} T_n}{\sum_{n=1}^N U_{nw}} \! [/math]
(10e)

Special Case for "Bang-Bang" HVAC Control

Nodes whose temperature is used to control [math]Q_n[/math] (typically heating or cooling from the HVAC system) must have a modified calculation procedure, because [math]Q_n[/math] is no longer a boundary condition but rather a function of the node temperature, [math]T_n[/math], and the thermostat setpoints [math]T_{cool,n}[/math] and [math]T_{heat,n}[/math].

Define [math]Q_n[/math] as the sum of the HVAC energy input (heating positive, cooling negative), and the sum of internal heat gains and solar heat gains, to Node [math]n[/math]:

[math]Q_n = Q_{hvac,n}+Q_{gains,n} \quad[/math]
(11)

Let [math]Q_{cool-cap,n}[/math] and [math]Q_{heat-cap,n}[/math] be the net cooling and heating capacity available to supply the zone, respectively (net of the fan power). Let [math]Q_{fan,n}[/math] be the fan power when the HVAC is off (i.e. if the fan runs continually then [math]Q_{fan,n} \gt 0[/math]).

Assume a heating and cooling thermostat with setpoints centered in a deadband with range [math]+\Delta T_n[/math] on either side of the setpoint. Note that setpoints must not overlap:

[math]T_{cool,n} - \Delta T_n \gt T_{heat,n} + \Delta T_n \quad[/math]
(12)

The thermostat sets [math] Q_{hvac,n} [/math], which persists in subsequent time steps until changed by the thermostat:

When [math] T_n \lt T_{heat,n} - \Delta T_n [/math]

[math] Mode_n = On \quad ; \quad Q_{hvac,n} = Q_{heat-cap,n} [/math].
(13)

When [math] T_n \gt T_{cool,n} + \Delta T_n [/math]

[math] Mode_n = On \quad ; \quad Q_{hvac,n} = Q_{cool-cap,n} [/math].
(14)

When [math] Mode_n=On [/math] and [math] T_{heat,n}+ \Delta T_n [/math][math] T_n [/math][math] T_{cool,n} - \Delta T_n [/math]

[math] Mode_n = Off \quad ; \quad Q_{hvac,n} = Q_{fan,n} [/math].
(15)

Then the node temperature at time [math]t+\Delta t[/math] can be calculated from Equation (6). Note that all HVAC control nodes with On/Off control must be massive. If one were massless, Equation (9) indicates there is no way to maintain a setpoint without proportional control of [math]Q_{hvac,n}[/math].

Special case for Proportional-Differential (PD) HVAC Control

In the case of proportional control for [math]Q_{hvac,n}[/math], then a proportional-differential control scheme:

When [math] T_{heat,n} - \Delta T_n \lt T_n(t) \lt T_{heat,n} + \Delta T_n \quad[/math]

[math] Q_{hvac,n}(t) = Q_{heat-cap,n}(t) \left[ \frac{T_{heat,n}+\Delta T_n-T_n(t)}{2\Delta T_n} - \frac{k}{2\Delta T_n} \left[ T_n(t)-T_n(t-\Delta t)\right]\right][/math].
(16)

When [math] T_{cool,n} - \Delta T_n \lt T_n(t) \lt T_{cool,n} + \Delta T_n \quad[/math]

[math] Q_{hvac,n}(t) = Q_{cool-cap,n}(t) \left[ \frac{T_n(t) - T_{cool,n} + \Delta T_n}{2 ∆T_n} - \frac{k}{2\Delta T_n} \left[ T_n(t-\Delta t)-T_n(t) \right] \right][/math].
(17)

When [math] T_{heat,n} + \Delta T_n \le T_n(t) \le T_{cool,n} - \Delta T_n[/math]

[math]Q_{hvac,n}(t) = Q_{fan,n} \quad[/math].
(18)

where [math]k[/math] is the proportional gain for the controller.

Step 3: Compute temperatures of reduced nodes

Massless nodes that have been reduced from the network are unnecessary to the simulation model of the network if

  1. the temperature of the node does not affect some non-linear aspect such as a thermostatic control, and
  2. the temperature is not needed as an output variable for some purpose.

In general, it may be best to assume that if the user specified “unneeded” nodes, that there was some purpose in mind, and their temperatures should be computed from Equation (9) as a final step.

Validation

 TODO:  Describe how to validate a numerical implementation of this method.

Authors

This method was developed by Robert G. Pratt and Lucy Huang at Pacific Northwest National Laboratory

See also

ETP Equation Solution Algorithm

Figure 2 - The general ETP equation

The evolution of air temperature as a function of time is the fundamental equation in ETP, as shown in Figure 2. It takes the following form

[math] f \left ( t \right ) = a e^{nt} + b e^{mt} + c [/math]

where both [math]m[/math] and [math]n[/math] are negative and equation [math]f(t) = 0[/math] must be solved for the first value of [math]t \gt 0[/math]. Unfortunately, this function has no closed-form solution and must be solved numerically.

When [math]a b \lt 0[/math], the function always has one extremum and one inflexion point, and the extremum time [math]t_m[/math]is always less than inflexion time [math]t_i[/math]. The extremum time is found at [math]t_m = \log (-a n / b m) / (m-n)[/math] and the value of the function at this point is defined as [math]f_m = f(t_m) [/math]. The inflexion point is found at [math]t_m = \log ( -a n^2 / b m^2) / (m-n)[/math] and the value of the function at this point is defined as [math]f_i = f(t_i)[/math]. The initial value at [math]t = 0[/math] if defined as [math]f_0 = f(0)[/math].


The simplest efficient numerical method is Newton's method, but the method will not converge under certain conditions that depend on when the extremum and the inflexion points occur. The following tests must be made before starting the numerical solution:

  • for [math]t_m \gt 0[/math], a solution exists when [math]f_0[/math]×[math]f_m \lt 0[/math] or [math]c[/math]×[math]f_m \lt 0[/math]
  • for [math]t_m = 0[/math] a solution only exists when [math]f_m = 0[/math]
  • for [math]t_m \lt 0[/math][math]ti[/math] a solution only exists when [math]f_m \lt 0 \lt f_i[/math]
  • for [math]t_i = 0[/math] a solution only exists when [math]f_i = 0[/math]
  • for [math]t_i \lt 0[/math] a solution only exists when [math]c[/math]×[math]f_i \lt 0[/math]

When a solution exists the starting point [math]t_0[/math] of the numerical solution must be chosen based on the values of [math]t_m[/math] and [math]t_i[/math]

  • [math]t_0 = 0[/math] should be used when [math]t_m \gt 0[/math] and [math]f_m[/math][math]f_0 \lt 0[/math]
  • [math]t_0 = t_i[/math] should be used for all other conditions for which a solution exists

References

  • Taylor, ZT and RG Pratt. 1988 "The effects of model simplifications on equivalent thermal parameters calculated from hourly building performance data." In proc. 1988 ACEEE Summer Study on Energy Efficiency in Buildings, pp. 10.268-10.285.

Authors: David Chassin and Ross Guttromson, Pacific Northwest National Laboratory, Richland Washington (USA), PNNL 17615, May 2008.

See also

  • Seem 1989