Could we install two different pumps (Q,H) in series or parallel? if we could what is the purpose?

Most people prefer to install pumps from the same type to Keep the similarity in Design calculations , Maintenance procedures and spare parts , even though  it is allowable to install different pumps in series / parallel arrangements taking in considerations the following :

If2 Different Pumps are connected in Series,

The Pump with the Lowest NPSH value should be installed as First Pump (for Critical Suction Conditions).

The Capacity in the Pump Installation should not    exceed the Max. Capacity of the Smallest Pump. Otherwise there will be a Pressure Drop   in the Smallest Pump.

For2 Different Pumps in parallel , If the Q1( pump1 flow rate )  is Smaller than the Q2 ( pump2 Flow rate ),  it is possible to install a Non-Return Valve   in the Discharge Line of Pump1

 to avoid Pump2 Pumping Fluid Back through Pump1.

In Parallel pumps can be different in term os Q and H. like in fire fighting jockey and main pump runs.

But in parallel both pumps should have same flow but head can be different depends on application and site requirements.

In parallel with  simultaneous operation with two or three different types of pumps  it is complicated, although we consider check valve to avoid reverse flow. Based on my experience I would suggest if we can avoid it is better to stay away.

But for series operation we may go for it considering same flow with different heads, but we have to take care of NPSH of the pumps. 

Series to add pressure to next pump suction pressure, Parallel to add flow rate in system manifold. Different (Q,H) will affect output of cumulative final H or Q values on the system curve of liquid pumping actual point that can be practically measured or theoretically calculated by advance software.

as per 2008 ASHRAE Handbook—HVAC Systems and Equipment:

Parallel Pumping

- pumps of equal size are recommended, and the parallel pump curve is established by doubling the flow of the single pump curve.

- Construction of the composite curve for two dissimilar parallel pumps requires special care; for example, note the shoulder in the composite pump curve in Figure 34.

check page 43.10

Both Arrangements are permitted as previously answered, but I'll go in details more.

First, lets agree that only (3) types of fire pumps drivers is acceptable to NFPA: Electric motor, Diesel engines, & steam turbines.

1- Due to the needs of a back-up fire pump(s) for the primary fire pump(s), the back-up fire pump(s) shall be installed in PARALLEL; both primary & back-up fire pump(s) shall of the same (Q, & H) OR in different (Q, & H) as in Pressure Maintenance (Jockey pump).

2- Due to the needs of a high pressure such as in high-rise building, fire pumps might be connected in SERIES to reach the required pressure in high level. To get that fire pumps in SERIES shall have same (Q) but different (H).

But this arrangement shall comply with NFPA -4..2.4 The pressure at any point in any pump in a series fire pump unit, with all pumps running at shutoff and rated speed at the maximum static suction supply, shall not exceed any pump suction, discharge, or case working pressure rating.

For simillarity of equipment selection once design calculation are done  it is better to have  same values of  Q &H . for  the pumps When connecting in parallel or series.

As we keep the pumps in parallel connection  in duty and standby it will make things complicated while shifting for duty to standby .but yes in multiples of the values say 20 gpm 40 gpm 60 gpm we can select  320 gpm or one 40 and one 20 gpm.

but selecting 12,48 etc is not advisable.

in series  we should only take care for the least NPSH VALUE  .and q is kept same be same.

 we can install two pumps in series at constant q, to increase the H

two pumps in parallel to increase Q