# The slope of the room ratio line

The room ratio line is the straight line, drawn on a psychrometric chart, joining the points representing the state maintained in the room and the initial condition of the air supplied to the room.

The slope of this line is an indication of the ratio of the latent and sensible heat exchanges taking place in the room, and the determination of its value plays a vital part in the selection of economical supply states.

Any supply state which lies on the room ratio line differs from the room state by a number of degrees of dry-bulb temperature and by a number of grams of moisture content. The values of this pair of differences are directly proportional to the mass of air supplied to the room for offsetting given sensible and latent gains or losses. Thus, in order to maintain a particular psychrometric state in a room, the state of the air supplied must always lie on the room ratio line.

For the room mentioned in example 6.6, calculate the condition which would be maintained therein if the supply air were not at a state of 16°C dry-bulb and 7.055 g kg-1 but at (a) 16°C dry-bulb and 7.986 g kg-1 and (b) 17°C dry-bulb and 7.055 g kg-1.

The heat gains are the same as in example 6.6.

Answer

(a) By equation (6.8)

GT = 7.986 + 0.321

= 8.307 g per kg dry air

The room state is therefore 20°C dry-bulb and 56 per cent saturation, by reference to CIBSE psychrometric tables.

*(b) *By equation (6.6)

T = 17 + 7.3 (273 + 16) _ 2j0q

R 1.473 358

The room condition is therefore 21 °C dry-bulb and 7.376 g kg-1, to give 47 per cent saturation, by reference to CIBSE tables. Note that the Charles’ law correction expression is still (273 + 16) because the volumetric flow rate used is 1.473 m3 s“1, which was evaluated at 16°C. Note also that the moisture content in the room remains at 7.376 g kg-1 because, by the calculation in example 6.6(b) the supply of 1.473 m3 s“1 at a moisture content of 7.055 g kg-1 offsets a latent gain of 1.4 kW and suffers a moisture pick-up of 0.321 g kg’1. In other words, this example assumes that some unmentioned device, such as a reheater battery, elevates the supply air temperature by 1°C but the supply air mass flow rate is unchanged.

The calculation of the slope of the room ratio line is clearly a matter of some importance since it appears that one has a choice of a variety of supply air states. It seems that if any state can be chosen and the corresponding supply air quantity calculated, then the correct conditions will be maintained in the room. This is not so. Economic pressures restrict the choice of supply air state to a value fairly close to the saturation curve. It must, of course, still lie on the room ratio line.

The calculation of the slope of the line can be done in one of two ways:

*(a) *By calculating it from the ratio of the sensible to total heat gains to the room,

*(b) *By making use of the ratio of the latent to sensible gains in the room.

Method (a) merely consists of making use of the protractor at the top left-hand comer of the chart.

EXAMPLE 6.8

Calculate the slope of the room ratio line if the sensible gains are 7.3 kW and the latent gains are 1.4 kW.

Answer

Total gain = 7.3 + 1.4 = 8.7 kW

= 0.84 |

The value can be marked on the outer scale of the protractor on the chart and with the aid of a parallel rule or a pair of set squares the same slope can be transferred to any position on the chart. Figure 6.2 illustrates this. The line O O’ in the protractor is parallel to lines drawn through Rj and R2. The important point to appreciate is that the room ratio line can be drawn anywhere on the chart. Its slope depends only on the heat gains occurring in the room and not on the particular room state.

—— |

O’

Fig. 6.2 The room ratio line. |

The slope of the line can also be expressed in terms of the ratio of latent gain to sensible gain. This, in itself, is correct but a slightly inaccurate version of this is to calculate the ratio in terms of the moisture picked up by the air supplied to the room and the rise in dry — bulb temperature which it suffers.

Making use of equations (6.8) and (6.6), we can write,

Latent gain in kW _ __Flow rate (m3 s"1 ) x (gr — gs ) x 856 x (273 + t)__ sensible gain in kW " flow rate (m3 s-1 ) x (оr — ts ) x 358 x (273 + t)

Hence

Gr ~ gs _ __Latent gain in kW__ ^ 358 tr — ts sensible gain in kW 856

Or

Ag __Latent gain__ 358

At sensible gain 856 ‘

This is slightly inaccurate because it expresses sensible changes in terms of dry-bulb temperature changes. Since the scale of dry-bulb temperature is not linear on the psychrometric chart, the linear displacement corresponding to a given change in dry-bulb temperature is not constant all over the chart. It follows that the angular displacement of the slope will vary slightly from place to place on the chart. This inaccuracy is small over the range of values of psychrometric state normally encountered and so the use of equation (6.9) is tolerated.

EXAMPLE 6.9

Calculate the slope of the room ratio line by evaluating Ag/At for sensible and latent heat gains of 7.3 kW and 1.4 kW respectively.

Answer

A/ = M X HI by ecluation(6-9)

= 0.0802 g kg"1 K1

The room ratio line does not necessarily have to slope downwards from right to left as in Figure 6.2. A slope of this kind indicates the presence of sensible and latent heat gains. Gains of both kinds do not always occur. In winter, for example, a room might have a sensible heat loss coupled with a latent heat gain. Under these circumstances, the line might slope downwards from left to right as in Figure 6.3(a). It is also possible, if a large amount of outside air having a moisture content lower than that in the room infiltrates, for the room to have a latent heat loss coupled with a sensible heat loss (or even with a sensible

(a) (b)

Fig. 6.3 Two possibilities for the slope of the room ratio line.

Heat gain). The room ratio line for both latent and sensible losses would slope upwards from left to right, as in Figure 6.3(b).

EXAMPLE 6.10

If the room mentioned in example 6.6 suffers a sensible heat loss of 3 kW and a latent heat gain of 1.2 kW in winter, calculate the necessary supply state of the air delivered to the room. It is assumed that the supply fan handles the same amount of air in winter as in summer and that the battery required to heat the air in winter is positioned on the suction side of the fan.

Answer

Supply air quantity = 1.473 m3 s-1.

This volumetric flow rate is the same in winter even though its temperature will be at a much higher value than the 16°C used for its expression in summer because the fan handles a constant amount of air (see section 15.16).

A diagram of the plant is shown in Figure 6.4(a) and of the psychrometry in Figure

**6.4 **(b).

Since a heat loss is to be offset, the air must be heated by the heater battery to temperature ts and the value of ts must exceed tr the room temperature.

From equation (6.6)

I 473 =_____ 3___ __(273 + ts)__

(?s — 20) 358

There are two ways of solving this linear equation:

(1) 1.473fs — 1.473 x 20 = 3 x (273 + fs)/358 whence

Ts = 21.68°C

This method is tedious and prone to error.

(2) Guess a value of t, say 25°C, and use in the expression (273 + t).

Then, from equation (6.6)

, — 20° i 3 x (273__ + 25)__ s 1.473 358

= 20° + 1.70°

= 21.70°C

Try again with t = 22°C in the expression (273 + t). Then

, — 20° + — J — x (273__ + 22)__ s 1.473 358

= 21.68°C

This method is preferred, provided that reasonable values are guessed for t. From equation (6.8)

= 7.376 — 0.280 = 7.096 g per kg dry air

Conditioned room —► R |

Dry steam is injected as necessary in winter when infiltration causes net latent heat losses |

Heater Battery |

(a) |

G, = 7.376 g/kg 9s •s Room ratio line |

Fr ts = 20°C (b) |

Fig. 6.4 (a) Plant arrangement for Example 6.10. (b) Psychrometry for Example 6.10. |

This establishes the supply state, S, as 21.68°C dry-bulb and 7.096 g per kg, moisture content.

S must lie on the room ratio line. This is self-evident since we have just calculated At = 1.68°C and Ag = 0.28 g kg“1

Thus

Ag 0.28

At 1.68

= 0.167 g kg"1 K“1

Strictly speaking, if the room ratio line is regarded as having a positive slope when both

Sensible and latent heat gains are occurring, its slope should be regarded as negative for this case. That is,

A? _ +0.28 _ n ifj7 p. w-i y-i

At ~ ~ ~°A61 gkg K

On the other hand, if both a latent heat loss and a sensible heat loss occurred, the slope would be positive again:

# = ^li = +0167sks-‘ k-1

It is best not to adhere to a sign convention but to express the slope always as positive and to use one’s understanding of fundamentals to decide on the way in which the line slopes.

It is just worth noting that the slope of the room ratio line is continuously changing as the heat gains vary. However, when designing a system, it is the slope of the room ratio line under summer design conditions that is relevant.

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